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

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(→‎Model Formulation)
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\begin{array}{cl}
 
\begin{array}{cl}
 
  \displaystyle \min_{x, G, F, u} & trace(F^{-1} (t_{t_f})) \\[1.5ex]
 
  \displaystyle \min_{x, G, F, u} & trace(F^{-1} (t_{t_f})) \\[1.5ex]
−
  \mbox{s.t.} & \dot{x}  =  f(x,u,p,t), \forall \, t \in I\\
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  \mbox{s.t.}  \dot{n_1}(t) & = -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}},  \\
−
& \dot{n_1}(t) = -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}},  \\
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  \dot{n_2}(t) & = -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
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  & \dot{n_2}(t) = -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
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  \dot{n_2}(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}}, \\
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  0 & = g(x(t_o),x(t_f),p) \\
−
  & 0 = g(x(t_o),x(t_f),p) \\
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  0 & \ge  c(x,u,p), \forall \, t \in I\\
−
  & 0  \ge  c(x,u,p), \forall \, t \in I\\
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  0 & =  h(x,u,p), \forall \, t \in I \\
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  & 0  =  h(x,u,p), \forall \, t \in I \\
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  x & \in \mathcal{X},\,u \in \mathcal{U},\, p \in P.
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  & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P.
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\end{array}  
 
\end{array}  
 
</math>
 
</math>

Revision as of 12:29, 4 December 2015

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Model Formulation

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_2}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \end{array}

Reaction velocity constant:

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


\begin{array}{cl} \displaystyle \min_{x, G, F, u} & trace(F^{-1} (t_{t_f})) \\[1.5ex] \mbox{s.t.} \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_2}(t) & = \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\ 0 & = g(x(t_o),x(t_f),p) \\ 0 & \ge c(x,u,p), \forall \, t \in I\\ 0 & = h(x,u,p), \forall \, t \in I \\ 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) = n_{a3}
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
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

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 3 n_{a3} [0.4,9.0]
Concentration of the catalyst c_{kat} [0.0,6.0]
Initial molar number 1 \vartheta(t) [20.0,100.0]

Parameters

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