Diels-Alder Reaction Experimental Design: Difference between revisions

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Model Formulation

Differential equation system:

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

Solvent:

$\begin{array}{rcl} n_{4} & = & n_{a 4} \end{array}$

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:

$T (t) = ϑ (t) + 273$

The ODE system is summarized to:

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

Optimum Experimental Design Problem

$\begin{array}{cll} \min_{x, G, F, u} & t r a c e (F^{- 1} (t_{t_{f}})) \\ s.t. \\ \dot{x} (t) & = & f (x (t), u (t), p), \\ \dot{h} (t) & = & \frac{n_{3} (t) \cdot M_{3}}{m_{t o t}} \cdot 100 \\ \dot{G} (t) & = & f_{x} (x (t), u (t), p) G (t) + f_{p} (x (t), u (t), p) \\ \dot{F} (t) & = & w (t) (h_{x} (x (t), u (t), p) G (t))^{T} (h_{x} (x (t), u (t), p) G (t)) \\ 0.1 & \leq & n_{a 1} \cdot M_{1} + n_{a 2} \cdot M_{2} + n_{a 4} \cdot M_{4}, \forall t \in I \\ 10 & \geq & n_{a 1} \cdot M_{1} + n_{a 2} \cdot M_{2} + n_{a 4} \cdot M_{4}, \forall t \in I \\ 0.1 & \leq & \frac{n_{a 1} \cdot M_{1} + n_{a 2} \cdot M_{2}}{n_{a 1} \cdot M_{1} + n_{a 2} \cdot M_{2} + n_{a 4} \cdot M_{4}}, \forall t \in I \\ 0.7 & \geq & \frac{n_{a 1} \cdot M_{1} + n_{a 2} \cdot M_{2}}{n_{a 1} \cdot M_{1} + n_{a 2} \cdot M_{2} + n_{a 4} \cdot M_{4}}, \forall t \in I \\ 0 & = & ϑ_{l o}, \forall t \in [t_{0}, 2] \\ 0 & = & ϑ_{l o} + \frac{t - 2}{6} (ϑ_{u p} - ϑ_{l o}), \forall t \in [2, 8] \\ 0 & = & ϑ_{u p}, \forall t \in [8, t_{e n d}] \\ x & \in & 𝒳, u \in 𝒰, p \in P . \end{array}$

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	$n_{3} (t)$	$n_{3} (t_{0}) = 0$

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	$T_{r e f}$	293
St.dev of measurement error	$σ$	1

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 4	$n_{a 4}$	[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 74: / Line 74: @@
   \\
 & = & \vartheta_{lo}, \forall \, t \in [t_0,2] \\
+ \\
 & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} )  , \forall \, t \in [2,8] \\
+ \\
 & = & \vartheta_{up}, \forall \, t \in [8,t_{end}] \\
+ \\
   x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
 \end{array}