Theoretical Analysis of Linear Two-Dimensional Single Component Reactive Liquid Chromatography Model

Theoretical Analysis of Linear Two-Dimensional Single Component Reactive Liquid Chromatography Model

Ugochukwu David Uche¹, Adetunji Kolawole Ilori², Toritsesan Brenda Batubo³

¹Mathematics Programme, National Mathematical Centre Abuja, Nigeria

²Department of Mathematics University of Abuja, Abuja, Nigeria

³Statistics Programme, National Mathematical Centre Abuja, Nigeria

⁴Department of Mathematics, University of Benin, Nigeria

DOI: https://doi.org/10.51584/IJRIAS.2024.910018

Received: 02 October 2024; Accepted: 17 September 2024; Published: 09 November 2024

ABSTRACT

Semi-analytical solutions for a linear two-dimensional reactive general rate model of liquid chromatography is presented in this work considering cylindrical geometry. The governing model equations comprises of a system of advection-diffusion-reaction partial differential equations together with an algebraic expression representing the isothermal relation. The solutions are obtained by the application of finite Hankel and Laplace transforms one after the other. Moments are numerically obtained and used to describe the shapes of the elution profiles. A high-resolution finite volume scheme is lastly applied to obtain approximate solutions of the governing model equations in order to verify the accuracy of the semi- analytical solutions. Several test cases are carried out and both the semi-analytical and numerical results are shown to have good agreements.

Key words: Reactive liquid chromatography, general rate model, semi-analytical solutions, numerical solutions.

INTRODUCTION

Liquid chromatography is a powerful technique widely used in the chemical and pharmaceutical industries for the separation and characterization of various substances. It is a method that allows the separation of a sample into its individual components based on their interactions with the mobile and stationary phases. The most common type of liquid chromatography is liquid-solid column chromatography, where a liquid mobile phase passes through a solid stationary phase to separate different components. Reactive liquid chromatography is a specialized technique that combines the principles of liquid chromatography with chemical reactions [1]. In this method, a reactive species is added to the mobile phase, which reacts with the analyte molecules as they pass through the column. This reaction can lead to changes in the analyte’s chemical properties, such as its charge or polarity, facilitating its separation from other components in the mixture.

Reactive liquid chromatography offers several advantages over traditional liquid chromatography methods. Firstly, it allows for selective and specific separation of analytes based on their reactivity. This is particularly useful when dealing with complex mixtures containing closely related compounds. By exploiting the differences in their reactivity, reactive liquid chromatography can achieve better resolution and separation efficiency [2, 3]. In this cur- rent work, we study the effects of the reaction coefficient on elution for a single component operating under an isothermal condition.

Mathematical modeling of reactive liquid chromatography plays a crucial role in under- standing and optimizing separation processes in the various industries where the techniques are applicable. This powerful analytical technique uses chemical reactions to enhance the separation of compounds, making it a valuable tool in pharmaceutical, environmental, and food analysis. By developing mathematical models, deeper insights into the complex interactions occurring between the solutes, solvents, and stationary phase in reactive liquid chromatography columns can be gained [4–6]. Various models which are characterized by their level of complexities and used to describe the reactive liquid chromatography process can be found in the literature [7–16]

This work extends the work recently done on one-dimensional reactive single component model by [17] to the two-dimensional model which considers both axial and radial concentration gradients. Analytical solutions of the governing model equations are derived by applying the Hankel and Laplace transformations successively together with the conventional solution technique for ODEs [18, 19]. These solutions are obtained for two kinds of boundary conditions namely Dirichlet and Danckwerts. It is not feasible to perform analytical inversions of the derived solutions because they contain complex functions. As a result, numerical Hankel and Laplace inversions are employed to convert the solutions back to their initial radial and time coordinates [20, 21]. In order to confirm the accuracy of the obtained solutions, numerical solutions are also acquired using a semi-discrete finite volume scheme (FVS) with high resolution, applied to the model governing equations [22, 23]. Test cases are carried out considering some kinetic parameters that influence the separation of solute in the column. The behavior of the chromatographic reactor is analyzed further by numerically calculating the moments of the concentration profile [24, 25].

The usefulness of the current two-dimensional model can be identified when at the time of solute injection at the column inlet, radial profile is introduced due to inexact injection, a non-homogeneously packed column and a larger column radius. These situations may occur in real life operations but are mostly neglected and thus, the application of a one- dimensional model. However, the current two-dimensional model can be a very reliable quantitative tool for more accurate predictions in liquid chromatography.

The article is further arranged in the following manner. In Section 2, a linear two- dimensional reactive general rate model of liquid chromatography is formulated and closed with the required initial and suggested boundary conditions. In Section 3, the analytical solutions for the two boundary conditions are obtained. Section 4 provides the test cases carried out and lastly in Section 5, the conclusion of the findings are given.

Linear two-dimensional reactive general rate model

An isothermal adsorption column filled with spherical particles of radius R_p is considered. Here, let t, z and r represent the time coordinate, axial coordinate along the column length and the radial coordinate along the column radius respectively.

Mass Balance Equations for the Mobile Phase of Adsorption

The solute is taken to travel in the z-direction by means of advection and axial dispersion, while being dispersed along the radius of the column. The mass balance equations for the mobile phase of adsorption are given as:

Document Content with Equations

The solute is taken to travel in the z-direction by means of advection and axial dispersion, while being dispersed along the radius of the column. The following injection types are assumed to project the radial mass transfer effects by dividing the inlet cross-section of the column into an inner cylindrical core and an outer annular ring with the introduction of a new parameter \( R_c \). This allows the solute injection through the inner core, the outer ring, or the whole cross-section.

If \( R_c \) is set equal to the radius of the column \( R_c \), then we have injection through the whole cross-section, representing the scenario where there are no radial gradients. Thus, the system can be represented by a simpler one-dimensional model (see [17]). The mass balance equations for the mobile phase of adsorption are given as:

$$ \frac{\partial c}{\partial t} = -u \frac{\partial c}{\partial z} + D_z \frac{\partial^2 c}{\partial z^2} + \frac{1}{r_p} \frac{\partial}{\partial r_p} \left( r_p D_r \frac{\partial c}{\partial r_p} \right) – k_e (c – c_p) \tag{1} $$

where:

• \( c \): Concentration in the bulk of fluid
• \( c_p \): Concentration in the particle pores
• \( u \): Interstitial velocity
• \( D_z \): Axial dispersion coefficient
• \( F = \frac{1 – \epsilon}{\epsilon} \): Phase ratio with \( \epsilon \) as the external porosity
• \( D_r \): Radial dispersion coefficient
• \( k_e \): External mass transfer coefficient
• \( r_p \): Radial coordinate of spherical particles of radius \( R_p \)

Mass Balance Equation Inside Particle Pores

The corresponding mass balance equation inside the pores of the particles is given as:

$$ \frac{\partial c_p}{\partial t} = D_p \frac{\partial^2 c_p}{\partial r_p^2} – v c_p \tag{2} $$

where:

• \( D_p \): Pore diffusivity
• \( \epsilon_p \): Internal porosity
• \( v \): Reaction rate constant of the solid phase

Equilibrium Linear Adsorption Isotherm

The Langmuir isotherm used in this study is expressed as:

$$ c_p = a c \tag{3} $$

where \( a \) denotes the linear adsorption or Henry’s constant.

Dimensionless Variables

The following dimensionless variables are introduced:

$$ \eta = \frac{z}{L}, \quad \xi = \frac{r_p}{R_p}, \quad \zeta = \frac{k_e L}{u}, \quad \omega = \frac{v L^2}{D_p} \tag{4} $$

Applying these dimensionless variables to equations (1) and (2) results in the following system of equations:

$$ \frac{\partial c}{\partial \tau} = -\frac{\partial c}{\partial \eta} + \frac{1}{Pe_z} \frac{\partial^2 c}{\partial \eta^2} + \frac{1}{\xi} \frac{\partial}{\partial \xi} \left( \xi \frac{\partial c}{\partial \xi} \right) – \zeta (c – c_p) \tag{5} $$

$$ \frac{\partial c_p}{\partial \tau} = \frac{1}{\omega} \frac{\partial^2 c_p}{\partial \xi^2} – c_p \tag{6} $$

where \( a^* = \epsilon_p + a(1 – \epsilon_p) \).

Initial Conditions

The following initial conditions are defined for the system of equations:

$$ c(\eta, 0) = 0, \quad c_p(\xi, 0) = 0 \tag{7} $$

Boundary Conditions for Radial Symmetry and Column Wall Impermeability

The boundary conditions for equation (5), which correspond to the symmetry of the radial profile and the impermeability of the column wall, are used along the radial coordinate of the column at \( \rho = 0 \) and \( \rho = 1 \):

$$ \frac{\partial c}{\partial \xi} \bigg|_{\xi=0} = 0, \quad \frac{\partial c}{\partial \xi} \bigg|_{\xi=1} = 0 \tag{9} $$

Boundary Conditions for Injection

For rectangular pulse injection through the inner cylindrical core, the boundary conditions are:

$$ c(\eta, 0) = c_{inj} \quad \text{for inner cylindrical core} \tag{10} $$

For injection through the outer ring:

$$ c(\eta, 1) = c_{inj} \quad \text{for outer ring} \tag{11} $$

Setting \( \rho = 1 \) in equation (10) or \( \rho = 0 \) in equation (11) leads to inlet injection through the whole cross-section.

Boundary Conditions at Outlet

The Neumann conditions at the outlet of a finite length column are:

$$ \frac{\partial c}{\partial \eta} \bigg|_{\eta=L} = 0 \tag{12} $$

Dirichlet Inlet Boundary Conditions

When the dispersion coefficient is small (i.e., for large axial Peclet number), the boundary conditions expressed in equations (10) to (12) reduce to the Dirichlet boundary conditions.

Boundary Conditions for Particle Radial Coordinates

The following boundary conditions for equation (6) are defined at the end of the radial coordinates of the particles \( \rho_p = 0 \) and \( \rho_p = 1 \):

$$ \frac{\partial c_p}{\partial \xi} \bigg|_{\xi=0} = 0, \quad c_p(1, \tau) = 0 \tag{13} $$

Semi-Analytical Solutions Using Hankel and Laplace Transforms

To obtain the solutions of the chromatographic model with the defined initial and boundary conditions, the finite Hankel and Laplace transforms are applied sequentially.

The zeroth-order finite Hankel transform of \( c(\rho, x, \tau) \) is defined as:

$$ c_H(\lambda_n, x, \tau) = \int_0^1 c(\xi, x, \tau) J_0(\lambda_n \xi) \, d\xi \tag{14} $$

The inverse Hankel transform is given as:

$$ c(\xi, x, \tau) = \sum_{n=1}^{\infty} c_H(\lambda_n, x, \tau) J_0(\lambda_n \xi) \tag{15} $$

Applying the above Hankel transformation on equation (5) with respect to the coordinate \( \rho \) gives:

$$ \frac{\partial c_H}{\partial \tau} = -u \frac{\partial c_H}{\partial x} + D_z \frac{\partial^2 c_H}{\partial x^2} – \lambda_n^2 D_r c_H \tag{16} $$

Next, the Laplace transformation of the Hankel-transformed function \( c_H \) is defined as:

$$ \tilde{c}_H(\lambda_n, x, s) = \int_0^{\infty} c_H(\lambda_n, x, \tau) e^{-s \tau} \, d\tau \tag{17} $$

Applying the Laplace transformation on equation (16) with respect to \( \tau \) while considering the initial concentration as zero gives:

$$ s \tilde{c}_H = -u \frac{\partial \tilde{c}_H}{\partial x} + D_z \frac{\partial^2 \tilde{c}_H}{\partial x^2} – \lambda_n^2 D_r \tilde{c}_H \tag{18} $$

The general solution of equation (19) is given as:

$$ \tilde{c}_p(\xi) = A_1 e^{\alpha \xi} + A_2 e^{-\alpha \xi} \tag{20} $$

where:

• \( A_1 \) and \( A_2 \) are constants determined by applying boundary conditions.

Thus, the solution in equation (17) at \( \rho_p = 1 \) becomes:

$$ c(\xi, x, \tau) = \text{Inverse Laplace and Hankel transforms of } \tilde{c}_H \tag{22} $$

The Hankel transform of Eq. (22) with respect to \( \rho \), gives

\[
\hat{f}(\nu) = \int_0^{\infty} f(\rho) J_0(\nu \rho) \, d\rho \tag{23}
\]

Putting Eq. (24) in Eq. (18), the following is obtained

\[
\tilde{f}(s) = \frac{1}{s} \int_0^{\infty} e^{-\rho s} f(\rho) \, d\rho \tag{24}
\]

where

\[
\tilde{f}(s) = \frac{1}{s} \left( f(\infty) – f(0) \right) \tag{25}
\]

The general solution of the above Eq. (22) is given below as

\[
f(\rho) = A J_0(\alpha \rho) + B Y_0(\alpha \rho) \tag{26}
\]

where

\[
\alpha = \sqrt{\frac{s}{D}} \tag{27}
\]

and \( A \) and \( B \) are constants to be determined from the following Hankel and Laplace transformed boundary conditions. \( A \) is obtained with the plus sign on top while \( B \) results from the minus sign below.

Next is to obtain the Hankel and Laplace transformed Danckwerts boundary conditions by taking the Hankel transformation of Eqs. (10) and (12), which gives

\[
\hat{f}(\nu) = \int_0^{\infty} f(\rho) J_0(\nu \rho) \, d\rho \tag{28}
\]

and

\[
\tilde{f}(s) = \frac{1}{s} \int_0^{\infty} e^{-\rho s} f(\rho) \, d\rho \tag{29}
\]

Where

\[
f_{\text{in}} = \frac{q_{\text{in}}}{\pi R_{\text{in}}^2}, \quad f_{\text{out}} = \frac{q_{\text{out}}}{\pi R_{\text{out}}^2} \tag{30}
\]

both represent inner and outer core injections, respectively.

Next, applying the Laplace transformation on the boundary conditions given in Eqs. (29) and (30), we obtain

\[
\tilde{f}(s) = \frac{1}{s} \left( f(\infty) – f(0) \right) \tag{31}
\]

and

\[
\tilde{g}(s) = \frac{1}{s} \left( g(\infty) – g(0) \right) \tag{32}
\]

Now, plugging in the values from these boundary conditions obtained in Eqs. (31) and (32) into Eq. (27), we obtain the solution as

\[
f(\rho) = A J_0(\alpha \rho) + B Y_0(\alpha \rho) \tag{33}
\]

Following a similar procedure, the solution obtained by considering the Dirichlet boundary condition is given as

\[
f(\rho) = A J_0(\alpha \rho) \tag{34}
\]

A numerical inversion approach is used to convert the above two solutions back into the original coordinates [20,21].

Numerical case studies

In this section, several test cases are studied to verify the accuracy of the semi-analytical solutions and to study the effects of some parameters on the concentration profile. To obtain approximate solutions for the governing model equations, the semi discrete high resolution finite volume scheme (HR-FVS) of Koren is implemented [22, 23, 25]. Parameters used for the test cases are taken from the ranges frequently used in liquid chromatography applications and are given in Table 1.

Figure 2: Effects of injection on the concentration profiles for 1d and 3d plots, through the inner core and outer ring for Dancwerts boundary condition

Figure 2 show one-dimensional (1d) and three-dimensional (3d) plots for the solutions resulting from injecting through the inner cylindrical core (Figures 2(a) and (b)) and via the outer annular ring (Figures 2(c) and (d)). The results were obtained using the Danckwerts boundary condition. The 1d plots show a good agreement between semi-analytical and the numerical solutions. The 3d plots are used to demonstrate the evolution of radial concentration profiles caused by slow radial dispersion. At the column outlet, due to slow radial dispersion (i.e. for Pe_ρ = 15), a higher concentration can be seen at the outer region as compared to the inner cylindrical region due to slow radial dispersion.

Figure 3: Effects of reaction rate constant ω and velocity u on the concentration profiles for inner core injection

Figure 3 display the effects of the reaction rate constant ω and velocity u on the concentration profiles for inner cylindrical core injection. The effect of ω as seen in Figure 3(a), shows that increasing the value of ω causes a reduced concentration profile. This shows that chemical reaction between sample components and the particle phase in the column increases. Hence the samples will be quickly converted to products as the reaction rate constant increases. Figure 3(b) gives the effects of increasing the velocity on the concentration profiles. It is evident that increasing the velocity leads to higher concentration profiles. This leads to faster elution and faster dispersion rates hence, causing less reaction due to less interaction time between the solute and the particle phase inside the column.

Figure 4: Effects of Peρ on the concentration profiles for injection through the inner core

In Figure 4, the radial dispersion coefficient Pe_ρ effect on the concentration profile is shown for inner cylindrical core injection. For small Pe_ρ = 1.5, which is the result shown in Figure 4(a), there is no visible effects on the concentration profile due to quicker radial dispersion. This case collapses the two-dimensional model to a one-dimensional model. On the other hand, the effects of slow radial dispersion (for Pe_ρ = 150) can be seen as the concentration profile reduces at the middle of the column in Figure 4(b).

Figure 5: Effects of Pez coupled with the BCs, η and ζ on the concentration profiles for inner core injection

Figure 5 gives the effects of the axial dispersion coefficient Pe_z, intraparticle diffusion co- efficient η and mass transfer coefficient ζ on the concentration profiles plotted for inner cylindrical core injections. In Figure 5(a), Pe_z is altered to see the effects on the con- centration profiles for the solutions obtained from the two types of boundary conditions considered. It is seen that for small axial Peclet number Pe_z = 1.5, the solutions of both boundary conditions tend to differ and the retention times of the solutes increases as broader profiles can be observed. The results in Figures 5(b) and 5(c) show similar behaviors in studying the effects of η and ζ respectively. It can be seen that reducing the values of each of η and ζ leads to broad tailed concentration profiles and hence leading to

Figure 6: Effects of Peρ on the local moments for inner core injection

increased retention times inside the column.

Figures 6 and 7 show the results of the obtained numerical moments for inner cylindrical core injections. Moments analysis is very useful tool in describing the shapes of the concentration profiles. These moments were calculated numerically by using the following expressions [5]

In Figure 6, the effect of the radial dispersion coefficient Pe_ρ on the local moments is shown. The results display that for the smallest value of Pe_ρ, the moments approach constant values along the radial coordinate. Due to the injection of the solute via the inner cylindrical core, there are no changes in the moments near the center of the column. The changes are however seen in the outer region of the column. An increase in Pe_ρ can be seen to cause larger changes in the higher moments. The results presented here are in good agreement with the results given in Figure 4.

Figure 7: Effects of Pez on the averaged central moments for inner core injection

Figure 7 shows the axial dispersion coefficient Pe_z effects on the averaged moments. The results show that no visible effect can be observed on the first moment. On the other hand, the effects can be seen on the other moments as the axial Peclet number is increased causing a reduction in the values of the moment. These results again are in good agreement with the results given in Figure 5(a).

Table 1: Standard parameters used in the test problems.

CONCLUSION

A linear two-dimensional reactive liquid chromatography model was solved analytically considering two types of boundary conditions. Semi-analytical solutions were obtained by the successive application of finite Hankel and Laplace transforms. The numerical inversions of Hankel and Laplace transforms were used to obtain the solutions in their true domains. A high resolution semi-discrete finite volume scheme was subsequently applied to obtain approximate solutions of the governing model equations in order to have confidence on the obtained semi-analytical solutions. Both the semi-analytical and the numerical solutions were seen to be in good agreement, this also validates the accuracy of numerical scheme used. The obtained two-dimensional model solutions was shown to capture the scenario where radial gradients are introduced during injection of solutes inside the chromatography columns and can play a significant role in further the developments of reactive chromatography processes.

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