J. Nig. Soc. Phys. Sci. 5 (2023) 1350 Journal of the Nigerian Society of Physical Sciences Numerical Simulation of Nonlinear and Non-Isothermal Liquid Chromatography for Studying Thermal Variations in Columns Packed with Core-Shell Particles Abdulaziz G. Ahmada,b,∗, Nnamdi F. Okechia, David U. Uchea,c, Abdulwasiu O. Salaudeend aDepartment of Mathematics Programme, National Mathematical Centre Abuja, Nigeria bDepartment of Applied Mathematics, Federal University of Technology Babura, Nigeria cDepartment of Mathematics, University of Abuja, Nigeria dDepartment of Applied Mathematics (Chemistry Unit) Programme, National Mathematical Centre Abuja, Nigeria Abstract A high-resolution flux-limiting semi-discrete finite volume scheme (HR-FVS) is applied in this study to numerically approximate the nonlinear and non-isothermal flow of one-dimensional lumped kinetic model (1D-LKM), for a fixed-bed column loaded with core-shell particles. The developed model comprise a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential equation and algebraic equation in the stationary phase. The solution of the model equations is obtained by utilizing a HR-FVS, the scheme has second-order accuracy even on the grid coarse and its explicit nature has the potential to resolve the arisen sharp discontinuities in the solution profiles. A second-order total variation diminishing (TVD) Runge-Kutta technique is used to solve the system of ODEs in time. Several forms of a single-solute mixture are produced to investigate the influences of the fractions of core radius on thermal waves and concentration fronts. Moreover, a particular criterion is introduced for analyzing the performance of the underlying process and to identify the optimal parameter values of the fraction of core radius. DOI:10.46481/jnsps.2023.1350 Keywords: Non-isothermal chromatography, Non-linear isotherm, One-dimensional lumped kinetic model, High-resolution scheme Article History : Received: 15 January 2023 Received in revised form: 06 March 2023 Accepted for publication: 10 March 2023 Published: 24 April 2023 © 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: J. Ndam 1. Introduction In recent years, chemical engineers and researchers are in- creasingly interested in High-performance liquid chromatogra- phy (HPLC) to further improve the performance of classical column. An innovative and valuable tool used for the sepa- ration and quantification of the multi-component mixture due ∗Corresponding author tel. no: +234 80326117615 Email address: agarbaahmad@yahoo.com (Abdulaziz G. Ahmad) to the different affinities of adsorption for the components is known as HPLC. This technique is commonly used in the chem- ical, pharmaceutical and food industries where the traditional operations in the thermal unit, such as distillation and extrac- tion, are unsuitable [1–3]. For both large and preparative scales this process is equally popular, especially for the purification of proteins and other higher valuable products. Figure 1 illus- trates a typical demonstration of a single-column HPLC. Here, the sample is injected via inner region, which propagates the 1 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 2 solute along z-direction of the column by advection and axial- dispersion. Separation efficiency of HPLC can be made better by using small diameter particles to be loaded into the column in order to reduced the resistance of intra-particulate mass transfer due to short diffusion distances [4–6]. These loaded core-shell par- ticles were designed and operated in the columns of HPLC to contempt the utilization of technical types of equipment. Re- cently, the use of core-shell particles has created significant in- terest in both analytical and preparative liquid chromatography. It was applied to segregate peptides and other molecules, such as nucleotides, and proteins [5, 7]. Aside from that, several the- oretical investigations involving the use of core-shell particles were made by analyzing models of liquid chromatography in one-dimensional (1D) forms [1, 3]. Kaczmarski and Guichon [8] consider the general rate model to substantiate the benefits of thin-shell coated beads. Also, the same general rate model is employed by Gu et al. [4] to optimize and examine the in- fluences of core radius fraction in isocratic elution of multi- component mixture. Several authors have recently emphasized the importance of studying the differences of core-shell and fully porous particles by formulating and solving a number of chromatographic models via core-shell particles [9–11]. Inspired by the work of Brandt, et al., [12]. Temperature has a wide impact on all chromatography processes, and there are several ways to analyze its significance [13, 14]. For example, increasing the temperature decreases viscosity while rising sol- ubility and diffusivity. Moreover, the peak shape, the efficiency of the column, complete-time analysis and retention time has temperature influences due to the thermodynamics, and adsorp- tion kinetics are temperature-dependent [15, 16]. Further, the use of a steady condition of temperature during the process in- creases reproducibility. Even though, effects of thermal prop- erty are typically ignored in the columns of liquid chromato- graphic because the effect of heat adsorption is assumed to be inconsiderable, many more chromatographers have figured out that temperature is critical to the process optimization [13]. On contrarily, a number of researchers have investigated thermal impacts in gas chromatography [17–19]. There are additional contributions in the literature that examine the thermal influ- ences in liquid chromatography columns, which are also acces- sible [3, 13, 14, 20–24]. A number of mathematical models for simulating the un- derlying chemical process at various extent of complexities are available in the literature to aid in the understanding of physical phenomena [15, 16, 25, 26]. This modeling approach allows us to comprehend what occurs within the column, and also during the separation process. The most influential and widely em- ployed models that exist in the literature include the equilib- rium dispersive model (EDM), the model based on linearized driving force, the lumped kinetic model (LKM), and the gen- eral rate model (GRM) [1–3, 27]. The mass transfer rate in the EDM is considered to be infinite, but the local concentration’s rate of change in the LKM is presumed to be finite [1–3]. GRM is a model that incorporates intra-particle diffusion which in- cludes mass transfer across the interface among the both phases (i.e., stationary and mobile). It is also referred to as the most comprehensive model [1]. A nonlinear 1D-LKM incorporating core-shell particles is numerically approximated in this article. This work extends the analysis of 1D-LKM non-isothermal model [28]. Unlike the past study, this prevailing non-isothermal 1D-LKM helps to examine the effect of core radius fractions on thermal and concentration fronts along the axial gradients of the eluent in the column. The formulated model comprises of a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential and al- gebraic equations in the stationary phase. The solution of the model is obtained by utilizing a HR-FVS. The scheme deals with integral form of conservation laws and also it has potential to produce numerical result that can resolve sharp discontinuity and achieve higher order accuracy. Afterwards, a Runge-Kutta approach for second-order TVD is used to simulate the system of ODEs in term of time [30]. Some certain test problems are considered to illustrate the simultaneous elution of thermal and concentration fronts. Furthermore, we hope that the key pa- rameters that influence the elution profiles within the column are identified. The article arranged as follows. In Section 2, a nonlinear and non-isothermal 1D-LKM incorporated with core-shell par- ticles is developed. In Section 3, the numerical simulation of the non-isothermal 1D-LKM is obtained for the boundary con- ditions discussed by Dankwerts condition. Section 4 outlines a number of simulated case problems, and ultimately, Section 5 contains the conclusions. 2. Mathematical model formulation of the process In this section, the following assumptions are employed to configure and develop the model equations: (i) the column is considered to be thermally insulated and homogeneously loaded with ore-shell particles, (ii) the fluid is incompressible with constant rate of volumetric flow, (iii) the stationary and mo- bile phases have negligible interaction between them, (iv) the axial heat conductivity coefficient are independent of the flow rate, (v) the temperature does not affect physical properties such as density, viscosity, heat capacity, or coefficients of transport (e.g. axial dispersion and heat conductivity), (vi) the linear driv- ing force model is utilized to determine the overall adsorption rate, and (vii) the heat transfer resistance of the solid phase is concentrated at the particle surface. Let t symbolize the coordinate time while the coordinate in axial direction alongside the column length is donated as z, the symbol Rp represents the unvarying size of cored beads parked in the column and fraction of core radius is ηcore = Rc/Rp where Rc symbolized the radius of the inert core. The solute will propagate via the column’s z-direction with advection and axial-dispersion. The one-dimensional equations for the bal- ance of mass and the heat for a solute mixture elution in the mobile phase via column loaded with spherical core beads are 2 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 3 expressed as ∂c ∂t + u ∂c ∂z − Dz ∂2c ∂z2 + F(1 −ηcore)k(φ ∗ −φ) = 0, (1) ∂T ∂t + u ∂T ∂z − λz ξ f ∂2T ∂z2 + F(1 −ηcore) 3hp Rpξ f (T − T s) = 0. (2) From the equations above, the symbol c represents the solute mixture in the mobile phase, the interstitial velocity is donated as u, the symbol Dz represents the axial-dispersion coefficient, while the symbol � denotes the external porosity and the phase ratio is expressed as F = 1−� � . Also, the symbol φ represents the non-equilibrium average loading of concentration in the partic- ular solid phase, in the mobile phase the temperature is indi- cated as T , and coefficients of heat conductivity along with the axial coordinates is represented as λz. Moreover, ξe = ρscsp, ξ f = ρ lclp, ρ s and ρl indicate the solid and liquid phases’ densi- ties per unit volume, respectively, T s stands for the temperature of the stationary phase, while csp and c l p symbolize the solid and liquid phases’ respective heat capacities. In the solid phase, the corresponding mass and heat balance equations are written as ∂φ ∂t =k(φ∗ −φ), (3) ∂T s ∂t = −∆HA ξe ∂φ ∂t + 3hp Rpξe (T − T s) . (4) From the equations above, the symbol k stands for the mass transfer rate coefficient, ∆HA represents the enthalpy of adsorp- tion and the coefficient for heat transfer among the mobile and solid phases is symbolized as hp. For the ι-th number of com- ponents in the mixture, the symbol φ∗(c, T s) shows a tempera- ture dependency relationship among the specific phase in solid part of concentration at equilibrium φ and the temperature, is defined as [16] φ∗(c, T s) = aref c exp [ −∆HA Rg ( 1 T s − 1 Tref )] 1 + brefι cι exp [ −∆HA Rg ( 1 T s − 1 Tref )] , ι = 1, · · · , Nc. (5) Here, the coefficient aref is symbolizing the Henry’s constant at a reference temperature, the symbol brefι is representing the nonlinearity coefficients, the symbol Tref represents the refer- ence temperature, and Rg is universal gas constant. In addition, the model includes the following additional dimensionless pa- rameters, which minimize the parameters in terms of number, this facilitates the analysis of the model. Pez,M = Lu Dz , Pez,H = ξ f Lu λz , x = z L , τ = ut L , κ = Lk u , βs = 3Lhp uRpξe , βL = 3Lhp uRpξ f , (6) The column length is represented by the symbol L, and the Peclet-numbers of both heat and the mass transfer via axial di- rection are indicated by the symbols Pez,H and Pez,M , accord- ingly. On utilizing Eq. (6) in Eqs. (1)-(4), we get the following equations below after some manipulations ∂c ∂τ = − ∂c ∂x + 1 Pez,M ∂2c ∂x2 − F(1 −ηcore)κ(φ ∗ −φ), (7) ∂T ∂τ = − ∂T ∂x + 1 Pez,H ∂2T ∂x2 − F(1 −ηcore)βL(T − T s), (8) ∂φ ∂τ =κ(φ∗ −φ), (9) ∂T s ∂τ = −∆HA ξe ∂φ ∂τ + βs(T − T s). (10) Appropriate initial conditions that are suitable for the computa- tional solution of the model equations (c.f. Eqs. (7)-(10)) in the range 0 ≤ x ≤ 1. The following are the initial conditions for an equilibrated column: c(x,τ = 0) = cinit, T (x,τ = 0) = Tinit, φ(x,τ = 0) = φ∗init, T s(x,τ = 0) = Tinit. (11) Here, the initial temperature is represented by Tinit, the symbol cinit is the initial equilibrated concentration of the single compo- nent solute, and φ∗init is obtained from Eq. (5). The inflow condi- tions which Dankwert’s boundary conditions (BCs) investigate at the column inlet are listed below [9, 31]. The injections in the inner circular region are described as follows: c(x = 0,τ) − 1 Pez,M ∂ci(x = 0,τ) ∂x = { cinj, if 0 ≤ τ ≤ τinj, 0, if τ > τinj, (12a) T (x = 0,τ) − 1 Pez,H ∂T (x = 0,τ) ∂x = { Tinj, if 0 ≤ τ ≤ τinj, Tref, if τ > τinj. (12b) In this Eq. (12a), cinj represents the concentration of the injected component, the dimensionless time of injection is represented by τinj, and Tinj represents the temperature of the injected com- ponent. Moreover, the thermal conditions at the boundary pro- vided by Eq. (12b) allow the temperature of the injection sam- ple to fluctuate. Further, the injected temperature donated by Tinj could be adjusted from the reference temperature of the bulk phase Tref . At the right end of the column, the Neuman conditions are considered: ∂c(x = 1,τ) ∂x = 0 , ∂T (x = 1,τ) ∂x = 0. (12c) The chromatographic process mathematical model is now com- plete. The subsequent task is to employ the suggested flux- limiting HR-FVS in order to solve the developed model equa- tions. 3. Numerical scheme In order to approximate the current model equations numer- ically, a flux-limiting semi-discrete method HR-FVS is applied, 3 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 4 Table 1: Values of the model parameters used in the test prob- lems Parameters Values Column length L = 4.0 cm Radius of the column R = 0.2 cm Radius of solid particle Rp = 0.004 cm Interstitial velocity u = 1.5 cm/min Porosity � = 0.4 Density of heat capacity of solid ξf = 4 kJ/l Density of heat capacity of liquid ξe = 4 kJ/l Axial dispersion coefficient Dz = 0.01 cm2/min Axial conductivity coefficient λz = 0.04 kJcm−1min−1 Mass transfer coefficient k = 1 cm/min Heat transfer coefficient hp = 1 W(cm2K)−1 Reference temperature Tref = 300 K Initial temperature Tinit = Tref Inlet temperature Tinj = Tref Initial concentration cinit = 0 mol/l Inlet concentration cinj = 1 mol/l Dimensionless injection time τinj = 1.5 Adsorption equilibrium constant aref = 1 Figure 1: Sketch of solute injection in the thermally insulated chromatographic column incorporating core-shell particles which have been widely discussed in the literature for approxi- mating 1D-models [9, 10]. To simulate the ODE system in term of time, the second-order TVD Runge-Kutta method is used. For the derivation of this numerical scheme, let us descritise the computational domain. Let N stand for the number descriti- sation, xl+ 12 to be the interval of left and right boundaries, ∆x is the cell width and xl symbolizes the cell center. Further, let us assign the following xN+ 12 = L, x 12 = 0, xl+ 12 = l∆xl, (13) xl = xl−12 − xl+ 12 2 , ∆xl = xl− 12 − xl+ 12 = L N + 1 . (14) The domain of cartesian grid [0, 1] that is fully covered by the cells Ψl = xl− 12 − xl+ 12 for l ≥ 1. Furthermore, the average initial data wl(0) in each interval are formulated as wl(0) = 1 ∆x ∫ x l+ 12 x l− 12 w(x, 0)d x, w ∈ {c, T, q, q∗}, l = 1, 2, · · · , N. (15) 0 2 4 6 8 10 12 τ 0 0.05 0.1 0.15 0.2 0.25 c (x ,τ ) [m o l/ l] η core = 0.0 η core = 0.4 η core = 0.8 (a) Figure 2: These display the non isothermal single-solute elu- tion profile for distinct values of fraction of core radius ηcore. Specifically, ∆HA = −10 kJ/mol. Table 1 lists all of the other parameter values that were chosen Once we discretized the computational domain and the associ- ated initial data for τ = 0 is specified for each mesh interval, the subsequent task is to employ the proposed scheme. Integrations of Eqs. (7)-(10) over Ψl give dcl dτ = − ( cl+ 12 − cl− 12 ) ∆xl + 1 ∆xlPez,M [ ( ∂c ∂x ) l+ 12 − ( ∂c ∂x ) l− 12 ] (16) − Fκ(φ∗l −φl), dTl dτ = − ( Tl+ 12 − Tl− 12 ) ∆xl + 1 ∆xlPez,H [ ( ∂T ∂x ) l+ 12 − ( ∂T ∂x ) l− 12 ] + (17) − FβL(Tl − T s,l), dφl dτ =κ(φ∗l −φl), (18) dT s,l dτ = −∆HA ξe κ(φ∗l, j −φ1,l, j) + βs(Tl − T s,l). (19) The derivatives appearing in Eqs. (16) and (17) are approxi- mated as [ ∂c ∂x ] l± 12 = ± [ cl±1 − cl ∆xl ] , (20)[ ∂T ∂x ] l± 12 = ± [ Tl±1 − Tl ∆xl ] . (21) 3.1. Koren HR-FVS According to the first order method, the values of concen- tration and with temperature of the cell-interface in Eqs. (16) 4 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 5 0 2 4 6 8 10 τ 0 0.02 0.04 0.06 0.08 0.1 0.12 c (x ,τ ) [m o l/ l] Pe z,H =600, η core =0.6 Pe z,M =50 Pe z,M =100 Pe z,M =600 (a) ∆ H A = -10 kJ/mol 0 2 4 6 8 10 τ 299.6 299.8 300 300.2 300.4 300.6 T (x ,τ ) [K ] Pe z,H =600, η core =0.6 Pe z,M =50 Pe z,M =100 Pe z,M =600 (b) ∆ H A = -10 kJ/mol Figure 3: Influence of Pez,M on the model of non isothermal for the single-component elution by utilising two different ηcore values. Table 1 lists all of the other parameter values that were considered and (17) are approximated as numerically cl+ 12 = cl, cl− 12 = cl−1, Tl+ 12 = Tl, Tl− 12 = Tl−1. (22) For the second-order HR-FVS, the developing flux-limiting for- mula is utilized in Eqs. (16) and (17) to numerically approx- imate the values of temperature and concentration of the cell interface [9, 30]: cl+ 12 = cl + 1 2 Φ1(Γl+ 12 )(cl − cl−1), (23) Tl+ 12 = Tl + 1 2 Φ2(Λl+ 12 )(Tl − Tl−1), (24) Here, the symbols Φ1 and Φ2 are the flux-limiting formula for the concentrations and temperature, respectively. Also, Γl+ 12 , j and Λl+ 12 , j are the concentration and temperature gradient ratios, accordingly. Γl+ 12 , j = cl+1 − cl + δ cl − cl−1 + δ , Λl+ 12 = Tl+1 − Tl + δ Tl − Tl−1 + δ . (25) In order to avoid division by zero, we have taken the value of δ = 10−10. Moreover, the limiting functions are expressed as Φ1(Γl+ 12 , j) = max [ 0, min ( 2Γl+ 12 , j, min ( 1 3 + 2 3 Γl+ 12 , j ))] , (26) Φ2(Λl+ 12 , j) = max [ 0, min ( 2Λl+ 12 , j, min ( 1 3 + 2 3 Λl+ 12 , j ))] , (27) where Φ1(Γl+ 12 , j) and Φ2(Λl+ 12 , j) are sequentially defined for the concentration and temperature. Likewise, cl− 12 and Tl− 12 can be evaluated by just replacing the index l by l − 1 in the equa- tions above. Finally, the built-in Rk-45 in Matlab is used to simulate the resulting ODE-system in Eqs. (16) and (17). The entire scheme described above was programmed and simulated in MATLAB. 4. Evaluating criterion for the process performance In this section, we presents an evaluation for the perfor- mance criterion that can be utilized to improve product qual- ity, according to the findings of Hováth and Fellinger [29] re- search. For applications to industries, preparative chromato- graphic methods required to be optimized in terms of their yield, productivity and efficiency. To describe this criterion proce- dure, we used a two-component mixture with component 2 hav- ing a higher reference affinity to the solid phase than component 1. i.e. a1ref < a 2 ref . Let ζ 1 represent the non-dimensional time when the fraction of component 1 be more than an appropriate benchmark (i.e, c1 < �c1,inj), where � = 10−6. In a similar way, let ζ2 stand for a non-dimensional time when the concentration of component 2 drops below a particular level (c2 < �c2,inj). The amount of time that has elapsed between the injections of consecutive two values, i.e. as cycle time, is represented by ζcyc and is written as ζcyc = ζ2 − ζ1. (28) The cut time is defined as the time it takes to complete compo- nent 1 fractionation Pur = ∫ ζcut ζ1 c1(x = 1,ζ)dζ∫ ζcut ζ1 [c1(x = 1,ζ)dζ + c2(x = 1,ζ)dζ] , (29) where ck(x = 1,ζ) = 2 ∫ 1 0 ck(x = 1,η,ζ)ηdη, k = 1, 2. (30) The productivity (Pr) is defined by the quantity of a required compound manufactured per cycle’s time. Also, the purity of 5 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 6 0 2 4 6 8 10 12 τ 0 0.1 0.2 0.3 0.4 0.5 c (x ,τ ) [m o l/ l] k = 1 min −1 k = 100 min −1 (a) 0 2 4 6 8 10 12 τ 299.4 299.6 299.8 300 300.2 300.4 300.6 300.8 T (x ,τ ) [K ] k = 1 min −1 k = 100 min −1 (b) 0 2 4 6 8 10 12 τ 0 0.05 0.1 0.15 0.2 0.25 0.3 c (x ,τ ) [m o l/ l] h p =0.5 W(cm 2 K) -1 h p =2.0 W(cm 2 K) -1 (c) 0 2 4 6 8 10 12 τ 299.7 299.8 299.9 300 300.1 300.2 300.3 T (x ,, τ ) [K ] h p =0.5 W(cm 2 K) -1 h p =2.0 W(cm 2 K) -1 (d) Figure 4: Impacts of both coefficients of mass as well as transfer of heat (k and hp) for unchanged ηcore = 0.8 and ∆HA = −10 kJ/mol. Table 1 lists all of the other parameter values that were considered the suitable peak area is placed equal to 99%. For component 1, it is obtained as Pr = ∫ ζcut ζ1 c1(x = 1,ζ)dζ ζcyc . (31) The proportion of considered component evaluated in segre- gated order and the complete amount of that injected compo- nent over the column inlet is referred as the quantity yield (Y). For the component 1 case, it is obtained as Y = ∫ ζcut ζ1 c1(x = 1,ζ)dζ∫ ζ2 ζ1 c1(x = 1,ζ)dζ . (32) 5. Numerical test cases Numerous numerical test cases for simulating the influences of ηcore, Pez,M , k and hp, on the elution profiles are presented in this section. These characterize the core radius fraction, the axial Peclet number, the mass transfer coefficient and the heat transfer coefficient, respectively. In addition, we have taken the nonlinearity coefficients zero (i.e., bι = 0 l/mol in Eq. (5)). Moreover, a process performance criterion for the parameters ζcyc, ζcut, Pr, and Y are simulated. Furthermore, the solute is injected through the column inner zone in all of the cases we investigated. In the plots in Figures 2-7, the solute component is denoted as c. While, the mobile phase temperature is denoted by the symbol T . The values of all needed parameters are taken from the scopes used in the applications of HPLC [32] and are classified in Table 1. In Figure 2, we display the elution profile plots of both con- centration and temperature of a single-component solute, for three distinct values of ηcore (i.e. ηcore = 0, 0.4, 0.8). Addition- ally, the value of adsorption enthalpy of the process is taken as ∆HA = −10 kJ/mol. It is obvious that operation of non- isothermal results in notable temperature differences within the column, as shown in given Figure 2(b). As shown in Figure 2(a), these temperature variations have no visible effect on the concentration profile due to the considered low value of ∆HA. It is also worth noting that as ηcore increases from 0 to 0.8 in- creases, the elution profiles become sharper, and as a result, their retention times reduce accordingly, i.e. By increasing the value of ηcore, the column efficiency gradually improves. At 6 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 7 0 2 4 6 8 10 12 τ 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 c (x ,τ ) [m o l/ l] η core = 0.0 ∆ H A = -10 kJ/mol ∆ H A = -20 kJ/mol ∆ H A = -40 kJ/mol (a) 0 2 4 6 8 10 12 τ 299 299.5 300 300.5 301 301.5 302 302.5 303 T (x ,τ ) [K ] η core = 0.0 ∆ H A = -10 kJ/mol ∆ H A = -20 kJ/mol ∆ H A = -40 kJ/mol (b) 0 2 4 6 8 10 12 τ 0 0.05 0.1 0.15 0.2 0.25 c (x ,τ ) [m o l/ l] η core = 0.8 ∆ H A = -10 kJ/mol ∆ H A = -20 kJ/mol ∆ H A = -40 kJ/mol (c) 0 2 4 6 8 10 12 τ 299 299.5 300 300.5 301 301.5 T (x ,τ ) [K ] η core = 0.8 ∆ H A = -10 kJ/mol ∆ H A = -20 kJ/mol ∆ H A = -40 kJ/mol (d) Figure 5: Effects of ∆HA on the model of non-isothermal for single-component elution including two different values of ξcore. Specifically, plots in 2-D for the single solute elutions profiles are presented at ξcore = 0 & ξcore = 0.8. Table 1 lists all of the other parameter values that were employed the same time, the column’s absorption capacity gradually de- creases due to a reduce in the thickness of the layer including in porous. Figure 3 demonstrates the impact of the non-dimension pa- rameter Pez,M of concentration and temperature on the profiles of elution by utilizing the value of ηcore = 0.6 and an unchanged value of Pez,M = 600. The lesser value of Pez,M develops wider (spread) peaks, and thus the column efficiency decreases. In contrast, a higher Pez,M value results in narrower peaks, which improves column performance. All plots clearly show analo- gous impacts of different Pez,M . Figure 4 illustrates the results of k and hp for an unchanged ηcore = 0.8. It is clear that lower values of these dimension pa- rameters result in broader elution profiles, while for high values of these parameters, the profiles are sharpened. The plots also show that the parameter hp has a minor influence on the given profiles, whereas k has a significant impact on both profiles. Figure 5 presents the influence of ∆HA on the eluent profile for two distinct ηcore values of a single-solute. As the operating condition for non-isothermal (∆HA = −10, −20 & −40 kJ/mol) produces remarkable temperature differences, as it’s clearly ob- served from Figure 5. Also, increasing the magnitude of ∆HA affects the solute profile as well, and causing the concentration profile to become sharper while the peak moves upward. Fur- ther, variation of the particles size of core-shell decreases the retention time of the solute profile by making them narrower and sharper. In Figure 6, for isothermal and non-isothermal conditions, we describe a process performance by simulating the following terms: ζcyc, ζcut, Pr, and Y (c.f. Eq. (28)-(29)) over ηcore for ∆HA = −10 kJ/mol. It can be clearly observed that the time cy- cle as well as the cut time is decreasing from 36 to 12 and 97 to 10, respectively, as varies from entirely particles that are porous i.e. ηcore=0.0 to the particles of core shell i.e. ηcore=0.8. More- over, a rise in the productivity can be seen up to ηcore = 0.74 and it gradually falls later. Furthermore, the Y continuously increases for an upward increment of the ηcore value. So, the cycle time, cut time, productivity and yield profiles are similar 7 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 8 0 0.2 0.4 0.6 0.8 1 η core 5 10 15 20 25 30 35 40 ζ c y c Isothermal Non-isothermal (a) 0 0.2 0.4 0.6 0.8 1 η core 0 20 40 60 80 100 ζ c u t Isothermal Non-isothermal (b) 0 0.2 0.4 0.6 0.8 1 η core 0 1 2 3 4 5 6 P r ×10 -3 Isothermal Non-isothermal (c) 0 0.2 0.4 0.6 0.8 1 η core 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Y Isothermal Non-isothermal (d) Figure 6: Isothermal and non-isothermal comparison for the process performance assessment. Here, the value for ∆HA = −40 kJ/mol is taken for the model of non isothermal process operating condition. Firstly, the plot (a) presents ζcyc, secondly, the plot (b) shows ζcut, thirdly the plot (c) depicts productivity (Pr) and the last plot (d) displays yield (Y) represents as a functions of ηcore for c1,inj = 1 = c2,inj. Table 1 lists all of the other parameter values that were considered for both isothermal and non-isothermal operations, as presented in Figure 5. In Figure 7, we discussed a performance process to the as- sessment for both conditions of isothermal and non-isothermal. However, here we change the magnitude of ∆HA from ∆HA = −10 kJ/mol to ∆HA = −40 kJ/mol over ηcore. The cycle and cut times are both decreasing, as can be seen from varying 36 to 12 and 97 to 10, respectively, when we get away from parti- cles that are completely porous (i.e. ηcore=0.0) to the particles of core-shell (i.e. ηcore=0.8). Similarly, the performance as- cends upward and drops thereafter around ηcore = 0.74. More- over, the yield rises upward continuously. However, significant deviations in the cycle time and productivity profiles of non- isothermal and isothermal operations are detected. In the case of non-isothermal operating conditions, the reduction in cycle time decelerates slightly, but yield and productivity improve. 6. Conclusion The effects of temperature variations were theoretically in- vestigated on fixed-bed columns filled with core-shell parti- cles. For that purpose, a nonlinear 1D-LKM was developed and solved numerically. It was found that rapid and better sep- arations of complex samples can be obtained in columns filled with core-shell particles. The findings showed that profiles with sharper peaks and shorter residence times are produced by core radius fractions with higher values. Thus, the diffusion path within the adsorbents was reduced, which boosted the column’s efficiency. Furthermore, the numerical results show that heat and concentration front interactions were investigated, which indicates an increase in the process performance assessment. Consequently, HPLC can be optimized by using core-shell par- ticles with a sufficient core radius fraction loaded in the column. 8 Ahmad et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1350 9 0 0.2 0.4 0.6 0.8 1 η core 5 10 15 20 25 30 35 40 ζ c y c Non-isothermal Isothermal (a) 0 0.2 0.4 0.6 0.8 1 η core 0 20 40 60 80 100 ζ c u t Non-isothermal Isothermal (b) 0 0.2 0.4 0.6 0.8 1 η core 0 1 2 3 4 5 6 P r ×10 -3 Non-isothermal Isothermal (c) 0 0.2 0.4 0.6 0.8 1 η core 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Y Non-Isothermal Isothermal (d) Figure 7: Isothermal and non-isothermal process comparison for the process performance assessment. Specifically, the value for ∆HA = −40 kJ/mol is taken for the process of non-isothermal operating condition. 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