Difference between revisions of "Diels-Alder Reaction Experimental Design"

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Revision as of 08:48, 9 December 2015

The Diels-Alder Reaction is an organic chemical reaction. A conjugated diene and a substituted alkene react and form a substituted cyclohexene system. Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.


Model Formulation

The reactionkinetics can be modelled by the following differential equation system:

\begin{array}{rcl} \dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\ & & \\ \dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\ & & \\ \dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\ & & \\ \dot{n_4}(t) &=& 0 \end{array}

The reaction velocity constant k consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

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}}) )

Total mass:

m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4

Temperature in Kelvin:

T(t) = \vartheta (t) + 273

The ODE system is summarized to:

\begin{array}{rcl} \dot{x}(t) &=& f(x(t), u(t), p) \end{array}

Constraints

The control variables are constrained with respect to the mass of sample weights:


\begin{array}{cll} 0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 \end{array}

and to the mass of active ingredient content:

\begin{array}{cll} 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 \end{array}

Optimum Experimental Design Problem

The aim is to compute an optimal experimental design \xi = (q,w) which minimizes the uncertainties of the parameters k_1, k_{cat}, E_1, E_{cat}, \lambda. So, we have to solve the following optimum experimental design problem:

<p> \begin{array}{cll} \displaystyle \min_{x, G, F, Tc, n_{a1}} && trace(F^{-1} (t_{end})) \\[1.5ex] \mbox{s.t.} \\ \dot{x}(t) & = & f(x(t), u(t),p), \\ \\ h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\ \\ \dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\ \\ \dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\ \\ 0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 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 } \\ \\ 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 } \\ \\ T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\ \\ T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\ \\ T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\ \\ x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P. \end{array}


State variables
Name Symbol Initial value (t_0)
Molar number 1 n_1(t) n_1(t_0) = n_{a1}
Molar number 2 n_2(t) n_2(t_0) = n_{a2}
Molar number 3 n_3(t) n_3(t_0) = 0
Solvent n_4(t) n_4(t_0) = n_{a4}


Constants
Name Symbol Value
Molar Mass M_1 0.1362
Molar Mass M_2 0.09806
Molar Mass M_3 0.23426
Molar Mass M_4 0.236
Universal gas constant R 8.314
Reference temperature T_{ref} 293
St.dev of measurement error \sigma 1
Parameters
Name Symbol Value
Steric factor k_1 p_1 \cdot 0.01
Steric factor k_{kat} p_2 \cdot 0.10
Activation energie E_1 p_3 \cdot 60000
Activation energie E_{kat} p_4 \cdot 40000
Catalyst deactivation coefficient \lambda p_5 \cdot 0.25

with p_j = 1, \ j =1, \dots, 5

Optimization/control variables
Name Symbol Interval
Initial molar number 1 n_{a1} [0.4,9.0]
Initial molar number 2 n_{a2} [0.4,9.0]
Initial molar number 4 n_{a4} [0.4,9.0]
Concentration of the catalyst c_{kat} [0.0,6.0]
Control function
Name Symbol Time interval Value interval Initial value
Initial molar number 1 \vartheta(t) [math>\t_{0}</math>,2] [20.0,100.0] 20.0
Initial molar number 1 \vartheta(t) [2,8] [20.0,100.0] 20.0
Initial molar number 1 \vartheta(t) [8,math>\t_{end}</math>] [20.0,100.0] 20.0

Measurement grid

\begin{array}{llll} t_0 = 0 & & & \\ t_{end} = 20 & & & \\ t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20. \end{array}

References

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002
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