Batch Chromatography - Revision history

Schuetz2 at 08:59, 11 June 2025

2025-06-11T08:59:17Z

← Older revision		Revision as of 08:59, 11 June 2025
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	title = {Batch Chromatography},		title = {Batch Chromatography},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/index.php/Batch_Chromatography}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/index.php/Batch_Chromatography}</nowiki>,
	year = {2018}		year = {2018}
	}		}

Mlinaric: Edit math, text fixes

2023-08-29T01:42:22Z

Edit math, text fixes

← Older revision		Revision as of 01:42, 29 August 2023
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	<figure id="fig:bach">[[File:Fig_BatchChrom.jpg\|490px\|thumb\|right\|<caption>Sketch of a batch chromatographic process for the separation of A and B.</caption>]]</figure>		<figure id="fig:bach">[[File:Fig_BatchChrom.jpg\|490px\|thumb\|right\|<caption>Sketch of a batch chromatographic process for the separation of A and B.</caption>]]</figure>

	Preparative liquid chromatography as a crucial separation and purification tool has been widely employed in food, fine chemical and pharmaceutical industries. Chromatographic separation at industry scale can be operated either discontinuously or in a continuous mode.		Preparative liquid chromatography as a crucial separation and purification tool has been widely employed in the food, fine chemical, and pharmaceutical industries. Chromatographic separation at the industry scale can be operated either discontinuously or in a continuous mode.
	The continuous case will be discussed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- [[wikipedia:Affinity_chromatography#Batch_and_column_setups\|batch chromatography]].		The continuous case will be discussed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- [[wikipedia:Affinity_chromatography#Batch_and_column_setups\|batch chromatography]].
	The principle of the batch chromatographic process for the binary separation is shown in Fig. 1.		The principle of the batch chromatographic process for the binary separation is shown in Fig. 1.
	During the injection period <math>t_{inj}</math>, a mixture of products A and B is injected at the inlet of the column packed with a suitable stationary phase.		During the injection period <math>t_{inj}</math>, a mixture of products A and B is injected at the inlet of the column packed with a suitable stationary phase.
	With the help of the mobile phase, the feed mixture flows through the column.		With the help of the mobile phase, the feed mixture flows through the column.
	Since the to-be-separated solutes exhibit different adsorption affinities to the stationary phase, they move at different velocities,		Since the to-be-separated solutes exhibit different adsorption affinities to the stationary phase, they move at different velocities
	and thus separate from each other when exiting the column.		and thus separate from each other when exiting the column.
	At the column outlet, component A is collected between <math>t_1</math> and <math>t_2</math>, and component B is collected between <math>t_3</math> and <math>t_4</math>.		At the column outlet, component A is collected between <math>t_1</math> and <math>t_2</math>, and component B is collected between <math>t_3</math> and <math>t_4</math>.
	Here the positions of <math>t_1</math> and <math>t_4</math> are determined by a minimum concentration threshold that the detector can resolve, and the positions of <math>t_2</math> and <math>t_3</math> are determined by the purity specifications imposed on the products.		Here the positions of <math>t_1</math> and <math>t_4</math> are determined by a minimum concentration threshold that the detector can resolve, and the positions of <math>t_2</math> and <math>t_3</math> are determined by the purity specifications imposed on the products.
	After the cycle period <math>t_{cyc}:=t_4-t_1</math>, the injection is repeated.		After the cycle period <math>t_{cyc}:=t_4-t_1</math>, the injection is repeated.
	The feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables.		The feed flow rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables.
	By properly choosing them, the process can achieve the desired performance criterion, such as production rate, while respecting the product specifications (e.g., purity, recovery yield).		By properly choosing them, the process can achieve the desired performance criterion, such as production rate, while respecting the product specifications (e.g., purity, recovery yield).

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	:<math>		:<math>
	\frac{\partial c_i}{\partial t}+\frac{1-\epsilon}{\epsilon}\frac{\partial q_i}{\partial t}+u\frac{\partial c_i}{\partial z}-D_i\frac{\partial^2 c_i}{\partial z^2}=0, \qquad z\in(0,\;L), \qquad (1)		\frac{\partial c_i}{\partial t}
			+ \frac{1 - \epsilon}{\epsilon} \frac{\partial q_i}{\partial t}
			+ u \frac{\partial c_i}{\partial z}
			- D_i \frac{\partial^2 c_i}{\partial z^2}
			= 0, \qquad
			z \in (0, L), \qquad (1)
	</math>		</math>

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	:<math>		:<math>
	\frac{\partial q_i}{\partial t} = \kappa_{i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L],		\frac{\partial q_i}{\partial t}
			= \kappa_{i} \left(q^{Eq}_i - q_i\right), \qquad
			z \in [0, L],
	</math>		</math>

Line 39:		Line 46:

	:<math>		:<math>
	q^{Eq}_i=\frac{H_{i,1}\,c_i}{1+\sum_{j=A,B}K_{j,1}\,c_j}+\frac{H_{i,2}\,c_i}{1+\sum_{j=A,B}K_{j,2}\,c_j},\; i=A,B,		q^{Eq}_i
			= \frac{H_{i,1}\,c_i}{1 + \sum_{j = A, B}K_{j,1}\,c_j}
			+ \frac{H_{i,2}\,c_i}{1 + \sum_{j = A, B}K_{j,2}\,c_j},\;
			i = A, B,
	</math>		</math>

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	:<math>		:<math>
	D_i\left.\frac{\partial c_i}{\partial z}\right\|_{z=0} = u\,(\left.c_i\right\|_{z=0}-c^{in}_i), \~~quad\quad~~ \left.\frac{\partial c_i}{\partial z}\right\|_{z=L}=0,		D_i \left.\frac{\partial c_i}{\partial z}\right\|_{z = 0}
			= u \left(\left.c_i\right\|_{z=0} - c^{in}_i\right), \qquad
			\left.\frac{\partial c_i}{\partial z}\right\|_{z = L}
			= 0,
	</math>		</math>

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	:<math>		:<math>
	c^{in}_i=~~\left \{~~ \begin{~~array~~}~~{cc}~~ c^F_i, &\text{if } t \le t_{inj};\\		c^{in}_i
	0, &\text{if } t > t_{inj}.		=
	\end{~~array~~}		\begin{cases}
	~~\right .~~		c^F_i, & \text{if } t \le t_{inj}, \\
			0, & \text{if } t > t_{inj}.
			\end{cases}
	</math>		</math>

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	:<math>		:<math>
	c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B.		c_i(t = 0, z)
			= q_i(t = 0,z)
			= 0, \quad
			z \in [0, L], \;
			i = A, B.
	</math>		</math>

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	:<math>		:<math>
	\left\{		\left\{
	\begin{~~array~~}{ll}		\begin{aligned}
	\mathbf{ A c}_i^{k+1} = \mathbf{Bc}_i^{k} + d_i^k		\mathbf{A} \mathbf{c}_i^{k+1}
	- \frac{1-\epsilon}{\epsilon} \Delta t \mathbf ~~h_i~~^k,\\		&= \mathbf{B} \mathbf{c}_i^{k}
	\mathbf{q}_i^{k+1} = \mathbf{q}_i^{k} + \Delta t \mathbf ~~h_i~~^k,		+ d_i^k
	\end{~~array~~}		- \frac{1 - \epsilon}{\epsilon} \Delta t \mathbf{h}_i^k, \\
	\right .		\mathbf{q}_i^{k + 1}
			&= \mathbf{q}_i^{k}
			+ \Delta t \mathbf{h}_i^k,
			\end{aligned}
			\right.
	</math>		</math>

	where <math> \mathbf {c}_i^k ,\mathbf ~~q_i~~^k \in \mathbb{R}^{\mathcal N}, i=A,B</math> are the solution vectors of <math> c_i</math> and <math>q_i</math> at the time instance <math>t=t^k, k = 0,1,\~~cdots~~,K, </math> respectively.		where <math>\mathbf{c}_i^k, \mathbf{q}_i^k \in \mathbb{R}^{\mathcal N}, i = A, B</math> are the solution vectors of <math>c_i</math> and <math>q_i</math> at the time instance <math>t = t^k, k = 0, 1, \ldots, K</math>, respectively.
	The time step <math>\Delta t </math> is determined by the stability condition.		The time step <math>\Delta t </math> is determined by the stability condition.
	The equation <math> \mathbf h_i^k = \kappa_{i} (\mathbf q^{Eq}_i - \mathbf q_i^k) </math>, is time- and parameter-dependent, the boldface <math>\mathbf{A,B}</math> are constant matrices. As a result, it is a nonlinear parametric system.		The equation <math> \mathbf h_i^k = \kappa_{i} (\mathbf q^{Eq}_i - \mathbf q_i^k) </math>, is time- and parameter-dependent, the boldface <math>\mathbf{A,B}</math> are constant matrices. As a result, it is a nonlinear parametric system.
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	:<math>		:<math>
	\left \{		\left\{
	\begin{~~array}{llll~~}		\begin{aligned}
	\hat{A}_{c_z} {a}_{c_z}^{n+1} =\hat{ B}_{c_z} {a}_{c_z}^{n} +		\hat{A}_{c_z} {a}_{c_z}^{n+1}
	d_0^n\hat{d}_{c_z} - \frac{1-\epsilon}{\epsilon} \Delta t \hat{H}_{c_z}\beta_z^n,\\		&= \hat{B}_{c_z} {a}_{c_z}^{n}
	{a}_{q_z}^{n+1} = {a}_{q_z}^{n} + \Delta t \hat{H}_{q_z} \beta_z^n,\\		+ d_0^n \hat{d}_{c_z}
	\end{~~array~~}		- \frac{1 - \epsilon}{\epsilon} \Delta t \hat{H}_{c_z} \beta_z^n, \\
	\right .		{a}_{q_z}^{n + 1}
			&= {a}_{q_z}^{n}
			+ \Delta t \hat{H}_{q_z} \beta_z^n,
			\end{aligned}
			\right.
	</math>		</math>

Himpe at 13:00, 25 August 2022

2022-08-25T13:00:22Z

← Older revision		Revision as of 13:00, 25 August 2022
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	[[Category:PDE]]		[[Category:PDE]]
	[[Category:nonlinear]]		[[Category:nonlinear]]
	[[Category:~~parametric 2-5 parameters~~]]		[[Category:Parametric]]

Himpe: Fix CrossRefs

2021-08-27T07:28:06Z

Fix CrossRefs

← Older revision		Revision as of 07:28, 27 August 2021
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	Preparative liquid chromatography as a crucial separation and purification tool has been widely employed in food, fine chemical and pharmaceutical industries. Chromatographic separation at industry scale can be operated either discontinuously or in a continuous mode.		Preparative liquid chromatography as a crucial separation and purification tool has been widely employed in food, fine chemical and pharmaceutical industries. Chromatographic separation at industry scale can be operated either discontinuously or in a continuous mode.
	The continuous case will be discussed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- [[wikipedia:Affinity_chromatography#Batch_and_column_setups\|batch chromatography]].		The continuous case will be discussed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- [[wikipedia:Affinity_chromatography#Batch_and_column_setups\|batch chromatography]].
	The principle of the batch chromatographic process for the binary separation is shown in ~~<xr id="fig:bach"/>~~.		The principle of the batch chromatographic process for the binary separation is shown in Fig. 1.
	During the injection period <math>t_{inj}</math>, a mixture of products A and B is injected at the inlet of the column packed with a suitable stationary phase.		During the injection period <math>t_{inj}</math>, a mixture of products A and B is injected at the inlet of the column packed with a suitable stationary phase.
	With the help of the mobile phase, the feed mixture flows through the column.		With the help of the mobile phase, the feed mixture flows through the column.
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	<figure id="fig:cabfr">[[File:cabfrA.png\|480px\|thumb\|right\|<caption>Concentrations at the outlet of the column.</caption>]]</figure>		<figure id="fig:cabfr">[[File:cabfrA.png\|480px\|thumb\|right\|<caption>Concentrations at the outlet of the column.</caption>]]</figure>

	~~<xr id="fig:cabfr"/>~~ shows the concentrations at the outlet of the column at a given parameter <math>\mu= (0.1018, 1.3487)</math>, which show that the ROM (<math>N=46, M=151</math>) reproduces the dynamics of the full order model (<math>\mathcal N=1000</math>).		Fig. 2 shows the concentrations at the outlet of the column at a given parameter <math>\mu= (0.1018, 1.3487)</math>, which show that the ROM (<math>N=46, M=151</math>) reproduces the dynamics of the full order model (<math>\mathcal N=1000</math>).

	==Citation==		==Citation==

Himpe: Add citation section and various other fixes.

2018-11-14T14:56:51Z

Add citation section and various other fixes.

Show changes

Himpe: Small changes

2018-02-16T10:12:59Z

Small changes

← Older revision		Revision as of 10:12, 16 February 2018
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	in the liquid phase can be written as:		in the liquid phase can be written as:

	<math>		:<math>
	\frac{\partial c_i}{\partial t}+\frac{1-\epsilon}{\epsilon}\frac{\partial q_i}{\partial t}+u\frac{\partial c_i}{\partial z}-D_i\frac{\partial^2 c_i}{\partial z^2}=0, \qquad z\in(0,\;L), \qquad (1)		\frac{\partial c_i}{\partial t}+\frac{1-\epsilon}{\epsilon}\frac{\partial q_i}{\partial t}+u\frac{\partial c_i}{\partial z}-D_i\frac{\partial^2 c_i}{\partial z^2}=0, \qquad z\in(0,\;L), \qquad (1)
	</math>		</math>

	where <math>c_i</math> and <math>q_i</math> are the concentrations of solute <math>i</math> in the liquid and solid phases, respectively, <math>u</math> the interstitial liquid velocity, <math>\epsilon</math> the column porosity, <math>t</math> the time coordinate, <math>z</math> the axial coordinate along the column, <math>L</math> the column length, <math>D_i=\frac{uL}{Pe}</math> the axial dispersion coefficient and <math>Pe</math> the Péclet number. The adsorption rate is modeled by the LDF approximation:		where <math>c_i</math> and <math>q_i</math> are the concentrations of solute <math>i</math> in the liquid and solid phases, respectively, <math>u</math> the interstitial liquid velocity, <math>\epsilon</math> the column porosity, <math>t</math> the time coordinate, <math>z</math> the axial coordinate along the column, <math>L</math> the column length, <math>D_i=\frac{uL}{Pe}</math> the axial dispersion coefficient and <math>Pe</math> the [[wikipedia:Péclet_number\|Péclet number]]. The adsorption rate is modeled by the LDF approximation:

	<math>		:<math>
	\frac{\partial q_i}{\partial t} = \kappa_{i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L],		\frac{\partial q_i}{\partial t} = \kappa_{i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L],
	</math>		</math>
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	where <math>\kappa_{i}</math> is the mass-transfer coefficient of component <math>i</math> and <math>q^{Eq}_i</math> is the adsorption equilibrium concentration calculated by the isotherm equation for component <math>i</math>. Here the bi-Langmuir isotherm model is used to describe the adsorption equilibrium:		where <math>\kappa_{i}</math> is the mass-transfer coefficient of component <math>i</math> and <math>q^{Eq}_i</math> is the adsorption equilibrium concentration calculated by the isotherm equation for component <math>i</math>. Here the bi-Langmuir isotherm model is used to describe the adsorption equilibrium:

	<math>		:<math>
	q^{Eq}_i=\frac{H_{i,1}\,c_i}{1+\sum_{j=A,B}K_{j,1}\,c_j}+\frac{H_{i,2}\,c_i}{1+\sum_{j=A,B}K_{j,2}\,c_j},\; i=A,B,		q^{Eq}_i=\frac{H_{i,1}\,c_i}{1+\sum_{j=A,B}K_{j,1}\,c_j}+\frac{H_{i,2}\,c_i}{1+\sum_{j=A,B}K_{j,2}\,c_j},\; i=A,B,
	</math>		</math>
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	The boundary conditions for (1) are specified by the Danckwerts relations:		The boundary conditions for (1) are specified by the Danckwerts relations:

	<math>		:<math>
	D_i\left.\frac{\partial c_i}{\partial z}\right\|_{z=0} = u\,(\left.c_i\right\|_{z=0}-c^{in}_i), \quad\quad \left.\frac{\partial c_i}{\partial z}\right\|_{z=L}=0,		D_i\left.\frac{\partial c_i}{\partial z}\right\|_{z=0} = u\,(\left.c_i\right\|_{z=0}-c^{in}_i), \quad\quad \left.\frac{\partial c_i}{\partial z}\right\|_{z=L}=0,
	</math>		</math>
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	where <math>c^{in}_i</math> is the concentration of component <math>i</math> at the inlet of the column. A rectangular injection is assumed for the system and thus		where <math>c^{in}_i</math> is the concentration of component <math>i</math> at the inlet of the column. A rectangular injection is assumed for the system and thus

	<math>		:<math>
	c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\		c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
	0, &\text{if } t > t_{inj}.		0, &\text{if } t > t_{inj}.
Line 50:		Line 50:
	Here <math>c^F_i</math> is the feed concentration for component <math>i</math> and <math>t_{inj}</math> is the injection period. In addition, the column is assumed unloaded initially:		Here <math>c^F_i</math> is the feed concentration for component <math>i</math> and <math>t_{inj}</math> is the injection period. In addition, the column is assumed unloaded initially:

	<math>		:<math>
	c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B.		c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B.
	</math>		</math>
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	In this model, the feed volumetric flow-rate <math>Q</math> and the injection period <math>t_{inj}</math> are considered as the operating parameters, and denoted as the parameter <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we get the full order model (FOM) as follows,		In this model, the feed volumetric flow-rate <math>Q</math> and the injection period <math>t_{inj}</math> are considered as the operating parameters, and denoted as the parameter <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we get the full order model (FOM) as follows,

	<math>		:<math>
	\left\{		\left\{
	\begin{array}{ll}		\begin{array}{ll}
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	the ROM can be formulated as:		the ROM can be formulated as:

	<math>		:<math>
	\left \{		\left \{
	\begin{array}{llll}		\begin{array}{llll}
Line 105:		Line 105:

	<ref name="guiochon2006"> G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, <span class="plainlinks">[http://books.google.com/books/about/Fundamentals_of_preparative_and_nonlinea.html?id=UjZRAAAAMAAJ Fundamentals of Preparative and Nonlinear Chromatography]</span>, 2nd Edition, Academic Press, 2006.</ref>		<ref name="guiochon2006"> G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, <span class="plainlinks">[http://books.google.com/books/about/Fundamentals_of_preparative_and_nonlinea.html?id=UjZRAAAAMAAJ Fundamentals of Preparative and Nonlinear Chromatography]</span>, 2nd Edition, Academic Press, 2006.</ref>


	<ref name="barrault04"> M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, "<span class="plainlinks">[http://dx.doi.org/10.1016/j.crma.2004.08.006 An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations]</span>", C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.</ref>		<ref name="barrault04"> M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, "<span class="plainlinks">[http://dx.doi.org/10.1016/j.crma.2004.08.006 An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations]</span>", C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.</ref>


	<ref name="haasdonk08"> B. Haasdonk and M. Ohlberger, "<span class="plainlinks">[http://www.agh.ians.uni-stuttgart.de/publications/2008/HO08b Reduced basis method for finite volume approximations of parameterized linear evolution equations]</span>", Mathematical Modeling and Numerical Analysis, 42 (2008), 277-302.</ref>		<ref name="haasdonk08"> B. Haasdonk and M. Ohlberger, "<span class="plainlinks">[http://www.agh.ians.uni-stuttgart.de/publications/2008/HO08b Reduced basis method for finite volume approximations of parameterized linear evolution equations]</span>", Mathematical Modeling and Numerical Analysis, 42 (2008), 277-302.</ref>

Zhangy: /* Generation of ROM */

2014-06-22T07:09:09Z

Generation of ROM

← Older revision		Revision as of 07:09, 22 June 2014
Line 74:		Line 74:

	The reduced order model (ROM) can be obtained by reduced bases methods, which is applicable for nonlinear parametric systems, see		The reduced order model (ROM) can be obtained by reduced bases methods, which is applicable for nonlinear parametric systems, see
	[[Reduced_Basis_PMOR_method\|Reduced Basis PMOR method]]. For parametrized time-dependent problems, the reduced basis ~~can~~ often ~~be obtained~~ by using POD-Greedy algorithm. Notice that the nonlinear functions <math>\mathbf h_i, i=A, B</math> can be approximated by the empirical interpolation method <ref name="barrault04"/>, such that the ROM can be obtained ~~more~~ efficiently by the offline-online ~~technique~~.		[[Reduced_Basis_PMOR_method\|Reduced Basis PMOR method]]. For parametrized time-dependent problems, the reduced basis is often generated by using the POD-Greedy algorithm <ref name="haasdonk08"/>. Notice that the nonlinear functions <math>\mathbf h_i, i=A, B</math> can be approximated by the empirical interpolation method <ref name="barrault04"/>, such that the ROM can be obtained efficiently by the strategy of offline-online decomposition.

	Assume <math>W_z</math> is the collateral reduced basis (CRB) for the nonlinear operator <math>h_z</math>, and <math>V_{c_z},V_{q_z}</math> are the RB for the field variables <math>c_z</math> and <math>q_z</math>, respectively. Applying Galerkin projection and empirical operator interpolation ~~<ref name="haasdonk08"/>~~,		Assume <math>W_z</math> is the collateral reduced basis (CRB) for the nonlinear operator <math>h_z</math>, and <math>V_{c_z},V_{q_z}</math> are the reduced bases for the field variables <math>c_z</math> and <math>q_z</math>, respectively. Applying Galerkin projection and empirical operator interpolation,
	the ROM ~~for the FOM~~ can be formulated as:		the ROM can be formulated as:

	<math>		<math>
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	<math>\hat{H}_{c_z}:= V_{c_z}^TW_z ,</math>		<math>\hat{H}_{c_z}:= V_{c_z}^TW_z ,</math>
	<math>\hat{H}_{q_z}:= V_{q_z}^TW_z </math> are the reduced matrices, <math>e_1:=(1,0,\cdots,0)</math>. <math>\beta_z^n \in \mathbb R^M</math> is the coefficients of the CRB <math>W_z</math> for the empirical interpolation.		<math>\hat{H}_{q_z}:= V_{q_z}^TW_z </math> are the reduced matrices, <math>e_1:=(1,0,\cdots,0)</math>. <math>\beta_z^n \in \mathbb R^M</math> is the coefficients of the CRB <math>W_z</math> for the empirical interpolation.



	==Data==		==Data==

Zhangy: /* Discretization */

2014-06-22T07:05:40Z

Discretization

← Older revision		Revision as of 07:05, 22 June 2014
Line 57:		Line 57:

	==Discretization==		==Discretization==
	In this model, the feed volumetric flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are considered as the operating parameters, and denoted as the parameter <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we ~~can~~ get the full order model (FOM) as follows,		In this model, the feed volumetric flow-rate <math>Q</math> and the injection period <math>t_{inj}</math> are considered as the operating parameters, and denoted as the parameter <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we get the full order model (FOM) as follows,

	<math>		<math>
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	</math>		</math>

	where <math> \mathbf {c}_i^k ,\mathbf q_i^k \in \mathbb{R}^{\mathcal N}, i=A,B</math> are the solution ~~vector~~ of <math> c_i</math> and <math>q_i</math> at the time instance <math>t=t^k, k = 0,1,\cdots,K, </math> respectively. The time step <math>\Delta t </math> is determined by the stability condition. <math> \mathbf h_i^k = \kappa_{i} (\mathbf q^{Eq}_i - \mathbf q_i^k) </math>, is time- and parameter-dependent, the boldface <math>\mathbf {A,B}</math> are constant matrices. As a result, it is a nonlinear parametric system.		where <math> \mathbf {c}_i^k ,\mathbf q_i^k \in \mathbb{R}^{\mathcal N}, i=A,B</math> are the solution vectors of <math> c_i</math> and <math>q_i</math> at the time instance <math>t=t^k, k = 0,1,\cdots,K, </math> respectively. The time step <math>\Delta t </math> is determined by the stability condition. <math> \mathbf h_i^k = \kappa_{i} (\mathbf q^{Eq}_i - \mathbf q_i^k) </math>, is time- and parameter-dependent, the boldface <math>\mathbf {A,B}</math> are constant matrices. As a result, it is a nonlinear parametric system.

	==Generation of ROM==		==Generation of ROM==

Himpe: book link

2013-06-11T23:59:19Z

book link

← Older revision		Revision as of 23:59, 11 June 2013
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	<references>		<references>

	<ref name="guiochon2006"> G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.</ref>		<ref name="guiochon2006"> G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, <span class="plainlinks">[http://books.google.com/books/about/Fundamentals_of_preparative_and_nonlinea.html?id=UjZRAAAAMAAJ Fundamentals of Preparative and Nonlinear Chromatography]</span>, 2nd Edition, Academic Press, 2006.</ref>

Zhangy: /* Data */

2013-06-05T18:17:16Z

Data

← Older revision		Revision as of 18:17, 5 June 2013
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	==Data==		==Data==
	<figure id="fig:cabfr">[[File:cabfrA.png\|~~300px~~\|thumb\|right\|<caption>Concentrations at the outlet of the column.</caption>]]</figure>		<figure id="fig:cabfr">[[File:cabfrA.png\|480px\|thumb\|right\|<caption>Concentrations at the outlet of the column.</caption>]]</figure>

	The <xr id="fig:cabfr"/> are the concentrations at the outlet of the column at a given parameter <math>\mu= (0.1018,1.3487)</math>, which show that the ROM (<math>N=46, M=151</math>) reproduces the dynamics of the FOM (<math>\mathcal N=1000</math>).		The <xr id="fig:cabfr"/> are the concentrations at the outlet of the column at a given parameter <math>\mu= (0.1018,1.3487)</math>, which show that the ROM (<math>N=46, M=151</math>) reproduces the dynamics of the FOM (<math>\mathcal N=1000</math>).