Diels-Alder Reaction Experimental Design

The Diels-Alder Reaction is an organic chemical reaction. A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.

More information about the reaction can be found in ...

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

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array}{cll} \displaystyle \min_{x, G, F, u} && trace(F^{-1} (t__{end})) \\[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)) \\ \\ 0.1 & \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 } \\ \\ 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	$n_{1} (t)$	$n_{1} (t_{0}) = n_{a 1}$
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]

References

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983 \\ Dissertation Stefan Körkel