Double diffusive Buoyancy‐driven flow in a fluid‐saturated elliptical annulus with a neural network‐based prediction of heat and mass transfer

This paper presents a numerical study of buoyancy‐driven double‐diffusive convection within an elliptical annulus enclosure filled with a saturated porous medium. An in‐house built FORTRAN code has been developed, and computations are carried out in a range of values of Darcy–Rayleigh number Ram (10 ≤ Ram ≤ 500), Lewis number Le (0.1 ≤ Le ≤ 10), and the ratio of buoyancy forces N (−5 ≤ N ≤ 5). In addition, three methods are used, namely the multi‐variable polynomial regression, the group method of data handling (GMDH), and the artificial neural network (ANN) for the predictions of heat and mass transfer rates. First, results are successfully validated with existing numerical and experimental data. Then, the results indicated that temperature and concentration distributions are sensitive to the Lewis number and thermal and mass plumes are developing in proportion to the Lewis number. Two particular values of Lewis number Le = 2.735 and Le = 2.75 captured the flow's transition toward an asymmetric structure with a bifurcation of convective cells. The average Nusselt number tends to have an almost asymptotic value for Le » 5. For the case of aiding buoyancies N > 1, the average Nusselt Number Nu¯ $\bar{{Nu}}$ decreased by 33% when the Lewis number increased to its maximum value. Then, it increased by 10% when the Lewis number increased to Le = 1 for the case of opposing buoyancies N < 1 and then decreased by 33% when the Lewis number increased to its maximum value., contrary to the behavior of the average Sherwood number Sh¯ $\bar{{Sh}}$ that increased by 700% for both cases N > 1 and N < 1. New correlations of Nu¯ $\bar{{Nu}}$ , and Sh¯ $\bar{{Sh}}$ as a function of Ram, Le, and N are derived and compared with GMDH and ANN methods, and the ANN method showed higher performance for the prediction of Nu¯ $\bar{{Nu}}$ and Sh¯ $\bar{{Sh}}$ with R2 exceeding 0.99.


| INTRODUCTION
Double diffusive natural convection has always been subject to theoretical, experimental, and numerical investigations within different enclosure geometries. For instance, for double diffusion, the heat and mass transfer processes are coupled together, leading to complex fluid behavior. This can be useful and enhance heat transfer rates. 1 The concentration difference parameter can also maximize the entropy and minimizes the dimensionless Bejan number. 2 The presence of porous media can significantly enhance the heat and mass transfer rate when different free or mixed convective mechanisms are considered with a combination of varying boundary conditions and fluid properties. [3][4][5][6][7] Natural convection in saturated porous media occurs when two independent mechanisms of diffusion are present in a porous medium, resulting in two different concentration gradients. This phenomenon has been extensively studied due to its relevance in various applications such as geothermal energy (heat and mass transfer in geothermal reservoirs), oil and gas production (extraction of oil from reservoirs and maximizing the production and recovery of oil), and materials processing (production of ceramic materials and the formation of a uniform and dense ceramic structure).
Double diffusive natural convection in saturated porous media has been extensively investigated in different geometries, and it is evident from the existing literature that most published studies related to this subject, particularly in the context of porous media, have focused on square or rectangular cavities with very few exploring circular annulus spaces. Moreover, to date, investigations into porous horizontal elliptical annuli are rare. The analysis of an annular elliptical enclosure was chosen due to the versatility of this geometry, which can take on circular, elliptical, or planar shapes depending on the values of the axis ratio.
A literature review has identified several comparable studies investigating natural convection in porous media-filled annular enclosures using different numerical techniques. The studies explored various aspects, such as thermal and solutal buoyancy, double-diffusive fluid flow, and the effect of other controlling parameters on heat transfer and flow structure. The numerical studies used different techniques such as Collocation-Chebyshev and Fourier-Galerkin, 8 the orthogonal system coordinates and finite differences method, 9,10 differential quadrature (DQ) and the formulation of the vorticity-stream function, 11,12 velocity-pressure formulation and Boussinesq's approximation, 13 finite volumes method and macroscopic k-ε model, 14 and the finite differences-based lattice BGK model formulated using the doubledistribution function approach, 15 to solve the dimensionless equations system. Results from the previous investigations and later studies [16][17][18][19] suggest the presence of different modes of natural convection and bicellular convection structures depending on the controlling parameters.
More recent investigations considered the effect of various parameters and driving mechanisms on heat and mass transfer characteristics. For instance, Tofaneli et al. 20 presented a numerical analysis of heat and mass transfer in an annular enclosure horizontal and cylindrical packed with saturated porous media. The driving mechanisms of thermal and solutal buoyancies contribute to the flow regime, which is considered laminar in both aiding and opposing modes. Mahfouz 21 investigated a fluid flow driven by the buoyancy effect coupled with heat transfer in a tilted elliptical enclosure representing the space between two confocal elliptical ducts. The mathematical model is framed under the stream function-vorticity formulation and solved using the pseudospectral method. The results showed that the elliptic ducts' inclination angle and aspect ratio significantly affect the flow regime and heat transfer. Cheddadi et al. and Belabid 22,23 presented a double-diffusive fluid flow computation throughout a porous horizontal annular enclosure using the finite difference method under the ADI scheme. The analysis considers the effect of different controlling parameters, and the study found that the onset of convection was delayed as the Lewis number decreased. The system exhibited complex flow structures at higher Rayleigh and Lewis numbers. Boulechfar et al. 24,25 conducted two numerical studies on natural convection in annular enclosures. The first study 24 examined the effects of inner eccentricity on flow structure and heat transfer in an annular elliptical space filled with a fluid-saturated porous medium. The results revealed two main modes of natural convection with two and multiple cells, depending on the free space within the enclosure linked to the value of the eccentricity of the inner ellipse. In the second study, 25 the authors analyzed the effects of the Rayleigh-Darcy number on heat and mass transfer in an annular cylindrical space filled with a fluid-saturated porous medium. The results showed the presence of two modes of heat and mass transfer, pseudo-conductive and convective, and a diffusive mode due to high solutal diffusivity Le = 0.1.
Currently, a more efficient approach for predicting thermal performance that is more precise and takes less time is gaining interest. It is known as machine learning (ML) prediction, particularly the Artificial Neural Network (ANN). The ANN works similarly to the human brain's neural network by anticipating outcomes within a reasonable range and learning from prior tasks. This system helps to reduce the time, cost, and effort involved with experimental and numerical investigations. ANN and other soft computing approaches are frequently used in thermal engineering systems to model, identify, and control various thermal processes. [26][27][28] For example, in the study by Yasin Varol et al., 29 the ANN and Adaptive-Network-Based Fuzzy Inference System (ANFIS) were used to predict natural convection variables in a triangular enclosure. The governing equations were solved using a finite difference technique, generating a database for ANN and ANFIS. The results were then compared with computational fluid dynamics (CFD) codes, showing that both soft programming codes could predict natural convection. Still, ANFIS had a more significant value compared with the actual value of ANN. Hyun Woo Cho et al. 30 examined the effect of variations in aspect ratio on three-dimensional (3D) mixed convection by numerical simulation at different Reynolds numbers, with a fixed Grashof number and Prandtl number. The heat transfer characteristics can be predicted accurately using an ANN, based on a few cases of direct numerical simulation, which otherwise would require days to calculate using DNS. Young Min Seo et al. 31 investigated the effect of a sinusoidal cylinder on natural convection in a rectangular enclosure, compared with a circular cylinder for different parameters, namely the mean radius and wavelength, and the Rayleigh number. Heat transfer characteristics at cylinder surfaces and enclosure walls were investigated, with improved performance of up to 27% at R mean = 0.4 L. An ANN was used to evaluate heat transfer performance, showing the Rayleigh number and mean cylinder radius have a significant impact on total heat transfer performance at cylinder surface and enclosure walls. Young Min Seo et al. 32 analyzed the flow and thermal fields of Rayleigh-Bénard convection in a rectangular channel with an internal circular cylinder, considering various parameters. Results are analyzed based on the iso-surface of temperature, vortical structure with orthogonal enstrophy distribution, and entropy generations. An ANN model accurately predicts the distribution of local Nusselt numbers with less computational time and cost than DNS. Youssef Tizakast et al. 33 used Machine Learning models to study double-diffusive natural convection in rectangular cavities filled with non-Newtonian fluids due to the complexity of the problem. Four models (ANN, RF, GBDT, and XGBoost) were employed to predict flow intensity, average Nusselt number, and average Sherwood number, based on four dimensionless parameters. The analysis confirms that the ANN model predicts the test data well, validates the choice of Machine Learning as a promising solution to model non-Newtonian doublediffusive fluid flows and confirms the nonlinear complexity of the problem.
Despite all these studies, double-diffusive buoyancy-driven convective flow in horizontal porous elliptical annulus has not so far been investigated as extensively as the case of studies in the vertical porous annulus. Therefore, the novelty of the present study is to examine the effect of different parameters such Darcy-Rayleigh number Ra m (10 ≤ Ra m ≤ 500), Lewis number Le (0.1 ≤ Le ≤ 10), and the ratio of buoyancy forces N (−5 ≤ N ≤ 5) for both aiding and opposing buoyancies in the annular elliptical geometry. An in-house-built FORTRAN code was developed to investigate numerically the effects mentioned above. The numerical analysis was carried out based on the conservative finite volumes method, the approximation scheme of power-law, and the algebraic system of equations obtained after discretization was solved by the Gauss-Seidel iterative method with an under-relaxation process. In addition to the numerical approach, three methods namely the multi-variable polynomial regression (MVPR), the group method of data handling (GMDH), and the ANN will be used for the predictions of heat and mass transfer rates. New correlations of the average Nusselt number Nu, and the average Sherwood number Sh as a function of Ra m , Le, and N will be derived and compared with GMDH and ANN methods.

| MODEL DESCRIPTION
The present study considers a saturated porous medium between 2D elliptical tubes to investigate natural convection with double diffusion phenomena, see Figure 1. The primary and minor axes of the two elliptical tubes are, respectively, A 1 , A 2 , and B 1 , B 2 . The elliptical walls are considered impermeable and maintained at constant concentrations and temperatures. The internal ellipse wall is characterized by T 1 , C 1, and the external wall by T 2 , C 2, considering that T 1 > T 2 and C 1 > C 2 .
Some hypotheses were retained to simplify the problem by considering a steady and laminar flow. Viscous dissipation and radiation are not taken into consideration. The interactions between heat and mass transfer are also neglected, and the porous medium is considered homogeneous and isotropic.

| MATHEMATICAL FORMULATION
The following equations represent the double-diffusive natural convection within a fluidsaturated porous annulus.
a. Equation of continuity: b. Momentum equation: The momentum equation is represented in our case by Darcy's model of the velocity equation in porous media: c. Equation of mass: The annular space geometry.
The Boussinesq approximation is represented by The transformation to the elliptic system of coordinates is completed by introducing the following: x c ch η θ y c sh η θ = · ( ) · cos( ), = · ( ) · sin( ).
As a result of the transformation, the metric coefficients are introduced as follows: The equations system is written in the elliptical system of coordinates and represented by the following system of equations: V η , V θ represent velocities in the elliptic coordinates, F(η, θ) and G(η, θ) are the transformation coefficients used in Equation (8) and defined as follows: The focal distance "c" is the characteristic length, while "a" represents fluid thermal diffusivity. The ratio (a/c) represents a characteristic velocity that is used in the nondimensional process.
The dimensionless system of equations is expressed as follows: V η + , V θ + represent the dimensionless velocities: Boundary conditions are summarized in the elliptic system as follows: The inner wall characterized by high temperature and high concentration: The outer wall characterized by low temperature and low concentration: With the introduction of Nusselt and Sherwood numbers which are defined locally as follows: The following relations express the average Nusselt and Sherwood numbers:

| NUMERICAL METHOD
A transformation of the coordinates to the elliptical system transforms the physical domain to a computational domain with a generated mesh using the elliptical method as shown in Figure 2.
The numerical analysis was carried out based on the conservative finite volume method and the approximation scheme of power-law to solve Equations (13)- (15) in the computational domain and take into consideration the associated boundary conditions. 34 Equation (16) was solved using the finite differences method. The algebraic equations obtained after discretization are solved by Gauss-Seidel iterative approach with an under-relaxation process. For better accuracy of the numerical results, the selected convergence criteria for the performed computations are defined for each variable as follows: A preliminary computation has been performed for different grid sizes to analyze the numerical solution sensitivity for the case of aiding buoyancies with N = 1 and Le = 0.1 for the ellipsis's eccentricities of e 1 = 0.9 and e 2 = 0.5 and the Rayleigh-Darcy number Ra m = 100 and α = 0°. Table 1 shows the mean Sherwood and Nusselt numbers and the maximum values of the stream function on the inner wall for different mesh sizes. Based on the preliminary analysis, the grid size of 51 × 71 demonstrated satisfaction for all further computations presented in this work.
Ensuring that the numerical calculation code can guarantee reliable and accurate results is essential at this stage. In this regard, the results of the developed in-house computational code were validated first with numerical data for the case of a pure natural convection problem for the elliptical space, which corresponds to a buoyancy ratio N = 0 and Pr = 7 that has been published by Mahfouz. 21 As illustrated in Table 2, a perfect agreement between the present results and results from Mahfouz 21 was noticed, especially for low Ra m values with an average relative error of 2% for the case considered. The comparison is encouraging and thereby indirectly validates our computational code.  Further validation is carried out with the experimental data of Kuehn and Goldstein, 35 which are often used as a benchmark for validating natural convection flow between two vertical coaxial cylinders. A qualitative and quantitative comparison between the present numerical results and the experimental data 35 is presented in Figures 3 and 4 for Ra = 4.7 × 10 4 , Pr = 0.706, and λ = 2.6. Figure 3 shows the resulting thermal field where the hot fluid moves upwards while the cold fluid moves downwards due to the buoyancy force. The temperature profiles for three different angles (θ = 0, 90, and 180) are shown in Figure 4. By carefully examining Figures 3 and 4, it is evident that the obtained results are in good agreement with the experimental data.

| RESULTS WITH DISCUSSION
The main purpose of the present study is to investigate two different approaches for the numerical analysis of double-diffusive heat and mass transfer. The first approach involves using a CFD method to analyze the impact of various parameters such as Darcy-Rayleigh number Ra m (10 ≤ Ra m ≤ 500), Lewis number Le (0.1 ≤ Le ≤ 10), and the ratio of buoyancy forces N (−5 ≤ N ≤ 5) when the fluid flow is driven by aiding and opposing thermal and mass buoyancies. The second approach utilizes an ANN model 36 to predict heat and mass transfer rates based on data collected from CFD results. A comparison is also made using the polynomial regression (MVPR) method 37 and the GMDH. 38 In this perspective, the distribution of stream function, isotherms, concentration isolines, and Nusselt and Sherwood numbers are presented. The computations were carried out using air as a Newtonian fluid, and the annular space was not inclined (α = 0°). However, there are currently no comparable results available in the literature that are satisfactory for comparison purposes. To perform this calculation, the annular spaces are characterized by the eccentricities of the inner e 1 = 0.9 and the outer ellipses e 2 = 0.5 and with Ra m = 100, and the annular space has a major horizontal axis (α = 0°). Multiple values of Lewis number were used within a range of [0. [1][2][3][4][5][6][7][8][9][10] for the case of cooperating buoyancies that generate the flow in the same direction and develop at the same rate when buoyancy ratio N = 1. Figure 5 represents the isolines of stream function, isotherms, and concentration which are symmetric concerning the ellipse's minor axis. The streamlines in this figure illustrate the fluid flow structure organized as two vortices spinning in opposite directions, mainly resulting from the upward and the downward forces. The upward movement is due to the buoyancy force caused by temperature and concentration gradient differences, and the downward movement is due to gravity. The distance between the counter-rotating cells in the upper region is reduced as the convective motion intensifies, while in the bottom space, the opposite is true Regarding the thermal field, the isotherms deform in the upper space due to the convection intensity compared with the lower region, where the isotherms are stratified near the inner ellipse, forming a boundary layer. This shape clearly demonstrates that two different modes of heat transfer are present: a convective mode leading the heat transfer in the upper space and a F I G U R E 4 Comparison of dimensionless temperature profiles in the radial direction at various angles between the present work and the experimental results.
pseudo-conductive mode widely present in the lower space. In the same figure, concentration isolines are characterized by concentric curves perfectly matching the inner and outer ellipses profiles, especially in the bottom region where a mass transfer is driven by diffusion. The concentration contours demonstrate a slight deformation announcing a transition to a convective mode in the upper space. Figure 5B,C illustrate that with an increase in Lewis number value, the flow remains structured in two main counter-rotating vortices, with a decrease in the stream function value due mainly to the thermal diffusion mechanism intensified by the increasing Lewis number. The isotherms in Figure 5B show a slight attenuation of the convection in the upper space where the thermal isolines are less distorted, but the convective mode remains dominant. Regarding the mass transfer in the same figure, concentration contours indicate that a convective mode is rising within the upper half of the enclosure due to the rise in Lewis number. In Figure 5C when Le = 1, both thermal and solutal stratifications are developing similarly, which is obvious because thermal and solutal diffusivities are equal.
This increase in the Lewis number attenuated fluid flow throughout the annular space due to thermal and solutal stratifications. Figure 5D shows that the stream function isolines changed significantly in their form, transitioning from a bean shape to a wing shape. As the gap width increased, the center of each vortex moved toward the extremities, and both vortices tend to become nonadjacent. The decrease in the stream function value is due to the attenuation of the flow caused by both thermal and solutal stratifications. As for the isotherms, their form changed from a dome-shaped curve to double-humped shape curves due to increased thermal diffusivity with an increase in Lewis number Le > 1. The concentration isolines indicate an intensification of solutal stratification in the middle region of the space. The distortion of the concentration curves in the bottom region clearly shows that mass transport is conducted by a convective mode taking place in the entire region due to low solutal diffusivity. Figure 5E, which corresponds to Le = 10 (thermal diffusivity 10 times larger than the solutal diffusivity), shows that the flow structure of the streamlines maintains unchanged. However, the gap width separating the counter-rotating vortices is more important due to the intensification of the thermal stratifications. Streamfunction isolines decline furthermore in the direction of the right and the left sides giving rise to a pseudo bifurcation in the upper space under the stratifications resulting from the preponderance of the thermal diffusion mechanism. On the one hand, isotherms in Figure 5E dramatically change in the upper region, becoming double-humped curves with a significant stratification between the two humps forming a boundary layer region. On the other hand, the thermal gradient is clearly decreasing in the bottom region, and the effect of the boundary layer is progressively vanishing. Moreover, the figure indicates that the solutal stratification is dramatically increasing within all regions of the enclosure, and the mass transport mechanism is driven mainly by a convective mode. This intensification gives rise to a skinny solutal boundary layer in the bottom region.

| The case of opposing buoyancies N < 0
For this case, where the thermal and mass gradients are opposed, it corresponds to a negative buoyancy ratio, N = −3.5 with Ra m = 50, which is chosen for the present calculation. Particular values of Lewis number were selected within a range of [1][2][3] to illustrate the flow structure's singularities which are completely different from the case of aiding buoyancies when N > 0. Figure 6 represents the isolines of stream function, isotherms, and concentrations. Compared to the previous case of aiding buoyancies, these isolines are not all characterized by symmetry with respect to the ellipse's minor axis. The streamlines in Figure 6 illustrate a fluid flow divided into two vortices spinning in opposite directions, and this flow structure mainly results from the competition between buoyancies. Isotherms, as well as the iso-concentrations, develop downwards in the antinatural direction of the convection. This is due to the effect of the negative buoyancy ratio N < 0, which drives the particles of the fluid in the direction of the gravitational force. For the case when the Lewis number is equal to unity Le = 1, the thermal and mass diffusivity are identical, which is interpreted by similar thermal and mass plumes developing with the same intensity. With an increase in Lewis number value to Le = 2.5, the streamlines in Figure 6 show that the two counter-rotating cells are getting closer to each other, and the gap in-between is decreasing.
Also, the value of the stream function decreased from ψ + = 13.15 to ψ + = 8.68. This indicates that changes in the buoyancy ratio and Lewis number values significantly impact the fluid flow and heat transfer in the annular space. Isotherms and concentration contours deform in the lower space instead of the upper space due to the negative value of the buoyancy ratio. Isotherms are not showing a significant change with the increase of Le compared with the concentration contours stratified near the inner ellipse in the upper space forming a convective boundary layer. By slightly increasing the value of the Lewis number, Figure 6C,D show that the values of the stream function decrease due to the mass plume's intensification, which slows the fluid flow. These two particular values Le = 2.735 and Le = 2.75 were chosen to capture the flow's transition toward an asymmetric structure and the appearance of other cells in the annular space under a bifurcation process. The streamlines show that the flow structure has undergone a significant change, where it has transited from a bicellular regime to a multicellular regime with the appearance of two additional cells in the upper part of the annular space. Figure 6 shows that with a further increase in Le, isolines regain their quasisymmetrical character with respect to the ellipses minor axis. The streamlines demonstrate that the convective flow is now organized in quadricellular that spins each in the opposite direction to the adjacent cells. The stratification of the mass plume is more significant compared with the thermal plume, resulting in a slight decrease in the value of the stream function. Overall, this behavior involves the interaction between buoyancy, heat transfer, and fluid flow, which results in complex flow structures in the annular space. Figure 7 illustrates Lewis number influence on heat and mass transfer interpreted respectively by Nusselt and Sherwood numbers on the inner ellipse wall for Ra m = 100 and buoyancy ratio of N = 1. Results related to the local Nusselt number show that when increasing the Lewis number value, the local Nusselt number decreases slightly in the upper space where the thermal stratification is developing. The local Nusselt variation shows for Le = 10 the existence of three maxima, two of which correspond to θ = 0°and θ = 180°represent the counter-rotating vortices that carry the fluid from the active exchange region to the region where heat and mass transfer are less active. For θ = 90°, the third maximum local Nusselt rate is interpreted as an additional heat exchange due to the conduction mode resulting from the thermal stratification in the middle-upper space. Results related to the local Sherwood number, which represents the mass transfer, allow us to note that the Sherwood number rises when the Lewis number is increased, and this is due to the attenuation of the solutal stratification while the mass transfer is enhanced due to the convection intensification, this behavior is representing the case when the solutal diffusivity is very low compared with the thermal diffusivity. Figure 8 illustrates Lewis's number influence on both heat and mass transfer interpreted respectively by the mean Nusselt and Sherwood numbers on the inner ellipse wall for Ra m = 100 and buoyancy ratio of N = 1. Lewis number effect on heat transfer is clearly opposite to its effect on mass transfer in this case. When the Lewis number increases, the mean Nusselt number slightly decreases while the mean Sherwood number considerably increases. For Le = 1, which corresponds to equal diffusivities, values of Sherwood and Nusselt numbers are identical due to the qualitative and quantitative similarity of thermal and solutal stratifications. The mean Nusselt number tends to have an almost asymptotic value for Le > 5, whereas the mean Sherwood number is almost linearly proportional. Local Sherwood and Nusselt numbers increase due to thermal and concentration stratifications in the lower region of the annular space where the convection is very active, contrary to the region corresponding to 45°<θ<145°w here both numbers are at their minimums.

|
The case of opposing buoyancies N < 0 Figure 9 demonstrates the Lewis number effect on heat and mass transfer represented respectively by Nusselt and Sherwood numbers on the inner ellipse wall for Ra m = 50 and buoyancy ratio of N = −3.5. As noticed in the case of aiding buoyancies, the local Nusselt number decreases slightly in the upper space where the thermal stratification is developing, contrary to the local Sherwood number increasing significantly. This behavior can be attributed to the attenuation of the solutal stratification while the mass transfer is enhanced due to the convection intensification. Both curves of local Nusselt and Sherwood numbers show for Le ≤ 2.5 the existence of one maximum that corresponds to θ = 90°and one minimum that corresponds to θ = 270°, which corresponds to a bicellular flow structure. For Le > 2.5, local Nusselt and Sherwood numbers increase with a slight increase in Le number; this unexpected behavior is due to the bifurcation phenomenon that improves heat and mass transfer locally near the active wall. Overall, the results suggest that the Lewis number plays a critical role in determining the flow structure and heat and mass transfer in the annular space. The findings also highlight the importance of selecting the proper flow and thermal characteristics for better designing efficient heat and mass transfer systems. Figure 10 illustrates the variation of the average Nusselt and Sherwood numbers on the inner ellipse wall in terms of Lewis number for Ra m = 50 and buoyancy ratio of N = −3.5. The heat transfer response is obviously the opposite of the mass transfer response in this case of opposing buoyancies, similar to the previous case of aiding buoyancies. When the Lewis number increases, the average Nu number slightly decreases while the average Sherwood number considerably increases except for the Lewis number Le « 1; its increase generates a slight increase in the average Nu number. For Le = 1, which corresponds to F I G U R E 9 Local internal Nusselt and Sherwood numbers at Ra m = 50, and N = −3.5 (N < 0). equal diffusivities, values of Sherwood and Nusselt numbers are identical due to the quantitative similarity of thermal and solutal stratifications. The average Nusselt number continues its slight decrease for Le > 1. In contrast, the average Sherwood number increases considerably with the increase of Le and then gradually slows under the effect of the mass plume, which slows down the convection. As the Lewis number increases, heat transfer decreases while mass transfer increases. As a result, the average Sherwood number also increases significantly as mass stratification decreases, allowing convection to develop further. However, for Lewis numbers less than 0.5, an increase in the Lewis number only leads to a slight increase in the mean Nusselt value on the inner wall, which decreases because of thermal stratification.

| Prediction of heat and mass transfer rates
In the present section, a comparison study is carried out between three models to predict the average Nusselt number Nu and the average Sherwood number Sh. In fact, three models are used to fit the data, namely, the ANN, 36 the MVPR, 37 To train the proposed neural network, a numerical data set composed of a total of 780 (input, target) pairs is considered in which 702 data were used for training (a ratio of 90%), and F I G U R E 11 The proposed architecture of the artificial neural network.  the remaining 78 data were used for testing (a ratio of 10%). Figure 11 shows more details about the architecture that has been used to simulate the proposed neural network estimator. The implementation of the three compared approaches is carried out based on the Matlab environment. To compare the obtained results, the following statistical evaluation metrics have been used: where y i , ŷ i , and N are the numerical data, predicted data, and the number of data points, respectively. Table 3 (Table 4, respectively) shows the parameters used to implement the three methods and the obtained evaluation metrics by applying the mentioned methods to predict the Nusselt number (the Sherwood number, respectively). The obtained results from Tables 3 and 4 show the efficiency of the proposed neural network in terms of all the evaluation metrics for the prediction of both the Nusselt number and the Sherwood number. Figures 12 and 13 present respectively the average Nusselt and Sherwood predictions. For the case where Le = 2, the predictions of the heat and mass transfer rates in terms of Rayleigh-Darcy number Ra m and the buoyancy ratio N give almost similar results between MVPR, GMDH, and ANN methods, except that the latter shows more details in the variation of Nu and Sh, especially near the critical value of the buoyancy ratio N = −1. In the case of N = −1.5, which represents a buoyancy ratio slightly close to the critical value, Figures 12 and 13 show different predictions between the three methods. The heat transfer rate Nu increases with increasing Ra m and shows a slight decrease for high values of the Lewis number Le. On the other hand, the mass transfer rate Sh increases drastically with the increase of Le for high values of Rayleigh-Darcy Ra m but remains insignificant for the low value of Le. It is also observed in these figures that the ANN method captures more details in the variation of transfer rates. Figure 12 for Ra m = 80 shows a particular behavior in the variation of Nu where the heat transfer rate reaches its maximum only for the limit values of the buoyancy ratio N = −5 and N = + 5 for low values of Le. On the other hand, Sh in this case of the low value of Ra m increases significantly with the increase of Le but far from the critical value of N = −1 where it decreases drastically according to the predictions MVPR and ANN. Compared to MVPR, GMDH, the ANN method is in very good agreement with the results of the CFD simulation and showed robust predictions of heat and mass transfer rates. The ANN captures more details in the variations of Nu and Sh especially near the critical value of buoyancy ratio N = −1. Figure 14 shows the regression plots obtained by applying the three considered methods, MVPR, GMDH, and ANN, to the training and test data sets. It can be observed from Figure 14C that the fit data line (blue line for training data and green line for test data) is superimposed on the ground truth training or test data lines (dotted line) for both average Nusselt and Sherwood numbers, with the R 2 metric exceeding the value of 0.99 in all the cases. However, a significant gap between the fit line and the ground truth data line for the prediction of both average Nusselt and Sherwood numbers was found in the plots in Figure 14A,B were obtained by applying the MVPR and GMDH methods (respectively), which means that the prediction errors resulting from the application of MVPR and GMDH methods surpass the prediction errors given by the ANN method. Finally, the regression plots in Figure 14 prove our hypothesis about the efficiency of the proposed ANN model in predicting both average Nusselt and Sherwood numbers. This conclusion can be affirmed by examining the values of the R 2 evaluation metric in Figure 14. Three methods (MVPR, GMDH, and ANN) were applied to training and test data sets to predict average Nusselt and Sherwood numbers. All three methods can make highly accurate predictions; however, the ANN outperformed MVPR and GMDH methods in predicting both numbers, with R 2 exceeding 0.99. In addition, the MVPR showed a better performance when compared with the GMDH method.