Batch Chromatography

Description of physical model

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 principle of batch elution chromatography for the binary separation is shown schematically in Fig.1 below. During the injection period $t_{inj}$, 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 $t_1$ and $t_2$, and component B is collected between $t_3$ and $t_4$. Here the positions of $t_1$ and $t_4$ are determined by a minimum concentration threshold that the detector can resolve. The positions of $t_2$ and $t_3$ are determined by the purity specifications imposed on the products. After the cycle period $t_{cyc}$, the injection is repeated. The feed flow-rate $Q$ and injection period $t_{inj}$ 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).

The batch chromatography can be described as the following convection-diffusion system,

$\frac{\partial c_i}{\partial t}+\frac{1-\epsilon}{\epsilon}\frac{\partial q_i}{\partial t}+u\frac{\partial c_i}{\partial z}-D_i\frac{\partial^2 c_i}{\partial z^2}=0, \qquad \qquad [1]$

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

where $c_z,q_z$ are concentrations of component $z$ in the liquid and solid phase, and $q_z^{Eq}$ is the adsorption equilibrium concentration defined as

$q^{Eq}_i=\frac{H_{i,1}\,c_i}{1+\sum_{j=A,B}K_{j,1}\,c_j}+\frac{H_{i,2}\,c_i}{1+\sum_{j=A,B}K_{j,2}\,c_j},\; i=A,B,$

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

$D_i\left.\frac{\partial c_i}{\partial z}\right|_{z=0} = u\,(\left.c_i\right|_{z=0}-c^{in}_i), \quad\quad \left.\frac{\partial c_i}{\partial z}\right|_{z=L}=0,$

where $c^{in}_i$ is the concentration of component $i$ at the inlet of the column. A rectangular injection is assumed for the system and thus

Failed to parse (syntax error): c^{in}_i = \begin{array}{cc} c^F_i, &\text{if} t \le t_{inj};}\\ 0, &\text{if} t > t_{inj}.} \end{array} <math> where <math>c^F_i<math> is the feed concentration for component <math>i and $t_{inj}$ is the injection period. In addition, the column is assumed unloaded initially:

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