View source for 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:

<math>
\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} 
</math>

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

<math>
 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}}) )
</math>

Total mass: 

<math>
 m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4 
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:
 
<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p) 
\end{array} 
</math>

== Constraints ==

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


<math>
\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} 
</math>

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

<p>
<math>
\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} 
</math>

== Optimum Experimental Design Problem ==

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


<math>
\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 & \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, \quad \forall \, t \in [t_0,2] \\
 \\
 T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273  , \quad \forall \, t \in [2,8] \\
 \\
 T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
 \\
& & x  \in  \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array} 
</math>
</p>



{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}


{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|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.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\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} 
</math>

== 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


[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]