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| − | [[Category:benchmark]] |
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| − | [[Category:nonlinear system]] |
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| − | [[Category:parametric system]] |
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| − | [[Category:time invariant]] |
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| − | [[Category:geometric parameters]] |
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| − | [[Category:two parameters]] |
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| − | [[Category:first order system]] |
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| − | ==Description of physical model== |
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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. The continuous case will be |
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| − | addressed in the benchmark ''[[SMB]]'', and here we focus on the discontinuous mode -- batch chromatography. |
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| − | The principle of batch elution chromatography for the binary separation is shown schematically in Fig.1 below. During the injection period |
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| − | <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. |
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| − | With the help of the mobile phase, the feed mixture then flows through the column. Since the solutes to be separated exhibit different |
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| − | adsorption affinities to the stationary phase, they move at different velocities in the column, and thus separate from each other |
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| − | when exiting the column. At the column outlet, component A is collected between cutting points <math>t_1</math> and <math>t_2</math>, |
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| − | 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> |
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| − | are determined by a minimum concentration threshold that the detector can resolve. The positions of <math>t_2</math> and <math>t_3</math> |
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| − | are determined by the purity specifications imposed on the products. After the cycle period <math>t_{cyc}</math>, the injection is repeated. |
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| − | The feed flow-rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables. |
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| − | By properly choosing them, the process can achieve the desired performance criterion, such as production rate, while respecting |
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| − | the product specifications (e.g., purity, recovery yield). |
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| − | The batch chromatography can be described as the following convection-diffusion system, |
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| − | <math> |
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| − | \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, |
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| − | \qquad \qquad [1] |
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| − | </math> |
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| − | <math> |
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| − | \frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in (0,\;L), \qquad \qquad [2] |
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| − | </math> |
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| − | |||
| − | where <math>c_z,q_z</math> are concentrations of component <math>z</math> in the liquid |
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| − | and solid phase, and <math>q_z^{Eq}</math> is the adsorption equilibrium concentration defined as |
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| − | |||
| − | <math> |
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| − | 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, |
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| − | </math> |
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| − | The boundary conditions for Eq. [1] are specified by the Danckwerts relations: |
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| − | |||
| − | <math> |
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| − | 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, |
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| − | </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 |
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| − | |||
| − | <math> |
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| − | c^{in}_i = \begin{array}{cc} c^F_i, &\text{if} t \le t_{inj};}\\ 0, &\text{if} t > t_{inj}.} \end{array} |
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| − | <math> |
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| − | |||
| − | 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: |
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| − | <math> |
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| − | c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. |
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| − | </math> |
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