Diels-Alder Reaction Experimental Design - Revision history

RobertLampel: /* Optimum Experimental Design Problem */

2025-10-09T08:51:55Z

Optimum Experimental Design Problem

← Older revision		Revision as of 08:51, 9 October 2025
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	== Optimum Experimental Design Problem ==		== Optimum Experimental Design Problem ==

	The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:		The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{\mathrm{cat}}, E_1, E_{\mathrm{cat}}, \lambda</math>. So, we have to solve the following optimum experimental design problem:


	<math>		<math>
	\begin{array}{cll}		\begin{array}{cll}
	\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{~~kat~~}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]		\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{\mathrm{cat}}^i,\ \vartheta(t)^i} && \mathrm{trace}(F^{-1} (t_{\mathrm{end}})) \\[1.5ex]
	\mbox{s.t.} \\		\mbox{s.t.} \\
	\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\		\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
	\\		\\
	h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\		h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{\mathrm{tot}}} \ \cdot \ 100 \\
	\\		\\
	\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\		\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
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	\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} & t \in [t_0,2] \\		\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} & t \in [t_0,2] \\
	\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) & t \in [2,8] \\		\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) & t \in [2,8] \\
	\vartheta_{up} & t \in [8,t_{end}]		\vartheta_{up} & t \in [8,t_{\mathrm{end}}]
	\end{array} \right. \\		\end{array} \right. \\
	& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\		& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
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	\|-		\|-
	\|Reference temperature		\|Reference temperature
	\|<math>T_{ref}</math>		\|<math>T_{\mathrm{ref}}</math>
	\|293		\|293
	\|-		\|-
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	\|-		\|-
	\|Steric factor		\|Steric factor
	\|<math>k_{~~kat~~}</math>		\|<math>k_{\mathrm{cat}}</math>
	\|<math>p_2 \cdot 0.10</math>		\|<math>p_2 \cdot 0.10</math>
	\|-		\|-
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	\|-		\|-
	\|Activation energie		\|Activation energie
	\|<math>E_{~~kat~~}</math>		\|<math>E_{\mathrm{cat}}</math>
	\|<math>p_4 \cdot 40000</math>		\|<math>p_4 \cdot 40000</math>
	\|-		\|-
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	\|-		\|-
	\|Concentration of the catalyst		\|Concentration of the catalyst
	\|<math>c_{~~kat~~}</math>		\|<math>c_{\mathrm{cat}}</math>
	\|[0,6.0]		\|[0,6.0]
	\|0.0		\|0.0
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	\|Temperature 3		\|Temperature 3
	\|<math>\vartheta(t)</math>		\|<math>\vartheta(t)</math>
	\|<math>[8,t_{end}]</math>		\|<math>[8,t_{\mathrm{end}}]</math>
	\|[20.0,100.0]		\|[20.0,100.0]
	\|20.0		\|20.0
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	\begin{array}{llll}		\begin{array}{llll}
	t_0 = 0 & & & \\		t_0 = 0 & & & \\
	t_{end} = 20 & & & \\		t_{\mathrm{end}} = 20 & & & \\
	t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.		t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
	\end{array}		\end{array}

RobertLampel: /* Model Formulation */

2025-10-09T08:44:58Z

Model Formulation

← Older revision		Revision as of 08:44, 9 October 2025
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	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\		\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{\mathrm{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_{\mathrm{tot}}}, \\
	& & \\		& & \\
	\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\		\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{\mathrm{tot}}} \\
	& & \\		& & \\
	\dot{n_4}(t) &=& 0		\dot{n_4}(t) &=& 0
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	<math>		<math>
	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}}) )		k = k_1 \ \cdot \ \exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{\mathrm{ref}}}) ) \ + \ k_{\mathrm{cat}} \ \cdot \ c_{\mathrm{cat}} \ \cdot \ \exp(-\lambda \ \cdot \ t) \ \cdot \ \exp( - \frac{E_{\mathrm{cat}}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{\mathrm{ref}}}) )
	</math>		</math>

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	<math>		<math>
	m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4		m_{\mathrm{tot}} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
	</math>		</math>

RobertLampel: /* Optimum Experimental Design Problem */

2025-10-09T08:34:48Z

Optimum Experimental Design Problem

← Older revision		Revision as of 08:34, 9 October 2025
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	\|Concentration of the catalyst		\|Concentration of the catalyst
	\|<math>c_{kat}</math>		\|<math>c_{kat}</math>
	\|[0,10.0]		\|[0,6.0]
	\|0.0		\|0.0
	\|1.0		\|1.0

RobertLampel: /* Optimum Experimental Design Problem */

2025-10-09T08:22:34Z

Optimum Experimental Design Problem

← Older revision		Revision as of 08:22, 9 October 2025
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	0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\		0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
	\\		\\
	\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} ~~+ 273~~ & t \in [t_0,2] \\		\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} & t \in [t_0,2] \\
	\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) ~~+ 273~~ & t \in [2,8] \\		\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) & t \in [2,8] \\
	\vartheta_{up} ~~+ 273~~ & t \in [8,t_{end}]		\vartheta_{up} & t \in [8,t_{end}]
	\end{array} \right. \\		\end{array} \right. \\
	& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\		& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\

RobertLampel: /* Optimum Experimental Design Problem */

2025-10-02T08:44:43Z

Optimum Experimental Design Problem

← Older revision		Revision as of 08:44, 2 October 2025
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	\|Initial value Exp 4		\|Initial value Exp 4
	\|-		\|-
	\|~~Initial molar number~~ 1		\|Temperature 1
	\|<math>\vartheta(t)</math>		\|<math>\vartheta(t)</math>
	\|<math>[t_0,2]</math>		\|<math>[t_0,2]</math>
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	\|20.0		\|20.0
	\|-		\|-
	\|~~Initial molar number 1~~		\|Temperature 2
	\|<math>\vartheta(t)</math>		\|<math>\vartheta(t)</math>
	\|<math>[2,8]</math>		\|<math>[2,8]</math>
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	\|20.0		\|20.0
	\|-		\|-
	\|~~Initial molar number 1~~		\|Temperature 3
	\|<math>\vartheta(t)</math>		\|<math>\vartheta(t)</math>
	\|<math>[8,t_{end}]</math>		\|<math>[8,t_{end}]</math>

SebastianSager at 12:49, 24 July 2025

2025-07-24T12:49:19Z

← Older revision		Revision as of 12:49, 24 July 2025
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			[[Category:MIOCP]]
	[[Category:Optimum Experimental Design]]		[[Category:Optimum Experimental Design]]
	[[Category:ODE model]]		[[Category:ODE model]]

FelixMueller: /* Source Code */

2016-07-29T14:17:49Z

Source Code

← Older revision		Revision as of 14:17, 29 July 2016
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	== Source Code ==		== Source Code ==

	* The VPLAN code using [[:Category: VPLAN \| VPLAN ~~code~~]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]		* The VPLAN code using [[:Category: VPLAN \| VPLAN]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

	== References ==		== References ==

FelixJost: /* Optimum Experimental Design Problem */

2016-02-01T13:04:07Z

Optimum Experimental Design Problem

← Older revision		Revision as of 13:04, 1 February 2016
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	\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\		\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
	\\		\\
	\dot{F}(t) & = & \~~sum_~~{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\		\dot{F}(t) & = & \sum\limits_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
	\\		\\
	0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\		0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\

FelixJost: /* Optimum Experimental Design Problem */

2016-02-01T13:00:57Z

Optimum Experimental Design Problem

← Older revision		Revision as of 13:00, 1 February 2016
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	\|Symbol		\|Symbol
	\|Interval		\|Interval
	\|Initial value (Exp 1)		\|Initial value Exp 1
	\|Initial value (Exp 2)		\|Initial value Exp 2
	\|Initial value (Exp 3)		\|Initial value Exp 3
	\|Initial value (Exp 4)		\|Initial value Exp 4
	\|-		\|-
	\|Initial molar number 1		\|Initial molar number 1
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	\|Time interval		\|Time interval
	\|Value interval		\|Value interval
	\|Initial value (Exp 1)		\|Initial value Exp 1
	\|Initial value (Exp 2)		\|Initial value Exp 2
	\|Initial value (Exp 3)		\|Initial value Exp 3
	\|Initial value (Exp 4)		\|Initial value Exp 4
	\|-		\|-
	\|Initial molar number 1		\|Initial molar number 1

FelixJost: /* Optimum Experimental Design Problem */

2016-02-01T13:00:16Z

Optimum Experimental Design Problem

← Older revision		Revision as of 13:00, 1 February 2016
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	\|Symbol		\|Symbol
	\|Interval		\|Interval
	\|Initial value Exp 1		\|Initial value (Exp 1)
	\|Initial value Exp 2		\|Initial value (Exp 2)
	\|Initial value Exp 3		\|Initial value (Exp 3)
	\|Initial value Exp 4		\|Initial value (Exp 4)
	\|-		\|-
	\|Initial molar number 1		\|Initial molar number 1