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 dynamics of the batch chromatographics 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 $i$ ( $i = A, B,$ ) in the liquid phase can be written as:

$\frac{\partial c_{i}}{\partial t} + \frac{1 - ϵ}{ϵ} \frac{\partial q_{i}}{\partial t} + u \frac{\partial c_{i}}{\partial z} - D_{i} \frac{\partial^{2} c_{i}}{\partial z^{2}} = 0, z \in (0, L), [1]$

where $c_{i}$ and $q_{i}$ are the concentrations of solute $i$ in the liquid and solid phases, respectively, $u$ the interstitial liquid velocity, $ϵ$ the column porosity, $t$ the time coordinate, $z$ the axial coordinate along the column, $L$ the column length, $D_{i} = u L / P e$ the axial dispersion coefficient and $P e$ the P\'{e}clet number. The adsorption rate is modeled by the LDF approximation:

$\frac{\partial q_{i}}{\partial t} = K_{m, i} (q_{i}^{E q} - q_{i}), z \in [0, L], [2]$

where $K_{m, i}$ is the mass-transfer coefficient of component $i$ and $q_{i}^{E q}$ is the adsorption equilibrium concentration calculated by the isotherm equation for component $i$ . Here the bi-Langmuir isotherm model is used to describe the adsorption equilibrium:

$q_{i}^{E q} = \frac{H_{i, 1} c_{i}}{1 + \sum_{j = A, B} K_{j, 1} c_{j}} + \frac{H_{i, 2} c_{i}}{1 + \sum_{j = A, B} K_{j, 2} c_{j}}, i = A, B, [3]$ where $H_{i, 1}$ and $H_{i, 2}$ are the Henry constants, and $K_{j, 1}$ and $K_{j, 2}$ the thermodynamic coefficients.

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

$D_{i} {\frac{\partial c_{i}}{\partial z} |}_{z = 0} = u ({c_{i} |}_{z = 0} - c_{i}^{i n}), {\frac{\partial c_{i}}{\partial z} |}_{z = L} = 0, [4]$

where $c_{i}^{i n}$ 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 (unknown function "\begin{array}"): {\displaystyle c^{in}_i=\left \{ \begin{array}{cc} c^F_i, &\text{if } t \le t_{inj};\\ 0, &\text{if } t > t_{inj}. \right . \end{array} }

where $c_{i}^{F}$ is the feed concentration for component $i$ and $t_{i n j}$ is the injection period. In addition, the column is assumed unloaded initially:

$c_{i} (t = 0, z) = q_{i} (t = 0, z) = 0, z \in [0, L], i = A, B . [5]$

Discretization