Difference between revisions of "Diels-Alder Reaction Experimental Design"

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  k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{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} \ - \ \frac{1}{T_{ref}}) )
+
  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}}) )
 
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T = \vartheta + 273
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T(t) = \vartheta (t) + 273
 
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  \displaystyle \min_{x, G, F, u} & trace(F^{-1} (t_{t_f})) \\[1.5ex]
 
  \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\\
 
  \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 = g(x(t_o),x(t_f),p) \\
 
  & 0  \ge  c(x,u,p), \forall \, t \in I\\
 
  & 0  \ge  c(x,u,p), \forall \, t \in I\\

Revision as of 12:26, 4 December 2015

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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]

Parameters

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