# 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 (initial mass):

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

and to the mass of active ingredient content (fraction of active substances):

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

## 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:

$\begin{array}{cll} \displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex] \mbox{s.t.} \\ \dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\ \\ h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\ \\ \dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\ \\ \dot{F}(t) & = & \sum\limits_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\ \\ 0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\ \\ 0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\ \\ \vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \right. \\ & & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\ \dot{z}^i(t) & = & w^i(t) \\ z(0) & = & 0 \\ w^i(t) &\in& [0,1] \\ 0 & \le & 4 - z^i(t_f). \\ \end{array}$

 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 $n_4(t)$ $n_4(t_0) = n_{a4}$

 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

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

 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 $p_j = 1, \ j =1, \dots, 5$

 Name Symbol Interval Initial value Exp 1 Initial value Exp 2 Initial value Exp 3 Initial value Exp 4 Initial molar number 1 $n_{a1}$ [0,10.0] 1.0 1.0 1.0 1.0 Initial molar number 2 $n_{a2}$ [0,10.0] 1.0 1.0 1.0 1.0 Initial molar number 4 $n_{a4}$ [0.4,9.0] 2.0 2.0 2.0 2.0 Concentration of the catalyst $c_{kat}$ [0,10.0] 0.0 1.0 2.0 3.0
 Name Symbol Time interval Value interval Initial value Exp 1 Initial value Exp 2 Initial value Exp 3 Initial value Exp 4 Initial molar number 1 $\vartheta(t)$ $[t_0,2]$ [20.0,100.0] 20.0 60.0 40.0 20.0 Initial molar number 1 $\vartheta(t)$ $[2,8]$ [20.0,100.0] 20.0 60.0 40.0 20.0 Initial molar number 1 $\vartheta(t)$ $[8,t_{end}]$ [20.0,100.0] 20.0 60.0 40.0 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