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:

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

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

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

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. [1] are specified by the Danckwerts relations:

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

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

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

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:

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

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,

$\left\{ \begin{array}{ll} \mathbf{ A c}_i^{k+1} = \mathbf{Bc}_i^{k} + \mathbf b_i^k - \frac{1-\epsilon}{\epsilon} \Delta t \mathbf h_i^k,\\ \mathbf{q}_i^{k+1} = \mathbf{q}_i^{k} + \Delta t \mathbf h_i^k, \end{array} \right .$ with $\mathbf h_i^k = K_{m,i} (\mathbf q^{Eq}_i - \mathbf q_i^k),\; i=A,B$, $k\in \mathbb K = \{0,1,\cdots,K\}$, the index for the time instance, and $\Delta t$ is the time step determined by the stability condition. Here, the bold capital $\mathbf {A,B}$ are constant matrices, and the bold $\mathbf {c}$ and

$\mathbf {q}$ are the solution vector in the high $\mathcal N-$dimensional space.