Batch Chromatography

Description of the process

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, $D_i=\frac{uL}{Pe}$ the axial dispersion coefficient and $Pe$ the Péclet number. The adsorption rate is modeled by the LDF approximation:

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

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

$q_i^{Eq}=\frac{H_{i,1}{c}_i}{1+\sum _{j=A,B}{K}_{j,1}{c}_j}+\frac{H_{i,2}{c}_i}{1+\sum _{j=A,B}{K}_{j,2}{c}_j},i=A,B,[3]$

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. [1] are specified by the Danckwerts relations:

$D_i{\frac{\partial c_i}{\partial z}|}_{z=0}=u({c_i|}_{z=0}-c_i^{in}),{\frac{\partial c_i}{\partial z}|}_{z=L}=0,[4]$

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

$c_i^{in}=\{\begin{array}{cc}{c}_i^F,& \text{if }t\le t_{inj};\\ 0,& \text{if }t>t_{inj}.\end{array}$

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

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

Discretization

In this model, the feed flow-rate $Q$ and injection period $t_{inj}$ are often considered as the operating variables, and will be parametrized as $\mu =(Q,t_{inj})$. Using the finite volume discretization, we can get the full order model,

$\{\begin{array}{l}{\mathbf{𝐀}\mathbf{𝐜}}_i^{k+1}={\mathbf{𝐁}\mathbf{𝐜}}_i^k+{\mathbf{𝐛}}_i^k-\frac{1-ϵ}{ϵ}\mathrm{\Delta }t{\mathbf{𝐡}}_i^k,\\ {\mathbf{𝐪}}_i^{k+1}={\mathbf{𝐪}}_i^k+\mathrm{\Delta }t{\mathbf{𝐡}}_i^k,\end{array}$

with ${\mathbf{𝐡}}_i^k=K_{m,i}({\mathbf{𝐪}}_i^{Eq}-{\mathbf{𝐪}}_i^k),i=A,B$, $k\in \mathbb{𝕂}=\{0,1,\cdots ,K\}$, the index for the time instance, and $\mathrm{\Delta }t$ is the time step determined by the stability condition. Here, the bold capital $\mathbf{𝐀},\mathbf{𝐁}$ are constant matrices, and the bold ${\mathbf{𝐜}}_i,{\mathbf{𝐪}}_i,{\mathbf{𝐛}}_i,{\mathbf{𝐡}}_i\in X_{{𝒩}}$. Note that ${\mathbf{𝐛}}_i,{\mathbf{𝐡}}_i$ are parameter-dependent.