MOR Wiki - User contributions [en]

User:Suzhou

2012-11-26T15:25:51Z

Suzhou:

Suzhou Li

Max Planck Institute for Dynamics of Complex Technical Systems

Sandtorstr. 1

39106 Magdeburg

Germany

Phone: +49 391 6110 367

Email: suzhou@mpi-magdeburg.mpg.de

www: http://www.mpi-magdeburg.mpg.de/mpcsc/suzhou

User:Suzhou

2012-11-26T15:25:29Z

Suzhou:

Suzhou Li

Max Planck Institute for Dynamics of Complex Technical Systems

Sandtorstr. 1

39106 Magdeburg

Germany

Phone: +49 391 6110 367
E-Mail: suzhou@mpi-magdeburg.mpg.de
www: http://www.mpi-magdeburg.mpg.de/mpcsc/suzhou

User:Suzhou

2012-11-26T15:24:52Z

Suzhou: Created page with 'Suzhou Li Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Magdeburg Germany Phone: +49 391 6110 367 E-Mail: suzhou@mpi-magdeburg.mpg.de www: h…'

Suzhou Li
Max Planck Institute for Dynamics of Complex Technical Systems
Sandtorstr. 1
39106 Magdeburg
Germany

Phone: +49 391 6110 367
E-Mail: suzhou@mpi-magdeburg.mpg.de
www: http://www.mpi-magdeburg.mpg.de/mpcsc/suzhou

Batch Chromatography

2012-11-22T11:45:09Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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 -- batch chromatography. The schematic diagram for the binary separation is shown below. During the injection period <math>t_{inj}</math>, a mixture consisting of 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 then flows through the column. Since the solutes to be separated exhibit different adsorption affinities to the stationary phase, they move at different velocities in the column, and thus separate from each other when exiting the column. At the column outlet, component A is collected between cutting points <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. 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}</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. 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).

[[File:Fig_BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are chosen as the operating parameters, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

'' [[User:Suzhou|Suzhou Li]]''

Batch Chromatography

2012-11-22T11:42:51Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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 -- batch chromatography. The schematic diagram for the binary separation is shown below. During the injection period <math>t_{inj}</math>, a mixture consisting of 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 then flows through the column. Since the solutes to be separated exhibit different adsorption affinities to the stationary phase, they move at different velocities in the column, and thus separate from each other when exiting the column. At the column outlet, component A is collected between cutting points <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. 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}</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. 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).

[[File:Fig_BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

'' [[User:Suzhou|Suzhou Li]]''

Batch Chromatography

2012-11-22T11:33:04Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography. The schematic diagram for the binary separation is shown below.

[[File:Fig_BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

'' [[User:Suzhou|Suzhou Li]]''

Batch Chromatography

2012-11-22T11:28:43Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

[[File:Fig_BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

'' [[User:Suzhou|Suzhou Li]]''

Batch Chromatography

2012-11-22T11:22:53Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

[[File:Fig_BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''
'' [[User:Suzhou|Suzhou Li]]''

Batch Chromatography

2012-11-22T11:22:03Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

[[File:Fig_BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

File:Fig BatchChrom.jpg

2012-11-22T11:21:29Z

Suzhou:

Batch Chromatography

2012-11-22T11:20:01Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

[[File:Example.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

Batch Chromatography

2012-11-22T11:15:40Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

[[File:BatchChrom.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

File:BatchChrom.jpg

2012-11-22T11:14:46Z

Suzhou:

Batch Chromatography

2012-11-22T11:04:28Z

Suzhou:

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

[[File:Example.jpg]]

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

Batch Chromatography

2012-11-22T09:48:33Z

Suzhou: /* Reference */

[[Category:benchmark]]
[[Category:nonlinear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:two parameters]]
[[Category:first order system]]

==Description of the process==
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
addressed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- batch chromatography.

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation. In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>)
in the liquid phase can be written as:

<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]
</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:

<math>
\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L], \qquad [2]
</math>

where <math>K_{m,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>
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, \qquad [3]
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for Eq. [1] are specified by the Danckwerts relations:

<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, \qquad [4]
</math>

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>
c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\
0, &\text{if } t > t_{inj}.

\end{array}
\right .
</math>

where <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>
c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]
</math>

==Discretization==
In this model, the feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables, and will be
parametrized as <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we can get the full order model,

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

with <math>\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B</math>,
<math>k\in \mathbb K = \{0,1,\cdots,K\}</math>, the index for the time instance, and <math>\Delta t</math> is the time step
determined by the stability condition.
Here, the bold capital <math>\mathbf {A,B}</math> are constant matrices, and the bold
<math>\mathbf {c}_i,\mathbf{q}_i, \mathbf{b}_i,\mathbf{h}_i \in \mathbb{R}^{\mathcal N}.</math> Note that <math> \mathbf{b}_i,\mathbf{h}_i</math> are parameter-dependent.

==Generation of ROM==

For parametrized time-dependent problems, the reduced basis can be often obtained by using POD-Greedy algorithm, see
[[Reduced_Basis_PMOR_method]]. Notice that the empirical interpolation technique can be exploited to get a more
efficient ROM, due to the nonlinearity of <math>\mathbf h_i</math> resulting from the nonlinear isotherm function
<math>q_i^{Eq}</math>.

==Reference==

G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, 2nd Edition, Academic Press, 2006.

Contact information:

'' [[User:Zhangy|Yongjin Zhang]]''

Batch Chromatography

2012-11-21T14:30:47Z

Suzhou: