Diels-Alder Reaction Experimental Design: Difference between revisions

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Model Formulation

Differential equation system:

$\begin{array}{rcl} \dot{n_{1}} (t) & = & - k \cdot \frac{n_{1} (t) \cdot n_{2} (t)}{m_{t o t}}, \\ \dot{n_{2}} (t) & = & - k \cdot \frac{n_{1} (t) \cdot n_{2} (t)}{m_{t o t}}, \\ \dot{n_{2}} (t) & = & k \cdot \frac{n_{1} (t) \cdot n_{2} (t)}{m_{t o t}} \end{array}$

Reaction velocity constant:

$k = k_{1} \cdot e x p (- \frac{E_{1}}{R} \cdot (\frac{1}{T (t)} - \frac{1}{T_{r e f}})) + k_{c a t} \cdot c_{c a t} \cdot e x p (- λ \cdot t) \cdot e x p (- \frac{E_{c a t}}{R} \cdot (\frac{1}{T (t)} - \frac{1}{T_{r e f}}))$

Total mass:

$m_{t o t} = n_{1} \cdot M_{1} + n_{2} \cdot M_{2} + n_{3} \cdot M_{3} + n_{4} \cdot M_{4}$

Temperature in Kelvin:

$T (t) = ϑ (t) + 273$

$\begin{array}{cl} \min_{x, G, F, u} & t r a c e (F^{- 1} (t_{t_{f}})) \\ s.t. & \dot{x} = f (x, u, p, t), \forall t \in I \\ \dot{n_{1}} (t) = - k \cdot \frac{n_{1} (t) \cdot n_{2} (t)}{m_{t o t}}, \\ \dot{n_{2}} (t) = - k \cdot \frac{n_{1} (t) \cdot n_{2} (t)}{m_{t o t}}, \\ \dot{n_{2}} (t) = k \cdot \frac{n_{1} (t) \cdot n_{2} (t)}{m_{t o t}}, \\ 0 = g (x (t_{o}), x (t_{f}), p) \\ 0 \geq c (x, u, p), \forall t \in I \\ 0 = h (x, u, p), \forall t \in I \\ x \in 𝒳, u \in 𝒰, p \in P . \end{array}$

State variables
Name	Symbol	Initial value ( $t_{0}$ )
Molar number 1	$n_{1} (t)$	$n_{1} (t_{0}) = n_{a 1}$
Molar number 2	$n_{2} (t)$	$n_{2} (t_{0}) = n_{a 2}$
Molar number 3	$n_{3} (t)$	$n_{3} (t_{0}) = n_{a 3}$

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	$T_{r e f}$	293

Parameters
Name	Symbol	Value
Steric factor	$k_{1}$	$p_{1} \cdot 0.01$
Steric factor	$k_{k a t}$	$p_{2} \cdot 0.10$
Activation energie	$E_{1}$	$p_{3} \cdot 60000$
Activation energie	$E_{k a t}$	$p_{4} \cdot 40000$
Catalyst deactivation coefficient	$λ$	$p_{5} \cdot 0.25$

with $p_{j} = 1, j = 1, \dots, 5$

Control variables
Name	Symbol	Interval
Initial molar number 1	$n_{a 1}$	[0.4,9.0]
Initial molar number 2	$n_{a 2}$	[0.4,9.0]
Initial molar number 3	$n_{a 3}$	[0.4,9.0]
Concentration of the catalyst	$c_{k a t}$	[0.0,6.0]
Initial molar number 1	$ϑ (t)$	[20.0,100.0]

Parameters

@@ Line 40: / Line 40: @@
   \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\\
-& \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}},   \\
-&                                                                \\
+ & \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}}, \\
+ & \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\\