## Abstract

The advection-convection models (ACM) have practical applications in the investigation of separation processes, where mass (heat) is transferred by convection and diffusion (dispersion) along mass/heat exchanger, eq. adsorption, chromatography column, tubular reactor, etc. The ACM consists of nonlinear partial differential equations which can be solved only with numerical methods. In the article, a comparison of the volume method (VM) and orthogonal collocation on finite elements (OCFE) is presented in terms of their reliability, accuracy of calculations, and speed of calculation. The OCFE proved to be more robust than VM.

The linear ACM model for the chromatography column has an analytical solution in the form of the equation for the number of theoretical plates (N). This equation is often applied in the interpretation and evaluation of model parameters. However, the versions of *N* equation published in the literature are not correct. The error can lead to significant imprecision for specific cases. Here, in the paper, the revised equations are presented and discussed for the most frequently used chromatography column models.

## 1 Introduction

The advection-convection model of chromatography column is a combination of the diffusion and convection (advection) equations that describe physical phenomena where molecules, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion (dispersion) and convection. In particular, the process can take place in an empty tube or tube filled with some packing, eq. adsorbent, catalyst [1–3].

The ACM consists of a mass transfer (heat transfer) equation for the moving phase and particle of packing. From the mathematical point of view, the ACM is described with parabolic partial differential equations (PDE). The ACM coupled with additional equations describing phenomena taking place on or inside particles of the packing can be used for modeling separation processes in chemical engineering, biotechnology, modeling chromatography columns, tubular reactors, and other processes [4, 5].

The ACMs include empirical parameters such as mass transfer coefficients dispersion coefficients and others. These parameters can be estimated based on experimental data and analytical solutions of the linear version of ACM as it is discussed in section 3.1.

In practice, almost always the ACMs are nonlinear and can be solved only with a numerical method [3].

In this paper, we tested the VM versus OCFE, solving chromatography column models, which belong to so-called stiff models. The difficulties in the solution of the chromatography column model are connected with a steep concentration front (called shock) which can appear inside the column [3]. The solution of these models in a reasonable time requires the application of sophisticated numerical procedures. The numerical procedures which are most robust in the case of solution of chromatography separation should be also robust in modeling other separation processes as well as for solving heat transfer problems (because the PDEs for heat and mass transfer are analogous).

In the following, we concentrate on chromatography column models.

In the general case of the processes that take place in packed beds, one PDE has to be formulated for the mobile phase percolating through the bed between the particles of adsorbent and a second PDE for mass balance for the particle. The PDEs are coupled with an ordinary differential equation describing the kinetics of the adsorption-desorption process or with an algebraic expression for the isotherm model.

The model that encompasses mass balances for mobile phase and adsorbent particle is called a general rate model (GRM). The GRM model was widely discussed in [1–3, 6].

When mass transfer resistances inside the particle and from the bulk phase to the particle are negligible then GRM can be simplified to an equilibrium-dispersive model (EDM) assuming that the adsorption-desorption process is instantaneous [3, 7]. For the EDM the set of PDE reduces to one PDE for the mobile phase for each component.

The GRM and EDM have no closed-form analytical solution in most cases and can be solved only with numerical methods. A variety of methods are available to derive numerical solutions. The simplest are finite-difference methods widely discussed in [3]. The principle of the finite difference methods consists of approximating the spatial and time derivatives with finite difference terms based on space Δz and time Δt increments. These methods are characterized by so-called numerical diffusion. The space Δz and time Δt increments are set in such a way that the numerical diffusion simulates the band dispersion in the column as is done in Rouchon et al. [8]. Similarly, choosing space and time increments for the Craig method is discussed in Czok and Guiochon [9]. However, with these methods, it is impossible to correctly simulate band dispersion for the species in a multicomponent mixture if the apparent dispersion for different species is not the same. Even for one-component chromatography and a more sophisticated isotherm, the Rouchon method can fail [10].

The overestimation or underestimation of the apparent dispersion may have a serious impact on the estimates of isotherm parameters with the inverse method and the optimization of the conditions of species separation.

More robust numerical algorithms are obtained when the finite volume method is used. Two volume methods are recommended in the literature to simulate chromatography separation. Von Lieres and Andersson [11] used the weighted essentially non-oscillatory (WENO) scheme [12, 13]. The WENO algorithm is used to approximate the convection term in PDE. The non-oscillatory character of this scheme enables to use of a coarser net of nodal points compared to the finite difference algorithm and decreases the time of computation. The second derivative term of PDE is calculated with a second-order centered difference approximation.

The WENO scheme of order 3 was applied in an open platform for integrated process modeling and simulation – CADET [14]. CADET is released under the GNU GPL v3. The latest version of CADET is v4.3.0 and can be copied from https://cadet.github.io/v4.3.0/index.html.

The second volume method called the Koren method [15] uses a robust upwind discretization method for advection term. It belongs also to non-oscillatory methods. The second derivative term of PDE is calculated with a second-order centered difference approximation. The Koren method in application to modeling of the chromatography process is discussed and recommended among others in the papers [16–19] Unfortunately, Danckwerts boundary condition was not implemented in this method.

In chromatography columns for concentration overload conditions, a very steep gradient can develop. One of the methods that can be effectively used to simulate steep gradients of concentration peak profiles is the orthogonal collocation on finite elements (OCFE). The OCFE method was widely discussed in the fundamental book of Villadsen and Michelsen [20]. In the case of chromatography separation simulation, the OCFE was initially applied to the solution of the EDM model by Ma and Guiochon [21] and next to gradient chromatography by Antia and Horváth [22].

In the OCFE method, space is divided into NS subdomains, commonly referred to as finite elements. In each subdomain, the NICP internal collocation points are chosen as the roots of the NICPth order Legendre polynomial. The unknown function is represented within each element by an interpolating polynomial which is continuous and has continuous derivatives. The details of approximate spatial derivatives by algebraic equations in the OCFE method are given in [23].

This work aimed to compare the accuracy, speed of calculation, and robustness of the Koren method, WENO methods, and CADET program with orthogonal collocation on a fixed finite element. To do it the PDEs were solved using the method of lines. The spatial derivatives were discretized according to the Koren, WENO, or OCFE method. The obtained set of ordinal differential equations was solved with the CVODE solver (SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic Equation Solvers, https://computing.llnl.gov/projects/sundials).

The BDF method was applied for all calculations. The Jacobian was calculated analytically in the case of the CADET program and numerically in other cases. The relative and absolute errors were set to 10^{−6} and 10^{−8} respectively.

The accuracy of the simulation results is usually evaluated by comparing solutions on computational grids with increasing numbers of elements (cells, nodal points). However, the correctness of the simulation code can only be checked by comparing the results with reference solutions that have been computed with a different code, ideally based on a fundamentally different mathematical algorithm for solving the model equations.

The accuracy of the numerical integration can be evaluated based on an analytical solution if available. In the case of chromatography column models the analytical solution is possible only for linear adsorption isotherm or linear adsorption-desorption kinetics. The analytical solution is expressed by moments of chromatographic peaks. The ratio of the square of the first absolute moment at the outlet of the column to the second central moment gives the equation for the number of theoretical plates, N. This expression can be used for the analysis of column efficiency, and estimation of the values of the model parameters.

The moment analysis for the general rate chromatography column model has been comprehensively discussed in the literature for fully porous particles, core-shell particles, for slow adsorption process as well as for linear adsorption isotherm – see for example [1, 3, 17, 18, 24–33].

However, comparing the calculated *N* value from formulae taken from literature with *N* calculated from the numerical solution the discrepancies between both values increasing with decreasing Peclet number (increasing dispersion) were observed. The reason for these discrepancies was no correct analytical solution of GRM and EDM coupled with the Danckwerts boundary conditions. So, the second aim of the work became the development of correct expression for *N* for fully porous adsorbent, core-shell adsorbent, slow or infinitely fast adsorption process, and rectangular injection profiles.

## 2 Mathematical models

In this section, the formulation of the GRM for core-shell adsorbent and finite adsorption rates are presented. We will also present an equation for a simple equilibrium dispersive model which is frequently applied for simulation band profiles. The models are transformed into dimensionless forms, which facilitates the analysis of investigated processes. Next, the analytical solution of these models, represented by equations for a retention time of sample band profile and equations for a number of theoretical plates are developed and discussed.

### 2.1 General rate and equilibrium dispersive model

The general rate model used in the paper is analogical to that analyzed in [3, 7, 26].

We assumed core-shell adsorbent and finite adsorption-desorption kinetic.

The Equations (1)–(4) have to be coupled with the initial and boundary conditions:

*t*= 0) are:

The meaning of the symbols used in the above equations is:

*c*and*c*_{p}– a concentration in the mobile phase or the stagnant fluid phase contained in the pores,*D*_{eff}– effective particle diffusivity,*D*_{L}– axial dispersion coefficient,*H = Kq*_{s}is Henry constant,*k*_{ads}*, k*_{des}– adsorption and desorption rate constant,*K*=*k*_{ads}*/k*_{des}– equilibrium constant,*k*_{ext}– external mass transfer coefficient,*L*– column length,*q*– concentration in the stationary phase,*q*_{s}– saturation capacity,*r*– radial coordinate,*R*_{i}– the radius of inert solid core,*R*_{p}– particle radius,*t*– time,*t*_{p}– time during the constant concentration is fed into the column,*u*– superficial velocity,*z*– axial coordinate,*ε*_{e}and*ε*_{p}– external and particle porosity,subscript f denotes the inlet value and the superscript

^{o}initial value.

*D*and

_{eff}*k*(external and internal mass transfer resistance decrease to zero) the concentration

_{ext}*c*in the mobile phase is equal to the concentration

*c*inside pores and the GRM reduces to the following EDM [7]:

_{p}*ε*

_{t}is the total column porosity.

Equation (9) is solved with Eq. (3) and the analogical to Eq. (6) and Eq. (7) initial and boundary conditions. The *D _{L}* denotes the dispersion coefficient. However, in practice, the

*D*is replaced by the apparent dispersion coefficient,

_{L}*D*in which the mass transfer resistances are lumped [26].

_{a}It will be convenient to rewrite these models in dimensionless form.

*c*, was assumed to be equal to 1.

_{r}The dimensionless modules characterize the external mass transfer resistances (St – Stanton number), dispersion (Pe – Peclet number), internal mass transfer resistances (Bi – Biot number), and rate of adsorption (

The dimensionless initial and boundary conditions are:

*τ*= 0):

The boundary and initial conditions are similar to those of the GR model.

### 2.2 Number of theoretical plates, general rate model, Danckwerts boundary conditions

*N*. The number of theoretical plates can be calculated analytically based on the solution of the chromatography column model in the Laplace domain. The

*N*is equal to the ratio of the square of the first absolute

The method of calculation for both moments applied in this work was described in the Appendix.

The first absolute moment µ_{1} is equal to the retention time τ_{r} of the peak center of gravity.

*N*are:

*ρ*= 0) equation (21) becomes:

### 2.3 Number of theoretical plates, general rate model, porous adsorbent particle, and Dirichlet boundary conditions

The development of the equations for *N* is the same as for Dankwerts conditions. Theoretically, the solution of the mass transfer model with the Danckwerts boundary conditions gives a more accurate solution compared with that obtained using the Dirichlet condition. However, the Dirichlet conditions can be used when Pe/2 is greater than about 100 [35].

### 2.4 Number of theoretical plates, equilibrium-dispersive model, Danckwerts boundary conditions, and fully porous adsorbent particle

*N*for the equilibrium-dispersive model are expressed by Eq. (26) and Eq. (28).

## 3 Results and discussion

This section consists of two subsections. First subsection 3.1 presents the validation of developed equations for dimensionless retention time, τ_{r,} and the number of theoretical plates, *N*, called in the following analytical solution. The validation is based on a comparison of analytical solutions with τ_{r} and *N* calculated from numerically obtained band profiles. The calculations were made using the OCFE method.

In section 3.2 the accuracy of computations using the OCFE method were compared with calculation based on volume methods.

In the case of OCFE, the total number of nodal points (NP) on the computational grid in the *z* direction was equal to NS*NICP. The NICP was always equal to 3. In the r direction, one subdomain with five collocation points was applied. The drawback of the OCFE method is the oscillatory character of the numerical solution. The oscillations decrease with increasing the number of subdomains, NS. In this work, the NS was chosen in such a way that the maximum amplitude of oscillations was less than about 10^{−5} of the maximum value of peak concentration at the column outlet.

### 3.1 Validation of the expressions on the number of theoretical plates

For

*N*is:

It is easy to check that for Pe = 1 the error is equal to about 100% but for Pe > 100 the error decreases below 1%.

*N*, is a comparison of

*N*values with the numerically calculated number of theoretical plates, N

_{nc}. To calculate N

_{nc}the GRM was solved with the method of orthogonal collocation on finite elements [20, 23] (see section 3.2) for several combinations of a dimensionless modulus: Bi, St, Pe, and Φ. Next, for the obtained band profiles the first and second moments were calculated according to the expressions [3]:

The integrals were calculated with the trapezoidal method and finally, the N_{nc} was estimated from Eq. (19).

Table 1 presents a comparison between τ_{r} and *N* calculated numerically and from formulas developed in section 2.2). The computations were performed for H = 3, y_{f} = 1, τ_{p} = 0.001, ε_{p} = 0.5, ε_{e} = 0.4. The values of dimensionless modulus were chosen to simulate the column work for very fast and very low mass transfer resistances, slow and fast adsorption kinetics, and extremely large and negligible axial dispersion.

Comparison of *N* and τ_{r} evaluated from the analytical and numerical solution. Danckwerts boundary conditions

Dimensionless modulus | Analytical solution, Eqs. (20, 21) | Numerical calculation – OCFE | ||||||

St | Bi | τ_{r} | N | τ_{r} | N_{nc} | CPU [s] and (NP) | ||

Pe | ρ = 0 | |||||||

0.1 | 1,000 | 1 | 1 | 4.0005 | 1.0302 | 3.9976 | 1.0211 | 1.2 (30) |

1 | 1,000 | 1 | 1 | 4.0005 | 1.3532 | 4.0003 | 1.3528 | 1.5 (30) |

10 | 1,000 | 1 | 1 | 4.0005 | 5.4529 | 4.0005 | 5.4530 | 1.5 (30) |

50 | 1,000 | 1 | 1 | 4.0005 | 23.463 | 4.0005 | 23.463 | <1 (30) |

100 | 1,000 | 1 | 1 | 4.0005 | 43.056 | 4.0005 | 43.053 | 2.6 (90) |

1,000 | 1,000 | 1 | 1 | 4.0005 | 184.30 | 4.0005 | 184.30 | 2.8 (90) |

10,000 | 10,000 | 0.1 | 1 | 4.0005 | 3314.3 | 4.0001 | 3314.1 | 58.7 (300) |

10,000 | 10,000 | 0.1 | 100 | 4.0005 | 3617.7 | 4.0005 | 3617.3 | 53.9 (300) |

10,000 | 10,000 | 1,000 | 100 | 4.0005 | 65.374 | 4.0005 | 65.372 | 1.5 (90) |

10,000 | 10,000 | 1,000 | 1 | 4.0005 | 3.7267 | 4.0003 | 3.7255 | 3.3 (90) |

100,000 | 100,000 | 0.1 | 100 | 4.0005 | 36,168 | 4.0005 | 36,165 | 109 (1,200) |

100,000 | 100,000 | 0.1 | 1 | 4.0005 | 33,136 | 4.0005 | 33,117 | 467 (1,200) |

100,000 | 100,000 | 0.1 | 0.01 | 4.0005 | 39.473 | 4.0004 | 39.484 | 4.7 (90) |

ρ | Pe = 1,000 | |||||||

0.3 | 1,000 | 1 | 1 | 3.9195 | 183.82 | 3.9283 | 183.90 | 5.6 (90) |

0.7 | 1,000 | 1 | 1 | 2.9715 | 176.85 | 2.9716 | 176.78 | 2.9 (90) |

0.98 | 1,000 | 1 | 1 | 1.1769 | 266.95 | 1.1769 | 266.95 | 2.3 (90) |

The difference between analytical and numerical solutions is in almost all cases less than about 0.05%. Only for Pe = 0.1 this difference is about 0.9%. These differences can be smaller for a lower value of relative error set in the CVODE solver. For example, for Pe = 0.1 and the relative and absolute error equal to 10^{−10} and 10^{−12} respectively the τ_{r} = 4.0004, *N* = 1.0301, and the difference between analytical and numerical solution is less than 0.01%.

However, an increase in accuracy is occupied by increasing CPU time which was equal to 3.3s in this time.

Table 2 presents a comparison between τ_{r} and *N* obtained when Dirichlet boundary conditions were assumed. The agreement between analytical and numerical solutions is the same as before. From the comparison of the *N* values presented in both Tables follows that difference between solutions obtained with simple Dirichlet conditions and Danckwerts conditions is less than 1% for Pe greater than about 100 – a fact known in the literature [3].

Comparison of *N* and τ_{r} evaluated from the analytical and numerical solution. Dirichlet boundary conditions. Calculations were done for St = 1,000, Bi = 1, *ρ* = 0

Pe | Analytical solution, Eqs. (25, 26) | Numerical calculation – OCFE | |||

τ_{r} | N | τ_{r} | N_{nc} | CPU [s] and (NP) | |

0.1 | 0.19399 | 1.3793 | 0.19399 | 1.3794 | <1 (30) |

1 | 1.4720 | 1.7006 | 1.4716 | 1.7036 | 1.3 (30) |

10 | 3.6005 | 5.2919 | 3.6005 | 5.2936 | <1 (30) |

50 | 3.9205 | 23.224 | 3.9206 | 23.232 | <1 (30) |

100 | 3.9605 | 42.815 | 3.9605 | 41.782 | <1 (30) |

1,000 | 3.9965 | 184.08 | 3.9964 | 184.04 | 2.0 (90) |

It is also worth noticing that the solution obtained with the Dirichlet condition predicts retention time decrease when Pe decreases. However, for Pe > 100 difference between τ_{r} calculated with Dirichlet or Danckwerts conditions is also less than 1%.

The obtained analytical expressions on *N* can be used to estimate the dimensionless modulus: St, Pr, Bi, and *N* from Eqs. (19, 30, 31) should be calculated and compared with estimated *N* values. The obtained set of algebraic equations can be solved with Excel.

### 3.2 Comparison of volume and OCFE methods

Table 3 presents a comparison of the accuracy of analytical and numerical solutions of EDM using OCFE, WENO order 5, Koren, and CADET programs. The CADET uses the WENO scheme order 3. As before the agreement between the analytical and the numerical solution obtained with OCFE is excellent. The other methods need much more nodal points and CPU time to attain a solution with the same accuracy as OCFE. Moreover, it should be noted that for all methods except OCFE, the retention times depend on the number of applied nodal points. It is especially visible for the Koren method and not relevant for CADET.

Comparison of *N* and τ_{r} evaluated from the analytical and numerical solution. Equilibrium-dispersive model, Danckwerts boundary conditions

Pe | Analytical solution, Eqs. (27, 29) | Numerical calculation – OCFE | Numerical calculation – CADET | |||||

τ_{r} | N | τ_{r} | N_{nc} | CPU [s] and (NP) | τ_{r} | N_{nc} | CPU [s] and (NP) | |

10 | 4.0005 | 5.5569 | 4.0003 | 5.5639 | <1 (30) | 3.9960 | 5.45 | <1 (30) |

3.9982 | 5.5973 | <1 (300) | ||||||

3.9978 | 5.6038 | 1.3 (1,000) | ||||||

500 | 4.0005 | 250.56 | 4.0005 | 250.54 | <1 (90) | 3.9942 | 182.66 | <1 (50) |

3.9958 | 227.43 | <1 (90) | ||||||

3.9980 | 248.22 | <1 (250) | ||||||

3.9989 | 249.91 | 2.2 (1,000) | ||||||

50,000 | 4.0005 | 25,002 | 4.0005 | 24,997 | 12 (900) | 3.9895 | 19,516 | 6.5 (900) |

3.9869 | 23,050 | 38 (2,500) | ||||||

3.9864 | 23,397 | 115 (5,000) | ||||||

Pe | Numerical calculation – WENO | Numerical calculation – Koren | ||||||

10 | 3.8691 | 5.3801 | <1 (30) | – | – | – | ||

3.9878 | 5.5265 | 1.1 (300) | – | – | – | |||

3.9932 | 5.6319 | 18.8 (1,000) | – | – | – | |||

500 | 3.9263 | 192.87 | <1 (50) | 3.8137 | 180.79 | <1 (50) | ||

3.9552 | 232.75 | <1 (90) | 3.8935 | 228.46 | <1 (90) | |||

3.9829 | 248.42 | <1 (250) | 3.9588 | 247.31 | 1.2 (250) | |||

3.9964 | 250.32 | 4.4 (1,000) | 3.9912 | 249.94 | 10.6 (1,000) | |||

50,000 | 3.9979 | 20,516 | 11 (900) | 3.9895 | 17,936 | 21 (900) | ||

3.9994 | 24,516 | 61 (2,500) | 3.9967 | 23,874 | 153 (2,500) | |||

3.9999 | 24,931 | 436 (5,000) | 3.9987 | 24,795 | 560 (5,000) |

The next comparison was made for the general rate model which was solved with the CADET program and a hybrid numerical method coupled with the WENO order 5 method for the mobile phase and OCFE for the adsorbent – see Table 4. The analytical and OCFE solutions are presented in Table 1.

Comparison of N_{nc} and τ_{r} evaluated from the CADET program and hybrid method. The calculation for GRM and Danckwerts boundary conditions

Dimensionless modulus | Numerical calculation – CADET | Numerical calculation – hybrid method | |||||||

Pe | St | Bi | τ_{r} | N_{nc} | CPU [s] and (NP) | τ_{r} | N_{nc} | CPU [s] and (NP) | |

10,000 | 10,000 | 0.1 | 100 | 3.9959 | 3,495 | 32 (300) | 3.9904 | 3,030 | 52 (300) |

10,000 | 10,000 | 0.1 | 100 | 3.9943 | 4,329 | 857 (3,600) | 3.9975 | 3,589 | 299 (1,200) |

10,000 | 10,000 | 0.1 | 100 | 3.9945 | 4,331 | 1,312 (5,000) | – | – | – |

10,000 | 10,000 | 1,000 | 100 | 3.9983 | 33.054* | 12 (300) | 3.9866 | 65.111 | 12 (300) |

^{*}Calculation made for 50 nodal points for particle.

The numerical solution should better approximate the analytical solution the larger the number of nodal points, NP. After reaching some value of NP for which the numerical solution is practically equal to analytical the next increase of NP should not change the solution. This is in the case for the hybrid method.

However, for the Cadet program, with the increase of NP, the calculated N_{nc} can remarkably overlap the value of theoretical plates obtained from the analytical solution – compare row 3 from Table 4 with row 8 from Table 1. This drawback can lead to misinterpretation of, for example, the analysis of mass transport. Moreover, for Bi = 1,000 (very slow diffusion inside adsorbent particles) the CADET program predicts N_{nc} = 33 assuming NP = 50 for adsorbent particle whereas the analytical solution predicts 65 theoretical plates. Further increase of NP does not influence the calculated value of N_{nc}. On the other hand, the solution obtained with the hybrid method agrees with the analytical solution.

The parameters of the isotherm model were chosen in such a way as to receive partly overlap concentration profiles. This example has practical meaning because the maximum productivity of the chromatography column is obtained for overlapping peaks of separated components [3]. The accuracy of the calculation of overlapping band profiles influences on accuracy of the determination of separation conditions. It also influences parameter estimation when the inverse method is used.

The accuracy of the numerical solution is increasing with increasing of the number of nodal points. However, when NP increases the CPU also increases. So, the NP should be as low as possible under conditions that the precision of computation is acceptable.

Figure 1 presents the comparison of the numerical solution of the isocratic EDM coupled with the competitive Langmuir adsorption model using OCFE, Koren, WENO order 5 methods, and the CADET program. The parameters were equal: Q_{s} = 8, K_{1} = 8, K_{2} = 10, S_{1} = S_{2} = 0, Pe = 1,000, inlet concentrations, y_{f} = 10, and τ_{p} = 0.025. The total number of nodal points for OCFE (solid lines) was equal to 225. Further increasing of NP did not change band profiles. The same number of NP was applied to three other methods. As can be seen, all peak profiles are shifted compared to the OCFE solution. The largest shift is observed for the Koren method and the smallest for CADET. The solutions of OCFE and CADET almost overlap in a given scale of the plot.

This observation correlates with differences in calculated retention times of analytical peaks. With increasing NP, the solution obtained with the WENO method overlaps the OCFE solution for NP equal to about 1,500. However, for the Koren method, this difference is still visible even for NP = 4,500.

The calculations for gradient mode were done for S_{1} = 10 and S_{2} = 15. Other parameters were unchanged. The conclusions are the same as for the isocratic mode. For the total number of NP greater than 180 the solutions obtained with the OCFE are not distinguishable in the presented plot scale. For the same number of NP, the solution obtained with the CADET program is closest to the OCFE result, however, both solutions are distinguishable – see Fig. 2. For NP greater than 600 the line representing concentration profiles calculated with CADET overlap band profiles calculated with OCFE.

## Conclusions

Available in the literature analytical solutions expressed by formulae for retention time, t_{r}, and the number of theoretical plates, N, of the linear general rate model and linear equilibrium dispersive model of chromatography column were re-examined and re-derived. The calculated values of *N* using expressions taken from literature well approximate the values obtained from newly developed expressions when Pe number is greater than about 100. However, for Pe value decreasing to 1, the differences in calculated *N* values can reach 100%.

The obtained new analytical solutions were validated by comparing them with the numerical solution of GRM and EDM with the OCFE method for a wide range of the changes of dimensionless modulus Pe, Bi, ST, and Φ which characterize mass transfer resistance and dispersion conditions in packed bed column. The analytical and numerical solutions agree in most cases up to four significant digits.

The next part of the work was devoted to the comparison of the efficiency and accuracy of numerical solution of chromatography column models using the popular volume method, namely Koren, WENO 5, and WENO 3 (the last one implemented in CADET) with OCFE.

We showed that all methods, in general, converge to analytical solutions when the number of nodal points is large enough. However, for the assumed accuracy of calculation, the OCFE required the smallest number of nodal points and the shortest calculation time in all the examples presented.

## Acknowledgments

This research was funded in whole by National Science Centre, Poland via grant 2022/45/B/ST8/00591.

## References

- 3.↑
Guiochon, G.; Felinger, A.; Shirazi, D. G.; Katti, A. M.

, Academic Press, 2006.*Fundamentals of Preparative and Nonlinear Chromatography* - 6.↑
Berninger, J. A.; Whitley, R. D.; Zhang, X.; Wang, N.-H. L. A versatile model for simulation of reaction and nonequilibrium dynamics in multicomponent fixed-bed adsorption processes.

1991, 15, 749–768. https://doi.org/10.1016/0098-1354(91)85020-U.*Comput. Chem. Eng.* - 7.↑
Kaczmarski, K.; Antos, D.; Sajonz, H.; Sajonz, P.; Guiochon, G. Comparative modeling of breakthrough curves of bovine serum albumin in anion-exchange chromatography.

2001, 925, 1–17. https://doi.org/10.1016/s0021-9673(01)01035-4.*J. Chromatogr. A.* - 8.↑
Rouchon, P.; Schonauer, M.; Valentin, P.; Guiochon, G. Numerical simulation of band propagation in nonlinear chromatography.

1987, 22, 1793–1833. https://doi.org/10.1080/01496398708057614.*Separation Sci. Technol.* - 9.↑
Czok, M.; Guiochon, G. The physical sense of simulation models of liquid chromatography: propagation through a grid or solution of the mass balance equation.

1990, 62, 189–200. https://doi.org/10.1021/ac00201a020.*Anal. Chem.* - 10.↑
Kaczmarski, K. Estimation of adsorption isotherm parameters with inverse method—possible problems.

2007, 1176, 57–68. https://doi.org/10.1016/j.chroma.2007.08.005.*J. Chromatogr. A.* - 11.↑
Von Lieres and, E.; Andersson, J. A fast and accurate solver for the general rate model of column liquid chromatography.

2010, 34, 1180–1191. https://doi.org/10.1016/j.compchemeng.2010.03.008.*Comput. Chem. Eng.* - 12.↑
Liu, S. X.-D.; Osher, S.; Chan, T. Weighted essentially non-oscillatory schemes.

1994, 115, 200–212. https://doi.org/10.1006/jcph.1994.1187.*J. Comput. Phys.* - 13.↑
Shu, C.-W.

*Essentially non-oscillatory and weighted essentially non oscillatory schemes for hyperbolic conservation laws*. Technical Report NASA/CR-97-206253. - 14.↑
Leweke, S.; von Lieres, E. Chromatography analysis and design toolkit (CADET).

2018, 113, 274–294. https://doi.org/10.1016/j.compchemeng.2018.02.025.*Comput. Chem. Eng.* - 15.↑
Koren, B. A robust upwind discretization method for advection, diffusion and source terms. In

*Numerical Methods for Advection-Diffusion Problems, Volume 45 of Notes on Numerical Fluid Mechanics*; Vreugdenhil, C. B., Koren, B., Eds. Vieweg Verlag: Braunschweig, 1993; pp 117–138. - 16.↑
Javeed, S.; Qamara, S.; Seidel-Morgenstern, A.; Warnecke, G. Efficient and accurate numerical simulation of nonlinear chromatographic processes.

2011, 35, 2294–2305. https://doi.org/10.1016/j.compchemeng.2010.10.002.*Comput. Chem. Eng.* - 17.↑
Qamar, S.; Abbasi, J. N.; Javeed, S.; Seidel-Morgenstern, A. Analytical solutions and moment analysis of general rate model for linear liquid chromatography.

2014, 107, 192–205. https://doi.org/10.1016/j.ces.2013.12.019.*Chem. Eng. Sci.* - 18.↑
Qamar, S.; Uche, D. U.; Khan, F. U.; Seidel-Morgenstern, A. Analysis of linear two-dimensional general rate model for chromatographic columns of cylindrical geometry.

2017, 1496, 92–104. https://doi.org/10.1016/j.chroma.2017.03.048.*J. Chromatogr. A.* - 19.
Khan, A.; Perveen, S.; Qamar, S. Discontinuous Galerkin scheme for solving a lumped kinetic model of non-isothermal liquid chromatography with Bi-Langmuir.

2021, 60, 12592–12601. https://doi.org/10.1021/acs.iecr.1c01074.*Ind. Eng. Chem. Res.* - 20.↑
Villadsen, J.; Michelsen, M. L.

*Solution of Differential Equation Models by Polynomial Approximation*; Prentice-Hall, 1978. - 21.↑
Ma, Z.; Guiochon, G. Application of orthogonal collocation on finite elements in the simulation of non-linear chromatography.

1991, 15, 415–426. https://doi.org/10.1016/0098-1354(91)87019-6.*Comput. Chem. Eng.* - 22.↑
Antia, F. D.; Horváth, C. Gradient elution in non-linear preparative liquid chromatography.

1989, 484, 1–27. https://doi.org/10.1016/S0021-9673(01)88960-3.*J. Chromatogr. A.* - 23.↑
Kaczmarski, K.; Mazzoti, M.; Storti, G.; Morbidelli, M. Modeling fixed-bed adsorption columns through orthogonal collocations on moving finite elements.

1997, 21, 641–660. https://doi.org/10.1016/S0098-1354(96)00300-6.*Comput. Chem. Eng.* - 24.↑
Kucera, E. Contribution to the theory of chromatography: linear non-equilibrium elution chromatography.

1965, 19, 237–248. https://doi.org/10.1016/S0021-9673(01)99457-9.*J. Chromatogr. A.* - 25.
Schneider, P.; Smith, J. M. Adsorption rate constants from chromatography.

1968, 14, 762–771.*AIChE J.* - 26.↑
Antos, D.; Kaczmarski, K.; Wojciech, P.; Seidel-Morgenstern, A. Concentration dependence of lumped mass transfer coefficients: linear versus non-linear chromatography and isocratic versus gradient operation.

2003, 1006, 61–76. https://doi.org/10.1016/S0021-9673(03)00948-8.*J. Chromatogr. A.* - 27.
Suzuki, M. Notes on determining the moments of the impulse response of the basic transformed equations.

1973, 6, 540–543.*J. Chem. Eng. Jpn.* - 28.
Kaczmarski, K.; Guiochon, G. Modeling of the mass-transfer kinetics in chromatographic columns packed with shell and pellicular particles.

2007, 79, 4648–4656. https://doi.org/10.1021/ac070209w.*Anal. Chem.* - 29.
Miyabe, K.; Guiochon, G. The moment equations of chromatography for monolithic stationary phases.

2002, 106, 8898–8909. https://doi.org/10.1021/jp020555b.*J. Phys. Chem. B* - 30.
Miyabe, K.; Guiochon, G. Measurement of the parameters of the mass transfer kinetics in high performance liquid chromatography.

2003, 26, 155–173. https://doi.org/10.1002/jssc.200390024.*J. Sep. Sci.* - 31.
Miyabe, K. Moment analysis of chromatographic behavior in reversed-phase liquid chromatography.

2009, 32, 757–770. https://doi.org/10.1002/jssc.200800607.*J. Sep. Sci.* - 32.
Miyabe, K. Moment equations for chromatography based on Langmuir type reaction kinetics.

2014, 1356, 171–179. https://doi.org/10.1016/j.chroma.2014.06.052.*J. Chromatogr. A.* - 33.
Qamar, S.; Abbasi, J. N.; Javeed, S.; Shah, M.; Khan, F. U.; Seidel-Morgenstern, A. Analytical solutions and moment analysis of chromatographic models for rectangular pulse injections.

2013, 1315, 92–106. https://doi.org/10.1016/j.chroma.2013.09.031.*J. Chromatogr. A.* - 34.↑
Qamar, S.; Abbasi, J.; Mehwish, A.; Seidel-Morgenstern, A. Linear general rate model of chromatography for core–shell particles: analytical solutions and moment analysis.

2015, 137, 352–363. https://doi.org/10.1016/j.ces.2015.06.053.*Chem. Eng. Sci.* - 35.↑
Seidel-Morgenstern, A. Analysis of boundary conditions in the axial dispersion model by application of numerical Laplace inversion.

1994, 46, 2567–2571. https://doi.org/10.1016/0009-2509(91)80049-5.*Chem. Eng. Sci.*

## Appendix

An analytical solution in the Laplace domain of the mass balance equations (11) (12) and (14) with appropriate initial and boundary conditions will be provided. The solution obtained will be used for derivation in the time domain of the moment's equations, which was applied for deducing important information presented in the main part of this work.

In the derivation presented below, some parameters and variables will be grouped for the brevity of the notation of mathematical equations and for the convenience of the reader. Additionally, the obvious functional dependence of the complex variables will be omitted; however, the groups that are a function of the complex variable will be indicated.

All mathematical manipulations presented below were performed using the computer algebra program MAPLE. For more complex relationships, such as those obtained, using computer algebra methods is advantageous.

*τ*= 0 and for 0 < R

_{m}< 1) are:

*τ*> 0 are:

Note that Eq. (14) does not change.

The set of equations (11), (A2), and (14) should be transformed into the complex domain in accordance with the commonly known Laplace transform and its properties.

All complex functions corresponding to real functions in the Laplace transform will be defined analogously and marked by a short line above the symbol.

_{m}= 1 to Eq. (A10) and calculate

Equation (A27) is a complex function that describes the concentration profile in a column. The solution of this equation is possible using any numerical inverse Laplace transform. However, this numerical solution can be helpful, but it is not necessary to realize the purposes of the present work. For that reason, we focus our attention on moment analysis.

Under this assumption, using moment definitions (Eqs. (A34)A36)–(A36)) and Final Value Theorem, appropriate moments can be derived.

The limits defined by Eq. (A34) were calculated using MAPLE program. The obtained equations are very long and complicated. However, after tedious algebraic manipulation and introduction variables defined by Eq. (21a) the expressions for the retention time (Eq. (20)), and number of theoretical plates (Eq. (21)) were obtained.