Analytical and numerical solutions of linear and nonlinear chromatography column models

Krzysztof Kaczmarski; Mirosław Krzysztof Szukiewicz

doi:10.1556/1326.2024.01199

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Acta Chromatographica

Volume/Issue:: Accepted Manuscript / Online First

Analytical and numerical solutions of linear and nonlinear chromatography column models

Authors:

Krzysztof Kaczmarski

Krzysztof Kaczmarski Department of Chemical and Process Engineering, Rzeszów University of Technology, 35-959, Rzeszów, Poland

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kkaczmarski@prz.edu.pl https://orcid.org/0000-0002-9209-1418

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Mirosław Krzysztof Szukiewicz

Mirosław Krzysztof Szukiewicz Department of Chemical and Process Engineering, Rzeszów University of Technology, 35-959, Rzeszów, Poland

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Online Publication Date:: 27 Mar 2024

Publication Date:: 27 Mar 2024

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1326.2024.01199

Keywords:: general rate model; equilibrium dispersive model; Koren method; OCFE; WENO method

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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.

Abstract

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 mass balance equation for the mobile phase can be written as follows:

ε_{e} \frac{\partial c}{\partial t} + u \frac{\partial c}{\partial z} = ε_{e} D_{L} \frac{\partial^{2} c}{\partial z^{2}} - ({1 - ε}_{e}) k_{e x t} \frac{3}{R_{p}} (c - c_{p} (r = R_{p}))

(1)

The mass balance equation for adsorbent is:

ε_{p} \frac{\partial c_{p}}{\partial t} + (1 - ε_{p}) \frac{\partial q}{\partial t} = D_{e f f} \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial c_{p}}{\partial r})

(2)

For infinitely fast adsorption-desorption process and linear adsorption model, we have:

q = q_{s} K c_{p} = H c_{p}

(3)

When the adsorption-desorption process is not infinitely fast then the second term of Eq. (3) should be expressed by the adsorption-desorption kinetic, which for linear adsorption is equal:

\frac{\partial q}{\partial t} = k_{d e s} (q_{s} K c_{p} - q) = k_{d e s} H (c_{p} - \frac{q}{H})

(4)

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

The initial conditions (at t = 0) are:

for 0 < z < L; c (0, z) = c^{0}

for 0 < z < L; Ri < r < R_{p};

c_{p} (0, r, z) = c_{p}^{o} (r, z); q (0, r, z) = q^{o} (r, z)

(5)

The boundary conditions for the first mass balance equation are:

for t > 0; z = 0; u_{f} c_{f}^{'} - u (0) c (t, 0) = - ε_{e} D_{L} \frac{\partial c (t, z)}{\partial z} for Danckwerts conditions

or

c (t, 0) = c_{f}^{'} for Dirichlet conditions

with c_{f}^{'} = c_{f} for t \in [0, t_{p}]

and with c_{f}^{'} = 0 for t > t_{p} .

(6)

For t > 0; z = L; \frac{\partial c (t, z)}{\partial z} = 0

(7)

The boundary conditions for the second mass balance equation are:

For t > 0; r = R_{p}; D_{e f f} \frac{\partial c_{p} (t, r)}{\partial r} = k_{e x t} [c - c_{p} (t, r)]

For t > 0; r = R_{i}; \frac{\partial c_{p} (t, r)}{\partial r} = 0

(8)

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.

For large enough D_eff and k_ext (external and internal mass transfer resistance decrease to zero) the concentration c in the mobile phase is equal to the concentration c_p inside pores and the GRM reduces to the following EDM [7]:

ε_{t} \frac{\partial c}{\partial t} + ({1 - ε}_{t}) \frac{\partial q}{\partial t} + u \frac{\partial c}{\partial z} = ε_{e} D_{L} \frac{\partial^{2} c}{\partial z^{2}}

(9)

where 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_L is replaced by the apparent dispersion coefficient, D_a in which the mass transfer resistances are lumped [26].

It will be convenient to rewrite these models in dimensionless form.

The dimensionless parameters are defined as follows:

x = \frac{z}{L}; τ = \frac{t u}{L ε_{e}}; R = \frac{r}{R_{p}}; y = \frac{C}{C_{r}}; y_{p} = \frac{C_{p}}{C_{r}}; ρ = \frac{R_{i}}{R_{p}}

Q = \frac{q}{C_{r}}; Q_{s} = \frac{q_{s}}{c_{r}}; P e = \frac{u L}{D_{L} ε_{e}}; St = \frac{3 k_{e x t} {L ε}_{e}}{R_{p} u}; Bi = \frac{k_{e x t} R_{p}}{D_{e f f}}; Φ^{2} = \frac{R_{p}^{2} k^{'}}{D_{e f f}}

(10)

where

k^{'} = k_{d e s} H

and reference concentration, c_r, was assumed to be equal to 1.

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 ( $Φ$ – Thiele modulus). For example, for St = 100 the mass transport is slow whereas for St = 100,000 the mass transport is extremely fast. Similarly, for Pe decreasing to zero the sample is instantaneously dispersed along a column, and the column becomes a perfect mixer. The larger the Peclete number, the more efficient the column and lower peak broadening due to dispersion. The Pe number can reach 100,000 and more. Biot number can be less than 1 (for very small molecules with high molecular diffusivity) and larger than 1,000 (this value can be obtained for large proteins and viruses). For Thiele modulus greater than 100, the adsorption process can be regarded as infinitely fast.

Introducing dimensionless parameters, the mass balance equation for the mobile phase can be written as follows:

\frac{\partial y}{\partial t} + \frac{\partial y}{\partial x} = \frac{1}{P e} \frac{\partial^{2} y}{\partial x^{2}} - \frac{(1 - ε_{e})}{ε_{e}} S t (y - y_{p} (R = 1))

(11)

The mass balance equation for adsorbent is:

ε_{p} \frac{\partial y_{p}}{\partial τ} + (1 - ε_{p}) \frac{\partial Q}{\partial τ} = \frac{S t}{3 B i} \frac{1}{R^{2}} \frac{\partial}{\partial R} (R^{2} \frac{\partial y_{p}}{\partial R})

(12)

and dimensionless form of Eq. (3) and Eq. (4) is

Q = H y_{p}

(13)

and

\frac{\partial Q}{\partial τ} = \frac{Φ^{2} S t}{3 B i} (y_{p} - \frac{Q}{H})

(14)

The dimensionless initial and boundary conditions are:

the initial conditions (at τ = 0):

for 0 < x < 1; y (0, x) = y^{0}

for 0 < x < 1; ρ < R < 1;

y_{p} (0, R, x) = y_{p}^{o} (R, x); Q (0, R, x) = Q^{o} (R, x)

(15)

The boundary conditions for the mass balance Eq. (11) are:

For τ > 0; x = 0; y_{f}^{'} - y (τ, 0) = - \frac{1}{P e} \frac{\partial y (τ, x)}{\partial x} for Danckwerts conditions

y (τ, 0) = y_{f}^{'} for Dirichlet conditions

with y_{f}^{'} = y_{f} for τ \in [0, τ_{p}]

and with y_{f}^{'} = 0 for τ > τ_{p}

For τ > 0; x = 1; \frac{\partial y (τ, x)}{\partial x} = 0

(16)

The boundary conditions for Eq. (12) are:

For τ > 0; R = 1; \frac{\partial y_{p} (τ, R)}{\partial R} = B i [y - y_{p} (τ, R)]

For τ > 0; R = ρ; \frac{\partial y_{p} (τ, R)}{\partial R} = 0

(17)

The dimensionless mass balance equation for the Equilibrium Dispersive Model is given by:

\frac{\partial y}{\partial τ} + \frac{(1 - ε_{t})}{ε_{t}} \frac{\partial Q}{\partial τ} + \frac{ε_{e}}{ε_{t}} \frac{\partial y}{\partial x} = \frac{ε_{e}}{ε_{t}} \frac{1}{P e} \frac{\partial^{2} y}{\partial x^{2}}

(18)

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

The performance of different chromatographic columns is conveniently compared based on the values of their number of theoretical plates 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

(µ_{1})

moment and the second central (

μ_{2}^{'}

) moment:

N = \frac{μ_{1}^{2}}{μ_{2}^{'}}

(19)

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

The first absolute moment µ₁ is equal to the retention time τ_r of the peak center of gravity.

For Danckwerts boundary conditions and the rectangular injection, the final equations for the first dimensionless absolute moment and N are:

μ_{1} = τ_{r} = 1 + k_{1} + \frac{τ_{p}}{2}

(20)

N = \frac{{(1 + k_{1} + \frac{τ_{p}}{2})}^{2}}{\frac{2}{{Pe}^{2}} (e^{- P e} + P e - 1) {(1 + k_{1})}^{2} + \frac{{τ_{p}}^{2}}{12} + \frac{2 {k_{1}}^{2}}{F^{'}} [\frac{1}{S t} + \frac{B i}{5 S t} * (\frac{1 + 2 ρ + 3 ρ^{2} - ρ^{3} - 5 ρ^{4}}{{(1 + ρ + ρ^{2})}^{2}}) + \frac{1}{(1 - ε_{p})} {(\frac{k_{p}}{1 + k_{p}})}^{2} \frac{3 B i}{S t Φ^{2}}]}

(21)

where:

k_{1} = F^{'} (ε_{p} + (1 - ε_{p}) H) (1 - ρ^{3}); F^{'} = \frac{1 - ε_{e}}{ε_{e}}; k_{p} = \frac{1 - ε_{p}}{ε_{p}} H

(21a)

For columns working in typical analytical conditions, the

τ_{p}

is very low and can be neglected. Assuming additionally that the adsorption-desorption process is very fast (

Φ \to \infty

), the adsorbent is a fully porous particle (ρ = 0) equation (21) becomes:

N^{'} = \frac{1}{\frac{2}{{Pe}^{2}} (e^{- P e} + P e - 1) + \frac{2}{F^{'}} \frac{{k_{1}}^{2}}{{(1 + k_{1})}^{2}} [\frac{1}{S t} + \frac{B i}{5 S t}]}

(22)

For the same condition in literature, Eq. (23) is given [3, 17, 34],

N^{″} = \frac{1}{\frac{2}{P e} + \frac{2}{F^{'}} \frac{{k_{1}}^{2}}{{(1 + k_{1})}^{2}} [\frac{1}{S t} + \frac{B i}{5 S t}]}

(23)

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

The development of the equations for $τ_{r}$ and 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].

The expression for retention time and the number of theoretical plates express Eqs. (24, 25).

μ_{1} = \frac{(1 + k_{1}) (e^{- P e} - 1)}{P e} + k_{1} + \frac{τ_{p}}{2} + 1

(24)

N = \frac{{P e}^{2} {(μ_{1})}^{2}}{2 (A + B)}

(25)

Where:

A = (P e - 2.5) ({k_{1}}^{2} + 2 k_{1}) + \frac{P e (P e - 1) (B i + 5)}{5 F^{'} S t} {k_{1}}^{2} + \frac{(T^{2} {P e}^{2} + 24 P e - 60)}{24} + \frac{3 B i H^{2} P e (P e - 1) F^{'} (1 - ε_{p})}{S t Φ^{2}}

B = ((\frac{2}{F^{'}} {k_{1}}^{2} ((P e + 1) F^{'} + \frac{P e (B i + 5)}{10 S t}) + 2 (P e + 1) (2 k_{1} + 1)) + \frac{3 B i H^{2}}{S t Φ^{2}} P e F^{'} (1 - ε_{p})) e^{- P e} + \frac{{(k_{1} + 1)}^{2}}{2} e^{- 2 P e}

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

The

τ_{r}

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

μ_{1} = \frac{τ_{p}}{2} + \frac{ε_{t}}{ε_{e}} (1 + \frac{1 - ε_{t}}{ε_{t}} H)

(26)

The above equation can be also directly obtained from Eq. (20) remembering that

ε_{t} = ε_{e} + ε_{p} (1 - ε_{e})

(27)

N = \frac{{(μ_{1})}^{2} b^{2} {P e}^{2}}{2 {(F H + 1)}^{2} (e^{- P e} + P e - 1) + {P e}^{2} b^{2} \frac{{τ_{p}}^{2}}{12}}

(28)

where:

F = \frac{1 - ε_{t}}{ε_{t}}

;

b = \frac{ε_{e}}{ε_{t}}

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

In the previous section, we presented two equations on the number of theoretical plates Eq. (22) developed in this work and Eq. (23) from the literature [3]. Both equations are recommended for the GR model for fully porous adsorbent particles and infinitely fast adsorption-desorption processes. Assuming that mass transfer resistances are negligible (

S t \to \infty, B i \to 0

) both equations can be rewritten as follows:

N^{'} = \frac{1}{\frac{2}{{Pe}^{2}} (e^{- P e} + P e - 1)}

(22a)

and

N^{″} = \frac{P e}{2}

(23a)

For $P e \to \infty$ the number of theoretical plates calculated from both equations is the same and is equal Pe/2. However, for $P e \to 0$ from Eq. (22a) we have N′ = 1 whereas from Eq. (23a) N″ = 0. Of course, the correct result is N′ = 1 because for Pe = 0 we have perfect mixing and the number of theoretical plates is equal to one.

The above comparison showed that Eq. (23) generates a wrong solution of the GR model for a low Pe number. The maximum error of calculation of N is:

E r r o r = \frac{N^{'} - N^{″}}{N^{'}} = \frac{1 - e^{- P e}}{P e}

(29)

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%.

The most robust validation of obtained analytical equations for the number of theoretical plates, 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]:

μ_{1} = \frac{\int C (t) t d t}{\int C (t) d t}

(30)

μ_{2}^{'} = \frac{\int C (t) {(t - μ_{1})}^{2} d t}{\int C (t) d t}

(31)

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.

Table 1.

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	$Φ$	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].

Table 2.

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, $Φ$ = 1, ρ = 0

Pe	Analytical solution, Eqs. (25, 26)		Numerical calculation – OCFE
Pe	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 $Φ$ . Assuming that values of the Henry constant and porosities are known at least 4 experiments should be done in linear conditions. For each experiment the 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.

Table 3.

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
Pe	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.

Table 4.

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	$Φ$	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 last comparison of analyzed numerical schemes was done for the two-component Langmuir isotherm model, Eq. (32), and column working in strongly overload conditions, for isocratic and gradient mode. The isotherm model reads:

Q_{i} = \frac{Q_{s} K_{i} \cdot \exp ({- S}_{i} \cdot φ) \cdot y_{i}}{1 + \sum_{j = 1}^{j = 2} K_{j} \cdot \exp ({- S}_{j} \cdot φ) \cdot y_{j}}

(32)

where

φ

is the volume fraction of the modifier in the mobile phase and S is a solvent strength parameter.

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, K₁ = 8, K₂ = 10, S₁ = S₂ = 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.

Fig. 1.

The comparison of the numerical solution of the EDM coupled with the competitive Langmuir adsorption model using OCFE (solid line), Koren (points), WENO order 5 (dash), and the CADET program (dash-dot). The CPU times were 6[s], 8[s], 5[s], and 4[s] respectively

Citation: Acta Chromatographica 2024; 10.1556/1326.2024.01199

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 S₁ = 10 and S₂ = 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.

Fig. 2.

The comparison of the numerical solutions of the EDM with the gradient of the modifier. The competitive Langmuir adsorption model. The dotted line represents the modifier. Solid line the OCFE solution, dashed line the CADET program solution. The CPU times were 10[s], and 3[s] respectively. The CPU = 12[s] for CADET program and NP = 600

Citation: Acta Chromatographica 2024; 10.1556/1326.2024.01199

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

1.↑
Ruthven, D. M. Principles of Adsorption and Adsorption Process, John Wiley & Sons, 1984.
- Search Google Scholar
- Export Citation
2.
Suzuki, M. Adsorption Engineering, Elsevier, 1990.
- Search Google Scholar
- Export Citation
3.↑
Guiochon, G.; Felinger, A.; Shirazi, D. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, 2006.
- Search Google Scholar
- Export Citation
4.↑
Stewart, W. E.; Bird, R. B.; Lightfoot, E. N. Transport Phenomena, Wiley, 2007.
- Search Google Scholar
- Export Citation
5.↑
Fogler, H. Elements of Chemical Reaction Engineering, PRENTICE HALL, 2020.
- Search Google Scholar
- Export Citation
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. Comput. Chem. Eng. 1991, 15, 749–768. https://doi.org/10.1016/0098-1354(91)85020-U.
- Search Google Scholar
- Export Citation
7.↑
Kaczmarski, K.; Antos, D.; Sajonz, H.; Sajonz, P.; Guiochon, G. Comparative modeling of breakthrough curves of bovine serum albumin in anion-exchange chromatography. J. Chromatogr. A. 2001, 925, 1–17. https://doi.org/10.1016/s0021-9673(01)01035-4.
- Search Google Scholar
- Export Citation
8.↑
Rouchon, P.; Schonauer, M.; Valentin, P.; Guiochon, G. Numerical simulation of band propagation in nonlinear chromatography. Separation Sci. Technol. 1987, 22, 1793–1833. https://doi.org/10.1080/01496398708057614.
- Search Google Scholar
- Export Citation
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. Anal. Chem. 1990, 62, 189–200. https://doi.org/10.1021/ac00201a020.
- Search Google Scholar
- Export Citation
10.↑
Kaczmarski, K. Estimation of adsorption isotherm parameters with inverse method—possible problems. J. Chromatogr. A. 2007, 1176, 57–68. https://doi.org/10.1016/j.chroma.2007.08.005.
- Search Google Scholar
- Export Citation
11.↑
Von Lieres and, E.; Andersson, J. A fast and accurate solver for the general rate model of column liquid chromatography. Comput. Chem. Eng. 2010, 34, 1180–1191. https://doi.org/10.1016/j.compchemeng.2010.03.008.
- Search Google Scholar
- Export Citation
12.↑
Liu, S. X.-D.; Osher, S.; Chan, T. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 1994, 115, 200–212. https://doi.org/10.1006/jcph.1994.1187.
- Search Google Scholar
- Export Citation
13.↑
Shu, C.-W. Essentially non-oscillatory and weighted essentially non oscillatory schemes for hyperbolic conservation laws. Technical Report NASA/CR-97-206253.
- Search Google Scholar
- Export Citation
14.↑
Leweke, S.; von Lieres, E. Chromatography analysis and design toolkit (CADET). Comput. Chem. Eng. 2018, 113, 274–294. https://doi.org/10.1016/j.compchemeng.2018.02.025.
- Search Google Scholar
- Export Citation
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.
- Search Google Scholar
- Export Citation
16.↑
Javeed, S.; Qamara, S.; Seidel-Morgenstern, A.; Warnecke, G. Efficient and accurate numerical simulation of nonlinear chromatographic processes. Comput. Chem. Eng. 2011, 35, 2294–2305. https://doi.org/10.1016/j.compchemeng.2010.10.002.
- Search Google Scholar
- Export Citation
17.↑
Qamar, S.; Abbasi, J. N.; Javeed, S.; Seidel-Morgenstern, A. Analytical solutions and moment analysis of general rate model for linear liquid chromatography. Chem. Eng. Sci. 2014, 107, 192–205. https://doi.org/10.1016/j.ces.2013.12.019.
- Search Google Scholar
- Export Citation
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. J. Chromatogr. A. 2017, 1496, 92–104. https://doi.org/10.1016/j.chroma.2017.03.048.
- Search Google Scholar
- Export Citation
19.
Khan, A.; Perveen, S.; Qamar, S. Discontinuous Galerkin scheme for solving a lumped kinetic model of non-isothermal liquid chromatography with Bi-Langmuir. Ind. Eng. Chem. Res. 2021, 60, 12592–12601. https://doi.org/10.1021/acs.iecr.1c01074.
- Search Google Scholar
- Export Citation
20.↑
Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice-Hall, 1978.
- Search Google Scholar
- Export Citation
21.↑
Ma, Z.; Guiochon, G. Application of orthogonal collocation on finite elements in the simulation of non-linear chromatography. Comput. Chem. Eng. 1991, 15, 415–426. https://doi.org/10.1016/0098-1354(91)87019-6.
- Search Google Scholar
- Export Citation
22.↑
Antia, F. D.; Horváth, C. Gradient elution in non-linear preparative liquid chromatography. J. Chromatogr. A. 1989, 484, 1–27. https://doi.org/10.1016/S0021-9673(01)88960-3.
- Search Google Scholar
- Export Citation
23.↑
Kaczmarski, K.; Mazzoti, M.; Storti, G.; Morbidelli, M. Modeling fixed-bed adsorption columns through orthogonal collocations on moving finite elements. Comput. Chem. Eng. 1997, 21, 641–660. https://doi.org/10.1016/S0098-1354(96)00300-6.
- Search Google Scholar
- Export Citation
24.↑
Kucera, E. Contribution to the theory of chromatography: linear non-equilibrium elution chromatography. J. Chromatogr. A. 1965, 19, 237–248. https://doi.org/10.1016/S0021-9673(01)99457-9.
- Search Google Scholar
- Export Citation
25.
Schneider, P.; Smith, J. M. Adsorption rate constants from chromatography. AIChE J. 1968, 14, 762–771.
- Search Google Scholar
- Export Citation
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. J. Chromatogr. A. 2003, 1006, 61–76. https://doi.org/10.1016/S0021-9673(03)00948-8.
- Search Google Scholar
- Export Citation
27.
Suzuki, M. Notes on determining the moments of the impulse response of the basic transformed equations. J. Chem. Eng. Jpn. 1973, 6, 540–543.
- Search Google Scholar
- Export Citation
28.
Kaczmarski, K.; Guiochon, G. Modeling of the mass-transfer kinetics in chromatographic columns packed with shell and pellicular particles. Anal. Chem. 2007, 79, 4648–4656. https://doi.org/10.1021/ac070209w.
- Search Google Scholar
- Export Citation
29.
Miyabe, K.; Guiochon, G. The moment equations of chromatography for monolithic stationary phases. J. Phys. Chem. B 2002, 106, 8898–8909. https://doi.org/10.1021/jp020555b.
- Search Google Scholar
- Export Citation
30.
Miyabe, K.; Guiochon, G. Measurement of the parameters of the mass transfer kinetics in high performance liquid chromatography. J. Sep. Sci. 2003, 26, 155–173. https://doi.org/10.1002/jssc.200390024.
- Search Google Scholar
- Export Citation
31.
Miyabe, K. Moment analysis of chromatographic behavior in reversed-phase liquid chromatography. J. Sep. Sci. 2009, 32, 757–770. https://doi.org/10.1002/jssc.200800607.
- Search Google Scholar
- Export Citation
32.
Miyabe, K. Moment equations for chromatography based on Langmuir type reaction kinetics. J. Chromatogr. A. 2014, 1356, 171–179. https://doi.org/10.1016/j.chroma.2014.06.052.
- Search Google Scholar
- Export Citation
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. J. Chromatogr. A. 2013, 1315, 92–106. https://doi.org/10.1016/j.chroma.2013.09.031.
- Search Google Scholar
- Export Citation
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. Chem. Eng. Sci. 2015, 137, 352–363. https://doi.org/10.1016/j.ces.2015.06.053.
- Search Google Scholar
- Export Citation
35.↑
Seidel-Morgenstern, A. Analysis of boundary conditions in the axial dispersion model by application of numerical Laplace inversion. Chem. Eng. Sci. 1994, 46, 2567–2571. https://doi.org/10.1016/0009-2509(91)80049-5.
- Search Google Scholar
- Export Citation

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.

In the first step, introducing a new space variable to the mass balance equations (12) and the appropriate boundary conditions is purposeful.

R_{m} = \frac{R - ρ}{1 - ρ}

(A1)

It results in the following equations

\frac{\partial y_{p}}{\partial τ} - \frac{S t}{3 B i ρ_{1}^{2} ε_{p}} (\frac{\partial^{2} y_{p}}{{\partial R_{m}}^{2}} + \frac{2}{R_{m} + ρ_{m}} \frac{\partial y_{p}}{\partial R_{m}}) + a_{p} \frac{\partial Q}{\partial τ} = 0

(A2)

where

ρ_{1} = 1 - ρ

(A3)

ρ_{m} = \frac{ρ}{ρ_{1}}

(A4)

a_{p} = \frac{1 - ε_{p}}{ε_{p}}

(A5)

The initial conditions (at τ = 0 and for 0 < R_m < 1) are:

y_{p} (0, R_{m}, x) = y_{p}^{o} (R_{m}, x)

Q (0, R_{m}, x) = Q^{o} (R_{m}, x)

(A6)

The boundary conditions τ > 0 are:

for R_{m} = 1;

\frac{\partial y_{p} (τ, R_{m})}{\partial R_{m}} = ρ_{1} B i [y - y_{p} (τ, R_{m})]

(A7)

for R_{m} = 0;

\frac{\partial y_{p} (τ, R_{m})}{\partial R_{m}} = 0

(A8)

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.

The definition of the Laplace transform presents

\bar{y} (s, x) = \int_{0}^{\infty} y (τ, x) e^{- s τ} d τ; τ \geq 0

(A9)

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

The mass balance for the mobile phase after transformation is given by

s \bar{y} + \frac{d \bar{y}}{d x} - \frac{1}{P e} \frac{d^{2} \bar{y}}{d x^{2}} + F^{'} S t (\bar{y} - {\bar{y}}_{p} (1))

(A10)

The boundary conditions are:

for x = 0;

\bar{y_{f}^{'}} - \bar{y} + \frac{1}{P e} \frac{d \bar{y}}{d x} = 0

(A11)

for x = 1;

\frac{d \bar{y}}{d x} = 0

(A12)

The mass balance for the adsorbent is:

s \bar{y_{p}} - \frac{S t}{3 B i ρ_{1}^{2} ε_{p}} (\frac{\partial^{2} \bar{y_{p}}}{{\partial R_{m}}^{2}} + \frac{2}{R_{m} + ρ_{m}} \frac{\partial \bar{y_{p}}}{\partial R_{m}}) + a_{p} s \bar{Q} = 0

(A13)

s \bar{Q} = \frac{Φ^{2} S t}{3 B i} (\bar{y_{p}} - \frac{\bar{Q}}{H})

(A14)

where a_{p} = \frac{1 - ε_{p}}{ε_{p}} .

(A15)

The boundary conditions are:

for R_{m} = 1;

\frac{\partial \bar{y_{p}}}{\partial R_{m}} = B i ρ_{1} [\bar{y} - \bar{y_{p}} (1)]

(A16)

for R_{m} = 0;

\frac{\partial \bar{y_{p}}}{\partial R_{m}} = 0

(A17)

The variable Q can be easily eliminated from Eq. (A13) using (A14). As a result, we obtain

s \bar{y_{p}} - \frac{S t}{3 B i ρ_{1}^{2} ε_{p}} (\frac{\partial^{2} \bar{y_{p}}}{{\partial R_{m}}^{2}} + \frac{2}{R_{m} + ρ_{m}} \frac{\partial \bar{y_{p}}}{\partial R_{m}}) + a_{p} s \bar{y_{p}} A (s) = 0

(A18)

where

A (s) = \frac{Φ^{2} S t}{3 B i (s + \frac{Φ^{2} S t}{3 B i H})}

(A19)

Equations (A10) and (A18) are not fully coupled and can be solved sequentially. And so, the solution of Eq. (A18) gives

\bar{y_{p}} = C_{1} \frac{\sinh (\sqrt{3 \frac{(a_{p} A (s) + 1) s B i ε_{p} ρ_{1}^{2}}{S t} R_{m}})}{R_{m} + ρ_{m}} + C_{2} \frac{\cosh (\sqrt{3 \frac{(a_{p} A (s) + 1) s B i ε_{p} ρ_{1}^{2}}{S t} R_{m}})}{R_{m} + ρ_{m}}

(A20)

where determined from boundary conditions integration constants are

C_{1} = \frac{ρ_{1} B i \bar{y}}{\frac{C_{3}}{{(1 + ρ_{m})}^{2}} + \frac{ρ_{1} B i (ρ_{m} \sqrt{3 B (s)} \cosh (\sqrt{3 B (s)}) + \sinh (\sqrt{3 B (s)}))}{1 + ρ_{m}}}

(A21)

C_{2} = C_{1} ρ_{m} \sqrt{3 B (s)}

(A22)

where

C_{3} = 3 \frac{(a_{p} A (s) + 1) s B i ε_{p} ρ_{1}^{2} \sinh (\sqrt{3 B (s)})}{S t} (ρ_{m}^{2} + ρ_{m}) + \sqrt{3 B (s)} \cosh (\sqrt{3 B (s)}) - \sinh (\sqrt{3 B (s)})

(A23)

B (s) = \frac{(a_{p} A (s) + 1) s B i ε_{p} ρ_{1}^{2}}{S t}

(A24)

Rearranging it gives

\bar{y_{p}} = \frac{(\frac{ρ_{m} \sqrt{3 B (s)} \cosh (\sqrt{3 B (s)}) + \sinh (\sqrt{3 B (s)})}{R_{m} + ρ_{m}}) ρ_{1} B i}{C (s)} \bar{y}

(A25)

where

C (s) = \frac{S t (\sqrt{3 B (s)} \cosh (\sqrt{3 B (s)}) - \sinh (\sqrt{3 B (s)})) + 3 B (s) ρ_{m} (1 + ρ_{m}) \sinh (\sqrt{3 B (s)})}{S t {(1 + ρ_{m})}^{2}} + \frac{ρ_{1} B i (\sqrt{3 B (s)} \cosh (\sqrt{3 B (s)}) ρ_{m} - \sinh (\sqrt{3 B (s)}))}{1 + ρ_{m}}

(A26)

Next, we substitute Eq. (A25) with R_m = 1 to Eq. (A10) and calculate

\bar{y}

obtaining:

\bar{y} = C_{11} \exp (\frac{P e}{2} + \frac{\sqrt{{P e}^{2} + P e E (s)}}{2}) + C_{21} \exp (\frac{P e}{2} - \frac{\sqrt{{P e}^{2} + P e E (s)}}{2})

(A27)

Integration constants can be evaluated using boundary conditions (A12) and (A13)

C_{11} = \frac{\bar{y_{f}^{'}} P e (F (s) - G (s))}{G (s) (P e + F (s) - G (s)) e^{F (s)} + (F (s) - G (s)) (P e - G (s))}

(A28)

C_{21} = \frac{C_{11} G (s) e^{F (s)}}{F (s) - G (s)}

(A29)

where

E (s) = s + a_{e} S t (1 - \frac{ρ_{m} \sqrt{3 B (s)} \cosh (\sqrt{3 B (s)}) + \sinh (\sqrt{3 B (s)})}{(1 + ρ_{m}) C (s)} ρ_{1} B i)

(A30)

F (s) = \sqrt{{P e}^{2} + 4 P e E (s)}

(A31)

G (s) = \frac{P e + F (s)}{2}

(A32)

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.

Let us assume that the inlet concentration is described by the rectangular impulse function (Eq. (16)), and that the Laplace transform is given by

\bar{y_{f}^{'}} = \frac{y_{f} (1 - e^{- s τ_{p}})}{s}

(A33)

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

n-th moment of a chromatographic peak is given by

m_{n} = \int_{0}^{\infty} y τ^{n} d τ = {(- 1)}^{n} \lim_{s \to 0} (\frac{d^{n} \bar{y}}{{d s}^{n}})

(A34)

The first absolute

(µ_{1})

and second central (

μ_{2}^{'}

) moments can be easily calculated from the well-known relationships:

μ_{1} = \frac{m_{1}}{m_{0}}

(A35)

μ_{2}^{'} = m_{2} - m_{1}^{2}

(A36)

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.

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Acta Chromatographica

Online ISSN:: 2083-5736

Senior editors

Editor(s)-in-Chief: Kowalska, Teresa (1946-2023)

Editor(s)-in-Chief: Sajewicz, Mieczyslaw, University of Silesia, Katowice, Poland

Editors(s)

Danica Agbaba, University of Belgrade, Belgrade, Serbia
Łukasz Komsta, Medical University of Lublin, Lublin, Poland
Ivana Stanimirova-Daszykowska, University of Silesia, Katowice, Poland
Monika Waksmundzka-Hajnos, Medical University of Lublin, Lublin, Poland

Editorial Board

Ravi Bhushan, The Indian Institute of Technology, Roorkee, India
Jacek Bojarski, Jagiellonian University, Kraków, Poland
Bezhan Chankvetadze, State University of Tbilisi, Tbilisi, Georgia
Michał Daszykowski, University of Silesia, Katowice, Poland
Tadeusz H. Dzido, Medical University of Lublin, Lublin, Poland
Attila Felinger, University of Pécs, Pécs, Hungary
Kazimierz Glowniak, Medical University of Lublin, Lublin, Poland
Bronisław Glód, Siedlce University of Natural Sciences and Humanities, Siedlce, Poland
Anna Gumieniczek, Medical University of Lublin, Lublin, Poland
Urszula Hubicka, Jagiellonian University, Kraków, Poland
Krzysztof Kaczmarski, Rzeszow University of Technology, Rzeszów, Poland
Huba Kalász, Semmelweis University, Budapest, Hungary
Katarina Karljiković Rajić, University of Belgrade, Belgrade, Serbia
Imre Klebovich, Semmelweis University, Budapest, Hungary
Angelika Koch, Private Pharmacy, Hamburg, Germany
Piotr Kus, Univerity of Silesia, Katowice, Poland
Debby Mangelings, Free University of Brussels, Brussels, Belgium
Emil Mincsovics, Corvinus University of Budapest, Budapest, Hungary
Ágnes M. Móricz, Centre for Agricultural Research, Budapest, Hungary
Gertrud Morlock, Giessen University, Giessen, Germany
Anna Petruczynik, Medical University of Lublin, Lublin, Poland
Robert Skibiński, Medical University of Lublin, Lublin, Poland
Bernd Spangenberg, Offenburg University of Applied Sciences, Germany
Tomasz Tuzimski, Medical University of Lublin, Lublin, Poland
Yvan Vander Heyden, Free University of Brussels, Brussels, Belgium
Adam Voelkel, Poznań University of Technology, Poznań, Poland
Beata Walczak, University of Silesia, Katowice, Poland
Wiesław Wasiak, Adam Mickiewicz University, Poznań, Poland
Igor G. Zenkevich, St. Petersburg State University, St. Petersburg, Russian Federation

KOWALSKA, TERESA (1946-2023)
E-mail: kowalska@us.edu.pl

SAJEWICZ, MIECZYSLAW
E-mail:msajewic@us.edu.pl

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2023
Web of Science
Journal Impact Factor	1.7
Rank by Impact Factor	Q3 (Chemistry, Analytical)
Journal Citation Indicator	0.43
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CiteScore	4.0
CiteScore rank	Q2 (General Chemistry)
SNIP	0.706
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SJR index	0.344
SJR Q rank	Q3

Acta Chromatographica
Publication Model	Online only Gold Open Access
Submission Fee	none
Article Processing Charge	400 EUR/article
Regional discounts on country of the funding agency	World Bank Lower-middle-income economies: 50% World Bank Low-income economies: 100%
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Acta Chromatographica
Language	English
Size	A4
Year of Foundation	1988
Volumes per Year	1
Issues per Year	4
Founder	Institute of Chemistry, University of Silesia
Founder's Address	PL-40-007 Katowice, Poland, Bankowa 12
Publisher	Akadémiai Kiadó
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	2083-5736 (Online)