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

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  \\
 
  \\
 
  0 & = & g(x(t_o),x(t_f),p) \\
 
  0 & = & g(x(t_o),x(t_f),p) \\
−
  0 & \ge &  c(x,u,p), \forall \, t \in I\\
+
  0.1 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4  , \forall \, t \in I \\
−
  0 & = & h(x,u,p), \forall \, t \in I \\
+
  10 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4    , \forall \, t \in I \\
 +
0.1 & \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 }  , \forall \, t \in I \\
 +
0.7 & \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 }  , \forall \, t \in I \\
 +
  0 & = & \vartheta_{lo}, \forall \, t \in [t_0,2] \\
 +
0 & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) , \forall \, t \in [2,8] \\
 +
0 & = & \vartheta_{up}, \forall \, t \in [8,t_{end}] \\
 
  x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
 
  x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
 
\end{array}  
 
\end{array}  

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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}

Solvent:

\begin{array}{rcl} n_4 &=& n_{a4} \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

The ODE system is summarized to:

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

Optimum Experimental Design Problem

\begin{array}{cll} \displaystyle \min_{x, G, F, u} && trace(F^{-1} (t_{t_f})) \\[1.5ex] \mbox{s.t.} \\ \dot{x}(t) & = & f(x(t), u(t),p), \\ \\ \dot{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 & = & g(x(t_o),x(t_f),p) \\ 0.1 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 , \forall \, t \in I \\ 10 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 , \forall \, t \in I \\ 0.1 & \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 } , \forall \, t \in I \\ 0.7 & \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 } , \forall \, t \in I \\ 0 & = & \vartheta_{lo}, \forall \, t \in [t_0,2] \\ 0 & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) , \forall \, t \in [2,8] \\ 0 & = & \vartheta_{up}, \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
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

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]
Initial molar number 1 \vartheta(t) [20.0,100.0]

Parameters

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