Batch reactor

Batch reactor
State dimension:	1
Differential states:	2
Continuous control functions:	1
Path constraints:	1
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 $x_2(t_f) = -0.611715$.

• Reference solution plots
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  Optimal states $x_1(t)$ and $x_2(t)$.

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  Optimal control $u(t)$.

Source Code

Model descriptions are available in

• JuMP code at Batch reactor (JuMP)

References

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