Difference between revisions of "Batch reactor"

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|nx        = 2
 
|nx        = 2
 
|nu        = 1
 
|nu        = 1
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|nc        = 2
 
|nre      = 2
 
|nre      = 2
 
}}
 
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<p>
 
<p>
 
<math>
 
<math>
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\begin{array}{llclr}
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\begin{array}{llcl}
 
  \displaystyle \max_{x, u} & x_2(t_f)  \\[1.5ex]
 
  \displaystyle \max_{x, u} & x_2(t_f)  \\[1.5ex]
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  \mbox{s.t.} & \dot{x}_1(t) & = & -k_1 x_1^2 \qquad & \forall t \in [t_0, t_f],  \\
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  \mbox{s.t.} & \dot{x}_1 & = & -k_1 x_1^2.\\
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  & \dot{x}_2(t) & = & k_1 x_1^2 - k_2 x_2 \qquad & \forall t \in [t_0, t_f],  \\
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  & \dot{x}_2 & = & k_1 x_1^2 - k_2 x_2,\\
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  & k_1 & = & 4000 \; e^{(-2500/T(t))} \qquad & \forall t \in [t_0,t_f], \\
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  & k_1 & = & 4000 \; e^{(-2500/T(t))}, \\
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  & k_2 & = & 620000 \; e^{(-5000/T(t))} \qquad & \forall t \in [t_0,t_f], \\[1.5ex]
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  & k_2 & = & 620000 \; e^{(-5000/T(t))}, \\[1.5ex]
 
  & x(0) &=& (1, 0)^T, \\
 
  & x(0) &=& (1, 0)^T, \\
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  & T(t) &\in&  [298, 398]  \qquad & \forall t \in [t_0, t_f].
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  & T(t) &\in&  [298, 398].
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
 
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<math> x_1(t) </math> and <math> x_2(t) </math> stand for the concentrations of A and B at timepoint <math> t </math> respectively. The control function <math> T(t) </math> represents the temperature.
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<math> x_1(t) </math> and <math> x_2(t) </math> represent the concentrations of A and B at timepoint <math> t </math> respectively. The control function <math> T(t) </math> represents the temperature.
  
 
== Parameters ==
 
== Parameters ==
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This solution was computed using JuMP with a collocation method and 300 discretization points. The differential equations were solved using the explicit Euler Method. The source code can be found at [[Batch reactor (JuMP)]].
 
This solution was computed using JuMP with a collocation method and 300 discretization points. The differential equations were solved using the explicit Euler Method. The source code can be found at [[Batch reactor (JuMP)]].
  
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The optimal objective value of the problem is <math> x_2(t_f) = -0.611715 </math>.
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The optimal objective value of the problem is x2(tf) = 0.611715.
  
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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Model descriptions are available in
 
Model descriptions are available in
 +
* [[:Category:Gekko | GEKKO Python code]] at [[Batch reactor (GEKKO)]]
 
* [[:Category: Julia/JuMP | JuMP code]] at [[Batch reactor (JuMP)]]
 
* [[:Category: Julia/JuMP | JuMP code]] at [[Batch reactor (JuMP)]]
  
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[[Category:MIOCP]]
 
[[Category:MIOCP]]
 
[[Category:ODE model]]
 
[[Category:ODE model]]
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[[Category:Chemical engineering]]

Latest revision as of 14:42, 11 April 2019

Batch reactor
State dimension: 1
Differential states: 2
Continuous control functions: 1
Path constraints: 2
Interior point equalities: 2


This batch reactor problem describes the consecutive reaction of some substance A via substance B into a desired product C.

The system is interacted with via the control function T(t) which stands for the temperature. The goal is to produce as much of substance B (which can then be converted into product C) as possible within the time limit.

Mathematical formulation

The optimal control problem is given by

\begin{array}{llcl} \displaystyle \max_{x, u} & x_2(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = & -k_1 x_1^2.\\ & \dot{x}_2 & = & k_1 x_1^2 - k_2 x_2,\\ & k_1 & = & 4000 \; e^{(-2500/T(t))}, \\ & k_2 & = & 620000 \; e^{(-5000/T(t))}, \\[1.5ex] & x(0) &=& (1, 0)^T, \\ & T(t) &\in& [298, 398]. \end{array}

x_1(t) and x_2(t) represent the concentrations of A and B at timepoint t respectively. The control function T(t) represents the temperature.

Parameters

The starting time and end time are given by [t_0, t_f] = [0, 1] .

Reference Solutions

This solution was computed using JuMP with a collocation method and 300 discretization points. The differential equations were solved using the explicit Euler Method. The source code can be found at Batch reactor (JuMP).

The optimal objective value of the problem is x2(tf) = 0.611715.

  • Reference solution plots

  • Optimal states x_1(t) and x_2(t).

  • Optimal control u(t).

Source Code

Model descriptions are available in

References

The problem can be found in the Tomlab PROPT guide or in the Dynopt guide.

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