2013-05-29T18:33:13Z

Zhangy:

[[Category:benchmark]]
[[Category:PDE]]
[[Category:nonlinear]]
[[Category:parametric 2-5 parameters]]

==Description==
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 in <xr id="fig:bach"/>. 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).

<figure id="fig:gyro">[[File:Fig_BatchChrom.jpg|600px|thumb |<caption>Schematic representation of the gyroscope.</caption>]]</figure>

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

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>
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>

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 (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,
</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>

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

More details about the mathematical modeling for batch chromatography can be found in the literature <ref name="guiochon2006"/>.

==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 as follows,

<math>
\left\{
\begin{array}{ll}
\mathbf{ A c}_i^{k+1} = \mathbf{Bc}_i^{k} + 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>

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.

==Generation of ROM==

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.

==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="barrault04"> M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, "[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]", C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.</ref>

</references>

==Contact==

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

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