Pre-ph.d. presentation on numeric study of some double diffusive convection reaction

PhD student presentation

1. Pre − Ph.D. Presentation Numerical Study of Some Double-Diffusive Convection-Reaction Flows in Porous Medium by Sunil Kumar Singh Roll no. 209211002 Su p ervis o r Co-Supervisor ( April 2025) Dr. Archana Dixit Dr. Abhishek Kumar Sharma Sunil Kumar Singh GLA University, Mathura 14 /11/202 5 1 / 1

2. T able of Content Objectives Literature Survey Outline of the Thesis Introduction Result & Discussion Conclusion References Sunil Kumar Singh GLA University, Mathura 14 /11/202 5 2 / 1

3. Objectives Sunil Kumar Singh GLA University, Mathura 14 /11/202 5 3 / 1 This PhD thesis explores porous media research, essential in geophysics, petroleum engineering, and environmental science. It examines mass, momentum, and energy transport through governing equations, hydrodynamic stability, and numerical modeling.

4. Sunil Kumar Singh GLA University, Mathura 14/11/2025 4 / 1 Outline of the Thesis Chapter 1 introduces fundamental concepts, including porous media definitions, conservation equations, and extended Darcy studies. Advanced topics like variable porosity, conjugate heat transfer, and double-diffusive convection are analysed. Special focus is given to inclined porous media, Soret and Dufour effects, and computational techniques like staggered grids and pressure correction. This work bridges theory and application, enhancing computational accuracy and serving as a resource for researchers and engineers

5. Contd ……… Sunil Kumar Singh GLA University, Mathura 14/11/2025 5 / 1 Literature Review Chapter 2 presents a comprehensive analysis of double-diffusive natural convection in two-dimensional porous cavities with constant or variable porosity in vertical and inclined configurations. Special attention is given to the influence of Soret and Dufour effects and the role of wall thickness in conjugate conduction-convection interactions. The review highlights the applicability of the extended Darcy model, incorporating Forchheimer and Brinkman corrections, under the Boussinesq approximation. Additionally, correlations between key dimensionless numbers, including Nusselt, Sherwood, Rayleigh, and Darcy numbers, provide insight into the thermal and mass transfer characteristics, offering a strong foundation for further research.

6. Chapter 3 focuses on the numerical study of double-diffusive reaction-convective flow in a square porous enclosure with fixed temperature and concentration at vertical walls, incorporating chemical reaction effects. Using the SIMPLER algorithm, results for streamlines, temperature, and concentration distributions are analyzed for varying parameters. The study highlights the influence of the Damköhler number on buoyancy, heat and mass transfer, and flow characteristics, demonstrating its role in reducing stream function intensity while enhancing mass transfer. Sunil Kumar Singh GLA University, Mathura 14/11/2025 6 / 1 Contd ……… Contd………

7. Chapter 4 presents a numerical investigation of double-diffusive reaction-convective flow in a square enclosure filled with anisotropic porous media. The study incorporates chemical reaction effects by maintaining fixed temperatures and concentrations at the vertical walls. Using the SIMPLE-revised method, the research examines streamline behaviour, temperature, and concentration distributions for various parameter sets. Results indicate that flow structure and transport characteristics are significantly influenced by permeability ratio and K* and Rayleigh-Darcy Ra′. Sunil Kumar Singh GLA University, Mathura 14/11/2025 7 / 1 Contd ……… Contd………

8. Chapter 5 examines double-diffusive convection-reaction flow in a square cavity filled with porous media under LTNE state. When the temperature differential between the two phases of the porous medium is substantial, the medium is said to be in a local thermal non-equilibrium (LTNE) state.

9. In the end, Chapter 6 presents the summary and concluding remarks of the thesis and some possible directions for future work.

10. Chapter 1 - Introduction A material with a solid structure and interconnected voids is called a porous medium. One or more fluids can pass through the material due to these interconnected vacuum spaces, also known as pores. In this study, solid matrix is considered rigid and incompressible By applying the continuum model assumption to a porous medium, the equation of continuity is derived by balancing the net mass. flow into a representative elementary volume (REV) with the rate of mass accumulation of the fluid within it. The SIMPLE algorithm has shown to be effective and widely utilised. For instance, this approach was used to complete all of the fluid-flow computations. Sunil Kumar Singh GLA University, Mathura 14/11/2025 10 / 1

11. Shashikant Upadhyay GLA University, Mathura 14/11/2025 11 / 1

12. Chapter 2 – Literature Review Sunil Kumar Singh GLA University, Mathura 14/11/2025 12 / 1 Over the past few years, there have seen a rise in interest in the effects of fluid and solid porous matrix characteristics, such as permeability, porosity and thermal conductivity, mass movement, and convectives heat transportation via porous materials . In addition to these characteristics, important research areas include the behaviour of Darcy and non-Darcy flows, as well as the consequences of Soret , Dufour, and conjugate problems. The research is divided into six categories: conjugate problems, variable porosity, inclined porous media, double-diffusive effects, Soret and Dufour effects, Darcy and extended Darcy flow in porous cavities, and variable porosity. Many studies has been conducting on convective heat transmission in porous media for more than a century using traditional Darcy formulas .

13. Conclusions Sunil Kumar Singh GLA University, Mathura 14/11/2025 13 / 1 In porous cavities that are two-dimensional and have either constant or variable porosity , vertical or inclined configurations, and the incorporation of Soret and Dufour effects, this thorough review concludes that investigating steady and unsteady natural DDC with heat and mass flow is a promising research direction. The Boussinesq approximation is used to model density variations. The proposed relationships between average Sherwood, Nusselt, Rayleigh, Darcy, and Lewis numbers, buoyancy ratio, Soret and Dufour coefficients, and thermal cony ratios encompass a broad variety of physical, geomductivitetrical , and material properties.

14. Chapter 3 Double-Diffusive Convection and Reactive Flow in a Porous Square Enclosure Sunil Kumar Singh GLA University, Mathura 14/11/2025 14 / 1

15. 66 th Congress of ISTAM Convection-reaction of binary mixture in a square enclosure filled with porous medium Presented by Sunil Kumar Singh*, Abhishek Kr. Sharma, Archana Dixit Department of Mathematics GLA University, Mathura

16. Outline: Double-diffusive convection-reaction of binary mixture in a square enclosure filled with a porous medium SIMPLE-R algorithm based on finite volume method References

17. Problem Definition: Figure 1: Schematic of the physical problem

18. Where do we encounter such type of problems? 1. 2. Figure 2: Fluid flow in porous medium

19. Mathematical Formulation of the Problem The two dimensional fluid flow, heat and mass transfer equations in Cartesian coordinate system ( x,y ) are given by:

20. In the above equations, are the velocity components along x axis y , respectively. Apart from these, p, T, C, ρ , β T and β C denote pressure, volume-averaged temperature, concentration, density of fluid, the volumetric thermal expansion coefficient, and volumetric solutal expansion coefficient, respectively. Also Ra, Pr, Le, Da , χ and ε are Rayleigh number, Prandtl number, Lewis number, Darcy number, Damkohler number, buoyancy ratio, and porosity, respectively.

21. The boundary conditions are given as: The overall heat transfer rate in terms of Nusselt number and mass transfer rate in terms of Sherwood number are given as:

22. Result and Conclusion The effects of reaction rate in terms of Damkohler number on the overall heat and mass transfer rates on the left wall in terms of Nusselt number and Sherwood number are plotted in figures 3 (for N= 1) and figure 4 (for = -1), Where is the buoyancy ratio. We have fixed the following values for other parameters: Ra = 100, Le = 10, Pt = 0.7.

23. Figure 3: The effect of reaction rate of Nu and Sh when N = 1

24. Figure 4: The effect of reaction rate of Nu and Sh when N = -1

25. It can be seen that Nu remains almost constant for the considered values of χ which is expected from the governing equations but shows a decreasing trend as the value of increases from 1 to 4. This is true for both values of considered permeability of porous medium in terms of Darcy number ( Da = 10 -2 to 10 -3 ) with porosities 0.9 and 0.4, respectively. This shows that as the reaction rate increases in the system the salt precipitation also increases causing a decrease in overall mass transfer rate.

26. On the other hand, the values of are large for Da = 10 -2 as compared to Da = 10 -3 due to higher permeability of the porous medium in the case of Da = 10 -2 . One interesting observation is that the graph of Sh is below the graph of Nu for Da = 10 -2 unlike the other cases where it is opposite. This phenomenon can be explained as for the considered values of χ , the combined effect of mass/salt diffusion and precipitation due to chemical reaction is less than the thermal diffusion in the case of opposing buoyancy forces when Da = 10 -2 .

27. Fig: Streamlines, contours of concentration and temperature. In order to study the dependency of flow dynamics as well as solute and thermal distributions on the reaction term, the above figure is plotted for The figure shows the flow pattern, concentration and temperature contours as a consequence of relative change in the strength of convective and diffusive flow in the system.

28. As can be observed only the clockwise rotating convective cells appear in the streamlines plot, which may be the consequence of relatively high thermal diffusivity of the fluid and left side heating of the cavity. The maximum value of convective cell in streamlines plot is almost zero. This indicates that the heat and solute distribution in the domain is primarily due to diffusion. As a consequence, the isotherms are straight lines only whereas the concentration contours show some advection. That is why variation in Sherwood number is expected, however, the variation of corresponding heat transfer rates (fluid as well as solid) depends on the dominance and nature (i.e., buoyancy forces are in favor of each other or against each other) of the buoyancy forces.

29. 29 / 31 Outline: Mathematical Formulation of the physical problem Numerical Solution using SIMPLE-R algorithm Results and Discussion References

30. 30 / 31 Introduction The effect of anisotropy of porous medium on double-diffusive convection-reaction flow in a square cavity filled with fluid-saturated porous medium has been addressed numerically. The two dimensional steady state flow is induced due to maintenance of constant temperature and concentration on the vertical walls and insulation of both horizontal walls of the cavity. Non-Darcy (Darcy–Lapwood-Brinkman) model has been taken and the complete governing equations are solved by standard finite volume based SIMPLE-R algorithm.

31. Physical problem We consider a two dimensional anisotropic fluid saturated porous medium enclosed in a square cavity as shown in Fig. 1. The sec on d order permeability tensor, K , is diagonal in the ( ox ∗ , oy ∗ ) coordinate system, with the diagonal components, K ∗ and K ∗ , respectively. x y It is assumed that the left and right walls of the cavity are maintained at temperatures T 1 and T 2 ( T 1 > T 2 ) while top and bottom walls are insulated. Figure 1: Schematic of the physical problem 31 / 31

32. 32 / 31 The following assumptions are made in the formulation of governing equations: The existence of local thermal equilibrium state. Anisotropic porous medium whose thermo- physical properties are constant except for the density. The thermal contribution of the reaction is ignored and the model is not restricted to a particular reaction rate. Chemical equilibrium at the boundaries is assumed [28]. The fluid is modelled as the Boussinesq fluid which saturates the porous matrix, whose density varies linearly with temperature and concentration as, ρ = ρ f [1 − β T ( T f − T )]. Here, T is mean temperature (( T 1 + T 2 ) / 2). The gravitational force is aligned in the negative y - direction.

33. Mathematical Formulation We have considered Darcy-Lapood-Brinkman model [4]. The governing equations are given as, Continuity equation: ∂x ∂y 33 / 31 ∂u + ∂v = (1) Momentum equations: ρ f ε ∂t ε 2 5 3 ∂x + u + v 1 ∂u 1 ∂u ∂u ∂y 46 ∂p ∂x = − + µ 2 µ f K ˜( ∇ u ) − u (2) ρ f 5 ε 2 3 1 ∂v 1 ∂v ε ∂t ∂x + u + v ∂v ∂y 46 ∂p ∂y = − + µ 2 µ f K ˜( ∇ v ) − v + ρ ⃗ g (3)

34. Heat Transport equations: ∂T p f ∂t p f 3 ε ( ρc ) + ( ρc ) u + v ∂T ∂T ∂x ∂y 4 f 2 = εk ( ∇ T ) , (4) Mass Transport equation: ∂t ∂x ∂y eq ∂c ∂c ∂c ε + ( u + v ) = D ( ∇ 2 c ) + k ˆ ( C − C ) (5) Here C eq ( T ) = C + γ ( T − T ), where γ = ( C 1 − C ) / ( T 1 − T ). The permeability tensor K is defined as, K = K 34 / 31 ∗ x a a 1 2 a 2 a 3 C D with a 1 = K ∗ sin 2 ϕ + cos 2 ϕ, a 2 = (1 − K ∗ ) sinϕ + cosϕ, a 3 = K ∗ cos 2 ϕ + sin 2 ϕ where K ∗ = K ∗ /K ∗ y x

35. Boundary conditions: v = , T f = T s = T 1 , C = C 1 at x = (6) v = , T f = T s = T 2 , C = C 2 at x = 1 (7) u = , ∂T f = ∂T s = ∂C = at y = , 1 (8) 35 / 31 ∂y ∂y ∂y The following non-dimensional quantities are used to obtain the non-dimensional governing equations: ( x, y ) ∗ = ( x, y ) /L , ∗ ∗ T T f 2 ∗ 2 2 T ( u, v ) = ( u, v ) L/εk , t = tk /L , p = pL /ρ k , f T = ( f 1 2 ∗ ∗ s T − T ) / ( T − T ), T = ( T − T ) / s 1 2 ( T − T ) and C ∗ = ( c − C ) / ( C 1 − C 2 ).

36. After dropping asterisk, the steady state non- dimensional governing equations can be written as ∂x ∂y ∂u + ∂v = (9) ∂u ∂u ∂p Pr u + v = − + εP r ( ∇ 2 u ) − ε ( a 3 u − a 2 v ) ∂x ∂y ∂x Da (10) ∂x ∂y ∂y ∂v ∂v ∂p Pr u + v = − + εP r ( ∇ 2 v ) − ε ( − a 2 u + a 1 v ) Da + εRaP r ( θ f + Nc ) (11) ∂x ∂y ∂θ ∂θ u + v = ( ∇ 2 θ ) (12) ∂x ∂y Le 36 / 31 ∂c ∂c 1 u + v = ( ∇ 2 c ) + k ( T − C ) (13)

37. 37 / 31 In the above equations, Ra = ρ f gβ T ( T 1 − T 2 ) L 3 /µ f α f is Rayleigh number, Le = εα f /D is Lewis number, N = β C ( C 1 − C 2 ) /β T ( T 1 − T 2 ) is buoyancy ratio, Pr = ν/α f is Prandtl number, Da = K/L 2 is Darcy number, γ = εk f / (1 − ε ) k s is thermal conductivity ratio. k = k ˆ L 2 /ε k f is Damkohler number Here, ν = µ f /ρ f and α f = k f / ( ρc p ) f .

38. The non-dimensional boundary conditions over the walls of the enclosure are θ f = θ s = c = . 5 , ψ = , at x = . (14) θ f = θ s = c = − . 5 , ψ = , at x = 1 . (15) ∂y ∂y ∂y 38 / 31 ∂θ f = ∂θ s = ∂c = ψ = , at y = , 1 . (16) The corresponding overall heat transfer rates in terms of Nusselt number for fluid ( Nu f ) as well as solid ( Nu s ) and mass transfer rate in terms of Sherwood number ( Sh ) are given by 1 Nu f = − ∂θ f ∂x ∫ 5 6 x =0 dy, (17) Nu s = 1 ∫ 5 − ∂θ s ∂x 6 x =0 dy, (18) Sh = 1 − ∂c ∫ 5 6 ∂x x =0 dy. (19)

39. Control Volume Formulation Figure 2: Control Volume for two-dimensional situation 39 / 31 Control volume formulation can be regarded as a special version of the method of weighted residuals, so that the integral of the residual over each control volume must become zero. The calculation domain is divided into a number of nonoverlapping control volumes such that there is one control volume surrounding each grid point.

40. 40 / 31 The differential equation is integrated over each control volume. Piecewise profiles expressing the variation of field variable lke velocity, temperature, etc (represented by ϕ ) between the grid points are used to evaluate the required integrals. The result is the discretization equation containing the values of ϕ for a group of grid points. The discretization equation obtained in this manner expresses the conservation principle for ϕ for the finite control volume, just as the differential equation expresses it for an infinitesimal control volume. Resulting solution would imply that the integral conservation of quantities such as mass, momentum, and energy is exactly satisfied over any group of CV.

41. Solution procedure The Revised Semi-Implicit Method for Pressure Linked Equations (SIMPLER) is used to solve the discretized equations. This Algorithm was given by S.V. Patankar in 1979. For 2- D case the discretization equations can be written as a i,j T i,j = a i − 1 ,j T i − 1 ,j + a i +1 ,j T i +1 ,j + a i,j − 1 T i,j − 1 + a i,j +1 T i,j +1 + d i,j (20) Figure 3: Staggered Grid used in SIMPLER 41 / 31

42. The steps involved in the numerical scheme that we have adopted are given below: Initially, a velocity field is guessed. Using the above velocity field the discretized pressure equation is solved to obtain the pressure field which is then used to solve the discretized momentum equations to obtain the velocity field. The discretized pressure correction equation, which is obtained from the continuity equation, is solved and the velocity field is corrected using this pressure correction. Using above corrected velocity field the discretized equation for other field variables i.e., concentration and temperature for fluid and solid are solved subsequently. The convergence criterion is based on the residue of the continuity equation on the whole domain called the mass source term ( b ) and the convergence is reached when b ≤ 10 − 10 , which indicates that we have acquired the correct pressure field and hence the correct velocity field. The steps 2- 4 are repeated until the convergence criterion is met. 15 / 31

43. 43 / 31 Code Validation Table 1: Variation in the Nusselt number and stream function (at the centre of the enclosure) with respect to the number of control volumes, when Le = 10, N = 1, k = 1, Pr = . 7, Ra = 10 5 and Da = 10 − 2 . Control Volumes Nu ψ c (0 . 5 , . 5) 20 × 20 5.1602 - 8.2214 30 × 30 4.6013 - 7.8924 40 × 40 4.5987 - 7.7689 50 × 50 4.5774 - 7.7348 60 × 60 4.5772 - 7.7345

44. 44 / 31 Results The effects media permeability, Lewis number Le , buoyancy ratio N and chemical reaction have been studied on the flow pattern and average heat as well as mass transfer rates Finally, the variation of Nu (for fluid as well as solid) and Sh has been investigated.

45. Figure 4: Effect of buoyancy ratio on streamlines when K ∗ = . 1, Ra ′ = 10 3 , k = 1, Le = 10, and Pr = . 7. 45 / 31

46. 46 / 31 The effect of buoyancy ratio ( N ) on the streamfunction, is displayed in figure 4 for three different values N = 1 , 2 and 3 of the buoyancy ratio when Ra ′ = 10 3 , k = 1, Le = 10, and Pr = . 7. As can be seen from the above figure that the pattern as well as the magnitude of ϕ undergo drastic changes. For K ∗ = . 1 the cells have tendency to tilt towards right side of the enclosure. This can be explained from the definition of K ∗ and fixing orientation angle of media permeability at 45 degree. Fixing ϕ at 45 implies that for K ∗ < 1, at any point in the enclosure, permeability along x ∗ is larger than the same in y ∗ direction.

47. Figure 5: Effect of buoyancy ratio on streamlines when K ∗ = 10, Ra ′ = 10 3 , k = 1, Le = 10, and Pr = . 7. 47 / 31

48. 48 / 31 for K ∗ = 10, cells have tendency to tilt towards left side of the enclosure. Fixing ϕ at 45 implies that for K ∗ < 1, at any point in the enclosure, permeability along x ∗ is larger than the same in y ∗ direction, whereas for K ∗ > 1, it is reverse. Also the flow is due to heating of the side wall which compels to tilt cells towards right for K ∗ < 1, and towards left for K ∗ > 1.

49. Figure 6: Variation of Nusselt number: (a) K ∗ = . 1; (b) K ∗ = 10; when Ra = 10 4 , Da = 10 − 2 49 / 31

50. 50 / 31 From the above Figure it can also be seen that similar to isotropic case, profile of Nu x is symmetric about x = 0.5 for odd values of N, whereas, for relatively low permeable case ( K ∗ = 10), the same is close to anti symmetric about x = 0.5 for even values of N. This indicates that as the flow strength is reduced by decreasing the media permeability, the anticipated result for even values of N dies out. It is important to mention here that Darcy model in general is not valid to capture the real physics of the problem of high convective flow, in which viscous shear and inertia are not negligible.

51. 51 / 31 Conclusion Two types of convective cells (rotating clockwise and counter clockwise) govern the flow for all values of N. Increasing of N increases the multiple cellular structures in the form of N + 1, which in turn, enhances the convection in the enclosure. For even values of N, cells rotating anticlockwise are dominated and covered the entire domain. Apart from these, for K ∗ equal to 0.1 and 10, cells are tilted respectively to right and left side of the enclosure for both even as well as odd values of N. In contrast to isotropic porous media, the effect of the permeability and permeability orientation angle on the flow is complex and often non intuitive.

52. 52 / 36 Outline: Present status of the first article Participation in Conference/Workshop A Discussion on the second problem: Mathematical Formulation of a 2- D double diffusive convection-reaction flow under local thermal non-equilibrium state Numerical Solution using SIMPLE-R algorithm Results and Discussion References

53. The First Article is Communicated to the International Journal "Special Topics and Reviews in Porous Media, Begell House" Present status: The article has been accepted with minor suggestions from the reviewers and soon it will get published. 53 / 36

54. Participation in Conference/Workshop 54 / 36

55. 55 / 36

56. 56 / 36 The second problem: A 2- D double diffusive convection- reaction flow under local thermal non- equilibrium state Abstract The influence of local thermal non-equilibrium state on double-diffusive convection-reaction flow in a square cavity filled with fluid- saturated porous medium has been addressed numerically. The two dimensional steady state flow is induced due to maintenance of constant temperature and concentration on the vertical walls and insulation of both horizontal walls of the cavity. Non-Darcy (Darcy–Lapwood-Brinkman) model has been taken and the complete governing equations are solved by standard finite volume based SIMPLE-R algorithm.

57. Physical problem We consider a two dimensional fluid saturated porous medium enclosed in a square cavity as shown in Fig. 1. It is assumed that the left and right walls of the cavity are maintained at temperatures T 1 and T 2 ( T 1 > T 2 ) and concentrations C 1 and C 2 , respectively, while top and bottom walls are insulated. L 57 / 36 L T 1 C 1 T 2 C 2 u x y v  g Figure 1: Schematic of the physical problem

58. 58 / 36 The following assumptions are made in the formulation of governing equations: The existence of local thermal non-equilibrium state. Isotropic porous medium whose thermo-physical properties are constant except for the density. The thermal contribution of the reaction is ignored and the model is not restricted to a particular reaction rate. Chemical equilibrium at the boundaries is assumed [28]. The fluid is modelled as the Boussinesq fluid which saturates the porous matrix, whose density varies linearly with temperature and concentration as, ρ = ρ f [1 − β T ( T f − T ) − β C ( C − C )]. Here, T and C are mean temperature (( T 1 + T 2 ) / 2) and mean concentration (( C 1 + C 2 ) / 2), respectively. The gravitational force is aligned in the negative y - direction.

59. Mathematical Formulation We have considered Darcy-Lapood-Brinkman model [4]. The governing equations are given as, Continuity equation: ∂x ∂y 59 / 36 ∂u + ∂v = (1) Momentum equations: f 5 ε 2 ρ ( 1 ∂u ∂u ∂x ∂y 6 ∂p ∂x u + v ) = − + µ 2 µ f K ˜( ∇ u ) − u (2) f 5 ε 2 ρ ( 1 ∂v ∂v ∂x ∂y 6 ∂p ∂y u + v ) = − + µ 2 µ f K ˜( ∇ v ) − v + ρ⃗g (3)

60. Heat Transport equations: ε ( ρC ) P f ∂t ∂T ∂T f f ∂x ∂T f ∂y f + ( u + v ) = εk ( 2 f ∇ T ) + h ( s f T − T ) (4) (1 − ε )( ρC ) ∂T s P s ∂t = (1 − ε ) 2 s s k ( ∇ T ) + h ( f s T − T ) (5) Mass Transport equation: ∂t ∂x ∂y 60 / 36 eq ∂c ∂c ∂c ε + ( u + v ) = D ( ∇ 2 c ) + k ˆ ( C − C ) (6) Here C eq ( T ) = C + γ ( T − T ), where γ = ( C 1 − C ) / ( T 1 − T ).

61. Boundary conditions: v = , T f = T s = T 1 , C = C 1 at x = (7) v = , T f = T s = T 2 , C = C 2 at x = 1 (8) u = , ∂T f = ∂T s = ∂C = at y = , 1 (9) 61 / 36 ∂y ∂y ∂y The following non-dimensional quantities are used to obtain the non-dimensional governing equations: ( x, y ) ∗ = ( x, y ) /L , ∗ ∗ T T f 2 ∗ 2 2 T ( u, v ) = ( u, v ) L/εk , t = tk /L , p = pL /ρ k , f T = ( f 1 2 ∗ ∗ s T − T ) / ( T − T ), T = ( T − T ) / s 1 2 ( T − T ) and C ∗ = ( c − C ) / ( C 1 − C 2 ).

62. After dropping asterisk, the steady state non- dimensional governing equations can be written as ∂x ∂y ∂u + ∂v = (10) ∂u ∂u ∂p Pr u + v = − + εP r ( ∇ 2 u ) − ε u ∂x ∂y ∂x Da (11) ∂v ∂v ∂p Pr u + v = − + εP r ( ∇ 2 v ) − ε v + εRaP r ( θ ∂x ∂y ∂y Da f + Nc ) (12) ∂θ f ∂x ∂θ f ∂y u + v = ( 2 f ∇ θ ) + H ( s f θ − θ ) (13) (14) (15) = ( ∇ 2 θ s ) + γH ( θ f − θ s ) ∂x ∂y Le 62 / 36 ∂c ∂c 1 u + v = ( ∇ 2 c ) + k ( T − C )

63. 63 / 36 In the above equations, Ra = ρ f gβ T ( T 1 − T 2 ) L 3 /µ f α f is Rayleigh number, H = hL 2 /εk f is interface heat transfer coefficient, Le = εα f /D is Lewis number, N = β C ( C 1 − C 2 ) /β T ( T 1 − T 2 ) is buoyancy ratio, Pr = ν/α f is Prandtl number, Da = K/L 2 is Darcy number, γ = εk f / (1 − ε ) k s is thermal conductivity ratio. k = k ˆ L 2 /ε k f is Damkohler number Here, ν = µ f /ρ f and α f = k f / ( ρc p ) f .

64. The non-dimensional boundary conditions over the walls of the enclosure are θ f = θ s = c = . 5 , ψ = , at x = . (16) θ f = θ s = c = − . 5 , ψ = , at x = 1 . (17) ∂y ∂y ∂y 64 / 36 ∂θ f = ∂θ s = ∂c = ψ = , at y = , 1 . (18) The corresponding overall heat transfer rates in terms of Nusselt number for fluid ( Nu f ) as well as solid ( Nu s ) and mass transfer rate in terms of Sherwood number ( Sh ) are given by 1 Nu f = − ∂θ f ∂x ∫ 5 6 x =0 dy, (19) Nu s = 1 ∫ 5 − ∂θ s ∂x 6 x =0 dy, (20) Sh = ∫ 1 − ∂c 5 6 ∂x x =0 dy. (21)

65. Control Volume Formulation Figure 2: Control Volume for two-dimensional situation 65 / 36 Control volume formulation can be regarded as a special version of the method of weighted residuals, so that the integral of the residual over each control volume must become zero. The calculation domain is divided into a number of nonoverlapping control volumes such that there is one control volume surrounding each grid point.

66. 66 / 36 The differential equation is integrated over each control volume. Piecewise profiles expressing the variation of field variable lke velocity, temperature, etc (represented by ϕ ) between the grid points are used to evaluate the required integrals. The result is the discretization equation containing the values of ϕ for a group of grid points. The discretization equation obtained in this manner expresses the conservation principle for ϕ for the finite control volume, just as the differential equation expresses it for an infinitesimal control volume. Resulting solution would imply that the integral conservation of quantities such as mass, momentum, and energy is exactly satisfied over any group of CV.

67. Solution procedure The Revised Semi-Implicit Method for Pressure Linked Equations (SIMPLER) is used to solve the discretized equations. This Algorithm was given by S.V. Patankar in 1979. For 2- D case the discretization equations can be written as a i,j T i,j = a i − 1 ,j T i − 1 ,j + a i +1 ,j T i +1 ,j + a i,j − 1 T i,j − 1 + a i,j +1 T i,j +1 + d i,j (22) Figure 3: Staggered Grid used in SIMPLER 67 / 36

68. The steps involved in the numerical scheme that we have adopted are given below: Initially, a velocity field is guessed. Using the above velocity field the discretized pressure equation is solved to obtain the pressure field which is then used to solve the discretized momentum equations to obtain the velocity field. The discretized pressure correction equation, which is obtained from the continuity equation, is solved and the velocity field is corrected using this pressure correction. Using above corrected velocity field the discretized equation for other field variables i.e., concentration and temperature for fluid and solid are solved subsequently. The convergence criterion is based on the residue of the continuity equation on the whole domain called the mass source term ( b ) and the convergence is reached when b ≤ 10 − 10 , which indicates that we have acquired the correct pressure field and hence the correct velocity field. The steps 2- 4 are repeated until the convergence criterion is met. 18 / 36

69. 69 / 36 Code Validation Table 1: Variation in the Nusselt number and stream function (at the centre of the enclosure) with respect to the number of control volumes, when Le = 10, N = 1, k = 1, Pr = . 7, Ra = 10 5 and Da = 10 − 2 . Control Volumes Nu ψ c (0 . 5 , . 5) 20 × 20 5.1602 - 8.2214 30 × 30 4.6013 - 7.8924 40 × 40 4.5987 - 7.7689 50 × 50 4.5774 - 7.7348 60 × 60 4.5772 - 7.7345

70. 70 / 36 Results The effects media permeability, Lewis number Le , buoyancy ratio N and chemical reaction have been studied on the flow pattern and average heat as well as mass transfer rates for different values of interphase heat transfer coefficient H and γ . Finally, the variation of Nu (for fluid as well as solid) and Sh as a function of H has been investigated.

71. (a)      (b)      (c)      - 6.85 - 6.85 - 6.85 - 4.57 - 4.57 - 4.57 - 4.57 - 4.57 - 2.28 - 2.28 - 2.28 - 2.28 - 2.28 - 2.28 - 7.35 - 7.35 - 7.35 - 4.90 - 7.35 - 4.90 - 4.90 - 4.90 - 2.45 - 2.45 - 2.45 - 2.45 - 2.45 - 0.01 - 0.01 - 0.01 0.30 0.30 0.30 0.30 0.30 0.61 0.61 0.30 0.61 0.61 0.61 0.92 0.92 0.92 0.92 0.92 0.06 0.06 0.38 0.38 0.69 0.69  0.06 0.3 0.38 0.38 .38 0.69 0.69 0.69  0.06 06 0.38 0.38 0.38 0.69 0.69 0.69  0.06 0.06 0.38 0.38 0.69 0.69 0.38 0.69 C 06 0.38 0.38 0.38 0.38 0.69 0.69 0.69 C 0.06 71 / 36 0.38 0.38 0.38 0.38 0.69 0.69 0.69 0.69 C Figure 4: Effect of buoyancy ratio on streamlines, contours of temperature and concentration for (a) N = − 2, (b) N = 0, and (c) N = 2 when Ra = 10 3 , k = 1, Le = 10, H = 10, γ = 10, and Pr = . 7.

72. 72 / 36 The effect of buoyancy ratio ( N ) on the streamfunction, contours of temperature and concentration is displayed in figure 4 for three different values N = − 2 , and 2 of the buoyancy ratio when Ra = 10 3 , k = 1, Le = 10, H = 10, γ = 10, and Pr = . 7. It can be seen that the streamlines show a transitional behavior from anticlockwise rotation to clockwise rotation. When N = − 2, the streamlines rotate in anticlockwise direction in the whole enclosure except in the bottom right and top left corner where two clockwise rotating vortices can be seen, athough their strength is very small. The maximum intensity of the flow is predicted for N = and it decreases as the value of N is either increased or decreased. The streamlines, isotherms and isoconcentration lines show that significant changes occur in the boundary-layer regions close to the walls of the enclosure. When N = 2, the streamlines, isotherms and isoconcentration lines are almost parallel to the horizontal walls in the core region of the enclosure.

73. N - 5 - 3 - 1 1 3 5 Sh 60 40 20 80 100 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 5 0.8 Nu s 1.2 1 1.4 1.6 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 5 2 8 Nu f 6 4 12 10 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 5 2 8 6 4 12 10 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 5 1.2 0.8 2.4 2 1.6 3.2 2.8 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 N 5 20 40 60 80 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 5 1 2 3 4 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 5 0.8 1.2 1.6 2 2.4 2.8 Le = 1 Le = 10 Le = 100 - 5 - 3 - 1 1 3 N 5 4 8 12 16 20 Le = 1 Le = 10 Le = 100 (a) 73 / 36 (b) (c) Figure 5: Variation of Nusselt numbers (for fluid (Nu f ) and solid (Nu s )) and Sherwood number ( Sh ) as a function of N for different values of Le :(a) RaDa = 10, (b) RaDa=100, (c) RaDa=1000 when H = γ = 10.

74. 74 / 36 In each case, depending on the value of Le , there is a point (now onwards it is referred to as the point of minimum of heat or mass transfer rate) in the domain of buoyancy ratio at which heat or mass transfer rate is minimum. In general, on increasing Le mass transfer rate increases, however, the corresponding effect on heat transfer rate is not straightforward. For both Nu f as well as Nu s there exist a sub-domain of N , starting from the point of minimum to a certain positive value of it depending on the choice of parameters, in which heat transfer rate increases on increasing Le . In the rest of the domain of N , Nu decreases on increasing Le .

75.    f      f      s      s      s      f    0.5  c  0.5  0.5  c  0.5  0.5  c  0.5 (a) (b) (c)                - 0.44 - 0.19 0.06 0.31 - 0.44 - 0.19 06 0.31 - 0.19 0.06 0.31 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 - 1.77 - 1.12 - 0.47 - 0.47 - 0.47 - 0.44 - 0.19 0.06 0.06 0.31 - 0.44 - 0.19 0.06 0.31 - 0.44 - 0.1 0.06 0.31 - 0.44 - 0.19 .06 0.31 - 0.44 - 0.19 0.0 .06 0.31 - 2.75 75 / 36 - 2.01 - 1.26 - 1.26 -0.52 - 0.52 - 0.44 - 0.19 0.06 0.31 Figure 6: Streamlines, contours of concentration and temperature (for fluid and solid phases): (a) Le = 1; (b) Le = 10; (c) Le = 100, when N = − 1, γ = 10, Ra = 10 4 , Da = 10 − 2 and H = γ = 10.

76. 76 / 36 As can be observed from Fig 6, for Le > 1( i.e., 10 , 100) only the clockwise rotating convective cells appear in the streamlines plot, which may be the consequence of relatively high thermal diffusivity of the fluid and heating left side of the cavity. For Le = 1, patterns of solute and thermal distribution are similar and the maximum value of convective cell in streamlines plot is almost zero. This indicates that the heat and solute distribution in the domain is primarily due to diffusion. As a consequence, both isotherm and iso- concentration lines are straight lines only.

77. Nu s Sh Nu f 25 50 75 100 1 1.4 1.8 2.2    25 50 75 100 1 1.5 2 2.5    25 50 75 100 Le 5 10 15    25 50 75 100 Le 5 10 15    25 50 75 100 1 1.5 2 2.5    (a) 25 50 75 100 1 1.4 1.8 2.2    (b) 77 / 36 Figure 7: Variation of Nusselt numbers (for fluid (Nu f ) and solid (Nu s )) and Sherwood number ( Sh ) as a function of Le : (a) N = 1, (b) N = − . 5 at H = 10, RaDa = 100.

78. 78 / 36 To discuss further in this direction, the variation of Nu and Sh are shown in Fig. 7(a) and (b) for N = 1 and N = − . 5, respectively. As can be seen from Fig. 7(a) and (b) that increasing of Le smoothly increases the value of Sh for both values of N . For N = 1, heat transfer rates decrease up to a certain threshold value L of Le , which is a function of γ , whereas for N = − . 5, they increase up to L . Nu remains almost constant as Le is increased further. Here, γ acts as a catalyst for heat transfer but remains inert for solute transfer (see Fig. 7(c)). The present study is in progress and it will be communicated soon for publication.

79. References I S.V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, USA, 1980. R.G. Carbonell and S. Whitaker. Heat and mass transfer in porous media. Fundamentals of Transport Phenomenon in Porous Media, pages 121- 198, USA, 1984. Martinus Nijhoff Publishers. M. Kaviany. Principles of Heat Transfer in Porous Media. Springer-Verlag, NY,USA, 1991. D.A. Nield and A. Bejan. Convection in Porous Media. Springer, NY,USA, 2013. K. Vafai. Handbook of Porous Media. Marcel Dekker, NY,USA, 2000. 79 / 31

80. References II D.A.S. Rees, A. Pssom, and P.G. Siddheshwar. Local thermal non-equilibrium effects arising from the injection of a hot uid into a porous medium. J. Fluid Mech., 594:379- 398, 2008. P. Nithiarasu, K.N. Seetharamu, and T. Sundarajana. Effect of porosity on natural convection heat transfer in a uid saturated porous media. Int. J. of Heat Fluid Flow, 19:56- 58, 1998. M.S. Phanikumar and R.L. Mahajan. Non-darcy natural convection in high porosity metal foams. International Journal of Heat and Mass Transfer, 45- 3781- 3793, 2002. 80 / 31

81. References III A.C. Baytas and I. Pop. Free convection in a square porous cavity using a thermal non-equilibrium model. International Journal of Thermal Sciences, 41:861- 870, 2002. I.A. Badruddin, A. Zainal, P.A.A. Narayana, and N. Seetharamu. Numerical analysis of convection conduction and radiation using a non-equilibrium model in a square porous cavity. International Journal of Thermal Sciences, 46:20- 29, 2007. Sarita Pippal and P. Bera. A thermal non-equilibrium approach for 2d natural convection due to lateral heat flux: Square as well as slender enclosure. International Journal of Heat and Mass Transfer, 56:501- 515, 2013. 81 / 31

82. References IV A.C. Baytas. Thermal non-equilibrium free convection in a cavity filled with a non-darcy porous medium. Emerging Technologies and Techniques in Porous Media, NATO Science Series, 134:247- 258, 2004. R. Helmig. Multiphase Flow and Transport Processes in the Subsurfaces. Springer-Verlag, Berlin, 1997. P.R. King, S.V. Buldyrev, N.V. Dokholyan, S. Havlin, Y. Lee, G. Paul, and H.E. Stanley. Applications of statistical physics to the oil industry: predicting oil recovery using percolation theory. Physica A, 274:6066, 1999. 82 / 31

83. References V P.R. Dando, D. Stuben, and S.P. Varnavas. Hydrothermalism in the mediterranean sea. Prog.Oceanogr., 44:333- 367, 1999. O.V. Trevisan and A. Bejan. Natural convection with combined heat and mass transfer buoyancy effects in porous medium. Interntional Journal of Heat and Mass Transfer, 28:1597- 1611, 1985. P. Nithiarasu, K.N. Seetharamu, and T. Sundararajan. Double-diffusive natural convection in an enclosure filled with uid-saturated porous medium: a generalized non- darcy approach. Numerical Heat Transfer : Part A: Appl., 30:413- 426, 1996. 83 / 31

84. References VI D. Angirasa, G.P. Peterson, and I. Pop. Combined heat and mass transfer by natural convection with opposing buoyancy effects in a uid saturated porous medium. International Journal of Heat Mass Transfer, 40:2755- 2773, 1997. M. Sankar, B. Kim, and J.M. Lopez. Thermosolutal convection from a discrete heat and solute source in a vertical porous annulus. International Journal of Heat and Mass Transfer, 55:4116- 4128, 2012. R. Bennacer, A. Tobbal, H. Beji, and P. Vasseur. Double diffusive convection in a vertical enclosure filled with anisotropic porous media. International Journal of thermal science, 40:30- 41, 2001. 84 / 31

85. References VII B. Goyeau, J.P. Songbe, and D. Gobin. Numerical study double-diffusive natural convection in a porous cavity using the darcy brinkman formulation. Interntional Journal of Heat and Mass Transfer, 39:1363- 1378, 1996. D.D. Dincov, K.A. Parrott, and K.A. Pericleous. Heat and mass transfer in two-phase porous materials under intensive microwave heating. Journal of Food Engineering, 65:403- 412, 2004. A.J. Chamkha. Double diffusive convection in porous enclosure with co- operating temperature and concentration gradients and heat generation or absorption effects. Numerical Heat Transfer A, 41:65- 87, 2002. 85 / 31

86. References VIII G. Degan, P. Vasseur, and E. Bilgen. Convective heat transfer in a vertical anisotropic porous layer. Int. J. of Heat and Mass Transfer, 32:115- 125, 1996. Nield DA, Bejan A. 2013 Convection in porous media . Springer, New York. Vafai K. 2000 Handbook of Porous Media . Marcel Dekker, New York. Kaviany M. 1991 Principles of Heat Transfer in Porous Media . Springer-Verlag, New York. Pritchard, D. and Richardson, C. N., 2007, "The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer", J. Fluid Mech.", 571, 59–95. 86 / 31

87. References I S.V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, USA, 1980. R.G. Carbonell and S. Whitaker. Heat and mass transfer in porous media. Fundamentals of Transport Phenomenon in Porous Media, pages 121- 198, USA, 1984. Martinus Nijhoff Publishers. M. Kaviany. Principles of Heat Transfer in Porous Media. Springer-Verlag, NY,USA, 1991. D.A. Nield and A. Bejan. Convection in Porous Media. Springer, NY,USA, 2013. K. Vafai. Handbook of Porous Media. Marcel Dekker, NY,USA, 2000. 87 / 36

88. References II D.A.S. Rees, A. Pssom, and P.G. Siddheshwar. Local thermal non-equilibrium effects arising from the injection of a hot uid into a porous medium. J. Fluid Mech., 594:379- 398, 2008. P. Nithiarasu, K.N. Seetharamu, and T. Sundarajana. Effect of porosity on natural convection heat transfer in a uid saturated porous media. Int. J. of Heat Fluid Flow, 19:56- 58, 1998. M.S. Phanikumar and R.L. Mahajan. Non-darcy natural convection in high porosity metal foams. International Journal of Heat and Mass Transfer, 45- 3781- 3793, 2002. 88 / 36

89. References III A.C. Baytas and I. Pop. Free convection in a square porous cavity using a thermal non-equilibrium model. International Journal of Thermal Sciences, 41:861- 870, 2002. I.A. Badruddin, A. Zainal, P.A.A. Narayana, and N. Seetharamu. Numerical analysis of convection conduction and radiation using a non-equilibrium model in a square porous cavity. International Journal of Thermal Sciences, 46:20- 29, 2007. Sarita Pippal and P. Bera. A thermal non-equilibrium approach for 2d natural convection due to lateral heat flux: Square as well as slender enclosure. International Journal of Heat and Mass Transfer, 56:501- 515, 2013. 89 / 36

90. References IV A.C. Baytas. Thermal non-equilibrium free convection in a cavity filled with a non-darcy porous medium. Emerging Technologies and Techniques in Porous Media, NATO Science Series, 134:247- 258, 2004. R. Helmig. Multiphase Flow and Transport Processes in the Subsurfaces. Springer-Verlag, Berlin, 1997. P.R. King, S.V. Buldyrev, N.V. Dokholyan, S. Havlin, Y. Lee, G. Paul, and H.E. Stanley. Applications of statistical physics to the oil industry: predicting oil recovery using percolation theory. Physica A, 274:6066, 1999. 90 / 36

91. References V P.R. Dando, D. Stuben, and S.P. Varnavas. Hydrothermalism in the mediterranean sea. Prog.Oceanogr., 44:333- 367, 1999. O.V. Trevisan and A. Bejan. Natural convection with combined heat and mass transfer buoyancy effects in porous medium. Interntional Journal of Heat and Mass Transfer, 28:1597- 1611, 1985. P. Nithiarasu, K.N. Seetharamu, and T. Sundararajan. Double-diffusive natural convection in an enclosure filled with uid-saturated porous medium: a generalized non- darcy approach. Numerical Heat Transfer : Part A: Appl., 30:413- 426, 1996. 91 / 36

92. References VI D. Angirasa, G.P. Peterson, and I. Pop. Combined heat and mass transfer by natural convection with opposing buoyancy effects in a uid saturated porous medium. International Journal of Heat Mass Transfer, 40:2755- 2773, 1997. M. Sankar, B. Kim, and J.M. Lopez. Thermosolutal convection from a discrete heat and solute source in a vertical porous annulus. International Journal of Heat and Mass Transfer, 55:4116- 4128, 2012. R. Bennacer, A. Tobbal, H. Beji, and P. Vasseur. Double diffusive convection in a vertical enclosure filled with anisotropic porous media. International Journal of thermal science, 40:30- 41, 2001. 92 / 36

93. References VII B. Goyeau, J.P. Songbe, and D. Gobin. Numerical study double-diffusive natural convection in a porous cavity using the darcy brinkman formulation. Interntional Journal of Heat and Mass Transfer, 39:1363- 1378, 1996. D.D. Dincov, K.A. Parrott, and K.A. Pericleous. Heat and mass transfer in two-phase porous materials under intensive microwave heating. Journal of Food Engineering, 65:403- 412, 2004. A.J. Chamkha. Double diffusive convection in porous enclosure with co- operating temperature and concentration gradients and heat generation or absorption effects. Numerical Heat Transfer A, 41:65- 87, 2002. 93 / 36

94. References VIII G. Degan, P. Vasseur, and E. Bilgen. Convective heat transfer in a vertical anisotropic porous layer. Int. J. of Heat and Mass Transfer, 32:115- 125, 1996. Nield DA, Bejan A. 2013 Convection in porous media . Springer, New York. Vafai K. 2000 Handbook of Porous Media . Marcel Dekker, New York. Kaviany M. 1991 Principles of Heat Transfer in Porous Media . Springer-Verlag, New York. Pritchard, D. and Richardson, C. N., 2007, "The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer", J. Fluid Mech.", 571, 59–95. 94 / 36

95. Sunil Kumar Singh GLA University, Mathura 14 /11/202 5 1 / 1