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.
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 Section 2, 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 radius and a core radius fraction , where 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 , and the coefficient of axial dispersion that is symbolized by . 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 species in the mobile phase are given as [28, 30]
The specific number in the mixture sample is indicated by in the equation above. For every component of , , , and 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. represents the coefficient of external mass transfer, represents the interstitial velocity, denotes the time, denotes the axial coordinate, and is the phase ratio, while 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:
An isotherm that is symbolized as 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:
Henry’s coefficient is represented by for the -th component. The degree of nonlinearity, which is connected with the isotherm for the -th number of components in the mixture, is quantified by . In most cases, experimentally determining the functional dependencies of the two isotherm parameters on 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
Here, , , and represent the reference values of the mass transfer coefficient, Henry’s constant, nonlinearity, and axial dispersion coefficient. The specific solvent strength is symbolized by , and 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]
Here, we defined the initial and boundary conditions for the model equations:
For a linear gradient approach, the following applies to the case:
In light of the above, the term is the slope (or steepness) of the gradient, and are the initial and final volume fraction of the modifier (in the mobile phase composition), represents the implemented profile of the gradient, and and 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:
These inlet BCs work with the following Neumann condition at the right end of the chromatographic column:where the injected concentration for the -th component is indicated by the symbol and the dimensionless time of injection is denoted by . Moreover, the injection becomes continuous when the time of the simulation process is less than the injection time or when .
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 by ; let be the large integer representing the number of grid points. The nodal point encircled by the cell for in the interval is as follows:
The averaged initial data are defined in each cell as
Here, .
This can help to simplify the cell averages at the next stage of time, that is, , if we know the cell averages at . Moreover, integrating (1) and (2) over gives
Volume fraction of the solvent is given as
The differential terms arise in the diffusion part of (15), which can be approximated as follows:
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
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, is defined asand
Furthermore, to control division by zero, we require . Here, (the flux-limiting functions) is utilized to keep the numerical scheme’s local monotonicity in (20), which can be described as [32]
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 faces , 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 and , the first-order approximation can be utilized to solve this problem. Let represent the injected flux, and then
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. -th temporal moment at the outlet of the chromatographic bed of length is confined as the -th temporal moment to obtain the simulated numerical moments.
The -th normalised temporal moment is donated as
The -th central moment is given as
Continuous injections might be carried out using derivatives of the given concentration profiles to transform stage responses that enable moments to be calculated [33]. The central moments used to analyze continuously successful curves are therefore presented as
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 coefficient , the mass transfer coefficients , the axial dispersion coefficients , and the nonlinearity coefficient . 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 (the modulator concentration) and (concentration of the solute) at the given outlet of the column , which are plotted against time . However, in practice, they may be different, and present model equations account for such variations.
In Figure 1, a case is investigated for distinct values of core radius fraction , 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 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 on both the single- and two-component plots. Similarly, the holding capacity of the column reduces gradually.
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Figure 2 demonstrates the effects of , , and for core radius fraction value . Figure 2(a) displays the impact on the strength of the solvent parameter examined on the nonlinear single-component profile. As 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 from the core-shell particles. Figure 2(b) shows the influence of the nonlinearity parameter 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 , 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 on the elution profiles for two distinct values. is not affecting the mean retention while it becomes peaked for increasing the value of . In Figure 2(d), this shows that elution profiles become more sharped and peaked for increasing the value of . Contrarily, as expected, the mass transfer coefficient does not affect the average retention time.
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In Figure 3, 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 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 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.
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Figure 4 presents the results of three different values of on temporal moments to check the influence of gradually changing values modulator concentration . For the higher value of , 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 . Similarly, the second and third moments can be seen with the same behavioral effect.
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Figure 5 demonstrates the plots of moments to analyze the influence of the beginning time of gradient for a fixed final time . Here, three different values of were utilized for core-shell particles. For the first moment, it described the meantime of retention. For the smaller value of , its value rises with increasing values of . 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 , the elution profile becomes peaked.
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Figure 6 analyzes the effect of ending time on the elution profile. Here, three different values of were utilized for the impact of core-shell particles. It is clear from the figures that a smaller value 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 , the elution profile become peaked.
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Figure 7 presents the numerically simulated results for distinct values of the parameter of solvent strength on the two-component elution profile. Here, the values of are used. When the value of is increased, the time spent by the pulses inside the column decreases. Once again, due to the impact of , the elution profile becomes peaked.
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Figure 8 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.
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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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Abdulaziz Garba Ahmad contributed to actualization, validation, methodology, formal analysis, investigation, and initial draft. Mohammed K. A. Kaabar contributed to actualization, methodology, formal analysis, validation, investigation, initial draft, and supervision of the original draft and editing. Saima Rashid contributed to actualization, methodology, formal analysis, validation, investigation, and initial draft. Muhammad Abid contributed to actualization, validation, methodology, formal analysis, investigation, and initial draft. All authors read and approved the final version.
Acknowledgments
The authors would like to express their sincere thanks to the National Mathematical Centre, Abuja, Nigeria, for supporting this study.