Batch Chromatography: Difference between revisions

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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:

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

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

$q_{i}^{E q} = \frac{H_{i, 1} c_{i}}{1 + \sum_{j = A, B} K_{j, 1} c_{j}} + \frac{H_{i, 2} c_{i}}{1 + \sum_{j = A, B} K_{j, 2} c_{j}}, i = A, B,$ 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:

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

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{cases}"): {\displaystyle c^{in}_i= \begin{cases} c^F_i, &\text{if $t \le t_{inj}$;}\\ 0, &\text{if $t > t_{inj}$.} \end{cases} } 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:

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

Discretization

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+[[Category:benchmark]]
+[[Category:nonlinear system]]
+[[Category:parametric system]]
+[[Category:time invariant]]
+[[Category:two parameters]]
+[[Category:first order system]]
+==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 <math>i</math> (<math>i=A,B,</math>)
+in the liquid phase can be written as:
+<math>
+\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),
+</math>
+where <math>c_i</math> and <math>q_i</math> are the concentrations of solute $i$ in the liquid and solid phases, respectively, $u$ the interstitial liquid velocity, <math>\epsilon</math> the column porosity, <math>t</math> the time coordinate, <math>z</math> the axial coordinate along the column, <math>L</math> the column length, <math>D_i=uL/Pe</math> the axial dispersion coefficient and <math>Pe</math> the P\'{e}clet number. The adsorption rate is modeled by the LDF approximation:
+<math>
+\frac{\partial q_i}{\partial t} = K_{m,i}\,(q^{Eq}_i-q_i), \qquad z\in[0,\;L],
+</math>
+where <math>K_{m,i}</math> is the mass-transfer coefficient of component <math>i</math> and <math>q^{Eq}_i</math> is the adsorption equilibrium concentration calculated by the isotherm equation for component <math>i</math>. Here the bi-Langmuir isotherm model is used to describe the adsorption equilibrium:
+<math>
+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,
+</math>
+where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.
+The boundary conditions for Eq. [] are specified by the Danckwerts relations:
+<math>
+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,
+</math>
+where <math>c^{in}_i</math> is the concentration of component <math>i</math> at the inlet of the column. A rectangular injection is assumed for the system and thus
+<math>
+c^{in}_i=
+\begin{cases}
+  c^F_i,  &\text{if $t \le t_{inj}$;}\\
+,  &\text{if $t >   t_{inj}$.}
+\end{cases}
+</math>
+where <math>c^F_i</math> is the feed concentration for component <math>i</math> and <math>t_{inj}</math> is the injection period. In addition, the column is assumed unloaded initially:
+<math>
+c_i(t=0,z)=q_i(t=0,z)=0,\quad z\in[0,\;L],\;i=A,B.
+</math>
+==Discretization==