On the Identification of Mobile and Stationary Zone Mass Transfer Resistances in Chromatography

Abstract

A robust and elegant approach, based on the Two-Zone Moment Analysis (TZMA) method, is proposed to assess the contributions of the mobile and stationary zones, $H_{Cm}$ and $H_{Cs}$, to the C term $H_C$ in the van Deemter equation for plate height. The TZMA method yields two formulations for $H_{Cm}$ and $H_{Cs}$, both fully equivalent in terms of $H_C$, yet offering different decompositions of the contributions from the mobile and stationary zones. The first formulation proposes an expression for the term $H_{Cs}$ that has strong similarities, but also significant differences, from the well-known and widely used one proposed by Giddings. While it addresses the inherent limitation of Giddings’ approach—namely, the complete decoupling of transport phenomena in the moving and stationary zones—it introduces the drawback of a non-unique decomposition of $H_C$. Despite this, it proves highly valuable in highlighting the limitations and flaws of Giddings’ method. In contrast, the second formulation not only properly accounts for the interaction between the moving and stationary zones, but provides a unique and consistent decomposition of $H_C$ into its components. Three different geometries are investigated in detail: the 2D triangular array of cylinders (pillar array columns), the 2D array of rectangular pillars (radially elongated pillar array columns) and the 3D face-centered cubic array of spheres. It is shown that Giddings’ approach significantly underestimates the $H_{Cs}$ term, especially for porous-shell particles. Its accuracy is limited, being reliable only when intra-particle diffusivity ($D_s$) and the zone retention factor ($k^{″}$) are very low, or when axially invariant systems are considered.

1. Introduction

Accurate models for band broadening are crucial for advancing the design of new stationary-phase materials and chromatographic columns, including 3D printed [1,2,3] and microfabricated pillar array columns [4,5,6]. In the context of particulate and monolithic materials, the General Plate Height Model (GPHM) is widely recognized [7,8] as the most reliable and widely accepted method for describing band broadening. According to the GPHM, the plate height H can be decomposed into four contributions:

$$H={\displaystyle \frac{2D^x}{U^x}}=H_A+H_B+\underset{H_C}{\underbrace{H_{Cm}+H_{Cs}}}$$

(1)

where the first term $H_A$ is associated with packing disorder, the second term $H_B$ relates to longitudinal diffusion, while the third $H_{Cm}$ and fourth $H_{Cs}$ terms account for mass transfer resistances in the mobile and stationary zones, respectively. The total plate height H can be experimentally and numerically evaluated from the axial dispersion coefficient $D^x$ and the effective axial velocity $U^x$ along the main fluid flow direction x. For ordered packings, such as the arrays of spheres and pillar array columns, $H_A$ is assumed to be negligible. The second term, $H_B$, also refereed to as the B term band broadening [9]

$$H_B={\displaystyle \frac{2D^{\mathrm{eff}}}{U^x}}$$

(2)

can be experimentally and numerically evaluated from peak parking experiments [10,11,12], allowing for the estimate of the longitudinal effective diffusion coefficient $D^{\mathrm{eff}}$, or equivalently the obstruction factor ${\gamma }_m^0=D^{\mathrm{eff}}/D_m$, with $D_m$ being the bare diffusion coefficient of the analyte in the elution solvent. Indeed, $D^{\mathrm{eff}}$ is the axial dispersion coefficient when the eluent flow is switched off, i.e., $D^{\mathrm{eff}}=D^x{|}_{\mathbf{v}=\mathbf{0}}$. Numerous reliable models exist in the literature [13,14,15,16] to express the B term as a function of the geometric parameters of the porous medium and the zone retention factor $k^{″}$. As far as it concerns the $H_C$ term, the most common approach [7,8] is to assume that mass transfer in the mobile and stationary zone are actually two independent processes so that the $H_{Cs}$ term, according to Giddings’ non-equilibrium theory [17], can be simply expressed as

$$H_{Cs}={\displaystyle \frac{q_f}{4}}{\displaystyle \frac{k^{″}}{{(1+k^{″})}^2}}{\displaystyle \frac{d_p^2}{D_s}}{U}_i$$

(3)

where $U_i$ is the interstitial velocity, $d_p$ is the particle/pillar diameter, $D_s$ is the diffusion coefficient in the meso-porous stationary zone and $q_f$ is a shape factor depending exclusively on the geometry of the stationary zone. Equation (3) for the $H_{Cs}$ term is fully equivalent to that obtained by the General Rate (GR) model [18] and the Parallel Zone model (PZ), as clearly discussed in [7]. By considering that H and $H_B$ can be experimentally or numerically evaluated, and adopting Equation (3) for $H_{Cs}$, the mobile zone contribution $H_{Cm}$ can be isolated [19,20,21,22,23]

$$H_{Cm}=H_C-H_{Cs}=(H-H_B)-H_{Cs}$$

(4)

and subsequently modelled via the GR or the PZ approaches as

$$H_{Cm}={\displaystyle \frac{2}{6}}{\displaystyle \frac{{k^{″}}^2}{{(1+k^{″})}^2}}{\displaystyle \frac{d_p}{S{h}_m}}\left({\displaystyle \frac{U_i{d}_p}{D_m}}\right)={\displaystyle \frac{2}{6}}{\displaystyle \frac{{k^{″}}^2}{{(1+k^{″})}^2}}{\displaystyle \frac{d_p}{S{h}_m}}PeS{h}_m=\mathrm{Sherwood}\mathrm{number}$$

(5)

or via more flexible expressions [21,22],

$$H_{Cm}={\displaystyle \frac{c_1+c_2{k}^{″}+c_3{k}^{″2}}{{(1+k^{″})}^2}}{\displaystyle \frac{Pe}{f\left(Pe\right)}}$$

(6)

or

$$H_{Cm}={\displaystyle \frac{f_1\left(Pe\right)+f_2\left(Pe\right)k^{″}+f_3\left(Pe\right)k^{″2}}{{(1+k^{″})}^2}}$$

(7)

accounting for the nonlinear dependence of $H_{Cm}$ on the Peclet number $Pe=U_i{d}_p/D_m$, also referred to as reduced velocity. It is therefore clear that all the models developed for the mobile $H_{Cm}$ and stationary $H_{Cs}$ mass transfer contributions are intrinsically based on the key assumption that mass transfers in the mobile and stationary zones are two independent processes, leading to Equations (3) and (4). However, this has already proven to be valid only for a few systems with relatively simple geometries, e.g., open-tubular liquid chromatography (OTLC) columns coated with a uniform meso-porous layer. Indeed, Huygens et al. [21] clearly showed a failure of the additivity principle by analyzing dispersion data for 2D- and 3D-ordered periodic media, such as 2D arrays of cylinders and 3D arrays of spheres. Moreover, the need to take into account the strong interdependence between the mass transfer processes in the two zones has already been amply demonstrated by the theoretical models for $D^{\mathrm{eff}}$ [14,15,16], particularly in the simplest case where there is no convection affecting solute transport in the mobile zone.

The Two-Zone Moment Analysis (TZMA) method [24,25,26], recently proposed as a generalization of the classical Brenner moment approach to hierarchical porous media [27,28], allows us to accurately estimate the axial dispersion coefficient $D^x$ in chromatographic systems with the further advantage that the final expression for $D^x$ is formulated such that its terms are clearly associated with either the mobile or stationary zone. This framework provides a more direct understanding of the relative contributions of various band broadening phenomena in different parts of the system, their actual interdependence, as well as their dependence on analyte properties such as retention factor and diffusion coefficients. The objectives of the present work are threefold: (1) to propose a completely general method for identifying the terms $H_{Cm}$ and $H_{Cs}$ in periodic means based on the TZMA method; (2) to analyze, both analytically and numerically, the limits of applicability of the Giddings’ approach; and (3) to quantify the error induced by the adoption of Equations (3) and (4) to the estimation of $H_{Cm}$ and $H_{Cs}$ for two-dimensional- and three-dimensional-ordered chromatographic systems such as cylindrical pillar arrays, elongated pillar arrays, and sphere arrays, all while varying the characteristic parameters such as the zone retention factor $k^{″}$ and the ratio of diffusivities $D_s/D_m$ in the two transport zones.

2. Materials and Methods

The Two-Zone Moment Analysis Approach

The TZMA approach allows us to model the transport of a single analyte through convection and diffusion within a fluid region ${\Sigma }_m$, in the presence of a meso-porous, adsorbing solid phase ${\Sigma }_s$. Let $\epsilon$ denote the external porosity of the chromatographic medium $\Sigma ={\Sigma }_m\cup {\Sigma }_s$. The solvent flow field $\mathbf{v}$, carrying the analyte under pressure-driven conditions, satisfies the Stokes equation with no-slip boundary conditions at the interface $\partial {\Sigma }_s$ between the mobile and stationary zones. Within the macro-porous fluid zone, solute transport occurs through both advective and diffusive processes, where $D_m$ is the diffusion coefficient in the mobile zone. When the analyte enters the meso-porous solid zone, where no fluid motion is present, transport occurs solely via hindered diffusion in the meso-pores, subject to adsorption boundary conditions. Both effects are accounted for in the effective diffusion coefficient $D_s$, according to the General Rate model [25,29,30]. The meso-porous zone is characterized by the internal porosity ${\epsilon }_{int}$, the specific surface area a [$m^2$/$m^3$], and the equilibrium adsorption constant $K_{eq}$ [m], all of which are quantities contributing to the zone retention factor:

$$k^{″}=k_p{\displaystyle \frac{V_s}{V_m}}={\epsilon }_{\mathrm{int}}(1+K_{eq}a){\displaystyle \frac{V_s}{V_m}}$$

(8)

where $V_m$ is the volume occupied by the mobile fluid zone ${\Sigma }_m$, $V_s$ the volume occupied by the stationary solid zone ${\Sigma }_s$ (total particle volume, including the volume of the meso-pores), and $k_p$ the partition coefficient related to the concentrations $c_m$ and $c_s$ in the mobile and stationary zones at the interface $\partial {\mathsf{\Omega }}_s$, i.e., $c_m{{|}_{\partial {\mathsf{\Omega }}_s}=k_p{c}_s|}_{\partial {\mathsf{\Omega }}_s}$.

The TZMA approach shares two notable advantages with the Volume Averaging (VA) approach [29,30]. First, all calculations are performed within the single-unit lattice cell $\mathsf{\Omega }$, from whose spatial repetition the entire chromatographic medium $\Sigma$ can be reproduced, i.e., $\Sigma ={\cup }_{i=1}^N{\mathsf{\Omega }}_i$. Second, the transport equations only need to be solved for lower-order moments in steady-state conditions. In contrast, Direct Numerical Simulations (DNSs) and Lattice Boltzmann (LB) methods [31,32,33,34,35] require solving the time-dependent advection-diffusion equation over a sufficiently large fluid domain, allowing chromatographic bands to reach asymptotic behavior, also referred to as macro-transport conditions [27].

Indeed, the TZMA method requires the solution of the following equations and boundary conditions for the stationary b-fields (so named after Brenner), namely the field $b_m$ [m] defined in the mobile zone ${\mathsf{\Omega }}_m$, and the field $b_s$ [m] defined in the stationary zone ${\mathsf{\Omega }}_s$:

$$\begin{array}{ccc}\hfill D_m{\nabla }^2{b}_m-\mathbf{v}·\nabla b_m& =& (U^x-v_x)\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill D_s{\nabla }^2{b}_s& =& U^x\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill b_m{|}_{\partial {\mathsf{\Omega }}_s}& =& b_s{|}_{\partial {\mathsf{\Omega }}_s}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill -D_m(\nabla b_m-{\mathbf{e}}_x)·\mathbf{n}{|}_{\partial {\mathsf{\Omega }}_s}& =& k_p[-D_s(\nabla b_s-{\mathbf{e}}_x)·\mathbf{n}]{|}_{\partial {\mathsf{\Omega }}_s}\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill (\nabla b_s-{\mathbf{e}}_x)·\mathbf{n}{|}_{\partial {\mathsf{\Omega }}_s^{\mathrm{imp}}}& =& 0\hfill \end{array}$$

(13)

where ${\mathsf{\Omega }}_m$ and ${\mathsf{\Omega }}_s$ are the mobile and stationary zones within the unit cell $\mathsf{\Omega }$, $\partial {\mathsf{\Omega }}_s$ is the boundary between the two zones, $U^x=U_i/(1+k^{″})$ is the effective axial velocity, $v_x$ is the axial velocity, $U_i={\langle v_x\rangle }_{{\mathsf{\Omega }}_m}$ is the interstitial axial velocity, ${\mathbf{e}}_x$ is the unit vector along x, and $\mathbf{n}$ is the normal vector pointing outward from the mobile zone. The boundary condition (13) applies if the stationary zone ${\mathsf{\Omega }}_s$ is composed of a meso-porous zone and an impermeable core with boundary $\partial {\mathsf{\Omega }}_s^{\mathrm{imp}}$, as in the case of porous-shell particles or porous-shell pillars [13,22,36,37,38]. Furthermore, periodic boundary conditions are enforced at the edges of the unit cell $\mathsf{\Omega }$. The physical meaning of $b_m$ and $b_s$ is not so straightforward. The b-fields represent the stationary time-independent part of the first-order spatial moments of the analyte particles moving within the chromatography bed. For a detailed analysis of the method of moments and the derivation of b-fields, the reader is referred to [24,25,27].

The transport Equations (9) and (10) for $b_m$ and $b_s$ are independent of one another. However, $b_m$ and $b_s$ are intrinsically linked to each other through the boundary conditions (11) and (12). Numerical details on the solution of the transport equations for $b_m$ and $b_s$ by the Finite Element Method (FEM) are given in Appendix A.

From the b-fields, the total plate height H can be simply evaluated as

$$\begin{array}{cc}\hfill H=H_m+H_s=& \underset{H_m}{\underbrace{{\displaystyle \frac{2D_m}{U_i}}\left(1-{\langle {\partial }_x{b}_m\rangle }_{{\mathsf{\Omega }}_m}\right)+{\displaystyle \frac{2}{U_i}}{〈(v_x-U^x)b_m〉}_{{\mathsf{\Omega }}_m}}}+\hfill \\ \hfill & \underset{H_s}{\underbrace{{\displaystyle \frac{2D_s{k}^{″}}{U_i}}\left(1-{\langle {\partial }_x{b}_s\rangle }_{{\mathsf{\Omega }}_s}\right)-{\displaystyle \frac{2k^{″}{U}^x}{U_i}}{〈b_s〉}_{{\mathsf{\Omega }}_s}}}\hfill \end{array}$$

(14)

or equivalently

$$H=H_m^g+H_s^g=\underset{H_m^g}{\underbrace{{\displaystyle \frac{2D_m}{U_i}}{〈\left|\right|\nabla b_m-{\mathbf{e}}_x{\left|\right|}^2〉}_{{\mathsf{\Omega }}_m}}}+\underset{H_s^g}{\underbrace{{\displaystyle \frac{2D_s{k}^{″}}{U_i}}{〈\left|\right|\nabla b_s-{\mathbf{e}}_x{\left|\right|}^2〉}_{{\mathsf{\Omega }}_s}}}$$

(15)

where the symbols ${\langle ·\rangle }_{{\mathsf{\Omega }}_m}$ and ${\langle ·\rangle }_{{\mathsf{\Omega }}_s}$ indicate the spatial average over the mobile and stationary zone, respectively. Equations (14) and (15) are fully equivalent, meaning that they furnish exactly the same result in terms of H but allow for a different decomposition of the two contributions of the mobile and stationary zones. Indeed, it can be shown [25] that $H_m+H_s=H_m^g+H_s^g$ but $H_m\ne H_m^g$ and $H_s\ne H_s^g$.

3. Results and Discussion

3.1. Analysis of the b-Field $b_s$ in the Stationary Zone

It must be observed that the b-fields $b_m$ and $b_s$, solutions of the transport Equations (9)–(13) are defined up to an additive constant C that is the same for both fields. In other words, if $b_m$ and $b_s$ are solutions of Equations (9)–(13), then $b_m+C$ and $b_s+C$ are also solutions of the same transport scheme. Indeed, by replacing $b_m+C$ and $b_s+C$ into Equation (14) for H, an extra term appears, namely

$${\displaystyle \frac{2C{\langle v_x-U^x\rangle }_{{\mathsf{\Omega }}_m}}{U_i}}-{\displaystyle \frac{2C{k}^{″}{U}^x}{U_i}}=2C\left(1-{\displaystyle \frac{1}{1+k^{″}}}-{\displaystyle \frac{k^{″}}{1+k^{″}}}\right)=0$$

whose final value is zero $\forall k^{″}$. Therefore, the value of the constant C is irrelevant to the evaluation of the total plate height H, but it affects the way the total H is partitioned into the two contributions $H_m$ and $H_s$ in Equation (14) because $H_m$ and $H_s$ depend explicitly on ${\langle b_m\rangle }_{{\mathsf{\Omega }}_m}$ and ${\langle b_s\rangle }_{{\mathsf{\Omega }}_s}$. On the contrary, $H_m^g$ and $H_s^g$ in Equation (15) are unaffected by the C value, as they depend exclusively on the gradients of the b-fields. This marks a profound difference between the two formulations, Equations (14) and (15), that aim to identify and separate the contribution of the moving zone from that of the stationary zone. By way of example, we analyzed the $H_s$ term in a square array of fully porous cylinders (see Figure 1a) to see how it is influenced by the choice of the C constant. Setting the C constant corresponds to setting a specific value of $b_m$ or $b_s$ at one point on the boundary $\partial {\mathsf{\Omega }}_s$. We analyzed three different cases, by setting $b_m=b_s=0$ in point 1, 2 or 3, as shown in Figure 1a.

Figure 1b–d show the $b_s$ field for $\alpha =D_s/D_m=0.3$, $Pe=U_i{d}_p/D_m=50$ and $k^{″}=4$ in the three distinct cases, where $d_p$ is the cylinder diameter. It is evident that the shape of $b_s$ is exactly the same in the three cases, but depending on the point at which the constraint $b_m=b_s=0$ is enforced, the $b_s$ field shifts towards positive or negative values. Consequently, since the $H_s$ term is dominated by the contribution $-2k^{″}{U}^x{\langle b_s\rangle }_{{\mathsf{\Omega }}_s}/U_i$, the curves $H_s$ vs. $Pe$ are significantly affected by the point at which the constraint is imposed, and there is nothing to prevent it from also obtaining negative values, as in the case where the constraint is enforced in point 1.

This is confirmed by dispersion data reported in Figure 2 showing the behavior of $h_s=H_s/d_p$ vs. $Pe$ in the three cases considered above, and the corresponding curves $h_m=H_m/d_p$ vs. $Pe$. Obviously, if $h_s$ changes with the point of constraint, $h_m$ also changes because their sum $h=H/d_p=h_m+h_s$ is not affected by the constraint point. This analysis clearly indicates that the identification of $H_m$ and $H_s$ based on Equation (14) cannot be adopted because it provides non-unique results, in contrast to the approach based on Equation (15), which instead allows for a unique decomposition of H into $H_m^g$ and $H_s^g$ that is independent of the constraint point. It is important to note, however, that although $H_m^g$ and $H_s^g$ allow for a unique decomposition of the contributions of the moving zone and the stationary zone, this does not imply that the two terms are independent since the b-fields are intrinsically interdependent via the boundary conditions, Equations (11) and (12). This mutual dependence underlies the limitations of the Giddings’ approach, as will be shown in Section 3.2.

3.2. Identification of $H_{Cm}$ and $H_{Cs}$

To isolate $H_{Cm}$ and $H_{Cs}$ from H, we first need to determine the $H_B$ contribution, to subsequently subtract it from H. By setting the eluent velocity field to zero in Equations (9) and (10), $\mathbf{v}=\mathbf{0}$, that implies $v_x=0$, $U_i=0$ and $U^x=0$, the b-fields $b_m^0=b_m{|}_{\mathbf{v}=\mathbf{0}}$ and $b_m^0=b_m{|}_{\mathbf{v}=\mathbf{0}}$, satisfy the diffusion equations

$$\begin{array}{ccc}\hfill D_m{\nabla }^2{b}_m^0& =& 0\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill D_s{\nabla }^2{b}_s^0& =& 0\hfill \end{array}$$

(17)

and boundary conditions of Equations (11)–(13). The two fields $b_m^0$ and $b_s^0$ allow for the estimate of the B term $H_B$ via the simplified form of Equations (14) and (15):

$$H_B=H_{Bm}+H_{Bs}=\underset{H_{Bm}}{\underbrace{{\displaystyle \frac{2D_m}{U_i}}\left(1-{\langle {\partial }_x{b}_m^0\rangle }_{{\mathsf{\Omega }}_m}\right)}}+\underset{H_{Bs}}{\underbrace{{\displaystyle \frac{2D_s{k}^{″}}{U_i}}\left(1-{\langle {\partial }_x{b}_s^0\rangle }_{{\mathsf{\Omega }}_s}\right)}}$$

(18)

or equivalently

$$H_B=H_{Bm}^g+H_{Bs}^g=\underset{H_{Bm}^g}{\underbrace{{\displaystyle \frac{2D_m}{U_i}}{〈\left|\right|\nabla b_m^0-{\mathbf{e}}_x{\left|\right|}^2〉}_{{\mathsf{\Omega }}_m}}}+\underset{H_{Bs}^g}{\underbrace{{\displaystyle \frac{2D_s{k}^{″}}{U_i}}{〈\left|\right|\nabla b_s^0-{\mathbf{e}}_x{\left|\right|}^2〉}_{{\mathsf{\Omega }}_s}}}$$

(19)

Also in this case, Equations (18) and (19) are fully equivalent, meaning that they furnish exactly the same result in terms of $H_B$ but allow for a different decomposition of the two contributions of the mobile and stationary zones, i.e., $H_{Bm}+H_{Bs}=H_{Bm}^g+H_{Bs}^g$ but $H_{Bm}\ne H_{Bm}^g$ and $H_{Bs}\ne H_{Bs}^g$.

The two different formulations of H and $H_B$ lead to the same $H_C$ term but to different splits of the two contributions $H_{Cm}$ and $H_{Cs}$. Indeed, from Equations (14) and (18), one obtains

$$\begin{array}{cc}\hfill H_C=& H-H_B=\underset{H_{Cm}}{\underbrace{(H_m-H_{Bm})}}+\underset{H_{Cs}}{\underbrace{(H_s-H_{Bs})}}=\hfill \\ \hfill & \underset{H_{Cm}}{\underbrace{{\displaystyle \frac{2D_m}{U_i}}\left({\langle {\partial }_x{b}_m^0-{\partial }_x{b}_m\rangle }_{{\mathsf{\Omega }}_m}\right)+{\displaystyle \frac{2}{U_i}}{〈(v_x-U^x)b_m〉}_{{\mathsf{\Omega }}_m}}}\hfill \\ \hfill +& \underset{H_{Cs}}{\underbrace{\stackrel{\stackrel{H_{Cs1}}{︷}}{{\displaystyle \frac{2D_s{k}^{″}}{U_i}}\left({\langle {\partial }_x{b}_s^0-{\partial }_x{b}_s\rangle }_{{\mathsf{\Omega }}_s}\right)}-\stackrel{\stackrel{H_{Cs2}}{︷}}{{\displaystyle \frac{2k^{″}{U}^x}{U_i}}{〈b_s〉}_{{\mathsf{\Omega }}_s}}}}\hfill \end{array}$$

(20)

whereas from Equations (15) and (19), one obtains

$$\begin{array}{cc}\hfill H_C=& H-H_B=\underset{H_{Cm}^g}{\underbrace{(H_m^g-H_{Bm}^g)}}+\underset{H_{Cs}^g}{\underbrace{(H_s^g-H_{Bs}^g)}}=\hfill \\ \hfill & \underset{H_{Cm}^g}{\underbrace{{\displaystyle \frac{2D_m}{U_i}}{〈\left|\right|\nabla b_m-{\mathbf{e}}_x{\left|\right|}^2-\left|\right|\nabla b_m^0-{\mathbf{e}}_x{\left|\right|}^2〉}_{{\mathsf{\Omega }}_m}}}+\hfill \\ \hfill & \underset{H_{Cs}^g}{\underbrace{{\displaystyle \frac{2D_s{k}^{″}}{U_i}}{〈\left|\right|\nabla b_s-{\mathbf{e}}_x{\left|\right|}^2-\left|\right|\nabla b_s^0-{\mathbf{e}}_x{\left|\right|}^2〉}_{{\mathsf{\Omega }}_s}}}\hfill \end{array}$$

(21)

The identification of $H_{Cm}$ and $H_{Cs}$ based on Equation (20) suffers from the same drawback as Equation (14), that is, it is not unique and depends on the point of constraint, unlike the formulation based on Equation (21), which is instead unique and independent of the point of constraint. Nevertheless, it is interesting to analyze Equation (20) in detail because it allows us to highlight the limitations and flaws of Giddings’ approach to the $H_{Cs}$ term.

3.3. Limitations and Flaws of Giddings’ Approach

The Giddings’ Equation (3) for the $H_{Cs}$ term requires the estimate of the shape factor $q_f$, defined as $q_f=2{\langle \theta \rangle }_{{\mathsf{\Omega }}_s}$, $\theta$ being a dimensionless field, defined in the stationary zone ${\mathsf{\Omega }}_s$, satisfying the following Poisson equation and boundary conditions:

$$\begin{array}{ccc}\hfill {\nabla }^2\theta & =& -1/L_r^2\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\theta |}_{\partial {\mathsf{\Omega }}_s}& =& 0\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\nabla \theta ·\mathbf{n}|}_{\partial {\mathsf{\Omega }}_s^{\mathrm{imp}}}& =& 0\hfill \end{array}$$

(24)

where $L_r$ is a reference length, usually the particle radius or the pillar radius $L_r=R_p=d_p/2$ for the packing of spheres or pillar array columns. The analytical solutions of Equations (22)–(24) for fully porous or porous-shell spherical and cylindrical particles are reported in Appendix B, together with the corresponding values of the shape factor $q_f$. The boundary condition Equation (23) at the interface $\partial {\mathsf{\Omega }}_s$ implies that the shape factor is totally independent of the transport process occurring in the mobile zone.

By comparing Equations (22) and (24) for $\theta$ with Equations (10) and (13) for $b_s$, it can be readily observed that $b_s$ and $\theta$ are linearly related as

$$b_s=-{\displaystyle \frac{L_r^2{U}^x}{D_s}}\theta +(x-x_0),x_0={\langle x\rangle }_{{\mathsf{\Omega }}_s}$$

(25)

By replacing Equation (25) for $b_s$ into Equation (20), one obtains that the $H_{Cs2}$ term attains the form

$$H_{Cs2}=-{\displaystyle \frac{2k^{″}{U}^x}{U_i}}{\langle b_s\rangle }_{{\mathsf{\Omega }}_s}={\displaystyle \frac{2k^{″}{U}^x}{U_i}}{\displaystyle \frac{L_r^2{U}_x}{D_s}}{\langle \theta \rangle }_{{\mathsf{\Omega }}_s}=\underset{q_f}{\underbrace{2{\langle \theta \rangle }_{{\mathsf{\Omega }}_s}}}{\displaystyle \frac{k^{″}}{{(1+k^{″})}^2}}{\displaystyle \frac{L_r^2}{D_s}}{U}_i$$

(26)

that coincides with Equation (3) for the $H_{Cs}$ term proposed by Giddings, by setting $L_r=d_p/2$ as reference length. This observation, which highlights the strong similarities between the $\theta$ field and the $b_s$ field, also allows us to point out the flaws in Giddings’ approach, namely the following:

1

The $H_{Cs1}$ term, which must sum in Equation (20) with the $H_{Cs2}$ term in order to complete the $H_{Cs}$ contribution of the stationary zone, is missing in Giddings’ expression, Equation (3). The term $H_{Cs1}$ plays a crucial role at low/intermediate Peclet values, and is absent only for axially invariant systems, for which $b_m$ and $b_s$ are independent of the axial coordinate x.

2

According to Equation (25), the boundary condition enforcing $\theta =0$ at the whole boundary $\partial {\mathsf{\Omega }}_s$, implies $b_s{{|}_{\partial {\mathsf{\Omega }}_s}=(x-x_0)|}_{\partial {\mathsf{\Omega }}_s}$, and therefore, is a very crude approximation of the actual boundary condition $b_s{{|}_{\partial {\mathsf{\Omega }}_s}=b_m|}_{\partial {\mathsf{\Omega }}_s}$, which properly accounts for the interaction between the mobile and stationary zones.

Figure 3 shows the $\theta$ field $\theta =(1-{(r/R_p)}^2)/4$ and the analytical solution of the Giddings’ Equation (22) with the boundary condition ${\theta |}_{\partial {\mathsf{\Omega }}_s}=0$ for the square array of fully porous cylinders, which is the same geometry analyzed in Figure 1 and Figure 2. For comparison, Figure 3 $b_m$ show the $\theta$ field—henceforth referred to as ${\theta }_{TZ}$, evaluated from Equation (25) and the $b_s$ field—solution of the transport scheme Equations (9)–(12), thus enforcing the proper boundary condition $b_s{{|}_{\partial {\mathsf{\Omega }}_s}=b_m|}_{\partial {\mathsf{\Omega }}_s}$. It can be observed that, for a very low value of the ratio $\alpha =D_s/D_m$, $\theta$ and ${\theta }_{TZ}$ are very similar. However, huge differences emerge for the highest value of $\alpha$, namely $\alpha =0.1$, throughout the range of Peclet values of practical interest, $Pe\in [10,200]$. The higher the $\alpha$ value, usually in the range $\alpha \in [0.1,0,5]$ for silica columns [39], the greater the inaccuracy of the Giddings’ prediction of the $H_{Cs}$ term. The inaccuracy is even larger if porous-shell particles are considered, as shown in Figure 4, where the comparison between $\theta$ (gray surface, Equation (A1)), and ${\theta }_{TZ}$ (colored surface) is shown in the case of a square array of porous-shell cylindrical particles (Figure 4a), where the ratio $\rho =R_i/R_p$ between the radius $R_i$ of the impermeable core and the particle radius $R_p$ is set to $\rho =0.7$. For an easy comparison between $\theta$ and ${\theta }_{TZ}$ surfaces, the constraint point in the solution of the $b_s$ field has been chosen in such a way that ${\langle \theta \rangle }_{{\mathsf{\Omega }}_s}={\langle {\theta }_{TZ}\rangle }_{{\mathsf{\Omega }}_s}$.

3.4. The Gradient-Based Approach to $H_{Cm}$ and $H_{Cs}$

In Section 3.2, we proposed to identify the terms $H_{Cm}$ and $H_{Cs}$ on the basis of Equation (21), i.e., through the use of the gradient-based $H_{Cs}^g$ and $H_{Cm}^g$ terms, because they are uniquely defined (independent of the constraint point) and account for the interplay between mass transfer in the mobile and stationary zones. In this section, we analyze in detail the numerical results for $H_{Cs}\equiv H_{Cs}^g$ and $H_{Cm}\equiv H_{Cm}^g$ for three different geometries, namely the 2D triangular array of cylinders (typical configuration of pillar array columns [4,22,23]), the 2D array of rectangular pillars (typical configuration of radially elongated pillar array columns [5,6]) and the 3D face-centered cubic array of spheres.

These three systems, shown in Figure 5, are investigated for the case of fully porous and porous-shell particles.

Figure 6a–f show the comparison of $H_{Cs}$ vs. $Pe$ curves, obtained by the TZMA approach, Equation (21), and Giddings’ expression Equation (3), for the 2D triangular array of cylinders. It can be observed that for fully porous particles, there is a general agreement between the two results. Significant deviations are observed only for high values of $\alpha$ and high values of $k^{″}$. On the contrary, for porous-shell particles, Giddings’ expression significantly underestimates the $H_{Cs}$ contribution, which actually strongly deviates from the linear behavior predicted via Equation (3). The inaccuracy of Giddings’ predictions is even larger for the 2D array of rectangular pillars, as shown in Figure 7a–d. For rectangular pillars, with aspect ratio $L_p/d_p=5$, the shape factor $q_f$ has been numerically evaluated by solving Equations (22)–(24), thus obtaining $q_f=0.562$ for fully porous pillars and $q_f=0.062$ for porous-shell pillars with $\rho =0.8$. Also for this system, the Giddings’ approach tends to underestimate the $H_{cs}$ term, especially for porous-shell particles and is unable to describe the non-monotonically increasing behavior of the $H_{Cs}$ vs. $Pe$ curves, which is particularly evident for porous-shell particles for low/intermediate values of $Pe$, as highlighted in Figure 7a’–d’. In general, all the curves $H_{Cs}$ vs. $Pe$, for porous-shell particles, exhibit a strong nonlinear behavior that cannot be described by Equation (3), which postulates that $H_{Cs}$ is a linear function of $Pe$.

Since Equation (3) significantly underestimates the $H_{Cs}$ term, consequently, the $H_{Cm}$ term, estimated as $H_C-H_{Cs}$, comes to be overestimated with the Giddings’ approach, as shown in Figure 8a,b for the 2D array of fully porous rectangular pillars and in Figure 9b–d for the face-centered cubic array of fully porous and porous-shell particles. The larger $\alpha$ and $k^{″}$, the greater the inaccuracy of Giddings’ predictions in the underestimation of $H_{Cs}$ and the consequent overestimation, by difference, of the $H_{Cm}$ term.

Large errors in the prediction of $H_{Cs}$ via Giddings’ expression Equation (3) for porous-shell particles are compensated by the fact that the mass transfer resistance in the shell-stationary zone is small with respect to that in the mobile zone, i.e., $H_{Cs}\ll H_{Cm}$. For this reason, the resulting error in the estimate of $H_{Cm}$ is around 15% for both fully porous and porous-shell particles for $Pe\in [10,100]$. It becomes definitely larger than 20% if larger values of Pe are considered, i.e., $Pe>100$.

The remarkable inaccuracy of Giddings’ predictions concerning the term $H_{Cs}$ is due not only to the boundary condition ${\theta |}_{\partial {\mathsf{\Omega }}_s}=0$ but also to the absence of the term $H_{Cs1}$, as discussed in Section 3.3. The $H_{Cs1}$ contributions is strictly zero for axially invariant systems. Indeed, when considering an axially invariant system, as the pillar array columns shown in Figure 10a, for which the main fluid flow component $v_x$ is parallel to the pillar’s symmetry axis, Giddings’ expression for $H_{Cs}$ is much more reliable in a wide range of $\alpha$ and $k^{″}$ values, even for porous-shell particles, as shown in Figure 10b. This confirms that the inaccuracy of Giddings’ predictions, for non-axially invariant systems, is due not only to the inexact boundary condition, which eliminates the coupling between zones, but also to the absence of the $H_{Cs1}$ term.

4. Conclusions

A refined approach using the Two-Zone Moment Analysis (TZMA) method is introduced to assess the contributions of the mobile and stationary zones, $H_{Cm}$ and $H_{Cs}$, to the van Deemter equation’s $H_C$ term. Two different formulations of $H_{Cm}$ and $H_{Cs}$ emerge from this method, offering different decompositions of $H_C$. The first formulation offers an expression for $H_{Cs}$ that has similarities with, yet also significant differences from, the widely used Giddings’ approach. While it addresses the limitation of Giddings’ formulation (which assumes complete decoupling of the moving and stationary zones), the first formulation lacks uniqueness in the decomposition of $H_C$. On the other hand, the second formulation captures the interaction between the two zones and provides a unique decomposition. The latter is the one we propose to investigate and quantify the contributions of the mobile and stationary zones to the total plate height.

Three geometries were examined: the 2D triangular array of cylinders, the 2D rectangular pillar array and the 3D face-centered cubic array of spheres. Numerical data for $H_{Cs}$ versus $Pe$ curves for these geometries demonstrate that while Giddings’ approach generally matches results for fully porous particles, it significantly underestimates $H_{Cs}$ for porous-shell particles, particularly at higher values of $\alpha$ and $k^{″}$. For porous-shell particles, Giddings’ method fails to capture the nonlinear $H_{Cs}$ versus $Pe$ behavior, as it assumes a linear relationship.

Since Giddings’ expression underestimates $H_{Cs}$, the $H_{Cm}$ term (calculated as $H_C-H_{Cs}$) is overestimated, with the discrepancies increasing as $\alpha$ and $k^{″}$ rise. The errors in predicting $H_{Cs}$ for porous-shell particles are compensated by the low mass transfer resistance in the shell-stationary zone, leading to low accuracy in the prediction of $H_{Cm}$ that is quantifiable with an error of around 15% for $Pe\in [10,100]$, and >20% for $Pe>100$, and is comparable in the fully porous and the porous-shell case.

The inaccuracies in Giddings’ predictions arise not only from boundary conditions, but also from the absence of the $H_{Cs1}$ term, which is zero in axially invariant systems. For the latter, Giddings’ expression for $H_{Cs}$ is more reliable across a wider range of $\alpha$ and $k^{″}$ values, even for porous-shell particles.