Hydrodynamic and Mass-Transfer Modeling of Uranium Recovery in a Packed Ion-Exchange Column with a Conical Flow Distributor

Abstract

Efficient uranium recovery from productive leaching solutions requires accurate prediction of hydrodynamic and mass-transfer processes in ion-exchange sorption columns. In this study, a coupled multidimensional hydrodynamic and mass-transfer model is developed to investigate uranium sorption in a packed ion-exchange column equipped with a conical flow distributor. Fluid flow in the porous resin bed is described using the Forchheimer filtration law combined with the mass conservation equation, while transport of dissolved uranium species is modeled using a convective–dispersion equation coupled with a linear driving force kinetic model. The numerical solution is obtained using the fictitious domain method, which enables accurate representation of complex column geometries. The results reveal pronounced radial flow non-uniformity, incomplete flow equalization, and the formation of a ring-shaped sorption zone, indicating uneven utilization of the sorbent bed. It is shown that under practical operating conditions, mass-transfer dynamics are governed primarily by hydrodynamics rather than intrinsic sorption kinetics. The proposed model provides a practical tool for analysis and optimization of industrial uranium recovery columns.

1. Introduction

Many deep or shallow deposits of rare and valuable metals are currently extracted using the in situ leaching (ISL) method because of its cost-effectiveness and relatively low environmental impact [1,2,3,4,5,6]. In this process, uranium-bearing minerals are dissolved underground and transported to the surface in productive solutions, commonly called pregnant leach solutions. During processing, sorption–precipitation methods are widely employed to concentrate valuable components directly at mining sites using ion-exchange resins [7,8,9].

Existing studies on uranium sorption mainly focus on modeling transport processes and sorption kinetics in fixed-bed columns, considering advection, hydrodynamic dispersion, and physicochemical interactions between uranium species and sorbents [10,11,12,13].

Despite extensive studies on uranium sorption in fixed-bed columns, most existing models assume one-dimensional plug-flow behavior and neglect radial hydrodynamic non-uniformities caused by realistic industrial column geometry. In particular, the influence of inlet distributor design on flow maldistribution, sorbent utilization, and mass-transfer zone development in large-scale uranium recovery columns remains insufficiently investigated. This limitation reduces the predictive capability of conventional fixed-bed models when applied to industrial sorption equipment. The present study addresses this gap by developing a multidimensional coupled hydrodynamic and reactive transport model for a full-scale packed ion-exchange column with a conical flow distributor. The model enables quantitative analysis of the influence of distributor geometry and operating conditions on hydrodynamics, sorption efficiency, and utilization of the sorbent bed.

This study examines the ion-exchange extraction dynamics of dissolved uranium species from pregnant leach solutions in a complex-geometry packed sorption column, using mathematical and numerical modeling. Such columns are commonly employed in the uranium industry for extracting uranium from productive solutions [11,12].

First, the design of the sorption column is described. Then, based on experimental data obtained in a packed column, the kinetic characteristics of the sorbent are determined. The corresponding kinetic coefficients are estimated by fitting the calculated and experimental concentration profiles of uranium in the liquid phase at different cross-sections of the column. These parameters are subsequently used in the mathematical model describing the transport of dissolved uranium species and their sorption in the porous sorbent bed.

In the final stage, the efficiency of the sorption column operation is analyzed using the fictitious domain method and numerical simulations in order to investigate the influence of the conical flow distributor geometry.

The present study considers the first cycle of the sorption column operation, in which the sorbent initially does not contain uranium species. Modeling of periodic operation of the column would require additional information on the residual uranium distribution in the sorbent after unloading the saturated layer.

2. Materials and Methods

2.1. Description of the Sorption Column

The design and operating scheme of the packed sorption column (hereafter referred to as the column or sorber), widely used in uranium mining enterprises of Kazatomprom (Kazakhstan), are shown in Figure 1. The column represents a cylindrical vessel of height H and radius R with a truncated conical bottom. In the lower part of the column, upstream of the feed pipe with diameter d, a conical flow distributor is installed. Its purpose is to reduce the non-uniformity of the hydrodynamic flow and the corresponding concentration distribution of uranium species over the cross-section of the column.

The column is densely packed with a fixed bed of granular ion-exchange resin (Purolite). The sorbent consists of porous spherical particles with diameter δ and a highly developed internal surface. The particle diameter is several orders of magnitude smaller than the diameter of the feed pipe and the column diameter D = 2R. Therefore, the sorbent bed can be treated as a porous medium with porosity ϕ.

The pregnant leach solution containing dissolved uranium species (primarily uranyl ions ${UO}_2^{2+}$ or their complexes) enters the column from below through the feed pipe at a constant volumetric flow rate Q₀. During the ion-exchange process, mobile hydrogen ions of the resin are replaced by uranium ions from the solution, resulting in the ion-exchange reaction:

$${2(R-H)+UO}_2^{2+ }\rightleftharpoons (R_2-U{O}_2)+2H^{+},$$

(1)

because the binding energy of uranium ions with the resin is higher than that of hydrogen ions:

$$E(resin-U)>E(resin-H).$$

Here, R denotes the resin matrix, $(R-H)$ represents the active functional group of the cation exchange resin, ${(R}_2-U{O}_2)$ is the stable resin-uranium complex, ${UO}_2^{2+}$ is the divalent uranium ion fixed on the resin, and $H^{+}$ is the hydrogen ion released into the solution.

Due to the developed internal structure of the resin granules and the small size of the capillaries inside them, the ion-exchange process between the solution and the sorbent is mainly controlled by intraparticle diffusion.

Before entering the sorbent bed, the flow passes through the conical distributor with opening angle α, defined as the angle between the column axis and the generatrix of the distributor.

During column operation, solution samples are periodically collected from the top outlet of the column and analyzed for uranium concentration. The sorption process continues until the average uranium concentration in the outlet solution reaches a specified limit value. At this stage, the lower part of the sorbent bed that has become saturated with uranium is removed, fresh sorbent is added from the top of the column, and the sorption cycle is repeated.

2.2. Mathematical Model

The mathematical model is developed under the following assumptions.

(i)

The pregnant leach solution containing dissolved uranium species consists predominantly of water and, under the considered conditions, can be approximated as an incompressible Newtonian fluid with density and viscosity close to those of water.

(ii)

Because the characteristic size of the resin particles is much smaller than the column diameter, the sorbent bed is densely packed and immobile, and the flow velocities are relatively low, the flow of the solution through the sorbent layer is considered as filtration flow in a porous medium.

(iii)

Due to the narrow particle size distribution and nearly spherical shape of the ion-exchange resin granules, and assuming uniform packing of the column, the sorbent bed can be treated in a macroscopic approximation as a homogeneous and isotropic porous medium characterized by effective filtration properties.

(iv)

Because the sorption column and the inlet pipe are axisymmetric, the fluid flow and the sorption process are also assumed to be axisymmetric. Consequently, the three-dimensional transport of the pregnant solution and the mass-transfer process in the column can be reduced in cylindrical coordinates (r, $\theta$, and z) to an axisymmetric two-dimensional problem (Figure 1b).

(v)

Resin swelling, structural deformation of the bed, and physicochemical aging of the sorbent are neglected.

(vi)

The analysis is restricted to the first sorption cycle with initially fresh sorbent.

Near the inlet pipe and the conical flow distributor, the Reynolds number may significantly exceed unity, whereas in the main part of the column, the flow retains a filtration character. Therefore, the motion of the fluid in the porous sorbent bed is described using the relations of filtration theory—the Forchheimer equation and the mass conservation equation for fluid in the pore space [14,15,16,17]:

$$\alpha \overrightarrow{u}\left(r,z,t\right)=\alpha \varphi \overrightarrow{v}\left(r,z,t\right)=-\nabla P\left(r,z,t\right)$$

(2)

$$\nabla ·\overrightarrow{u}=\nabla ·\left(\varphi \overrightarrow{v} \right)=0,$$

(3)

where $\overrightarrow{u}\left(r,z,t\right)$ is the Darcy velocity, (m/h), $\varphi$ is the bed porosity, $\overrightarrow{v}\left(r,z,t\right)$ is the average fluid velocity in the pores,${\alpha =\alpha }_1+{\alpha }_2|\overrightarrow{u}|$, $P=p/\rho +\overrightarrow{g}·\overrightarrow{x}$, p is the fluid pressure, $\rho$ is the fluid density, $\overrightarrow{g}$ is the gravitational acceleration, $\overrightarrow{x}$ is the position vector, and $\mu$ is the dynamic viscosity of the solution:

$${\alpha }_1=633\mu \left(1-\varphi \right)/\rho {\delta }^2 , {\alpha }_2= 3(1-\varphi )\varphi /2\theta \delta ,$$

where $\mathrm{\delta }$ is the diameter of spherical sorbent particles (l).

For typical porosity values $\varphi \le 0.4$, it is assumed that the empirical coefficients satisfy the relationship given by [16]:

$$\theta =0.508-0.56\left(1-\varphi \right).$$

The transport of dissolved uranium species in the pregnant leach solution and the ion-exchange process between the liquid solution and the sorbent are assumed to be isothermal and are described by the following partial differential equation [12,13,16,17,18,19]:

$$\partial C/\partial t + \nabla \cdot (\overrightarrow{v} C)=\nabla \cdot \left(D\nabla C\right)-\left({\phi }^{-1}-1\right) \partial \overline{C}/\partial t,$$

(4)

where D is the hydrodynamic dispersion tensor, C is the concentration of dissolved uranium species in the solution (g/L), and $\overline{C}$ is the concentration of uranium species adsorbed by the sorbent (g/L).

Because the porous medium is assumed to be homogeneous and isotropic, the dispersion tensor is symmetric and has three non-zero components [14,15]:

$$D_{rr}={\alpha }_L {\displaystyle \frac{v_r^2}{|\overrightarrow{v}|}}+{\alpha }_T{\displaystyle \frac{v_z^2}{|\overrightarrow{v}|}} +D_m,$$

$$D_{zz}={\alpha }_T {\displaystyle \frac{v_r^2}{|\overrightarrow{v}|}}+{\alpha }_L{\displaystyle \frac{v_z^2}{|\overrightarrow{v}|}}+D_m,$$

$$D_{rz}=D_{zr}=\left({\alpha }_T-{\alpha }_L\right){\displaystyle \frac{v_r{v}_z}{|\overrightarrow{v}|}} ,$$

where ${\alpha }_L$ is the longitudinal dispersion coefficient proportional to the sorbent particle diameter (l),${ \alpha }_T$ is the radial dispersion coefficient (l), and ${\alpha }_L/{\alpha }_T\approx 5, D_m$ is the molecular diffusion coefficient in the liquid phase (l²/h).

The fluid velocity field appearing in the transport equation is determined as follows.From the Forchheimer equation, the magnitude of the velocity is expressed as a function of the pressure gradient magnitude. The velocity vector can therefore be written as:

$$\overrightarrow{u}=-\nabla P/({\alpha }_1+{\alpha }_{2}f(\left|\nabla P\right|) .$$

(2’)

Substituting this expression into the mass conservation equation described by Equation (3) in cylindrical coordinates yields a nonlinear elliptic equation for the pressure field:

$${\displaystyle \frac{\partial }{\partial r}}\left(rA{\displaystyle \frac{\partial P}{\partial r}}\right)+ {\displaystyle \frac{\partial }{\partial z}}\left(rA{\displaystyle \frac{\partial P}{\partial z}}\right)=0$$

(5)

where

$$A\left(r,z,t\right)=1/({\alpha }_1+{\alpha }_{2}f(\left|\nabla P\right|).$$

After solving Equation (5), the velocity field in the sorption column is obtained from Equation (2’).

The ion-exchange process between the solution and the sorbent according to scheme (1) is assumed to be controlled by intraparticle diffusion and is described by the pseudo-first-order kinetics equation [9,18,20,21]:

$$\partial \overline{C}/\partial t=\mathrm{\beta }(C-\overline{C}/K_d)$$

(6)

where β is the kinetic coefficient of sorption characterizing the rate of intraparticle mass transfer (h⁻¹), and $K_d$ is the equilibrium distribution coefficient. In the present model, $K_d$ is defined as the ratio of equilibrium concentrations expressed in the same units; therefore, it is dimensionless.

Boundary conditions were imposed as follows.

For the fluid velocity and pressure at the solid walls of the column, the no-flux boundary condition is imposed:

$${\left.\alpha \overrightarrow{u}·\overrightarrow{n}\right|}_{\mathit{S}}={\left.-\partial P/\partial n\right|}_{\mathit{S}}=0,$$

(7)

where $\overrightarrow{n}$ is the unit normal vector to the column wall.

At the column inlet, the cross-sectional area of the feed pipe, the volumetric flow rate, and the inlet concentration of dissolved uranium species are specified:

$${\left.\overrightarrow{u}·\overrightarrow{n}\right|}_{\mathit{i}\mathit{n}}={\left.\partial P/\partial z\right|}_{\mathit{i}\mathit{n}}=V_0=Q_0/S_{in}, {\left.C\right|}_{\mathit{i}\mathit{n}}=C_{in}.$$

(8)

At the column outlet, the pressure is assumed equal to the ambient pressure, and soft boundary conditions are imposed on the concentration distribution:

$${\left.P\right|}_{\mathit{o}\mathit{u}\mathit{t}}=P_{out}, {\left.\partial C/\partial z\right|}_{\mathit{o}\mathit{u}\mathit{t}}=0.$$

(9)

At the initial time, the column is filled with liquid and fresh sorbent that does not contain uranium species:

$$C(\overrightarrow{x},t=0)=0, \overline{C}(\overrightarrow{x},t=0)=0.$$

(10)

The kinetic parameters $\beta$ and K_d in Equation (6) were determined by fitting the numerical results to experimental sorption data obtained for Purolite Ltd. (Llantrisant, Wales, UK) ion-exchange resin. The sorption kinetic curves obtained from experimental measurements were used to determine the kinetic characteristics of the sorbent, including the equilibrium time, sorbent capacity at equilibrium, the equilibrium distribution coefficient, and the kinetic sorption coefficient (Figure 2). Considering the sorption process in the experiment as a mass-transfer process occurring in an ideal reactor, the following parameter values were obtained: $\beta \approx$ 120 h⁻¹ and K_d = 286.

2.3. Numerical Method and Implementation

Among numerical methods for solving problems of mathematical physics, the finite-difference method is one of the most universal approaches. From a computational point of view, this method is most efficiently implemented on regular grids in domains with simple geometries, such as rectangles or their combinations. However, in numerical studies of mass-transfer processes accompanied by physicochemical reactions in domains of complex shape, the finite-element method or the finite-volume method is more commonly applied [22,23,24].

The use of general-purpose commercial software (e.g., COMSOL Multiphysics and OpenFOAM) and specialized hydrogeological simulators was not adopted in the present study due to their limited flexibility in implementing the fictitious domain method (FDM) on structured grids [25,26,27,28]. The problem under consideration requires an accurate representation of the curvilinear geometry of the conical distributor without resorting to computationally expensive unstructured tetrahedral meshes. This requirement is satisfied by employing the FDM formulation combined with a Heaviside function to represent internal boundaries [25,26,27,28]. Furthermore, the developed numerical code allows explicit control of the nonlinear iterative solution procedure, including periodic storage of intermediate computational states and controlled interruption of the simulation. Such an approach ensures enhanced numerical stability and flexibility compared to closed-source commercial solvers.

In the present study, the numerical investigation of the sorption process in a complex-shaped column was carried out using the fictitious domain method (see Appendix A). The effects of the conical flow distributor geometry and the volumetric solution flow rate on the characteristics and efficiency of the sorption process were investigated.

Due to the axial symmetry of the problem with respect to the Oz axis, the calculations were performed in a rectangular computational domain defined by:

$$0<r<R, 0<z<H$$

using a finite-difference grid of size N_r × N_z = 150 × 600.

The numerical study was performed for industrially relevant geometric and operating conditions representative of uranium recovery columns used in commercial ISL operations. The principal geometric dimensions, transport parameters, and operating conditions adopted in the simulations are summarized in

Table 1. Main geometric and operating parameters used in simulations.

Symbol	Parameter	Dimension, Value
$\overrightarrow{u}\left(r,z,t\right)$	Darcy’s velocity	m/h
p	Fluid pressure	(kPa)
$\rho$	Water density	kg/m3
$\mu$	Dynamic viscosity of the water	g/(cm × sec)
$D_m$	Molecular diffusion coefficient in the liquid phase	l2/h
D	Hydrodynamic dispersion tensor	l2/h
C	Concentration of dissolved uranium species in the solution	g/L
$\overline{C}$	Concentration of uranium species adsorbed by the sorbent	g/L
$\varphi$	Bed porosity	$0.36$
$C_{in}$	Inlet uranium concentration	$0.085 \mathrm{g}/\mathrm{L}$
H	Column height	6 m
R	Column radius	1.5 m
d	Inlet pipe diameter	0.4 m
$\delta$	Resin particle diameter	0.4 mm
2α	Distributor angle	90°, 120°
$\beta$	Sorption kinetic coefficient	120 h−1
Kd	Equilibrium distribution coefficient	286
Q0	Inlet flow rate	0.35 m3/h, 0.7 m3/h
Nr × Nz	Computational grid	150 × 600

.

The influence of the conical distributor geometry and the solution flow rate on the efficiency of the sorption column and the completeness of the ion-exchange process was investigated for the following cases:

• In the absence of a conical flow distributor, with volumetric flow rate Q₀ = 0.35 m³/h;
• In a fixed apex position of the conical distributor, with opening angles α = 45° and 60°, Q₀ = 0.35 m³/h;
• At different flow rates of the solution, Q₀ = 0.35 m³/h and Q₀ = 0.7 m³/h, with the same distributor angle α = 45°.

3. Results

3.1. Flow Structure Without a Conical Distributor

Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 show the calculated spatial distributions of pressure, flow velocity, and uranium concentration in the immobile phase (sorbent) in the absence of the conical distributor at the flow rate Q₀ = 0.35 m³/h.

Figure 3 presents the calculated pressure distribution (kPa) and the Reynolds number distribution. The Reynolds number was calculated using the diameter of the inlet pipe $Re=V_0\rho d/\mu$. When calculated based on the diameter of the resin granules, its value is three orders of magnitude smaller. As expected, near the inlet pipe, whose cross-sectional area is (D/d)² times smaller than that of the column, both the pressure and flow velocity are significantly higher than in regions farther away from the inlet.

Figure 4 shows the distributions of uranium concentration in the immobile phase after 10 and 40 h of sorber operation. It can be seen that the main flow containing dissolved uranium species develops in the central part of the column, whereas the near-wall region of the sorbent is more weakly involved in the sorption process.

This effect is particularly evident in Figure 5, which shows the distributions of uranium concentration in the immobile phase across the column cross-section at the level z = 1.5 m at different times after the beginning of the sorption process. The results indicate that the average uranium concentration across the column cross-section, given by:

$${\overline{C}}_Z\left(Z,t\right)={\int }_0^{R\left(Z\right)}r\overline{C}\left(z,r,t\right)dr/(\pi R^2(Z)),$$

does not exceed 60% of the maximum possible concentration (23 g/L) (see Figure 5 and Figure 6).

To assess the efficiency of sorbent utilization, the degree of sorbent involvement in the sorption process was analyzed. Particular attention was given to the uniformity of sorption across the column cross-section, which represents a standard criterion used in the operational analysis of industrial sorption columns.

For this purpose, the ratio between the total mass of sorbed mineral in the sorbent layer of height:

$$M\left(z,t\right)= {\int }_0^z{\int }_0^{R\left(z\right)}r\overline{C}(z,r,t)drdz$$

and the maximum possible mass that can be retained by the sorbent in the same volume:

$$M_m\left(z,t\right)= {\int }_0^z{\int }_0^{R\left(z\right)}r{\overline{C}}_{max}\left(r,z,t\right)drdz=V_C{K}_d{C}_{in}$$

were calculated. Here, V_C = V_C(z) denotes the volume of the column segment of height z. Figure 7 shows the dependence of the ratio:

$$\mu \left(z,t\right)= M\left(z,t\right)/M_m\left(z,t\right)$$

on the height of the column and time.

Thus, in the absence of a conical distributor, the main flow containing dissolved uranium species is concentrated in the central region of the column, whereas the near-wall sorbent zone participates significantly less in the sorption process. As a result, the average uranium concentration across the column cross-section does not exceed two-thirds of the equilibrium concentration.

Analysis of the sorbent utilization factor also indicates low efficiency of sorbent usage in the column under these conditions.

3.2. Influence of the Conical Distributor

Further analysis considered sorber designs with conical distributors having angles α = 45° and 60° at the flow rate Q₀ = 0.35 m³/h.

Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 show the calculated distributions of pressure, filtration velocity, and uranium concentration in the immobile phase.

Figure 8 illustrates the detailed velocity and pressure fields near the inlet pipe and the conical distributor for the opening angle 2α = 90°.

Figure 9 and Figure 10 present the distributions of pressure and the local Reynolds number in the column. The Reynolds number is defined based on the inlet pipe diameter: $Re=Vd\rho /\mu$.

The results show that both pressure and the local Reynolds number decrease rapidly in the vicinity of the conical distributor. For larger opening angles, the pressure drop near the cone is slightly greater; however, farther downstream, the pressure fields in both cases become nearly identical.

Figure 11 and Figure 12 show the distributions of uranium concentration in the solid phase after 10 and 40 h of column operation. The results indicate that for larger distributor opening angles the saturation zone extends farther along the column. At the same time, the region directly behind the distributor that is not involved in sorption becomes wider compared to the case of smaller angles.

The uranium concentration in each cross-section of the column gradually increases with time until reaching the equilibrium concentration, which follows directly from the sorption kinetics described by Equation (6).

Analysis of the concentration profiles shows that the distributions of uranium concentration in the solid phase form a sorption saturation wave propagating along the column in the direction of the flow. The front of this wave gradually broadens due to longitudinal hydrodynamic dispersion in the porous sorbent bed.

Figure 13 shows the cross-sectional distributions of uranium concentration in the solid phase at the level z = 1.5 m from the inlet of the sorption column at different times and for different opening angles of the distributor. It can be seen that for the angle 2α = 120°, the saturation wave propagates farther along the column than in the case of 2α = 90°.

The distribution of the cross-sectionally averaged concentration of uranium adsorbed by the sorbent along the column height is determined by:

$${\overline{C}}_z(z,t)={\displaystyle \frac{1}{\pi R^2(z)}}{\int }_0^{R\left(z\right)}r\overline{C}\left(z,r,t\right)dr.$$

Figure 14 shows the distributions of the averaged concentration ${\overline{C}}_z$ at different times for different opening angles of the distributor. The influence of the distributor opening angle on the overall character of the concentration distribution is relatively small and becomes noticeable mainly at the initial stage of the process. In the case of a narrower distributor (2α = 90°), a more intensive adsorption of uranium is observed. This effect is associated with the higher filtration velocity of the solution in the corresponding region of the sorption column (see Figure 10).

This feature is also clearly illustrated in Figure 15, which shows the distributions of the degree of sorbent involvement in the sorption process along the column height for different opening angles of the conical distributor.

Based on the performed analysis, it can be concluded that the non-uniform distribution of dissolved uranium species across the column cross-section indicates incomplete sorption and therefore insufficient efficiency of the column operation. In particular,

• The main volume of uranium in both the mobile and immobile phases forms a ring-shaped cylindrical region whose thickness is approximately two times smaller than the column radius;
• The mass concentration of uranium in the central part of the column in both phases is significantly lower than the equilibrium saturation concentration of the sorbent;
• The conical flow distributor installed at the inlet of the column does not ensure a sufficiently uniform distribution of the solution across the column cross-section.

These results indicate that the efficiency of sorbent utilization in the column is largely controlled by the hydrodynamic structure of the flow, which determines the spatial distribution of mass transfer in the porous sorbent bed.

3.3. Influence of Flow Rate

Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 present the results of a comparative analysis of the sorption column operation at two solution flow rates: Q₀ = 0.35 m³/h and Q₀ = 0.7 m³/h. In all cases, the opening angle of the conical distributor is 2α = 90°.

With increasing solution flow rate, the pressure gradient and therefore the hydraulic resistance of the resin bed increase significantly. A considerable part of the pressure drop occurs in the region between the inlet pipe and the flow distributor, where the liquid velocity is the highest (Figure 16). The local Reynolds number decreases sharply behind the conical distributor in both columns (Figure 16 and Figure 17).

For efficient operation of the sorption column, the Damköhler number $Da=\beta H/V$ must satisfy the condition $Da\ge 1$ which ensures the formation of a narrow mass-transfer zone and efficient column operation [19], where H is the column height and V is the superficial velocity.

For practical values of the system parameters

$$\beta =120 {\mu }^{-1}, K_d=286, \phi =0.36, H=6 \mathrm{m}, R=1.5 \mathrm{m}, S=\pi R^2,Q > 0.35 {\mathrm{m}}^3/\mathrm{h}, V=Q/\left(\phi S\right)$$

The hydrodynamic residence time of the solution in the column is much larger than the characteristic sorption time; therefore, $Da\gg 1$. This means that sorption occurs much faster than convective transport. Consequently, the structure of the mass-transfer zone is mainly determined by flow hydrodynamics and longitudinal dispersion.

Figure 18 and Figure 19 illustrate the distributions of uranium concentration in the mobile (liquid) and immobile (adsorbed) phases at two different times. For the considered parameters of the sorption column, the ion-exchange sorption of dissolved uranium species occurs almost instantaneously compared to convective transport. Therefore, the ion-exchange process is primarily hydrodynamically controlled. The calculations also show that doubling the solution flow rate (i.e., the average linear velocity) leads to an increase in the width of the sorption front (the mass-transfer zone) by approximately 1.5 times (Figure 20). This effect is caused by the increase in the longitudinal dispersion coefficient, which is proportional to the flow velocity.

At the same time, excessive increase in the solution flow rate results in a substantial rise in the hydraulic resistance of the resin bed and consequently increases energy consumption.

4. Discussion

The obtained results demonstrate that hydrodynamic non-uniformity is a key factor governing the performance of industrial-scale packed ion-exchange columns. In contrast to conventional one-dimensional fixed-bed models, where the superficial velocity is treated as a cross-sectionally averaged quantity and radial gradients are neglected, the present multidimensional formulation reveals pronounced radial variations in the filtration velocity induced by the inlet geometry and distributor configuration. Similar limitations of one-dimensional approaches have been widely recognized in the analysis of packed-bed reactors and adsorption systems, where such models tend to overestimate bed utilization and mass-transfer efficiency under non-uniform flow conditions [18,29].

The predicted concentration of the main flow in the central region of the column in the absence of a distributor is consistent with well-known channeling and preferential flow effects reported for packed beds. Such flow structures lead to ineffective utilization of the near-wall regions of the bed and reduction of overall mass-transfer efficiency. The relatively low average sorbent loading obtained in the simulations, despite high equilibrium capacity, is therefore in agreement with experimental and theoretical studies showing that hydrodynamic maldistribution can become a dominant limiting factor in large-diameter adsorption columns.

The introduction of a conical distributor substantially modifies the internal flow structure and improves the radial redistribution of the incoming solution. However, the simulations indicate that complete flow equalization is not achieved even with the distributor. This observation is consistent with previous studies on flow distributors in packed columns, which show that single-stage distributors often provide only partial homogenization of the velocity field, especially at industrial scales [29]. This suggests that further optimization may require multi-stage or more advanced distributor designs.

An important feature revealed by the present model is the formation of a ring-shaped active sorption zone within the column cross-section. This effect arises from the combined influence of radial velocity gradients and longitudinal dispersion. Similar multidimensional structures of the mass-transfer zone have been reported in studies of reactive transport in porous media, although they cannot be captured by one-dimensional models. This highlights the necessity of multidimensional modeling for accurate prediction of industrial sorption column performance when complex inlet geometries are involved.

The calculated Damköhler numbers significantly exceed unity under all considered operating conditions, indicating that ion-exchange kinetics are much faster than convective transport. This places the system in a transport-controlled regime, where the overall process is governed primarily by hydrodynamic dispersion and flow structure rather than by intrinsic sorption kinetics. Such behavior is consistent with the classical theory of reactive transport in porous media [14,30]. Therefore, further improvements in column performance should focus on hydrodynamic optimization rather than on modification of sorbent kinetic properties.

An increase in the solution flow rate leads to a broadening of the mass-transfer zone despite the increase in axial transport velocity. This effect can be explained by enhanced longitudinal dispersion and mixing within the porous medium and is in agreement with classical correlations for axial dispersion in packed beds [31]. At the same time, the corresponding increase in hydraulic resistance results in higher pressure losses, indicating a trade-off between throughput and energy consumption that must be considered in industrial operation.

Unlike most previous studies, which rely on one-dimensional models or assume uniform flow distribution, the present work explicitly resolves radial hydrodynamic non-uniformities induced by the inlet geometry. This allows the capturing of effects such as preferential flow paths, incomplete radial equalization, and ring-shaped sorption zones. At the same time, the general trends obtained in this study—such as the dominant role of hydrodynamics at high Damköhler numbers and the broadening of the mass-transfer front with increasing flow rate—are in qualitative agreement with existing theoretical and experimental studies of transport and adsorption in porous media.

It should be noted that the present model assumes a homogeneous porous sorbent bed with effective averaged transport parameters and does not explicitly account for local heterogeneity, particle-scale diffusion effects, or possible physicochemical changes in resin properties during long-term operation. These limitations are consistent with commonly used continuum models and should be addressed in future studies for more detailed system-specific predictions.

The reliability of the numerical approach is further supported by its prior validation for more complex flow and transport problems involving Navier–Stokes equations and complex geometries [32].

5. Conclusions

A multidimensional coupled hydrodynamic and mass-transfer model of uranium sorption in a packed ion-exchange column equipped with a conical flow distributor has been developed. The model combines the Forchheimer filtration law, convective–dispersion transport, and sorption kinetics, enabling a detailed analysis of coupled flow and mass-transfer processes in industrial sorption columns with complex geometries.

The main conclusions of the study are as follows:

• Flow maldistribution significantly affects sorption performance. In the absence of a distributor, the main solution flow concentrates in the central region of the column, while the near-wall sorbent layer remains underutilized.
• A conical distributor improves but does not fully eliminate radial non-uniformity. The distributor substantially redistributes the incoming flow; however, complete radial equalization is not achieved under the investigated conditions.
• Mass-transfer dynamics are governed primarily by hydrodynamics rather than kinetics. For the considered operating conditions, the Damköhler number is significantly greater than unity, indicating that sorption kinetics are much faster than convective transport.
• Increasing the flow rate broadens the mass-transfer zone and increases hydraulic resistance. Doubling the solution flow rate increases the width of the sorption front by approximately 1.5 times due to enhanced longitudinal dispersion, while simultaneously increasing pressure losses.

The obtained results are consistent with established theoretical and experimental observations on packed-bed hydrodynamics and transport processes, while providing new insight into multidimensional flow and mass-transfer structures that are not captured by conventional one-dimensional models.