# A Novel Numerical Treatment of Nonlinear and Nonequilibrium Model of Gradient Elution Chromatography considering Core-Shell Particles in the Column

Authors: Abdulaziz Garba Ahmad; Mohammed K. A. Kaabar; Saima Rashid; Muhammad Abid
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

An extended method of semidiscrete high-resolution finite volume is used in this paper to obtain numerical solutions for a formulated nonlinear lumped kinetic model of liquid chromatographic process to examine the effect of chromatographic column overloading gradient elution considering core-shell particles. The model constitutes linear solvent strength (LSS), Henry’s constant, coefficient of nonlinearity, and coefficient of axial dispersion. The effects of modulator concentration changes for the elution of single and two components are analyzed. The advantages of introducing gradient elution against isocratic elution in terms of core radius fraction are investigated intensively. Numerical temporal moments are generated from the solutions obtained for a more in-depth examination of the considered model. Moreover, multiple forms of a single- and two-component mixture are generated to analyze the influences of core radius fractions on gradient elution. For example, the obtained results are utilized to investigate the effects of the slope of gradient, concentration of modulator, solvent strength parameter, coefficient of nonlinearity, coefficient of mass transfer, and coefficient of axial dispersion on the profiles of concentration in order to improve the process performance using optimal core radius fraction parameter values.


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## Body

## 1. Introduction

Chromatography is a laboratory technique that has become a widespread and popular tool for separating and purifying the mixture into its components. This technique can be utilized to separate chemical compounds that other traditional methods such as distillation, filtration, or extraction cannot in chemical analysis. It is used in chemical engineering and forensic science and has many other diverse industrial applications. Also, this method is used by environmental protection agencies to test the purity of drinking water and the quality of the air [1–3].Gradient elution in liquid chromatography is a process that is generated by changing the configuration of the mobile phase over time while keeping the flow rate constant during the gradient run [4]. So, the composition of the liquid phase remains fixed (constant) throughout the process of high-performance liquid chromatographic (HPLC) separation, and the operation is referred to as an isocratic elution. Gradient elution chromatography can be used to separate molecules with widespread shifting elution peaks [5, 6]. This chromatography method was introduced in the 1950s and had a wide range of analytical and preparation applications. In this type of chromatography, the strength of elution can be increased or decreased by changing the mobile phase composition. The technique significantly improves analysis time, peak resolution, and peak detection capabilities, which has piqued the interest among researchers in gradient elution chromatography [7]. Because the retention behavior of the solvent in this operating state changes with the composition of the mobile phase, theoretical investigation for gradient elution appears to be more complicated than those for isocratic elution. Snyder [8, 9] is acknowledged for contributing significantly to gradient elution theory by formulating an extensively utilized concept of linear solvent strength (LSS) gradients. The gradient program is modified in an LSS-gradient approach so that the logarithms of the sample compounds’ retention factors decrease linearly [10]. Snyder and Dolan presented a comprehensive investigation of the linearized solvent strength gradient approach [11]. LSS gradients are employed for predicting results in the chromatographic reverse phase, where the gradient elution technique is widely used. Non-LSS highly curved plots of retention factor over modulator concentration, on the other hand, have been implemented in various chromatographic methods, specifically in ion-exchange chromatography.In order to increase the effectiveness of HPLC columns, different packing materials have been investigated [12–15]. The utilization of core-shell particles has increasingly captivated the interest of many researchers in both the analytical and preparatory liquid chromatography separation processes [16–18]. These were used to separate peptides as well as other molecules like nucleotides and proteins [12, 16]. Aside from that, some theoretical findings on the use of core-shell particles were made by analyzing one-dimensional (1D) liquid chromatography models. Kaczmarski and Guichon [19] have published a detailed analysis of the general rate model, which was used to prove the benefits of the thin-shell coated beads. Also, Wang et al. [15] investigated the relationship between the pressure flow ion-exchange resin and the core-shell particles. Further, Luo et al. [20] analyzed that when compared to completely porous homogeneous agarose beads, core beads increased vital stiffness and permeability. The most completed model, that is, the general rate model (GRM), was used to investigate and optimize the effects on multicomponent isocratic elution by using the core radius fraction by Gu et al. [21]. Qamar et al. and his fellow researchers have analyzed the effectiveness of fully porous particles and also some with the core-shell particles by developing and solving several chromatographic separation models [18, 22].To understand the chromatographic separation processes, a number of mathematical models existing in the literature might be employed [3, 23, 24]. Each model has its own complexity level in describing the chromatography processes. The intricacies of these existing mathematical models are based on the suppositions made about the behavior of the thermodynamic system, adsorption-desorption kinetics, kinds of packing particles used, and also the dispersion and mass transfer mechanisms. The general rate model (GRM), the lumped kinetic model, and the dispersive equilibrium model are the most frequently and commonly utilized models in liquid chromatography. In the literature, multiple numerical techniques like finite volumes method, finite-difference method, and finite element methods have been employed to simulate the chromatographic models [25–27].In this work, a one-dimensional nonlinear lumped kinetic model of gradient elution chromatography packed with core-shell particles in the column is formulated and numerically approximated in order to theoretically investigate the effect of core radius fraction in column overloading gradient elution. In the developed model, Henry’s constants parameter, coefficients of nonlinearity, coefficients of mass transfer, and the coefficients of axial dispersion rely on the composition of the solvent. The present study extends our previous analysis [28], which is based on nonlinear (1D) LKM of gradient elution chromatography considering fully porous particles. Contrary to the previous analysis, this current research considered core-shell particles loaded in the column. Analytical solutions of the model equations for nonlinear adsorption isotherm are not possible to acquire. Numerical simulations are required for this purpose to precisely predict the behavior of a chromatographic column dynamically. Several techniques have been discussed in the literature to obtain the approximate solution of nonlinear partial differential equations [29]. To solve the governing equations arising in the given model, the high-resolution finite volume method (HR-FVM) is developed and implemented. As it is illustrated in our last research article, this method allows second-to-third-order accuracy in the solution. Furthermore, numerical temporal moments are derived from the derived solutions for a more in-depth examination of the considered model. The implications of different parameters on process performance are investigated, as well as the advantages of gradient elution instead of isocratic elution in the simulation analysis. The obtained results also demonstrate the model’s and numerical scheme’s strength and potential in situations more similar to real-world applications.The remainder of this presentation is divided into several parts. In Section2, we formulate the model’s specification and description of nonlinear 1D-LKM incorporated with core-shell particles and overloading gradient elution in the chromatographic column. In Section 3, we present the numerical simulation of the nonisothermal 1D-LKM with the considered Dirichlet boundary conditions. In Section 4, we describe the approximate results obtained. In the last section, we summarize the concluding remarks.


## 2. Model Configuration and Formulation

A nonlinear 1D-LKM of gradient elution chromatography is formulated for an isothermal column filled with core-shell particles as packing material. The process of adsorption in the column is controlled through axial dispersions and the resistance to mass transfer phenomena. The LKM uses a driving force that is linear in the solid phase and only investigates one extra parameter to supplement the coefficient of axial dispersion. Also, the coefficient of mass transfer combines the effects of mass transport resistances both internally and externally. The model equations are derived based on the following assumptions, which are considered in the model formulations. The mobile phase is perfectly incompressible. Furthermore, the cored beads have a uniform radiusRp and a core radius fraction ζcore=Rc/Rp, where Rc is the radius of the inert core. The total mass transfer rate in a column is connected with two kinetic parameters, the mass transfer rate coefficient that is represented by KL, and the coefficient of axial dispersion that is symbolized by Dz,i. This lumped kinetic model 1D-LKM is employed to study the dynamic behavior in the chromatographic separation process considering core-shell particles and overloading gradient elution in the column. Considering the above assumptions, the mass balance and kinetic equations for the movement of Nc species in the mobile phase are given as [28, 30](1)∂ui∂t+v∂ui∂z=Dz,iϕ∂2ui∂z2−F1−ζcoreKLθϕi∗−ϕi,i=1,2,…,Nc.The specific number in the mixture sample is indicated byNc in the equation above. For every component of i, ui, ϕi, and ϕi∗ denote the solute concentration, the solute concentration in the immobilize phase, and the concentration of solute adsorbed by the stationary phase, respectively, in the bulk of the fluid. KL represents the coefficient of external mass transfer, v represents the interstitial velocity, t denotes the time, z denotes the axial coordinate, and F=1−ε/ε is the phase ratio, while ϵ is the external porosity.The diffusion and adsorption-desorption processes in the stagnant mobile phase are assumed to be infinitely fast in this model. As a result, in order to complete the model, the following basic linear driving force method is applied to assess the accumulation in the solid phase:(2)∂ϕi∂t=F1−ζcoreKLqϕi∗−ϕi,i=1,2,…,Nc.An isotherm that is symbolized asϕi∗ describes the relationship between the equilibrium adsorbate concentrations in the liquid phase and the equilibrium adsorption quantities in the solid phase. The isotherms can be used to simulate the equilibrium adsorption data. Also, it is employed by researchers to examine adsorption information such as adsorption mechanisms, adsorbent properties, and maximum adsorption capacity. A variety of adsorption isotherm models, including linear, Langmuir, and Freundlich isotherms, have been established in the literature. In the following study, the adsorption equilibrium was modelled using the multicomponent Langmuir equation [3]. The composition of the mobile phase varies throughout the gradient elution process due to changes in a modifier concentration. The following is a formulation of the local equilibrium:(3)ϕi∗ui,q=KH,iqui1+∑j=1Ncbjquj.Henry’s coefficient is represented byKH,i for the i-th component. The degree of nonlinearity, which is connected with the isotherm for the i-th number of components in the mixture, is quantified by br,i. In most cases, experimentally determining the functional dependencies of the two isotherm parameters on c is required. These two strategies are simultaneously utilized to model the retention of analytes as a function of the organic solvent fraction. The mathematical description of the LSS model is(4)biq=br,ie−αq,Dz,iq=Dzr,ie−αq,KL,iq,=KLr,ie−αq,&KH,iq=KHr,ie−αq.Here,KHr,i,KLr,i, biref, and Dzr,i represent the reference values of the mass transfer coefficient, Henry’s constant, nonlinearity, and axial dispersion coefficient. The specific solvent strength is symbolized by q, and α is the volume-fraction parameter of the updating nonretained solvent.It turns out that determining the distribution of the mobile phase’s strong solvent across the column is required before investigating solute retention behavior during elution because it calculates the solute’s migration velocity at any time and as well as at any point within the column. Based on the preceding assumptions, the optimal model for calculating the change in the volume fraction of the strong solvent parameter in the mobile phase is [1](5)∂q∂t+v∂q∂z=0.Here, we defined the initial and boundary conditions for the model equations:(6)qz,0=q0,0≤z≤ℒ,(7)q0,t=q0,ift<ts,qt−ts,ifts≤t≤te,qeif,t>te.For a linear gradient approach, the following applies to the case:(8)qt=q0+βt,β=qe−q0te−ts.In light of the above, the termβ is the slope (or steepness) of the gradient, q0 and qe are the initial and final volume fraction of the modifier (in the mobile phase composition), q represents the implemented profile of the gradient, and ts and te are the starting and ending time of the gradient.The initial and boundary conditions are added to complete the gradient elution model for lumped kinetic considering core-shell particles in the column:(9)uiz,0=0,ϕiz,0=0,i=1,2,3,…,Nc,(10)ui0,t=ui,inj,if0≤t≤tinj,0,t>tinj,i=1,2,3,…,Nc.These inlet BCs work with the following Neumann condition at the right end of the chromatographic column:(11)∂ui∂zL,t=0,i=1,2,…,Nc,where the injected concentration for the i-th component is indicated by the symbol ui,inj and the dimensionless time of injection is denoted by tinj. Moreover, the injection becomes continuous when the time of the simulation process is less than the injection time tinj or when tinj⟶∞.


## 3. Numerical Method

Numerous numerical procedures for approximating chromatographic models are known in literature [1, 3, 25]. The model equations are solved in this section using a semidiscrete high-resolution flux-limiting finite volume procedure. Moreover, this procedure was recently employed on gradient elution chromatography models [28, 31].


### 3.1. Domain Discretization

The first stage in executing the scheme into practice is to discretize the computing domain. The primary and essential goal of the following discretization procedure is to generate a set of time-coordinated coupled ODEs.The mesh interval width is constant and represented byΔξ; let Nξ be the large integer representing the number of grid points. The nodal point ξj encircled by the cell for 1≤h≤Nξ in the interval Ωh≡ξh−1/2,ξh+1/2 is as follows:(12)ξ1/2=0,ξh+1/2=hΔξ,ξN+1/2=ℒ,(13)ξh=ξh−1/2+ξh+1/22,Δξh=ξh+1/2−ξh−1/2=ℒNξ+1.The averaged initial dataλh0 are defined in each cell as(14)λh0=λht=1Δξh∫Ωhλξ,0dξ,h=1,2,…,N.Here,λ∈ui,ϕi,ϕi∗.This can help to simplify the cell averagesλhn+1 at the next stage of time, that is, tn+1, if we know the cell averages at tn. Moreover, integrating (1) and (2) over Ωh gives(15)dui,hdt=−vui,h+1/2−ui,h−1/2Δz+Dz,iqhtΔz∂ui∂zi,h+1/2−∂ui∂zi,h−1/2−F1−ζcoreKLqhtϕi∗−ϕi,(16)dϕi,hdt=F1−ζcoreKLqhtϕi∗−ϕi,i=1,2,…,Nc.Volume fraction of the solventqht is given as(17)qht=q0,ift−zhu<ts,qt−zhu−ts,ifts≤t−zhu≤te,qe,ift−zhu>te.The differential terms arise in the diffusion part of (15), which can be approximated as follows:(18)∂ui∂zh±1/2=ui,h±1−ui,hΔz.The further step is to calculate concentrations at interfaces of the cell. There are numerous techniques for approximating these fluxes, resulting in a wide range of numerical systems. The first- and second-order approximations are presented here.


### 3.2. First-Order Approximation

Concentration values are estimated at interfaces of the cell in (18) by employing the formula of backward difference. Also, for the term of concentration, the first-order approximation is represented as(19)ui,h+1/2=ui,h,ui,h−1/2=ui,h−1.


### 3.3. Second-Order Approximation

The flux-limiting formulas described in the following are used to approximate concentration values at the cell’s interface [32]. Now, ui,h+1/2 is defined as(20)ui,h+1/2=ui,h+12Ψri,h+1/2ui,h−ui−1,h,and(21)rh+1/2=ui,h+1−ui,h+ηui,h−ui,h−1+η.Furthermore, to control division by zero, we requireη=10−10. Here, Ψ (the flux-limiting functions) is utilized to keep the numerical scheme’s local monotonicity in (20), which can be described as [32](22)Ψrh+1/2=max0,min2rh+1/2,min13+23ri+1/2,2.The flux-limiting approximations do not cover the boundary intervals. Take into account the left boundary, which also has an in-flow boundary condition. This interval facesξ1/2, and the in-flow boundary is in the same place. Consequently, cell-interface concentration is simulated using the first-order backward procedure. The fluxes at the boundaries of inner cells can be calculated using the previously described (second-order accurate) HR-FVM. At the cell boundaries ξ3/2 and ξN+1/2, the first-order approximation can be utilized to solve this problem. Let ui,inj represent the injected flux, and then(23)ui,1/2=ui,inj,ui,1/2=ui,1,ui,N+1/2=ui,N.


## 3.1. Domain Discretization

The first stage in executing the scheme into practice is to discretize the computing domain. The primary and essential goal of the following discretization procedure is to generate a set of time-coordinated coupled ODEs.The mesh interval width is constant and represented byΔξ; let Nξ be the large integer representing the number of grid points. The nodal point ξj encircled by the cell for 1≤h≤Nξ in the interval Ωh≡ξh−1/2,ξh+1/2 is as follows:(12)ξ1/2=0,ξh+1/2=hΔξ,ξN+1/2=ℒ,(13)ξh=ξh−1/2+ξh+1/22,Δξh=ξh+1/2−ξh−1/2=ℒNξ+1.The averaged initial dataλh0 are defined in each cell as(14)λh0=λht=1Δξh∫Ωhλξ,0dξ,h=1,2,…,N.Here,λ∈ui,ϕi,ϕi∗.This can help to simplify the cell averagesλhn+1 at the next stage of time, that is, tn+1, if we know the cell averages at tn. Moreover, integrating (1) and (2) over Ωh gives(15)dui,hdt=−vui,h+1/2−ui,h−1/2Δz+Dz,iqhtΔz∂ui∂zi,h+1/2−∂ui∂zi,h−1/2−F1−ζcoreKLqhtϕi∗−ϕi,(16)dϕi,hdt=F1−ζcoreKLqhtϕi∗−ϕi,i=1,2,…,Nc.Volume fraction of the solventqht is given as(17)qht=q0,ift−zhu<ts,qt−zhu−ts,ifts≤t−zhu≤te,qe,ift−zhu>te.The differential terms arise in the diffusion part of (15), which can be approximated as follows:(18)∂ui∂zh±1/2=ui,h±1−ui,hΔz.The further step is to calculate concentrations at interfaces of the cell. There are numerous techniques for approximating these fluxes, resulting in a wide range of numerical systems. The first- and second-order approximations are presented here.


## 3.2. First-Order Approximation

Concentration values are estimated at interfaces of the cell in (18) by employing the formula of backward difference. Also, for the term of concentration, the first-order approximation is represented as(19)ui,h+1/2=ui,h,ui,h−1/2=ui,h−1.


## 3.3. Second-Order Approximation

The flux-limiting formulas described in the following are used to approximate concentration values at the cell’s interface [32]. Now, ui,h+1/2 is defined as(20)ui,h+1/2=ui,h+12Ψri,h+1/2ui,h−ui−1,h,and(21)rh+1/2=ui,h+1−ui,h+ηui,h−ui,h−1+η.Furthermore, to control division by zero, we requireη=10−10. Here, Ψ (the flux-limiting functions) is utilized to keep the numerical scheme’s local monotonicity in (20), which can be described as [32](22)Ψrh+1/2=max0,min2rh+1/2,min13+23ri+1/2,2.The flux-limiting approximations do not cover the boundary intervals. Take into account the left boundary, which also has an in-flow boundary condition. This interval facesξ1/2, and the in-flow boundary is in the same place. Consequently, cell-interface concentration is simulated using the first-order backward procedure. The fluxes at the boundaries of inner cells can be calculated using the previously described (second-order accurate) HR-FVM. At the cell boundaries ξ3/2 and ξN+1/2, the first-order approximation can be utilized to solve this problem. Let ui,inj represent the injected flux, and then(23)ui,1/2=ui,inj,ui,1/2=ui,1,ui,N+1/2=ui,N.


## 4. Numerical Temporal Moments

Temporal profiles of the moments can be utilized to demonstrate the graphical findings obtained in the chromatogram for the duration between the moving phases of the chromatographic process. It is an effective approach for collecting fundamental informative details regarding liquid chromatography elution in a column. Chromatographic phenomena are affected by both retention equilibrium and kinetic mass transfer inside the column, for instance, column efficiency, the profile of the elution peak, sample retention, separation performance, band broadening, and other variables. The simulated moments are generated using the numerical methods we proposed.n-th temporal moment at the outlet of the chromatographic bed of length z=ℒ is confined as the n-th temporal moment to obtain the simulated numerical moments.(24)ℳn=∫0∞z=ℒ,ttndt.Then-th normalised temporal moment is donated as(25)μn=ℳnℳ0=∫0∞uz=ℒ,ttndt∫0∞uz=ℒ,tdt.Then-th central moment is given as(26)μi∗=∫0∞uz=ℒ,tt−μ1ndt∫0∞uz=ℒ,tdt.Continuous injections might be carried out using derivatives of the given concentration profilestinj⟶∞ to transform stage responses that enable moments to be calculated [33]. The central moments used to analyze continuously successful curves are therefore presented as(27)μi∗=∫0∞u˙z=ℒ,tt−μ1ndt∫0∞u˙z=ℒ,tdt,whereu˙=dudt.The elution curve, also known as the probability density curve, is utilized in liquid chromatography to characterize the concentration distribution concerning time. These density curves are determined by the statistical moments, including zeroth, first, second, and third moments. The zero moments, which are indicated by the variations in the concentration profiles, are used to determine the area of peak or total mass of the solvent. The third moment includes information regarding the peak’s asymmetry or skewness.


## 5. Numerical Results and Discussion

The suggested HR-FVM is used in this section to simulate the impacts of various gradient elution chromatographic separation procedures. Here, in this numerical case study, the chromatographic processes of single- and two-component solutes are investigated. To facilitate the simulation process, some assumptions have been made, the Henry coefficientKH,i=kHr,i, the mass transfer coefficients KL,i=kLr, the axial dispersion coefficients Dz,i=Dzr, and the nonlinearity coefficient bi=bbref. The rest of all other remaining parameters are listed in Tables 1 and 2. A nonlinear 1D LKM given by equations (1)–(4) was analyzed. The plots show q (the modulator concentration) and u (concentration of the solute) at the given outlet of the column z=ℒ, which are plotted against time t. However, in practice, they may be different, and present model equations account for such variations.Table 1
Used parameters for linear solvent strength case for single component.
Parameters nameNumeric valuesLength of chromatographic columnℒ=10cmPorosityε=0.4Interstitial velocityv=1.0cm/minReference mass transfer coefficientkLr=10min−1Reference Henry’s coefficientkHr=7Reference coefficient of axial dispersionDzr=0.002cm2/minReference nonlinearity coefficientbbref=2.0Initial concentrationq0=0.1Final concentrationqe=0.95Gradient starting timets=5minGradient ending timete=120minSolvent strength parameterα=0.90Table 2
Used parameters for linear solvent strength case for two components.
Parameters nameNumeric valuesLength of chromatographic columnℒ=10cmInterstitial velocityv=0.6cm/minPorosityε=0.4Reference coefficient of mass transferkLr=10min−1Reference coefficient of axial dispersionDzr=0.0002cm2/minReference Henry’s constant for component-IkHr,1=10Reference Henry’s constant for component-IIkHr,2=15Injected concentrationuinj,1=1.0mol/lInjected concentrationuinj,2=1.0mol/lInjection timetinj=10.0minSolvent strength parameterα1=0.90Solvent strength parameterα2=0.40Solvent strength parameterγ1=0.90Solvent strength parameterγ2=0.40Reference nonlinearity coefficient for component-Ib1bref=1Reference nonlinearity coefficient for component-IIb2bref=2Gradient starting timets=5minGradient ending timete=120minInitial concentrationq0=0.10Final concentrationqe=0.95In Figure1, a case is investigated for distinct values of core radius fraction ζcore, for the solution profile of (a) nonlinear single-component elution profile and (b) nonlinear two-component elution profile. It can be deeply evaluated that the profiles of elution become sharper by increasing ζcore from 0 to 0.8 and decrease their times of retention accordingly; that is, the efficiency of the column gradually increases by enlargement of the value of ζcore on both the single- and two-component plots. Similarly, the holding capacity of the column reduces gradually.Figure 1
The elution profile of single- and two-component for distinct values of core radius fractionζcore. Table 1 lists all of the other parameter values that were chosen.
(a)(b)Figure2 demonstrates the effects of α, bbref,Dzr, and kLr for core radius fraction value ζcore=0.5. Figure 2(a) displays the impact on the strength of the solvent parameter α examined on the nonlinear single-component profile. As α changes from 0.1 to 1.0, the retention time decreases, that is, as the sensitivities of Henry’s constant to significant difference increases in the modulator concentration profile. Also, the peak of the profile increases higher due to the influence of ζcore from the core-shell particles. Figure 2(b) shows the influence of the nonlinearity parameter bbref on the elution profile. For the increasing value of the coefficient of nonlinearity, the elution peaks become tailed, and the retention time decreases. However, for bbref=0, the elution profile has a representation of Gaussian shape. This describes the Langmuir behavior that is normally noticed in liquid chromatography processes. Figure 2(c) depicts the influence of the parameter of coefficient of axial dispersion Dzr on the elution profiles for two distinct values. Dzr is not affecting the mean retention while it becomes peaked for increasing the value of Dzr. In Figure 2(d), this shows that elution profiles become more sharped and peaked for increasing the value of kLr. Contrarily, as expected, the mass transfer coefficient does not affect the average retention time.Figure 2
Here, the value ofζcore=0.5 is applied to check the effects of (a) parameter of specific component α, (b) reference coefficient of nonlinearity bref, (c) reference coefficient of axial dispersion Dzr , and (d) reference coefficient external mass transfer KLr on nonlinear one-dimensional single-component elution profiles. Table 1 lists all of the other parameter values that were chosen.
(a)(b)(c)In Figure3, it demonstrates the comparison among the elution profiles for isocratic and gradient for both positive and negative gradients’ profiles sequentially for the elution of a single component. Here, the value of the core radius fraction ζcore=0.5 is utilized. In Figure 3(a), it can be seen that isocratic elution produces a bigger profile with more asymmetry and a longer transportation time. However, the isocratic profile with q=qe generates a narrower profile, which is symmetric. Figure 3(b) displays the elution profiles generated by the positive value of gradient elution, which are symmetric, narrower, and most importantly asymmetric. In next Figure 3(c), the impact of negative gradient elution results in a smaller and peaked elution profile.Figure 3
Here, in this case, the parameterζcore=0.5 is used to show the comparison of elution profiles for isocratic and gradient, including positive and negative profiles of the gradients utilizing the elution for a single component. Table 1 contains a list of other parameters.
(a)(b)(c)Figure4 presents the results of three different values of ζcore on temporal moments to check the influence of gradually changing values modulator concentration qe. For the higher value of ζcore, the profile of the temporal moment has a high magnitude which produces the gradual reduction of the holding capacity of the column. For the first moment, it described the mean retention time. Also, its value reduces with higher values of qe. Similarly, the second and third moments can be seen with the same behavioral effect.Figure 4
Here, three different values ofζcore are used to check the effect of modulator concentration qe on temporal profile of moments. qe has been significantly updated, while the other parameters remain unchanged as shown in Table 1.
(a)(b)(c)Figure5 demonstrates the plots of moments to analyze the influence of the beginning time of gradient ts for a fixed final time te. Here, three different values of ζcore were utilized for core-shell particles. For the first moment, it described the meantime of retention. For the smaller value of ζcore, its value rises with increasing values of ts. A similar behavioral effect can be noticed for the moment plots of the second as well as the third moment, which described the variance and asymmetry of the eluent profile. In addition, due to the impact of ζcore, the elution profile becomes peaked.Figure 5
Here, three different values ofζcore are used to check the effect of gradient initial time ts on numerical temporal moments. While other parameters are fixed, ts has been gradually modified. Table 1 displays all of the other parameter values that were chosen.
(a)(b)(c)Figure6 analyzes the effect of ending time te on the elution profile. Here, three different values of ζcore were utilized for the impact of core-shell particles. It is clear from the figures that a smaller value ζcore results in a larger slope of the gradient and consequently results in shortened residence time of the elution profile in the chromatographic column. For the plots of the second and third moments, a similar behavioral impact can be observed. Also, due to the impact of ζcore, the elution profile become peaked.Figure 6
Here, three different values ofζcore are used to check the effect of gradient final time te on the temporal profile of moments. te has been steadily updated, although other parameters have stayed unchanged, as seen in Table 1, which includes all of the other parameter values that were selected.
(a)(b)(c)Figure7 presents the numerically simulated results for distinct values of the parameter of solvent strength α on the two-component elution profile. Here, the values of ζcore=0.5 are used. When the value of α is increased, the time spent by the pulses inside the column decreases. Once again, due to the impact of ζcore, the elution profile becomes peaked.Figure 7
Here, the value ofζcore=0.5 is employed to check the effect of α (i.e., the specific parameter of solvent strength on profiles of elution). Table 2 illustrates how α has increased steadily while other parameters have stayed unchanged.
(a)(b)(c)Figure8 uses a two-component elution profile to analyze the eluent profiles of isocratic and gradient elution for both positive and negative gradients. Here, a similar behavioral effect can be noticed, like the one we discussed earlier in Figure 3.Figure 8
Here, the value ofζcore=0.5 is employed for comparing the elution profile of isocratic and gradient for both positive and negative profiles of the gradients sequentially utilizing the elution profile for two components. Table 2 has a range of required parameters.
(a)(b)(c)


## 6. Conclusion

In the present paper, a nonlinear 1D-LKM of gradient elution chromatography packed with core-shell particles in the column was developed and numerically approximated to analyze the effect of core radius fraction in column overloading gradient elution. The model solution was obtained numerically using an HR-FVM considering a suitable flux limiter. The nonlinear isotherm was analyzed in which the Henry’s constant and the coefficients of nonlinearity both depended on the composition of the solvent. The study showed that larger values of core radius fraction generate profiles of sharper peaks and small residence times. Thus, the efficiency of the column was increased because of a reduction in the diffusion path inside the adsorbents. Additionally, the other parameters’ effects on the separate components that behave differently inside the chromatographic column were also investigated. Moreover, the numerical solutions obtained illustrate that core-shell particles of suitable core radius fraction can be used for optimizing the liquid chromatography processes.



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