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:

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

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:

$\begin{array}{cll} \displaystyle \min_{x, G, F, u} && trace(F^{-1} (t_{end})) \\[1.5ex] \mbox{s.t.} \\ \dot{x}(t) & = & f(x(t), u(t),p), \\ \\ h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\ \\ \dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\ \\ \dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\ \\ 0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\ \\ 0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\ \\ 0 & = & \vartheta_{lo}, \quad \forall \, t \in [t_0,2] \\ \\ 0 & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) , \quad \forall \, t \in [2,8] \\ \\ 0 & = & \vartheta_{up}, \quad \forall \, t \in [8,t_{end}] \\ \\ x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P. \end{array}$

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

$\begin{array}{llll} t_0 = 0 & & & \\ t_{end} = 20 & & & \\ t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20. \end{array}$

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:

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

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