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### A numerical analysis of three dimensional darcy model in an inclined rect

1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME554A NUMERICAL ANALYSIS OF THREE-DIMENSIONAL DARCYMODEL IN AN INCLINED RECTANGULAR POROUS BOX USING S AR TECHNIQUEDr. R. P. SharmaDept. of Mechanical Engineering, Birla Institute of Technology, Mesra, Ranchi, 835215IndiaABSTRACTIn this paper, numerical studies on three- dimensional natural convection in aninclined differentially heated porous box employing Darcy flow model are presented. Therelative effects of inertia and viscous forces on natural convection in porous media areexamined. The governing equations for the present studies are obtained by setting Da=0 andFc/Pr = 0 in the general governing equations for Darcy flow description. The system ischaracterized by Rayleigh number (Ra), two aspect ratios (ARY, ARZ), and angle ofinclination (φ). Numerical solutions have been obtained by employing the S A R scheme fordifferent values of Rayleigh number (Ra), aspect ratio and angle of inclination. It is foundthat for the Darcy flow model Nusselt number for 3-D and 2-D are same. Also there exists acritical angle of inclination of the porous box at which the average Nusselt number becomemaximum i.e. 30° in Darcy flow model.Keywords: Darcy, SAR, Porous, critical angle of inclination etc.1.0 INTRODUCTIONOwing to the relevance in many physical system of interest, investigation on fluidflow and heat transfer in porous media are being widely reported in the literature. Gill [1]reported on the stability aspects of an infinite vertical porous layer subjected to a temperaturedifference. Gill employed the Darcy flow model and assumed constant fluid properties exceptfor density variation in evaluating the buoyancy force. This feature of accounting for thevariation in density in evaluating the buoyancy force only is commonly referred to asBoussinesq approximation. The study [1] concluded that the thermal convection generatedINTERNATIONAL JOURNAL OF MECHANICAL ENGINEERINGAND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online)Volume 4, Issue 3, May - June (2013), pp. 554-561© IAEME: www.iaeme.com/ijmet.aspJournal Impact Factor (2013): 5.7731 (Calculated by GISI)www.jifactor.comIJMET© I A E M E
2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME555flow is always stable. Experimental studies along with integral solutions for the rectangularporous slabs have been reported by Klarsfeld [2]. Numerical results for Darcy flow modelalso have been reported by Vlasuk [3]. Bories and Combarnous [4] reported experimentalresults for Rayleigh numbers ranging from 100 to 1000. According to the numerical studiesof Bankvall [5], onset of boundary layer type flow occurred when the Rayleigh number isgreater than 200. Weber [6] developed an Oseen linearized solution for the boundary layerregime for high aspect ratios. Weber’s study accounted for the variation in viscosity using anaverage of hot and cold values. Seki, Fukusako and Inaba [7] correlated the average Nusseltnumber with Ra, Pr and AR from the experimental results obtained with different solid fluidcombinations in the range 1 < Ra < 105, 1< Pr < 200 {Pr(= ν/α) is the Prandtl number} forAR = 5,10 and 26. Comparison with the numerical results of Vlasuk [3] established that theDarcy model is adequate to describe the fluid flow for Ra < 1000. Bejan and Tien [8] andWalker and Homsy [9] obtained analytical results for small aspect ratios employing Darcyflow model. Walker and Homsy [9] also presented numerical solutions for boundary layerequations. Isothermal boundary conditions are considered, where two opposite walls are keptat constant but different temperatures and the other two are thermally insulated. Three mainconvective modes are found, conduction single and multiple cell convection and theirfeatures described in detail. Local and global Nusselt numbers are presents as function of theexternal parameters. Moya, et, al., [10] analyzed two-dimensional natural convective flow ina titled rectangular porous material saturated with fluid by solving numerically the mass,momentum and energy balance equations using Darcys law and the Boussinesqapproximation. Sharma R P & Sharma R V has worked on modelling &simulation of three –dimensional natural convection in a porous box and concluded that three-dimensionalaverage Nusselt values are lower than two-dimensional values. [11]2.0 MATHEMATICAL MODELLING2.1 Governing Equation2.2Fig. 1 Physical model and co-ordinate system
3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME556The physical model is shown in Fig. 1 is a parallelepiped box of length L, width B and heightH filled with fluid saturated porous medium.Governing dimensionless equations for natural convection in the porous box comprising ofconservation of mass, momentum and energy are as follows:0=∂∂+∂∂+∂∂ZWYVXU(1)U+ φρβsin||Pr TRaXPUVFc∆−∂∂−= + Da ∂∂+∂∂+∂∂222222ZUYUXU(2)V+ φρβcos||Pr TRaYPVVFc∆−∂∂−= + Da ∂∂+∂∂+∂∂222222ZVYVXV(3)W+ZPWVFc∂∂−=||Pr+ Da ∂∂+∂∂+∂∂222222ZWYWXW(4)222222ZYXZWYVXU∂∂+∂∂+∂∂=∂∂+∂∂+∂∂ θθθθθθ(5)ρ = 1 - β∆T (θ-0.5) (6)Non-dimensional parameters Ra, the Rayleigh number, Fc, the Forchheimer number, Pr, thePrandtl number and Da, the Darcy number are defined by –ναβ TKgLRa∆= ;LKFc′= ;αv=Pr ; 2LKDa = (7)Hydrodynamic Boundary Conditions(i) Without Brinkman termsU = 0 at X = 0, 1 for 0 ≤ Y ≤ ARY and 0 ≤ Z ≤ ARZV = 0 at Y = 0, ARY for 0 ≤ X ≤ 1 and 0 ≤ Z ≤ ARZ (8)W = 0 at Z = 0, ARY for 0 ≤ X ≤ 1 and 0 ≤ Y ≤ ARYThermal Boundary Conditionsθ = 0 at X = 0, for 0 ≤ Y ≤ ARY and 0 ≤ Z ≤ ARZθ = 1 at X = 1, for 0 ≤ Y ≤ ARY and 0 ≤ Z ≤ ARZ (9)0Y=∂θ∂at Y = 0, ARY for 0 ≤ X ≤ 1 and 0 ≤ Z ≤ ARZ0Z=∂θ∂at Z = 0, ARZ for 0 ≤ X ≤ 1 and 0 ≤ Y ≤ ARYWhere ARY, the vertical aspect ratio and ARZ the horizontal aspect ratio are defined as
4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME557ARY = H/L (10)ARZ = B/L (11)Boundary conditions on temperature (θ) are the same as given by Eq. (9). The averageNusselt number based on the characteristic length, L of the box is defined as,kLhNu = (12)The average Nusselt number at X = 0 and X = 1 is obtained by numerical integrationaccording to,∫ ∫ =∂∂−=Y ZAR ARxZYhXARARNu0 001 θdY dz (13)∫ ∫ =∂∂−=Y ZAR ARxZYcXARARNu0 011 θdY dZ (14)In order to obtain the numerical solution of the above equations along with boundaryconditions, the Successive Accelerated Replacement (S A R) scheme has been employed andresults are obtained.The successive accelerated replacement (S A R) scheme is a point iterative scheme.The basic philosophy of the S A R scheme is to guess a profile for each variable whichsatisfies the boundary conditions. Each dependent variable is associated with one governingequation. It is natural to associate the equation for a variable which contains the highest orderderivative of that variable. For example, conservation of energy equation is associated fortemperature. The non-dimensionalised form of the governing equations with boundaryconditions is written in finite difference form using central differencing scheme. Let the finitedifference equation governing a variable φ be given by J.,IΦ = 0 at any mesh point (I, J)corresponding to (X, Y) position. The guessed profile will not in general satisfy the equation.The error arising out of the guessed profile is evaluated. Let the error in the equation at (I, J)and at the nth iteration be nJ,IΦ . The (n+l)thapproximation to the variable φ is obtained from,J,InJ,I~nJ,I~nJ,IlnJ,I/ φ∂Φ∂Φω−φ=φ +(15)where, ω is an acceleration factor which varies between 0 and 2. The procedure of correctingthe variable at every mesh point in the entire region is repeated until a set of convergencecriteria are satisfied. The criterion is,lnJ,InJ,IlnJ,I++φφ−φ< ∈where, ∈, the error tolerance limit is a prescribed small positive number. The feature of usingthe corrected value of the variable immediately upon becoming available is inherent in thismethod.
5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME558The average Nusselt number at X = 0 and X = 1 is obtained by numerical integrationaccording to,∫ ∫ =∂∂−=Y ZAR ARxZYhXARARNu0 001 θdY dZ (16)∫ ∫ =∂∂−=Y ZAR ARxZYcXARARNu0 011 θdY dZ (17)RESULTS & DISCUSSIONFigs. 2 and 3 show iso-vector-potential lines (Ψz) and Figs. 4 and 5 show isotherms(ө) for inclined porous box. For ϕ= - 45°, a multi-cellular flow is observed (see Fig. 2). Forpositive angle of inclination (ϕ = 45°), unicellular flow exists, (Fig. 3). Compared to verticalporous box, there is drastic change in temperature field particularly for negative angle ofinclination (Fig. 4). It can be seen from Fig. 4 that isotherms are confined around diagonal inthe X-Y plane of the porous box. Fig. 6 shows the variation of average Nusselt number (Nu)with vertical and horizontal aspect ratios (ARY and ARZ) for ϕ = 45° at Ra=1000. It can benoted from the above surface plots that average Nusselt number is independent of horizontalaspect ratio (ARz). Hence, for inclined porous box, the heat transfer is two-dimensional likethe vertical porous box. Flow field, temperature field and heat transfer for natural convectionin porous box depends on the angle of inclination. There exists a critical angle of inclinationat which the average Nusselt number value is maximum (+30°) and then decreases forRa=200, 500, 1000 and 2000 (Fig. 7). When the Rayleigh number increases, the buoyancyforces increases. The physical mechanism behind it is that when a horizontal layer of aviscous fluid is heated, a temperature gradient is set up. Convective motions will commenceas soon as the vertical temperature gradient is greater than a given critical value.7.3 CONCLUSIONSNumerical solutions to the equations governing natural convection heat transfer in aninclined porous box have been obtained using the SAR scheme. The flow description iswithin the framework of Darcy’s law. The results obtained within the framework of Darcyflow description are the same as the corresponding two-dimensional system. The non-dimensional parameters needed to describe the system are Ra, ARy, ARz and ϕ. Thenumerical solutions obtained include flow and temperature field and average Nusseltnumber.Iso-vector potential lines and isotherms are similar to streamlines and isotherms fortwo-dimensional system. Average Nusselt number values are independent of horizontalaspect ratio (ARz). There exists a critical angle of inclination at which the average Nusseltnumber is maximum. The critical angle of inclination is independent of Rayleigh number andit depends on vertical aspect ratio (ARy). As ARy increases, the critical angle of inclination(ϕ) decreases. For ARy=1.0, critical angle of inclination is 30°.
6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME5590.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.000.100.200.300.400.500.600.700.800.901.000.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.000.100.200.300.400.500.600.700.800.901.00Fig. 2: Iso-vector-potential ( ) for Ra=1000, Fig. 3: Iso-vector-potential ( ) for Ra=1000,ARY=1.0, ARZ=1.0 and ϕ= -450 ARY=1.0, ARZ=1.0Fig. 4: Isotherms for Ra=1000, ARY=1.0, Fig. 5 : Isotherms for Ra=1000, ARY=1.0,ARZ=1.0 and ϕ= -450 ARZ=1.0 and ϕ= 450
7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME560Fig. 6: Variation of Nu with ARY and Fig. 7: Variation of Nu with ϕ forARZ for Ra=1000 and ϕ= -450 ARY=1.0 and ARZ=1.0REFERENCES[1] A.E. Gill, A proof that convection in a porous vertical slab is stable, Journal of FluidMechanics,Vol. 35 pp.545-547, 1969.[2] S. Klarsfeld, Champs de temperature associes aux mouvements de convectionnaturelle dans un milien poreux limite, Revue Gen, Thermique, Vol. 9 pp.1403-1424, 1970.[3] M.P. Vlasuk, Transfer de chaleur par convection dans une couche poreuse, InProccedings, 4thAll-Union Heat and Mass Transfer Conference, Minsk, USSR,1972.[4] S.A. Bories and M.A. Combarnous, Natural convection in a sloping porous layer,Journal of Fluid Mechanics, Vol. 57 pp.63-79,1973.[5] C.G. Bankvall, Natural convection in vertical permeable space, Warme-andStoffubertagung, Vol. 7 pp.22-30, 1974.[6] J.E. Weber, The boundary layer regime for convection in a vertical porous layer,International Journal of Heat and Mass Transfer, Vol. 18 pp.569-573, 1975.[7] N.Seki,S. Fukusako, and H. Inaba, Heat transfer in a confined cavity packed withporous media, International Journal of Heat and Mass Transfer, Vol. 21 pp.985-989,1978.[8] A. Bejan and C. L. Tien, Natural convection in a horizontal porous medium subjectedto end-to-end temperature difference, ASME Trans., Journal of Heat Transfer,Vol.100 pp.191-198, 1978.[9] K. L. Walker and G.M. Homsy, Convection in a porous cavity, Journal of FluidMechanics, Vol. 97 pp.449-474, 1978.[10] Sara. L. Moya, Eduardo Ramos and Mihir Sen, Numerical study of natural convectionin a tilted rectangular porous material , International Journal of Heat and MassTransfer, Vol. 30 pp.741-755, 1987.
8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME561[11] R.P. Sharma, R.V. Sharma, “Modelling & simulation of three-dimensional naturalconvection in a porous media”, International Journal of Mechanical Engineering andTechnology (IJMET), Volume 3, Issue 2, 2012, pp. 712-721, ISSN Print:0976 – 6340, ISSN Online: 0976 – 6359.[12] Dr. R. P. Sharma and Dr. R. V. Sharma, “A Numerical Study of Three-DimensionalDarcybrinkman-Forchheimer (Dbf) Model in a Inclined Rectangular Porous Box”,International Journal of Mechanical Engineering & Technology (IJMET), Volume 3,Issue 2, 2012, pp. 702 - 711, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.