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https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1656
Diels-Alder Reaction Experimental Design
2016-02-01T13:04:07Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum\limits_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \le & 4 - z^i(t_f). \\<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|Initial value Exp 1<br />
|Initial value Exp 2<br />
|Initial value Exp 3<br />
|Initial value Exp 4<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|2.0<br />
|2.0<br />
|2.0<br />
|2.0<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0,10.0]<br />
|0.0<br />
|1.0<br />
|2.0<br />
|3.0<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value Exp 1<br />
|Initial value Exp 2<br />
|Initial value Exp 3<br />
|Initial value Exp 4<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1655
Diels-Alder Reaction Experimental Design
2016-02-01T13:00:57Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \le & 4 - z^i(t_f). \\<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|Initial value Exp 1<br />
|Initial value Exp 2<br />
|Initial value Exp 3<br />
|Initial value Exp 4<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|2.0<br />
|2.0<br />
|2.0<br />
|2.0<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0,10.0]<br />
|0.0<br />
|1.0<br />
|2.0<br />
|3.0<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value Exp 1<br />
|Initial value Exp 2<br />
|Initial value Exp 3<br />
|Initial value Exp 4<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1654
Diels-Alder Reaction Experimental Design
2016-02-01T13:00:16Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \le & 4 - z^i(t_f). \\<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|Initial value (Exp 1)<br />
|Initial value (Exp 2)<br />
|Initial value (Exp 3)<br />
|Initial value (Exp 4)<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|2.0<br />
|2.0<br />
|2.0<br />
|2.0<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0,10.0]<br />
|0.0<br />
|1.0<br />
|2.0<br />
|3.0<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value (Exp 1)<br />
|Initial value (Exp 2)<br />
|Initial value (Exp 3)<br />
|Initial value (Exp 4)<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1653
Diels-Alder Reaction Experimental Design
2016-02-01T12:57:10Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \le & 4 - z^i(t_f). \\<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|Initial value Exp 1<br />
|Initial value Exp 2<br />
|Initial value Exp 3<br />
|Initial value Exp 4<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0,10.0]<br />
|1.0<br />
|1.0<br />
|1.0<br />
|1.0<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|2.0<br />
|2.0<br />
|2.0<br />
|2.0<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0,10.0]<br />
|0.0<br />
|1.0<br />
|2.0<br />
|3.0<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value (Exp 1)<br />
|Initial value (Exp 2)<br />
|Initial value (Exp 3)<br />
|Initial value (Exp 4)<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|60.0<br />
|40.0<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1652
Diels-Alder Reaction Experimental Design
2016-02-01T12:47:56Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \le & 4 - z^i(t_f). \\<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1651
Diels-Alder Reaction Experimental Design
2016-02-01T12:47:38Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \le & 4 - z^i(t_f) \\.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1650
Diels-Alder Reaction Experimental Design
2016-02-01T12:46:43Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\<br />
\\<br />
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\<br />
\\<br />
\dot{z}^i(t) & = & w^i(t) \\<br />
z(0) & = & 0 \\<br />
w^i(t) &\in& [0,1] \\<br />
0 & \ge & 4 - z^i(t_f) \\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1649
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:36:10Z
<p>FelixJost: </p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 M1, M2, M3, M4, mR<br />
<br />
c Berechnung der Reaktormasse<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)<br />
<br />
c Messwert: Massenprozent<br />
<br />
h = M3*x(3) * 100.0d+0/mR <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
Standard deviation of measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
Experiment 1<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 0.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
Experiment 2<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 1.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=60 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=60 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=60 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
Experiment 3<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 2.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=40 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=40 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=40 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
<br />
</source><br />
<br />
Experiment 4<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 3.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
ini-file for running VPLAN:<br />
<source lang="optimica"><br />
<br />
[Aktion]<br />
;aktion=Integration<br />
;aktion=Simulationsumgebung<br />
;aktion=Parameterschaetzung<br />
;aktion=Versuchsplanung<br />
;aktion=ObjectiveTest<br />
;aktion=DerivativeTest<br />
;aktion={CS}<br />
aktion={CSVCS}<br />
<br />
[Pfade]<br />
problempath=vpbimolkat_origin<br />
inpath=in<br />
outpath=out<br />
messpath=mess<br />
plotpath=plot<br />
fortranpath=fortran<br />
<br />
[Parameter]<br />
pAnzahl=5<br />
p1=kr1 1 -1e+10 1e+10 0<br />
p2=e 1 -1e+10 1e+10 0<br />
p3=k1 1 -1e+10 1e+10 0<br />
p4=ekat 1 -1e+10 1e+10 0<br />
p5=lambda 1 -1e+10 1e+10 0<br />
<br />
[Versuchsplan]<br />
expAnzahl=4<br />
exp1=exp1.ini exp1.ini<br />
exp2=exp2.ini exp2.ini<br />
exp3=exp3.ini exp3.ini<br />
exp4=exp4.ini exp4.ini<br />
<br />
[Guetekriterium]<br />
Optimierungskriterium=A<br />
AKriterium=-1<br />
DKriterium=-1<br />
EKriterium=-1<br />
MKriterium=-1<br />
covmat=covmat.m<br />
jacmat=jacmat.m<br />
status=undefiniert<br />
<br />
[Residuum]<br />
res=0<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Messdatenfiles]<br />
mess1=mess1.dat<br />
mess2=mess2.dat<br />
mess3=mess3.dat<br />
mess4=mess4.dat<br />
<br />
[Outputfiles]<br />
out1=plot 0.05 integ.plt.1<br />
out2=plot 0.05 integ.plt.2<br />
out3=plot 0.05 integ.plt.3<br />
out4=plot 0.05 integ.plt.4<br />
<br />
[Residuenfiles]<br />
rsd1=res1.txt<br />
rsd2=res2.txt<br />
rsd3=res3.txt<br />
rsd4=res4.txt<br />
<br />
[ExtensionFlags]<br />
experimenttype=0<br />
integrator=0<br />
dmode=0<br />
pdeFlag=0<br />
<br />
[OptionenAllgemein]<br />
visflag=0<br />
messfileflag=0<br />
seed=-1<br />
numberofthreads=1<br />
robustflag=0<br />
epsmach=0<br />
infinity=1e+10<br />
epsilon=1e-08<br />
conflevel=0.95<br />
hrobust=1e-05<br />
computesigma=0<br />
exitonFPE=1<br />
iniprecision=6<br />
clipboardflag=0<br />
printxi=0<br />
printconstr=0<br />
printcolorful=-1<br />
<br />
[OptionenParameterschaetzung]<br />
eps=0.001<br />
itmax=50<br />
cond=10000<br />
condflag=1<br />
boundcheck=0<br />
startflag=0<br />
index1=1e-08<br />
fashort=0.8<br />
fa0=0.01<br />
farel=0.1<br />
famax=1.0<br />
realworkspace=10000<br />
integerworkspace=1000<br />
printlevel=2<br />
method=3<br />
<br />
[OptionenVersuchsplanung]<br />
maxit=300<br />
opttol=1e-06<br />
funcprec=1e-07<br />
linfeas=1e-07<br />
nlinfeas=0.01<br />
maxitQP=300<br />
maxitgesQP=10000<br />
opttolQP=1e-06<br />
pivottolQP=3.7e-11<br />
steplimitLS=2<br />
tolLS=0.9<br />
crashtol=0.0001<br />
elasticweight=100<br />
superbasics=1<br />
scaling=1<br />
sconstraints=0<br />
realworkspace=300000<br />
integerworkspace=300000<br />
charworkspace=500<br />
printlevel=10<br />
method=1<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1648
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:32:17Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 M1, M2, M3, M4, mR<br />
<br />
c Berechnung der Reaktormasse<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)<br />
<br />
c Messwert: Massenprozent<br />
<br />
h = M3*x(3) * 100.0d+0/mR <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
Standard deviation of measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
Experiment 1<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 0.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
Experiment 2<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 1.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=60 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=60 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=60 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
Experiment 3<br />
<source lang="optimica"><br />
<br />
<br />
</source><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 2.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=40 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=40 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=40 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
<br />
</source><br />
<br />
Experiment 4<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 3.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
ini-file for running VPLAN:<br />
<source lang="optimica"><br />
<br />
[Aktion]<br />
;aktion=Integration<br />
;aktion=Simulationsumgebung<br />
;aktion=Parameterschaetzung<br />
;aktion=Versuchsplanung<br />
;aktion=ObjectiveTest<br />
;aktion=DerivativeTest<br />
;aktion={CS}<br />
aktion={CSVCS}<br />
<br />
[Pfade]<br />
problempath=vpbimolkat_origin<br />
inpath=in<br />
outpath=out<br />
messpath=mess<br />
plotpath=plot<br />
fortranpath=fortran<br />
<br />
[Parameter]<br />
pAnzahl=5<br />
p1=kr1 1 -1e+10 1e+10 0<br />
p2=e 1 -1e+10 1e+10 0<br />
p3=k1 1 -1e+10 1e+10 0<br />
p4=ekat 1 -1e+10 1e+10 0<br />
p5=lambda 1 -1e+10 1e+10 0<br />
<br />
[Versuchsplan]<br />
expAnzahl=4<br />
exp1=exp1.ini exp1.ini<br />
exp2=exp2.ini exp2.ini<br />
exp3=exp3.ini exp3.ini<br />
exp4=exp4.ini exp4.ini<br />
<br />
[Guetekriterium]<br />
Optimierungskriterium=A<br />
AKriterium=-1<br />
DKriterium=-1<br />
EKriterium=-1<br />
MKriterium=-1<br />
covmat=covmat.m<br />
jacmat=jacmat.m<br />
status=undefiniert<br />
<br />
[Residuum]<br />
res=0<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Messdatenfiles]<br />
mess1=mess1.dat<br />
mess2=mess2.dat<br />
mess3=mess3.dat<br />
mess4=mess4.dat<br />
<br />
[Outputfiles]<br />
out1=plot 0.05 integ.plt.1<br />
out2=plot 0.05 integ.plt.2<br />
out3=plot 0.05 integ.plt.3<br />
out4=plot 0.05 integ.plt.4<br />
<br />
[Residuenfiles]<br />
rsd1=res1.txt<br />
rsd2=res2.txt<br />
rsd3=res3.txt<br />
rsd4=res4.txt<br />
<br />
[ExtensionFlags]<br />
experimenttype=0<br />
integrator=0<br />
dmode=0<br />
pdeFlag=0<br />
<br />
[OptionenAllgemein]<br />
visflag=0<br />
messfileflag=0<br />
seed=-1<br />
numberofthreads=1<br />
robustflag=0<br />
epsmach=0<br />
infinity=1e+10<br />
epsilon=1e-08<br />
conflevel=0.95<br />
hrobust=1e-05<br />
computesigma=0<br />
exitonFPE=1<br />
iniprecision=6<br />
clipboardflag=0<br />
printxi=0<br />
printconstr=0<br />
printcolorful=-1<br />
<br />
[OptionenParameterschaetzung]<br />
eps=0.001<br />
itmax=50<br />
cond=10000<br />
condflag=1<br />
boundcheck=0<br />
startflag=0<br />
index1=1e-08<br />
fashort=0.8<br />
fa0=0.01<br />
farel=0.1<br />
famax=1.0<br />
realworkspace=10000<br />
integerworkspace=1000<br />
printlevel=2<br />
method=3<br />
<br />
[OptionenVersuchsplanung]<br />
maxit=300<br />
opttol=1e-06<br />
funcprec=1e-07<br />
linfeas=1e-07<br />
nlinfeas=0.01<br />
maxitQP=300<br />
maxitgesQP=10000<br />
opttolQP=1e-06<br />
pivottolQP=3.7e-11<br />
steplimitLS=2<br />
tolLS=0.9<br />
crashtol=0.0001<br />
elasticweight=100<br />
superbasics=1<br />
scaling=1<br />
sconstraints=0<br />
realworkspace=300000<br />
integerworkspace=300000<br />
charworkspace=500<br />
printlevel=10<br />
method=1<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1647
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:30:58Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 M1, M2, M3, M4, mR<br />
<br />
c Berechnung der Reaktormasse<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)<br />
<br />
c Messwert: Massenprozent<br />
<br />
h = M3*x(3) * 100.0d+0/mR <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
Standard deviation of measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
Experiment 1<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 0.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
Experiment 2<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 1.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=60 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=60 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=60 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
Experiment 3<br />
<source lang="optimica"><br />
<br />
<br />
</source><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 2.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=40 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=40 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=40 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
<br />
</source><br />
<br />
Experiment 4<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 3.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1646
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:29:50Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 M1, M2, M3, M4, mR<br />
<br />
c Berechnung der Reaktormasse<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)<br />
<br />
c Messwert: Massenprozent<br />
<br />
h = M3*x(3) * 100.0d+0/mR <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
Standard deviation of measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
Experiment 1<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 0.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
Experiment 2<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 1.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=60 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=60 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=60 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
Experiment 3<br />
<source lang="optimica"><br />
<br />
<br />
</source><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 2.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=40 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=40 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=40 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
<br />
Experiment 4<br />
<source lang="optimica"><br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=10<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=4<br />
y1=n1 na1 -1e+10 1e+10<br />
y2=n2 na2 -1e+10 1e+10<br />
y3=n3 0 -1e+10 1e+10<br />
y4=n4 na4 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=0<br />
t1=0<br />
t2=0.33<br />
t3=0.66<br />
t4=1<br />
t5=1.33<br />
t6=1.66<br />
t7=2<br />
t8=2.33<br />
t9=2.66<br />
t10=3<br />
t11=3.33<br />
t12=3.66<br />
t13=4<br />
t14=4.33<br />
t15=4.66<br />
t16=5<br />
t17=6<br />
t18=7<br />
t19=9<br />
t20=10<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=4<br />
q1=na1 1.0 0 10 0 -1<br />
q2=na2 1.0 0 10 0 -1<br />
q3=na4 2.0 0.4 9 0 -1<br />
q4=Ckat 3.0 0 6 0 -1<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=Tc 3 20 100<br />
u1tAnzahl=3<br />
u1t0=t0<br />
u1t1q=20 20 100 0 0 0 0<br />
u1t1=2<br />
u1t2q=20 20 100 0.0 -1e+10 1e+10 <br />
u1t2=8<br />
u1t3q=20 20 100 0 0 0<br />
u1t3=tend<br />
<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=2<br />
c1=cfcn1 1<br />
c1bnd1=0.1 0.7<br />
c2=cfcn2 1<br />
c2bnd1=0.1 10<br />
<br />
[Messverfahren]<br />
mAnzahl=1<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
mminmaxges=0 6<br />
<br />
[Messungen]<br />
tAnzahl=20<br />
<br />
t1=0.33<br />
t1Anzahl=1<br />
t1m1=mfcn1 0.3 1e-06 1 <br />
t1minmax=0 1e+10<br />
<br />
t2=0.66<br />
t2Anzahl=1<br />
t2m1=mfcn1 0.3 1e-06 1 <br />
t2minmax=0 1e+10<br />
<br />
t3=1<br />
t3Anzahl=1<br />
t3m1=mfcn1 0.3 1e-06 1 <br />
t3minmax=0 1e+10<br />
<br />
t4=1.33<br />
t4Anzahl=1<br />
t4m1=mfcn1 0.3 1e-06 1 <br />
t4minmax=0 1e+10<br />
<br />
t5=1.66<br />
t5Anzahl=1<br />
t5m1=mfcn1 0.3 1e-06 1 <br />
t5minmax=0 1e+10<br />
<br />
t6=2<br />
t6Anzahl=1<br />
t6m1=mfcn1 0.3 1e-06 1 <br />
t6minmax=0 1e+10<br />
<br />
t7=2.33<br />
t7Anzahl=1<br />
t7m1=mfcn1 0.3 1e-06 1 <br />
t7minmax=0 1e+10<br />
<br />
t8=2.66<br />
t8Anzahl=1<br />
t8m1=mfcn1 0.3 1e-06 1 <br />
t8minmax=0 1e+10<br />
<br />
t9=3<br />
t9Anzahl=1<br />
t9m1=mfcn1 0.3 1e-06 1 <br />
t9minmax=0 1e+10<br />
<br />
t10=3.33<br />
t10Anzahl=1<br />
t10m1=mfcn1 0.3 1e-06 1 <br />
t10minmax=0 1e+10<br />
<br />
t11=3.66<br />
t11Anzahl=1<br />
t11m1=mfcn1 0.3 1e-06 1 <br />
t11minmax=0 1e+10<br />
<br />
t12=4<br />
t12Anzahl=1<br />
t12m1=mfcn1 0.3 1e-06 1 <br />
t12minmax=0 1e+10<br />
<br />
t13=4.33<br />
t13Anzahl=1<br />
t13m1=mfcn1 0.3 1e-06 1 <br />
t13minmax=0 1e+10<br />
<br />
t14=4.66<br />
t14Anzahl=1<br />
t14m1=mfcn1 0.3 1e-06 1 <br />
t14minmax=0 1e+10<br />
<br />
t15=5<br />
t15Anzahl=1<br />
t15m1=mfcn1 0.3 1e-06 1 <br />
t15minmax=0 1e+10<br />
<br />
t16=6<br />
t16Anzahl=1<br />
t16m1=mfcn1 0.3 1e-06 1 <br />
t16minmax=0 1e+10<br />
<br />
t17=7<br />
t17Anzahl=1<br />
t17m1=mfcn1 0.3 1e-06 1 <br />
t17minmax=0 1e+10<br />
<br />
t18=8<br />
t18Anzahl=1<br />
t18m1=mfcn1 0.3 1e-06 1 <br />
t18minmax=0 1e+10<br />
<br />
t19=9<br />
t19Anzahl=1<br />
t19m1=mfcn1 0.3 1e-06 1 <br />
t19minmax=0 1e+10<br />
<br />
t20=10<br />
t20Anzahl=1<br />
t20m1=mfcn1 0.3 1e-06 1 <br />
t20minmax=0 1e+10<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
noerrorcontrol=0<br />
realworkspace=170000<br />
integerworkspace=500<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1645
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:26:14Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 M1, M2, M3, M4, mR<br />
<br />
c Berechnung der Reaktormasse<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)<br />
<br />
c Messwert: Massenprozent<br />
<br />
h = M3*x(3) * 100.0d+0/mR <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
Standard deviation of measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1644
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:23:38Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 M1, M2, M3, M4, mR<br />
<br />
c Berechnung der Reaktormasse<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)<br />
<br />
c Messwert: Massenprozent<br />
<br />
h = M3*x(3) * 100.0d+0/mR <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design_(VPLAN)&diff=1643
Diels-Alder Reaction Experimental Design (VPLAN)
2016-02-01T12:22:07Z
<p>FelixJost: Created page with "== VPLAN == Differential equations: <source lang="fortran"> c RHS of the differential equations subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag ) implic..."</p>
<hr />
<div>== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 n1, n2, n3, n4<br />
real*8 na1, na2, na4<br />
real*8 fg, Temp, E , Rg , T1, Tc<br />
real*8 r1, mR<br />
real*8 kr1, kkat, Ckat, Ekat<br />
real*8 k1, lambda<br />
real*8 M1, M2, M3, M4<br />
real*8 dm<br />
<br />
c State variables<br />
<br />
n1 = x(1)<br />
n2 = x(2)<br />
n3 = x(3)<br />
n4 = x(4)<br />
<br />
c Control variables<br />
<br />
na1 = q(1) <br />
na2 = q(2)<br />
na4 = q(3)<br />
Ckat = q(4)<br />
<br />
c Control function<br />
<br />
c DISCRETIZE1( Tc, rwh, iwh )<br />
<br />
c Parameters<br />
<br />
kr1 = p(1) * 1.0d-2 <br />
E = p(2) * 60000.0d+0<br />
k1 = p(3) * 0.10d+0 <br />
Ekat = p(4) * 40000.0d0<br />
lambda = p(5) * 0.25d+0<br />
<br />
c Molar masses (in kg/mol)<br />
<br />
M1 = 0.1362d+0<br />
M2 = 0.09806d+0<br />
M3 = M1 + M2<br />
M4 = 0.236d+0<br />
<br />
Temp = Tc + 273.0d+0<br />
Rg = 8.314d+0<br />
T1 = 293.0d+0<br />
<br />
c Reaction rates<br />
<br />
mR = n1*M1 + n2*M2 +n3*M3 + n4*M4<br />
<br />
kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) ) <br />
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )<br />
& * Ckat * dexp( -lambda * t )<br />
<br />
r1 = kkat * n1 * n2 / mR<br />
<br />
f(1) = -r1 <br />
f(2) = -r1 <br />
f(3) = r1 <br />
f(4) = 0.0d0<br />
<br />
end<br />
<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1642
Diels-Alder Reaction Experimental Design
2016-02-01T12:19:14Z
<p>FelixJost: </p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== Source Code ==<br />
<br />
* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]<br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1641
Diels-Alder Reaction Experimental Design
2016-02-01T12:15:37Z
<p>FelixJost: </p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=1640
Diels-Alder Reaction Experimental Design
2016-02-01T12:11:59Z
<p>FelixJost: </p>
<hr />
<div>{{Dimensions<br />
|nd = 1<br />
|nx = 2<br />
|nu = 1<br />
|nre = 4<br />
}}The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1346
Lotka Experimental Design (VPLAN)
2016-01-19T16:29:14Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function:<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<source lang="fortran"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
<source lang="optimica"><br />
<br />
; ini-File fuer Experiment<br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=12<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=2<br />
y1=x1 0.5 -1e+10 1e+10<br />
y2=x2 0.7 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=12<br />
t1=1<br />
t2=2<br />
t3=3<br />
t4=4<br />
t5=5<br />
t6=6<br />
t7=7<br />
t8=8<br />
t9=9<br />
t10=10<br />
t11=11<br />
t12=12<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=0<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=myu 0 0.0 1.0<br />
u1tAnzahl=500<br />
u1t0=t0<br />
u1t1q=0.3 0.0 1.0 0 0<br />
u1t1=0.024<br />
u1t2q=0.3 0.0 1.0 0 0<br />
u1t2=0.048<br />
u1t3q=0.3 0.0 1.0 0 0<br />
u1t3=0.072<br />
u1t4q=0.3 0.0 1.0 0 0<br />
u1t4=0.096<br />
u1t5q=0.3 0.0 1.0 0 0<br />
u1t5=0.12<br />
u1t6q=0.3 0.0 1.0 0 0<br />
u1t6=0.144<br />
u1t7q=0.3 0.0 1.0 0 0<br />
u1t7=0.168<br />
u1t8q=0.3 0.0 1.0 0 0<br />
u1t8=0.192<br />
u1t9q=0.3 0.0 1.0 0 0<br />
u1t9=0.216<br />
u1t10q=0.3 0.0 1.0 0 0<br />
u1t10=0.24<br />
u1t11q=0.3 0.0 1.0 0 0<br />
u1t11=0.264<br />
u1t12q=0.3 0.0 1.0 0 0<br />
u1t12=0.288<br />
u1t13q=0.3 0.0 1.0 0 0<br />
u1t13=0.312<br />
u1t14q=0.3 0.0 1.0 0 0<br />
u1t14=0.336<br />
u1t15q=0.3 0.0 1.0 0 0<br />
u1t15=0.36<br />
u1t16q=0.3 0.0 1.0 0 0<br />
u1t16=0.384<br />
u1t17q=0.3 0.0 1.0 0 0<br />
u1t17=0.408<br />
u1t18q=0.3 0.0 1.0 0 0<br />
u1t18=0.432<br />
u1t19q=0.3 0.0 1.0 0 0<br />
u1t19=0.456<br />
u1t20q=0.3 0.0 1.0 0 0<br />
u1t20=0.48<br />
u1t21q=0.3 0.0 1.0 0 0<br />
u1t21=0.504<br />
u1t22q=0.3 0.0 1.0 0 0<br />
u1t22=0.528<br />
u1t23q=0.3 0.0 1.0 0 0<br />
u1t23=0.552<br />
u1t24q=0.3 0.0 1.0 0 0<br />
u1t24=0.576<br />
u1t25q=0.3 0.0 1.0 0 0<br />
u1t25=0.6<br />
u1t26q=0.3 0.0 1.0 0 0<br />
u1t26=0.624<br />
u1t27q=0.3 0.0 1.0 0 0<br />
u1t27=0.648<br />
u1t28q=0.3 0.0 1.0 0 0<br />
u1t28=0.672<br />
u1t29q=0.3 0.0 1.0 0 0<br />
u1t29=0.696<br />
u1t30q=0.3 0.0 1.0 0 0<br />
u1t30=0.72<br />
u1t31q=0.3 0.0 1.0 0 0<br />
u1t31=0.744<br />
u1t32q=0.3 0.0 1.0 0 0<br />
u1t32=0.768<br />
u1t33q=0.3 0.0 1.0 0 0<br />
u1t33=0.792<br />
u1t34q=0.3 0.0 1.0 0 0<br />
u1t34=0.816<br />
u1t35q=0.3 0.0 1.0 0 0<br />
u1t35=0.84<br />
u1t36q=0.3 0.0 1.0 0 0<br />
u1t36=0.864<br />
u1t37q=0.3 0.0 1.0 0 0<br />
u1t37=0.888<br />
u1t38q=0.3 0.0 1.0 0 0<br />
u1t38=0.912<br />
u1t39q=0.3 0.0 1.0 0 0<br />
u1t39=0.936<br />
u1t40q=0.3 0.0 1.0 0 0<br />
u1t40=0.96<br />
u1t41q=0.3 0.0 1.0 0 0<br />
u1t41=0.984<br />
u1t42q=0.3 0.0 1.0 0 0<br />
u1t42=1.008<br />
u1t43q=0.3 0.0 1.0 0 0<br />
u1t43=1.032<br />
u1t44q=0.3 0.0 1.0 0 0<br />
u1t44=1.056<br />
u1t45q=0.3 0.0 1.0 0 0<br />
u1t45=1.08<br />
u1t46q=0.3 0.0 1.0 0 0<br />
u1t46=1.104<br />
u1t47q=0.3 0.0 1.0 0 0<br />
u1t47=1.128<br />
u1t48q=0.3 0.0 1.0 0 0<br />
u1t48=1.152<br />
u1t49q=0.3 0.0 1.0 0 0<br />
u1t49=1.176<br />
u1t50q=0.3 0.0 1.0 0 0<br />
u1t50=1.2<br />
u1t51q=0.3 0.0 1.0 0 0<br />
u1t51=1.224<br />
u1t52q=0.3 0.0 1.0 0 0<br />
u1t52=1.248<br />
u1t53q=0.3 0.0 1.0 0 0<br />
u1t53=1.272<br />
u1t54q=0.3 0.0 1.0 0 0<br />
u1t54=1.296<br />
u1t55q=0.3 0.0 1.0 0 0<br />
u1t55=1.32<br />
u1t56q=0.3 0.0 1.0 0 0<br />
u1t56=1.344<br />
u1t57q=0.3 0.0 1.0 0 0<br />
u1t57=1.368<br />
u1t58q=0.3 0.0 1.0 0 0<br />
u1t58=1.392<br />
u1t59q=0.3 0.0 1.0 0 0<br />
u1t59=1.416<br />
u1t60q=0.3 0.0 1.0 0 0<br />
u1t60=1.44<br />
u1t61q=0.3 0.0 1.0 0 0<br />
u1t61=1.464<br />
u1t62q=0.3 0.0 1.0 0 0<br />
u1t62=1.488<br />
u1t63q=0.3 0.0 1.0 0 0<br />
u1t63=1.512<br />
u1t64q=0.3 0.0 1.0 0 0<br />
u1t64=1.536<br />
u1t65q=0.3 0.0 1.0 0 0<br />
u1t65=1.56<br />
u1t66q=0.3 0.0 1.0 0 0<br />
u1t66=1.584<br />
u1t67q=0.3 0.0 1.0 0 0<br />
u1t67=1.608<br />
u1t68q=0.3 0.0 1.0 0 0<br />
u1t68=1.632<br />
u1t69q=0.3 0.0 1.0 0 0<br />
u1t69=1.656<br />
u1t70q=0.3 0.0 1.0 0 0<br />
u1t70=1.68<br />
u1t71q=0.3 0.0 1.0 0 0<br />
u1t71=1.704<br />
u1t72q=0.3 0.0 1.0 0 0<br />
u1t72=1.728<br />
u1t73q=0.3 0.0 1.0 0 0<br />
u1t73=1.752<br />
u1t74q=0.3 0.0 1.0 0 0<br />
u1t74=1.776<br />
u1t75q=0.3 0.0 1.0 0 0<br />
u1t75=1.8<br />
u1t76q=0.3 0.0 1.0 0 0<br />
u1t76=1.824<br />
u1t77q=0.3 0.0 1.0 0 0<br />
u1t77=1.848<br />
u1t78q=0.3 0.0 1.0 0 0<br />
u1t78=1.872<br />
u1t79q=0.3 0.0 1.0 0 0<br />
u1t79=1.896<br />
u1t80q=0.3 0.0 1.0 0 0<br />
u1t80=1.92<br />
u1t81q=0.3 0.0 1.0 0 0<br />
u1t81=1.944<br />
u1t82q=0.3 0.0 1.0 0 0<br />
u1t82=1.968<br />
u1t83q=0.3 0.0 1.0 0 0<br />
u1t83=1.992<br />
u1t84q=0.3 0.0 1.0 0 0<br />
u1t84=2.016<br />
u1t85q=0.3 0.0 1.0 0 0<br />
u1t85=2.04<br />
u1t86q=0.3 0.0 1.0 0 0<br />
u1t86=2.064<br />
u1t87q=0.3 0.0 1.0 0 0<br />
u1t87=2.088<br />
u1t88q=0.3 0.0 1.0 0 0<br />
u1t88=2.112<br />
u1t89q=0.3 0.0 1.0 0 0<br />
u1t89=2.136<br />
u1t90q=0.3 0.0 1.0 0 0<br />
u1t90=2.16<br />
u1t91q=0.3 0.0 1.0 0 0<br />
u1t91=2.184<br />
u1t92q=0.3 0.0 1.0 0 0<br />
u1t92=2.208<br />
u1t93q=0.3 0.0 1.0 0 0<br />
u1t93=2.232<br />
u1t94q=0.3 0.0 1.0 0 0<br />
u1t94=2.256<br />
u1t95q=0.3 0.0 1.0 0 0<br />
u1t95=2.28<br />
u1t96q=0.3 0.0 1.0 0 0<br />
u1t96=2.304<br />
u1t97q=0.3 0.0 1.0 0 0<br />
u1t97=2.328<br />
u1t98q=0.3 0.0 1.0 0 0<br />
u1t98=2.352<br />
u1t99q=0.3 0.0 1.0 0 0<br />
u1t99=2.376<br />
u1t100q=0.3 0.0 1.0 0 0<br />
u1t100=2.4<br />
u1t101q=0.3 0.0 1.0 0 0<br />
u1t101=2.424<br />
u1t102q=0.3 0.0 1.0 0 0<br />
u1t102=2.448<br />
u1t103q=0.3 0.0 1.0 0 0<br />
u1t103=2.472<br />
u1t104q=0.3 0.0 1.0 0 0<br />
u1t104=2.496<br />
u1t105q=0.3 0.0 1.0 0 0<br />
u1t105=2.52<br />
u1t106q=0.3 0.0 1.0 0 0<br />
u1t106=2.544<br />
u1t107q=0.3 0.0 1.0 0 0<br />
u1t107=2.568<br />
u1t108q=0.3 0.0 1.0 0 0<br />
u1t108=2.592<br />
u1t109q=0.3 0.0 1.0 0 0<br />
u1t109=2.616<br />
u1t110q=0.3 0.0 1.0 0 0<br />
u1t110=2.64<br />
u1t111q=0.3 0.0 1.0 0 0<br />
u1t111=2.664<br />
u1t112q=0.3 0.0 1.0 0 0<br />
u1t112=2.688<br />
u1t113q=0.3 0.0 1.0 0 0<br />
u1t113=2.712<br />
u1t114q=0.3 0.0 1.0 0 0<br />
u1t114=2.736<br />
u1t115q=0.3 0.0 1.0 0 0<br />
u1t115=2.76<br />
u1t116q=0.3 0.0 1.0 0 0<br />
u1t116=2.784<br />
u1t117q=0.3 0.0 1.0 0 0<br />
u1t117=2.808<br />
u1t118q=0.3 0.0 1.0 0 0<br />
u1t118=2.832<br />
u1t119q=0.3 0.0 1.0 0 0<br />
u1t119=2.856<br />
u1t120q=0.3 0.0 1.0 0 0<br />
u1t120=2.88<br />
u1t121q=0.3 0.0 1.0 0 0<br />
u1t121=2.904<br />
u1t122q=0.3 0.0 1.0 0 0<br />
u1t122=2.928<br />
u1t123q=0.3 0.0 1.0 0 0<br />
u1t123=2.952<br />
u1t124q=0.3 0.0 1.0 0 0<br />
u1t124=2.976<br />
u1t125q=0.3 0.0 1.0 0 0<br />
u1t125=3.0<br />
u1t126q=0.3 0.0 1.0 0 0<br />
u1t126=3.024<br />
u1t127q=0.3 0.0 1.0 0 0<br />
u1t127=3.048<br />
u1t128q=0.3 0.0 1.0 0 0<br />
u1t128=3.072<br />
u1t129q=0.3 0.0 1.0 0 0<br />
u1t129=3.096<br />
u1t130q=0.3 0.0 1.0 0 0<br />
u1t130=3.12<br />
u1t131q=0.3 0.0 1.0 0 0<br />
u1t131=3.144<br />
u1t132q=0.3 0.0 1.0 0 0<br />
u1t132=3.168<br />
u1t133q=0.3 0.0 1.0 0 0<br />
u1t133=3.192<br />
u1t134q=0.3 0.0 1.0 0 0<br />
u1t134=3.216<br />
u1t135q=0.3 0.0 1.0 0 0<br />
u1t135=3.24<br />
u1t136q=0.3 0.0 1.0 0 0<br />
u1t136=3.264<br />
u1t137q=0.3 0.0 1.0 0 0<br />
u1t137=3.288<br />
u1t138q=0.3 0.0 1.0 0 0<br />
u1t138=3.312<br />
u1t139q=0.3 0.0 1.0 0 0<br />
u1t139=3.336<br />
u1t140q=0.3 0.0 1.0 0 0<br />
u1t140=3.36<br />
u1t141q=0.3 0.0 1.0 0 0<br />
u1t141=3.384<br />
u1t142q=0.3 0.0 1.0 0 0<br />
u1t142=3.408<br />
u1t143q=0.3 0.0 1.0 0 0<br />
u1t143=3.432<br />
u1t144q=0.3 0.0 1.0 0 0<br />
u1t144=3.456<br />
u1t145q=0.3 0.0 1.0 0 0<br />
u1t145=3.48<br />
u1t146q=0.3 0.0 1.0 0 0<br />
u1t146=3.504<br />
u1t147q=0.3 0.0 1.0 0 0<br />
u1t147=3.528<br />
u1t148q=0.3 0.0 1.0 0 0<br />
u1t148=3.552<br />
u1t149q=0.3 0.0 1.0 0 0<br />
u1t149=3.576<br />
u1t150q=0.3 0.0 1.0 0 0<br />
u1t150=3.6<br />
u1t151q=0.3 0.0 1.0 0 0<br />
u1t151=3.624<br />
u1t152q=0.3 0.0 1.0 0 0<br />
u1t152=3.648<br />
u1t153q=0.3 0.0 1.0 0 0<br />
u1t153=3.672<br />
u1t154q=0.3 0.0 1.0 0 0<br />
u1t154=3.696<br />
u1t155q=0.3 0.0 1.0 0 0<br />
u1t155=3.72<br />
u1t156q=0.3 0.0 1.0 0 0<br />
u1t156=3.744<br />
u1t157q=0.3 0.0 1.0 0 0<br />
u1t157=3.768<br />
u1t158q=0.3 0.0 1.0 0 0<br />
u1t158=3.792<br />
u1t159q=0.3 0.0 1.0 0 0<br />
u1t159=3.816<br />
u1t160q=0.3 0.0 1.0 0 0<br />
u1t160=3.84<br />
u1t161q=0.3 0.0 1.0 0 0<br />
u1t161=3.864<br />
u1t162q=0.3 0.0 1.0 0 0<br />
u1t162=3.888<br />
u1t163q=0.3 0.0 1.0 0 0<br />
u1t163=3.912<br />
u1t164q=0.3 0.0 1.0 0 0<br />
u1t164=3.936<br />
u1t165q=0.3 0.0 1.0 0 0<br />
u1t165=3.96<br />
u1t166q=0.3 0.0 1.0 0 0<br />
u1t166=3.984<br />
u1t167q=0.3 0.0 1.0 0 0<br />
u1t167=4.008<br />
u1t168q=0.3 0.0 1.0 0 0<br />
u1t168=4.032<br />
u1t169q=0.3 0.0 1.0 0 0<br />
u1t169=4.056<br />
u1t170q=0.3 0.0 1.0 0 0<br />
u1t170=4.08<br />
u1t171q=0.3 0.0 1.0 0 0<br />
u1t171=4.104<br />
u1t172q=0.3 0.0 1.0 0 0<br />
u1t172=4.128<br />
u1t173q=0.3 0.0 1.0 0 0<br />
u1t173=4.152<br />
u1t174q=0.3 0.0 1.0 0 0<br />
u1t174=4.176<br />
u1t175q=0.3 0.0 1.0 0 0<br />
u1t175=4.2<br />
u1t176q=0.3 0.0 1.0 0 0<br />
u1t176=4.224<br />
u1t177q=0.3 0.0 1.0 0 0<br />
u1t177=4.248<br />
u1t178q=0.3 0.0 1.0 0 0<br />
u1t178=4.272<br />
u1t179q=0.3 0.0 1.0 0 0<br />
u1t179=4.296<br />
u1t180q=0.3 0.0 1.0 0 0<br />
u1t180=4.32<br />
u1t181q=0.3 0.0 1.0 0 0<br />
u1t181=4.344<br />
u1t182q=0.3 0.0 1.0 0 0<br />
u1t182=4.368<br />
u1t183q=0.3 0.0 1.0 0 0<br />
u1t183=4.392<br />
u1t184q=0.3 0.0 1.0 0 0<br />
u1t184=4.416<br />
u1t185q=0.3 0.0 1.0 0 0<br />
u1t185=4.44<br />
u1t186q=0.3 0.0 1.0 0 0<br />
u1t186=4.464<br />
u1t187q=0.3 0.0 1.0 0 0<br />
u1t187=4.488<br />
u1t188q=0.3 0.0 1.0 0 0<br />
u1t188=4.512<br />
u1t189q=0.3 0.0 1.0 0 0<br />
u1t189=4.536<br />
u1t190q=0.3 0.0 1.0 0 0<br />
u1t190=4.56<br />
u1t191q=0.3 0.0 1.0 0 0<br />
u1t191=4.584<br />
u1t192q=0.3 0.0 1.0 0 0<br />
u1t192=4.608<br />
u1t193q=0.3 0.0 1.0 0 0<br />
u1t193=4.632<br />
u1t194q=0.3 0.0 1.0 0 0<br />
u1t194=4.656<br />
u1t195q=0.3 0.0 1.0 0 0<br />
u1t195=4.68<br />
u1t196q=0.3 0.0 1.0 0 0<br />
u1t196=4.704<br />
u1t197q=0.3 0.0 1.0 0 0<br />
u1t197=4.728<br />
u1t198q=0.3 0.0 1.0 0 0<br />
u1t198=4.752<br />
u1t199q=0.3 0.0 1.0 0 0<br />
u1t199=4.776<br />
u1t200q=0.3 0.0 1.0 0 0<br />
u1t200=4.8<br />
u1t201q=0.3 0.0 1.0 0 0<br />
u1t201=4.824<br />
u1t202q=0.3 0.0 1.0 0 0<br />
u1t202=4.848<br />
u1t203q=0.3 0.0 1.0 0 0<br />
u1t203=4.872<br />
u1t204q=0.3 0.0 1.0 0 0<br />
u1t204=4.896<br />
u1t205q=0.3 0.0 1.0 0 0<br />
u1t205=4.92<br />
u1t206q=0.3 0.0 1.0 0 0<br />
u1t206=4.944<br />
u1t207q=0.3 0.0 1.0 0 0<br />
u1t207=4.968<br />
u1t208q=0.3 0.0 1.0 0 0<br />
u1t208=4.992<br />
u1t209q=0.3 0.0 1.0 0 0<br />
u1t209=5.016<br />
u1t210q=0.3 0.0 1.0 0 0<br />
u1t210=5.04<br />
u1t211q=0.3 0.0 1.0 0 0<br />
u1t211=5.064<br />
u1t212q=0.3 0.0 1.0 0 0<br />
u1t212=5.088<br />
u1t213q=0.3 0.0 1.0 0 0<br />
u1t213=5.112<br />
u1t214q=0.3 0.0 1.0 0 0<br />
u1t214=5.136<br />
u1t215q=0.3 0.0 1.0 0 0<br />
u1t215=5.16<br />
u1t216q=0.3 0.0 1.0 0 0<br />
u1t216=5.184<br />
u1t217q=0.3 0.0 1.0 0 0<br />
u1t217=5.208<br />
u1t218q=0.3 0.0 1.0 0 0<br />
u1t218=5.232<br />
u1t219q=0.3 0.0 1.0 0 0<br />
u1t219=5.256<br />
u1t220q=0.3 0.0 1.0 0 0<br />
u1t220=5.28<br />
u1t221q=0.3 0.0 1.0 0 0<br />
u1t221=5.304<br />
u1t222q=0.3 0.0 1.0 0 0<br />
u1t222=5.328<br />
u1t223q=0.3 0.0 1.0 0 0<br />
u1t223=5.352<br />
u1t224q=0.3 0.0 1.0 0 0<br />
u1t224=5.376<br />
u1t225q=0.3 0.0 1.0 0 0<br />
u1t225=5.4<br />
u1t226q=0.3 0.0 1.0 0 0<br />
u1t226=5.424<br />
u1t227q=0.3 0.0 1.0 0 0<br />
u1t227=5.448<br />
u1t228q=0.3 0.0 1.0 0 0<br />
u1t228=5.472<br />
u1t229q=0.3 0.0 1.0 0 0<br />
u1t229=5.496<br />
u1t230q=0.3 0.0 1.0 0 0<br />
u1t230=5.52<br />
u1t231q=0.3 0.0 1.0 0 0<br />
u1t231=5.544<br />
u1t232q=0.3 0.0 1.0 0 0<br />
u1t232=5.568<br />
u1t233q=0.3 0.0 1.0 0 0<br />
u1t233=5.592<br />
u1t234q=0.3 0.0 1.0 0 0<br />
u1t234=5.616<br />
u1t235q=0.3 0.0 1.0 0 0<br />
u1t235=5.64<br />
u1t236q=0.3 0.0 1.0 0 0<br />
u1t236=5.664<br />
u1t237q=0.3 0.0 1.0 0 0<br />
u1t237=5.688<br />
u1t238q=0.3 0.0 1.0 0 0<br />
u1t238=5.712<br />
u1t239q=0.3 0.0 1.0 0 0<br />
u1t239=5.736<br />
u1t240q=0.3 0.0 1.0 0 0<br />
u1t240=5.76<br />
u1t241q=0.3 0.0 1.0 0 0<br />
u1t241=5.784<br />
u1t242q=0.3 0.0 1.0 0 0<br />
u1t242=5.808<br />
u1t243q=0.3 0.0 1.0 0 0<br />
u1t243=5.832<br />
u1t244q=0.3 0.0 1.0 0 0<br />
u1t244=5.856<br />
u1t245q=0.3 0.0 1.0 0 0<br />
u1t245=5.88<br />
u1t246q=0.3 0.0 1.0 0 0<br />
u1t246=5.904<br />
u1t247q=0.3 0.0 1.0 0 0<br />
u1t247=5.928<br />
u1t248q=0.3 0.0 1.0 0 0<br />
u1t248=5.952<br />
u1t249q=0.3 0.0 1.0 0 0<br />
u1t249=5.976<br />
u1t250q=0.3 0.0 1.0 0 0<br />
u1t250=6.0<br />
u1t251q=0.3 0.0 1.0 0 0<br />
u1t251=6.024<br />
u1t252q=0.3 0.0 1.0 0 0<br />
u1t252=6.048<br />
u1t253q=0.3 0.0 1.0 0 0<br />
u1t253=6.072<br />
u1t254q=0.3 0.0 1.0 0 0<br />
u1t254=6.096<br />
u1t255q=0.3 0.0 1.0 0 0<br />
u1t255=6.12<br />
u1t256q=0.3 0.0 1.0 0 0<br />
u1t256=6.144<br />
u1t257q=0.3 0.0 1.0 0 0<br />
u1t257=6.168<br />
u1t258q=0.3 0.0 1.0 0 0<br />
u1t258=6.192<br />
u1t259q=0.3 0.0 1.0 0 0<br />
u1t259=6.216<br />
u1t260q=0.3 0.0 1.0 0 0<br />
u1t260=6.24<br />
u1t261q=0.3 0.0 1.0 0 0<br />
u1t261=6.264<br />
u1t262q=0.3 0.0 1.0 0 0<br />
u1t262=6.288<br />
u1t263q=0.3 0.0 1.0 0 0<br />
u1t263=6.312<br />
u1t264q=0.3 0.0 1.0 0 0<br />
u1t264=6.336<br />
u1t265q=0.3 0.0 1.0 0 0<br />
u1t265=6.36<br />
u1t266q=0.3 0.0 1.0 0 0<br />
u1t266=6.384<br />
u1t267q=0.3 0.0 1.0 0 0<br />
u1t267=6.408<br />
u1t268q=0.3 0.0 1.0 0 0<br />
u1t268=6.432<br />
u1t269q=0.3 0.0 1.0 0 0<br />
u1t269=6.456<br />
u1t270q=0.3 0.0 1.0 0 0<br />
u1t270=6.48<br />
u1t271q=0.3 0.0 1.0 0 0<br />
u1t271=6.504<br />
u1t272q=0.3 0.0 1.0 0 0<br />
u1t272=6.528<br />
u1t273q=0.3 0.0 1.0 0 0<br />
u1t273=6.552<br />
u1t274q=0.3 0.0 1.0 0 0<br />
u1t274=6.576<br />
u1t275q=0.3 0.0 1.0 0 0<br />
u1t275=6.6<br />
u1t276q=0.3 0.0 1.0 0 0<br />
u1t276=6.624<br />
u1t277q=0.3 0.0 1.0 0 0<br />
u1t277=6.648<br />
u1t278q=0.3 0.0 1.0 0 0<br />
u1t278=6.672<br />
u1t279q=0.3 0.0 1.0 0 0<br />
u1t279=6.696<br />
u1t280q=0.3 0.0 1.0 0 0<br />
u1t280=6.72<br />
u1t281q=0.3 0.0 1.0 0 0<br />
u1t281=6.744<br />
u1t282q=0.3 0.0 1.0 0 0<br />
u1t282=6.768<br />
u1t283q=0.3 0.0 1.0 0 0<br />
u1t283=6.792<br />
u1t284q=0.3 0.0 1.0 0 0<br />
u1t284=6.816<br />
u1t285q=0.3 0.0 1.0 0 0<br />
u1t285=6.84<br />
u1t286q=0.3 0.0 1.0 0 0<br />
u1t286=6.864<br />
u1t287q=0.3 0.0 1.0 0 0<br />
u1t287=6.888<br />
u1t288q=0.3 0.0 1.0 0 0<br />
u1t288=6.912<br />
u1t289q=0.3 0.0 1.0 0 0<br />
u1t289=6.936<br />
u1t290q=0.3 0.0 1.0 0 0<br />
u1t290=6.96<br />
u1t291q=0.3 0.0 1.0 0 0<br />
u1t291=6.984<br />
u1t292q=0.3 0.0 1.0 0 0<br />
u1t292=7.008<br />
u1t293q=0.3 0.0 1.0 0 0<br />
u1t293=7.032<br />
u1t294q=0.3 0.0 1.0 0 0<br />
u1t294=7.056<br />
u1t295q=0.3 0.0 1.0 0 0<br />
u1t295=7.08<br />
u1t296q=0.3 0.0 1.0 0 0<br />
u1t296=7.104<br />
u1t297q=0.3 0.0 1.0 0 0<br />
u1t297=7.128<br />
u1t298q=0.3 0.0 1.0 0 0<br />
u1t298=7.152<br />
u1t299q=0.3 0.0 1.0 0 0<br />
u1t299=7.176<br />
u1t300q=0.3 0.0 1.0 0 0<br />
u1t300=7.2<br />
u1t301q=0.3 0.0 1.0 0 0<br />
u1t301=7.224<br />
u1t302q=0.3 0.0 1.0 0 0<br />
u1t302=7.248<br />
u1t303q=0.3 0.0 1.0 0 0<br />
u1t303=7.272<br />
u1t304q=0.3 0.0 1.0 0 0<br />
u1t304=7.296<br />
u1t305q=0.3 0.0 1.0 0 0<br />
u1t305=7.32<br />
u1t306q=0.3 0.0 1.0 0 0<br />
u1t306=7.344<br />
u1t307q=0.3 0.0 1.0 0 0<br />
u1t307=7.368<br />
u1t308q=0.3 0.0 1.0 0 0<br />
u1t308=7.392<br />
u1t309q=0.3 0.0 1.0 0 0<br />
u1t309=7.416<br />
u1t310q=0.3 0.0 1.0 0 0<br />
u1t310=7.44<br />
u1t311q=0.3 0.0 1.0 0 0<br />
u1t311=7.464<br />
u1t312q=0.3 0.0 1.0 0 0<br />
u1t312=7.488<br />
u1t313q=0.3 0.0 1.0 0 0<br />
u1t313=7.512<br />
u1t314q=0.3 0.0 1.0 0 0<br />
u1t314=7.536<br />
u1t315q=0.3 0.0 1.0 0 0<br />
u1t315=7.56<br />
u1t316q=0.3 0.0 1.0 0 0<br />
u1t316=7.584<br />
u1t317q=0.3 0.0 1.0 0 0<br />
u1t317=7.608<br />
u1t318q=0.3 0.0 1.0 0 0<br />
u1t318=7.632<br />
u1t319q=0.3 0.0 1.0 0 0<br />
u1t319=7.656<br />
u1t320q=0.3 0.0 1.0 0 0<br />
u1t320=7.68<br />
u1t321q=0.3 0.0 1.0 0 0<br />
u1t321=7.704<br />
u1t322q=0.3 0.0 1.0 0 0<br />
u1t322=7.728<br />
u1t323q=0.3 0.0 1.0 0 0<br />
u1t323=7.752<br />
u1t324q=0.3 0.0 1.0 0 0<br />
u1t324=7.776<br />
u1t325q=0.3 0.0 1.0 0 0<br />
u1t325=7.8<br />
u1t326q=0.3 0.0 1.0 0 0<br />
u1t326=7.824<br />
u1t327q=0.3 0.0 1.0 0 0<br />
u1t327=7.848<br />
u1t328q=0.3 0.0 1.0 0 0<br />
u1t328=7.872<br />
u1t329q=0.3 0.0 1.0 0 0<br />
u1t329=7.896<br />
u1t330q=0.3 0.0 1.0 0 0<br />
u1t330=7.92<br />
u1t331q=0.3 0.0 1.0 0 0<br />
u1t331=7.944<br />
u1t332q=0.3 0.0 1.0 0 0<br />
u1t332=7.968<br />
u1t333q=0.3 0.0 1.0 0 0<br />
u1t333=7.992<br />
u1t334q=0.3 0.0 1.0 0 0<br />
u1t334=8.016<br />
u1t335q=0.3 0.0 1.0 0 0<br />
u1t335=8.04<br />
u1t336q=0.3 0.0 1.0 0 0<br />
u1t336=8.064<br />
u1t337q=0.3 0.0 1.0 0 0<br />
u1t337=8.088<br />
u1t338q=0.3 0.0 1.0 0 0<br />
u1t338=8.112<br />
u1t339q=0.3 0.0 1.0 0 0<br />
u1t339=8.136<br />
u1t340q=0.3 0.0 1.0 0 0<br />
u1t340=8.16<br />
u1t341q=0.3 0.0 1.0 0 0<br />
u1t341=8.184<br />
u1t342q=0.3 0.0 1.0 0 0<br />
u1t342=8.208<br />
u1t343q=0.3 0.0 1.0 0 0<br />
u1t343=8.232<br />
u1t344q=0.3 0.0 1.0 0 0<br />
u1t344=8.256<br />
u1t345q=0.3 0.0 1.0 0 0<br />
u1t345=8.28<br />
u1t346q=0.3 0.0 1.0 0 0<br />
u1t346=8.304<br />
u1t347q=0.3 0.0 1.0 0 0<br />
u1t347=8.328<br />
u1t348q=0.3 0.0 1.0 0 0<br />
u1t348=8.352<br />
u1t349q=0.3 0.0 1.0 0 0<br />
u1t349=8.376<br />
u1t350q=0.3 0.0 1.0 0 0<br />
u1t350=8.4<br />
u1t351q=0.3 0.0 1.0 0 0<br />
u1t351=8.424<br />
u1t352q=0.3 0.0 1.0 0 0<br />
u1t352=8.448<br />
u1t353q=0.3 0.0 1.0 0 0<br />
u1t353=8.472<br />
u1t354q=0.3 0.0 1.0 0 0<br />
u1t354=8.496<br />
u1t355q=0.3 0.0 1.0 0 0<br />
u1t355=8.52<br />
u1t356q=0.3 0.0 1.0 0 0<br />
u1t356=8.544<br />
u1t357q=0.3 0.0 1.0 0 0<br />
u1t357=8.568<br />
u1t358q=0.3 0.0 1.0 0 0<br />
u1t358=8.592<br />
u1t359q=0.3 0.0 1.0 0 0<br />
u1t359=8.616<br />
u1t360q=0.3 0.0 1.0 0 0<br />
u1t360=8.64<br />
u1t361q=0.3 0.0 1.0 0 0<br />
u1t361=8.664<br />
u1t362q=0.3 0.0 1.0 0 0<br />
u1t362=8.688<br />
u1t363q=0.3 0.0 1.0 0 0<br />
u1t363=8.712<br />
u1t364q=0.3 0.0 1.0 0 0<br />
u1t364=8.736<br />
u1t365q=0.3 0.0 1.0 0 0<br />
u1t365=8.76<br />
u1t366q=0.3 0.0 1.0 0 0<br />
u1t366=8.784<br />
u1t367q=0.3 0.0 1.0 0 0<br />
u1t367=8.808<br />
u1t368q=0.3 0.0 1.0 0 0<br />
u1t368=8.832<br />
u1t369q=0.3 0.0 1.0 0 0<br />
u1t369=8.856<br />
u1t370q=0.3 0.0 1.0 0 0<br />
u1t370=8.88<br />
u1t371q=0.3 0.0 1.0 0 0<br />
u1t371=8.904<br />
u1t372q=0.3 0.0 1.0 0 0<br />
u1t372=8.928<br />
u1t373q=0.3 0.0 1.0 0 0<br />
u1t373=8.952<br />
u1t374q=0.3 0.0 1.0 0 0<br />
u1t374=8.976<br />
u1t375q=0.3 0.0 1.0 0 0<br />
u1t375=9.0<br />
u1t376q=0.3 0.0 1.0 0 0<br />
u1t376=9.024<br />
u1t377q=0.3 0.0 1.0 0 0<br />
u1t377=9.048<br />
u1t378q=0.3 0.0 1.0 0 0<br />
u1t378=9.072<br />
u1t379q=0.3 0.0 1.0 0 0<br />
u1t379=9.096<br />
u1t380q=0.3 0.0 1.0 0 0<br />
u1t380=9.12<br />
u1t381q=0.3 0.0 1.0 0 0<br />
u1t381=9.144<br />
u1t382q=0.3 0.0 1.0 0 0<br />
u1t382=9.168<br />
u1t383q=0.3 0.0 1.0 0 0<br />
u1t383=9.192<br />
u1t384q=0.3 0.0 1.0 0 0<br />
u1t384=9.216<br />
u1t385q=0.3 0.0 1.0 0 0<br />
u1t385=9.24<br />
u1t386q=0.3 0.0 1.0 0 0<br />
u1t386=9.264<br />
u1t387q=0.3 0.0 1.0 0 0<br />
u1t387=9.288<br />
u1t388q=0.3 0.0 1.0 0 0<br />
u1t388=9.312<br />
u1t389q=0.3 0.0 1.0 0 0<br />
u1t389=9.336<br />
u1t390q=0.3 0.0 1.0 0 0<br />
u1t390=9.36<br />
u1t391q=0.3 0.0 1.0 0 0<br />
u1t391=9.384<br />
u1t392q=0.3 0.0 1.0 0 0<br />
u1t392=9.408<br />
u1t393q=0.3 0.0 1.0 0 0<br />
u1t393=9.432<br />
u1t394q=0.3 0.0 1.0 0 0<br />
u1t394=9.456<br />
u1t395q=0.3 0.0 1.0 0 0<br />
u1t395=9.48<br />
u1t396q=0.3 0.0 1.0 0 0<br />
u1t396=9.504<br />
u1t397q=0.3 0.0 1.0 0 0<br />
u1t397=9.528<br />
u1t398q=0.3 0.0 1.0 0 0<br />
u1t398=9.552<br />
u1t399q=0.3 0.0 1.0 0 0<br />
u1t399=9.576<br />
u1t400q=0.3 0.0 1.0 0 0<br />
u1t400=9.6<br />
u1t401q=0.3 0.0 1.0 0 0<br />
u1t401=9.624<br />
u1t402q=0.3 0.0 1.0 0 0<br />
u1t402=9.648<br />
u1t403q=0.3 0.0 1.0 0 0<br />
u1t403=9.672<br />
u1t404q=0.3 0.0 1.0 0 0<br />
u1t404=9.696<br />
u1t405q=0.3 0.0 1.0 0 0<br />
u1t405=9.72<br />
u1t406q=0.3 0.0 1.0 0 0<br />
u1t406=9.744<br />
u1t407q=0.3 0.0 1.0 0 0<br />
u1t407=9.768<br />
u1t408q=0.3 0.0 1.0 0 0<br />
u1t408=9.792<br />
u1t409q=0.3 0.0 1.0 0 0<br />
u1t409=9.816<br />
u1t410q=0.3 0.0 1.0 0 0<br />
u1t410=9.84<br />
u1t411q=0.3 0.0 1.0 0 0<br />
u1t411=9.864<br />
u1t412q=0.3 0.0 1.0 0 0<br />
u1t412=9.888<br />
u1t413q=0.3 0.0 1.0 0 0<br />
u1t413=9.912<br />
u1t414q=0.3 0.0 1.0 0 0<br />
u1t414=9.936<br />
u1t415q=0.3 0.0 1.0 0 0<br />
u1t415=9.96<br />
u1t416q=0.3 0.0 1.0 0 0<br />
u1t416=9.984<br />
u1t417q=0.3 0.0 1.0 0 0<br />
u1t417=10.008<br />
u1t418q=0.3 0.0 1.0 0 0<br />
u1t418=10.032<br />
u1t419q=0.3 0.0 1.0 0 0<br />
u1t419=10.056<br />
u1t420q=0.3 0.0 1.0 0 0<br />
u1t420=10.08<br />
u1t421q=0.3 0.0 1.0 0 0<br />
u1t421=10.104<br />
u1t422q=0.3 0.0 1.0 0 0<br />
u1t422=10.128<br />
u1t423q=0.3 0.0 1.0 0 0<br />
u1t423=10.152<br />
u1t424q=0.3 0.0 1.0 0 0<br />
u1t424=10.176<br />
u1t425q=0.3 0.0 1.0 0 0<br />
u1t425=10.2<br />
u1t426q=0.3 0.0 1.0 0 0<br />
u1t426=10.224<br />
u1t427q=0.3 0.0 1.0 0 0<br />
u1t427=10.248<br />
u1t428q=0.3 0.0 1.0 0 0<br />
u1t428=10.272<br />
u1t429q=0.3 0.0 1.0 0 0<br />
u1t429=10.296<br />
u1t430q=0.3 0.0 1.0 0 0<br />
u1t430=10.32<br />
u1t431q=0.3 0.0 1.0 0 0<br />
u1t431=10.344<br />
u1t432q=0.3 0.0 1.0 0 0<br />
u1t432=10.368<br />
u1t433q=0.3 0.0 1.0 0 0<br />
u1t433=10.392<br />
u1t434q=0.3 0.0 1.0 0 0<br />
u1t434=10.416<br />
u1t435q=0.3 0.0 1.0 0 0<br />
u1t435=10.44<br />
u1t436q=0.3 0.0 1.0 0 0<br />
u1t436=10.464<br />
u1t437q=0.3 0.0 1.0 0 0<br />
u1t437=10.488<br />
u1t438q=0.3 0.0 1.0 0 0<br />
u1t438=10.512<br />
u1t439q=0.3 0.0 1.0 0 0<br />
u1t439=10.536<br />
u1t440q=0.3 0.0 1.0 0 0<br />
u1t440=10.56<br />
u1t441q=0.3 0.0 1.0 0 0<br />
u1t441=10.584<br />
u1t442q=0.3 0.0 1.0 0 0<br />
u1t442=10.608<br />
u1t443q=0.3 0.0 1.0 0 0<br />
u1t443=10.632<br />
u1t444q=0.3 0.0 1.0 0 0<br />
u1t444=10.656<br />
u1t445q=0.3 0.0 1.0 0 0<br />
u1t445=10.68<br />
u1t446q=0.3 0.0 1.0 0 0<br />
u1t446=10.704<br />
u1t447q=0.3 0.0 1.0 0 0<br />
u1t447=10.728<br />
u1t448q=0.3 0.0 1.0 0 0<br />
u1t448=10.752<br />
u1t449q=0.3 0.0 1.0 0 0<br />
u1t449=10.776<br />
u1t450q=0.3 0.0 1.0 0 0<br />
u1t450=10.8<br />
u1t451q=0.3 0.0 1.0 0 0<br />
u1t451=10.824<br />
u1t452q=0.3 0.0 1.0 0 0<br />
u1t452=10.848<br />
u1t453q=0.3 0.0 1.0 0 0<br />
u1t453=10.872<br />
u1t454q=0.3 0.0 1.0 0 0<br />
u1t454=10.896<br />
u1t455q=0.3 0.0 1.0 0 0<br />
u1t455=10.92<br />
u1t456q=0.3 0.0 1.0 0 0<br />
u1t456=10.944<br />
u1t457q=0.3 0.0 1.0 0 0<br />
u1t457=10.968<br />
u1t458q=0.3 0.0 1.0 0 0<br />
u1t458=10.992<br />
u1t459q=0.3 0.0 1.0 0 0<br />
u1t459=11.016<br />
u1t460q=0.3 0.0 1.0 0 0<br />
u1t460=11.04<br />
u1t461q=0.3 0.0 1.0 0 0<br />
u1t461=11.064<br />
u1t462q=0.3 0.0 1.0 0 0<br />
u1t462=11.088<br />
u1t463q=0.3 0.0 1.0 0 0<br />
u1t463=11.112<br />
u1t464q=0.3 0.0 1.0 0 0<br />
u1t464=11.136<br />
u1t465q=0.3 0.0 1.0 0 0<br />
u1t465=11.16<br />
u1t466q=0.3 0.0 1.0 0 0<br />
u1t466=11.184<br />
u1t467q=0.3 0.0 1.0 0 0<br />
u1t467=11.208<br />
u1t468q=0.3 0.0 1.0 0 0<br />
u1t468=11.232<br />
u1t469q=0.3 0.0 1.0 0 0<br />
u1t469=11.256<br />
u1t470q=0.3 0.0 1.0 0 0<br />
u1t470=11.28<br />
u1t471q=0.3 0.0 1.0 0 0<br />
u1t471=11.304<br />
u1t472q=0.3 0.0 1.0 0 0<br />
u1t472=11.328<br />
u1t473q=0.3 0.0 1.0 0 0<br />
u1t473=11.352<br />
u1t474q=0.3 0.0 1.0 0 0<br />
u1t474=11.376<br />
u1t475q=0.3 0.0 1.0 0 0<br />
u1t475=11.4<br />
u1t476q=0.3 0.0 1.0 0 0<br />
u1t476=11.424<br />
u1t477q=0.3 0.0 1.0 0 0<br />
u1t477=11.448<br />
u1t478q=0.3 0.0 1.0 0 0<br />
u1t478=11.472<br />
u1t479q=0.3 0.0 1.0 0 0<br />
u1t479=11.496<br />
u1t480q=0.3 0.0 1.0 0 0<br />
u1t480=11.52<br />
u1t481q=0.3 0.0 1.0 0 0<br />
u1t481=11.544<br />
u1t482q=0.3 0.0 1.0 0 0<br />
u1t482=11.568<br />
u1t483q=0.3 0.0 1.0 0 0<br />
u1t483=11.592<br />
u1t484q=0.3 0.0 1.0 0 0<br />
u1t484=11.616<br />
u1t485q=0.3 0.0 1.0 0 0<br />
u1t485=11.64<br />
u1t486q=0.3 0.0 1.0 0 0<br />
u1t486=11.664<br />
u1t487q=0.3 0.0 1.0 0 0<br />
u1t487=11.688<br />
u1t488q=0.3 0.0 1.0 0 0<br />
u1t488=11.712<br />
u1t489q=0.3 0.0 1.0 0 0<br />
u1t489=11.736<br />
u1t490q=0.3 0.0 1.0 0 0<br />
u1t490=11.76<br />
u1t491q=0.3 0.0 1.0 0 0<br />
u1t491=11.784<br />
u1t492q=0.3 0.0 1.0 0 0<br />
u1t492=11.808<br />
u1t493q=0.3 0.0 1.0 0 0<br />
u1t493=11.832<br />
u1t494q=0.3 0.0 1.0 0 0<br />
u1t494=11.856<br />
u1t495q=0.3 0.0 1.0 0 0<br />
u1t495=11.88<br />
u1t496q=0.3 0.0 1.0 0 0<br />
u1t496=11.904<br />
u1t497q=0.3 0.0 1.0 0 0<br />
u1t497=11.928<br />
u1t498q=0.3 0.0 1.0 0 0<br />
u1t498=11.952<br />
u1t499q=0.3 0.0 1.0 0 0<br />
u1t499=11.976<br />
u1t500q=0.3 0 0 0 0<br />
u1t500=tend<br />
<br />
[Messungen]<br />
tAnzahl=12<br />
<br />
t1=1<br />
t1Anzahl=2<br />
t1m1=mfcn1 1.0 1e-06 1<br />
t1m2=mfcn2 1.0 1e-06 1<br />
t1minmax=0 1e+10<br />
<br />
t2=2.000<br />
t2Anzahl=2<br />
t2m1=mfcn1 1.0 1e-06 1<br />
t2m2=mfcn2 1.0 1e-06 1<br />
t2minmax=0 1e+10<br />
<br />
t3=3<br />
t3Anzahl=2<br />
t3m1=mfcn1 1.0 1e-06 1<br />
t3m2=mfcn2 1.0 1e-06 1<br />
t3minmax=0 1e+10<br />
<br />
t4=4<br />
t4Anzahl=2<br />
t4m1=mfcn1 1.0 1e-06 1<br />
t4m2=mfcn2 1.0 1e-06 1<br />
t4minmax=0 1e+10<br />
<br />
t5=5<br />
t5Anzahl=2<br />
t5m1=mfcn1 1.0 1e-06 1<br />
t5m2=mfcn2 1.0 1e-06 1<br />
t5minmax=0 1e+10<br />
<br />
t6=6<br />
t6Anzahl=2<br />
t6m1=mfcn1 1.0 1e-06 1<br />
t6m2=mfcn2 1.0 1e-06 1<br />
t6minmax=0 1e+10<br />
<br />
t7=7<br />
t7Anzahl=2<br />
t7m1=mfcn1 1.0 1e-06 1<br />
t7m2=mfcn2 1.0 1e-06 1<br />
t7minmax=0 1e+10<br />
<br />
t8=8<br />
t8Anzahl=2<br />
t8m1=mfcn1 1.0 1e-06 1<br />
t8m2=mfcn2 1.0 1e-06 1<br />
t8minmax=0 1e+10<br />
<br />
t9=9<br />
t9Anzahl=2<br />
t9m1=mfcn1 1.0 1e-06 1<br />
t9m2=mfcn2 1.0 1e-06 1<br />
t9minmax=0 1e+10<br />
<br />
t10=10.000<br />
t10Anzahl=2<br />
t10m1=mfcn1 1.0 1e-06 1<br />
t10m2=mfcn2 1.0 1e-06 1<br />
t10minmax=0 1e+10<br />
<br />
t11=11.000<br />
t11Anzahl=2<br />
t11m1=mfcn1 1.0 1e-06 1<br />
t11m2=mfcn2 1.0 1e-06 1<br />
t11minmax=0 1e+10<br />
<br />
t12=12.000<br />
t12Anzahl=2<br />
t12m1=mfcn1 1.0 1e-06 1<br />
t12m2=mfcn2 1.0 1e-06 1<br />
t12minmax=0 1e+10<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=0<br />
[Messverfahren]<br />
mAnzahl=2<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
m2=mfcn2 1 0 1e+10 0<br />
m2f1=mess4 sigma4 1<br />
mminmaxges=0 8<br />
<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
realworkspace=1700000<br />
integerworkspace=5000<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
ini-file for running VPLAN:<br />
<source lang="optimica"><br />
<br />
; ini-File fuer VPLAN<br />
<br />
[Aktion]<br />
;aktion=Integration<br />
;aktion=Simulationsumgebung<br />
;aktion=Parameterschaetzung<br />
;aktion=Versuchsplanung<br />
;aktion=ObjectiveTest<br />
;aktion=DerivativeTest<br />
;aktion={ISPS}<br />
aktion={ISCVCS}<br />
<br />
[Pfade]<br />
problempath=lotka_seminar <br />
inpath=in<br />
outpath=simu<br />
messpath=mess<br />
plotpath=plot<br />
fortranpath=fortran<br />
<br />
[Parameter]<br />
pAnzahl=2<br />
p1=p2 1.0 -1e+10 1e+10 0<br />
p2=p4 1.0 -1e+10 1e+10 0<br />
[Versuchsplan]<br />
expAnzahl=1<br />
exp1=exp1.ini exp1.ini<br />
<br />
[Guetekriterium]<br />
Optimierungskriterium=A<br />
AKriterium=-1<br />
DKriterium=-1<br />
EKriterium=-1<br />
MKriterium=-1<br />
covmat=covmat.m<br />
jacmat=jacmat.m<br />
status=undefiniert<br />
<br />
[Residuum]<br />
res=0<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Messdatenfiles]<br />
mess1=mess1.dat <br />
<br />
[Outputfiles]<br />
out1=plot2 0.05 integ.plt.1<br />
<br />
[Residuenfiles]<br />
rsd1=res1.txt<br />
<br />
[ExtensionFlags]<br />
experimenttype=0<br />
integrator=0<br />
dmode=0<br />
pdeFlag=0<br />
<br />
[OptionenAllgemein]<br />
visflag=0<br />
messfileflag=0<br />
seed=-1<br />
numberofthreads=1<br />
robustflag=0<br />
epsmach=0<br />
infinity=1e+10<br />
epsilon=1e-08<br />
conflevel=0.95<br />
hrobust=1e-05<br />
computesigma=0<br />
exitonFPE=1<br />
iniprecision=6<br />
clipboardflag=0<br />
printxi=0<br />
printconstr=0<br />
printcolorful=-1<br />
<br />
[OptionenParameterschaetzung]<br />
eps=0.001<br />
itmax=50<br />
cond=10000<br />
condflag=1<br />
boundcheck=0<br />
startflag=0<br />
index1=1e-08<br />
fashort=0.8<br />
fa0=0.01<br />
farel=0.1<br />
famax=1.0<br />
realworkspace=1000000<br />
integerworkspace=1000000<br />
printlevel=2<br />
method=0<br />
<br />
[OptionenVersuchsplanung]<br />
maxit=300<br />
opttol=1e-06<br />
funcprec=1e-07<br />
linfeas=1e-07<br />
nlinfeas=0.01<br />
maxitQP=300<br />
maxitgesQP=10000<br />
opttolQP=1e-06<br />
pivottolQP=3.7e-11<br />
steplimitLS=2<br />
tolLS=0.9<br />
crashtol=0.0001<br />
elasticweight=100<br />
superbasics=1<br />
scaling=1<br />
sconstraints=0<br />
realworkspace=3000000<br />
integerworkspace=3000000<br />
charworkspace=500<br />
printlevel=10<br />
method=2<br />
<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1345
Lotka Experimental Design (VPLAN)
2016-01-19T16:28:25Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function<br />
<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<br />
<source lang="fortran"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
<br />
<br />
<source lang="optimica"><br />
<br />
; ini-File fuer Experiment<br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=12<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=2<br />
y1=x1 0.5 -1e+10 1e+10<br />
y2=x2 0.7 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=12<br />
t1=1<br />
t2=2<br />
t3=3<br />
t4=4<br />
t5=5<br />
t6=6<br />
t7=7<br />
t8=8<br />
t9=9<br />
t10=10<br />
t11=11<br />
t12=12<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=0<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=myu 0 0.0 1.0<br />
u1tAnzahl=500<br />
u1t0=t0<br />
u1t1q=0.3 0.0 1.0 0 0<br />
u1t1=0.024<br />
u1t2q=0.3 0.0 1.0 0 0<br />
u1t2=0.048<br />
u1t3q=0.3 0.0 1.0 0 0<br />
u1t3=0.072<br />
u1t4q=0.3 0.0 1.0 0 0<br />
u1t4=0.096<br />
u1t5q=0.3 0.0 1.0 0 0<br />
u1t5=0.12<br />
u1t6q=0.3 0.0 1.0 0 0<br />
u1t6=0.144<br />
u1t7q=0.3 0.0 1.0 0 0<br />
u1t7=0.168<br />
u1t8q=0.3 0.0 1.0 0 0<br />
u1t8=0.192<br />
u1t9q=0.3 0.0 1.0 0 0<br />
u1t9=0.216<br />
u1t10q=0.3 0.0 1.0 0 0<br />
u1t10=0.24<br />
u1t11q=0.3 0.0 1.0 0 0<br />
u1t11=0.264<br />
u1t12q=0.3 0.0 1.0 0 0<br />
u1t12=0.288<br />
u1t13q=0.3 0.0 1.0 0 0<br />
u1t13=0.312<br />
u1t14q=0.3 0.0 1.0 0 0<br />
u1t14=0.336<br />
u1t15q=0.3 0.0 1.0 0 0<br />
u1t15=0.36<br />
u1t16q=0.3 0.0 1.0 0 0<br />
u1t16=0.384<br />
u1t17q=0.3 0.0 1.0 0 0<br />
u1t17=0.408<br />
u1t18q=0.3 0.0 1.0 0 0<br />
u1t18=0.432<br />
u1t19q=0.3 0.0 1.0 0 0<br />
u1t19=0.456<br />
u1t20q=0.3 0.0 1.0 0 0<br />
u1t20=0.48<br />
u1t21q=0.3 0.0 1.0 0 0<br />
u1t21=0.504<br />
u1t22q=0.3 0.0 1.0 0 0<br />
u1t22=0.528<br />
u1t23q=0.3 0.0 1.0 0 0<br />
u1t23=0.552<br />
u1t24q=0.3 0.0 1.0 0 0<br />
u1t24=0.576<br />
u1t25q=0.3 0.0 1.0 0 0<br />
u1t25=0.6<br />
u1t26q=0.3 0.0 1.0 0 0<br />
u1t26=0.624<br />
u1t27q=0.3 0.0 1.0 0 0<br />
u1t27=0.648<br />
u1t28q=0.3 0.0 1.0 0 0<br />
u1t28=0.672<br />
u1t29q=0.3 0.0 1.0 0 0<br />
u1t29=0.696<br />
u1t30q=0.3 0.0 1.0 0 0<br />
u1t30=0.72<br />
u1t31q=0.3 0.0 1.0 0 0<br />
u1t31=0.744<br />
u1t32q=0.3 0.0 1.0 0 0<br />
u1t32=0.768<br />
u1t33q=0.3 0.0 1.0 0 0<br />
u1t33=0.792<br />
u1t34q=0.3 0.0 1.0 0 0<br />
u1t34=0.816<br />
u1t35q=0.3 0.0 1.0 0 0<br />
u1t35=0.84<br />
u1t36q=0.3 0.0 1.0 0 0<br />
u1t36=0.864<br />
u1t37q=0.3 0.0 1.0 0 0<br />
u1t37=0.888<br />
u1t38q=0.3 0.0 1.0 0 0<br />
u1t38=0.912<br />
u1t39q=0.3 0.0 1.0 0 0<br />
u1t39=0.936<br />
u1t40q=0.3 0.0 1.0 0 0<br />
u1t40=0.96<br />
u1t41q=0.3 0.0 1.0 0 0<br />
u1t41=0.984<br />
u1t42q=0.3 0.0 1.0 0 0<br />
u1t42=1.008<br />
u1t43q=0.3 0.0 1.0 0 0<br />
u1t43=1.032<br />
u1t44q=0.3 0.0 1.0 0 0<br />
u1t44=1.056<br />
u1t45q=0.3 0.0 1.0 0 0<br />
u1t45=1.08<br />
u1t46q=0.3 0.0 1.0 0 0<br />
u1t46=1.104<br />
u1t47q=0.3 0.0 1.0 0 0<br />
u1t47=1.128<br />
u1t48q=0.3 0.0 1.0 0 0<br />
u1t48=1.152<br />
u1t49q=0.3 0.0 1.0 0 0<br />
u1t49=1.176<br />
u1t50q=0.3 0.0 1.0 0 0<br />
u1t50=1.2<br />
u1t51q=0.3 0.0 1.0 0 0<br />
u1t51=1.224<br />
u1t52q=0.3 0.0 1.0 0 0<br />
u1t52=1.248<br />
u1t53q=0.3 0.0 1.0 0 0<br />
u1t53=1.272<br />
u1t54q=0.3 0.0 1.0 0 0<br />
u1t54=1.296<br />
u1t55q=0.3 0.0 1.0 0 0<br />
u1t55=1.32<br />
u1t56q=0.3 0.0 1.0 0 0<br />
u1t56=1.344<br />
u1t57q=0.3 0.0 1.0 0 0<br />
u1t57=1.368<br />
u1t58q=0.3 0.0 1.0 0 0<br />
u1t58=1.392<br />
u1t59q=0.3 0.0 1.0 0 0<br />
u1t59=1.416<br />
u1t60q=0.3 0.0 1.0 0 0<br />
u1t60=1.44<br />
u1t61q=0.3 0.0 1.0 0 0<br />
u1t61=1.464<br />
u1t62q=0.3 0.0 1.0 0 0<br />
u1t62=1.488<br />
u1t63q=0.3 0.0 1.0 0 0<br />
u1t63=1.512<br />
u1t64q=0.3 0.0 1.0 0 0<br />
u1t64=1.536<br />
u1t65q=0.3 0.0 1.0 0 0<br />
u1t65=1.56<br />
u1t66q=0.3 0.0 1.0 0 0<br />
u1t66=1.584<br />
u1t67q=0.3 0.0 1.0 0 0<br />
u1t67=1.608<br />
u1t68q=0.3 0.0 1.0 0 0<br />
u1t68=1.632<br />
u1t69q=0.3 0.0 1.0 0 0<br />
u1t69=1.656<br />
u1t70q=0.3 0.0 1.0 0 0<br />
u1t70=1.68<br />
u1t71q=0.3 0.0 1.0 0 0<br />
u1t71=1.704<br />
u1t72q=0.3 0.0 1.0 0 0<br />
u1t72=1.728<br />
u1t73q=0.3 0.0 1.0 0 0<br />
u1t73=1.752<br />
u1t74q=0.3 0.0 1.0 0 0<br />
u1t74=1.776<br />
u1t75q=0.3 0.0 1.0 0 0<br />
u1t75=1.8<br />
u1t76q=0.3 0.0 1.0 0 0<br />
u1t76=1.824<br />
u1t77q=0.3 0.0 1.0 0 0<br />
u1t77=1.848<br />
u1t78q=0.3 0.0 1.0 0 0<br />
u1t78=1.872<br />
u1t79q=0.3 0.0 1.0 0 0<br />
u1t79=1.896<br />
u1t80q=0.3 0.0 1.0 0 0<br />
u1t80=1.92<br />
u1t81q=0.3 0.0 1.0 0 0<br />
u1t81=1.944<br />
u1t82q=0.3 0.0 1.0 0 0<br />
u1t82=1.968<br />
u1t83q=0.3 0.0 1.0 0 0<br />
u1t83=1.992<br />
u1t84q=0.3 0.0 1.0 0 0<br />
u1t84=2.016<br />
u1t85q=0.3 0.0 1.0 0 0<br />
u1t85=2.04<br />
u1t86q=0.3 0.0 1.0 0 0<br />
u1t86=2.064<br />
u1t87q=0.3 0.0 1.0 0 0<br />
u1t87=2.088<br />
u1t88q=0.3 0.0 1.0 0 0<br />
u1t88=2.112<br />
u1t89q=0.3 0.0 1.0 0 0<br />
u1t89=2.136<br />
u1t90q=0.3 0.0 1.0 0 0<br />
u1t90=2.16<br />
u1t91q=0.3 0.0 1.0 0 0<br />
u1t91=2.184<br />
u1t92q=0.3 0.0 1.0 0 0<br />
u1t92=2.208<br />
u1t93q=0.3 0.0 1.0 0 0<br />
u1t93=2.232<br />
u1t94q=0.3 0.0 1.0 0 0<br />
u1t94=2.256<br />
u1t95q=0.3 0.0 1.0 0 0<br />
u1t95=2.28<br />
u1t96q=0.3 0.0 1.0 0 0<br />
u1t96=2.304<br />
u1t97q=0.3 0.0 1.0 0 0<br />
u1t97=2.328<br />
u1t98q=0.3 0.0 1.0 0 0<br />
u1t98=2.352<br />
u1t99q=0.3 0.0 1.0 0 0<br />
u1t99=2.376<br />
u1t100q=0.3 0.0 1.0 0 0<br />
u1t100=2.4<br />
u1t101q=0.3 0.0 1.0 0 0<br />
u1t101=2.424<br />
u1t102q=0.3 0.0 1.0 0 0<br />
u1t102=2.448<br />
u1t103q=0.3 0.0 1.0 0 0<br />
u1t103=2.472<br />
u1t104q=0.3 0.0 1.0 0 0<br />
u1t104=2.496<br />
u1t105q=0.3 0.0 1.0 0 0<br />
u1t105=2.52<br />
u1t106q=0.3 0.0 1.0 0 0<br />
u1t106=2.544<br />
u1t107q=0.3 0.0 1.0 0 0<br />
u1t107=2.568<br />
u1t108q=0.3 0.0 1.0 0 0<br />
u1t108=2.592<br />
u1t109q=0.3 0.0 1.0 0 0<br />
u1t109=2.616<br />
u1t110q=0.3 0.0 1.0 0 0<br />
u1t110=2.64<br />
u1t111q=0.3 0.0 1.0 0 0<br />
u1t111=2.664<br />
u1t112q=0.3 0.0 1.0 0 0<br />
u1t112=2.688<br />
u1t113q=0.3 0.0 1.0 0 0<br />
u1t113=2.712<br />
u1t114q=0.3 0.0 1.0 0 0<br />
u1t114=2.736<br />
u1t115q=0.3 0.0 1.0 0 0<br />
u1t115=2.76<br />
u1t116q=0.3 0.0 1.0 0 0<br />
u1t116=2.784<br />
u1t117q=0.3 0.0 1.0 0 0<br />
u1t117=2.808<br />
u1t118q=0.3 0.0 1.0 0 0<br />
u1t118=2.832<br />
u1t119q=0.3 0.0 1.0 0 0<br />
u1t119=2.856<br />
u1t120q=0.3 0.0 1.0 0 0<br />
u1t120=2.88<br />
u1t121q=0.3 0.0 1.0 0 0<br />
u1t121=2.904<br />
u1t122q=0.3 0.0 1.0 0 0<br />
u1t122=2.928<br />
u1t123q=0.3 0.0 1.0 0 0<br />
u1t123=2.952<br />
u1t124q=0.3 0.0 1.0 0 0<br />
u1t124=2.976<br />
u1t125q=0.3 0.0 1.0 0 0<br />
u1t125=3.0<br />
u1t126q=0.3 0.0 1.0 0 0<br />
u1t126=3.024<br />
u1t127q=0.3 0.0 1.0 0 0<br />
u1t127=3.048<br />
u1t128q=0.3 0.0 1.0 0 0<br />
u1t128=3.072<br />
u1t129q=0.3 0.0 1.0 0 0<br />
u1t129=3.096<br />
u1t130q=0.3 0.0 1.0 0 0<br />
u1t130=3.12<br />
u1t131q=0.3 0.0 1.0 0 0<br />
u1t131=3.144<br />
u1t132q=0.3 0.0 1.0 0 0<br />
u1t132=3.168<br />
u1t133q=0.3 0.0 1.0 0 0<br />
u1t133=3.192<br />
u1t134q=0.3 0.0 1.0 0 0<br />
u1t134=3.216<br />
u1t135q=0.3 0.0 1.0 0 0<br />
u1t135=3.24<br />
u1t136q=0.3 0.0 1.0 0 0<br />
u1t136=3.264<br />
u1t137q=0.3 0.0 1.0 0 0<br />
u1t137=3.288<br />
u1t138q=0.3 0.0 1.0 0 0<br />
u1t138=3.312<br />
u1t139q=0.3 0.0 1.0 0 0<br />
u1t139=3.336<br />
u1t140q=0.3 0.0 1.0 0 0<br />
u1t140=3.36<br />
u1t141q=0.3 0.0 1.0 0 0<br />
u1t141=3.384<br />
u1t142q=0.3 0.0 1.0 0 0<br />
u1t142=3.408<br />
u1t143q=0.3 0.0 1.0 0 0<br />
u1t143=3.432<br />
u1t144q=0.3 0.0 1.0 0 0<br />
u1t144=3.456<br />
u1t145q=0.3 0.0 1.0 0 0<br />
u1t145=3.48<br />
u1t146q=0.3 0.0 1.0 0 0<br />
u1t146=3.504<br />
u1t147q=0.3 0.0 1.0 0 0<br />
u1t147=3.528<br />
u1t148q=0.3 0.0 1.0 0 0<br />
u1t148=3.552<br />
u1t149q=0.3 0.0 1.0 0 0<br />
u1t149=3.576<br />
u1t150q=0.3 0.0 1.0 0 0<br />
u1t150=3.6<br />
u1t151q=0.3 0.0 1.0 0 0<br />
u1t151=3.624<br />
u1t152q=0.3 0.0 1.0 0 0<br />
u1t152=3.648<br />
u1t153q=0.3 0.0 1.0 0 0<br />
u1t153=3.672<br />
u1t154q=0.3 0.0 1.0 0 0<br />
u1t154=3.696<br />
u1t155q=0.3 0.0 1.0 0 0<br />
u1t155=3.72<br />
u1t156q=0.3 0.0 1.0 0 0<br />
u1t156=3.744<br />
u1t157q=0.3 0.0 1.0 0 0<br />
u1t157=3.768<br />
u1t158q=0.3 0.0 1.0 0 0<br />
u1t158=3.792<br />
u1t159q=0.3 0.0 1.0 0 0<br />
u1t159=3.816<br />
u1t160q=0.3 0.0 1.0 0 0<br />
u1t160=3.84<br />
u1t161q=0.3 0.0 1.0 0 0<br />
u1t161=3.864<br />
u1t162q=0.3 0.0 1.0 0 0<br />
u1t162=3.888<br />
u1t163q=0.3 0.0 1.0 0 0<br />
u1t163=3.912<br />
u1t164q=0.3 0.0 1.0 0 0<br />
u1t164=3.936<br />
u1t165q=0.3 0.0 1.0 0 0<br />
u1t165=3.96<br />
u1t166q=0.3 0.0 1.0 0 0<br />
u1t166=3.984<br />
u1t167q=0.3 0.0 1.0 0 0<br />
u1t167=4.008<br />
u1t168q=0.3 0.0 1.0 0 0<br />
u1t168=4.032<br />
u1t169q=0.3 0.0 1.0 0 0<br />
u1t169=4.056<br />
u1t170q=0.3 0.0 1.0 0 0<br />
u1t170=4.08<br />
u1t171q=0.3 0.0 1.0 0 0<br />
u1t171=4.104<br />
u1t172q=0.3 0.0 1.0 0 0<br />
u1t172=4.128<br />
u1t173q=0.3 0.0 1.0 0 0<br />
u1t173=4.152<br />
u1t174q=0.3 0.0 1.0 0 0<br />
u1t174=4.176<br />
u1t175q=0.3 0.0 1.0 0 0<br />
u1t175=4.2<br />
u1t176q=0.3 0.0 1.0 0 0<br />
u1t176=4.224<br />
u1t177q=0.3 0.0 1.0 0 0<br />
u1t177=4.248<br />
u1t178q=0.3 0.0 1.0 0 0<br />
u1t178=4.272<br />
u1t179q=0.3 0.0 1.0 0 0<br />
u1t179=4.296<br />
u1t180q=0.3 0.0 1.0 0 0<br />
u1t180=4.32<br />
u1t181q=0.3 0.0 1.0 0 0<br />
u1t181=4.344<br />
u1t182q=0.3 0.0 1.0 0 0<br />
u1t182=4.368<br />
u1t183q=0.3 0.0 1.0 0 0<br />
u1t183=4.392<br />
u1t184q=0.3 0.0 1.0 0 0<br />
u1t184=4.416<br />
u1t185q=0.3 0.0 1.0 0 0<br />
u1t185=4.44<br />
u1t186q=0.3 0.0 1.0 0 0<br />
u1t186=4.464<br />
u1t187q=0.3 0.0 1.0 0 0<br />
u1t187=4.488<br />
u1t188q=0.3 0.0 1.0 0 0<br />
u1t188=4.512<br />
u1t189q=0.3 0.0 1.0 0 0<br />
u1t189=4.536<br />
u1t190q=0.3 0.0 1.0 0 0<br />
u1t190=4.56<br />
u1t191q=0.3 0.0 1.0 0 0<br />
u1t191=4.584<br />
u1t192q=0.3 0.0 1.0 0 0<br />
u1t192=4.608<br />
u1t193q=0.3 0.0 1.0 0 0<br />
u1t193=4.632<br />
u1t194q=0.3 0.0 1.0 0 0<br />
u1t194=4.656<br />
u1t195q=0.3 0.0 1.0 0 0<br />
u1t195=4.68<br />
u1t196q=0.3 0.0 1.0 0 0<br />
u1t196=4.704<br />
u1t197q=0.3 0.0 1.0 0 0<br />
u1t197=4.728<br />
u1t198q=0.3 0.0 1.0 0 0<br />
u1t198=4.752<br />
u1t199q=0.3 0.0 1.0 0 0<br />
u1t199=4.776<br />
u1t200q=0.3 0.0 1.0 0 0<br />
u1t200=4.8<br />
u1t201q=0.3 0.0 1.0 0 0<br />
u1t201=4.824<br />
u1t202q=0.3 0.0 1.0 0 0<br />
u1t202=4.848<br />
u1t203q=0.3 0.0 1.0 0 0<br />
u1t203=4.872<br />
u1t204q=0.3 0.0 1.0 0 0<br />
u1t204=4.896<br />
u1t205q=0.3 0.0 1.0 0 0<br />
u1t205=4.92<br />
u1t206q=0.3 0.0 1.0 0 0<br />
u1t206=4.944<br />
u1t207q=0.3 0.0 1.0 0 0<br />
u1t207=4.968<br />
u1t208q=0.3 0.0 1.0 0 0<br />
u1t208=4.992<br />
u1t209q=0.3 0.0 1.0 0 0<br />
u1t209=5.016<br />
u1t210q=0.3 0.0 1.0 0 0<br />
u1t210=5.04<br />
u1t211q=0.3 0.0 1.0 0 0<br />
u1t211=5.064<br />
u1t212q=0.3 0.0 1.0 0 0<br />
u1t212=5.088<br />
u1t213q=0.3 0.0 1.0 0 0<br />
u1t213=5.112<br />
u1t214q=0.3 0.0 1.0 0 0<br />
u1t214=5.136<br />
u1t215q=0.3 0.0 1.0 0 0<br />
u1t215=5.16<br />
u1t216q=0.3 0.0 1.0 0 0<br />
u1t216=5.184<br />
u1t217q=0.3 0.0 1.0 0 0<br />
u1t217=5.208<br />
u1t218q=0.3 0.0 1.0 0 0<br />
u1t218=5.232<br />
u1t219q=0.3 0.0 1.0 0 0<br />
u1t219=5.256<br />
u1t220q=0.3 0.0 1.0 0 0<br />
u1t220=5.28<br />
u1t221q=0.3 0.0 1.0 0 0<br />
u1t221=5.304<br />
u1t222q=0.3 0.0 1.0 0 0<br />
u1t222=5.328<br />
u1t223q=0.3 0.0 1.0 0 0<br />
u1t223=5.352<br />
u1t224q=0.3 0.0 1.0 0 0<br />
u1t224=5.376<br />
u1t225q=0.3 0.0 1.0 0 0<br />
u1t225=5.4<br />
u1t226q=0.3 0.0 1.0 0 0<br />
u1t226=5.424<br />
u1t227q=0.3 0.0 1.0 0 0<br />
u1t227=5.448<br />
u1t228q=0.3 0.0 1.0 0 0<br />
u1t228=5.472<br />
u1t229q=0.3 0.0 1.0 0 0<br />
u1t229=5.496<br />
u1t230q=0.3 0.0 1.0 0 0<br />
u1t230=5.52<br />
u1t231q=0.3 0.0 1.0 0 0<br />
u1t231=5.544<br />
u1t232q=0.3 0.0 1.0 0 0<br />
u1t232=5.568<br />
u1t233q=0.3 0.0 1.0 0 0<br />
u1t233=5.592<br />
u1t234q=0.3 0.0 1.0 0 0<br />
u1t234=5.616<br />
u1t235q=0.3 0.0 1.0 0 0<br />
u1t235=5.64<br />
u1t236q=0.3 0.0 1.0 0 0<br />
u1t236=5.664<br />
u1t237q=0.3 0.0 1.0 0 0<br />
u1t237=5.688<br />
u1t238q=0.3 0.0 1.0 0 0<br />
u1t238=5.712<br />
u1t239q=0.3 0.0 1.0 0 0<br />
u1t239=5.736<br />
u1t240q=0.3 0.0 1.0 0 0<br />
u1t240=5.76<br />
u1t241q=0.3 0.0 1.0 0 0<br />
u1t241=5.784<br />
u1t242q=0.3 0.0 1.0 0 0<br />
u1t242=5.808<br />
u1t243q=0.3 0.0 1.0 0 0<br />
u1t243=5.832<br />
u1t244q=0.3 0.0 1.0 0 0<br />
u1t244=5.856<br />
u1t245q=0.3 0.0 1.0 0 0<br />
u1t245=5.88<br />
u1t246q=0.3 0.0 1.0 0 0<br />
u1t246=5.904<br />
u1t247q=0.3 0.0 1.0 0 0<br />
u1t247=5.928<br />
u1t248q=0.3 0.0 1.0 0 0<br />
u1t248=5.952<br />
u1t249q=0.3 0.0 1.0 0 0<br />
u1t249=5.976<br />
u1t250q=0.3 0.0 1.0 0 0<br />
u1t250=6.0<br />
u1t251q=0.3 0.0 1.0 0 0<br />
u1t251=6.024<br />
u1t252q=0.3 0.0 1.0 0 0<br />
u1t252=6.048<br />
u1t253q=0.3 0.0 1.0 0 0<br />
u1t253=6.072<br />
u1t254q=0.3 0.0 1.0 0 0<br />
u1t254=6.096<br />
u1t255q=0.3 0.0 1.0 0 0<br />
u1t255=6.12<br />
u1t256q=0.3 0.0 1.0 0 0<br />
u1t256=6.144<br />
u1t257q=0.3 0.0 1.0 0 0<br />
u1t257=6.168<br />
u1t258q=0.3 0.0 1.0 0 0<br />
u1t258=6.192<br />
u1t259q=0.3 0.0 1.0 0 0<br />
u1t259=6.216<br />
u1t260q=0.3 0.0 1.0 0 0<br />
u1t260=6.24<br />
u1t261q=0.3 0.0 1.0 0 0<br />
u1t261=6.264<br />
u1t262q=0.3 0.0 1.0 0 0<br />
u1t262=6.288<br />
u1t263q=0.3 0.0 1.0 0 0<br />
u1t263=6.312<br />
u1t264q=0.3 0.0 1.0 0 0<br />
u1t264=6.336<br />
u1t265q=0.3 0.0 1.0 0 0<br />
u1t265=6.36<br />
u1t266q=0.3 0.0 1.0 0 0<br />
u1t266=6.384<br />
u1t267q=0.3 0.0 1.0 0 0<br />
u1t267=6.408<br />
u1t268q=0.3 0.0 1.0 0 0<br />
u1t268=6.432<br />
u1t269q=0.3 0.0 1.0 0 0<br />
u1t269=6.456<br />
u1t270q=0.3 0.0 1.0 0 0<br />
u1t270=6.48<br />
u1t271q=0.3 0.0 1.0 0 0<br />
u1t271=6.504<br />
u1t272q=0.3 0.0 1.0 0 0<br />
u1t272=6.528<br />
u1t273q=0.3 0.0 1.0 0 0<br />
u1t273=6.552<br />
u1t274q=0.3 0.0 1.0 0 0<br />
u1t274=6.576<br />
u1t275q=0.3 0.0 1.0 0 0<br />
u1t275=6.6<br />
u1t276q=0.3 0.0 1.0 0 0<br />
u1t276=6.624<br />
u1t277q=0.3 0.0 1.0 0 0<br />
u1t277=6.648<br />
u1t278q=0.3 0.0 1.0 0 0<br />
u1t278=6.672<br />
u1t279q=0.3 0.0 1.0 0 0<br />
u1t279=6.696<br />
u1t280q=0.3 0.0 1.0 0 0<br />
u1t280=6.72<br />
u1t281q=0.3 0.0 1.0 0 0<br />
u1t281=6.744<br />
u1t282q=0.3 0.0 1.0 0 0<br />
u1t282=6.768<br />
u1t283q=0.3 0.0 1.0 0 0<br />
u1t283=6.792<br />
u1t284q=0.3 0.0 1.0 0 0<br />
u1t284=6.816<br />
u1t285q=0.3 0.0 1.0 0 0<br />
u1t285=6.84<br />
u1t286q=0.3 0.0 1.0 0 0<br />
u1t286=6.864<br />
u1t287q=0.3 0.0 1.0 0 0<br />
u1t287=6.888<br />
u1t288q=0.3 0.0 1.0 0 0<br />
u1t288=6.912<br />
u1t289q=0.3 0.0 1.0 0 0<br />
u1t289=6.936<br />
u1t290q=0.3 0.0 1.0 0 0<br />
u1t290=6.96<br />
u1t291q=0.3 0.0 1.0 0 0<br />
u1t291=6.984<br />
u1t292q=0.3 0.0 1.0 0 0<br />
u1t292=7.008<br />
u1t293q=0.3 0.0 1.0 0 0<br />
u1t293=7.032<br />
u1t294q=0.3 0.0 1.0 0 0<br />
u1t294=7.056<br />
u1t295q=0.3 0.0 1.0 0 0<br />
u1t295=7.08<br />
u1t296q=0.3 0.0 1.0 0 0<br />
u1t296=7.104<br />
u1t297q=0.3 0.0 1.0 0 0<br />
u1t297=7.128<br />
u1t298q=0.3 0.0 1.0 0 0<br />
u1t298=7.152<br />
u1t299q=0.3 0.0 1.0 0 0<br />
u1t299=7.176<br />
u1t300q=0.3 0.0 1.0 0 0<br />
u1t300=7.2<br />
u1t301q=0.3 0.0 1.0 0 0<br />
u1t301=7.224<br />
u1t302q=0.3 0.0 1.0 0 0<br />
u1t302=7.248<br />
u1t303q=0.3 0.0 1.0 0 0<br />
u1t303=7.272<br />
u1t304q=0.3 0.0 1.0 0 0<br />
u1t304=7.296<br />
u1t305q=0.3 0.0 1.0 0 0<br />
u1t305=7.32<br />
u1t306q=0.3 0.0 1.0 0 0<br />
u1t306=7.344<br />
u1t307q=0.3 0.0 1.0 0 0<br />
u1t307=7.368<br />
u1t308q=0.3 0.0 1.0 0 0<br />
u1t308=7.392<br />
u1t309q=0.3 0.0 1.0 0 0<br />
u1t309=7.416<br />
u1t310q=0.3 0.0 1.0 0 0<br />
u1t310=7.44<br />
u1t311q=0.3 0.0 1.0 0 0<br />
u1t311=7.464<br />
u1t312q=0.3 0.0 1.0 0 0<br />
u1t312=7.488<br />
u1t313q=0.3 0.0 1.0 0 0<br />
u1t313=7.512<br />
u1t314q=0.3 0.0 1.0 0 0<br />
u1t314=7.536<br />
u1t315q=0.3 0.0 1.0 0 0<br />
u1t315=7.56<br />
u1t316q=0.3 0.0 1.0 0 0<br />
u1t316=7.584<br />
u1t317q=0.3 0.0 1.0 0 0<br />
u1t317=7.608<br />
u1t318q=0.3 0.0 1.0 0 0<br />
u1t318=7.632<br />
u1t319q=0.3 0.0 1.0 0 0<br />
u1t319=7.656<br />
u1t320q=0.3 0.0 1.0 0 0<br />
u1t320=7.68<br />
u1t321q=0.3 0.0 1.0 0 0<br />
u1t321=7.704<br />
u1t322q=0.3 0.0 1.0 0 0<br />
u1t322=7.728<br />
u1t323q=0.3 0.0 1.0 0 0<br />
u1t323=7.752<br />
u1t324q=0.3 0.0 1.0 0 0<br />
u1t324=7.776<br />
u1t325q=0.3 0.0 1.0 0 0<br />
u1t325=7.8<br />
u1t326q=0.3 0.0 1.0 0 0<br />
u1t326=7.824<br />
u1t327q=0.3 0.0 1.0 0 0<br />
u1t327=7.848<br />
u1t328q=0.3 0.0 1.0 0 0<br />
u1t328=7.872<br />
u1t329q=0.3 0.0 1.0 0 0<br />
u1t329=7.896<br />
u1t330q=0.3 0.0 1.0 0 0<br />
u1t330=7.92<br />
u1t331q=0.3 0.0 1.0 0 0<br />
u1t331=7.944<br />
u1t332q=0.3 0.0 1.0 0 0<br />
u1t332=7.968<br />
u1t333q=0.3 0.0 1.0 0 0<br />
u1t333=7.992<br />
u1t334q=0.3 0.0 1.0 0 0<br />
u1t334=8.016<br />
u1t335q=0.3 0.0 1.0 0 0<br />
u1t335=8.04<br />
u1t336q=0.3 0.0 1.0 0 0<br />
u1t336=8.064<br />
u1t337q=0.3 0.0 1.0 0 0<br />
u1t337=8.088<br />
u1t338q=0.3 0.0 1.0 0 0<br />
u1t338=8.112<br />
u1t339q=0.3 0.0 1.0 0 0<br />
u1t339=8.136<br />
u1t340q=0.3 0.0 1.0 0 0<br />
u1t340=8.16<br />
u1t341q=0.3 0.0 1.0 0 0<br />
u1t341=8.184<br />
u1t342q=0.3 0.0 1.0 0 0<br />
u1t342=8.208<br />
u1t343q=0.3 0.0 1.0 0 0<br />
u1t343=8.232<br />
u1t344q=0.3 0.0 1.0 0 0<br />
u1t344=8.256<br />
u1t345q=0.3 0.0 1.0 0 0<br />
u1t345=8.28<br />
u1t346q=0.3 0.0 1.0 0 0<br />
u1t346=8.304<br />
u1t347q=0.3 0.0 1.0 0 0<br />
u1t347=8.328<br />
u1t348q=0.3 0.0 1.0 0 0<br />
u1t348=8.352<br />
u1t349q=0.3 0.0 1.0 0 0<br />
u1t349=8.376<br />
u1t350q=0.3 0.0 1.0 0 0<br />
u1t350=8.4<br />
u1t351q=0.3 0.0 1.0 0 0<br />
u1t351=8.424<br />
u1t352q=0.3 0.0 1.0 0 0<br />
u1t352=8.448<br />
u1t353q=0.3 0.0 1.0 0 0<br />
u1t353=8.472<br />
u1t354q=0.3 0.0 1.0 0 0<br />
u1t354=8.496<br />
u1t355q=0.3 0.0 1.0 0 0<br />
u1t355=8.52<br />
u1t356q=0.3 0.0 1.0 0 0<br />
u1t356=8.544<br />
u1t357q=0.3 0.0 1.0 0 0<br />
u1t357=8.568<br />
u1t358q=0.3 0.0 1.0 0 0<br />
u1t358=8.592<br />
u1t359q=0.3 0.0 1.0 0 0<br />
u1t359=8.616<br />
u1t360q=0.3 0.0 1.0 0 0<br />
u1t360=8.64<br />
u1t361q=0.3 0.0 1.0 0 0<br />
u1t361=8.664<br />
u1t362q=0.3 0.0 1.0 0 0<br />
u1t362=8.688<br />
u1t363q=0.3 0.0 1.0 0 0<br />
u1t363=8.712<br />
u1t364q=0.3 0.0 1.0 0 0<br />
u1t364=8.736<br />
u1t365q=0.3 0.0 1.0 0 0<br />
u1t365=8.76<br />
u1t366q=0.3 0.0 1.0 0 0<br />
u1t366=8.784<br />
u1t367q=0.3 0.0 1.0 0 0<br />
u1t367=8.808<br />
u1t368q=0.3 0.0 1.0 0 0<br />
u1t368=8.832<br />
u1t369q=0.3 0.0 1.0 0 0<br />
u1t369=8.856<br />
u1t370q=0.3 0.0 1.0 0 0<br />
u1t370=8.88<br />
u1t371q=0.3 0.0 1.0 0 0<br />
u1t371=8.904<br />
u1t372q=0.3 0.0 1.0 0 0<br />
u1t372=8.928<br />
u1t373q=0.3 0.0 1.0 0 0<br />
u1t373=8.952<br />
u1t374q=0.3 0.0 1.0 0 0<br />
u1t374=8.976<br />
u1t375q=0.3 0.0 1.0 0 0<br />
u1t375=9.0<br />
u1t376q=0.3 0.0 1.0 0 0<br />
u1t376=9.024<br />
u1t377q=0.3 0.0 1.0 0 0<br />
u1t377=9.048<br />
u1t378q=0.3 0.0 1.0 0 0<br />
u1t378=9.072<br />
u1t379q=0.3 0.0 1.0 0 0<br />
u1t379=9.096<br />
u1t380q=0.3 0.0 1.0 0 0<br />
u1t380=9.12<br />
u1t381q=0.3 0.0 1.0 0 0<br />
u1t381=9.144<br />
u1t382q=0.3 0.0 1.0 0 0<br />
u1t382=9.168<br />
u1t383q=0.3 0.0 1.0 0 0<br />
u1t383=9.192<br />
u1t384q=0.3 0.0 1.0 0 0<br />
u1t384=9.216<br />
u1t385q=0.3 0.0 1.0 0 0<br />
u1t385=9.24<br />
u1t386q=0.3 0.0 1.0 0 0<br />
u1t386=9.264<br />
u1t387q=0.3 0.0 1.0 0 0<br />
u1t387=9.288<br />
u1t388q=0.3 0.0 1.0 0 0<br />
u1t388=9.312<br />
u1t389q=0.3 0.0 1.0 0 0<br />
u1t389=9.336<br />
u1t390q=0.3 0.0 1.0 0 0<br />
u1t390=9.36<br />
u1t391q=0.3 0.0 1.0 0 0<br />
u1t391=9.384<br />
u1t392q=0.3 0.0 1.0 0 0<br />
u1t392=9.408<br />
u1t393q=0.3 0.0 1.0 0 0<br />
u1t393=9.432<br />
u1t394q=0.3 0.0 1.0 0 0<br />
u1t394=9.456<br />
u1t395q=0.3 0.0 1.0 0 0<br />
u1t395=9.48<br />
u1t396q=0.3 0.0 1.0 0 0<br />
u1t396=9.504<br />
u1t397q=0.3 0.0 1.0 0 0<br />
u1t397=9.528<br />
u1t398q=0.3 0.0 1.0 0 0<br />
u1t398=9.552<br />
u1t399q=0.3 0.0 1.0 0 0<br />
u1t399=9.576<br />
u1t400q=0.3 0.0 1.0 0 0<br />
u1t400=9.6<br />
u1t401q=0.3 0.0 1.0 0 0<br />
u1t401=9.624<br />
u1t402q=0.3 0.0 1.0 0 0<br />
u1t402=9.648<br />
u1t403q=0.3 0.0 1.0 0 0<br />
u1t403=9.672<br />
u1t404q=0.3 0.0 1.0 0 0<br />
u1t404=9.696<br />
u1t405q=0.3 0.0 1.0 0 0<br />
u1t405=9.72<br />
u1t406q=0.3 0.0 1.0 0 0<br />
u1t406=9.744<br />
u1t407q=0.3 0.0 1.0 0 0<br />
u1t407=9.768<br />
u1t408q=0.3 0.0 1.0 0 0<br />
u1t408=9.792<br />
u1t409q=0.3 0.0 1.0 0 0<br />
u1t409=9.816<br />
u1t410q=0.3 0.0 1.0 0 0<br />
u1t410=9.84<br />
u1t411q=0.3 0.0 1.0 0 0<br />
u1t411=9.864<br />
u1t412q=0.3 0.0 1.0 0 0<br />
u1t412=9.888<br />
u1t413q=0.3 0.0 1.0 0 0<br />
u1t413=9.912<br />
u1t414q=0.3 0.0 1.0 0 0<br />
u1t414=9.936<br />
u1t415q=0.3 0.0 1.0 0 0<br />
u1t415=9.96<br />
u1t416q=0.3 0.0 1.0 0 0<br />
u1t416=9.984<br />
u1t417q=0.3 0.0 1.0 0 0<br />
u1t417=10.008<br />
u1t418q=0.3 0.0 1.0 0 0<br />
u1t418=10.032<br />
u1t419q=0.3 0.0 1.0 0 0<br />
u1t419=10.056<br />
u1t420q=0.3 0.0 1.0 0 0<br />
u1t420=10.08<br />
u1t421q=0.3 0.0 1.0 0 0<br />
u1t421=10.104<br />
u1t422q=0.3 0.0 1.0 0 0<br />
u1t422=10.128<br />
u1t423q=0.3 0.0 1.0 0 0<br />
u1t423=10.152<br />
u1t424q=0.3 0.0 1.0 0 0<br />
u1t424=10.176<br />
u1t425q=0.3 0.0 1.0 0 0<br />
u1t425=10.2<br />
u1t426q=0.3 0.0 1.0 0 0<br />
u1t426=10.224<br />
u1t427q=0.3 0.0 1.0 0 0<br />
u1t427=10.248<br />
u1t428q=0.3 0.0 1.0 0 0<br />
u1t428=10.272<br />
u1t429q=0.3 0.0 1.0 0 0<br />
u1t429=10.296<br />
u1t430q=0.3 0.0 1.0 0 0<br />
u1t430=10.32<br />
u1t431q=0.3 0.0 1.0 0 0<br />
u1t431=10.344<br />
u1t432q=0.3 0.0 1.0 0 0<br />
u1t432=10.368<br />
u1t433q=0.3 0.0 1.0 0 0<br />
u1t433=10.392<br />
u1t434q=0.3 0.0 1.0 0 0<br />
u1t434=10.416<br />
u1t435q=0.3 0.0 1.0 0 0<br />
u1t435=10.44<br />
u1t436q=0.3 0.0 1.0 0 0<br />
u1t436=10.464<br />
u1t437q=0.3 0.0 1.0 0 0<br />
u1t437=10.488<br />
u1t438q=0.3 0.0 1.0 0 0<br />
u1t438=10.512<br />
u1t439q=0.3 0.0 1.0 0 0<br />
u1t439=10.536<br />
u1t440q=0.3 0.0 1.0 0 0<br />
u1t440=10.56<br />
u1t441q=0.3 0.0 1.0 0 0<br />
u1t441=10.584<br />
u1t442q=0.3 0.0 1.0 0 0<br />
u1t442=10.608<br />
u1t443q=0.3 0.0 1.0 0 0<br />
u1t443=10.632<br />
u1t444q=0.3 0.0 1.0 0 0<br />
u1t444=10.656<br />
u1t445q=0.3 0.0 1.0 0 0<br />
u1t445=10.68<br />
u1t446q=0.3 0.0 1.0 0 0<br />
u1t446=10.704<br />
u1t447q=0.3 0.0 1.0 0 0<br />
u1t447=10.728<br />
u1t448q=0.3 0.0 1.0 0 0<br />
u1t448=10.752<br />
u1t449q=0.3 0.0 1.0 0 0<br />
u1t449=10.776<br />
u1t450q=0.3 0.0 1.0 0 0<br />
u1t450=10.8<br />
u1t451q=0.3 0.0 1.0 0 0<br />
u1t451=10.824<br />
u1t452q=0.3 0.0 1.0 0 0<br />
u1t452=10.848<br />
u1t453q=0.3 0.0 1.0 0 0<br />
u1t453=10.872<br />
u1t454q=0.3 0.0 1.0 0 0<br />
u1t454=10.896<br />
u1t455q=0.3 0.0 1.0 0 0<br />
u1t455=10.92<br />
u1t456q=0.3 0.0 1.0 0 0<br />
u1t456=10.944<br />
u1t457q=0.3 0.0 1.0 0 0<br />
u1t457=10.968<br />
u1t458q=0.3 0.0 1.0 0 0<br />
u1t458=10.992<br />
u1t459q=0.3 0.0 1.0 0 0<br />
u1t459=11.016<br />
u1t460q=0.3 0.0 1.0 0 0<br />
u1t460=11.04<br />
u1t461q=0.3 0.0 1.0 0 0<br />
u1t461=11.064<br />
u1t462q=0.3 0.0 1.0 0 0<br />
u1t462=11.088<br />
u1t463q=0.3 0.0 1.0 0 0<br />
u1t463=11.112<br />
u1t464q=0.3 0.0 1.0 0 0<br />
u1t464=11.136<br />
u1t465q=0.3 0.0 1.0 0 0<br />
u1t465=11.16<br />
u1t466q=0.3 0.0 1.0 0 0<br />
u1t466=11.184<br />
u1t467q=0.3 0.0 1.0 0 0<br />
u1t467=11.208<br />
u1t468q=0.3 0.0 1.0 0 0<br />
u1t468=11.232<br />
u1t469q=0.3 0.0 1.0 0 0<br />
u1t469=11.256<br />
u1t470q=0.3 0.0 1.0 0 0<br />
u1t470=11.28<br />
u1t471q=0.3 0.0 1.0 0 0<br />
u1t471=11.304<br />
u1t472q=0.3 0.0 1.0 0 0<br />
u1t472=11.328<br />
u1t473q=0.3 0.0 1.0 0 0<br />
u1t473=11.352<br />
u1t474q=0.3 0.0 1.0 0 0<br />
u1t474=11.376<br />
u1t475q=0.3 0.0 1.0 0 0<br />
u1t475=11.4<br />
u1t476q=0.3 0.0 1.0 0 0<br />
u1t476=11.424<br />
u1t477q=0.3 0.0 1.0 0 0<br />
u1t477=11.448<br />
u1t478q=0.3 0.0 1.0 0 0<br />
u1t478=11.472<br />
u1t479q=0.3 0.0 1.0 0 0<br />
u1t479=11.496<br />
u1t480q=0.3 0.0 1.0 0 0<br />
u1t480=11.52<br />
u1t481q=0.3 0.0 1.0 0 0<br />
u1t481=11.544<br />
u1t482q=0.3 0.0 1.0 0 0<br />
u1t482=11.568<br />
u1t483q=0.3 0.0 1.0 0 0<br />
u1t483=11.592<br />
u1t484q=0.3 0.0 1.0 0 0<br />
u1t484=11.616<br />
u1t485q=0.3 0.0 1.0 0 0<br />
u1t485=11.64<br />
u1t486q=0.3 0.0 1.0 0 0<br />
u1t486=11.664<br />
u1t487q=0.3 0.0 1.0 0 0<br />
u1t487=11.688<br />
u1t488q=0.3 0.0 1.0 0 0<br />
u1t488=11.712<br />
u1t489q=0.3 0.0 1.0 0 0<br />
u1t489=11.736<br />
u1t490q=0.3 0.0 1.0 0 0<br />
u1t490=11.76<br />
u1t491q=0.3 0.0 1.0 0 0<br />
u1t491=11.784<br />
u1t492q=0.3 0.0 1.0 0 0<br />
u1t492=11.808<br />
u1t493q=0.3 0.0 1.0 0 0<br />
u1t493=11.832<br />
u1t494q=0.3 0.0 1.0 0 0<br />
u1t494=11.856<br />
u1t495q=0.3 0.0 1.0 0 0<br />
u1t495=11.88<br />
u1t496q=0.3 0.0 1.0 0 0<br />
u1t496=11.904<br />
u1t497q=0.3 0.0 1.0 0 0<br />
u1t497=11.928<br />
u1t498q=0.3 0.0 1.0 0 0<br />
u1t498=11.952<br />
u1t499q=0.3 0.0 1.0 0 0<br />
u1t499=11.976<br />
u1t500q=0.3 0 0 0 0<br />
u1t500=tend<br />
<br />
[Messungen]<br />
tAnzahl=12<br />
<br />
t1=1<br />
t1Anzahl=2<br />
t1m1=mfcn1 1.0 1e-06 1<br />
t1m2=mfcn2 1.0 1e-06 1<br />
t1minmax=0 1e+10<br />
<br />
t2=2.000<br />
t2Anzahl=2<br />
t2m1=mfcn1 1.0 1e-06 1<br />
t2m2=mfcn2 1.0 1e-06 1<br />
t2minmax=0 1e+10<br />
<br />
t3=3<br />
t3Anzahl=2<br />
t3m1=mfcn1 1.0 1e-06 1<br />
t3m2=mfcn2 1.0 1e-06 1<br />
t3minmax=0 1e+10<br />
<br />
t4=4<br />
t4Anzahl=2<br />
t4m1=mfcn1 1.0 1e-06 1<br />
t4m2=mfcn2 1.0 1e-06 1<br />
t4minmax=0 1e+10<br />
<br />
t5=5<br />
t5Anzahl=2<br />
t5m1=mfcn1 1.0 1e-06 1<br />
t5m2=mfcn2 1.0 1e-06 1<br />
t5minmax=0 1e+10<br />
<br />
t6=6<br />
t6Anzahl=2<br />
t6m1=mfcn1 1.0 1e-06 1<br />
t6m2=mfcn2 1.0 1e-06 1<br />
t6minmax=0 1e+10<br />
<br />
t7=7<br />
t7Anzahl=2<br />
t7m1=mfcn1 1.0 1e-06 1<br />
t7m2=mfcn2 1.0 1e-06 1<br />
t7minmax=0 1e+10<br />
<br />
t8=8<br />
t8Anzahl=2<br />
t8m1=mfcn1 1.0 1e-06 1<br />
t8m2=mfcn2 1.0 1e-06 1<br />
t8minmax=0 1e+10<br />
<br />
t9=9<br />
t9Anzahl=2<br />
t9m1=mfcn1 1.0 1e-06 1<br />
t9m2=mfcn2 1.0 1e-06 1<br />
t9minmax=0 1e+10<br />
<br />
t10=10.000<br />
t10Anzahl=2<br />
t10m1=mfcn1 1.0 1e-06 1<br />
t10m2=mfcn2 1.0 1e-06 1<br />
t10minmax=0 1e+10<br />
<br />
t11=11.000<br />
t11Anzahl=2<br />
t11m1=mfcn1 1.0 1e-06 1<br />
t11m2=mfcn2 1.0 1e-06 1<br />
t11minmax=0 1e+10<br />
<br />
t12=12.000<br />
t12Anzahl=2<br />
t12m1=mfcn1 1.0 1e-06 1<br />
t12m2=mfcn2 1.0 1e-06 1<br />
t12minmax=0 1e+10<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=0<br />
[Messverfahren]<br />
mAnzahl=2<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
m2=mfcn2 1 0 1e+10 0<br />
m2f1=mess4 sigma4 1<br />
mminmaxges=0 8<br />
<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
realworkspace=1700000<br />
integerworkspace=5000<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source><br />
<br />
<br />
ini-file for running VPLAN:<br />
<br />
<br />
<source lang="optimica"><br />
<br />
; ini-File fuer VPLAN<br />
<br />
[Aktion]<br />
;aktion=Integration<br />
;aktion=Simulationsumgebung<br />
;aktion=Parameterschaetzung<br />
;aktion=Versuchsplanung<br />
;aktion=ObjectiveTest<br />
;aktion=DerivativeTest<br />
;aktion={ISPS}<br />
aktion={ISCVCS}<br />
<br />
[Pfade]<br />
problempath=lotka_seminar <br />
inpath=in<br />
outpath=simu<br />
messpath=mess<br />
plotpath=plot<br />
fortranpath=fortran<br />
<br />
[Parameter]<br />
pAnzahl=2<br />
p1=p2 1.0 -1e+10 1e+10 0<br />
p2=p4 1.0 -1e+10 1e+10 0<br />
[Versuchsplan]<br />
expAnzahl=1<br />
exp1=exp1.ini exp1.ini<br />
<br />
[Guetekriterium]<br />
Optimierungskriterium=A<br />
AKriterium=-1<br />
DKriterium=-1<br />
EKriterium=-1<br />
MKriterium=-1<br />
covmat=covmat.m<br />
jacmat=jacmat.m<br />
status=undefiniert<br />
<br />
[Residuum]<br />
res=0<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Messdatenfiles]<br />
mess1=mess1.dat <br />
<br />
[Outputfiles]<br />
out1=plot2 0.05 integ.plt.1<br />
<br />
[Residuenfiles]<br />
rsd1=res1.txt<br />
<br />
[ExtensionFlags]<br />
experimenttype=0<br />
integrator=0<br />
dmode=0<br />
pdeFlag=0<br />
<br />
[OptionenAllgemein]<br />
visflag=0<br />
messfileflag=0<br />
seed=-1<br />
numberofthreads=1<br />
robustflag=0<br />
epsmach=0<br />
infinity=1e+10<br />
epsilon=1e-08<br />
conflevel=0.95<br />
hrobust=1e-05<br />
computesigma=0<br />
exitonFPE=1<br />
iniprecision=6<br />
clipboardflag=0<br />
printxi=0<br />
printconstr=0<br />
printcolorful=-1<br />
<br />
[OptionenParameterschaetzung]<br />
eps=0.001<br />
itmax=50<br />
cond=10000<br />
condflag=1<br />
boundcheck=0<br />
startflag=0<br />
index1=1e-08<br />
fashort=0.8<br />
fa0=0.01<br />
farel=0.1<br />
famax=1.0<br />
realworkspace=1000000<br />
integerworkspace=1000000<br />
printlevel=2<br />
method=0<br />
<br />
[OptionenVersuchsplanung]<br />
maxit=300<br />
opttol=1e-06<br />
funcprec=1e-07<br />
linfeas=1e-07<br />
nlinfeas=0.01<br />
maxitQP=300<br />
maxitgesQP=10000<br />
opttolQP=1e-06<br />
pivottolQP=3.7e-11<br />
steplimitLS=2<br />
tolLS=0.9<br />
crashtol=0.0001<br />
elasticweight=100<br />
superbasics=1<br />
scaling=1<br />
sconstraints=0<br />
realworkspace=3000000<br />
integerworkspace=3000000<br />
charworkspace=500<br />
printlevel=10<br />
method=2<br />
<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1344
Lotka Experimental Design (VPLAN)
2016-01-19T16:26:17Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function<br />
<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<br />
<source lang="fortran"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
<br />
<br />
<source lang="optimica"><br />
<br />
; ini-File fuer Experiment<br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=12<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=2<br />
y1=x1 0.5 -1e+10 1e+10<br />
y2=x2 0.7 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=12<br />
t1=1<br />
t2=2<br />
t3=3<br />
t4=4<br />
t5=5<br />
t6=6<br />
t7=7<br />
t8=8<br />
t9=9<br />
t10=10<br />
t11=11<br />
t12=12<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=0<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=myu 0 0.0 1.0<br />
u1tAnzahl=500<br />
u1t0=t0<br />
u1t1q=0.3 0.0 1.0 0 0<br />
u1t1=0.024<br />
u1t2q=0.3 0.0 1.0 0 0<br />
u1t2=0.048<br />
u1t3q=0.3 0.0 1.0 0 0<br />
u1t3=0.072<br />
u1t4q=0.3 0.0 1.0 0 0<br />
u1t4=0.096<br />
u1t5q=0.3 0.0 1.0 0 0<br />
u1t5=0.12<br />
u1t6q=0.3 0.0 1.0 0 0<br />
u1t6=0.144<br />
u1t7q=0.3 0.0 1.0 0 0<br />
u1t7=0.168<br />
u1t8q=0.3 0.0 1.0 0 0<br />
u1t8=0.192<br />
u1t9q=0.3 0.0 1.0 0 0<br />
u1t9=0.216<br />
u1t10q=0.3 0.0 1.0 0 0<br />
u1t10=0.24<br />
u1t11q=0.3 0.0 1.0 0 0<br />
u1t11=0.264<br />
u1t12q=0.3 0.0 1.0 0 0<br />
u1t12=0.288<br />
u1t13q=0.3 0.0 1.0 0 0<br />
u1t13=0.312<br />
u1t14q=0.3 0.0 1.0 0 0<br />
u1t14=0.336<br />
u1t15q=0.3 0.0 1.0 0 0<br />
u1t15=0.36<br />
u1t16q=0.3 0.0 1.0 0 0<br />
u1t16=0.384<br />
u1t17q=0.3 0.0 1.0 0 0<br />
u1t17=0.408<br />
u1t18q=0.3 0.0 1.0 0 0<br />
u1t18=0.432<br />
u1t19q=0.3 0.0 1.0 0 0<br />
u1t19=0.456<br />
u1t20q=0.3 0.0 1.0 0 0<br />
u1t20=0.48<br />
u1t21q=0.3 0.0 1.0 0 0<br />
u1t21=0.504<br />
u1t22q=0.3 0.0 1.0 0 0<br />
u1t22=0.528<br />
u1t23q=0.3 0.0 1.0 0 0<br />
u1t23=0.552<br />
u1t24q=0.3 0.0 1.0 0 0<br />
u1t24=0.576<br />
u1t25q=0.3 0.0 1.0 0 0<br />
u1t25=0.6<br />
u1t26q=0.3 0.0 1.0 0 0<br />
u1t26=0.624<br />
u1t27q=0.3 0.0 1.0 0 0<br />
u1t27=0.648<br />
u1t28q=0.3 0.0 1.0 0 0<br />
u1t28=0.672<br />
u1t29q=0.3 0.0 1.0 0 0<br />
u1t29=0.696<br />
u1t30q=0.3 0.0 1.0 0 0<br />
u1t30=0.72<br />
u1t31q=0.3 0.0 1.0 0 0<br />
u1t31=0.744<br />
u1t32q=0.3 0.0 1.0 0 0<br />
u1t32=0.768<br />
u1t33q=0.3 0.0 1.0 0 0<br />
u1t33=0.792<br />
u1t34q=0.3 0.0 1.0 0 0<br />
u1t34=0.816<br />
u1t35q=0.3 0.0 1.0 0 0<br />
u1t35=0.84<br />
u1t36q=0.3 0.0 1.0 0 0<br />
u1t36=0.864<br />
u1t37q=0.3 0.0 1.0 0 0<br />
u1t37=0.888<br />
u1t38q=0.3 0.0 1.0 0 0<br />
u1t38=0.912<br />
u1t39q=0.3 0.0 1.0 0 0<br />
u1t39=0.936<br />
u1t40q=0.3 0.0 1.0 0 0<br />
u1t40=0.96<br />
u1t41q=0.3 0.0 1.0 0 0<br />
u1t41=0.984<br />
u1t42q=0.3 0.0 1.0 0 0<br />
u1t42=1.008<br />
u1t43q=0.3 0.0 1.0 0 0<br />
u1t43=1.032<br />
u1t44q=0.3 0.0 1.0 0 0<br />
u1t44=1.056<br />
u1t45q=0.3 0.0 1.0 0 0<br />
u1t45=1.08<br />
u1t46q=0.3 0.0 1.0 0 0<br />
u1t46=1.104<br />
u1t47q=0.3 0.0 1.0 0 0<br />
u1t47=1.128<br />
u1t48q=0.3 0.0 1.0 0 0<br />
u1t48=1.152<br />
u1t49q=0.3 0.0 1.0 0 0<br />
u1t49=1.176<br />
u1t50q=0.3 0.0 1.0 0 0<br />
u1t50=1.2<br />
u1t51q=0.3 0.0 1.0 0 0<br />
u1t51=1.224<br />
u1t52q=0.3 0.0 1.0 0 0<br />
u1t52=1.248<br />
u1t53q=0.3 0.0 1.0 0 0<br />
u1t53=1.272<br />
u1t54q=0.3 0.0 1.0 0 0<br />
u1t54=1.296<br />
u1t55q=0.3 0.0 1.0 0 0<br />
u1t55=1.32<br />
u1t56q=0.3 0.0 1.0 0 0<br />
u1t56=1.344<br />
u1t57q=0.3 0.0 1.0 0 0<br />
u1t57=1.368<br />
u1t58q=0.3 0.0 1.0 0 0<br />
u1t58=1.392<br />
u1t59q=0.3 0.0 1.0 0 0<br />
u1t59=1.416<br />
u1t60q=0.3 0.0 1.0 0 0<br />
u1t60=1.44<br />
u1t61q=0.3 0.0 1.0 0 0<br />
u1t61=1.464<br />
u1t62q=0.3 0.0 1.0 0 0<br />
u1t62=1.488<br />
u1t63q=0.3 0.0 1.0 0 0<br />
u1t63=1.512<br />
u1t64q=0.3 0.0 1.0 0 0<br />
u1t64=1.536<br />
u1t65q=0.3 0.0 1.0 0 0<br />
u1t65=1.56<br />
u1t66q=0.3 0.0 1.0 0 0<br />
u1t66=1.584<br />
u1t67q=0.3 0.0 1.0 0 0<br />
u1t67=1.608<br />
u1t68q=0.3 0.0 1.0 0 0<br />
u1t68=1.632<br />
u1t69q=0.3 0.0 1.0 0 0<br />
u1t69=1.656<br />
u1t70q=0.3 0.0 1.0 0 0<br />
u1t70=1.68<br />
u1t71q=0.3 0.0 1.0 0 0<br />
u1t71=1.704<br />
u1t72q=0.3 0.0 1.0 0 0<br />
u1t72=1.728<br />
u1t73q=0.3 0.0 1.0 0 0<br />
u1t73=1.752<br />
u1t74q=0.3 0.0 1.0 0 0<br />
u1t74=1.776<br />
u1t75q=0.3 0.0 1.0 0 0<br />
u1t75=1.8<br />
u1t76q=0.3 0.0 1.0 0 0<br />
u1t76=1.824<br />
u1t77q=0.3 0.0 1.0 0 0<br />
u1t77=1.848<br />
u1t78q=0.3 0.0 1.0 0 0<br />
u1t78=1.872<br />
u1t79q=0.3 0.0 1.0 0 0<br />
u1t79=1.896<br />
u1t80q=0.3 0.0 1.0 0 0<br />
u1t80=1.92<br />
u1t81q=0.3 0.0 1.0 0 0<br />
u1t81=1.944<br />
u1t82q=0.3 0.0 1.0 0 0<br />
u1t82=1.968<br />
u1t83q=0.3 0.0 1.0 0 0<br />
u1t83=1.992<br />
u1t84q=0.3 0.0 1.0 0 0<br />
u1t84=2.016<br />
u1t85q=0.3 0.0 1.0 0 0<br />
u1t85=2.04<br />
u1t86q=0.3 0.0 1.0 0 0<br />
u1t86=2.064<br />
u1t87q=0.3 0.0 1.0 0 0<br />
u1t87=2.088<br />
u1t88q=0.3 0.0 1.0 0 0<br />
u1t88=2.112<br />
u1t89q=0.3 0.0 1.0 0 0<br />
u1t89=2.136<br />
u1t90q=0.3 0.0 1.0 0 0<br />
u1t90=2.16<br />
u1t91q=0.3 0.0 1.0 0 0<br />
u1t91=2.184<br />
u1t92q=0.3 0.0 1.0 0 0<br />
u1t92=2.208<br />
u1t93q=0.3 0.0 1.0 0 0<br />
u1t93=2.232<br />
u1t94q=0.3 0.0 1.0 0 0<br />
u1t94=2.256<br />
u1t95q=0.3 0.0 1.0 0 0<br />
u1t95=2.28<br />
u1t96q=0.3 0.0 1.0 0 0<br />
u1t96=2.304<br />
u1t97q=0.3 0.0 1.0 0 0<br />
u1t97=2.328<br />
u1t98q=0.3 0.0 1.0 0 0<br />
u1t98=2.352<br />
u1t99q=0.3 0.0 1.0 0 0<br />
u1t99=2.376<br />
u1t100q=0.3 0.0 1.0 0 0<br />
u1t100=2.4<br />
u1t101q=0.3 0.0 1.0 0 0<br />
u1t101=2.424<br />
u1t102q=0.3 0.0 1.0 0 0<br />
u1t102=2.448<br />
u1t103q=0.3 0.0 1.0 0 0<br />
u1t103=2.472<br />
u1t104q=0.3 0.0 1.0 0 0<br />
u1t104=2.496<br />
u1t105q=0.3 0.0 1.0 0 0<br />
u1t105=2.52<br />
u1t106q=0.3 0.0 1.0 0 0<br />
u1t106=2.544<br />
u1t107q=0.3 0.0 1.0 0 0<br />
u1t107=2.568<br />
u1t108q=0.3 0.0 1.0 0 0<br />
u1t108=2.592<br />
u1t109q=0.3 0.0 1.0 0 0<br />
u1t109=2.616<br />
u1t110q=0.3 0.0 1.0 0 0<br />
u1t110=2.64<br />
u1t111q=0.3 0.0 1.0 0 0<br />
u1t111=2.664<br />
u1t112q=0.3 0.0 1.0 0 0<br />
u1t112=2.688<br />
u1t113q=0.3 0.0 1.0 0 0<br />
u1t113=2.712<br />
u1t114q=0.3 0.0 1.0 0 0<br />
u1t114=2.736<br />
u1t115q=0.3 0.0 1.0 0 0<br />
u1t115=2.76<br />
u1t116q=0.3 0.0 1.0 0 0<br />
u1t116=2.784<br />
u1t117q=0.3 0.0 1.0 0 0<br />
u1t117=2.808<br />
u1t118q=0.3 0.0 1.0 0 0<br />
u1t118=2.832<br />
u1t119q=0.3 0.0 1.0 0 0<br />
u1t119=2.856<br />
u1t120q=0.3 0.0 1.0 0 0<br />
u1t120=2.88<br />
u1t121q=0.3 0.0 1.0 0 0<br />
u1t121=2.904<br />
u1t122q=0.3 0.0 1.0 0 0<br />
u1t122=2.928<br />
u1t123q=0.3 0.0 1.0 0 0<br />
u1t123=2.952<br />
u1t124q=0.3 0.0 1.0 0 0<br />
u1t124=2.976<br />
u1t125q=0.3 0.0 1.0 0 0<br />
u1t125=3.0<br />
u1t126q=0.3 0.0 1.0 0 0<br />
u1t126=3.024<br />
u1t127q=0.3 0.0 1.0 0 0<br />
u1t127=3.048<br />
u1t128q=0.3 0.0 1.0 0 0<br />
u1t128=3.072<br />
u1t129q=0.3 0.0 1.0 0 0<br />
u1t129=3.096<br />
u1t130q=0.3 0.0 1.0 0 0<br />
u1t130=3.12<br />
u1t131q=0.3 0.0 1.0 0 0<br />
u1t131=3.144<br />
u1t132q=0.3 0.0 1.0 0 0<br />
u1t132=3.168<br />
u1t133q=0.3 0.0 1.0 0 0<br />
u1t133=3.192<br />
u1t134q=0.3 0.0 1.0 0 0<br />
u1t134=3.216<br />
u1t135q=0.3 0.0 1.0 0 0<br />
u1t135=3.24<br />
u1t136q=0.3 0.0 1.0 0 0<br />
u1t136=3.264<br />
u1t137q=0.3 0.0 1.0 0 0<br />
u1t137=3.288<br />
u1t138q=0.3 0.0 1.0 0 0<br />
u1t138=3.312<br />
u1t139q=0.3 0.0 1.0 0 0<br />
u1t139=3.336<br />
u1t140q=0.3 0.0 1.0 0 0<br />
u1t140=3.36<br />
u1t141q=0.3 0.0 1.0 0 0<br />
u1t141=3.384<br />
u1t142q=0.3 0.0 1.0 0 0<br />
u1t142=3.408<br />
u1t143q=0.3 0.0 1.0 0 0<br />
u1t143=3.432<br />
u1t144q=0.3 0.0 1.0 0 0<br />
u1t144=3.456<br />
u1t145q=0.3 0.0 1.0 0 0<br />
u1t145=3.48<br />
u1t146q=0.3 0.0 1.0 0 0<br />
u1t146=3.504<br />
u1t147q=0.3 0.0 1.0 0 0<br />
u1t147=3.528<br />
u1t148q=0.3 0.0 1.0 0 0<br />
u1t148=3.552<br />
u1t149q=0.3 0.0 1.0 0 0<br />
u1t149=3.576<br />
u1t150q=0.3 0.0 1.0 0 0<br />
u1t150=3.6<br />
u1t151q=0.3 0.0 1.0 0 0<br />
u1t151=3.624<br />
u1t152q=0.3 0.0 1.0 0 0<br />
u1t152=3.648<br />
u1t153q=0.3 0.0 1.0 0 0<br />
u1t153=3.672<br />
u1t154q=0.3 0.0 1.0 0 0<br />
u1t154=3.696<br />
u1t155q=0.3 0.0 1.0 0 0<br />
u1t155=3.72<br />
u1t156q=0.3 0.0 1.0 0 0<br />
u1t156=3.744<br />
u1t157q=0.3 0.0 1.0 0 0<br />
u1t157=3.768<br />
u1t158q=0.3 0.0 1.0 0 0<br />
u1t158=3.792<br />
u1t159q=0.3 0.0 1.0 0 0<br />
u1t159=3.816<br />
u1t160q=0.3 0.0 1.0 0 0<br />
u1t160=3.84<br />
u1t161q=0.3 0.0 1.0 0 0<br />
u1t161=3.864<br />
u1t162q=0.3 0.0 1.0 0 0<br />
u1t162=3.888<br />
u1t163q=0.3 0.0 1.0 0 0<br />
u1t163=3.912<br />
u1t164q=0.3 0.0 1.0 0 0<br />
u1t164=3.936<br />
u1t165q=0.3 0.0 1.0 0 0<br />
u1t165=3.96<br />
u1t166q=0.3 0.0 1.0 0 0<br />
u1t166=3.984<br />
u1t167q=0.3 0.0 1.0 0 0<br />
u1t167=4.008<br />
u1t168q=0.3 0.0 1.0 0 0<br />
u1t168=4.032<br />
u1t169q=0.3 0.0 1.0 0 0<br />
u1t169=4.056<br />
u1t170q=0.3 0.0 1.0 0 0<br />
u1t170=4.08<br />
u1t171q=0.3 0.0 1.0 0 0<br />
u1t171=4.104<br />
u1t172q=0.3 0.0 1.0 0 0<br />
u1t172=4.128<br />
u1t173q=0.3 0.0 1.0 0 0<br />
u1t173=4.152<br />
u1t174q=0.3 0.0 1.0 0 0<br />
u1t174=4.176<br />
u1t175q=0.3 0.0 1.0 0 0<br />
u1t175=4.2<br />
u1t176q=0.3 0.0 1.0 0 0<br />
u1t176=4.224<br />
u1t177q=0.3 0.0 1.0 0 0<br />
u1t177=4.248<br />
u1t178q=0.3 0.0 1.0 0 0<br />
u1t178=4.272<br />
u1t179q=0.3 0.0 1.0 0 0<br />
u1t179=4.296<br />
u1t180q=0.3 0.0 1.0 0 0<br />
u1t180=4.32<br />
u1t181q=0.3 0.0 1.0 0 0<br />
u1t181=4.344<br />
u1t182q=0.3 0.0 1.0 0 0<br />
u1t182=4.368<br />
u1t183q=0.3 0.0 1.0 0 0<br />
u1t183=4.392<br />
u1t184q=0.3 0.0 1.0 0 0<br />
u1t184=4.416<br />
u1t185q=0.3 0.0 1.0 0 0<br />
u1t185=4.44<br />
u1t186q=0.3 0.0 1.0 0 0<br />
u1t186=4.464<br />
u1t187q=0.3 0.0 1.0 0 0<br />
u1t187=4.488<br />
u1t188q=0.3 0.0 1.0 0 0<br />
u1t188=4.512<br />
u1t189q=0.3 0.0 1.0 0 0<br />
u1t189=4.536<br />
u1t190q=0.3 0.0 1.0 0 0<br />
u1t190=4.56<br />
u1t191q=0.3 0.0 1.0 0 0<br />
u1t191=4.584<br />
u1t192q=0.3 0.0 1.0 0 0<br />
u1t192=4.608<br />
u1t193q=0.3 0.0 1.0 0 0<br />
u1t193=4.632<br />
u1t194q=0.3 0.0 1.0 0 0<br />
u1t194=4.656<br />
u1t195q=0.3 0.0 1.0 0 0<br />
u1t195=4.68<br />
u1t196q=0.3 0.0 1.0 0 0<br />
u1t196=4.704<br />
u1t197q=0.3 0.0 1.0 0 0<br />
u1t197=4.728<br />
u1t198q=0.3 0.0 1.0 0 0<br />
u1t198=4.752<br />
u1t199q=0.3 0.0 1.0 0 0<br />
u1t199=4.776<br />
u1t200q=0.3 0.0 1.0 0 0<br />
u1t200=4.8<br />
u1t201q=0.3 0.0 1.0 0 0<br />
u1t201=4.824<br />
u1t202q=0.3 0.0 1.0 0 0<br />
u1t202=4.848<br />
u1t203q=0.3 0.0 1.0 0 0<br />
u1t203=4.872<br />
u1t204q=0.3 0.0 1.0 0 0<br />
u1t204=4.896<br />
u1t205q=0.3 0.0 1.0 0 0<br />
u1t205=4.92<br />
u1t206q=0.3 0.0 1.0 0 0<br />
u1t206=4.944<br />
u1t207q=0.3 0.0 1.0 0 0<br />
u1t207=4.968<br />
u1t208q=0.3 0.0 1.0 0 0<br />
u1t208=4.992<br />
u1t209q=0.3 0.0 1.0 0 0<br />
u1t209=5.016<br />
u1t210q=0.3 0.0 1.0 0 0<br />
u1t210=5.04<br />
u1t211q=0.3 0.0 1.0 0 0<br />
u1t211=5.064<br />
u1t212q=0.3 0.0 1.0 0 0<br />
u1t212=5.088<br />
u1t213q=0.3 0.0 1.0 0 0<br />
u1t213=5.112<br />
u1t214q=0.3 0.0 1.0 0 0<br />
u1t214=5.136<br />
u1t215q=0.3 0.0 1.0 0 0<br />
u1t215=5.16<br />
u1t216q=0.3 0.0 1.0 0 0<br />
u1t216=5.184<br />
u1t217q=0.3 0.0 1.0 0 0<br />
u1t217=5.208<br />
u1t218q=0.3 0.0 1.0 0 0<br />
u1t218=5.232<br />
u1t219q=0.3 0.0 1.0 0 0<br />
u1t219=5.256<br />
u1t220q=0.3 0.0 1.0 0 0<br />
u1t220=5.28<br />
u1t221q=0.3 0.0 1.0 0 0<br />
u1t221=5.304<br />
u1t222q=0.3 0.0 1.0 0 0<br />
u1t222=5.328<br />
u1t223q=0.3 0.0 1.0 0 0<br />
u1t223=5.352<br />
u1t224q=0.3 0.0 1.0 0 0<br />
u1t224=5.376<br />
u1t225q=0.3 0.0 1.0 0 0<br />
u1t225=5.4<br />
u1t226q=0.3 0.0 1.0 0 0<br />
u1t226=5.424<br />
u1t227q=0.3 0.0 1.0 0 0<br />
u1t227=5.448<br />
u1t228q=0.3 0.0 1.0 0 0<br />
u1t228=5.472<br />
u1t229q=0.3 0.0 1.0 0 0<br />
u1t229=5.496<br />
u1t230q=0.3 0.0 1.0 0 0<br />
u1t230=5.52<br />
u1t231q=0.3 0.0 1.0 0 0<br />
u1t231=5.544<br />
u1t232q=0.3 0.0 1.0 0 0<br />
u1t232=5.568<br />
u1t233q=0.3 0.0 1.0 0 0<br />
u1t233=5.592<br />
u1t234q=0.3 0.0 1.0 0 0<br />
u1t234=5.616<br />
u1t235q=0.3 0.0 1.0 0 0<br />
u1t235=5.64<br />
u1t236q=0.3 0.0 1.0 0 0<br />
u1t236=5.664<br />
u1t237q=0.3 0.0 1.0 0 0<br />
u1t237=5.688<br />
u1t238q=0.3 0.0 1.0 0 0<br />
u1t238=5.712<br />
u1t239q=0.3 0.0 1.0 0 0<br />
u1t239=5.736<br />
u1t240q=0.3 0.0 1.0 0 0<br />
u1t240=5.76<br />
u1t241q=0.3 0.0 1.0 0 0<br />
u1t241=5.784<br />
u1t242q=0.3 0.0 1.0 0 0<br />
u1t242=5.808<br />
u1t243q=0.3 0.0 1.0 0 0<br />
u1t243=5.832<br />
u1t244q=0.3 0.0 1.0 0 0<br />
u1t244=5.856<br />
u1t245q=0.3 0.0 1.0 0 0<br />
u1t245=5.88<br />
u1t246q=0.3 0.0 1.0 0 0<br />
u1t246=5.904<br />
u1t247q=0.3 0.0 1.0 0 0<br />
u1t247=5.928<br />
u1t248q=0.3 0.0 1.0 0 0<br />
u1t248=5.952<br />
u1t249q=0.3 0.0 1.0 0 0<br />
u1t249=5.976<br />
u1t250q=0.3 0.0 1.0 0 0<br />
u1t250=6.0<br />
u1t251q=0.3 0.0 1.0 0 0<br />
u1t251=6.024<br />
u1t252q=0.3 0.0 1.0 0 0<br />
u1t252=6.048<br />
u1t253q=0.3 0.0 1.0 0 0<br />
u1t253=6.072<br />
u1t254q=0.3 0.0 1.0 0 0<br />
u1t254=6.096<br />
u1t255q=0.3 0.0 1.0 0 0<br />
u1t255=6.12<br />
u1t256q=0.3 0.0 1.0 0 0<br />
u1t256=6.144<br />
u1t257q=0.3 0.0 1.0 0 0<br />
u1t257=6.168<br />
u1t258q=0.3 0.0 1.0 0 0<br />
u1t258=6.192<br />
u1t259q=0.3 0.0 1.0 0 0<br />
u1t259=6.216<br />
u1t260q=0.3 0.0 1.0 0 0<br />
u1t260=6.24<br />
u1t261q=0.3 0.0 1.0 0 0<br />
u1t261=6.264<br />
u1t262q=0.3 0.0 1.0 0 0<br />
u1t262=6.288<br />
u1t263q=0.3 0.0 1.0 0 0<br />
u1t263=6.312<br />
u1t264q=0.3 0.0 1.0 0 0<br />
u1t264=6.336<br />
u1t265q=0.3 0.0 1.0 0 0<br />
u1t265=6.36<br />
u1t266q=0.3 0.0 1.0 0 0<br />
u1t266=6.384<br />
u1t267q=0.3 0.0 1.0 0 0<br />
u1t267=6.408<br />
u1t268q=0.3 0.0 1.0 0 0<br />
u1t268=6.432<br />
u1t269q=0.3 0.0 1.0 0 0<br />
u1t269=6.456<br />
u1t270q=0.3 0.0 1.0 0 0<br />
u1t270=6.48<br />
u1t271q=0.3 0.0 1.0 0 0<br />
u1t271=6.504<br />
u1t272q=0.3 0.0 1.0 0 0<br />
u1t272=6.528<br />
u1t273q=0.3 0.0 1.0 0 0<br />
u1t273=6.552<br />
u1t274q=0.3 0.0 1.0 0 0<br />
u1t274=6.576<br />
u1t275q=0.3 0.0 1.0 0 0<br />
u1t275=6.6<br />
u1t276q=0.3 0.0 1.0 0 0<br />
u1t276=6.624<br />
u1t277q=0.3 0.0 1.0 0 0<br />
u1t277=6.648<br />
u1t278q=0.3 0.0 1.0 0 0<br />
u1t278=6.672<br />
u1t279q=0.3 0.0 1.0 0 0<br />
u1t279=6.696<br />
u1t280q=0.3 0.0 1.0 0 0<br />
u1t280=6.72<br />
u1t281q=0.3 0.0 1.0 0 0<br />
u1t281=6.744<br />
u1t282q=0.3 0.0 1.0 0 0<br />
u1t282=6.768<br />
u1t283q=0.3 0.0 1.0 0 0<br />
u1t283=6.792<br />
u1t284q=0.3 0.0 1.0 0 0<br />
u1t284=6.816<br />
u1t285q=0.3 0.0 1.0 0 0<br />
u1t285=6.84<br />
u1t286q=0.3 0.0 1.0 0 0<br />
u1t286=6.864<br />
u1t287q=0.3 0.0 1.0 0 0<br />
u1t287=6.888<br />
u1t288q=0.3 0.0 1.0 0 0<br />
u1t288=6.912<br />
u1t289q=0.3 0.0 1.0 0 0<br />
u1t289=6.936<br />
u1t290q=0.3 0.0 1.0 0 0<br />
u1t290=6.96<br />
u1t291q=0.3 0.0 1.0 0 0<br />
u1t291=6.984<br />
u1t292q=0.3 0.0 1.0 0 0<br />
u1t292=7.008<br />
u1t293q=0.3 0.0 1.0 0 0<br />
u1t293=7.032<br />
u1t294q=0.3 0.0 1.0 0 0<br />
u1t294=7.056<br />
u1t295q=0.3 0.0 1.0 0 0<br />
u1t295=7.08<br />
u1t296q=0.3 0.0 1.0 0 0<br />
u1t296=7.104<br />
u1t297q=0.3 0.0 1.0 0 0<br />
u1t297=7.128<br />
u1t298q=0.3 0.0 1.0 0 0<br />
u1t298=7.152<br />
u1t299q=0.3 0.0 1.0 0 0<br />
u1t299=7.176<br />
u1t300q=0.3 0.0 1.0 0 0<br />
u1t300=7.2<br />
u1t301q=0.3 0.0 1.0 0 0<br />
u1t301=7.224<br />
u1t302q=0.3 0.0 1.0 0 0<br />
u1t302=7.248<br />
u1t303q=0.3 0.0 1.0 0 0<br />
u1t303=7.272<br />
u1t304q=0.3 0.0 1.0 0 0<br />
u1t304=7.296<br />
u1t305q=0.3 0.0 1.0 0 0<br />
u1t305=7.32<br />
u1t306q=0.3 0.0 1.0 0 0<br />
u1t306=7.344<br />
u1t307q=0.3 0.0 1.0 0 0<br />
u1t307=7.368<br />
u1t308q=0.3 0.0 1.0 0 0<br />
u1t308=7.392<br />
u1t309q=0.3 0.0 1.0 0 0<br />
u1t309=7.416<br />
u1t310q=0.3 0.0 1.0 0 0<br />
u1t310=7.44<br />
u1t311q=0.3 0.0 1.0 0 0<br />
u1t311=7.464<br />
u1t312q=0.3 0.0 1.0 0 0<br />
u1t312=7.488<br />
u1t313q=0.3 0.0 1.0 0 0<br />
u1t313=7.512<br />
u1t314q=0.3 0.0 1.0 0 0<br />
u1t314=7.536<br />
u1t315q=0.3 0.0 1.0 0 0<br />
u1t315=7.56<br />
u1t316q=0.3 0.0 1.0 0 0<br />
u1t316=7.584<br />
u1t317q=0.3 0.0 1.0 0 0<br />
u1t317=7.608<br />
u1t318q=0.3 0.0 1.0 0 0<br />
u1t318=7.632<br />
u1t319q=0.3 0.0 1.0 0 0<br />
u1t319=7.656<br />
u1t320q=0.3 0.0 1.0 0 0<br />
u1t320=7.68<br />
u1t321q=0.3 0.0 1.0 0 0<br />
u1t321=7.704<br />
u1t322q=0.3 0.0 1.0 0 0<br />
u1t322=7.728<br />
u1t323q=0.3 0.0 1.0 0 0<br />
u1t323=7.752<br />
u1t324q=0.3 0.0 1.0 0 0<br />
u1t324=7.776<br />
u1t325q=0.3 0.0 1.0 0 0<br />
u1t325=7.8<br />
u1t326q=0.3 0.0 1.0 0 0<br />
u1t326=7.824<br />
u1t327q=0.3 0.0 1.0 0 0<br />
u1t327=7.848<br />
u1t328q=0.3 0.0 1.0 0 0<br />
u1t328=7.872<br />
u1t329q=0.3 0.0 1.0 0 0<br />
u1t329=7.896<br />
u1t330q=0.3 0.0 1.0 0 0<br />
u1t330=7.92<br />
u1t331q=0.3 0.0 1.0 0 0<br />
u1t331=7.944<br />
u1t332q=0.3 0.0 1.0 0 0<br />
u1t332=7.968<br />
u1t333q=0.3 0.0 1.0 0 0<br />
u1t333=7.992<br />
u1t334q=0.3 0.0 1.0 0 0<br />
u1t334=8.016<br />
u1t335q=0.3 0.0 1.0 0 0<br />
u1t335=8.04<br />
u1t336q=0.3 0.0 1.0 0 0<br />
u1t336=8.064<br />
u1t337q=0.3 0.0 1.0 0 0<br />
u1t337=8.088<br />
u1t338q=0.3 0.0 1.0 0 0<br />
u1t338=8.112<br />
u1t339q=0.3 0.0 1.0 0 0<br />
u1t339=8.136<br />
u1t340q=0.3 0.0 1.0 0 0<br />
u1t340=8.16<br />
u1t341q=0.3 0.0 1.0 0 0<br />
u1t341=8.184<br />
u1t342q=0.3 0.0 1.0 0 0<br />
u1t342=8.208<br />
u1t343q=0.3 0.0 1.0 0 0<br />
u1t343=8.232<br />
u1t344q=0.3 0.0 1.0 0 0<br />
u1t344=8.256<br />
u1t345q=0.3 0.0 1.0 0 0<br />
u1t345=8.28<br />
u1t346q=0.3 0.0 1.0 0 0<br />
u1t346=8.304<br />
u1t347q=0.3 0.0 1.0 0 0<br />
u1t347=8.328<br />
u1t348q=0.3 0.0 1.0 0 0<br />
u1t348=8.352<br />
u1t349q=0.3 0.0 1.0 0 0<br />
u1t349=8.376<br />
u1t350q=0.3 0.0 1.0 0 0<br />
u1t350=8.4<br />
u1t351q=0.3 0.0 1.0 0 0<br />
u1t351=8.424<br />
u1t352q=0.3 0.0 1.0 0 0<br />
u1t352=8.448<br />
u1t353q=0.3 0.0 1.0 0 0<br />
u1t353=8.472<br />
u1t354q=0.3 0.0 1.0 0 0<br />
u1t354=8.496<br />
u1t355q=0.3 0.0 1.0 0 0<br />
u1t355=8.52<br />
u1t356q=0.3 0.0 1.0 0 0<br />
u1t356=8.544<br />
u1t357q=0.3 0.0 1.0 0 0<br />
u1t357=8.568<br />
u1t358q=0.3 0.0 1.0 0 0<br />
u1t358=8.592<br />
u1t359q=0.3 0.0 1.0 0 0<br />
u1t359=8.616<br />
u1t360q=0.3 0.0 1.0 0 0<br />
u1t360=8.64<br />
u1t361q=0.3 0.0 1.0 0 0<br />
u1t361=8.664<br />
u1t362q=0.3 0.0 1.0 0 0<br />
u1t362=8.688<br />
u1t363q=0.3 0.0 1.0 0 0<br />
u1t363=8.712<br />
u1t364q=0.3 0.0 1.0 0 0<br />
u1t364=8.736<br />
u1t365q=0.3 0.0 1.0 0 0<br />
u1t365=8.76<br />
u1t366q=0.3 0.0 1.0 0 0<br />
u1t366=8.784<br />
u1t367q=0.3 0.0 1.0 0 0<br />
u1t367=8.808<br />
u1t368q=0.3 0.0 1.0 0 0<br />
u1t368=8.832<br />
u1t369q=0.3 0.0 1.0 0 0<br />
u1t369=8.856<br />
u1t370q=0.3 0.0 1.0 0 0<br />
u1t370=8.88<br />
u1t371q=0.3 0.0 1.0 0 0<br />
u1t371=8.904<br />
u1t372q=0.3 0.0 1.0 0 0<br />
u1t372=8.928<br />
u1t373q=0.3 0.0 1.0 0 0<br />
u1t373=8.952<br />
u1t374q=0.3 0.0 1.0 0 0<br />
u1t374=8.976<br />
u1t375q=0.3 0.0 1.0 0 0<br />
u1t375=9.0<br />
u1t376q=0.3 0.0 1.0 0 0<br />
u1t376=9.024<br />
u1t377q=0.3 0.0 1.0 0 0<br />
u1t377=9.048<br />
u1t378q=0.3 0.0 1.0 0 0<br />
u1t378=9.072<br />
u1t379q=0.3 0.0 1.0 0 0<br />
u1t379=9.096<br />
u1t380q=0.3 0.0 1.0 0 0<br />
u1t380=9.12<br />
u1t381q=0.3 0.0 1.0 0 0<br />
u1t381=9.144<br />
u1t382q=0.3 0.0 1.0 0 0<br />
u1t382=9.168<br />
u1t383q=0.3 0.0 1.0 0 0<br />
u1t383=9.192<br />
u1t384q=0.3 0.0 1.0 0 0<br />
u1t384=9.216<br />
u1t385q=0.3 0.0 1.0 0 0<br />
u1t385=9.24<br />
u1t386q=0.3 0.0 1.0 0 0<br />
u1t386=9.264<br />
u1t387q=0.3 0.0 1.0 0 0<br />
u1t387=9.288<br />
u1t388q=0.3 0.0 1.0 0 0<br />
u1t388=9.312<br />
u1t389q=0.3 0.0 1.0 0 0<br />
u1t389=9.336<br />
u1t390q=0.3 0.0 1.0 0 0<br />
u1t390=9.36<br />
u1t391q=0.3 0.0 1.0 0 0<br />
u1t391=9.384<br />
u1t392q=0.3 0.0 1.0 0 0<br />
u1t392=9.408<br />
u1t393q=0.3 0.0 1.0 0 0<br />
u1t393=9.432<br />
u1t394q=0.3 0.0 1.0 0 0<br />
u1t394=9.456<br />
u1t395q=0.3 0.0 1.0 0 0<br />
u1t395=9.48<br />
u1t396q=0.3 0.0 1.0 0 0<br />
u1t396=9.504<br />
u1t397q=0.3 0.0 1.0 0 0<br />
u1t397=9.528<br />
u1t398q=0.3 0.0 1.0 0 0<br />
u1t398=9.552<br />
u1t399q=0.3 0.0 1.0 0 0<br />
u1t399=9.576<br />
u1t400q=0.3 0.0 1.0 0 0<br />
u1t400=9.6<br />
u1t401q=0.3 0.0 1.0 0 0<br />
u1t401=9.624<br />
u1t402q=0.3 0.0 1.0 0 0<br />
u1t402=9.648<br />
u1t403q=0.3 0.0 1.0 0 0<br />
u1t403=9.672<br />
u1t404q=0.3 0.0 1.0 0 0<br />
u1t404=9.696<br />
u1t405q=0.3 0.0 1.0 0 0<br />
u1t405=9.72<br />
u1t406q=0.3 0.0 1.0 0 0<br />
u1t406=9.744<br />
u1t407q=0.3 0.0 1.0 0 0<br />
u1t407=9.768<br />
u1t408q=0.3 0.0 1.0 0 0<br />
u1t408=9.792<br />
u1t409q=0.3 0.0 1.0 0 0<br />
u1t409=9.816<br />
u1t410q=0.3 0.0 1.0 0 0<br />
u1t410=9.84<br />
u1t411q=0.3 0.0 1.0 0 0<br />
u1t411=9.864<br />
u1t412q=0.3 0.0 1.0 0 0<br />
u1t412=9.888<br />
u1t413q=0.3 0.0 1.0 0 0<br />
u1t413=9.912<br />
u1t414q=0.3 0.0 1.0 0 0<br />
u1t414=9.936<br />
u1t415q=0.3 0.0 1.0 0 0<br />
u1t415=9.96<br />
u1t416q=0.3 0.0 1.0 0 0<br />
u1t416=9.984<br />
u1t417q=0.3 0.0 1.0 0 0<br />
u1t417=10.008<br />
u1t418q=0.3 0.0 1.0 0 0<br />
u1t418=10.032<br />
u1t419q=0.3 0.0 1.0 0 0<br />
u1t419=10.056<br />
u1t420q=0.3 0.0 1.0 0 0<br />
u1t420=10.08<br />
u1t421q=0.3 0.0 1.0 0 0<br />
u1t421=10.104<br />
u1t422q=0.3 0.0 1.0 0 0<br />
u1t422=10.128<br />
u1t423q=0.3 0.0 1.0 0 0<br />
u1t423=10.152<br />
u1t424q=0.3 0.0 1.0 0 0<br />
u1t424=10.176<br />
u1t425q=0.3 0.0 1.0 0 0<br />
u1t425=10.2<br />
u1t426q=0.3 0.0 1.0 0 0<br />
u1t426=10.224<br />
u1t427q=0.3 0.0 1.0 0 0<br />
u1t427=10.248<br />
u1t428q=0.3 0.0 1.0 0 0<br />
u1t428=10.272<br />
u1t429q=0.3 0.0 1.0 0 0<br />
u1t429=10.296<br />
u1t430q=0.3 0.0 1.0 0 0<br />
u1t430=10.32<br />
u1t431q=0.3 0.0 1.0 0 0<br />
u1t431=10.344<br />
u1t432q=0.3 0.0 1.0 0 0<br />
u1t432=10.368<br />
u1t433q=0.3 0.0 1.0 0 0<br />
u1t433=10.392<br />
u1t434q=0.3 0.0 1.0 0 0<br />
u1t434=10.416<br />
u1t435q=0.3 0.0 1.0 0 0<br />
u1t435=10.44<br />
u1t436q=0.3 0.0 1.0 0 0<br />
u1t436=10.464<br />
u1t437q=0.3 0.0 1.0 0 0<br />
u1t437=10.488<br />
u1t438q=0.3 0.0 1.0 0 0<br />
u1t438=10.512<br />
u1t439q=0.3 0.0 1.0 0 0<br />
u1t439=10.536<br />
u1t440q=0.3 0.0 1.0 0 0<br />
u1t440=10.56<br />
u1t441q=0.3 0.0 1.0 0 0<br />
u1t441=10.584<br />
u1t442q=0.3 0.0 1.0 0 0<br />
u1t442=10.608<br />
u1t443q=0.3 0.0 1.0 0 0<br />
u1t443=10.632<br />
u1t444q=0.3 0.0 1.0 0 0<br />
u1t444=10.656<br />
u1t445q=0.3 0.0 1.0 0 0<br />
u1t445=10.68<br />
u1t446q=0.3 0.0 1.0 0 0<br />
u1t446=10.704<br />
u1t447q=0.3 0.0 1.0 0 0<br />
u1t447=10.728<br />
u1t448q=0.3 0.0 1.0 0 0<br />
u1t448=10.752<br />
u1t449q=0.3 0.0 1.0 0 0<br />
u1t449=10.776<br />
u1t450q=0.3 0.0 1.0 0 0<br />
u1t450=10.8<br />
u1t451q=0.3 0.0 1.0 0 0<br />
u1t451=10.824<br />
u1t452q=0.3 0.0 1.0 0 0<br />
u1t452=10.848<br />
u1t453q=0.3 0.0 1.0 0 0<br />
u1t453=10.872<br />
u1t454q=0.3 0.0 1.0 0 0<br />
u1t454=10.896<br />
u1t455q=0.3 0.0 1.0 0 0<br />
u1t455=10.92<br />
u1t456q=0.3 0.0 1.0 0 0<br />
u1t456=10.944<br />
u1t457q=0.3 0.0 1.0 0 0<br />
u1t457=10.968<br />
u1t458q=0.3 0.0 1.0 0 0<br />
u1t458=10.992<br />
u1t459q=0.3 0.0 1.0 0 0<br />
u1t459=11.016<br />
u1t460q=0.3 0.0 1.0 0 0<br />
u1t460=11.04<br />
u1t461q=0.3 0.0 1.0 0 0<br />
u1t461=11.064<br />
u1t462q=0.3 0.0 1.0 0 0<br />
u1t462=11.088<br />
u1t463q=0.3 0.0 1.0 0 0<br />
u1t463=11.112<br />
u1t464q=0.3 0.0 1.0 0 0<br />
u1t464=11.136<br />
u1t465q=0.3 0.0 1.0 0 0<br />
u1t465=11.16<br />
u1t466q=0.3 0.0 1.0 0 0<br />
u1t466=11.184<br />
u1t467q=0.3 0.0 1.0 0 0<br />
u1t467=11.208<br />
u1t468q=0.3 0.0 1.0 0 0<br />
u1t468=11.232<br />
u1t469q=0.3 0.0 1.0 0 0<br />
u1t469=11.256<br />
u1t470q=0.3 0.0 1.0 0 0<br />
u1t470=11.28<br />
u1t471q=0.3 0.0 1.0 0 0<br />
u1t471=11.304<br />
u1t472q=0.3 0.0 1.0 0 0<br />
u1t472=11.328<br />
u1t473q=0.3 0.0 1.0 0 0<br />
u1t473=11.352<br />
u1t474q=0.3 0.0 1.0 0 0<br />
u1t474=11.376<br />
u1t475q=0.3 0.0 1.0 0 0<br />
u1t475=11.4<br />
u1t476q=0.3 0.0 1.0 0 0<br />
u1t476=11.424<br />
u1t477q=0.3 0.0 1.0 0 0<br />
u1t477=11.448<br />
u1t478q=0.3 0.0 1.0 0 0<br />
u1t478=11.472<br />
u1t479q=0.3 0.0 1.0 0 0<br />
u1t479=11.496<br />
u1t480q=0.3 0.0 1.0 0 0<br />
u1t480=11.52<br />
u1t481q=0.3 0.0 1.0 0 0<br />
u1t481=11.544<br />
u1t482q=0.3 0.0 1.0 0 0<br />
u1t482=11.568<br />
u1t483q=0.3 0.0 1.0 0 0<br />
u1t483=11.592<br />
u1t484q=0.3 0.0 1.0 0 0<br />
u1t484=11.616<br />
u1t485q=0.3 0.0 1.0 0 0<br />
u1t485=11.64<br />
u1t486q=0.3 0.0 1.0 0 0<br />
u1t486=11.664<br />
u1t487q=0.3 0.0 1.0 0 0<br />
u1t487=11.688<br />
u1t488q=0.3 0.0 1.0 0 0<br />
u1t488=11.712<br />
u1t489q=0.3 0.0 1.0 0 0<br />
u1t489=11.736<br />
u1t490q=0.3 0.0 1.0 0 0<br />
u1t490=11.76<br />
u1t491q=0.3 0.0 1.0 0 0<br />
u1t491=11.784<br />
u1t492q=0.3 0.0 1.0 0 0<br />
u1t492=11.808<br />
u1t493q=0.3 0.0 1.0 0 0<br />
u1t493=11.832<br />
u1t494q=0.3 0.0 1.0 0 0<br />
u1t494=11.856<br />
u1t495q=0.3 0.0 1.0 0 0<br />
u1t495=11.88<br />
u1t496q=0.3 0.0 1.0 0 0<br />
u1t496=11.904<br />
u1t497q=0.3 0.0 1.0 0 0<br />
u1t497=11.928<br />
u1t498q=0.3 0.0 1.0 0 0<br />
u1t498=11.952<br />
u1t499q=0.3 0.0 1.0 0 0<br />
u1t499=11.976<br />
u1t500q=0.3 0 0 0 0<br />
u1t500=tend<br />
<br />
[Messungen]<br />
tAnzahl=12<br />
<br />
t1=1<br />
t1Anzahl=2<br />
t1m1=mfcn1 1.0 1e-06 1<br />
t1m2=mfcn2 1.0 1e-06 1<br />
t1minmax=0 1e+10<br />
<br />
t2=2.000<br />
t2Anzahl=2<br />
t2m1=mfcn1 1.0 1e-06 1<br />
t2m2=mfcn2 1.0 1e-06 1<br />
t2minmax=0 1e+10<br />
<br />
t3=3<br />
t3Anzahl=2<br />
t3m1=mfcn1 1.0 1e-06 1<br />
t3m2=mfcn2 1.0 1e-06 1<br />
t3minmax=0 1e+10<br />
<br />
t4=4<br />
t4Anzahl=2<br />
t4m1=mfcn1 1.0 1e-06 1<br />
t4m2=mfcn2 1.0 1e-06 1<br />
t4minmax=0 1e+10<br />
<br />
t5=5<br />
t5Anzahl=2<br />
t5m1=mfcn1 1.0 1e-06 1<br />
t5m2=mfcn2 1.0 1e-06 1<br />
t5minmax=0 1e+10<br />
<br />
t6=6<br />
t6Anzahl=2<br />
t6m1=mfcn1 1.0 1e-06 1<br />
t6m2=mfcn2 1.0 1e-06 1<br />
t6minmax=0 1e+10<br />
<br />
t7=7<br />
t7Anzahl=2<br />
t7m1=mfcn1 1.0 1e-06 1<br />
t7m2=mfcn2 1.0 1e-06 1<br />
t7minmax=0 1e+10<br />
<br />
t8=8<br />
t8Anzahl=2<br />
t8m1=mfcn1 1.0 1e-06 1<br />
t8m2=mfcn2 1.0 1e-06 1<br />
t8minmax=0 1e+10<br />
<br />
t9=9<br />
t9Anzahl=2<br />
t9m1=mfcn1 1.0 1e-06 1<br />
t9m2=mfcn2 1.0 1e-06 1<br />
t9minmax=0 1e+10<br />
<br />
t10=10.000<br />
t10Anzahl=2<br />
t10m1=mfcn1 1.0 1e-06 1<br />
t10m2=mfcn2 1.0 1e-06 1<br />
t10minmax=0 1e+10<br />
<br />
t11=11.000<br />
t11Anzahl=2<br />
t11m1=mfcn1 1.0 1e-06 1<br />
t11m2=mfcn2 1.0 1e-06 1<br />
t11minmax=0 1e+10<br />
<br />
t12=12.000<br />
t12Anzahl=2<br />
t12m1=mfcn1 1.0 1e-06 1<br />
t12m2=mfcn2 1.0 1e-06 1<br />
t12minmax=0 1e+10<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=0<br />
[Messverfahren]<br />
mAnzahl=2<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
m2=mfcn2 1 0 1e+10 0<br />
m2f1=mess4 sigma4 1<br />
mminmaxges=0 8<br />
<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
realworkspace=1700000<br />
integerworkspace=5000<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1343
Lotka Experimental Design (VPLAN)
2016-01-19T16:25:27Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function<br />
<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<br />
<br />
<source lang="optimica"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
<br />
<br />
<source lang="vplan"><br />
<br />
; ini-File fuer Experiment<br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=12<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=2<br />
y1=x1 0.5 -1e+10 1e+10<br />
y2=x2 0.7 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=12<br />
t1=1<br />
t2=2<br />
t3=3<br />
t4=4<br />
t5=5<br />
t6=6<br />
t7=7<br />
t8=8<br />
t9=9<br />
t10=10<br />
t11=11<br />
t12=12<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=0<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=myu 0 0.0 1.0<br />
u1tAnzahl=500<br />
u1t0=t0<br />
u1t1q=0.3 0.0 1.0 0 0<br />
u1t1=0.024<br />
u1t2q=0.3 0.0 1.0 0 0<br />
u1t2=0.048<br />
u1t3q=0.3 0.0 1.0 0 0<br />
u1t3=0.072<br />
u1t4q=0.3 0.0 1.0 0 0<br />
u1t4=0.096<br />
u1t5q=0.3 0.0 1.0 0 0<br />
u1t5=0.12<br />
u1t6q=0.3 0.0 1.0 0 0<br />
u1t6=0.144<br />
u1t7q=0.3 0.0 1.0 0 0<br />
u1t7=0.168<br />
u1t8q=0.3 0.0 1.0 0 0<br />
u1t8=0.192<br />
u1t9q=0.3 0.0 1.0 0 0<br />
u1t9=0.216<br />
u1t10q=0.3 0.0 1.0 0 0<br />
u1t10=0.24<br />
u1t11q=0.3 0.0 1.0 0 0<br />
u1t11=0.264<br />
u1t12q=0.3 0.0 1.0 0 0<br />
u1t12=0.288<br />
u1t13q=0.3 0.0 1.0 0 0<br />
u1t13=0.312<br />
u1t14q=0.3 0.0 1.0 0 0<br />
u1t14=0.336<br />
u1t15q=0.3 0.0 1.0 0 0<br />
u1t15=0.36<br />
u1t16q=0.3 0.0 1.0 0 0<br />
u1t16=0.384<br />
u1t17q=0.3 0.0 1.0 0 0<br />
u1t17=0.408<br />
u1t18q=0.3 0.0 1.0 0 0<br />
u1t18=0.432<br />
u1t19q=0.3 0.0 1.0 0 0<br />
u1t19=0.456<br />
u1t20q=0.3 0.0 1.0 0 0<br />
u1t20=0.48<br />
u1t21q=0.3 0.0 1.0 0 0<br />
u1t21=0.504<br />
u1t22q=0.3 0.0 1.0 0 0<br />
u1t22=0.528<br />
u1t23q=0.3 0.0 1.0 0 0<br />
u1t23=0.552<br />
u1t24q=0.3 0.0 1.0 0 0<br />
u1t24=0.576<br />
u1t25q=0.3 0.0 1.0 0 0<br />
u1t25=0.6<br />
u1t26q=0.3 0.0 1.0 0 0<br />
u1t26=0.624<br />
u1t27q=0.3 0.0 1.0 0 0<br />
u1t27=0.648<br />
u1t28q=0.3 0.0 1.0 0 0<br />
u1t28=0.672<br />
u1t29q=0.3 0.0 1.0 0 0<br />
u1t29=0.696<br />
u1t30q=0.3 0.0 1.0 0 0<br />
u1t30=0.72<br />
u1t31q=0.3 0.0 1.0 0 0<br />
u1t31=0.744<br />
u1t32q=0.3 0.0 1.0 0 0<br />
u1t32=0.768<br />
u1t33q=0.3 0.0 1.0 0 0<br />
u1t33=0.792<br />
u1t34q=0.3 0.0 1.0 0 0<br />
u1t34=0.816<br />
u1t35q=0.3 0.0 1.0 0 0<br />
u1t35=0.84<br />
u1t36q=0.3 0.0 1.0 0 0<br />
u1t36=0.864<br />
u1t37q=0.3 0.0 1.0 0 0<br />
u1t37=0.888<br />
u1t38q=0.3 0.0 1.0 0 0<br />
u1t38=0.912<br />
u1t39q=0.3 0.0 1.0 0 0<br />
u1t39=0.936<br />
u1t40q=0.3 0.0 1.0 0 0<br />
u1t40=0.96<br />
u1t41q=0.3 0.0 1.0 0 0<br />
u1t41=0.984<br />
u1t42q=0.3 0.0 1.0 0 0<br />
u1t42=1.008<br />
u1t43q=0.3 0.0 1.0 0 0<br />
u1t43=1.032<br />
u1t44q=0.3 0.0 1.0 0 0<br />
u1t44=1.056<br />
u1t45q=0.3 0.0 1.0 0 0<br />
u1t45=1.08<br />
u1t46q=0.3 0.0 1.0 0 0<br />
u1t46=1.104<br />
u1t47q=0.3 0.0 1.0 0 0<br />
u1t47=1.128<br />
u1t48q=0.3 0.0 1.0 0 0<br />
u1t48=1.152<br />
u1t49q=0.3 0.0 1.0 0 0<br />
u1t49=1.176<br />
u1t50q=0.3 0.0 1.0 0 0<br />
u1t50=1.2<br />
u1t51q=0.3 0.0 1.0 0 0<br />
u1t51=1.224<br />
u1t52q=0.3 0.0 1.0 0 0<br />
u1t52=1.248<br />
u1t53q=0.3 0.0 1.0 0 0<br />
u1t53=1.272<br />
u1t54q=0.3 0.0 1.0 0 0<br />
u1t54=1.296<br />
u1t55q=0.3 0.0 1.0 0 0<br />
u1t55=1.32<br />
u1t56q=0.3 0.0 1.0 0 0<br />
u1t56=1.344<br />
u1t57q=0.3 0.0 1.0 0 0<br />
u1t57=1.368<br />
u1t58q=0.3 0.0 1.0 0 0<br />
u1t58=1.392<br />
u1t59q=0.3 0.0 1.0 0 0<br />
u1t59=1.416<br />
u1t60q=0.3 0.0 1.0 0 0<br />
u1t60=1.44<br />
u1t61q=0.3 0.0 1.0 0 0<br />
u1t61=1.464<br />
u1t62q=0.3 0.0 1.0 0 0<br />
u1t62=1.488<br />
u1t63q=0.3 0.0 1.0 0 0<br />
u1t63=1.512<br />
u1t64q=0.3 0.0 1.0 0 0<br />
u1t64=1.536<br />
u1t65q=0.3 0.0 1.0 0 0<br />
u1t65=1.56<br />
u1t66q=0.3 0.0 1.0 0 0<br />
u1t66=1.584<br />
u1t67q=0.3 0.0 1.0 0 0<br />
u1t67=1.608<br />
u1t68q=0.3 0.0 1.0 0 0<br />
u1t68=1.632<br />
u1t69q=0.3 0.0 1.0 0 0<br />
u1t69=1.656<br />
u1t70q=0.3 0.0 1.0 0 0<br />
u1t70=1.68<br />
u1t71q=0.3 0.0 1.0 0 0<br />
u1t71=1.704<br />
u1t72q=0.3 0.0 1.0 0 0<br />
u1t72=1.728<br />
u1t73q=0.3 0.0 1.0 0 0<br />
u1t73=1.752<br />
u1t74q=0.3 0.0 1.0 0 0<br />
u1t74=1.776<br />
u1t75q=0.3 0.0 1.0 0 0<br />
u1t75=1.8<br />
u1t76q=0.3 0.0 1.0 0 0<br />
u1t76=1.824<br />
u1t77q=0.3 0.0 1.0 0 0<br />
u1t77=1.848<br />
u1t78q=0.3 0.0 1.0 0 0<br />
u1t78=1.872<br />
u1t79q=0.3 0.0 1.0 0 0<br />
u1t79=1.896<br />
u1t80q=0.3 0.0 1.0 0 0<br />
u1t80=1.92<br />
u1t81q=0.3 0.0 1.0 0 0<br />
u1t81=1.944<br />
u1t82q=0.3 0.0 1.0 0 0<br />
u1t82=1.968<br />
u1t83q=0.3 0.0 1.0 0 0<br />
u1t83=1.992<br />
u1t84q=0.3 0.0 1.0 0 0<br />
u1t84=2.016<br />
u1t85q=0.3 0.0 1.0 0 0<br />
u1t85=2.04<br />
u1t86q=0.3 0.0 1.0 0 0<br />
u1t86=2.064<br />
u1t87q=0.3 0.0 1.0 0 0<br />
u1t87=2.088<br />
u1t88q=0.3 0.0 1.0 0 0<br />
u1t88=2.112<br />
u1t89q=0.3 0.0 1.0 0 0<br />
u1t89=2.136<br />
u1t90q=0.3 0.0 1.0 0 0<br />
u1t90=2.16<br />
u1t91q=0.3 0.0 1.0 0 0<br />
u1t91=2.184<br />
u1t92q=0.3 0.0 1.0 0 0<br />
u1t92=2.208<br />
u1t93q=0.3 0.0 1.0 0 0<br />
u1t93=2.232<br />
u1t94q=0.3 0.0 1.0 0 0<br />
u1t94=2.256<br />
u1t95q=0.3 0.0 1.0 0 0<br />
u1t95=2.28<br />
u1t96q=0.3 0.0 1.0 0 0<br />
u1t96=2.304<br />
u1t97q=0.3 0.0 1.0 0 0<br />
u1t97=2.328<br />
u1t98q=0.3 0.0 1.0 0 0<br />
u1t98=2.352<br />
u1t99q=0.3 0.0 1.0 0 0<br />
u1t99=2.376<br />
u1t100q=0.3 0.0 1.0 0 0<br />
u1t100=2.4<br />
u1t101q=0.3 0.0 1.0 0 0<br />
u1t101=2.424<br />
u1t102q=0.3 0.0 1.0 0 0<br />
u1t102=2.448<br />
u1t103q=0.3 0.0 1.0 0 0<br />
u1t103=2.472<br />
u1t104q=0.3 0.0 1.0 0 0<br />
u1t104=2.496<br />
u1t105q=0.3 0.0 1.0 0 0<br />
u1t105=2.52<br />
u1t106q=0.3 0.0 1.0 0 0<br />
u1t106=2.544<br />
u1t107q=0.3 0.0 1.0 0 0<br />
u1t107=2.568<br />
u1t108q=0.3 0.0 1.0 0 0<br />
u1t108=2.592<br />
u1t109q=0.3 0.0 1.0 0 0<br />
u1t109=2.616<br />
u1t110q=0.3 0.0 1.0 0 0<br />
u1t110=2.64<br />
u1t111q=0.3 0.0 1.0 0 0<br />
u1t111=2.664<br />
u1t112q=0.3 0.0 1.0 0 0<br />
u1t112=2.688<br />
u1t113q=0.3 0.0 1.0 0 0<br />
u1t113=2.712<br />
u1t114q=0.3 0.0 1.0 0 0<br />
u1t114=2.736<br />
u1t115q=0.3 0.0 1.0 0 0<br />
u1t115=2.76<br />
u1t116q=0.3 0.0 1.0 0 0<br />
u1t116=2.784<br />
u1t117q=0.3 0.0 1.0 0 0<br />
u1t117=2.808<br />
u1t118q=0.3 0.0 1.0 0 0<br />
u1t118=2.832<br />
u1t119q=0.3 0.0 1.0 0 0<br />
u1t119=2.856<br />
u1t120q=0.3 0.0 1.0 0 0<br />
u1t120=2.88<br />
u1t121q=0.3 0.0 1.0 0 0<br />
u1t121=2.904<br />
u1t122q=0.3 0.0 1.0 0 0<br />
u1t122=2.928<br />
u1t123q=0.3 0.0 1.0 0 0<br />
u1t123=2.952<br />
u1t124q=0.3 0.0 1.0 0 0<br />
u1t124=2.976<br />
u1t125q=0.3 0.0 1.0 0 0<br />
u1t125=3.0<br />
u1t126q=0.3 0.0 1.0 0 0<br />
u1t126=3.024<br />
u1t127q=0.3 0.0 1.0 0 0<br />
u1t127=3.048<br />
u1t128q=0.3 0.0 1.0 0 0<br />
u1t128=3.072<br />
u1t129q=0.3 0.0 1.0 0 0<br />
u1t129=3.096<br />
u1t130q=0.3 0.0 1.0 0 0<br />
u1t130=3.12<br />
u1t131q=0.3 0.0 1.0 0 0<br />
u1t131=3.144<br />
u1t132q=0.3 0.0 1.0 0 0<br />
u1t132=3.168<br />
u1t133q=0.3 0.0 1.0 0 0<br />
u1t133=3.192<br />
u1t134q=0.3 0.0 1.0 0 0<br />
u1t134=3.216<br />
u1t135q=0.3 0.0 1.0 0 0<br />
u1t135=3.24<br />
u1t136q=0.3 0.0 1.0 0 0<br />
u1t136=3.264<br />
u1t137q=0.3 0.0 1.0 0 0<br />
u1t137=3.288<br />
u1t138q=0.3 0.0 1.0 0 0<br />
u1t138=3.312<br />
u1t139q=0.3 0.0 1.0 0 0<br />
u1t139=3.336<br />
u1t140q=0.3 0.0 1.0 0 0<br />
u1t140=3.36<br />
u1t141q=0.3 0.0 1.0 0 0<br />
u1t141=3.384<br />
u1t142q=0.3 0.0 1.0 0 0<br />
u1t142=3.408<br />
u1t143q=0.3 0.0 1.0 0 0<br />
u1t143=3.432<br />
u1t144q=0.3 0.0 1.0 0 0<br />
u1t144=3.456<br />
u1t145q=0.3 0.0 1.0 0 0<br />
u1t145=3.48<br />
u1t146q=0.3 0.0 1.0 0 0<br />
u1t146=3.504<br />
u1t147q=0.3 0.0 1.0 0 0<br />
u1t147=3.528<br />
u1t148q=0.3 0.0 1.0 0 0<br />
u1t148=3.552<br />
u1t149q=0.3 0.0 1.0 0 0<br />
u1t149=3.576<br />
u1t150q=0.3 0.0 1.0 0 0<br />
u1t150=3.6<br />
u1t151q=0.3 0.0 1.0 0 0<br />
u1t151=3.624<br />
u1t152q=0.3 0.0 1.0 0 0<br />
u1t152=3.648<br />
u1t153q=0.3 0.0 1.0 0 0<br />
u1t153=3.672<br />
u1t154q=0.3 0.0 1.0 0 0<br />
u1t154=3.696<br />
u1t155q=0.3 0.0 1.0 0 0<br />
u1t155=3.72<br />
u1t156q=0.3 0.0 1.0 0 0<br />
u1t156=3.744<br />
u1t157q=0.3 0.0 1.0 0 0<br />
u1t157=3.768<br />
u1t158q=0.3 0.0 1.0 0 0<br />
u1t158=3.792<br />
u1t159q=0.3 0.0 1.0 0 0<br />
u1t159=3.816<br />
u1t160q=0.3 0.0 1.0 0 0<br />
u1t160=3.84<br />
u1t161q=0.3 0.0 1.0 0 0<br />
u1t161=3.864<br />
u1t162q=0.3 0.0 1.0 0 0<br />
u1t162=3.888<br />
u1t163q=0.3 0.0 1.0 0 0<br />
u1t163=3.912<br />
u1t164q=0.3 0.0 1.0 0 0<br />
u1t164=3.936<br />
u1t165q=0.3 0.0 1.0 0 0<br />
u1t165=3.96<br />
u1t166q=0.3 0.0 1.0 0 0<br />
u1t166=3.984<br />
u1t167q=0.3 0.0 1.0 0 0<br />
u1t167=4.008<br />
u1t168q=0.3 0.0 1.0 0 0<br />
u1t168=4.032<br />
u1t169q=0.3 0.0 1.0 0 0<br />
u1t169=4.056<br />
u1t170q=0.3 0.0 1.0 0 0<br />
u1t170=4.08<br />
u1t171q=0.3 0.0 1.0 0 0<br />
u1t171=4.104<br />
u1t172q=0.3 0.0 1.0 0 0<br />
u1t172=4.128<br />
u1t173q=0.3 0.0 1.0 0 0<br />
u1t173=4.152<br />
u1t174q=0.3 0.0 1.0 0 0<br />
u1t174=4.176<br />
u1t175q=0.3 0.0 1.0 0 0<br />
u1t175=4.2<br />
u1t176q=0.3 0.0 1.0 0 0<br />
u1t176=4.224<br />
u1t177q=0.3 0.0 1.0 0 0<br />
u1t177=4.248<br />
u1t178q=0.3 0.0 1.0 0 0<br />
u1t178=4.272<br />
u1t179q=0.3 0.0 1.0 0 0<br />
u1t179=4.296<br />
u1t180q=0.3 0.0 1.0 0 0<br />
u1t180=4.32<br />
u1t181q=0.3 0.0 1.0 0 0<br />
u1t181=4.344<br />
u1t182q=0.3 0.0 1.0 0 0<br />
u1t182=4.368<br />
u1t183q=0.3 0.0 1.0 0 0<br />
u1t183=4.392<br />
u1t184q=0.3 0.0 1.0 0 0<br />
u1t184=4.416<br />
u1t185q=0.3 0.0 1.0 0 0<br />
u1t185=4.44<br />
u1t186q=0.3 0.0 1.0 0 0<br />
u1t186=4.464<br />
u1t187q=0.3 0.0 1.0 0 0<br />
u1t187=4.488<br />
u1t188q=0.3 0.0 1.0 0 0<br />
u1t188=4.512<br />
u1t189q=0.3 0.0 1.0 0 0<br />
u1t189=4.536<br />
u1t190q=0.3 0.0 1.0 0 0<br />
u1t190=4.56<br />
u1t191q=0.3 0.0 1.0 0 0<br />
u1t191=4.584<br />
u1t192q=0.3 0.0 1.0 0 0<br />
u1t192=4.608<br />
u1t193q=0.3 0.0 1.0 0 0<br />
u1t193=4.632<br />
u1t194q=0.3 0.0 1.0 0 0<br />
u1t194=4.656<br />
u1t195q=0.3 0.0 1.0 0 0<br />
u1t195=4.68<br />
u1t196q=0.3 0.0 1.0 0 0<br />
u1t196=4.704<br />
u1t197q=0.3 0.0 1.0 0 0<br />
u1t197=4.728<br />
u1t198q=0.3 0.0 1.0 0 0<br />
u1t198=4.752<br />
u1t199q=0.3 0.0 1.0 0 0<br />
u1t199=4.776<br />
u1t200q=0.3 0.0 1.0 0 0<br />
u1t200=4.8<br />
u1t201q=0.3 0.0 1.0 0 0<br />
u1t201=4.824<br />
u1t202q=0.3 0.0 1.0 0 0<br />
u1t202=4.848<br />
u1t203q=0.3 0.0 1.0 0 0<br />
u1t203=4.872<br />
u1t204q=0.3 0.0 1.0 0 0<br />
u1t204=4.896<br />
u1t205q=0.3 0.0 1.0 0 0<br />
u1t205=4.92<br />
u1t206q=0.3 0.0 1.0 0 0<br />
u1t206=4.944<br />
u1t207q=0.3 0.0 1.0 0 0<br />
u1t207=4.968<br />
u1t208q=0.3 0.0 1.0 0 0<br />
u1t208=4.992<br />
u1t209q=0.3 0.0 1.0 0 0<br />
u1t209=5.016<br />
u1t210q=0.3 0.0 1.0 0 0<br />
u1t210=5.04<br />
u1t211q=0.3 0.0 1.0 0 0<br />
u1t211=5.064<br />
u1t212q=0.3 0.0 1.0 0 0<br />
u1t212=5.088<br />
u1t213q=0.3 0.0 1.0 0 0<br />
u1t213=5.112<br />
u1t214q=0.3 0.0 1.0 0 0<br />
u1t214=5.136<br />
u1t215q=0.3 0.0 1.0 0 0<br />
u1t215=5.16<br />
u1t216q=0.3 0.0 1.0 0 0<br />
u1t216=5.184<br />
u1t217q=0.3 0.0 1.0 0 0<br />
u1t217=5.208<br />
u1t218q=0.3 0.0 1.0 0 0<br />
u1t218=5.232<br />
u1t219q=0.3 0.0 1.0 0 0<br />
u1t219=5.256<br />
u1t220q=0.3 0.0 1.0 0 0<br />
u1t220=5.28<br />
u1t221q=0.3 0.0 1.0 0 0<br />
u1t221=5.304<br />
u1t222q=0.3 0.0 1.0 0 0<br />
u1t222=5.328<br />
u1t223q=0.3 0.0 1.0 0 0<br />
u1t223=5.352<br />
u1t224q=0.3 0.0 1.0 0 0<br />
u1t224=5.376<br />
u1t225q=0.3 0.0 1.0 0 0<br />
u1t225=5.4<br />
u1t226q=0.3 0.0 1.0 0 0<br />
u1t226=5.424<br />
u1t227q=0.3 0.0 1.0 0 0<br />
u1t227=5.448<br />
u1t228q=0.3 0.0 1.0 0 0<br />
u1t228=5.472<br />
u1t229q=0.3 0.0 1.0 0 0<br />
u1t229=5.496<br />
u1t230q=0.3 0.0 1.0 0 0<br />
u1t230=5.52<br />
u1t231q=0.3 0.0 1.0 0 0<br />
u1t231=5.544<br />
u1t232q=0.3 0.0 1.0 0 0<br />
u1t232=5.568<br />
u1t233q=0.3 0.0 1.0 0 0<br />
u1t233=5.592<br />
u1t234q=0.3 0.0 1.0 0 0<br />
u1t234=5.616<br />
u1t235q=0.3 0.0 1.0 0 0<br />
u1t235=5.64<br />
u1t236q=0.3 0.0 1.0 0 0<br />
u1t236=5.664<br />
u1t237q=0.3 0.0 1.0 0 0<br />
u1t237=5.688<br />
u1t238q=0.3 0.0 1.0 0 0<br />
u1t238=5.712<br />
u1t239q=0.3 0.0 1.0 0 0<br />
u1t239=5.736<br />
u1t240q=0.3 0.0 1.0 0 0<br />
u1t240=5.76<br />
u1t241q=0.3 0.0 1.0 0 0<br />
u1t241=5.784<br />
u1t242q=0.3 0.0 1.0 0 0<br />
u1t242=5.808<br />
u1t243q=0.3 0.0 1.0 0 0<br />
u1t243=5.832<br />
u1t244q=0.3 0.0 1.0 0 0<br />
u1t244=5.856<br />
u1t245q=0.3 0.0 1.0 0 0<br />
u1t245=5.88<br />
u1t246q=0.3 0.0 1.0 0 0<br />
u1t246=5.904<br />
u1t247q=0.3 0.0 1.0 0 0<br />
u1t247=5.928<br />
u1t248q=0.3 0.0 1.0 0 0<br />
u1t248=5.952<br />
u1t249q=0.3 0.0 1.0 0 0<br />
u1t249=5.976<br />
u1t250q=0.3 0.0 1.0 0 0<br />
u1t250=6.0<br />
u1t251q=0.3 0.0 1.0 0 0<br />
u1t251=6.024<br />
u1t252q=0.3 0.0 1.0 0 0<br />
u1t252=6.048<br />
u1t253q=0.3 0.0 1.0 0 0<br />
u1t253=6.072<br />
u1t254q=0.3 0.0 1.0 0 0<br />
u1t254=6.096<br />
u1t255q=0.3 0.0 1.0 0 0<br />
u1t255=6.12<br />
u1t256q=0.3 0.0 1.0 0 0<br />
u1t256=6.144<br />
u1t257q=0.3 0.0 1.0 0 0<br />
u1t257=6.168<br />
u1t258q=0.3 0.0 1.0 0 0<br />
u1t258=6.192<br />
u1t259q=0.3 0.0 1.0 0 0<br />
u1t259=6.216<br />
u1t260q=0.3 0.0 1.0 0 0<br />
u1t260=6.24<br />
u1t261q=0.3 0.0 1.0 0 0<br />
u1t261=6.264<br />
u1t262q=0.3 0.0 1.0 0 0<br />
u1t262=6.288<br />
u1t263q=0.3 0.0 1.0 0 0<br />
u1t263=6.312<br />
u1t264q=0.3 0.0 1.0 0 0<br />
u1t264=6.336<br />
u1t265q=0.3 0.0 1.0 0 0<br />
u1t265=6.36<br />
u1t266q=0.3 0.0 1.0 0 0<br />
u1t266=6.384<br />
u1t267q=0.3 0.0 1.0 0 0<br />
u1t267=6.408<br />
u1t268q=0.3 0.0 1.0 0 0<br />
u1t268=6.432<br />
u1t269q=0.3 0.0 1.0 0 0<br />
u1t269=6.456<br />
u1t270q=0.3 0.0 1.0 0 0<br />
u1t270=6.48<br />
u1t271q=0.3 0.0 1.0 0 0<br />
u1t271=6.504<br />
u1t272q=0.3 0.0 1.0 0 0<br />
u1t272=6.528<br />
u1t273q=0.3 0.0 1.0 0 0<br />
u1t273=6.552<br />
u1t274q=0.3 0.0 1.0 0 0<br />
u1t274=6.576<br />
u1t275q=0.3 0.0 1.0 0 0<br />
u1t275=6.6<br />
u1t276q=0.3 0.0 1.0 0 0<br />
u1t276=6.624<br />
u1t277q=0.3 0.0 1.0 0 0<br />
u1t277=6.648<br />
u1t278q=0.3 0.0 1.0 0 0<br />
u1t278=6.672<br />
u1t279q=0.3 0.0 1.0 0 0<br />
u1t279=6.696<br />
u1t280q=0.3 0.0 1.0 0 0<br />
u1t280=6.72<br />
u1t281q=0.3 0.0 1.0 0 0<br />
u1t281=6.744<br />
u1t282q=0.3 0.0 1.0 0 0<br />
u1t282=6.768<br />
u1t283q=0.3 0.0 1.0 0 0<br />
u1t283=6.792<br />
u1t284q=0.3 0.0 1.0 0 0<br />
u1t284=6.816<br />
u1t285q=0.3 0.0 1.0 0 0<br />
u1t285=6.84<br />
u1t286q=0.3 0.0 1.0 0 0<br />
u1t286=6.864<br />
u1t287q=0.3 0.0 1.0 0 0<br />
u1t287=6.888<br />
u1t288q=0.3 0.0 1.0 0 0<br />
u1t288=6.912<br />
u1t289q=0.3 0.0 1.0 0 0<br />
u1t289=6.936<br />
u1t290q=0.3 0.0 1.0 0 0<br />
u1t290=6.96<br />
u1t291q=0.3 0.0 1.0 0 0<br />
u1t291=6.984<br />
u1t292q=0.3 0.0 1.0 0 0<br />
u1t292=7.008<br />
u1t293q=0.3 0.0 1.0 0 0<br />
u1t293=7.032<br />
u1t294q=0.3 0.0 1.0 0 0<br />
u1t294=7.056<br />
u1t295q=0.3 0.0 1.0 0 0<br />
u1t295=7.08<br />
u1t296q=0.3 0.0 1.0 0 0<br />
u1t296=7.104<br />
u1t297q=0.3 0.0 1.0 0 0<br />
u1t297=7.128<br />
u1t298q=0.3 0.0 1.0 0 0<br />
u1t298=7.152<br />
u1t299q=0.3 0.0 1.0 0 0<br />
u1t299=7.176<br />
u1t300q=0.3 0.0 1.0 0 0<br />
u1t300=7.2<br />
u1t301q=0.3 0.0 1.0 0 0<br />
u1t301=7.224<br />
u1t302q=0.3 0.0 1.0 0 0<br />
u1t302=7.248<br />
u1t303q=0.3 0.0 1.0 0 0<br />
u1t303=7.272<br />
u1t304q=0.3 0.0 1.0 0 0<br />
u1t304=7.296<br />
u1t305q=0.3 0.0 1.0 0 0<br />
u1t305=7.32<br />
u1t306q=0.3 0.0 1.0 0 0<br />
u1t306=7.344<br />
u1t307q=0.3 0.0 1.0 0 0<br />
u1t307=7.368<br />
u1t308q=0.3 0.0 1.0 0 0<br />
u1t308=7.392<br />
u1t309q=0.3 0.0 1.0 0 0<br />
u1t309=7.416<br />
u1t310q=0.3 0.0 1.0 0 0<br />
u1t310=7.44<br />
u1t311q=0.3 0.0 1.0 0 0<br />
u1t311=7.464<br />
u1t312q=0.3 0.0 1.0 0 0<br />
u1t312=7.488<br />
u1t313q=0.3 0.0 1.0 0 0<br />
u1t313=7.512<br />
u1t314q=0.3 0.0 1.0 0 0<br />
u1t314=7.536<br />
u1t315q=0.3 0.0 1.0 0 0<br />
u1t315=7.56<br />
u1t316q=0.3 0.0 1.0 0 0<br />
u1t316=7.584<br />
u1t317q=0.3 0.0 1.0 0 0<br />
u1t317=7.608<br />
u1t318q=0.3 0.0 1.0 0 0<br />
u1t318=7.632<br />
u1t319q=0.3 0.0 1.0 0 0<br />
u1t319=7.656<br />
u1t320q=0.3 0.0 1.0 0 0<br />
u1t320=7.68<br />
u1t321q=0.3 0.0 1.0 0 0<br />
u1t321=7.704<br />
u1t322q=0.3 0.0 1.0 0 0<br />
u1t322=7.728<br />
u1t323q=0.3 0.0 1.0 0 0<br />
u1t323=7.752<br />
u1t324q=0.3 0.0 1.0 0 0<br />
u1t324=7.776<br />
u1t325q=0.3 0.0 1.0 0 0<br />
u1t325=7.8<br />
u1t326q=0.3 0.0 1.0 0 0<br />
u1t326=7.824<br />
u1t327q=0.3 0.0 1.0 0 0<br />
u1t327=7.848<br />
u1t328q=0.3 0.0 1.0 0 0<br />
u1t328=7.872<br />
u1t329q=0.3 0.0 1.0 0 0<br />
u1t329=7.896<br />
u1t330q=0.3 0.0 1.0 0 0<br />
u1t330=7.92<br />
u1t331q=0.3 0.0 1.0 0 0<br />
u1t331=7.944<br />
u1t332q=0.3 0.0 1.0 0 0<br />
u1t332=7.968<br />
u1t333q=0.3 0.0 1.0 0 0<br />
u1t333=7.992<br />
u1t334q=0.3 0.0 1.0 0 0<br />
u1t334=8.016<br />
u1t335q=0.3 0.0 1.0 0 0<br />
u1t335=8.04<br />
u1t336q=0.3 0.0 1.0 0 0<br />
u1t336=8.064<br />
u1t337q=0.3 0.0 1.0 0 0<br />
u1t337=8.088<br />
u1t338q=0.3 0.0 1.0 0 0<br />
u1t338=8.112<br />
u1t339q=0.3 0.0 1.0 0 0<br />
u1t339=8.136<br />
u1t340q=0.3 0.0 1.0 0 0<br />
u1t340=8.16<br />
u1t341q=0.3 0.0 1.0 0 0<br />
u1t341=8.184<br />
u1t342q=0.3 0.0 1.0 0 0<br />
u1t342=8.208<br />
u1t343q=0.3 0.0 1.0 0 0<br />
u1t343=8.232<br />
u1t344q=0.3 0.0 1.0 0 0<br />
u1t344=8.256<br />
u1t345q=0.3 0.0 1.0 0 0<br />
u1t345=8.28<br />
u1t346q=0.3 0.0 1.0 0 0<br />
u1t346=8.304<br />
u1t347q=0.3 0.0 1.0 0 0<br />
u1t347=8.328<br />
u1t348q=0.3 0.0 1.0 0 0<br />
u1t348=8.352<br />
u1t349q=0.3 0.0 1.0 0 0<br />
u1t349=8.376<br />
u1t350q=0.3 0.0 1.0 0 0<br />
u1t350=8.4<br />
u1t351q=0.3 0.0 1.0 0 0<br />
u1t351=8.424<br />
u1t352q=0.3 0.0 1.0 0 0<br />
u1t352=8.448<br />
u1t353q=0.3 0.0 1.0 0 0<br />
u1t353=8.472<br />
u1t354q=0.3 0.0 1.0 0 0<br />
u1t354=8.496<br />
u1t355q=0.3 0.0 1.0 0 0<br />
u1t355=8.52<br />
u1t356q=0.3 0.0 1.0 0 0<br />
u1t356=8.544<br />
u1t357q=0.3 0.0 1.0 0 0<br />
u1t357=8.568<br />
u1t358q=0.3 0.0 1.0 0 0<br />
u1t358=8.592<br />
u1t359q=0.3 0.0 1.0 0 0<br />
u1t359=8.616<br />
u1t360q=0.3 0.0 1.0 0 0<br />
u1t360=8.64<br />
u1t361q=0.3 0.0 1.0 0 0<br />
u1t361=8.664<br />
u1t362q=0.3 0.0 1.0 0 0<br />
u1t362=8.688<br />
u1t363q=0.3 0.0 1.0 0 0<br />
u1t363=8.712<br />
u1t364q=0.3 0.0 1.0 0 0<br />
u1t364=8.736<br />
u1t365q=0.3 0.0 1.0 0 0<br />
u1t365=8.76<br />
u1t366q=0.3 0.0 1.0 0 0<br />
u1t366=8.784<br />
u1t367q=0.3 0.0 1.0 0 0<br />
u1t367=8.808<br />
u1t368q=0.3 0.0 1.0 0 0<br />
u1t368=8.832<br />
u1t369q=0.3 0.0 1.0 0 0<br />
u1t369=8.856<br />
u1t370q=0.3 0.0 1.0 0 0<br />
u1t370=8.88<br />
u1t371q=0.3 0.0 1.0 0 0<br />
u1t371=8.904<br />
u1t372q=0.3 0.0 1.0 0 0<br />
u1t372=8.928<br />
u1t373q=0.3 0.0 1.0 0 0<br />
u1t373=8.952<br />
u1t374q=0.3 0.0 1.0 0 0<br />
u1t374=8.976<br />
u1t375q=0.3 0.0 1.0 0 0<br />
u1t375=9.0<br />
u1t376q=0.3 0.0 1.0 0 0<br />
u1t376=9.024<br />
u1t377q=0.3 0.0 1.0 0 0<br />
u1t377=9.048<br />
u1t378q=0.3 0.0 1.0 0 0<br />
u1t378=9.072<br />
u1t379q=0.3 0.0 1.0 0 0<br />
u1t379=9.096<br />
u1t380q=0.3 0.0 1.0 0 0<br />
u1t380=9.12<br />
u1t381q=0.3 0.0 1.0 0 0<br />
u1t381=9.144<br />
u1t382q=0.3 0.0 1.0 0 0<br />
u1t382=9.168<br />
u1t383q=0.3 0.0 1.0 0 0<br />
u1t383=9.192<br />
u1t384q=0.3 0.0 1.0 0 0<br />
u1t384=9.216<br />
u1t385q=0.3 0.0 1.0 0 0<br />
u1t385=9.24<br />
u1t386q=0.3 0.0 1.0 0 0<br />
u1t386=9.264<br />
u1t387q=0.3 0.0 1.0 0 0<br />
u1t387=9.288<br />
u1t388q=0.3 0.0 1.0 0 0<br />
u1t388=9.312<br />
u1t389q=0.3 0.0 1.0 0 0<br />
u1t389=9.336<br />
u1t390q=0.3 0.0 1.0 0 0<br />
u1t390=9.36<br />
u1t391q=0.3 0.0 1.0 0 0<br />
u1t391=9.384<br />
u1t392q=0.3 0.0 1.0 0 0<br />
u1t392=9.408<br />
u1t393q=0.3 0.0 1.0 0 0<br />
u1t393=9.432<br />
u1t394q=0.3 0.0 1.0 0 0<br />
u1t394=9.456<br />
u1t395q=0.3 0.0 1.0 0 0<br />
u1t395=9.48<br />
u1t396q=0.3 0.0 1.0 0 0<br />
u1t396=9.504<br />
u1t397q=0.3 0.0 1.0 0 0<br />
u1t397=9.528<br />
u1t398q=0.3 0.0 1.0 0 0<br />
u1t398=9.552<br />
u1t399q=0.3 0.0 1.0 0 0<br />
u1t399=9.576<br />
u1t400q=0.3 0.0 1.0 0 0<br />
u1t400=9.6<br />
u1t401q=0.3 0.0 1.0 0 0<br />
u1t401=9.624<br />
u1t402q=0.3 0.0 1.0 0 0<br />
u1t402=9.648<br />
u1t403q=0.3 0.0 1.0 0 0<br />
u1t403=9.672<br />
u1t404q=0.3 0.0 1.0 0 0<br />
u1t404=9.696<br />
u1t405q=0.3 0.0 1.0 0 0<br />
u1t405=9.72<br />
u1t406q=0.3 0.0 1.0 0 0<br />
u1t406=9.744<br />
u1t407q=0.3 0.0 1.0 0 0<br />
u1t407=9.768<br />
u1t408q=0.3 0.0 1.0 0 0<br />
u1t408=9.792<br />
u1t409q=0.3 0.0 1.0 0 0<br />
u1t409=9.816<br />
u1t410q=0.3 0.0 1.0 0 0<br />
u1t410=9.84<br />
u1t411q=0.3 0.0 1.0 0 0<br />
u1t411=9.864<br />
u1t412q=0.3 0.0 1.0 0 0<br />
u1t412=9.888<br />
u1t413q=0.3 0.0 1.0 0 0<br />
u1t413=9.912<br />
u1t414q=0.3 0.0 1.0 0 0<br />
u1t414=9.936<br />
u1t415q=0.3 0.0 1.0 0 0<br />
u1t415=9.96<br />
u1t416q=0.3 0.0 1.0 0 0<br />
u1t416=9.984<br />
u1t417q=0.3 0.0 1.0 0 0<br />
u1t417=10.008<br />
u1t418q=0.3 0.0 1.0 0 0<br />
u1t418=10.032<br />
u1t419q=0.3 0.0 1.0 0 0<br />
u1t419=10.056<br />
u1t420q=0.3 0.0 1.0 0 0<br />
u1t420=10.08<br />
u1t421q=0.3 0.0 1.0 0 0<br />
u1t421=10.104<br />
u1t422q=0.3 0.0 1.0 0 0<br />
u1t422=10.128<br />
u1t423q=0.3 0.0 1.0 0 0<br />
u1t423=10.152<br />
u1t424q=0.3 0.0 1.0 0 0<br />
u1t424=10.176<br />
u1t425q=0.3 0.0 1.0 0 0<br />
u1t425=10.2<br />
u1t426q=0.3 0.0 1.0 0 0<br />
u1t426=10.224<br />
u1t427q=0.3 0.0 1.0 0 0<br />
u1t427=10.248<br />
u1t428q=0.3 0.0 1.0 0 0<br />
u1t428=10.272<br />
u1t429q=0.3 0.0 1.0 0 0<br />
u1t429=10.296<br />
u1t430q=0.3 0.0 1.0 0 0<br />
u1t430=10.32<br />
u1t431q=0.3 0.0 1.0 0 0<br />
u1t431=10.344<br />
u1t432q=0.3 0.0 1.0 0 0<br />
u1t432=10.368<br />
u1t433q=0.3 0.0 1.0 0 0<br />
u1t433=10.392<br />
u1t434q=0.3 0.0 1.0 0 0<br />
u1t434=10.416<br />
u1t435q=0.3 0.0 1.0 0 0<br />
u1t435=10.44<br />
u1t436q=0.3 0.0 1.0 0 0<br />
u1t436=10.464<br />
u1t437q=0.3 0.0 1.0 0 0<br />
u1t437=10.488<br />
u1t438q=0.3 0.0 1.0 0 0<br />
u1t438=10.512<br />
u1t439q=0.3 0.0 1.0 0 0<br />
u1t439=10.536<br />
u1t440q=0.3 0.0 1.0 0 0<br />
u1t440=10.56<br />
u1t441q=0.3 0.0 1.0 0 0<br />
u1t441=10.584<br />
u1t442q=0.3 0.0 1.0 0 0<br />
u1t442=10.608<br />
u1t443q=0.3 0.0 1.0 0 0<br />
u1t443=10.632<br />
u1t444q=0.3 0.0 1.0 0 0<br />
u1t444=10.656<br />
u1t445q=0.3 0.0 1.0 0 0<br />
u1t445=10.68<br />
u1t446q=0.3 0.0 1.0 0 0<br />
u1t446=10.704<br />
u1t447q=0.3 0.0 1.0 0 0<br />
u1t447=10.728<br />
u1t448q=0.3 0.0 1.0 0 0<br />
u1t448=10.752<br />
u1t449q=0.3 0.0 1.0 0 0<br />
u1t449=10.776<br />
u1t450q=0.3 0.0 1.0 0 0<br />
u1t450=10.8<br />
u1t451q=0.3 0.0 1.0 0 0<br />
u1t451=10.824<br />
u1t452q=0.3 0.0 1.0 0 0<br />
u1t452=10.848<br />
u1t453q=0.3 0.0 1.0 0 0<br />
u1t453=10.872<br />
u1t454q=0.3 0.0 1.0 0 0<br />
u1t454=10.896<br />
u1t455q=0.3 0.0 1.0 0 0<br />
u1t455=10.92<br />
u1t456q=0.3 0.0 1.0 0 0<br />
u1t456=10.944<br />
u1t457q=0.3 0.0 1.0 0 0<br />
u1t457=10.968<br />
u1t458q=0.3 0.0 1.0 0 0<br />
u1t458=10.992<br />
u1t459q=0.3 0.0 1.0 0 0<br />
u1t459=11.016<br />
u1t460q=0.3 0.0 1.0 0 0<br />
u1t460=11.04<br />
u1t461q=0.3 0.0 1.0 0 0<br />
u1t461=11.064<br />
u1t462q=0.3 0.0 1.0 0 0<br />
u1t462=11.088<br />
u1t463q=0.3 0.0 1.0 0 0<br />
u1t463=11.112<br />
u1t464q=0.3 0.0 1.0 0 0<br />
u1t464=11.136<br />
u1t465q=0.3 0.0 1.0 0 0<br />
u1t465=11.16<br />
u1t466q=0.3 0.0 1.0 0 0<br />
u1t466=11.184<br />
u1t467q=0.3 0.0 1.0 0 0<br />
u1t467=11.208<br />
u1t468q=0.3 0.0 1.0 0 0<br />
u1t468=11.232<br />
u1t469q=0.3 0.0 1.0 0 0<br />
u1t469=11.256<br />
u1t470q=0.3 0.0 1.0 0 0<br />
u1t470=11.28<br />
u1t471q=0.3 0.0 1.0 0 0<br />
u1t471=11.304<br />
u1t472q=0.3 0.0 1.0 0 0<br />
u1t472=11.328<br />
u1t473q=0.3 0.0 1.0 0 0<br />
u1t473=11.352<br />
u1t474q=0.3 0.0 1.0 0 0<br />
u1t474=11.376<br />
u1t475q=0.3 0.0 1.0 0 0<br />
u1t475=11.4<br />
u1t476q=0.3 0.0 1.0 0 0<br />
u1t476=11.424<br />
u1t477q=0.3 0.0 1.0 0 0<br />
u1t477=11.448<br />
u1t478q=0.3 0.0 1.0 0 0<br />
u1t478=11.472<br />
u1t479q=0.3 0.0 1.0 0 0<br />
u1t479=11.496<br />
u1t480q=0.3 0.0 1.0 0 0<br />
u1t480=11.52<br />
u1t481q=0.3 0.0 1.0 0 0<br />
u1t481=11.544<br />
u1t482q=0.3 0.0 1.0 0 0<br />
u1t482=11.568<br />
u1t483q=0.3 0.0 1.0 0 0<br />
u1t483=11.592<br />
u1t484q=0.3 0.0 1.0 0 0<br />
u1t484=11.616<br />
u1t485q=0.3 0.0 1.0 0 0<br />
u1t485=11.64<br />
u1t486q=0.3 0.0 1.0 0 0<br />
u1t486=11.664<br />
u1t487q=0.3 0.0 1.0 0 0<br />
u1t487=11.688<br />
u1t488q=0.3 0.0 1.0 0 0<br />
u1t488=11.712<br />
u1t489q=0.3 0.0 1.0 0 0<br />
u1t489=11.736<br />
u1t490q=0.3 0.0 1.0 0 0<br />
u1t490=11.76<br />
u1t491q=0.3 0.0 1.0 0 0<br />
u1t491=11.784<br />
u1t492q=0.3 0.0 1.0 0 0<br />
u1t492=11.808<br />
u1t493q=0.3 0.0 1.0 0 0<br />
u1t493=11.832<br />
u1t494q=0.3 0.0 1.0 0 0<br />
u1t494=11.856<br />
u1t495q=0.3 0.0 1.0 0 0<br />
u1t495=11.88<br />
u1t496q=0.3 0.0 1.0 0 0<br />
u1t496=11.904<br />
u1t497q=0.3 0.0 1.0 0 0<br />
u1t497=11.928<br />
u1t498q=0.3 0.0 1.0 0 0<br />
u1t498=11.952<br />
u1t499q=0.3 0.0 1.0 0 0<br />
u1t499=11.976<br />
u1t500q=0.3 0 0 0 0<br />
u1t500=tend<br />
<br />
[Messungen]<br />
tAnzahl=12<br />
<br />
t1=1<br />
t1Anzahl=2<br />
t1m1=mfcn1 1.0 1e-06 1<br />
t1m2=mfcn2 1.0 1e-06 1<br />
t1minmax=0 1e+10<br />
<br />
t2=2.000<br />
t2Anzahl=2<br />
t2m1=mfcn1 1.0 1e-06 1<br />
t2m2=mfcn2 1.0 1e-06 1<br />
t2minmax=0 1e+10<br />
<br />
t3=3<br />
t3Anzahl=2<br />
t3m1=mfcn1 1.0 1e-06 1<br />
t3m2=mfcn2 1.0 1e-06 1<br />
t3minmax=0 1e+10<br />
<br />
t4=4<br />
t4Anzahl=2<br />
t4m1=mfcn1 1.0 1e-06 1<br />
t4m2=mfcn2 1.0 1e-06 1<br />
t4minmax=0 1e+10<br />
<br />
t5=5<br />
t5Anzahl=2<br />
t5m1=mfcn1 1.0 1e-06 1<br />
t5m2=mfcn2 1.0 1e-06 1<br />
t5minmax=0 1e+10<br />
<br />
t6=6<br />
t6Anzahl=2<br />
t6m1=mfcn1 1.0 1e-06 1<br />
t6m2=mfcn2 1.0 1e-06 1<br />
t6minmax=0 1e+10<br />
<br />
t7=7<br />
t7Anzahl=2<br />
t7m1=mfcn1 1.0 1e-06 1<br />
t7m2=mfcn2 1.0 1e-06 1<br />
t7minmax=0 1e+10<br />
<br />
t8=8<br />
t8Anzahl=2<br />
t8m1=mfcn1 1.0 1e-06 1<br />
t8m2=mfcn2 1.0 1e-06 1<br />
t8minmax=0 1e+10<br />
<br />
t9=9<br />
t9Anzahl=2<br />
t9m1=mfcn1 1.0 1e-06 1<br />
t9m2=mfcn2 1.0 1e-06 1<br />
t9minmax=0 1e+10<br />
<br />
t10=10.000<br />
t10Anzahl=2<br />
t10m1=mfcn1 1.0 1e-06 1<br />
t10m2=mfcn2 1.0 1e-06 1<br />
t10minmax=0 1e+10<br />
<br />
t11=11.000<br />
t11Anzahl=2<br />
t11m1=mfcn1 1.0 1e-06 1<br />
t11m2=mfcn2 1.0 1e-06 1<br />
t11minmax=0 1e+10<br />
<br />
t12=12.000<br />
t12Anzahl=2<br />
t12m1=mfcn1 1.0 1e-06 1<br />
t12m2=mfcn2 1.0 1e-06 1<br />
t12minmax=0 1e+10<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=0<br />
[Messverfahren]<br />
mAnzahl=2<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
m2=mfcn2 1 0 1e+10 0<br />
m2f1=mess4 sigma4 1<br />
mminmaxges=0 8<br />
<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
realworkspace=1700000<br />
integerworkspace=5000<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1342
Lotka Experimental Design (VPLAN)
2016-01-19T16:23:19Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
Algebraic equations:<br />
<br />
<source lang="fortran"><br />
<br />
c Dummyfunction for RHS of algebraic equations<br />
<br />
subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), g(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
iflag=0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function<br />
<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<br />
<br />
<source lang="fortran"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
<br />
<br />
<source lang="vplan"><br />
<br />
; ini-File fuer Experiment<br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=12<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=2<br />
y1=x1 0.5 -1e+10 1e+10<br />
y2=x2 0.7 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=12<br />
t1=1<br />
t2=2<br />
t3=3<br />
t4=4<br />
t5=5<br />
t6=6<br />
t7=7<br />
t8=8<br />
t9=9<br />
t10=10<br />
t11=11<br />
t12=12<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=0<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=myu 0 0.0 1.0<br />
u1tAnzahl=500<br />
u1t0=t0<br />
u1t1q=0.3 0.0 1.0 0 0<br />
u1t1=0.024<br />
u1t2q=0.3 0.0 1.0 0 0<br />
u1t2=0.048<br />
u1t3q=0.3 0.0 1.0 0 0<br />
u1t3=0.072<br />
u1t4q=0.3 0.0 1.0 0 0<br />
u1t4=0.096<br />
u1t5q=0.3 0.0 1.0 0 0<br />
u1t5=0.12<br />
u1t6q=0.3 0.0 1.0 0 0<br />
u1t6=0.144<br />
u1t7q=0.3 0.0 1.0 0 0<br />
u1t7=0.168<br />
u1t8q=0.3 0.0 1.0 0 0<br />
u1t8=0.192<br />
u1t9q=0.3 0.0 1.0 0 0<br />
u1t9=0.216<br />
u1t10q=0.3 0.0 1.0 0 0<br />
u1t10=0.24<br />
u1t11q=0.3 0.0 1.0 0 0<br />
u1t11=0.264<br />
u1t12q=0.3 0.0 1.0 0 0<br />
u1t12=0.288<br />
u1t13q=0.3 0.0 1.0 0 0<br />
u1t13=0.312<br />
u1t14q=0.3 0.0 1.0 0 0<br />
u1t14=0.336<br />
u1t15q=0.3 0.0 1.0 0 0<br />
u1t15=0.36<br />
u1t16q=0.3 0.0 1.0 0 0<br />
u1t16=0.384<br />
u1t17q=0.3 0.0 1.0 0 0<br />
u1t17=0.408<br />
u1t18q=0.3 0.0 1.0 0 0<br />
u1t18=0.432<br />
u1t19q=0.3 0.0 1.0 0 0<br />
u1t19=0.456<br />
u1t20q=0.3 0.0 1.0 0 0<br />
u1t20=0.48<br />
u1t21q=0.3 0.0 1.0 0 0<br />
u1t21=0.504<br />
u1t22q=0.3 0.0 1.0 0 0<br />
u1t22=0.528<br />
u1t23q=0.3 0.0 1.0 0 0<br />
u1t23=0.552<br />
u1t24q=0.3 0.0 1.0 0 0<br />
u1t24=0.576<br />
u1t25q=0.3 0.0 1.0 0 0<br />
u1t25=0.6<br />
u1t26q=0.3 0.0 1.0 0 0<br />
u1t26=0.624<br />
u1t27q=0.3 0.0 1.0 0 0<br />
u1t27=0.648<br />
u1t28q=0.3 0.0 1.0 0 0<br />
u1t28=0.672<br />
u1t29q=0.3 0.0 1.0 0 0<br />
u1t29=0.696<br />
u1t30q=0.3 0.0 1.0 0 0<br />
u1t30=0.72<br />
u1t31q=0.3 0.0 1.0 0 0<br />
u1t31=0.744<br />
u1t32q=0.3 0.0 1.0 0 0<br />
u1t32=0.768<br />
u1t33q=0.3 0.0 1.0 0 0<br />
u1t33=0.792<br />
u1t34q=0.3 0.0 1.0 0 0<br />
u1t34=0.816<br />
u1t35q=0.3 0.0 1.0 0 0<br />
u1t35=0.84<br />
u1t36q=0.3 0.0 1.0 0 0<br />
u1t36=0.864<br />
u1t37q=0.3 0.0 1.0 0 0<br />
u1t37=0.888<br />
u1t38q=0.3 0.0 1.0 0 0<br />
u1t38=0.912<br />
u1t39q=0.3 0.0 1.0 0 0<br />
u1t39=0.936<br />
u1t40q=0.3 0.0 1.0 0 0<br />
u1t40=0.96<br />
u1t41q=0.3 0.0 1.0 0 0<br />
u1t41=0.984<br />
u1t42q=0.3 0.0 1.0 0 0<br />
u1t42=1.008<br />
u1t43q=0.3 0.0 1.0 0 0<br />
u1t43=1.032<br />
u1t44q=0.3 0.0 1.0 0 0<br />
u1t44=1.056<br />
u1t45q=0.3 0.0 1.0 0 0<br />
u1t45=1.08<br />
u1t46q=0.3 0.0 1.0 0 0<br />
u1t46=1.104<br />
u1t47q=0.3 0.0 1.0 0 0<br />
u1t47=1.128<br />
u1t48q=0.3 0.0 1.0 0 0<br />
u1t48=1.152<br />
u1t49q=0.3 0.0 1.0 0 0<br />
u1t49=1.176<br />
u1t50q=0.3 0.0 1.0 0 0<br />
u1t50=1.2<br />
u1t51q=0.3 0.0 1.0 0 0<br />
u1t51=1.224<br />
u1t52q=0.3 0.0 1.0 0 0<br />
u1t52=1.248<br />
u1t53q=0.3 0.0 1.0 0 0<br />
u1t53=1.272<br />
u1t54q=0.3 0.0 1.0 0 0<br />
u1t54=1.296<br />
u1t55q=0.3 0.0 1.0 0 0<br />
u1t55=1.32<br />
u1t56q=0.3 0.0 1.0 0 0<br />
u1t56=1.344<br />
u1t57q=0.3 0.0 1.0 0 0<br />
u1t57=1.368<br />
u1t58q=0.3 0.0 1.0 0 0<br />
u1t58=1.392<br />
u1t59q=0.3 0.0 1.0 0 0<br />
u1t59=1.416<br />
u1t60q=0.3 0.0 1.0 0 0<br />
u1t60=1.44<br />
u1t61q=0.3 0.0 1.0 0 0<br />
u1t61=1.464<br />
u1t62q=0.3 0.0 1.0 0 0<br />
u1t62=1.488<br />
u1t63q=0.3 0.0 1.0 0 0<br />
u1t63=1.512<br />
u1t64q=0.3 0.0 1.0 0 0<br />
u1t64=1.536<br />
u1t65q=0.3 0.0 1.0 0 0<br />
u1t65=1.56<br />
u1t66q=0.3 0.0 1.0 0 0<br />
u1t66=1.584<br />
u1t67q=0.3 0.0 1.0 0 0<br />
u1t67=1.608<br />
u1t68q=0.3 0.0 1.0 0 0<br />
u1t68=1.632<br />
u1t69q=0.3 0.0 1.0 0 0<br />
u1t69=1.656<br />
u1t70q=0.3 0.0 1.0 0 0<br />
u1t70=1.68<br />
u1t71q=0.3 0.0 1.0 0 0<br />
u1t71=1.704<br />
u1t72q=0.3 0.0 1.0 0 0<br />
u1t72=1.728<br />
u1t73q=0.3 0.0 1.0 0 0<br />
u1t73=1.752<br />
u1t74q=0.3 0.0 1.0 0 0<br />
u1t74=1.776<br />
u1t75q=0.3 0.0 1.0 0 0<br />
u1t75=1.8<br />
u1t76q=0.3 0.0 1.0 0 0<br />
u1t76=1.824<br />
u1t77q=0.3 0.0 1.0 0 0<br />
u1t77=1.848<br />
u1t78q=0.3 0.0 1.0 0 0<br />
u1t78=1.872<br />
u1t79q=0.3 0.0 1.0 0 0<br />
u1t79=1.896<br />
u1t80q=0.3 0.0 1.0 0 0<br />
u1t80=1.92<br />
u1t81q=0.3 0.0 1.0 0 0<br />
u1t81=1.944<br />
u1t82q=0.3 0.0 1.0 0 0<br />
u1t82=1.968<br />
u1t83q=0.3 0.0 1.0 0 0<br />
u1t83=1.992<br />
u1t84q=0.3 0.0 1.0 0 0<br />
u1t84=2.016<br />
u1t85q=0.3 0.0 1.0 0 0<br />
u1t85=2.04<br />
u1t86q=0.3 0.0 1.0 0 0<br />
u1t86=2.064<br />
u1t87q=0.3 0.0 1.0 0 0<br />
u1t87=2.088<br />
u1t88q=0.3 0.0 1.0 0 0<br />
u1t88=2.112<br />
u1t89q=0.3 0.0 1.0 0 0<br />
u1t89=2.136<br />
u1t90q=0.3 0.0 1.0 0 0<br />
u1t90=2.16<br />
u1t91q=0.3 0.0 1.0 0 0<br />
u1t91=2.184<br />
u1t92q=0.3 0.0 1.0 0 0<br />
u1t92=2.208<br />
u1t93q=0.3 0.0 1.0 0 0<br />
u1t93=2.232<br />
u1t94q=0.3 0.0 1.0 0 0<br />
u1t94=2.256<br />
u1t95q=0.3 0.0 1.0 0 0<br />
u1t95=2.28<br />
u1t96q=0.3 0.0 1.0 0 0<br />
u1t96=2.304<br />
u1t97q=0.3 0.0 1.0 0 0<br />
u1t97=2.328<br />
u1t98q=0.3 0.0 1.0 0 0<br />
u1t98=2.352<br />
u1t99q=0.3 0.0 1.0 0 0<br />
u1t99=2.376<br />
u1t100q=0.3 0.0 1.0 0 0<br />
u1t100=2.4<br />
u1t101q=0.3 0.0 1.0 0 0<br />
u1t101=2.424<br />
u1t102q=0.3 0.0 1.0 0 0<br />
u1t102=2.448<br />
u1t103q=0.3 0.0 1.0 0 0<br />
u1t103=2.472<br />
u1t104q=0.3 0.0 1.0 0 0<br />
u1t104=2.496<br />
u1t105q=0.3 0.0 1.0 0 0<br />
u1t105=2.52<br />
u1t106q=0.3 0.0 1.0 0 0<br />
u1t106=2.544<br />
u1t107q=0.3 0.0 1.0 0 0<br />
u1t107=2.568<br />
u1t108q=0.3 0.0 1.0 0 0<br />
u1t108=2.592<br />
u1t109q=0.3 0.0 1.0 0 0<br />
u1t109=2.616<br />
u1t110q=0.3 0.0 1.0 0 0<br />
u1t110=2.64<br />
u1t111q=0.3 0.0 1.0 0 0<br />
u1t111=2.664<br />
u1t112q=0.3 0.0 1.0 0 0<br />
u1t112=2.688<br />
u1t113q=0.3 0.0 1.0 0 0<br />
u1t113=2.712<br />
u1t114q=0.3 0.0 1.0 0 0<br />
u1t114=2.736<br />
u1t115q=0.3 0.0 1.0 0 0<br />
u1t115=2.76<br />
u1t116q=0.3 0.0 1.0 0 0<br />
u1t116=2.784<br />
u1t117q=0.3 0.0 1.0 0 0<br />
u1t117=2.808<br />
u1t118q=0.3 0.0 1.0 0 0<br />
u1t118=2.832<br />
u1t119q=0.3 0.0 1.0 0 0<br />
u1t119=2.856<br />
u1t120q=0.3 0.0 1.0 0 0<br />
u1t120=2.88<br />
u1t121q=0.3 0.0 1.0 0 0<br />
u1t121=2.904<br />
u1t122q=0.3 0.0 1.0 0 0<br />
u1t122=2.928<br />
u1t123q=0.3 0.0 1.0 0 0<br />
u1t123=2.952<br />
u1t124q=0.3 0.0 1.0 0 0<br />
u1t124=2.976<br />
u1t125q=0.3 0.0 1.0 0 0<br />
u1t125=3.0<br />
u1t126q=0.3 0.0 1.0 0 0<br />
u1t126=3.024<br />
u1t127q=0.3 0.0 1.0 0 0<br />
u1t127=3.048<br />
u1t128q=0.3 0.0 1.0 0 0<br />
u1t128=3.072<br />
u1t129q=0.3 0.0 1.0 0 0<br />
u1t129=3.096<br />
u1t130q=0.3 0.0 1.0 0 0<br />
u1t130=3.12<br />
u1t131q=0.3 0.0 1.0 0 0<br />
u1t131=3.144<br />
u1t132q=0.3 0.0 1.0 0 0<br />
u1t132=3.168<br />
u1t133q=0.3 0.0 1.0 0 0<br />
u1t133=3.192<br />
u1t134q=0.3 0.0 1.0 0 0<br />
u1t134=3.216<br />
u1t135q=0.3 0.0 1.0 0 0<br />
u1t135=3.24<br />
u1t136q=0.3 0.0 1.0 0 0<br />
u1t136=3.264<br />
u1t137q=0.3 0.0 1.0 0 0<br />
u1t137=3.288<br />
u1t138q=0.3 0.0 1.0 0 0<br />
u1t138=3.312<br />
u1t139q=0.3 0.0 1.0 0 0<br />
u1t139=3.336<br />
u1t140q=0.3 0.0 1.0 0 0<br />
u1t140=3.36<br />
u1t141q=0.3 0.0 1.0 0 0<br />
u1t141=3.384<br />
u1t142q=0.3 0.0 1.0 0 0<br />
u1t142=3.408<br />
u1t143q=0.3 0.0 1.0 0 0<br />
u1t143=3.432<br />
u1t144q=0.3 0.0 1.0 0 0<br />
u1t144=3.456<br />
u1t145q=0.3 0.0 1.0 0 0<br />
u1t145=3.48<br />
u1t146q=0.3 0.0 1.0 0 0<br />
u1t146=3.504<br />
u1t147q=0.3 0.0 1.0 0 0<br />
u1t147=3.528<br />
u1t148q=0.3 0.0 1.0 0 0<br />
u1t148=3.552<br />
u1t149q=0.3 0.0 1.0 0 0<br />
u1t149=3.576<br />
u1t150q=0.3 0.0 1.0 0 0<br />
u1t150=3.6<br />
u1t151q=0.3 0.0 1.0 0 0<br />
u1t151=3.624<br />
u1t152q=0.3 0.0 1.0 0 0<br />
u1t152=3.648<br />
u1t153q=0.3 0.0 1.0 0 0<br />
u1t153=3.672<br />
u1t154q=0.3 0.0 1.0 0 0<br />
u1t154=3.696<br />
u1t155q=0.3 0.0 1.0 0 0<br />
u1t155=3.72<br />
u1t156q=0.3 0.0 1.0 0 0<br />
u1t156=3.744<br />
u1t157q=0.3 0.0 1.0 0 0<br />
u1t157=3.768<br />
u1t158q=0.3 0.0 1.0 0 0<br />
u1t158=3.792<br />
u1t159q=0.3 0.0 1.0 0 0<br />
u1t159=3.816<br />
u1t160q=0.3 0.0 1.0 0 0<br />
u1t160=3.84<br />
u1t161q=0.3 0.0 1.0 0 0<br />
u1t161=3.864<br />
u1t162q=0.3 0.0 1.0 0 0<br />
u1t162=3.888<br />
u1t163q=0.3 0.0 1.0 0 0<br />
u1t163=3.912<br />
u1t164q=0.3 0.0 1.0 0 0<br />
u1t164=3.936<br />
u1t165q=0.3 0.0 1.0 0 0<br />
u1t165=3.96<br />
u1t166q=0.3 0.0 1.0 0 0<br />
u1t166=3.984<br />
u1t167q=0.3 0.0 1.0 0 0<br />
u1t167=4.008<br />
u1t168q=0.3 0.0 1.0 0 0<br />
u1t168=4.032<br />
u1t169q=0.3 0.0 1.0 0 0<br />
u1t169=4.056<br />
u1t170q=0.3 0.0 1.0 0 0<br />
u1t170=4.08<br />
u1t171q=0.3 0.0 1.0 0 0<br />
u1t171=4.104<br />
u1t172q=0.3 0.0 1.0 0 0<br />
u1t172=4.128<br />
u1t173q=0.3 0.0 1.0 0 0<br />
u1t173=4.152<br />
u1t174q=0.3 0.0 1.0 0 0<br />
u1t174=4.176<br />
u1t175q=0.3 0.0 1.0 0 0<br />
u1t175=4.2<br />
u1t176q=0.3 0.0 1.0 0 0<br />
u1t176=4.224<br />
u1t177q=0.3 0.0 1.0 0 0<br />
u1t177=4.248<br />
u1t178q=0.3 0.0 1.0 0 0<br />
u1t178=4.272<br />
u1t179q=0.3 0.0 1.0 0 0<br />
u1t179=4.296<br />
u1t180q=0.3 0.0 1.0 0 0<br />
u1t180=4.32<br />
u1t181q=0.3 0.0 1.0 0 0<br />
u1t181=4.344<br />
u1t182q=0.3 0.0 1.0 0 0<br />
u1t182=4.368<br />
u1t183q=0.3 0.0 1.0 0 0<br />
u1t183=4.392<br />
u1t184q=0.3 0.0 1.0 0 0<br />
u1t184=4.416<br />
u1t185q=0.3 0.0 1.0 0 0<br />
u1t185=4.44<br />
u1t186q=0.3 0.0 1.0 0 0<br />
u1t186=4.464<br />
u1t187q=0.3 0.0 1.0 0 0<br />
u1t187=4.488<br />
u1t188q=0.3 0.0 1.0 0 0<br />
u1t188=4.512<br />
u1t189q=0.3 0.0 1.0 0 0<br />
u1t189=4.536<br />
u1t190q=0.3 0.0 1.0 0 0<br />
u1t190=4.56<br />
u1t191q=0.3 0.0 1.0 0 0<br />
u1t191=4.584<br />
u1t192q=0.3 0.0 1.0 0 0<br />
u1t192=4.608<br />
u1t193q=0.3 0.0 1.0 0 0<br />
u1t193=4.632<br />
u1t194q=0.3 0.0 1.0 0 0<br />
u1t194=4.656<br />
u1t195q=0.3 0.0 1.0 0 0<br />
u1t195=4.68<br />
u1t196q=0.3 0.0 1.0 0 0<br />
u1t196=4.704<br />
u1t197q=0.3 0.0 1.0 0 0<br />
u1t197=4.728<br />
u1t198q=0.3 0.0 1.0 0 0<br />
u1t198=4.752<br />
u1t199q=0.3 0.0 1.0 0 0<br />
u1t199=4.776<br />
u1t200q=0.3 0.0 1.0 0 0<br />
u1t200=4.8<br />
u1t201q=0.3 0.0 1.0 0 0<br />
u1t201=4.824<br />
u1t202q=0.3 0.0 1.0 0 0<br />
u1t202=4.848<br />
u1t203q=0.3 0.0 1.0 0 0<br />
u1t203=4.872<br />
u1t204q=0.3 0.0 1.0 0 0<br />
u1t204=4.896<br />
u1t205q=0.3 0.0 1.0 0 0<br />
u1t205=4.92<br />
u1t206q=0.3 0.0 1.0 0 0<br />
u1t206=4.944<br />
u1t207q=0.3 0.0 1.0 0 0<br />
u1t207=4.968<br />
u1t208q=0.3 0.0 1.0 0 0<br />
u1t208=4.992<br />
u1t209q=0.3 0.0 1.0 0 0<br />
u1t209=5.016<br />
u1t210q=0.3 0.0 1.0 0 0<br />
u1t210=5.04<br />
u1t211q=0.3 0.0 1.0 0 0<br />
u1t211=5.064<br />
u1t212q=0.3 0.0 1.0 0 0<br />
u1t212=5.088<br />
u1t213q=0.3 0.0 1.0 0 0<br />
u1t213=5.112<br />
u1t214q=0.3 0.0 1.0 0 0<br />
u1t214=5.136<br />
u1t215q=0.3 0.0 1.0 0 0<br />
u1t215=5.16<br />
u1t216q=0.3 0.0 1.0 0 0<br />
u1t216=5.184<br />
u1t217q=0.3 0.0 1.0 0 0<br />
u1t217=5.208<br />
u1t218q=0.3 0.0 1.0 0 0<br />
u1t218=5.232<br />
u1t219q=0.3 0.0 1.0 0 0<br />
u1t219=5.256<br />
u1t220q=0.3 0.0 1.0 0 0<br />
u1t220=5.28<br />
u1t221q=0.3 0.0 1.0 0 0<br />
u1t221=5.304<br />
u1t222q=0.3 0.0 1.0 0 0<br />
u1t222=5.328<br />
u1t223q=0.3 0.0 1.0 0 0<br />
u1t223=5.352<br />
u1t224q=0.3 0.0 1.0 0 0<br />
u1t224=5.376<br />
u1t225q=0.3 0.0 1.0 0 0<br />
u1t225=5.4<br />
u1t226q=0.3 0.0 1.0 0 0<br />
u1t226=5.424<br />
u1t227q=0.3 0.0 1.0 0 0<br />
u1t227=5.448<br />
u1t228q=0.3 0.0 1.0 0 0<br />
u1t228=5.472<br />
u1t229q=0.3 0.0 1.0 0 0<br />
u1t229=5.496<br />
u1t230q=0.3 0.0 1.0 0 0<br />
u1t230=5.52<br />
u1t231q=0.3 0.0 1.0 0 0<br />
u1t231=5.544<br />
u1t232q=0.3 0.0 1.0 0 0<br />
u1t232=5.568<br />
u1t233q=0.3 0.0 1.0 0 0<br />
u1t233=5.592<br />
u1t234q=0.3 0.0 1.0 0 0<br />
u1t234=5.616<br />
u1t235q=0.3 0.0 1.0 0 0<br />
u1t235=5.64<br />
u1t236q=0.3 0.0 1.0 0 0<br />
u1t236=5.664<br />
u1t237q=0.3 0.0 1.0 0 0<br />
u1t237=5.688<br />
u1t238q=0.3 0.0 1.0 0 0<br />
u1t238=5.712<br />
u1t239q=0.3 0.0 1.0 0 0<br />
u1t239=5.736<br />
u1t240q=0.3 0.0 1.0 0 0<br />
u1t240=5.76<br />
u1t241q=0.3 0.0 1.0 0 0<br />
u1t241=5.784<br />
u1t242q=0.3 0.0 1.0 0 0<br />
u1t242=5.808<br />
u1t243q=0.3 0.0 1.0 0 0<br />
u1t243=5.832<br />
u1t244q=0.3 0.0 1.0 0 0<br />
u1t244=5.856<br />
u1t245q=0.3 0.0 1.0 0 0<br />
u1t245=5.88<br />
u1t246q=0.3 0.0 1.0 0 0<br />
u1t246=5.904<br />
u1t247q=0.3 0.0 1.0 0 0<br />
u1t247=5.928<br />
u1t248q=0.3 0.0 1.0 0 0<br />
u1t248=5.952<br />
u1t249q=0.3 0.0 1.0 0 0<br />
u1t249=5.976<br />
u1t250q=0.3 0.0 1.0 0 0<br />
u1t250=6.0<br />
u1t251q=0.3 0.0 1.0 0 0<br />
u1t251=6.024<br />
u1t252q=0.3 0.0 1.0 0 0<br />
u1t252=6.048<br />
u1t253q=0.3 0.0 1.0 0 0<br />
u1t253=6.072<br />
u1t254q=0.3 0.0 1.0 0 0<br />
u1t254=6.096<br />
u1t255q=0.3 0.0 1.0 0 0<br />
u1t255=6.12<br />
u1t256q=0.3 0.0 1.0 0 0<br />
u1t256=6.144<br />
u1t257q=0.3 0.0 1.0 0 0<br />
u1t257=6.168<br />
u1t258q=0.3 0.0 1.0 0 0<br />
u1t258=6.192<br />
u1t259q=0.3 0.0 1.0 0 0<br />
u1t259=6.216<br />
u1t260q=0.3 0.0 1.0 0 0<br />
u1t260=6.24<br />
u1t261q=0.3 0.0 1.0 0 0<br />
u1t261=6.264<br />
u1t262q=0.3 0.0 1.0 0 0<br />
u1t262=6.288<br />
u1t263q=0.3 0.0 1.0 0 0<br />
u1t263=6.312<br />
u1t264q=0.3 0.0 1.0 0 0<br />
u1t264=6.336<br />
u1t265q=0.3 0.0 1.0 0 0<br />
u1t265=6.36<br />
u1t266q=0.3 0.0 1.0 0 0<br />
u1t266=6.384<br />
u1t267q=0.3 0.0 1.0 0 0<br />
u1t267=6.408<br />
u1t268q=0.3 0.0 1.0 0 0<br />
u1t268=6.432<br />
u1t269q=0.3 0.0 1.0 0 0<br />
u1t269=6.456<br />
u1t270q=0.3 0.0 1.0 0 0<br />
u1t270=6.48<br />
u1t271q=0.3 0.0 1.0 0 0<br />
u1t271=6.504<br />
u1t272q=0.3 0.0 1.0 0 0<br />
u1t272=6.528<br />
u1t273q=0.3 0.0 1.0 0 0<br />
u1t273=6.552<br />
u1t274q=0.3 0.0 1.0 0 0<br />
u1t274=6.576<br />
u1t275q=0.3 0.0 1.0 0 0<br />
u1t275=6.6<br />
u1t276q=0.3 0.0 1.0 0 0<br />
u1t276=6.624<br />
u1t277q=0.3 0.0 1.0 0 0<br />
u1t277=6.648<br />
u1t278q=0.3 0.0 1.0 0 0<br />
u1t278=6.672<br />
u1t279q=0.3 0.0 1.0 0 0<br />
u1t279=6.696<br />
u1t280q=0.3 0.0 1.0 0 0<br />
u1t280=6.72<br />
u1t281q=0.3 0.0 1.0 0 0<br />
u1t281=6.744<br />
u1t282q=0.3 0.0 1.0 0 0<br />
u1t282=6.768<br />
u1t283q=0.3 0.0 1.0 0 0<br />
u1t283=6.792<br />
u1t284q=0.3 0.0 1.0 0 0<br />
u1t284=6.816<br />
u1t285q=0.3 0.0 1.0 0 0<br />
u1t285=6.84<br />
u1t286q=0.3 0.0 1.0 0 0<br />
u1t286=6.864<br />
u1t287q=0.3 0.0 1.0 0 0<br />
u1t287=6.888<br />
u1t288q=0.3 0.0 1.0 0 0<br />
u1t288=6.912<br />
u1t289q=0.3 0.0 1.0 0 0<br />
u1t289=6.936<br />
u1t290q=0.3 0.0 1.0 0 0<br />
u1t290=6.96<br />
u1t291q=0.3 0.0 1.0 0 0<br />
u1t291=6.984<br />
u1t292q=0.3 0.0 1.0 0 0<br />
u1t292=7.008<br />
u1t293q=0.3 0.0 1.0 0 0<br />
u1t293=7.032<br />
u1t294q=0.3 0.0 1.0 0 0<br />
u1t294=7.056<br />
u1t295q=0.3 0.0 1.0 0 0<br />
u1t295=7.08<br />
u1t296q=0.3 0.0 1.0 0 0<br />
u1t296=7.104<br />
u1t297q=0.3 0.0 1.0 0 0<br />
u1t297=7.128<br />
u1t298q=0.3 0.0 1.0 0 0<br />
u1t298=7.152<br />
u1t299q=0.3 0.0 1.0 0 0<br />
u1t299=7.176<br />
u1t300q=0.3 0.0 1.0 0 0<br />
u1t300=7.2<br />
u1t301q=0.3 0.0 1.0 0 0<br />
u1t301=7.224<br />
u1t302q=0.3 0.0 1.0 0 0<br />
u1t302=7.248<br />
u1t303q=0.3 0.0 1.0 0 0<br />
u1t303=7.272<br />
u1t304q=0.3 0.0 1.0 0 0<br />
u1t304=7.296<br />
u1t305q=0.3 0.0 1.0 0 0<br />
u1t305=7.32<br />
u1t306q=0.3 0.0 1.0 0 0<br />
u1t306=7.344<br />
u1t307q=0.3 0.0 1.0 0 0<br />
u1t307=7.368<br />
u1t308q=0.3 0.0 1.0 0 0<br />
u1t308=7.392<br />
u1t309q=0.3 0.0 1.0 0 0<br />
u1t309=7.416<br />
u1t310q=0.3 0.0 1.0 0 0<br />
u1t310=7.44<br />
u1t311q=0.3 0.0 1.0 0 0<br />
u1t311=7.464<br />
u1t312q=0.3 0.0 1.0 0 0<br />
u1t312=7.488<br />
u1t313q=0.3 0.0 1.0 0 0<br />
u1t313=7.512<br />
u1t314q=0.3 0.0 1.0 0 0<br />
u1t314=7.536<br />
u1t315q=0.3 0.0 1.0 0 0<br />
u1t315=7.56<br />
u1t316q=0.3 0.0 1.0 0 0<br />
u1t316=7.584<br />
u1t317q=0.3 0.0 1.0 0 0<br />
u1t317=7.608<br />
u1t318q=0.3 0.0 1.0 0 0<br />
u1t318=7.632<br />
u1t319q=0.3 0.0 1.0 0 0<br />
u1t319=7.656<br />
u1t320q=0.3 0.0 1.0 0 0<br />
u1t320=7.68<br />
u1t321q=0.3 0.0 1.0 0 0<br />
u1t321=7.704<br />
u1t322q=0.3 0.0 1.0 0 0<br />
u1t322=7.728<br />
u1t323q=0.3 0.0 1.0 0 0<br />
u1t323=7.752<br />
u1t324q=0.3 0.0 1.0 0 0<br />
u1t324=7.776<br />
u1t325q=0.3 0.0 1.0 0 0<br />
u1t325=7.8<br />
u1t326q=0.3 0.0 1.0 0 0<br />
u1t326=7.824<br />
u1t327q=0.3 0.0 1.0 0 0<br />
u1t327=7.848<br />
u1t328q=0.3 0.0 1.0 0 0<br />
u1t328=7.872<br />
u1t329q=0.3 0.0 1.0 0 0<br />
u1t329=7.896<br />
u1t330q=0.3 0.0 1.0 0 0<br />
u1t330=7.92<br />
u1t331q=0.3 0.0 1.0 0 0<br />
u1t331=7.944<br />
u1t332q=0.3 0.0 1.0 0 0<br />
u1t332=7.968<br />
u1t333q=0.3 0.0 1.0 0 0<br />
u1t333=7.992<br />
u1t334q=0.3 0.0 1.0 0 0<br />
u1t334=8.016<br />
u1t335q=0.3 0.0 1.0 0 0<br />
u1t335=8.04<br />
u1t336q=0.3 0.0 1.0 0 0<br />
u1t336=8.064<br />
u1t337q=0.3 0.0 1.0 0 0<br />
u1t337=8.088<br />
u1t338q=0.3 0.0 1.0 0 0<br />
u1t338=8.112<br />
u1t339q=0.3 0.0 1.0 0 0<br />
u1t339=8.136<br />
u1t340q=0.3 0.0 1.0 0 0<br />
u1t340=8.16<br />
u1t341q=0.3 0.0 1.0 0 0<br />
u1t341=8.184<br />
u1t342q=0.3 0.0 1.0 0 0<br />
u1t342=8.208<br />
u1t343q=0.3 0.0 1.0 0 0<br />
u1t343=8.232<br />
u1t344q=0.3 0.0 1.0 0 0<br />
u1t344=8.256<br />
u1t345q=0.3 0.0 1.0 0 0<br />
u1t345=8.28<br />
u1t346q=0.3 0.0 1.0 0 0<br />
u1t346=8.304<br />
u1t347q=0.3 0.0 1.0 0 0<br />
u1t347=8.328<br />
u1t348q=0.3 0.0 1.0 0 0<br />
u1t348=8.352<br />
u1t349q=0.3 0.0 1.0 0 0<br />
u1t349=8.376<br />
u1t350q=0.3 0.0 1.0 0 0<br />
u1t350=8.4<br />
u1t351q=0.3 0.0 1.0 0 0<br />
u1t351=8.424<br />
u1t352q=0.3 0.0 1.0 0 0<br />
u1t352=8.448<br />
u1t353q=0.3 0.0 1.0 0 0<br />
u1t353=8.472<br />
u1t354q=0.3 0.0 1.0 0 0<br />
u1t354=8.496<br />
u1t355q=0.3 0.0 1.0 0 0<br />
u1t355=8.52<br />
u1t356q=0.3 0.0 1.0 0 0<br />
u1t356=8.544<br />
u1t357q=0.3 0.0 1.0 0 0<br />
u1t357=8.568<br />
u1t358q=0.3 0.0 1.0 0 0<br />
u1t358=8.592<br />
u1t359q=0.3 0.0 1.0 0 0<br />
u1t359=8.616<br />
u1t360q=0.3 0.0 1.0 0 0<br />
u1t360=8.64<br />
u1t361q=0.3 0.0 1.0 0 0<br />
u1t361=8.664<br />
u1t362q=0.3 0.0 1.0 0 0<br />
u1t362=8.688<br />
u1t363q=0.3 0.0 1.0 0 0<br />
u1t363=8.712<br />
u1t364q=0.3 0.0 1.0 0 0<br />
u1t364=8.736<br />
u1t365q=0.3 0.0 1.0 0 0<br />
u1t365=8.76<br />
u1t366q=0.3 0.0 1.0 0 0<br />
u1t366=8.784<br />
u1t367q=0.3 0.0 1.0 0 0<br />
u1t367=8.808<br />
u1t368q=0.3 0.0 1.0 0 0<br />
u1t368=8.832<br />
u1t369q=0.3 0.0 1.0 0 0<br />
u1t369=8.856<br />
u1t370q=0.3 0.0 1.0 0 0<br />
u1t370=8.88<br />
u1t371q=0.3 0.0 1.0 0 0<br />
u1t371=8.904<br />
u1t372q=0.3 0.0 1.0 0 0<br />
u1t372=8.928<br />
u1t373q=0.3 0.0 1.0 0 0<br />
u1t373=8.952<br />
u1t374q=0.3 0.0 1.0 0 0<br />
u1t374=8.976<br />
u1t375q=0.3 0.0 1.0 0 0<br />
u1t375=9.0<br />
u1t376q=0.3 0.0 1.0 0 0<br />
u1t376=9.024<br />
u1t377q=0.3 0.0 1.0 0 0<br />
u1t377=9.048<br />
u1t378q=0.3 0.0 1.0 0 0<br />
u1t378=9.072<br />
u1t379q=0.3 0.0 1.0 0 0<br />
u1t379=9.096<br />
u1t380q=0.3 0.0 1.0 0 0<br />
u1t380=9.12<br />
u1t381q=0.3 0.0 1.0 0 0<br />
u1t381=9.144<br />
u1t382q=0.3 0.0 1.0 0 0<br />
u1t382=9.168<br />
u1t383q=0.3 0.0 1.0 0 0<br />
u1t383=9.192<br />
u1t384q=0.3 0.0 1.0 0 0<br />
u1t384=9.216<br />
u1t385q=0.3 0.0 1.0 0 0<br />
u1t385=9.24<br />
u1t386q=0.3 0.0 1.0 0 0<br />
u1t386=9.264<br />
u1t387q=0.3 0.0 1.0 0 0<br />
u1t387=9.288<br />
u1t388q=0.3 0.0 1.0 0 0<br />
u1t388=9.312<br />
u1t389q=0.3 0.0 1.0 0 0<br />
u1t389=9.336<br />
u1t390q=0.3 0.0 1.0 0 0<br />
u1t390=9.36<br />
u1t391q=0.3 0.0 1.0 0 0<br />
u1t391=9.384<br />
u1t392q=0.3 0.0 1.0 0 0<br />
u1t392=9.408<br />
u1t393q=0.3 0.0 1.0 0 0<br />
u1t393=9.432<br />
u1t394q=0.3 0.0 1.0 0 0<br />
u1t394=9.456<br />
u1t395q=0.3 0.0 1.0 0 0<br />
u1t395=9.48<br />
u1t396q=0.3 0.0 1.0 0 0<br />
u1t396=9.504<br />
u1t397q=0.3 0.0 1.0 0 0<br />
u1t397=9.528<br />
u1t398q=0.3 0.0 1.0 0 0<br />
u1t398=9.552<br />
u1t399q=0.3 0.0 1.0 0 0<br />
u1t399=9.576<br />
u1t400q=0.3 0.0 1.0 0 0<br />
u1t400=9.6<br />
u1t401q=0.3 0.0 1.0 0 0<br />
u1t401=9.624<br />
u1t402q=0.3 0.0 1.0 0 0<br />
u1t402=9.648<br />
u1t403q=0.3 0.0 1.0 0 0<br />
u1t403=9.672<br />
u1t404q=0.3 0.0 1.0 0 0<br />
u1t404=9.696<br />
u1t405q=0.3 0.0 1.0 0 0<br />
u1t405=9.72<br />
u1t406q=0.3 0.0 1.0 0 0<br />
u1t406=9.744<br />
u1t407q=0.3 0.0 1.0 0 0<br />
u1t407=9.768<br />
u1t408q=0.3 0.0 1.0 0 0<br />
u1t408=9.792<br />
u1t409q=0.3 0.0 1.0 0 0<br />
u1t409=9.816<br />
u1t410q=0.3 0.0 1.0 0 0<br />
u1t410=9.84<br />
u1t411q=0.3 0.0 1.0 0 0<br />
u1t411=9.864<br />
u1t412q=0.3 0.0 1.0 0 0<br />
u1t412=9.888<br />
u1t413q=0.3 0.0 1.0 0 0<br />
u1t413=9.912<br />
u1t414q=0.3 0.0 1.0 0 0<br />
u1t414=9.936<br />
u1t415q=0.3 0.0 1.0 0 0<br />
u1t415=9.96<br />
u1t416q=0.3 0.0 1.0 0 0<br />
u1t416=9.984<br />
u1t417q=0.3 0.0 1.0 0 0<br />
u1t417=10.008<br />
u1t418q=0.3 0.0 1.0 0 0<br />
u1t418=10.032<br />
u1t419q=0.3 0.0 1.0 0 0<br />
u1t419=10.056<br />
u1t420q=0.3 0.0 1.0 0 0<br />
u1t420=10.08<br />
u1t421q=0.3 0.0 1.0 0 0<br />
u1t421=10.104<br />
u1t422q=0.3 0.0 1.0 0 0<br />
u1t422=10.128<br />
u1t423q=0.3 0.0 1.0 0 0<br />
u1t423=10.152<br />
u1t424q=0.3 0.0 1.0 0 0<br />
u1t424=10.176<br />
u1t425q=0.3 0.0 1.0 0 0<br />
u1t425=10.2<br />
u1t426q=0.3 0.0 1.0 0 0<br />
u1t426=10.224<br />
u1t427q=0.3 0.0 1.0 0 0<br />
u1t427=10.248<br />
u1t428q=0.3 0.0 1.0 0 0<br />
u1t428=10.272<br />
u1t429q=0.3 0.0 1.0 0 0<br />
u1t429=10.296<br />
u1t430q=0.3 0.0 1.0 0 0<br />
u1t430=10.32<br />
u1t431q=0.3 0.0 1.0 0 0<br />
u1t431=10.344<br />
u1t432q=0.3 0.0 1.0 0 0<br />
u1t432=10.368<br />
u1t433q=0.3 0.0 1.0 0 0<br />
u1t433=10.392<br />
u1t434q=0.3 0.0 1.0 0 0<br />
u1t434=10.416<br />
u1t435q=0.3 0.0 1.0 0 0<br />
u1t435=10.44<br />
u1t436q=0.3 0.0 1.0 0 0<br />
u1t436=10.464<br />
u1t437q=0.3 0.0 1.0 0 0<br />
u1t437=10.488<br />
u1t438q=0.3 0.0 1.0 0 0<br />
u1t438=10.512<br />
u1t439q=0.3 0.0 1.0 0 0<br />
u1t439=10.536<br />
u1t440q=0.3 0.0 1.0 0 0<br />
u1t440=10.56<br />
u1t441q=0.3 0.0 1.0 0 0<br />
u1t441=10.584<br />
u1t442q=0.3 0.0 1.0 0 0<br />
u1t442=10.608<br />
u1t443q=0.3 0.0 1.0 0 0<br />
u1t443=10.632<br />
u1t444q=0.3 0.0 1.0 0 0<br />
u1t444=10.656<br />
u1t445q=0.3 0.0 1.0 0 0<br />
u1t445=10.68<br />
u1t446q=0.3 0.0 1.0 0 0<br />
u1t446=10.704<br />
u1t447q=0.3 0.0 1.0 0 0<br />
u1t447=10.728<br />
u1t448q=0.3 0.0 1.0 0 0<br />
u1t448=10.752<br />
u1t449q=0.3 0.0 1.0 0 0<br />
u1t449=10.776<br />
u1t450q=0.3 0.0 1.0 0 0<br />
u1t450=10.8<br />
u1t451q=0.3 0.0 1.0 0 0<br />
u1t451=10.824<br />
u1t452q=0.3 0.0 1.0 0 0<br />
u1t452=10.848<br />
u1t453q=0.3 0.0 1.0 0 0<br />
u1t453=10.872<br />
u1t454q=0.3 0.0 1.0 0 0<br />
u1t454=10.896<br />
u1t455q=0.3 0.0 1.0 0 0<br />
u1t455=10.92<br />
u1t456q=0.3 0.0 1.0 0 0<br />
u1t456=10.944<br />
u1t457q=0.3 0.0 1.0 0 0<br />
u1t457=10.968<br />
u1t458q=0.3 0.0 1.0 0 0<br />
u1t458=10.992<br />
u1t459q=0.3 0.0 1.0 0 0<br />
u1t459=11.016<br />
u1t460q=0.3 0.0 1.0 0 0<br />
u1t460=11.04<br />
u1t461q=0.3 0.0 1.0 0 0<br />
u1t461=11.064<br />
u1t462q=0.3 0.0 1.0 0 0<br />
u1t462=11.088<br />
u1t463q=0.3 0.0 1.0 0 0<br />
u1t463=11.112<br />
u1t464q=0.3 0.0 1.0 0 0<br />
u1t464=11.136<br />
u1t465q=0.3 0.0 1.0 0 0<br />
u1t465=11.16<br />
u1t466q=0.3 0.0 1.0 0 0<br />
u1t466=11.184<br />
u1t467q=0.3 0.0 1.0 0 0<br />
u1t467=11.208<br />
u1t468q=0.3 0.0 1.0 0 0<br />
u1t468=11.232<br />
u1t469q=0.3 0.0 1.0 0 0<br />
u1t469=11.256<br />
u1t470q=0.3 0.0 1.0 0 0<br />
u1t470=11.28<br />
u1t471q=0.3 0.0 1.0 0 0<br />
u1t471=11.304<br />
u1t472q=0.3 0.0 1.0 0 0<br />
u1t472=11.328<br />
u1t473q=0.3 0.0 1.0 0 0<br />
u1t473=11.352<br />
u1t474q=0.3 0.0 1.0 0 0<br />
u1t474=11.376<br />
u1t475q=0.3 0.0 1.0 0 0<br />
u1t475=11.4<br />
u1t476q=0.3 0.0 1.0 0 0<br />
u1t476=11.424<br />
u1t477q=0.3 0.0 1.0 0 0<br />
u1t477=11.448<br />
u1t478q=0.3 0.0 1.0 0 0<br />
u1t478=11.472<br />
u1t479q=0.3 0.0 1.0 0 0<br />
u1t479=11.496<br />
u1t480q=0.3 0.0 1.0 0 0<br />
u1t480=11.52<br />
u1t481q=0.3 0.0 1.0 0 0<br />
u1t481=11.544<br />
u1t482q=0.3 0.0 1.0 0 0<br />
u1t482=11.568<br />
u1t483q=0.3 0.0 1.0 0 0<br />
u1t483=11.592<br />
u1t484q=0.3 0.0 1.0 0 0<br />
u1t484=11.616<br />
u1t485q=0.3 0.0 1.0 0 0<br />
u1t485=11.64<br />
u1t486q=0.3 0.0 1.0 0 0<br />
u1t486=11.664<br />
u1t487q=0.3 0.0 1.0 0 0<br />
u1t487=11.688<br />
u1t488q=0.3 0.0 1.0 0 0<br />
u1t488=11.712<br />
u1t489q=0.3 0.0 1.0 0 0<br />
u1t489=11.736<br />
u1t490q=0.3 0.0 1.0 0 0<br />
u1t490=11.76<br />
u1t491q=0.3 0.0 1.0 0 0<br />
u1t491=11.784<br />
u1t492q=0.3 0.0 1.0 0 0<br />
u1t492=11.808<br />
u1t493q=0.3 0.0 1.0 0 0<br />
u1t493=11.832<br />
u1t494q=0.3 0.0 1.0 0 0<br />
u1t494=11.856<br />
u1t495q=0.3 0.0 1.0 0 0<br />
u1t495=11.88<br />
u1t496q=0.3 0.0 1.0 0 0<br />
u1t496=11.904<br />
u1t497q=0.3 0.0 1.0 0 0<br />
u1t497=11.928<br />
u1t498q=0.3 0.0 1.0 0 0<br />
u1t498=11.952<br />
u1t499q=0.3 0.0 1.0 0 0<br />
u1t499=11.976<br />
u1t500q=0.3 0 0 0 0<br />
u1t500=tend<br />
<br />
[Messungen]<br />
tAnzahl=12<br />
<br />
t1=1<br />
t1Anzahl=2<br />
t1m1=mfcn1 1.0 1e-06 1<br />
t1m2=mfcn2 1.0 1e-06 1<br />
t1minmax=0 1e+10<br />
<br />
t2=2.000<br />
t2Anzahl=2<br />
t2m1=mfcn1 1.0 1e-06 1<br />
t2m2=mfcn2 1.0 1e-06 1<br />
t2minmax=0 1e+10<br />
<br />
t3=3<br />
t3Anzahl=2<br />
t3m1=mfcn1 1.0 1e-06 1<br />
t3m2=mfcn2 1.0 1e-06 1<br />
t3minmax=0 1e+10<br />
<br />
t4=4<br />
t4Anzahl=2<br />
t4m1=mfcn1 1.0 1e-06 1<br />
t4m2=mfcn2 1.0 1e-06 1<br />
t4minmax=0 1e+10<br />
<br />
t5=5<br />
t5Anzahl=2<br />
t5m1=mfcn1 1.0 1e-06 1<br />
t5m2=mfcn2 1.0 1e-06 1<br />
t5minmax=0 1e+10<br />
<br />
t6=6<br />
t6Anzahl=2<br />
t6m1=mfcn1 1.0 1e-06 1<br />
t6m2=mfcn2 1.0 1e-06 1<br />
t6minmax=0 1e+10<br />
<br />
t7=7<br />
t7Anzahl=2<br />
t7m1=mfcn1 1.0 1e-06 1<br />
t7m2=mfcn2 1.0 1e-06 1<br />
t7minmax=0 1e+10<br />
<br />
t8=8<br />
t8Anzahl=2<br />
t8m1=mfcn1 1.0 1e-06 1<br />
t8m2=mfcn2 1.0 1e-06 1<br />
t8minmax=0 1e+10<br />
<br />
t9=9<br />
t9Anzahl=2<br />
t9m1=mfcn1 1.0 1e-06 1<br />
t9m2=mfcn2 1.0 1e-06 1<br />
t9minmax=0 1e+10<br />
<br />
t10=10.000<br />
t10Anzahl=2<br />
t10m1=mfcn1 1.0 1e-06 1<br />
t10m2=mfcn2 1.0 1e-06 1<br />
t10minmax=0 1e+10<br />
<br />
t11=11.000<br />
t11Anzahl=2<br />
t11m1=mfcn1 1.0 1e-06 1<br />
t11m2=mfcn2 1.0 1e-06 1<br />
t11minmax=0 1e+10<br />
<br />
t12=12.000<br />
t12Anzahl=2<br />
t12m1=mfcn1 1.0 1e-06 1<br />
t12m2=mfcn2 1.0 1e-06 1<br />
t12minmax=0 1e+10<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=0<br />
[Messverfahren]<br />
mAnzahl=2<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
m2=mfcn2 1 0 1e+10 0<br />
m2f1=mess4 sigma4 1<br />
mminmaxges=0 8<br />
<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
realworkspace=1700000<br />
integerworkspace=5000<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1341
Lotka Experimental Design (VPLAN)
2016-01-19T16:20:46Z
<p>FelixJost: /* VPLAN */</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function<br />
<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<br />
<br />
<source lang="fortran"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source><br />
<br />
<br />
VPLAN specific experimental setup:<br />
<br />
<br />
<source lang="vplan"><br />
<br />
; ini-File fuer Experiment<br />
<br />
[Flags]<br />
switch=1<br />
<br />
[Kosten]<br />
costs=0 -1e+10 1e+10<br />
<br />
[Integrationsintervall]<br />
t0=0<br />
tend=12<br />
<br />
[Modellfunktionen]<br />
ffcn=ffcn<br />
gfcn=gfcn<br />
<br />
[Zustandsvariablen]<br />
yAnzahl=2<br />
y1=x1 0.5 -1e+10 1e+10<br />
y2=x2 0.7 -1e+10 1e+10<br />
<br />
zAnzahl=0<br />
<br />
[Mehrzielknoten]<br />
tAnzahl=12<br />
t1=1<br />
t2=2<br />
t3=3<br />
t4=4<br />
t5=5<br />
t6=6<br />
t7=7<br />
t8=8<br />
t9=9<br />
t10=10<br />
t11=11<br />
t12=12<br />
<br />
[DynamischeNebenbedingungen]<br />
bAnzahl=0<br />
<br />
[GitterUeberpruefungNebenbedingungen]<br />
tAnzahl=0<br />
<br />
[Steuergroessen]<br />
qAnzahl=0<br />
<br />
[Steuerfunktionen]<br />
uAnzahl=1<br />
u1=myu 0 0.0 1.0<br />
u1tAnzahl=500<br />
u1t0=t0<br />
u1t1q=0.3 0.0 1.0 0 0<br />
u1t1=0.024<br />
u1t2q=0.3 0.0 1.0 0 0<br />
u1t2=0.048<br />
u1t3q=0.3 0.0 1.0 0 0<br />
u1t3=0.072<br />
u1t4q=0.3 0.0 1.0 0 0<br />
u1t4=0.096<br />
u1t5q=0.3 0.0 1.0 0 0<br />
u1t5=0.12<br />
u1t6q=0.3 0.0 1.0 0 0<br />
u1t6=0.144<br />
u1t7q=0.3 0.0 1.0 0 0<br />
u1t7=0.168<br />
u1t8q=0.3 0.0 1.0 0 0<br />
u1t8=0.192<br />
u1t9q=0.3 0.0 1.0 0 0<br />
u1t9=0.216<br />
u1t10q=0.3 0.0 1.0 0 0<br />
u1t10=0.24<br />
u1t11q=0.3 0.0 1.0 0 0<br />
u1t11=0.264<br />
u1t12q=0.3 0.0 1.0 0 0<br />
u1t12=0.288<br />
u1t13q=0.3 0.0 1.0 0 0<br />
u1t13=0.312<br />
u1t14q=0.3 0.0 1.0 0 0<br />
u1t14=0.336<br />
u1t15q=0.3 0.0 1.0 0 0<br />
u1t15=0.36<br />
u1t16q=0.3 0.0 1.0 0 0<br />
u1t16=0.384<br />
u1t17q=0.3 0.0 1.0 0 0<br />
u1t17=0.408<br />
u1t18q=0.3 0.0 1.0 0 0<br />
u1t18=0.432<br />
u1t19q=0.3 0.0 1.0 0 0<br />
u1t19=0.456<br />
u1t20q=0.3 0.0 1.0 0 0<br />
u1t20=0.48<br />
u1t21q=0.3 0.0 1.0 0 0<br />
u1t21=0.504<br />
u1t22q=0.3 0.0 1.0 0 0<br />
u1t22=0.528<br />
u1t23q=0.3 0.0 1.0 0 0<br />
u1t23=0.552<br />
u1t24q=0.3 0.0 1.0 0 0<br />
u1t24=0.576<br />
u1t25q=0.3 0.0 1.0 0 0<br />
u1t25=0.6<br />
u1t26q=0.3 0.0 1.0 0 0<br />
u1t26=0.624<br />
u1t27q=0.3 0.0 1.0 0 0<br />
u1t27=0.648<br />
u1t28q=0.3 0.0 1.0 0 0<br />
u1t28=0.672<br />
u1t29q=0.3 0.0 1.0 0 0<br />
u1t29=0.696<br />
u1t30q=0.3 0.0 1.0 0 0<br />
u1t30=0.72<br />
u1t31q=0.3 0.0 1.0 0 0<br />
u1t31=0.744<br />
u1t32q=0.3 0.0 1.0 0 0<br />
u1t32=0.768<br />
u1t33q=0.3 0.0 1.0 0 0<br />
u1t33=0.792<br />
u1t34q=0.3 0.0 1.0 0 0<br />
u1t34=0.816<br />
u1t35q=0.3 0.0 1.0 0 0<br />
u1t35=0.84<br />
u1t36q=0.3 0.0 1.0 0 0<br />
u1t36=0.864<br />
u1t37q=0.3 0.0 1.0 0 0<br />
u1t37=0.888<br />
u1t38q=0.3 0.0 1.0 0 0<br />
u1t38=0.912<br />
u1t39q=0.3 0.0 1.0 0 0<br />
u1t39=0.936<br />
u1t40q=0.3 0.0 1.0 0 0<br />
u1t40=0.96<br />
u1t41q=0.3 0.0 1.0 0 0<br />
u1t41=0.984<br />
u1t42q=0.3 0.0 1.0 0 0<br />
u1t42=1.008<br />
u1t43q=0.3 0.0 1.0 0 0<br />
u1t43=1.032<br />
u1t44q=0.3 0.0 1.0 0 0<br />
u1t44=1.056<br />
u1t45q=0.3 0.0 1.0 0 0<br />
u1t45=1.08<br />
u1t46q=0.3 0.0 1.0 0 0<br />
u1t46=1.104<br />
u1t47q=0.3 0.0 1.0 0 0<br />
u1t47=1.128<br />
u1t48q=0.3 0.0 1.0 0 0<br />
u1t48=1.152<br />
u1t49q=0.3 0.0 1.0 0 0<br />
u1t49=1.176<br />
u1t50q=0.3 0.0 1.0 0 0<br />
u1t50=1.2<br />
u1t51q=0.3 0.0 1.0 0 0<br />
u1t51=1.224<br />
u1t52q=0.3 0.0 1.0 0 0<br />
u1t52=1.248<br />
u1t53q=0.3 0.0 1.0 0 0<br />
u1t53=1.272<br />
u1t54q=0.3 0.0 1.0 0 0<br />
u1t54=1.296<br />
u1t55q=0.3 0.0 1.0 0 0<br />
u1t55=1.32<br />
u1t56q=0.3 0.0 1.0 0 0<br />
u1t56=1.344<br />
u1t57q=0.3 0.0 1.0 0 0<br />
u1t57=1.368<br />
u1t58q=0.3 0.0 1.0 0 0<br />
u1t58=1.392<br />
u1t59q=0.3 0.0 1.0 0 0<br />
u1t59=1.416<br />
u1t60q=0.3 0.0 1.0 0 0<br />
u1t60=1.44<br />
u1t61q=0.3 0.0 1.0 0 0<br />
u1t61=1.464<br />
u1t62q=0.3 0.0 1.0 0 0<br />
u1t62=1.488<br />
u1t63q=0.3 0.0 1.0 0 0<br />
u1t63=1.512<br />
u1t64q=0.3 0.0 1.0 0 0<br />
u1t64=1.536<br />
u1t65q=0.3 0.0 1.0 0 0<br />
u1t65=1.56<br />
u1t66q=0.3 0.0 1.0 0 0<br />
u1t66=1.584<br />
u1t67q=0.3 0.0 1.0 0 0<br />
u1t67=1.608<br />
u1t68q=0.3 0.0 1.0 0 0<br />
u1t68=1.632<br />
u1t69q=0.3 0.0 1.0 0 0<br />
u1t69=1.656<br />
u1t70q=0.3 0.0 1.0 0 0<br />
u1t70=1.68<br />
u1t71q=0.3 0.0 1.0 0 0<br />
u1t71=1.704<br />
u1t72q=0.3 0.0 1.0 0 0<br />
u1t72=1.728<br />
u1t73q=0.3 0.0 1.0 0 0<br />
u1t73=1.752<br />
u1t74q=0.3 0.0 1.0 0 0<br />
u1t74=1.776<br />
u1t75q=0.3 0.0 1.0 0 0<br />
u1t75=1.8<br />
u1t76q=0.3 0.0 1.0 0 0<br />
u1t76=1.824<br />
u1t77q=0.3 0.0 1.0 0 0<br />
u1t77=1.848<br />
u1t78q=0.3 0.0 1.0 0 0<br />
u1t78=1.872<br />
u1t79q=0.3 0.0 1.0 0 0<br />
u1t79=1.896<br />
u1t80q=0.3 0.0 1.0 0 0<br />
u1t80=1.92<br />
u1t81q=0.3 0.0 1.0 0 0<br />
u1t81=1.944<br />
u1t82q=0.3 0.0 1.0 0 0<br />
u1t82=1.968<br />
u1t83q=0.3 0.0 1.0 0 0<br />
u1t83=1.992<br />
u1t84q=0.3 0.0 1.0 0 0<br />
u1t84=2.016<br />
u1t85q=0.3 0.0 1.0 0 0<br />
u1t85=2.04<br />
u1t86q=0.3 0.0 1.0 0 0<br />
u1t86=2.064<br />
u1t87q=0.3 0.0 1.0 0 0<br />
u1t87=2.088<br />
u1t88q=0.3 0.0 1.0 0 0<br />
u1t88=2.112<br />
u1t89q=0.3 0.0 1.0 0 0<br />
u1t89=2.136<br />
u1t90q=0.3 0.0 1.0 0 0<br />
u1t90=2.16<br />
u1t91q=0.3 0.0 1.0 0 0<br />
u1t91=2.184<br />
u1t92q=0.3 0.0 1.0 0 0<br />
u1t92=2.208<br />
u1t93q=0.3 0.0 1.0 0 0<br />
u1t93=2.232<br />
u1t94q=0.3 0.0 1.0 0 0<br />
u1t94=2.256<br />
u1t95q=0.3 0.0 1.0 0 0<br />
u1t95=2.28<br />
u1t96q=0.3 0.0 1.0 0 0<br />
u1t96=2.304<br />
u1t97q=0.3 0.0 1.0 0 0<br />
u1t97=2.328<br />
u1t98q=0.3 0.0 1.0 0 0<br />
u1t98=2.352<br />
u1t99q=0.3 0.0 1.0 0 0<br />
u1t99=2.376<br />
u1t100q=0.3 0.0 1.0 0 0<br />
u1t100=2.4<br />
u1t101q=0.3 0.0 1.0 0 0<br />
u1t101=2.424<br />
u1t102q=0.3 0.0 1.0 0 0<br />
u1t102=2.448<br />
u1t103q=0.3 0.0 1.0 0 0<br />
u1t103=2.472<br />
u1t104q=0.3 0.0 1.0 0 0<br />
u1t104=2.496<br />
u1t105q=0.3 0.0 1.0 0 0<br />
u1t105=2.52<br />
u1t106q=0.3 0.0 1.0 0 0<br />
u1t106=2.544<br />
u1t107q=0.3 0.0 1.0 0 0<br />
u1t107=2.568<br />
u1t108q=0.3 0.0 1.0 0 0<br />
u1t108=2.592<br />
u1t109q=0.3 0.0 1.0 0 0<br />
u1t109=2.616<br />
u1t110q=0.3 0.0 1.0 0 0<br />
u1t110=2.64<br />
u1t111q=0.3 0.0 1.0 0 0<br />
u1t111=2.664<br />
u1t112q=0.3 0.0 1.0 0 0<br />
u1t112=2.688<br />
u1t113q=0.3 0.0 1.0 0 0<br />
u1t113=2.712<br />
u1t114q=0.3 0.0 1.0 0 0<br />
u1t114=2.736<br />
u1t115q=0.3 0.0 1.0 0 0<br />
u1t115=2.76<br />
u1t116q=0.3 0.0 1.0 0 0<br />
u1t116=2.784<br />
u1t117q=0.3 0.0 1.0 0 0<br />
u1t117=2.808<br />
u1t118q=0.3 0.0 1.0 0 0<br />
u1t118=2.832<br />
u1t119q=0.3 0.0 1.0 0 0<br />
u1t119=2.856<br />
u1t120q=0.3 0.0 1.0 0 0<br />
u1t120=2.88<br />
u1t121q=0.3 0.0 1.0 0 0<br />
u1t121=2.904<br />
u1t122q=0.3 0.0 1.0 0 0<br />
u1t122=2.928<br />
u1t123q=0.3 0.0 1.0 0 0<br />
u1t123=2.952<br />
u1t124q=0.3 0.0 1.0 0 0<br />
u1t124=2.976<br />
u1t125q=0.3 0.0 1.0 0 0<br />
u1t125=3.0<br />
u1t126q=0.3 0.0 1.0 0 0<br />
u1t126=3.024<br />
u1t127q=0.3 0.0 1.0 0 0<br />
u1t127=3.048<br />
u1t128q=0.3 0.0 1.0 0 0<br />
u1t128=3.072<br />
u1t129q=0.3 0.0 1.0 0 0<br />
u1t129=3.096<br />
u1t130q=0.3 0.0 1.0 0 0<br />
u1t130=3.12<br />
u1t131q=0.3 0.0 1.0 0 0<br />
u1t131=3.144<br />
u1t132q=0.3 0.0 1.0 0 0<br />
u1t132=3.168<br />
u1t133q=0.3 0.0 1.0 0 0<br />
u1t133=3.192<br />
u1t134q=0.3 0.0 1.0 0 0<br />
u1t134=3.216<br />
u1t135q=0.3 0.0 1.0 0 0<br />
u1t135=3.24<br />
u1t136q=0.3 0.0 1.0 0 0<br />
u1t136=3.264<br />
u1t137q=0.3 0.0 1.0 0 0<br />
u1t137=3.288<br />
u1t138q=0.3 0.0 1.0 0 0<br />
u1t138=3.312<br />
u1t139q=0.3 0.0 1.0 0 0<br />
u1t139=3.336<br />
u1t140q=0.3 0.0 1.0 0 0<br />
u1t140=3.36<br />
u1t141q=0.3 0.0 1.0 0 0<br />
u1t141=3.384<br />
u1t142q=0.3 0.0 1.0 0 0<br />
u1t142=3.408<br />
u1t143q=0.3 0.0 1.0 0 0<br />
u1t143=3.432<br />
u1t144q=0.3 0.0 1.0 0 0<br />
u1t144=3.456<br />
u1t145q=0.3 0.0 1.0 0 0<br />
u1t145=3.48<br />
u1t146q=0.3 0.0 1.0 0 0<br />
u1t146=3.504<br />
u1t147q=0.3 0.0 1.0 0 0<br />
u1t147=3.528<br />
u1t148q=0.3 0.0 1.0 0 0<br />
u1t148=3.552<br />
u1t149q=0.3 0.0 1.0 0 0<br />
u1t149=3.576<br />
u1t150q=0.3 0.0 1.0 0 0<br />
u1t150=3.6<br />
u1t151q=0.3 0.0 1.0 0 0<br />
u1t151=3.624<br />
u1t152q=0.3 0.0 1.0 0 0<br />
u1t152=3.648<br />
u1t153q=0.3 0.0 1.0 0 0<br />
u1t153=3.672<br />
u1t154q=0.3 0.0 1.0 0 0<br />
u1t154=3.696<br />
u1t155q=0.3 0.0 1.0 0 0<br />
u1t155=3.72<br />
u1t156q=0.3 0.0 1.0 0 0<br />
u1t156=3.744<br />
u1t157q=0.3 0.0 1.0 0 0<br />
u1t157=3.768<br />
u1t158q=0.3 0.0 1.0 0 0<br />
u1t158=3.792<br />
u1t159q=0.3 0.0 1.0 0 0<br />
u1t159=3.816<br />
u1t160q=0.3 0.0 1.0 0 0<br />
u1t160=3.84<br />
u1t161q=0.3 0.0 1.0 0 0<br />
u1t161=3.864<br />
u1t162q=0.3 0.0 1.0 0 0<br />
u1t162=3.888<br />
u1t163q=0.3 0.0 1.0 0 0<br />
u1t163=3.912<br />
u1t164q=0.3 0.0 1.0 0 0<br />
u1t164=3.936<br />
u1t165q=0.3 0.0 1.0 0 0<br />
u1t165=3.96<br />
u1t166q=0.3 0.0 1.0 0 0<br />
u1t166=3.984<br />
u1t167q=0.3 0.0 1.0 0 0<br />
u1t167=4.008<br />
u1t168q=0.3 0.0 1.0 0 0<br />
u1t168=4.032<br />
u1t169q=0.3 0.0 1.0 0 0<br />
u1t169=4.056<br />
u1t170q=0.3 0.0 1.0 0 0<br />
u1t170=4.08<br />
u1t171q=0.3 0.0 1.0 0 0<br />
u1t171=4.104<br />
u1t172q=0.3 0.0 1.0 0 0<br />
u1t172=4.128<br />
u1t173q=0.3 0.0 1.0 0 0<br />
u1t173=4.152<br />
u1t174q=0.3 0.0 1.0 0 0<br />
u1t174=4.176<br />
u1t175q=0.3 0.0 1.0 0 0<br />
u1t175=4.2<br />
u1t176q=0.3 0.0 1.0 0 0<br />
u1t176=4.224<br />
u1t177q=0.3 0.0 1.0 0 0<br />
u1t177=4.248<br />
u1t178q=0.3 0.0 1.0 0 0<br />
u1t178=4.272<br />
u1t179q=0.3 0.0 1.0 0 0<br />
u1t179=4.296<br />
u1t180q=0.3 0.0 1.0 0 0<br />
u1t180=4.32<br />
u1t181q=0.3 0.0 1.0 0 0<br />
u1t181=4.344<br />
u1t182q=0.3 0.0 1.0 0 0<br />
u1t182=4.368<br />
u1t183q=0.3 0.0 1.0 0 0<br />
u1t183=4.392<br />
u1t184q=0.3 0.0 1.0 0 0<br />
u1t184=4.416<br />
u1t185q=0.3 0.0 1.0 0 0<br />
u1t185=4.44<br />
u1t186q=0.3 0.0 1.0 0 0<br />
u1t186=4.464<br />
u1t187q=0.3 0.0 1.0 0 0<br />
u1t187=4.488<br />
u1t188q=0.3 0.0 1.0 0 0<br />
u1t188=4.512<br />
u1t189q=0.3 0.0 1.0 0 0<br />
u1t189=4.536<br />
u1t190q=0.3 0.0 1.0 0 0<br />
u1t190=4.56<br />
u1t191q=0.3 0.0 1.0 0 0<br />
u1t191=4.584<br />
u1t192q=0.3 0.0 1.0 0 0<br />
u1t192=4.608<br />
u1t193q=0.3 0.0 1.0 0 0<br />
u1t193=4.632<br />
u1t194q=0.3 0.0 1.0 0 0<br />
u1t194=4.656<br />
u1t195q=0.3 0.0 1.0 0 0<br />
u1t195=4.68<br />
u1t196q=0.3 0.0 1.0 0 0<br />
u1t196=4.704<br />
u1t197q=0.3 0.0 1.0 0 0<br />
u1t197=4.728<br />
u1t198q=0.3 0.0 1.0 0 0<br />
u1t198=4.752<br />
u1t199q=0.3 0.0 1.0 0 0<br />
u1t199=4.776<br />
u1t200q=0.3 0.0 1.0 0 0<br />
u1t200=4.8<br />
u1t201q=0.3 0.0 1.0 0 0<br />
u1t201=4.824<br />
u1t202q=0.3 0.0 1.0 0 0<br />
u1t202=4.848<br />
u1t203q=0.3 0.0 1.0 0 0<br />
u1t203=4.872<br />
u1t204q=0.3 0.0 1.0 0 0<br />
u1t204=4.896<br />
u1t205q=0.3 0.0 1.0 0 0<br />
u1t205=4.92<br />
u1t206q=0.3 0.0 1.0 0 0<br />
u1t206=4.944<br />
u1t207q=0.3 0.0 1.0 0 0<br />
u1t207=4.968<br />
u1t208q=0.3 0.0 1.0 0 0<br />
u1t208=4.992<br />
u1t209q=0.3 0.0 1.0 0 0<br />
u1t209=5.016<br />
u1t210q=0.3 0.0 1.0 0 0<br />
u1t210=5.04<br />
u1t211q=0.3 0.0 1.0 0 0<br />
u1t211=5.064<br />
u1t212q=0.3 0.0 1.0 0 0<br />
u1t212=5.088<br />
u1t213q=0.3 0.0 1.0 0 0<br />
u1t213=5.112<br />
u1t214q=0.3 0.0 1.0 0 0<br />
u1t214=5.136<br />
u1t215q=0.3 0.0 1.0 0 0<br />
u1t215=5.16<br />
u1t216q=0.3 0.0 1.0 0 0<br />
u1t216=5.184<br />
u1t217q=0.3 0.0 1.0 0 0<br />
u1t217=5.208<br />
u1t218q=0.3 0.0 1.0 0 0<br />
u1t218=5.232<br />
u1t219q=0.3 0.0 1.0 0 0<br />
u1t219=5.256<br />
u1t220q=0.3 0.0 1.0 0 0<br />
u1t220=5.28<br />
u1t221q=0.3 0.0 1.0 0 0<br />
u1t221=5.304<br />
u1t222q=0.3 0.0 1.0 0 0<br />
u1t222=5.328<br />
u1t223q=0.3 0.0 1.0 0 0<br />
u1t223=5.352<br />
u1t224q=0.3 0.0 1.0 0 0<br />
u1t224=5.376<br />
u1t225q=0.3 0.0 1.0 0 0<br />
u1t225=5.4<br />
u1t226q=0.3 0.0 1.0 0 0<br />
u1t226=5.424<br />
u1t227q=0.3 0.0 1.0 0 0<br />
u1t227=5.448<br />
u1t228q=0.3 0.0 1.0 0 0<br />
u1t228=5.472<br />
u1t229q=0.3 0.0 1.0 0 0<br />
u1t229=5.496<br />
u1t230q=0.3 0.0 1.0 0 0<br />
u1t230=5.52<br />
u1t231q=0.3 0.0 1.0 0 0<br />
u1t231=5.544<br />
u1t232q=0.3 0.0 1.0 0 0<br />
u1t232=5.568<br />
u1t233q=0.3 0.0 1.0 0 0<br />
u1t233=5.592<br />
u1t234q=0.3 0.0 1.0 0 0<br />
u1t234=5.616<br />
u1t235q=0.3 0.0 1.0 0 0<br />
u1t235=5.64<br />
u1t236q=0.3 0.0 1.0 0 0<br />
u1t236=5.664<br />
u1t237q=0.3 0.0 1.0 0 0<br />
u1t237=5.688<br />
u1t238q=0.3 0.0 1.0 0 0<br />
u1t238=5.712<br />
u1t239q=0.3 0.0 1.0 0 0<br />
u1t239=5.736<br />
u1t240q=0.3 0.0 1.0 0 0<br />
u1t240=5.76<br />
u1t241q=0.3 0.0 1.0 0 0<br />
u1t241=5.784<br />
u1t242q=0.3 0.0 1.0 0 0<br />
u1t242=5.808<br />
u1t243q=0.3 0.0 1.0 0 0<br />
u1t243=5.832<br />
u1t244q=0.3 0.0 1.0 0 0<br />
u1t244=5.856<br />
u1t245q=0.3 0.0 1.0 0 0<br />
u1t245=5.88<br />
u1t246q=0.3 0.0 1.0 0 0<br />
u1t246=5.904<br />
u1t247q=0.3 0.0 1.0 0 0<br />
u1t247=5.928<br />
u1t248q=0.3 0.0 1.0 0 0<br />
u1t248=5.952<br />
u1t249q=0.3 0.0 1.0 0 0<br />
u1t249=5.976<br />
u1t250q=0.3 0.0 1.0 0 0<br />
u1t250=6.0<br />
u1t251q=0.3 0.0 1.0 0 0<br />
u1t251=6.024<br />
u1t252q=0.3 0.0 1.0 0 0<br />
u1t252=6.048<br />
u1t253q=0.3 0.0 1.0 0 0<br />
u1t253=6.072<br />
u1t254q=0.3 0.0 1.0 0 0<br />
u1t254=6.096<br />
u1t255q=0.3 0.0 1.0 0 0<br />
u1t255=6.12<br />
u1t256q=0.3 0.0 1.0 0 0<br />
u1t256=6.144<br />
u1t257q=0.3 0.0 1.0 0 0<br />
u1t257=6.168<br />
u1t258q=0.3 0.0 1.0 0 0<br />
u1t258=6.192<br />
u1t259q=0.3 0.0 1.0 0 0<br />
u1t259=6.216<br />
u1t260q=0.3 0.0 1.0 0 0<br />
u1t260=6.24<br />
u1t261q=0.3 0.0 1.0 0 0<br />
u1t261=6.264<br />
u1t262q=0.3 0.0 1.0 0 0<br />
u1t262=6.288<br />
u1t263q=0.3 0.0 1.0 0 0<br />
u1t263=6.312<br />
u1t264q=0.3 0.0 1.0 0 0<br />
u1t264=6.336<br />
u1t265q=0.3 0.0 1.0 0 0<br />
u1t265=6.36<br />
u1t266q=0.3 0.0 1.0 0 0<br />
u1t266=6.384<br />
u1t267q=0.3 0.0 1.0 0 0<br />
u1t267=6.408<br />
u1t268q=0.3 0.0 1.0 0 0<br />
u1t268=6.432<br />
u1t269q=0.3 0.0 1.0 0 0<br />
u1t269=6.456<br />
u1t270q=0.3 0.0 1.0 0 0<br />
u1t270=6.48<br />
u1t271q=0.3 0.0 1.0 0 0<br />
u1t271=6.504<br />
u1t272q=0.3 0.0 1.0 0 0<br />
u1t272=6.528<br />
u1t273q=0.3 0.0 1.0 0 0<br />
u1t273=6.552<br />
u1t274q=0.3 0.0 1.0 0 0<br />
u1t274=6.576<br />
u1t275q=0.3 0.0 1.0 0 0<br />
u1t275=6.6<br />
u1t276q=0.3 0.0 1.0 0 0<br />
u1t276=6.624<br />
u1t277q=0.3 0.0 1.0 0 0<br />
u1t277=6.648<br />
u1t278q=0.3 0.0 1.0 0 0<br />
u1t278=6.672<br />
u1t279q=0.3 0.0 1.0 0 0<br />
u1t279=6.696<br />
u1t280q=0.3 0.0 1.0 0 0<br />
u1t280=6.72<br />
u1t281q=0.3 0.0 1.0 0 0<br />
u1t281=6.744<br />
u1t282q=0.3 0.0 1.0 0 0<br />
u1t282=6.768<br />
u1t283q=0.3 0.0 1.0 0 0<br />
u1t283=6.792<br />
u1t284q=0.3 0.0 1.0 0 0<br />
u1t284=6.816<br />
u1t285q=0.3 0.0 1.0 0 0<br />
u1t285=6.84<br />
u1t286q=0.3 0.0 1.0 0 0<br />
u1t286=6.864<br />
u1t287q=0.3 0.0 1.0 0 0<br />
u1t287=6.888<br />
u1t288q=0.3 0.0 1.0 0 0<br />
u1t288=6.912<br />
u1t289q=0.3 0.0 1.0 0 0<br />
u1t289=6.936<br />
u1t290q=0.3 0.0 1.0 0 0<br />
u1t290=6.96<br />
u1t291q=0.3 0.0 1.0 0 0<br />
u1t291=6.984<br />
u1t292q=0.3 0.0 1.0 0 0<br />
u1t292=7.008<br />
u1t293q=0.3 0.0 1.0 0 0<br />
u1t293=7.032<br />
u1t294q=0.3 0.0 1.0 0 0<br />
u1t294=7.056<br />
u1t295q=0.3 0.0 1.0 0 0<br />
u1t295=7.08<br />
u1t296q=0.3 0.0 1.0 0 0<br />
u1t296=7.104<br />
u1t297q=0.3 0.0 1.0 0 0<br />
u1t297=7.128<br />
u1t298q=0.3 0.0 1.0 0 0<br />
u1t298=7.152<br />
u1t299q=0.3 0.0 1.0 0 0<br />
u1t299=7.176<br />
u1t300q=0.3 0.0 1.0 0 0<br />
u1t300=7.2<br />
u1t301q=0.3 0.0 1.0 0 0<br />
u1t301=7.224<br />
u1t302q=0.3 0.0 1.0 0 0<br />
u1t302=7.248<br />
u1t303q=0.3 0.0 1.0 0 0<br />
u1t303=7.272<br />
u1t304q=0.3 0.0 1.0 0 0<br />
u1t304=7.296<br />
u1t305q=0.3 0.0 1.0 0 0<br />
u1t305=7.32<br />
u1t306q=0.3 0.0 1.0 0 0<br />
u1t306=7.344<br />
u1t307q=0.3 0.0 1.0 0 0<br />
u1t307=7.368<br />
u1t308q=0.3 0.0 1.0 0 0<br />
u1t308=7.392<br />
u1t309q=0.3 0.0 1.0 0 0<br />
u1t309=7.416<br />
u1t310q=0.3 0.0 1.0 0 0<br />
u1t310=7.44<br />
u1t311q=0.3 0.0 1.0 0 0<br />
u1t311=7.464<br />
u1t312q=0.3 0.0 1.0 0 0<br />
u1t312=7.488<br />
u1t313q=0.3 0.0 1.0 0 0<br />
u1t313=7.512<br />
u1t314q=0.3 0.0 1.0 0 0<br />
u1t314=7.536<br />
u1t315q=0.3 0.0 1.0 0 0<br />
u1t315=7.56<br />
u1t316q=0.3 0.0 1.0 0 0<br />
u1t316=7.584<br />
u1t317q=0.3 0.0 1.0 0 0<br />
u1t317=7.608<br />
u1t318q=0.3 0.0 1.0 0 0<br />
u1t318=7.632<br />
u1t319q=0.3 0.0 1.0 0 0<br />
u1t319=7.656<br />
u1t320q=0.3 0.0 1.0 0 0<br />
u1t320=7.68<br />
u1t321q=0.3 0.0 1.0 0 0<br />
u1t321=7.704<br />
u1t322q=0.3 0.0 1.0 0 0<br />
u1t322=7.728<br />
u1t323q=0.3 0.0 1.0 0 0<br />
u1t323=7.752<br />
u1t324q=0.3 0.0 1.0 0 0<br />
u1t324=7.776<br />
u1t325q=0.3 0.0 1.0 0 0<br />
u1t325=7.8<br />
u1t326q=0.3 0.0 1.0 0 0<br />
u1t326=7.824<br />
u1t327q=0.3 0.0 1.0 0 0<br />
u1t327=7.848<br />
u1t328q=0.3 0.0 1.0 0 0<br />
u1t328=7.872<br />
u1t329q=0.3 0.0 1.0 0 0<br />
u1t329=7.896<br />
u1t330q=0.3 0.0 1.0 0 0<br />
u1t330=7.92<br />
u1t331q=0.3 0.0 1.0 0 0<br />
u1t331=7.944<br />
u1t332q=0.3 0.0 1.0 0 0<br />
u1t332=7.968<br />
u1t333q=0.3 0.0 1.0 0 0<br />
u1t333=7.992<br />
u1t334q=0.3 0.0 1.0 0 0<br />
u1t334=8.016<br />
u1t335q=0.3 0.0 1.0 0 0<br />
u1t335=8.04<br />
u1t336q=0.3 0.0 1.0 0 0<br />
u1t336=8.064<br />
u1t337q=0.3 0.0 1.0 0 0<br />
u1t337=8.088<br />
u1t338q=0.3 0.0 1.0 0 0<br />
u1t338=8.112<br />
u1t339q=0.3 0.0 1.0 0 0<br />
u1t339=8.136<br />
u1t340q=0.3 0.0 1.0 0 0<br />
u1t340=8.16<br />
u1t341q=0.3 0.0 1.0 0 0<br />
u1t341=8.184<br />
u1t342q=0.3 0.0 1.0 0 0<br />
u1t342=8.208<br />
u1t343q=0.3 0.0 1.0 0 0<br />
u1t343=8.232<br />
u1t344q=0.3 0.0 1.0 0 0<br />
u1t344=8.256<br />
u1t345q=0.3 0.0 1.0 0 0<br />
u1t345=8.28<br />
u1t346q=0.3 0.0 1.0 0 0<br />
u1t346=8.304<br />
u1t347q=0.3 0.0 1.0 0 0<br />
u1t347=8.328<br />
u1t348q=0.3 0.0 1.0 0 0<br />
u1t348=8.352<br />
u1t349q=0.3 0.0 1.0 0 0<br />
u1t349=8.376<br />
u1t350q=0.3 0.0 1.0 0 0<br />
u1t350=8.4<br />
u1t351q=0.3 0.0 1.0 0 0<br />
u1t351=8.424<br />
u1t352q=0.3 0.0 1.0 0 0<br />
u1t352=8.448<br />
u1t353q=0.3 0.0 1.0 0 0<br />
u1t353=8.472<br />
u1t354q=0.3 0.0 1.0 0 0<br />
u1t354=8.496<br />
u1t355q=0.3 0.0 1.0 0 0<br />
u1t355=8.52<br />
u1t356q=0.3 0.0 1.0 0 0<br />
u1t356=8.544<br />
u1t357q=0.3 0.0 1.0 0 0<br />
u1t357=8.568<br />
u1t358q=0.3 0.0 1.0 0 0<br />
u1t358=8.592<br />
u1t359q=0.3 0.0 1.0 0 0<br />
u1t359=8.616<br />
u1t360q=0.3 0.0 1.0 0 0<br />
u1t360=8.64<br />
u1t361q=0.3 0.0 1.0 0 0<br />
u1t361=8.664<br />
u1t362q=0.3 0.0 1.0 0 0<br />
u1t362=8.688<br />
u1t363q=0.3 0.0 1.0 0 0<br />
u1t363=8.712<br />
u1t364q=0.3 0.0 1.0 0 0<br />
u1t364=8.736<br />
u1t365q=0.3 0.0 1.0 0 0<br />
u1t365=8.76<br />
u1t366q=0.3 0.0 1.0 0 0<br />
u1t366=8.784<br />
u1t367q=0.3 0.0 1.0 0 0<br />
u1t367=8.808<br />
u1t368q=0.3 0.0 1.0 0 0<br />
u1t368=8.832<br />
u1t369q=0.3 0.0 1.0 0 0<br />
u1t369=8.856<br />
u1t370q=0.3 0.0 1.0 0 0<br />
u1t370=8.88<br />
u1t371q=0.3 0.0 1.0 0 0<br />
u1t371=8.904<br />
u1t372q=0.3 0.0 1.0 0 0<br />
u1t372=8.928<br />
u1t373q=0.3 0.0 1.0 0 0<br />
u1t373=8.952<br />
u1t374q=0.3 0.0 1.0 0 0<br />
u1t374=8.976<br />
u1t375q=0.3 0.0 1.0 0 0<br />
u1t375=9.0<br />
u1t376q=0.3 0.0 1.0 0 0<br />
u1t376=9.024<br />
u1t377q=0.3 0.0 1.0 0 0<br />
u1t377=9.048<br />
u1t378q=0.3 0.0 1.0 0 0<br />
u1t378=9.072<br />
u1t379q=0.3 0.0 1.0 0 0<br />
u1t379=9.096<br />
u1t380q=0.3 0.0 1.0 0 0<br />
u1t380=9.12<br />
u1t381q=0.3 0.0 1.0 0 0<br />
u1t381=9.144<br />
u1t382q=0.3 0.0 1.0 0 0<br />
u1t382=9.168<br />
u1t383q=0.3 0.0 1.0 0 0<br />
u1t383=9.192<br />
u1t384q=0.3 0.0 1.0 0 0<br />
u1t384=9.216<br />
u1t385q=0.3 0.0 1.0 0 0<br />
u1t385=9.24<br />
u1t386q=0.3 0.0 1.0 0 0<br />
u1t386=9.264<br />
u1t387q=0.3 0.0 1.0 0 0<br />
u1t387=9.288<br />
u1t388q=0.3 0.0 1.0 0 0<br />
u1t388=9.312<br />
u1t389q=0.3 0.0 1.0 0 0<br />
u1t389=9.336<br />
u1t390q=0.3 0.0 1.0 0 0<br />
u1t390=9.36<br />
u1t391q=0.3 0.0 1.0 0 0<br />
u1t391=9.384<br />
u1t392q=0.3 0.0 1.0 0 0<br />
u1t392=9.408<br />
u1t393q=0.3 0.0 1.0 0 0<br />
u1t393=9.432<br />
u1t394q=0.3 0.0 1.0 0 0<br />
u1t394=9.456<br />
u1t395q=0.3 0.0 1.0 0 0<br />
u1t395=9.48<br />
u1t396q=0.3 0.0 1.0 0 0<br />
u1t396=9.504<br />
u1t397q=0.3 0.0 1.0 0 0<br />
u1t397=9.528<br />
u1t398q=0.3 0.0 1.0 0 0<br />
u1t398=9.552<br />
u1t399q=0.3 0.0 1.0 0 0<br />
u1t399=9.576<br />
u1t400q=0.3 0.0 1.0 0 0<br />
u1t400=9.6<br />
u1t401q=0.3 0.0 1.0 0 0<br />
u1t401=9.624<br />
u1t402q=0.3 0.0 1.0 0 0<br />
u1t402=9.648<br />
u1t403q=0.3 0.0 1.0 0 0<br />
u1t403=9.672<br />
u1t404q=0.3 0.0 1.0 0 0<br />
u1t404=9.696<br />
u1t405q=0.3 0.0 1.0 0 0<br />
u1t405=9.72<br />
u1t406q=0.3 0.0 1.0 0 0<br />
u1t406=9.744<br />
u1t407q=0.3 0.0 1.0 0 0<br />
u1t407=9.768<br />
u1t408q=0.3 0.0 1.0 0 0<br />
u1t408=9.792<br />
u1t409q=0.3 0.0 1.0 0 0<br />
u1t409=9.816<br />
u1t410q=0.3 0.0 1.0 0 0<br />
u1t410=9.84<br />
u1t411q=0.3 0.0 1.0 0 0<br />
u1t411=9.864<br />
u1t412q=0.3 0.0 1.0 0 0<br />
u1t412=9.888<br />
u1t413q=0.3 0.0 1.0 0 0<br />
u1t413=9.912<br />
u1t414q=0.3 0.0 1.0 0 0<br />
u1t414=9.936<br />
u1t415q=0.3 0.0 1.0 0 0<br />
u1t415=9.96<br />
u1t416q=0.3 0.0 1.0 0 0<br />
u1t416=9.984<br />
u1t417q=0.3 0.0 1.0 0 0<br />
u1t417=10.008<br />
u1t418q=0.3 0.0 1.0 0 0<br />
u1t418=10.032<br />
u1t419q=0.3 0.0 1.0 0 0<br />
u1t419=10.056<br />
u1t420q=0.3 0.0 1.0 0 0<br />
u1t420=10.08<br />
u1t421q=0.3 0.0 1.0 0 0<br />
u1t421=10.104<br />
u1t422q=0.3 0.0 1.0 0 0<br />
u1t422=10.128<br />
u1t423q=0.3 0.0 1.0 0 0<br />
u1t423=10.152<br />
u1t424q=0.3 0.0 1.0 0 0<br />
u1t424=10.176<br />
u1t425q=0.3 0.0 1.0 0 0<br />
u1t425=10.2<br />
u1t426q=0.3 0.0 1.0 0 0<br />
u1t426=10.224<br />
u1t427q=0.3 0.0 1.0 0 0<br />
u1t427=10.248<br />
u1t428q=0.3 0.0 1.0 0 0<br />
u1t428=10.272<br />
u1t429q=0.3 0.0 1.0 0 0<br />
u1t429=10.296<br />
u1t430q=0.3 0.0 1.0 0 0<br />
u1t430=10.32<br />
u1t431q=0.3 0.0 1.0 0 0<br />
u1t431=10.344<br />
u1t432q=0.3 0.0 1.0 0 0<br />
u1t432=10.368<br />
u1t433q=0.3 0.0 1.0 0 0<br />
u1t433=10.392<br />
u1t434q=0.3 0.0 1.0 0 0<br />
u1t434=10.416<br />
u1t435q=0.3 0.0 1.0 0 0<br />
u1t435=10.44<br />
u1t436q=0.3 0.0 1.0 0 0<br />
u1t436=10.464<br />
u1t437q=0.3 0.0 1.0 0 0<br />
u1t437=10.488<br />
u1t438q=0.3 0.0 1.0 0 0<br />
u1t438=10.512<br />
u1t439q=0.3 0.0 1.0 0 0<br />
u1t439=10.536<br />
u1t440q=0.3 0.0 1.0 0 0<br />
u1t440=10.56<br />
u1t441q=0.3 0.0 1.0 0 0<br />
u1t441=10.584<br />
u1t442q=0.3 0.0 1.0 0 0<br />
u1t442=10.608<br />
u1t443q=0.3 0.0 1.0 0 0<br />
u1t443=10.632<br />
u1t444q=0.3 0.0 1.0 0 0<br />
u1t444=10.656<br />
u1t445q=0.3 0.0 1.0 0 0<br />
u1t445=10.68<br />
u1t446q=0.3 0.0 1.0 0 0<br />
u1t446=10.704<br />
u1t447q=0.3 0.0 1.0 0 0<br />
u1t447=10.728<br />
u1t448q=0.3 0.0 1.0 0 0<br />
u1t448=10.752<br />
u1t449q=0.3 0.0 1.0 0 0<br />
u1t449=10.776<br />
u1t450q=0.3 0.0 1.0 0 0<br />
u1t450=10.8<br />
u1t451q=0.3 0.0 1.0 0 0<br />
u1t451=10.824<br />
u1t452q=0.3 0.0 1.0 0 0<br />
u1t452=10.848<br />
u1t453q=0.3 0.0 1.0 0 0<br />
u1t453=10.872<br />
u1t454q=0.3 0.0 1.0 0 0<br />
u1t454=10.896<br />
u1t455q=0.3 0.0 1.0 0 0<br />
u1t455=10.92<br />
u1t456q=0.3 0.0 1.0 0 0<br />
u1t456=10.944<br />
u1t457q=0.3 0.0 1.0 0 0<br />
u1t457=10.968<br />
u1t458q=0.3 0.0 1.0 0 0<br />
u1t458=10.992<br />
u1t459q=0.3 0.0 1.0 0 0<br />
u1t459=11.016<br />
u1t460q=0.3 0.0 1.0 0 0<br />
u1t460=11.04<br />
u1t461q=0.3 0.0 1.0 0 0<br />
u1t461=11.064<br />
u1t462q=0.3 0.0 1.0 0 0<br />
u1t462=11.088<br />
u1t463q=0.3 0.0 1.0 0 0<br />
u1t463=11.112<br />
u1t464q=0.3 0.0 1.0 0 0<br />
u1t464=11.136<br />
u1t465q=0.3 0.0 1.0 0 0<br />
u1t465=11.16<br />
u1t466q=0.3 0.0 1.0 0 0<br />
u1t466=11.184<br />
u1t467q=0.3 0.0 1.0 0 0<br />
u1t467=11.208<br />
u1t468q=0.3 0.0 1.0 0 0<br />
u1t468=11.232<br />
u1t469q=0.3 0.0 1.0 0 0<br />
u1t469=11.256<br />
u1t470q=0.3 0.0 1.0 0 0<br />
u1t470=11.28<br />
u1t471q=0.3 0.0 1.0 0 0<br />
u1t471=11.304<br />
u1t472q=0.3 0.0 1.0 0 0<br />
u1t472=11.328<br />
u1t473q=0.3 0.0 1.0 0 0<br />
u1t473=11.352<br />
u1t474q=0.3 0.0 1.0 0 0<br />
u1t474=11.376<br />
u1t475q=0.3 0.0 1.0 0 0<br />
u1t475=11.4<br />
u1t476q=0.3 0.0 1.0 0 0<br />
u1t476=11.424<br />
u1t477q=0.3 0.0 1.0 0 0<br />
u1t477=11.448<br />
u1t478q=0.3 0.0 1.0 0 0<br />
u1t478=11.472<br />
u1t479q=0.3 0.0 1.0 0 0<br />
u1t479=11.496<br />
u1t480q=0.3 0.0 1.0 0 0<br />
u1t480=11.52<br />
u1t481q=0.3 0.0 1.0 0 0<br />
u1t481=11.544<br />
u1t482q=0.3 0.0 1.0 0 0<br />
u1t482=11.568<br />
u1t483q=0.3 0.0 1.0 0 0<br />
u1t483=11.592<br />
u1t484q=0.3 0.0 1.0 0 0<br />
u1t484=11.616<br />
u1t485q=0.3 0.0 1.0 0 0<br />
u1t485=11.64<br />
u1t486q=0.3 0.0 1.0 0 0<br />
u1t486=11.664<br />
u1t487q=0.3 0.0 1.0 0 0<br />
u1t487=11.688<br />
u1t488q=0.3 0.0 1.0 0 0<br />
u1t488=11.712<br />
u1t489q=0.3 0.0 1.0 0 0<br />
u1t489=11.736<br />
u1t490q=0.3 0.0 1.0 0 0<br />
u1t490=11.76<br />
u1t491q=0.3 0.0 1.0 0 0<br />
u1t491=11.784<br />
u1t492q=0.3 0.0 1.0 0 0<br />
u1t492=11.808<br />
u1t493q=0.3 0.0 1.0 0 0<br />
u1t493=11.832<br />
u1t494q=0.3 0.0 1.0 0 0<br />
u1t494=11.856<br />
u1t495q=0.3 0.0 1.0 0 0<br />
u1t495=11.88<br />
u1t496q=0.3 0.0 1.0 0 0<br />
u1t496=11.904<br />
u1t497q=0.3 0.0 1.0 0 0<br />
u1t497=11.928<br />
u1t498q=0.3 0.0 1.0 0 0<br />
u1t498=11.952<br />
u1t499q=0.3 0.0 1.0 0 0<br />
u1t499=11.976<br />
u1t500q=0.3 0 0 0 0<br />
u1t500=tend<br />
<br />
[Messungen]<br />
tAnzahl=12<br />
<br />
t1=1<br />
t1Anzahl=2<br />
t1m1=mfcn1 1.0 1e-06 1<br />
t1m2=mfcn2 1.0 1e-06 1<br />
t1minmax=0 1e+10<br />
<br />
t2=2.000<br />
t2Anzahl=2<br />
t2m1=mfcn1 1.0 1e-06 1<br />
t2m2=mfcn2 1.0 1e-06 1<br />
t2minmax=0 1e+10<br />
<br />
t3=3<br />
t3Anzahl=2<br />
t3m1=mfcn1 1.0 1e-06 1<br />
t3m2=mfcn2 1.0 1e-06 1<br />
t3minmax=0 1e+10<br />
<br />
t4=4<br />
t4Anzahl=2<br />
t4m1=mfcn1 1.0 1e-06 1<br />
t4m2=mfcn2 1.0 1e-06 1<br />
t4minmax=0 1e+10<br />
<br />
t5=5<br />
t5Anzahl=2<br />
t5m1=mfcn1 1.0 1e-06 1<br />
t5m2=mfcn2 1.0 1e-06 1<br />
t5minmax=0 1e+10<br />
<br />
t6=6<br />
t6Anzahl=2<br />
t6m1=mfcn1 1.0 1e-06 1<br />
t6m2=mfcn2 1.0 1e-06 1<br />
t6minmax=0 1e+10<br />
<br />
t7=7<br />
t7Anzahl=2<br />
t7m1=mfcn1 1.0 1e-06 1<br />
t7m2=mfcn2 1.0 1e-06 1<br />
t7minmax=0 1e+10<br />
<br />
t8=8<br />
t8Anzahl=2<br />
t8m1=mfcn1 1.0 1e-06 1<br />
t8m2=mfcn2 1.0 1e-06 1<br />
t8minmax=0 1e+10<br />
<br />
t9=9<br />
t9Anzahl=2<br />
t9m1=mfcn1 1.0 1e-06 1<br />
t9m2=mfcn2 1.0 1e-06 1<br />
t9minmax=0 1e+10<br />
<br />
t10=10.000<br />
t10Anzahl=2<br />
t10m1=mfcn1 1.0 1e-06 1<br />
t10m2=mfcn2 1.0 1e-06 1<br />
t10minmax=0 1e+10<br />
<br />
t11=11.000<br />
t11Anzahl=2<br />
t11m1=mfcn1 1.0 1e-06 1<br />
t11m2=mfcn2 1.0 1e-06 1<br />
t11minmax=0 1e+10<br />
<br />
t12=12.000<br />
t12Anzahl=2<br />
t12m1=mfcn1 1.0 1e-06 1<br />
t12m2=mfcn2 1.0 1e-06 1<br />
t12minmax=0 1e+10<br />
[NebenbedingungenSteuergroessen]<br />
cAnzahl=0<br />
[Messverfahren]<br />
mAnzahl=2<br />
m1=mfcn1 1 0 1e+10 0<br />
m1f1=mess3 sigma3 1<br />
m2=mfcn2 1 0 1e+10 0<br />
m2f1=mess4 sigma4 1<br />
mminmaxges=0 8<br />
<br />
<br />
[OptionenIntegration]<br />
teps=1e-08<br />
rtol=1e-08<br />
atol=1e-07<br />
stepsize=0.0001<br />
maxorder=6<br />
maxstepnumber=4000<br />
minstepsize=-1<br />
maxstepsize=-1<br />
maxitNewton=3<br />
realworkspace=1700000<br />
integerworkspace=5000<br />
printlevel=0<br />
mcnonlinearflag=0<br />
mcDAEflag=0<br />
mctol=1e-07<br />
mcmaxit=50<br />
mclinesearch=1<br />
mcalpha0=1<br />
rndmethod=-1<br />
rndeps=1e-05<br />
rndverbose=0<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Lotka_Experimental_Design_(VPLAN)&diff=1340
Lotka Experimental Design (VPLAN)
2016-01-19T16:18:02Z
<p>FelixJost: Created page with " == VPLAN == Differential equations: <source lang="fortran"> c RHS of the differential equations subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag ) impl..."</p>
<hr />
<div><br />
<br />
== VPLAN ==<br />
<br />
<br />
Differential equations:<br />
<br />
<source lang="fortran"><br />
<br />
c RHS of the differential equations<br />
<br />
subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 x(*), f(*), p(*), q(*), rwh(*), t<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 p1,p3,p5,p6, myu<br />
<br />
c fixed parameters<br />
p1 = 1.0<br />
p3 = 1.0<br />
p5 = 0.4<br />
p6 = 0.2<br />
<br />
c DISCRETIZE1( myu, rwh, iwh )<br />
<br />
<br />
f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1) <br />
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
First Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(1) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
<br />
<br />
Second Measurement function:<br />
<br />
<source lang="fortran"><br />
<br />
c Messfunktion<br />
<br />
subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 t, x(*), h, p(*), q(*), rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
h = x(2) <br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
<br />
<br />
</source><br />
<br />
Standard deviation of first measurement function<br />
<br />
<br />
<source lang="fortran"><br />
<br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s<br />
real*8 h<br />
<br />
s = 1.0d+0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
<br />
</source><br />
<br />
Standard deviation of second measurement function:<br />
<br />
<br />
<source lang="fortran"><br />
c Standardabweichung der Messfunktion<br />
<br />
subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )<br />
implicit none<br />
<br />
real*8 rwh(*)<br />
integer*4 iwh(*), iflag<br />
<br />
real*8 t, x(*), p(*), q(*)<br />
real*8 s(*)<br />
<br />
s(1) = 1.0<br />
<br />
iflag = 0<br />
<br />
end<br />
<br />
</source></div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=968
Diels-Alder Reaction Experimental Design
2015-12-09T08:14:49Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ <br />
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\<br />
\vartheta_{up} + 273 & t \in [8,t_{end}]<br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=967
Diels-Alder Reaction Experimental Design
2015-12-09T08:13:18Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
L(x, 1) &=& \left\{ \begin{array}{cl} b & b \\ <br />
b & b \\<br />
& <br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=966
Diels-Alder Reaction Experimental Design
2015-12-09T08:13:03Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \\<br />
\\<br />
<br />
<br />
L(x, 1) &=& \left\{ \begin{array}{cl} b & b \\ <br />
b & b \\<br />
& <br />
\end{array} \right. \\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=965
Diels-Alder Reaction Experimental Design
2015-12-09T08:10:50Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \\<br />
\\<br />
<br />
<br />
<br />
<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=964
Diels-Alder Reaction Experimental Design
2015-12-09T08:10:39Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=963
Diels-Alder Reaction Experimental Design
2015-12-09T08:10:23Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=962
Diels-Alder Reaction Experimental Design
2015-12-09T08:10:13Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \right. \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=961
Diels-Alder Reaction Experimental Design
2015-12-09T08:08:42Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
L(x, 1) &=& \left\{ \begin{array}{cl} e \; p_1 & \mbox{if } \sigma_1 \ge 0 \\ e \; p_2 & \mbox{else if } \sigma_2 \ge 0 \\ e \; \sum_{i=0}^{5} c_i (1) \left( \frac{1}{10} \gamma\ x_1 \right)^{-i} \quad & \mbox{else} \end{array} \right. \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=960
Diels-Alder Reaction Experimental Design
2015-12-09T08:07:25Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=959
Diels-Alder Reaction Experimental Design
2015-12-09T08:07:09Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
\begin{equation}<br />
\begin{cases}<br />
2x^{2} & \text{f"ur } x \textless 4 \\<br />
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\<br />
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27 <br />
\end{cases}<br />
\end{equation}<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=958
Diels-Alder Reaction Experimental Design
2015-12-09T08:06:56Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<math><br />
\begin{equation}<br />
\begin{cases}<br />
2x^{2} & \text{f"ur } x \textless 4 \\<br />
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\<br />
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27 <br />
\end{cases}<br />
\end{equation}<br />
</math><br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=957
Diels-Alder Reaction Experimental Design
2015-12-09T08:06:21Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<math><br />
\begin{cases}<br />
2x^{2} & \text{f"ur } x \textless 4 \\<br />
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\<br />
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27 <br />
\end{cases}<br />
</math><br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=956
Diels-Alder Reaction Experimental Design
2015-12-09T08:06:03Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) & = & \begin{cases}<br />
x(n),\\<br />
x(n-1)\\<br />
x(n-1)<br />
\end{cases}<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<math><br />
\begin{cases}<br />
2x^{2} & \text{f"ur } x \textless 4 \\<br />
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\<br />
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27 <br />
\end{cases}<br />
</math><br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=955
Diels-Alder Reaction Experimental Design
2015-12-09T08:05:05Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) & = & \begin{cases}<br />
x(n),\\<br />
x(n-1)\\<br />
x(n-1)<br />
\end{cases}<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=954
Diels-Alder Reaction Experimental Design
2015-12-09T08:04:32Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
\vartheta(t) & = & \begin{cases}<br />
test \\<br />
test<br />
\end{cases} <br />
<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=953
Diels-Alder Reaction Experimental Design
2015-12-09T08:03:19Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=952
Diels-Alder Reaction Experimental Design
2015-12-09T08:02:03Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Fixed parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=951
Diels-Alder Reaction Experimental Design
2015-12-09T08:01:25Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=950
Diels-Alder Reaction Experimental Design
2015-12-09T08:01:10Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P & &.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=949
Diels-Alder Reaction Experimental Design
2015-12-09T08:00:35Z
<p>FelixJost: /* Constraints */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights (initial mass):<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content (fraction of active substances):<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=948
Diels-Alder Reaction Experimental Design
2015-12-09T07:56:56Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content:<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
<br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[2,8]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[8,t_{end}]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=947
Diels-Alder Reaction Experimental Design
2015-12-09T07:55:59Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content:<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
<br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[t_0,2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[2,8]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[8,math>\t_{end}</math>]<br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=946
Diels-Alder Reaction Experimental Design
2015-12-09T07:55:38Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content:<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
<br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>t_0,2</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[2,8]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[8,math>\t_{end}</math>]<br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=945
Diels-Alder Reaction Experimental Design
2015-12-09T07:54:51Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content:<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
<br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|<math>[\t_{0},2]</math><br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[2,8]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[8,math>\t_{end}</math>]<br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=944
Diels-Alder Reaction Experimental Design
2015-12-09T07:53:23Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content:<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
<br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[math>\t_{0}</math>,2]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[2,8]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[8,math>\t_{end}</math>]<br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost
https://mintoc.de/index.php?title=Diels-Alder_Reaction_Experimental_Design&diff=943
Diels-Alder Reaction Experimental Design
2015-12-09T07:52:53Z
<p>FelixJost: /* Optimum Experimental Design Problem */</p>
<hr />
<div>The '''Diels-Alder Reaction''' is an organic chemical reaction. <br />
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.<br />
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.<br />
<br />
<br />
== Model Formulation ==<br />
<br />
The reactionkinetics can be modelled by the following differential equation system:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\<br />
& & \\<br />
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\<br />
& & \\<br />
\dot{n_4}(t) &=& 0<br />
\end{array} <br />
</math><br />
<br />
The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation<br />
<br />
<math><br />
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )<br />
</math><br />
<br />
Total mass: <br />
<br />
<math><br />
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 <br />
</math><br />
<br />
Temperature in Kelvin:<br />
<br />
<math><br />
T(t) = \vartheta (t) + 273<br />
</math><br />
<br />
The ODE system is summarized to:<br />
<br />
<math><br />
\begin{array}{rcl}<br />
\dot{x}(t) &=& f(x(t), u(t), p) <br />
\end{array} <br />
</math><br />
<br />
== Constraints ==<br />
<br />
The control variables are constrained with respect to the mass of sample weights:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 <br />
\end{array} <br />
</math><br />
<br />
and to the mass of active ingredient content:<br />
<br />
<p><br />
<math><br />
\begin{array}{cll}<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7<br />
\end{array} <br />
</math><br />
<br />
<br />
<br />
== Optimum Experimental Design Problem ==<br />
<br />
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:<br />
<br />
<br />
<math><br />
\begin{array}{cll}<br />
\displaystyle \min_{x, G, F, Tc, n_{a1}, n_{a2}, n_{a4}, c_{kat}, \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]<br />
\mbox{s.t.} \\<br />
\dot{x}(t) & = & f(x(t), u(t),p), \\<br />
\\<br />
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\<br />
\\<br />
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\<br />
\\<br />
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\<br />
\\<br />
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\<br />
\\<br />
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\<br />
\\<br />
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\<br />
\\<br />
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\<br />
\\<br />
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
<br />
<br />
{| class="wikitable"<br />
|+State variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Initial value (<math>t_0</math>)<br />
|-<br />
|Molar number 1<br />
|<math>n_1(t)</math><br />
|<math>n_1(t_0) = n_{a1} </math><br />
|-<br />
|Molar number 2<br />
|<math>n_2(t)</math><br />
|<math>n_2(t_0) = n_{a2} </math><br />
|-<br />
|Molar number 3<br />
|<math>n_3(t)</math><br />
|<math>n_3(t_0) = 0 </math><br />
|-<br />
|Solvent<br />
|<math>n_4(t)</math><br />
|<math>n_4(t_0) = n_{a4} </math><br />
|}<br />
<br />
<br />
{| class="wikitable"<br />
|+Constants<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Molar Mass<br />
|<math>M_1</math><br />
|0.1362<br />
|-<br />
|Molar Mass<br />
|<math>M_2</math><br />
|0.09806<br />
|-<br />
|Molar Mass<br />
|<math>M_3</math><br />
|0.23426<br />
|-<br />
|Molar Mass<br />
|<math>M_4</math><br />
|0.236<br />
|-<br />
|Universal gas constant<br />
|<math>R</math><br />
|8.314<br />
|-<br />
|Reference temperature<br />
|<math>T_{ref}</math><br />
|293<br />
|-<br />
|St.dev of measurement error<br />
|<math>\sigma</math><br />
|1<br />
|}<br />
<br />
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.<br />
<br />
{| class="wikitable"<br />
|+Parameters<br />
|-<br />
|Name<br />
|Symbol<br />
|Value<br />
|-<br />
|Steric factor<br />
|<math>k_1</math><br />
|<math>p_1 \cdot 0.01</math><br />
|-<br />
|Steric factor<br />
|<math>k_{kat}</math><br />
|<math>p_2 \cdot 0.10</math><br />
|-<br />
|Activation energie<br />
|<math>E_1</math><br />
|<math>p_3 \cdot 60000</math><br />
|-<br />
|Activation energie<br />
|<math>E_{kat}</math><br />
|<math>p_4 \cdot 40000</math><br />
|-<br />
|Catalyst deactivation coefficient<br />
|<math>\lambda</math><br />
|<math>p_5 \cdot 0.25</math><br />
|}<br />
with <math>p_j = 1, \ j =1, \dots, 5</math><br />
<br />
{| class="wikitable"<br />
|+Optimization/control variables<br />
|-<br />
|Name<br />
|Symbol<br />
|Interval<br />
|-<br />
|Initial molar number 1<br />
|<math>n_{a1}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 2<br />
|<math>n_{a2}</math><br />
|[0.4,9.0]<br />
|-<br />
|Initial molar number 4<br />
|<math>n_{a4}</math><br />
|[0.4,9.0]<br />
|-<br />
|Concentration of the catalyst<br />
|<math>c_{kat}</math><br />
|[0.0,6.0]<br />
|}<br />
<br />
{| class="wikitable"<br />
|+Control function<br />
|-<br />
|Name<br />
|Symbol<br />
|Time interval<br />
|Value interval<br />
|Initial value<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[math>\t_{0}</math>,2]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[2,8]<br />
|[20.0,100.0]<br />
|20.0<br />
|-<br />
|Initial molar number 1<br />
|<math>\vartheta(t)</math><br />
|[8,math>\t_{end}</math>]<br />
|[20.0,100.0]<br />
|20.0<br />
|}<br />
<br />
'''Measurement grid'''<br />
<br />
<math><br />
\begin{array}{llll}<br />
t_0 = 0 & & & \\<br />
t_{end} = 20 & & & \\<br />
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.<br />
\end{array} <br />
</math><br />
<br />
== References ==<br />
<br />
R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983<br />
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002<br />
<br />
<br />
[[Category:VPLAN]]<br />
[[Category:Optimum Experimental Design]]<br />
[[Category:ODE model]]</div>
FelixJost