Theoretical Hydrodynamic Modeling of the Fluidized Bed Photoreactor (FBP) Using Computational Fluid Dynamics (CFD): Fluidization Conditions for TiO 2 -CuO Immobilized on Beach Sand Granules

. The ﬂ ow regime is essential in the photoreactor ’ s performance in pollutant degradation in the aqueous medium, especially in ﬂ uidized systems. Therefore, this study is focused on determining the ﬂ uidization conditions of a granular catalyst based on TiO 2 -CuO nanoparticles (1 wt.% CuO) immobilized on beach sand granules using an FBP photoreactor. COMSOL Multiphysics 6.0 was employed for inlet velocities between 0.1m/s and 1.0m/s, mainly from the Reynolds averaged Navier – Stokes (RANS) turbulence model and the Stokes drag law. The results indicated that the average velocities in the annular section are much higher ( 4 11 u t and 5 42 u t ) than the required particle terminal velocity. Moreover, the pressure contour lines revealed that these ﬂ ow velocities do not represent excessive pressures in the concentric cylinders, with maximum gauge pressures of 740.52 Pa and 1310 Pa for inlet velocities U o = 0 75 and 1.0 m/s, respectively. Finally, it was determined that the Reynolds number adjusted ( Re pf ) values lower than or equal to 1 37 × 10 − 3 allow high ﬂ uidization after 2 seconds. This information makes it possible to adapt and assemble the FBP equipment for future photocatalytic evaluation.


Introduction
Commercial computational fluid dynamic (CFD) packages, such as COMSOL Multiphysics and ANSYS Fluent, have been successfully used in the simulation of advanced photochemical and nonphotochemical oxidation processes, specifically for the modeling of hydrodynamic [1][2][3][4][5][6], radiative [7][8][9][10][11], mass transfer [12][13][14], and kinetic [4,[15][16][17][18] phenomena, establishing the optimal values of typical dimensionless numbers, such as Reynolds (Re), Hatta (Ha), Schmidt (Sc), and Sherwood (Sh), among others, being helpful for the scaling process of this technology.According to the bibliometric study (2010-2023) carried out in this work using the Scopus database (http://www.scopus.com/), Figure 1(a) shows a growing interest of the international scientific community in topics related to the modeling and mathematical simulation of photoreactors used in advanced oxidation processes (AOPs).Moreover, leading researchers (Figure 1(b)) in the area were identified, among which Alfano, O. M.; Satuf, M. L.; Li Puma, G.; Marugán, J.; and Machuca-Martínez, F. stand out.Further, according to Figure 1(c), the National Natural Science Foundation of China and the Universidad Nacional del Litoral (Argentina) are the primary funding sponsors worldwide for research focused on reactor 2 Modelling and Simulation in Engineering modeling for AOPs, which agrees with the major countries' top (China, Canada, France, United States, Argentina, and Spain) and universities/affiliations, depicted in Figures 1(d) and 1(e), respectively.Besides, the most significant publications are scientific (85.4%) and review articles (5.9%) in chemical engineering, environmental sciences, and chemistry (see Figure 1(f)).Furthermore, a map based on bibliographic data from Scopus was created in the VOSviewer (Figure 2), a software tool for constructing and visualizing bibliometric networks.This map confirmed the wide use of CFD programs and other modeling and simulation strategies for advanced oxidation processes, obtaining essential information, such as velocity and radiation fields, and kinetic models to study the mass transfer phenomena for both suspended and immobilized materials in packed and fluidized beds.
For example, Asgharian et al. [19] performed CFD modeling and validation in the COMSOL software of a Where , is the density, U es velocity, P es pressure, es la viscosity, I is unit tensor.

Drag law: Stokes' law
Where, u' y are the velocity field of the fluid and the particle, respectively.
Particle properties : Step 1. Terminal velocity: Step 2. Average fluid velocity: Step 3. Reynolds number (Re): Where U 0 , is the input velocity, and d boqi is the inlet nozzles diameter. Newtonian. Incompressible.
Isothermal.Non-reactive.Constant physical properties.Laminar steady state flow.Modelling and Simulation in Engineering code FLUENT in 3D.Also, Khataee et al. [20] reported photocatalytic ozonation for the anthelmintic drug degradation using ceramic-coated TiO 2 NPs through CFD simulation coupled with kinetic mechanisms.This study represents a stage before the photocatalytic activity evaluation of the granular catalysts based on TiO 2 -CuO immobilized on beach sand granules, whose objective is to determine the fluidization conditions in an annular photoreactor using the CFD software of COMSOL Multiphysics 6.0.The required flow regime analysis for fluidization represents an essential step in the concentric cylinder photoreactor's assembly before the photocatalytic evaluation of the granular catalyst with potential easy separation, reuse, and toxicological effect inhibition associated with

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Modelling and Simulation in Engineering nanomaterials remaining.Also, this research represents a prestudy to the photocatalytic degradation of metformin using TiO 2 -CuO heterojunctions synthesized by green chemistry and immobilized on beach sand granules in a fluidized bed annular photoreactor.Moreover, these conditions have allowed the composite material fluidization to favor its dispersion and photoactivation throughout the entire flow reaction system through the annular section.Therefore, the experimental design for drug degradation in an aqueous medium has considered this flow rate range.

Methodology
2.1.Velocity Profiles in the Annular Region and Particle Trajectory.Hydrodynamic modeling was performed in the finite element analysis and resolution software COMSOL 6.0 (licensed through the University of Cartagena), following the stages described by Memon et al. [21].Figure 3 describes each stage for generating the velocity profiles and estimating the trajectory of the particles dragged by the fluid flow from the physical model shown in Figure 4.The apparent density (ρ apparent ) of the granular photocatalyst was determined by the graduated cylinder method using the following equation [22]: where m tot is the total mass of the granular catalyst and V app is the apparent volume including solids and internal pores.Besides, an intermediate size range was selected after the sieving process in an Orto Alresa machine (Model: VIBRO), corresponding to the fine sand type.Moreover, Table 1 summarizes the granular catalyst's physical properties, dimensions, and specifications of the fluidized bed annular photoreactor, and parameters considered in the fluidization conditions of the granular catalyst, which are  Modelling and Simulation in Engineering indispensable for hydrodynamic modeling.Furthermore, the immobilization process was performed using the twostep method of immersion/heat treatment, reported in a recent publication by our research group, which focuses on the optical, morphological, and structural characterization of TiO 2 -CuO heterojunction nanoparticles synthesized by green chemistry supported on beach sand granules [23].The average velocity (v, m/s) was calculated by applying the continuity equation; the flow regime was also determined.The values obtained were compared with the estimated data (see Table 2) from COMSOL CFD software using the standard deviation of the residuals (prediction errors) by root mean square deviation (RMSD, Equation ( 2)) and the average absolute deviation (AAD, Equation ( 3)) [25].Both statistical indicators have values close to 0 (<0.003), which indicates that the CFD hydrodynamic model adequately predicts the velocity field.Table 3 presents a compilation of information relevant to the velocity profile obtained by CFD modeling for three tubular photoreactors reported in recent research.However, CFD modeling has been developed for other photocatalytic reactors, such as the raceway pond reactor [26], stirred tank [20], packing bed [27], compound parabolic collector [28,29], parallel-channel microreactor [16], curved channel [30], photo impinging streams cyclone [31], cross-flow [32], flat plate [33], baffled flat-plate [34], and static mixer [35], among others.In other work, Liu et al. [36] have developed a CFD modeling in gas-liquid-solid minifluidized beds.However, the dragging of solid particles was caused by the air distribution at the photoreactor's bottom.The research in Table 3 corresponds to photoreactors' CFD modeling with similar geometries as the FBP proposed in this study without including solid particles' fluidization through the drag phenomenon.

Results and Discussion
3.2.Pressure Profile.The pressure's contour lines were modeled with up-flow for four input velocities (U o = 0 1, 0.5, 0.75, and 1.0 m/s).These pressure profiles (plane YZ) are illustrated in Figures 7(a)-7(d).In these contour lines, max-imum gauge pressures of 14.75 Pa, 331.12 Pa, 740.52 Pa, and 1310 Pa were identified at the bottom of the fluidized bed annular photoreactor for inlet velocities, U o = 0 1, 0.5, 0.75, and 1.0 m/s, respectively; while the lowest pressures are in the discharge nozzles.
Most CFD studies of photoreactors do not report on pressure profiles; despite this, three recent publications were found that included this variable in their CFD analysis, as is the case of Asgharian et al. [19], who obtained the contour lines for pressure gauge inside a stirred tank photoreactor equipped with 8-blade backswept impeller for tetracycline degradation.These authors informed that after 0.5 min, the pressure increased and eventually reached about 4 Pa.Also, the pressure was negative behind the mixer blades and positive in front of them.
The research in Table 4 corresponds to the photoreactors' pressure profile obtained by CFD modeling with similar geometries but with a different orientation (horizontal) to the proposal in this study (vertical).

Particle's
Trajectory.This section shows the particle trajectory results in the fluidized bed annular photoreactor for the four study velocities (U o = 0 1, 0.5, 0.75, and 1.0 m/s), corresponding to 0 66u t , 2 79u t , 4 11u t , and 5 42u t ,   Modelling and Simulation in Engineering       Modelling and Simulation in Engineering respectively.In the case of U o = 0 1 m/s (0 66u t ), the particles do not leave the bottom (see Figure 8) because the fluid's velocity is lower than the terminal velocity (u t = 0 030 m/s).This phenomenon was reported by Picabea et al. [42], who found negligible fluidization for velocities below 0.021 m/s in a liquid-solid fluidized bed (LSFB) system with calcium alginate spheres.However, for velocities higher than terminal velocity (2 79u t , 4 11u t , and 5 42u t ), considerable fluidization is achieved due to drag, as shown in Figures 9-11, respectively.In a similar study, Abdulrahman et al. [43] demonstrated the influence of diameter (0.003 m, 0.004 m, and 0.006 m with a density of 2500 kg/m 3 ) of glass spheres and fluid velocity on fluidization velocity in a liquid-solid fluidized bed system; from results, authors informed that the expansion ratio is proportional to the liquid velocity and inversely proportional to the diameter of the beads.
Other research has also analyzed fluidized systems such as bubbling [44], droplet injection [45], and gas-liquid-solid flow [46].Still, studies have not been focused on photocatalytic applications in vertical concentric cylindrical equipment.
To ensure the granular catalyst's fluidization, a potential scaling-up was found through dimensional analysis-Buckingham π theorem, a Reynolds number adjusted (Re pf ) (Equation ( 4)) to a liquid-solid fluidized system, which relates the particle's Reynolds number (Re pf ) [15] with the flow developed Reynolds number (Re f ) in the annular region for the average velocity (v CFD i ).
Besides, it is proposed to select the Re pf values (see Figure 12) that allow exceeding half the reactor's length after two seconds (2 s) since rapid fluidization of the heterogeneous photocatalytic system is required.This analysis indicates that the velocities are U o = 0 75 m/s (4 11u t ) and U o = 1 0 m/s (5 42u t ), i.e., a velocity field with an average value greater than 4.11 times the particle's terminal velocity.

Conclusions
In this research, the fluidized bed annular photoreactor's hydrodynamics was successfully modeled before the adaptation and assembly process for the photocatalytic evaluation of the granular photocatalyst based on TiO 2 -CuO heterojunctions immobilized on beach sand granules.The results indicate that inlet velocities of U o = 0 75 m/s and U o = 1 0 m/s are required, generating average velocities in the annular section much higher than the particle terminal velocity, corresponding to 4 11u t and 5 42u t , respectively.Moreover, contour lines were obtained for the gauge pressure in the concentric cylinders, with maximum gauge pressures of 14.75 Pa, 331.12 Pa, 740.52 Pa, and 1310 Pa at the photoreactor's bottom for inlet velocities U o = 0 1, 0.5, 0.75, and 1.0 m/s, respectively.Finally, it was determined that Reynolds number adjusted (Re pf ) values lower than or equal to 1 37 × 10 −3 allow high flu-idization after 2 seconds, corresponding to a short time frame to ensure fluidization from the beginning of the pilot scale photoreactor operation.This information makes it possible to adapt and assemble the FBP equipment for the photocatalytic evaluation of the granular catalyst.Finally, the findings facilitated the identification of the best photoreactor configuration, the inlet velocities range to guarantee the fluidized regime, as well as the proposal of an adjusted dimensionless number for replication in studies operating conditions standardization in fluidized photoreactors for wastewater treatment by heterogeneous photocatalysis with reusable granular catalysts.

3. 1 .
photoreactor for subsequent stabilization (see Figure6(b)) with a piston-type flow regime in the annular region at the top of the photoreactor (see plane ZX for U o : 1.0 m/s in Figure6(a)).The average velocity (v, m/s) was calculated by applying the continuity equation; the flow regime was also determined.The values obtained were compared with the estimated data (see Table2) from COMSOL CFD software using the standard deviation of the residuals (prediction errors) by root mean square deviation (RMSD, Equation (2)) and the average absolute deviation (AAD, Equation (3))[25].

Table 1 :
Granular catalyst's physical properties and the dimensions and specifications of the fluidized bed annular photoreactor.

Table 3 :
Photoreactors' CFD modeling data for similar geometries from recent literature.

Table 2 :
CFD and calculated velocities for different inlet velocities.