Denbigh Reaction - Revision history

RobertLampel: /* Parameters */

2026-01-29T12:56:24Z

Parameters

← Older revision		Revision as of 12:56, 29 January 2026
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	\|-		\|-
	\|<math>E_1</math>		\|<math>E_1</math>
	\|<math>10^3</math>		\|<math>3 \cdot 10^3</math>
	\|-		\|-
	\|<math>E_2</math>		\|<math>E_2</math>
	\|<math>10^7</math>		\|<math>6 \cdot 10^3</math>
	\|-		\|-
	\|<math>E_3</math>		\|<math>E_3</math>
	\|<math>10</math>		\|<math>3 \cdot 10^3</math>
	\|-		\|-
	\|<math>E_4</math>		\|<math>E_4</math>
	\|<math>~~10^{-3}~~</math>		\|<math>0</math>
	\|-		\|-
	\|<math>k_1^*</math>		\|<math>k_1^*</math>
	\|<math>~~3 \cdot~~ 10^3</math>		\|<math>10^3</math>
	\|-		\|-
	\|<math>k_2^*</math>		\|<math>k_2^*</math>
	\|<math>~~6 \cdot~~ 10^3</math>		\|<math>10^7</math>
	\|-		\|-
	\|<math>k_3^*</math>		\|<math>k_3^*</math>
	\|<math>~~3 \cdot~~ 10^3</math>		\|<math>10</math>
	\|-		\|-
	\|<math>k_4^*</math>		\|<math>k_4^*</math>
	\|<math>0</math>		\|<math>10^{-3}</math>
	\|-		\|-
	\|<math>t_f</math>		\|<math>t_f</math>
	\|<math>10^3</math>		\|<math>10^3</math>
	\|}		\|}



	== Reference Solutions ==		== Reference Solutions ==

RobertLampel: /* Reference Solutions */

2025-11-28T13:46:17Z

Reference Solutions

← Older revision		Revision as of 13:46, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Denbigh.png\| States and discretized control for a local optimum.		Image:Denbigh.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2025-08-22T14:18:26Z

Mathematical formulation

← Older revision		Revision as of 14:18, 22 August 2025
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	\quad \dot{x_3}(t) & = & k_3(t) \cdot x_2(t),\\		\quad \dot{x_3}(t) & = & k_3(t) \cdot x_2(t),\\
	\quad k_i(t) & = & k_i^* \cdot \exp\left( \frac{-E_i}{T(t)} \right), \ i=1,\ldots,4, \\		\quad k_i(t) & = & k_i^* \cdot \exp\left( \frac{-E_i}{T(t)} \right), \ i=1,\ldots,4, \\
	\quad T(t) & \in & [273, 415] \ \quad \forall t \in [0,t_f] \\		\quad T(t) & \in & [273, 415] \ \quad \forall t \in [0,t_f], \\
	\quad x(0) &=& (1, 0, 0)^T		\quad x(0) &=& (1, 0, 0)^T.
	\end{array}		\end{array}
	</math>		</math>

RobertLampel at 11:10, 21 August 2025

2025-08-21T11:10:01Z

← Older revision		Revision as of 11:10, 21 August 2025
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	}}		}}

	The '''~~Mountain Car~~ problem''' s based on the system of chemical reactions initially considered by Denbigh [[#Denbigh \| [1]]], which was also studied by Aris [[#Aris \| [2]]] and more recently by Luus [[#Luus \| [3]]]:		The '''Denbigh Reaction problem''' is based on the system of chemical reactions initially considered by Denbigh [[#Denbigh \| [1]]], which was also studied by Aris [[#Aris \| [2]]] and more recently by Luus [[#Luus \| [3]]]:
	<p>		<p>
	<math>		<math>

RobertLampel: /* References */

2025-08-21T11:09:37Z

References

← Older revision		Revision as of 11:09, 21 August 2025
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	[[Category:MIOCP]]		[[Category:MIOCP]]
	~~[[Category:Path-constrained arc]]~~

RobertLampel: /* References */

2025-08-21T11:05:30Z

References

← Older revision		Revision as of 11:05, 21 August 2025
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	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:~~Bang bang~~]]		[[Category:Path-constrained arc]]

RobertLampel at 11:04, 21 August 2025

2025-08-21T11:04:20Z

← Older revision		Revision as of 11:04, 21 August 2025
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	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>.		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 control functions is given by a [[:Path-constrained arcs\|path-constrained arc]].		Its dynamics are given by a three-dimensional [[:Category:ODE model\|ODE model]]. The optimal control functions is given by a [[:Category:Path-constrained arcs\|path-constrained arc]].

	== Mathematical formulation ==		== Mathematical formulation ==

RobertLampel at 11:03, 21 August 2025

2025-08-21T11:03:56Z

← Older revision		Revision as of 11:03, 21 August 2025
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	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>.		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~~.		Its dynamics are given by a three-dimensional [[:Category:ODE model\|ODE model]]. The optimal control functions is given by a [[:Path-constrained arcs\|path-constrained arc]].

	== Mathematical formulation ==		== Mathematical formulation ==

RobertLampel at 11:02, 21 August 2025

2025-08-21T11:02:03Z

← Older revision		Revision as of 11:02, 21 August 2025
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	}}		}}

	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)~~:		The '''Mountain Car problem''' s based on the system of chemical reactions initially considered by Denbigh [[#Denbigh \| [1]]], which was also studied by Aris [[#Aris \| [2]]] and more recently by Luus [[#Luus \| [3]]]:
	<p>		<p>
	<math>		<math>
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	== References ==		== References ==
	<span id="Aris">[1]</span> Rutherford Aris. The Optimal Design of Chemical Reactors A Study in Dynamic Programming. Academic Press, London, 1961.<br>		<span id="Denbigh">[1]</span> Kenneth Denbigh, Chemical Reactor Theory an Introduction, Cambridge University Press, London, 1965.<br>
	<span id="Luus">[2]</span> Rein Luus, Iterative Dynamic Programming. CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics, New York, 2000~~.<br>~~		<span id="Aris">[2]</span> Rutherford Aris. The Optimal Design of Chemical Reactors A Study in Dynamic Programming. Academic Press, London, 1961.<br>
	~~<span id="Denbigh">[3]</span> Kenneth Denbigh, Chemical Reactor Theory an Introduction, Cambridge University Press, London, 1965~~.<br>		<span id="Luus">[3]</span> Rein Luus, Iterative Dynamic Programming. CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics, New York, 2000.<br>
	<span id="Tomlab">[4]</span> Tomlab optimization: https://tomopt.com/docs/propt/tomlab_propt030.php<br>		<span id="Tomlab">[4]</span> Tomlab optimization: https://tomopt.com/docs/propt/tomlab_propt030.php<br>

	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:Bang bang]]		[[Category:Bang bang]]

RobertLampel: /* Miscellaneous and Further Reading */

2025-08-21T10:59:57Z

Miscellaneous and Further Reading

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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	This formulation and a detailed description can be found in [[#~~tomlab~~\|[1]]].		This formulation and a detailed description can be found in [[#Tomlab\|[1]]].

	== References ==		== References ==