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:

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

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

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

where $K_{m,i}$ is the mass-transfer coefficient of component $i$ and $q^{Eq}_i$ 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:

$q^{Eq}_i=\frac{H_{i,1}\,c_i}{1+\sum_{j=A,B}K_{j,1}\,c_j}+\frac{H_{i,2}\,c_i}{1+\sum_{j=A,B}K_{j,2}\,c_j},\; i=A,B, \qquad [3]$ 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:

$D_i\left.\frac{\partial c_i}{\partial z}\right|_{z=0} = u\,(\left.c_i\right|_{z=0}-c^{in}_i), \quad\quad \left.\frac{\partial c_i}{\partial z}\right|_{z=L}=0, \qquad [4]$

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

where $c^F_i$ is the feed concentration for component $i$ and $t_{inj}$ is the injection period. In addition, the column is assumed unloaded initially:

$c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B. \qquad [5]$