Diels-Alder Reaction Experimental Design

This page can now be filled with content.

Model Formulation

Differential equation system:

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

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:

$T(t) = \vartheta (t) + 273$

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): {\begin{array}{cl}\displaystyle \min _{{x,G,F,u}}&trace(F^{{-1}}(t_{{t_{f}}}))\\[1.5ex]{\mbox{s.t.}}&{\dot {x}}=f(x,u,p,t),\forall \,t\in I\\&{\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_{2}}}(t)=\ \ k\cdot {\frac {n_{1}(t)\ \cdot \ n_{2}(t)}{m_{{tot}}}}&0=g(x(t_{o}),x(t_{f}),p)\\&0\geq c(x,u,p),\forall \,t\in I\\&0=h(x,u,p),\forall \,t\in I\\&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	$n_3(t)$	$n_3(t_0) = n_{a3}$

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

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 3	$n_{a3}$	[0.4,9.0]
Concentration of the catalyst	$c_{kat}$	[0.0,6.0]
Initial molar number 1	$\vartheta(t)$	[20.0,100.0]