Numerical Solution of Non-Darcian Effects on Natural Convection in a Rectangular Porous Enclosure with Heated Walls

Transcription

1 International Journal of Advanced Mechanical Engineering. ISSN Volume 8, Number 1 (18), pp Research India Publications Numerical Solution of Non-Darcian Effects on Natural Convection in a Rectangular Porous Enclosure with Heated Walls Vikash Kumar* 1, Abha Rani 2 and Aja Kumar Singh 3 *1, 2 Department of Applied Mathematics, Indian Institute of Technolog(ISM) Dhanbad , India. 3 CSIR-Central Institute of Mining and Fuel Research, Dhanbad , India. * Corresponding author; vikashiitism@gmail.com Abstract In the present stud Numerical investigation of natural convection flow in a rectangular porous enclosure with non-linear and inertia effects were considered. The dimensionless non-linear coupled partial differential equations are solved numericall with an appropriate set of boundar conditions using finite difference method. Two vertical side walls of the rectangular cavit are maintained at the same temperature and the top and bottom horizontal walls are taken at two different temperatures. Results are presented in terms of Nusselt number and average Nusselt numbers for various parameters such as Raleigh-Darc number, Darc number, aspect ratio, top wall temperature and nonlinear coefficient. Kewords: Finite difference approach, non-darcian model, Nusselt number, Raleigh-Darc number, aspect ratio, rectangular porous enclosure 1. INTRODUCTION Natural convection in a fluid saturated porous enclosure has received considerable attention for its application in ground water flow, convection in the goaf areas in underground coal mines, oil recover, cooling of radioactive waste container, to name just a few. A number of analtical, numerical and eperimental studies have been performed to investigate natural convection in vertical rectangular porous enclosures with isothermal vertical walls and adiabatic horizontal walls. Two dimensional fluid flow and heat transfer in a saturated porous medium enclosed b four isothermal and impermeable walls have been analzed in this paper. The two vertical side walls are maintained at the same temperature and the top and bottom horizontal walls are kept

2 72 Vikash Kumar, Abha Rani and Aja Kumar Singh at two different temperatures. The assumption of insulated horizontal walls ma be of fundamental interest but there are some phsical situations in which vertical heat transfer through horizontal boundaries is not negligible. Investigations that focus on natural convection in rectangular enclosure wherein the side walls, top wall and the bottom wall are constant but maintained at different temperatures are few in number. This condition permits heat transfer across all the boundaries, which results in considerable variation of the flow field. Eamples of such situations include geothermal applications and coal mine fires in the goaf areas. Christopher [1] has analzed transient natural convection for the case when all the four walls are isothermal. Furthermore, in most of these studies, it is assumed that Darc law is applicable. Analtical studies b Weber [2], Walker and Homs [3], Bejan [4] and Simkins and Blthe [5] and numerical investigations b Holst and Aziz [6], Hicko and Gartling [7], Shiralkar et al. [8], Prsasd and Kilacki [9] based on Darc law. The Brinkman-etended Darc model which included the viscous force terms and the noslip boundar conditions was used b Chan et al. [10], Tong and Subramanian [11] and Tong and Orangi [12]. Poulikakos and Bejan [] and Prasad and Tuntom [14] have analzed the inertia effects b using Forchheimer-etended Darc equation of motion and the significance of Forchheimer s velocit square term are eamined. Lauriat and Prasad [15] have performed a numerical stud for a generalized momentum equation and have included viscous force terms and inertia force simultaneousl. Nithiarasu et al. [] have analzed the natural convective heat transfer in a fluid saturated variable porosit medium taking into account linear and non-linear matri drag components as well as the inertial and viscous forces within the fluid. Saeid et al. [] have investigated numerical solution of non-darc natural convection in a square cavit filled with a porous medium. Basak et al. [18] have performed numerical solution using Galerkin finite element method with penalt parameter to analze stead laminar convection flow in square cavit with uniform and non-uniforml heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical wall. Oztop [19] has performed numerical investigation based on finite volume based finite difference method using the SIMPLE algorithm to analze natural convection in a partiall cooled and inclined porous rectangular enclosure with one of the side walls having constant hot temperature, another side wall is partiall cooled and the vertical walls are adiabatic. Chen et al. [] have obtained numerical solutions for free convection in a porous wav cavit based on the Darc-Brinkman-Forchheimer etended model. Natural convection in a rectangular cavit consisting of two insulated horizontal walls of finite thickness and two vertical walls which are maintained at constant but different temperatures has been investigated numericall Al-Amiri et al. [] using Forchheimer Brinkman-etended Darc model. Unstead conjugate natural convection in a square enclosure filled with a porous medium has been analzed b Aleshkova et al. [22]. Oztop et al. [23] have discussed the problem based on natural convection heat transfer in a partiall opened cavit filled with porous media. Babu at al. [24] have analzed non-darcian free and forced convection flow through a porous medium in a coaial duct with radiation derived from Brinkmann-Forchhheimer etended Darc model. Recentl, Khanafer [25] studied fluid structure interaction analsis of non-darcian effects on natural

3 Numerical Solution of Non-Darcian Effects on Natural Convection 73 convection in a porous enclosure. Wu et al. [26] have studied natural convection in a cavit filled with porous medium with partiall thermal active sidewalls under local thermal non-equilibrium conditions. In another paper, Wu et al. [27] have analzed natural convection in a porous rectangular enclosure with sinusoidal temperature distributions on both side walls using a thermal non-equilibrium model. The present investigation is concerned with the stead, a two dimensional buoanc-induced convection in a vertical porous rectangular enclosure with specified temperature on all the four surfaces. The boundaries are impermeable and the results are based on inclusion of boundar and inertia effects as well as nonlinear effect. The pressure field appearing in the momentum equation is eliminated and the stream function vorticit equation together with the energ equation is solved b the finite difference method. This method leads to a simple and straight forward analsis and possesses a highl desirable propert of numerical stabilit. The stream function vorticit and temperature fields and the local and average Nusselt number are obtained for five dimensionless parameters, namel, the Raleigh-Darc number, Darc number, aspect ratio, top wall temperature and non-linear coefficient CF. 2. MATHEMATICAL FORMULATION A two dimensional fluid saturated porous medium enclosed b four isothermal and impermeable walls as shown in Figure 1, has been considered. The two vertical side walls are maintained at the same temperature T s and the top and bottom horizontal walls are kept at two different temperatures T t and T b respectivel. Subject to the Boussinesq approimation, the continuit, momentum and energ equations for the flow of a Newtonian fluid are, Y,V B U=0, V=0, T=T t U=0 V=0 T=T S POROUS MEDIA U=0 V=0 T=T S 0 0 U=0, V=0, T=T b A X,U Figure 1: Geometr of the problem

4 74 Vikash Kumar, Abha Rani and Aja Kumar Singh U + V = 0 (1) X Y ρ (U U U + V ) = P + μ U X Y X ( U X 2 Y 2) μu C FρU 2 K K ρ (U V V + V ) = P + μ V X Y Y ( 2 U T T + V = α T X Y ( T X 2 Y + 2 V X 2 Y μv 2) + ρgβ(t T K s) C FρV 2 K (2) (3) 2) (4) The boundar conditions are: X = 0: U = 0, V = 0, T = T s X = A: U = 0, V = 0, T = T s Y = 0: U = 0, V = 0, T = T b Y = B: U = 0, V = 0, T = T t (5) Where U andv are the horizontal and vertical velocit components in the X and Y directions respectivel, P is the pressure, T is the temperature, ρ is the densit of the fluid, μ is the viscosit, α is the thermal diffusivit, β is the coefficient thermal epansion, g is the acceleration due to gravit, K is the permeabilit of the porous medium and A and B are width and height of the enclosure respectivel. 3. DIMENSIONLESS EQUATIONS The following dimensionless variables are introduced to normalize the governing equations: = X A, = Y B, u = UA α As = B A, Ra = ρgβ(t b T s )KA μα VA PA2 v =, p = α ρα 2, θ = T T s, θ T b T t = T t T s, Pr = μ, Da = K s T b T s ρα A 2, (6) The governing equations (1)-(4) are reduced to the following dimensionless form where, Da and Ra are Darc number and Raleigh-Darc number respectivel, Pr is the Prandtl number and As is the aspect ratio: u + v = 0 (7) u u u + v = P + Pr u ( 2 u v v + v = P + Pr v ( u v 2 2) Pr u u2 C F Da Da Pr v 2) Da (Ra θ v) v2 C F Da (8) (9)

5 Numerical Solution of Non-Darcian Effects on Natural Convection 75 u θ θ + v = 2 θ + 2 θ (10) 2 2 The boundar conditions assume the following dimensionless form: = 0: u = 0, v = 0, θ = 0 = 1: u = 0, v = 0, θ = 0 = 0: u = 0, v = 0, θ = 1 = As: u = 0, v = 0, θ = θ t (11) Introducing stream function ψ and vorticit ω, as u = ψ ψ, v = ω = u v and (12) The equation of continuit (7) is automaticall satisfied b (12) and the vorticit can be written in terms of ψ as, () ω = 2 ψ ψ 2 (14) The momentum equation (8) and (9) are transformed into the vorticit equation, ψ ω ψ ω = Pr ω ( ω 2 2) Pr Da The energ equation (10) now takes the form, ψ θ ψ θ = 2 θ + 2 θ 2 (ω + Ra θ ) 2C Fω Da. (15) 2. () The boundar conditions to be satisfied b ψ and θ are, = 0: ψ = 0, ψ = 1: ψ = 0, ψ = 0, θ = 0 = 0, θ = 0 = 0: ψ = 0, ψ = 0, θ = 1 = As: ψ = 0, ψ = 0, θ = θ t. ()

6 76 Vikash Kumar, Abha Rani and Aja Kumar Singh 4. NUMERICAL PROCEDURE The governing equations are coupled non-linear partial differential equations for which an iterative procedure of solution is natural. Finite difference equations are derived using upwind differences for convective terms and central difference for other terms. The upwind difference approimation of the convective terms depends on the flow directions in which a derivative is replaced b a backward difference if the velocit in the concerned direction is positive and b a forward difference if that is negative. The resulting finite difference equation is onl first order accurate but leads to a diagonall dominant matri and to a numericall stable scheme independent of the step sizes. The governing equations (14)-() are approimated in the following difference forms: ψ i 1,j 2ψ i,j +ψ i+1,j (Δ) 2 + ψ i,j 1 2ψ i,j +ψ i,j+1 (Δ) 2 = ω i,j (18) Δψ j 2Δ (ω i,j ω i±1,j ) + Δψ i Δ 2Δ (ω i,j ω i,j±1 ) Δ Δψ j 2Δ (θ i,j θ i±1,j Δ = Pr ( ω i 1,j 2ω i,j + ω i+1,j (Δ) 2 Pr + ω i,j 1 2ω i,j + ω i,j+1 (Δ) 2 ) (ω Da i,j + Ra θ i+1,j θ i,j±1 ) 2 C F ω Δ Da i,j (19) ) + Δψ i 2Δ (θ i,j θ i,j±1 Δ ) = ( θ i 1,j 2θ i,j +θ i+1,j (Δ) 2 + θ i,j 1 2θ i,j +θ i,j+1 (Δ) 2 ) () The difference equations (18)-() are solved b line-b-line method presented b Patankar (1980). Firstl the local Nusselt number along the surfaces are determined, the average Nusselt number along each surface is evaluated b Simpson s rule of integration. 5. RESULTS AND DISCUSSION To test the validit of the numerical procedure, the average Nusselt number along the side wall obtained b the present method are compared with the results of Chen et al. [28] in Table 1, for Pr = 7.0, As = and 10 4 Ra 10 6 in the absence of porous matri, where Ra is the Raleigh number. The results tabulated in Table 1 are obtained for dimensionless side wall temperature =, top wall temperature = and bottom wall temperature =. The agreement is satisfactor, which substantiates the correctness of the computational work.

7 Numerical Solution of Non-Darcian Effects on Natural Convection 77 Table 1: Average Nusselt number along the side wall for Pr = 7.0, As = and 10 4 Ra 10 6 Ra Comparison Left wall Right wall 10 4 Chen et al. [28] Present 10 5 Chen et al. [28] Present 10 6 Chen et al. [28] Present At the second stage the boundar conditions for the present work have been imposed b setting the side wall temperature =, bottom wall temperature = and three different values of the top wall temperature =, 0.5,. Computations for different grid sizes were carried out and 61 61, and grid fields showed insignificant changes in the results. It was observed that a grid sstem is a good choice. The phsical domain is limited b 0 1 and 0 As. Numerical results have been obtained b using the following three models: (A) (B) Darc s model Non-Darcian model with non linear inertia term and viscous term (C) Non Darcian model with viscous term, non linear inertia term and velocit square term Local Nusselt numbers along the side, top and bottom walls have been computed usingthe models A, B and C and are presented in Tables 2, 3 and 4 respectivel. Table 2: Local Nusselt number along the side wall for Pr =, As = 1. θ t= 0.5, Ra = 10, CF=0.55 =0.5 Da Model(A) Model(B) Model(C)

8 78 Vikash Kumar, Abha Rani and Aja Kumar Singh Table 3: Local Nusselt number along the top wall for Pr =, As = 1. θ t= 0.5, Ra = 10, CF=0.55 =0.5 Da Model(A) Model(B) Model(C) Table 4: Local Nusselt number along the bottom wall for Pr =, As = 1. θ t= 0.5, Ra = 10, CF=0.55 =0.5 Da Model(A) Model(B) Model(C) It is observed that while the local Nusselt number remains unchanged in the case of Darc s model, significant variation is noticed when computations are made using Model B and Model C. It decreases with increase of Darc number along the side walls but a reverse trend is observed near top and bottom walls. It is interesting to note that the percentage difference is insignificant for low values of Darc number and this difference is considerable for higher values of Darc number. The difference between the local Nusselt number values for model A and model C near the bottom wall is merel 0.3 percent at Da = 10-5 which graduall increases to almost 11 percent at Da = This confirms the limitation of Darc s law at higher values of permeabilit. It is imperative therefore to use non-darcian model at higher values of permeabilit. Flow patterns and isotherms for various parameters are shown in Figures 2 to 9. There is smmetr in the vorte pattern about the mid-plane of the enclosure, due to smmetr nature of the boundar conditions prescribed in the problem. The flow consists of a single cell filling the entire half of the enclosure rotating in the clockwise direction but anticlockwise in the other half. The velocities are higher near the wall and lower along the line of smmetr. It is observed the flow is almost stagnant near the corner. The circulation strength depends on Ra, Da, Asand θ t. An increase in Raleigh-Darc number increases the circulation vorte. It is seen that this change in the vorte strength is nearl proportional to the Raleigh-Darc number. As the Darc number decreases, the strength of the vorte decreases due to increasing resistance of the porous matri. In order to stud the effect of aspect ratio, three different sizes of the enclosures are chosen. The patterns are ver similar for As = 0.5, 1 and 2 ecept that the values of

9 Numerical Solution of Non-Darcian Effects on Natural Convection 79 stream function and temperature are larger for large aspect ratio. Streamlines and isotherms are also plotted for two different values of top wall temperature =0.5, and. The change in the top wall temperature also alters the flow structure. The circulation pattern is smoother in the case of θ t = when compared to the case θ t = 0.5. Furthermore, for small values of Ra, a pure conduction solution with parallel isotherms were obtained. As the Raleigh-Darc number increases, the effects of convection are noticed. For moderate values of Ra, the isotherms are evenl spaced curves indicating that the heat transfer is dominated b conduction and the convection effects are ver less. The isotherm patterns are more distorted in the case of large Raleigh-Darc number when the convection is dominant Figure-2 Streamlines and isotherm for θ t = 0.5, Pr = 1, As = 1, Ra = 10 2, C F = 0.55, Da = Figure-3 Streamlines and isotherm for θ t = 0.5, Pr = 1, As = 1, Ra = 10 3, C F =.55, Da = 10 3

10 80 Vikash Kumar, Abha Rani and Aja Kumar Singh Figure-4 Streamlines and isotherm for θ t = 1, Pr = 1, As = 1, Ra = 10 2, C F =.55, Da = Figure-5 Streamlines and isotherm for θ t = 1, Pr = 1, As = 1, Ra = 10 3, C F =.55, Da = Figure-6 Streamlines and isotherm for θ t = 1, Pr = 1, As = 1, Ra = 10 3, C F =.55, Da = 10 2

11 Numerical Solution of Non-Darcian Effects on Natural Convection Figure-7 Streamlines and isotherm for θ t = 1, Pr = 1, As = 1, Ra = 10 4, C F =.55, Da = Figure-8 Streamlines and isotherm for θ t = 1, Pr = 1, As = 2, Ra = 10 3, C F =.55, Da = 10 3 Figure-9 Streamlines and isotherm for θ t = 1, Pr = 1, As = 1/2, Ra = 10 3, C F =.55, Da = 10 3

12 82 Vikash Kumar, Abha Rani and Aja Kumar Singh Figure 10 and 11 shows the local Nusselt number along the bottom surface as a function of Raleigh-Darc number for different values of Da and θ t. The Nusselt numbers along each surface are found to increase with Raleigh-Darc number but decreases considerabl as Da increases. Nusselt number along the bottom surface is largest and it is the least along the top surface. Increasing θ t increases the Nusselt number along the top surface. Increasing the aspect ratio, decreases the Nusselt number along top and bottom surfaces. Along the side wall, increase in the aspect ratio brings a large increase in the Nusselet number for θ t = 0. for large magnitude of θ t, changing the aspect ratio had little effect on the Nusselt number Da Da * 4 Nu * 4 * 3 * 2 Nu Ra* 10 2 Ra* Figure Local Nusselt number along the bottom wall for θ t = 0.5,Pr = 1, C F =.55, As = t t Nu * 4 * 3 * Nu * 4 * 3 * Figure-11 Local Nusselt number along the bottom wall for Pr = 1, C F =.55, As = 1, Da = CONCLUSION Natural convection in a rectangular fluid saturated porous cavit with isothermal horizontal and vertical walls is investigated numericall in a non-darcian regime. Results have been obtained using three different models viz. Darc s model, non-

13 Numerical Solution of Non-Darcian Effects on Natural Convection 83 Darcian model with non linear inertia term and viscous term and non-darcian model with viscous term, non linear inertia term and velocit square term. While the difference in the local Nusselt number results at lower values of Darc number for all the three models is insignificant, the difference increases with increase of Darc number. This is attributed to the deviation from Darc regime at higher values of permeabilit. Flow patterns and isotherms are smmetr in the vorte pattern about the mid-plane of the enclosure. The flow consists of a single cell filling the entire half of the enclosure rotating in the clockwise direction but anticlockwise in the other half. Also we have observed the velocities are higher near the wall and lower along the line of smmetr. It is observed the flow is almost stagnant near the corner. The circulation strength strongl depends on Ra, Da, As, C F and θ t. An increase in Raleigh-Darc number leads to increases the circulation vorte. NOMENCLATURE Ra -Raleigh-Darc number As -aspect ratio Da -Darc number K -permeabilit of a porous medium K -thermal conductivit, [Wm -1 K -1 ] Nu -Nusselt number Pr -Prandtl number Ra -Raleigh number u, v -velocit components along, directions, [ms -1 ], -Cartesian coordinates A, B -width and height of the enclosure respectivel. P -pressure of the fluid, [Nm -2 ] T -temperature of the fluid, [K] C F -non linear coefficent Greek smbols θ t -top wall temperature, [K] g -acceleration due to gravit, [ms 2 ] α -thermal diffusivit, [m 2 s -1 ]

14 84 Vikash Kumar, Abha Rani and Aja Kumar Singh β -coefficient thermal epansion, [K -1 ] θ -fluid temperature, [K] μ -viscosit, [kgm -1 s -1 ] ρ -densit of the fluid, [kgm -1 ] ψ -stream function ω -vorticit REFERENCES [1] Christopher, D. M. (1986), Transient natural convection heat transfer in a cavit filled with a fluid-saturated, porous medium. In Proc. Int. Heat Transfer Conf, 1986 (Vol. 5, pp ). [2] Weber, J. E., The boundar-laer regime for convection in a vertical porous laer. International Journal of Heat and Mass Transfer, 18(1975), [3] Walker, K. L., Homs, G. M., Convection in a porous cavit. Journal of Fluid Mechanics, 87(1978), [4] Bejan, A., On the boundar laer regime in a vertical enclosure filled with a porous medium. Letters in Heat and Mass Transfer, 6(1979), 932. [5] Simpkins, P. G., Blthe, P. A., Convection in a porous laer. International Journal of Heat and Mass Transfer, 23(1980), [6] Holst, P. H., Aziz, K., A theoretical and eperimental stud of natural convection in a confined porous medium. The Canadian Journal of Chemical Engineering, 50(1972), [7] Hicko, C. E., Gartling, D. K., A numerical stud of natural convection in a horizontal porous laer subjected to an end-to-end temperature difference. Journal of Heat Transfer, 103(1981), [8] Shiralkar, G. S., et al., Numerical stud of high Raleigh number convection in a vertical porous enclosure. Numerical Heat Transfer, 6(1983), [9] Kulacki, F. A., Corrective Heat Transfer in a Rectangular Porous Cawitw Effect of Aspect Ratio on Flow Structure and Heat Transfer. transfer, 1984,1000, s2. [10] Chan, B. K. C., et al., Natural convection in enclosed porous media with rectangular boundaries. Journal of heat transfer, 92(1970), -27. [11] Tong, T. W., Subramanian, E., A boundar-laer analsis for natural convection in vertical porous enclosures use of the Brinkman-etended Darc model. International journal of heat and mass transfer, 28(1985),

15 Numerical Solution of Non-Darcian Effects on Natural Convection 85 [12] Tong, T. W., Orangi, S., A numerical analsis for high modified Raleigh number natural convection in enclosures containing a porous medium. Continuit, 3, 3(1986). [] Poulikalos, D., A departure from the Darc model in boundar laer natural convection in a vertical porous laer with uniform heat flu from the side. Journal of heat transfer, 107(1985), 7-7. [14] Prasad, V., Tuntomo, A., Inertia effects on natural convection in a vertical porous cavit. Numerical Heat Transfer, Part A Applications, 11(1987), [15] Lauriat, G., Prasad, V., Natural convection in a vertical porous cavit: a numerical stud for Brinkman-etended Darc formulation. J. Heat Trans, 295(1987). [] Nithiarasu, P., et al., Natural convective heat transfer in a fluid saturated variable porosit medium. International Journal of Heat and Mass Transfer, 40(1997), [] Saeid, N., Pop, I., Natural convection from a discrete heater in a square cavit filled with a porous medium. Journal of Porous Media, 8(05). [18] Basak, T., et al., Natural convection in a square cavit filled with a porous medium: effects of various thermal boundar conditions. International Journal of Heat and Mass Transfer, 49(06), [19] Oztop, H. F., Natural convection in partiall cooled and inclined porous rectangular enclosures. International Journal of Thermal Sciences, 46(07), [] Chen, X. B., et al., Free convection in a porous wav cavit based on the Darc-Brinkman-Forchheimer etended model. Numerical Heat Transfer, Part A: Applications, 52(07), [] Al-Amiri, A., et al., Stead-state conjugate natural convection in a fluidsaturated porous cavit. International Journal of Heat and Mass Transfer, 51(08), [22] Aleshkova, I. A., Sheremet, M. А., Unstead conjugate natural convection in a square enclosure filled with a porous medium. International Journal of Heat and Mass Transfer, 53(10), [23] Oztop, H. F., et al., Natural convection heat transfer in a partiall opened cavit filled with porous media. International Journal of Heat and Mass Transfer, 54(11), [24] Babu, D. C., Rao, D. P., Non-Darcian Free and Forced Convection Flow Through a Porous Medium in a Coaial Duct with Radiation. Journal of Computer and Mathematical Sciences Vol, 3(12),

16 86 Vikash Kumar, Abha Rani and Aja Kumar Singh [25] Khanafer, K., Fluid structure interaction analsis of non-darcian effects on natural convection in a porous enclosure. International Journal of Heat and Mass Transfer, 58(), [26] Wu, F., et al., Natural convection in a cavit filled with porous medium with partiall thermal active sidewalls under local thermal nonequilibrium conditions. Journal of Porous Media, (14). [27] Wu, F., et al., Natural convection in a porous rectangular enclosure with sinusoidal temperature distributions on both side walls using a thermal nonequilibrium model. International Journal of Heat and Mass Transfer, 85, 15, [28] Chen, K. S., et al., Stead, two-dimensional, natural convection in rectangular enclosures with differentl heated walls. Journal of heat transfer, 109(1987),