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:

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): {\begin{array}{cll}\displaystyle \min _{{x,\ G,\ F,\ Tc,\ n_{{a1}},\ n_{{a2}},\ n_{{a4}},\ c_{{kat}},\ \vartheta (t)}}&&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&\leq &n_{{a1}}\ \cdot \ M_{1}\ +\ n_{{a2}}\ \cdot \ M_{2}\ +\ n_{{a4}}\ \cdot \ M_{4}\\\\10&\geq &n_{{a1}}\ \cdot \ M_{1}\ +\ n_{{a2}}\ \cdot \ M_{2}\ +\ n_{{a4}}\ \cdot \ M_{4}\\\\0.1&\leq &{\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&\geq &{\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)&=&\vartheta _{{lo}}+273,\quad \forall \,t\in [t_{0},2]\\\\\vartheta (t)&=&\vartheta _{{lo}}+{\frac {t-2}{6}}(\vartheta _{{up}}-\vartheta _{{lo}})+273,\quad \forall \,t\in [2,8]\\\\\vartheta (t)&=&\vartheta _{{up}}+273,\quad \forall \,t\in [8,t_{{end}}]\\\\\vartheta (t)&=&{\begin{cases}x(n),\\x(n-1)\\x(n-1)\end{cases}}&&x\in {\mathcal {X}},\,u\in {\mathcal {U}},\,p\in P.\end{array}}

Failed to parse (lexing error): \begin{cases} 2x^{2} & \text{f"ur } x \textless 4 \\ 2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\ 3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27 \end{cases}

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

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.

Fixed 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	$\vartheta(t)$	$[t_0,2]$	[20.0,100.0]	20.0
Initial molar number 1	$\vartheta(t)$	$[2,8]$	[20.0,100.0]	20.0
Initial molar number 1	$\vartheta(t)$	$[8,t_{end}]$	[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