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:

$\begin{array}{rcl}\dot{n_1}(t)& =& -k\cdot \frac{n_1(t)\cdot n_2(t)}{m_{tot}},\\ & & \\ \dot{n_2}(t)& =& -k\cdot \frac{n_1(t)\cdot n_2(t)}{m_{tot}},\\ & & \\ \dot{n_3}(t)& =& k\cdot \frac{n_1(t)\cdot n_2(t)}{m_{tot}}\\ & & \\ \dot{n_4}(t)& =& 0\end{array}$

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

The aim is to compute an optimal experimental design Failed to parse (unknown function "\math"): {\displaystyle \psi<\math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda<\math>. So, we have to solve the following optimum experimental design problem: <p> <math> \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}, \quad \forall \, t \in [t_0,2] \\ \\ 0 & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) , \quad \forall \, t \in [2,8] \\ \\ 0 & = & \vartheta_{up}, \quad \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	$n_3(t)$	$n_3(t_0)=0$
Solvent	$n_4(t)$	$n_4(t_0)=n_{a4}$

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]

Measurement grid

$\begin{array}{llll}{t}_0=0& & & \\ t_{end}=20& & & \\ t_j=j/3,& j=1,\dots ,15,& t_j=j-10,& j=16,\dots ,20.\end{array}$

References

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