Denbigh Reaction: Difference between revisions

Denbigh Reaction
State dimension:	1
Differential states:	3
Discrete control functions:	1

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Revision as of 09:31, 21 August 2025

The Mountain Car problem s based on the system of chemical reactions initially considered by Denbigh (1958), which was also studied by Aris (1960) and more recently by Luus (1994):

$\begin{aligned} A + B & \to X \\ X & \to Q \\ X & \to Y \\ A + X & \to P \end{aligned}$

where $X$ is an intermediate, $Y$ is the desired product, and $P$ and $Q$ are waste products. The optimal control problem is to find $T (t)$ (the temperature of the reactor as a function of time) so that the yield of $Y$ is maximized at the end of the given batch time $t_{f}$ .

Its dynamics are given by a three-dimensional ODE model. The optimal integer control functions exhibits a bang bang structure.

Mathematical formulation

$\begin{array}{lll} \max_{u} & x_{3} (t_{f}) \\ subject to \\ \dot{x_{1}} (t) & = & - k_{1} (t) \cdot x_{1} (t) - k_{2} (t) \cdot x_{1} (t), \\ \dot{x_{2}} (t) & = & k_{1} (t) \cdot x_{1} (t) - k_{3} (t) + k_{4} (t) \cdot x_{2} (t), \\ \dot{x_{3}} (t) & = & k_{3} (t) \cdot x_{2} (t), \\ k_{i} (t) & = & k_{i}^{*} \cdot \exp (\frac{- E_{i}}{T (t)}), i = 1, \dots, 4, \\ T (t) & \in & [273, 415] \forall t \in [0, t_{f}] \\ x (0) & = & (1, 0, 0)^{T} \end{array}$

Parameters

Parameters
Symbol	Value
$E_{1}$	$1 0^{3}$
$E_{2}$	$1 0^{7}$
$E_{3}$	$10$
$E_{4}$	$1 0^{- 3}$
$k_{1}^{*}$	$3 \cdot 1 0^{3}$
$k_{2}^{*}$	$6 \cdot 1 0^{3}$
$k_{3}^{*}$	$3 \cdot 1 0^{3}$
$k_{4}^{*}$	$0$
$t_{f}$	$1 0^{3}$

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and discretized control for a local optimum.

Miscellaneous and Further Reading

This formulation and a detailed description can be found in [1].

References

[1] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html
[2] Andrew William Moore. Efficient memory-based learning for robot control. Technical Report UCAM-CL-TR-209, University of Cambridge, Computer Laboratory, November 1990. URL: https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-209.pdf, doi:10.48456/tr-209.
[3] Alexey A Melnikov, Adi Makmal, and Hans J Briegel. Projective simulation applied to the grid-world and the mountain-car problem. arXiv preprint arXiv:1405.5459, 2014.

@@ Line 16: / Line 16: @@
 </math>
 </p>
-where X is an intermediate, Y is the desired product, and P and Q are waste products.
+where <math>X</math> is an intermediate, <math>Y</math> is the desired product, and <math>P</math> and <math>Q</math> are waste products. The optimal control problem is to find <math>T(t)</math> (the temperature of the reactor as a function of time) so that the yield of <math>Y</math> is maximized at the end of the given batch time <math>t_f</math>.
 Its dynamics are given by a three-dimensional [[:Category:ODE model|ODE model]]. The optimal integer control functions exhibits a [[:Category:Bang bang|bang bang]] structure.