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. Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

Model Formulation

The reactionkinetics can be modelled by the following differential equation system:

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

The reaction velocity constant $k$ consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

$k=k_1\cdot exp(-\frac{E_1}{R}\cdot (\frac{1}{T(t)}-\frac{1}{T_{ref}}))+k_{cat}\cdot c_{cat}\cdot exp(-\lambda \cdot t)\cdot exp(-\frac{E_{cat}}{R}\cdot (\frac{1}{T(t)}-\frac{1}{T_{ref}}))$

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:

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

Constraints

The control variables are constrained with respect to the mass of sample weights:

$\begin{array}{cll}0.1& \le & n_{a1}\cdot M_1+n_{a2}\cdot M_2+n_{a4}\cdot M_4\le 10\end{array}$

and to the mass of active ingredient content:

$\begin{array}{cll}0.1& \le & \frac{n_{a1}\cdot M_1+n_{a2}\cdot M_2}{n_{a1}\cdot M_1+n_{a2}\cdot M_2+n_{a4}\cdot M_4}\le 0.7\end{array}$

Optimum Experimental Design Problem

The aim is to compute an optimal experimental design $\xi =(q,w)$ which minimizes the uncertainties of the parameters $k_1,k_{cat},E_1,E_{cat},\lambda$. So, we have to solve the following optimum experimental design problem:

$\begin{array}{cll}\underset{x,G,F,Tc,n_{a1}}{\min }& & trace(F^{-1}(t_{end}))\\ \text{s.t.}\\ \dot{x}(t)& =& f(x(t),u(t),p),\\ \\ h(t)& =& \frac{n_3(t)\cdot M_3}{m_{tot}}\cdot 100\\ \\ \dot{G}(t)& =& f_x(x(t),u(t),p)G(t)+f_p(x(t),u(t),p)\\ \\ \dot{F}(t)& =& w(t)(h_x(x(t),u(t),p)G(t){)}^T(h_x(x(t),u(t),p)G(t))\\ \\ 0.1& \le & n_{a1}\cdot M_1+n_{a2}\cdot M_2+n_{a4}\cdot M_4\\ \\ 10& \ge & n_{a1}\cdot M_1+n_{a2}\cdot M_2+n_{a4}\cdot M_4\\ \\ 0.1& \le & \frac{n_{a1}\cdot M_1+n_{a2}\cdot M_2}{n_{a1}\cdot M_1+n_{a2}\cdot M_2+n_{a4}\cdot M_4}\\ \\ 0.7& \ge & \frac{n_{a1}\cdot M_1+n_{a2}\cdot M_2}{n_{a1}\cdot M_1+n_{a2}\cdot M_2+n_{a4}\cdot M_4}\\ \\ T(t)& =& {\vartheta }_{lo}+273,\mathrm{\forall }t\in [t_0,2]\\ \\ T(t)& =& {\vartheta }_{lo}+\frac{t-2}{6}({\vartheta }_{up}-{\vartheta }_{lo})+273,\mathrm{\forall }t\in [2,8]\\ \\ T(t)& =& {\vartheta }_{up}+273,\mathrm{\forall }t\in [8,t_{end}]\\ \\ x& \in & {𝒳},u\in {𝒰},p\in P.\end{array}$

State variables

Name	Symbol	Initial value ($t_0$)
Molar number 1	$n_1(t)$	$n_1(t_0)=n_{a1}$
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 $p_j=1,j=1,\dots ,5$

Optimization/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]

Control function

Name	Symbol	Time interval	Value interval	Initial value
Initial molar number 1	$\vartheta (t)$	[math>\t_{0}</math>,2]	[20.0,100.0]	20.0
Initial molar number 1	$\vartheta (t)$	[2,8]	[20.0,100.0]	20.0
Initial molar number 1	$\vartheta (t)$	[8,math>\t_{end}</math>]	[20.0,100.0]	20.0

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
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002