A numerical study of three dimensional darcy- forchheimer d-f- model in an

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A numerical study of three dimensional darcy- forchheimer d-f- model in an

  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) © IAEME259A NUMERICAL STUDY OF THREE-DIMENSIONAL DARCY-FORCHHEIMER (D-F) MODEL IN AN INCLINED RECTANGULARPOROUS BOXDr. 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- Forchheimer (D-F) flow modelare presented. The effects of non-linear inertia forces on natural convection in porous mediaare examined. When nonlinear inertial terms are included in the momentum equation, no-slipconditions for velocity at the walls are satisfied. The governing equations for the presentstudies are obtained by setting Da=0 and Fc/Pr ≠ 0 in the general governing equations for D-Fflow description. The system is characterized by Rayleigh Number (Ra), two aspect ratios(ARY, ARZ), ratio of Forchheimer number to Prandtl number (Fc/Pr) and the angle ofinclination φ. Numerical solutions have been obtained employing S A R scheme for 200< Ra< 2000, 0.2 < ARY < 5, 0.2 < ARZ < 5 10-5< Fc/Pr < 10-2and –60o≤ φ < 60o.Keywords: non-linear inertia, natural convection, Forchheimmer number& Prandtl numberefc1.0 INTRODUCTIONPoulikakos and Bejan [1] employed Forchheimer extended Darcy flow model andobtained boundary layer solutions for a tall cavity and the agreement with the numericalresults for Ra = 2000 and AR = 2 has been found to be good. Based on these studies,Poulikakos and Bejan classified the flow regimes as Darcy, intermediate and non-Darcy.Tong and Subramaniam [2] developed boundary layer solutions to Brinkman extended Darcyflow model based on the modified Oseen technique. Beckermann et al. [3] demonstrated thatINTERNATIONAL JOURNAL OF MECHANICAL ENGINEERINGAND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online)Volume 4, Issue 3, May - June (2013), pp. 259-265© 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) © IAEME260the inclusion of both the inertia and boundary effects is important for natural convection in arectangular cavity packed with spherical particles. Prasad and Tuntomo [4] reportednumerical results for Forchheimer extended Darcy flow model for AR > 1. The study [4]concluded that the inclusion of Forchheimer inertial term does not render boundary layer typesimplification and the influence on the flow field and heat transfer is significant. Satyamurtyand Rao [5] examined the relative influence of variable fluid properties and Forchheimerextended Darcy flow model on average Nusselt number. Zebib and Kassoy [6] analyticallystudied, using weakly nonlinear theory, the possibility of two and three-dimensional flowpatterns to exist in a rectangular parallelepiped. Holst [7] numerically solved transient, three-dimensional natural convection in a porous box. They concluded that, compared to the two-dimensional values, the three-dimensional heat flow under certain instances is higher at highRayleigh numbers Horne [8] investigated the tendency of the flow to be two or three-dimensional. Chan and Bannerjee [9] studied three-dimensional natural convection using afinite difference scheme based on the simplified marker and cell technique Dawood andBurns [10] studied steady three-dimensional convective heat transfer in a porous box withside heating numerically using multigrid method, which accelerated the convergence.Dawood and Burns [10] studied steady three-dimensional convective heat transfer in a porousbox with side heating numerically using multigrid method, which accelerated theconvergence. Dawood and Burns reported numerical results have been reported for horizontaland vertical aspect ratios, 0.5 < ARz, ARy < 20 at a Rayleigh number of 200. Vasseur et al.,[11] studied analytically and numerically the thermally driven flow in a thin, inclined,rectangular cavity filled with a fluid saturated porous layer. Sharma R P & Sharma R V hasworked on modelling &simulation of three –dimensional natural convection in a porous boxand concluded that three-dimensional average Nusselt values are lower than two-dimensionalvalues. [14]2.0 MATHEMATICAL MODELLING2.1 Governing EquationFig. 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) © IAEME261The physical model is shown in Fig. 1 is a parallelepiped box of length L, width B and heightH filled with fluid saturated porous medium. The dimensionless conservation of mass,momentum and energy are derived and non-dimensional parameters Ra, the Rayleighnumber, Fc, the Forchheimer number, Pr, the Prandtl number and Da, the Darcy number aredefined by -ρ = 1 - β∆T (θ-0.5) ;ναβ TKgLRa∆= ;LKFc′= ;αv=Pr ; 2LKDa = (1)After eliminating the pressure term from the derived non-dimensional momentum equations,vector potential formulation equations are as follows –x2xx DacosZRa|V|Z|V|VY|V|WPrFcΩ∇+φ∂θ∂−=Ω+∂∂−∂∂+Ω (2)y2yy DasinZRa|V|X|V|WZ|V|UPrFcΩ∇+φ∂θ∂=Ω+∂∂−∂∂+Ω (3)z2zzDacosXsinYRa|V|Y|V|UX|V|VPrFcΩ∇+φ∂θ∂−φ∂θ∂−=Ω+∂∂−∂∂+ΩEnergy equation can be rewritten as, (4)222222ZYXZWYVXU∂θ∂+∂θ∂+∂θ∂=∂θ∂+∂θ∂+∂θ∂ (5)The velocity components U, V and W are related to the components of the vector-potentialby,ZYU yz∂ψ∂−∂ψ∂= (6)XZV zx∂ψ∂−∂ψ∂= (7)YXW xy∂ψ∂−∂ψ∂= (8)Boundary Conditions on Vector Potential Ψ( )Boundary condition on vector-potential Ψ( ) due to Hirasaki and Hellums [12] are given by,0Xzyx=Ψ=Ψ=∂Ψ∂ at X = 0, 10Yxzy=Ψ=Ψ=∂ψ∂ at Y = 0, ARY0Zyxz=Ψ=Ψ=∂ψ∂ at Z = 0, ARZ (9)
  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) © IAEME262Boundary Conditions on Vorticity-Vector (Ω )(i) Vorticity vector (Ω ) at the walls for no-slip condition due to Aziz and Hellums [13] aregiven by,xΩ = 0; ;XWy∂∂−=ΩXVz∂∂=Ω at X = 0, 1;YWx∂∂=Ω ;0y =ΩYUz∂∂−=Ω at Y = 0, ARYZVx∂∂−=Ω ;ZUy∂∂=Ω 0x=Ω at Z = 0, ARZ (10)(ii) Vorticity vector Ω( ) at the walls when velocity slip is allowed are given by,;ZVYWx∂∂−∂∂=Ω ;XWy∂∂−=ΩXVz∂∂=Ω at X = 0,1;YWx∂∂=Ω ;XWZUy∂∂−∂∂=ΩYUz∂∂−=Ω at Y = 0, ARY;ZVx∂∂−=Ω ;ZUy∂∂=ΩYUXVz∂∂−∂∂=Ω at Z = 0, ARZ (11)Thermal boundary conditions on temperature (θ) are applied and the average Nusselt numberbased 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,∫ ∫=∂θ∂−=Y ZAR0AR00xZYhXARAR1Nu dY dZ (13)∫ ∫=∂θ∂−=Y ZAR0AR01xZYcXARAR1Nu dY dZ (14)RESULTS & DISCUSSIONWhen the non-linear inertial terms due to Forchheimer are included in the porous boxinclined at φ = -30o, the strength of convection increases as can be seen from Fig. 2. There issignificant change in isotherms particularly in the core region which gives rise to higher heattransfer. Variation of average Nusselt number with angle of inclination of porous box (φ) isshown in Figs. 3 and 4.In the above plots as the angle of inclination increases from –60oto +60o, the average Nusselt number first increases attains a peak (φ = -30o) and then decreases.For a given angle as Fc/Pr increases, average Nusselt number decreases. The change inaverage Nusselt number is more pronounced when the angle of inclination is negative (-60oto0o). For positive angle of inclination, the effect of Fc/Pr is negligible. As expected, theaverage Nusselt number (Nu) increases with increasing Ra but the effect of Fc/Pr is just thereverse due to increasing effect of inertial forces which increase with increase in Fc/Pr. In
  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) © IAEME2630.00 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. 4, as the angle of inclination (φ) increases from –60oto 60o, the average Nusselt number(Nu) values first increases and reaches peak at φ = -30oand then decreases. The physicalreason/mechanism can be explained on the basis of flow fields and temperature fields shownin Fig. 2 for φ = -30o. It can be seen from the flow and temperature field that the naturalconvection is more pronounced in case of φ = -30o. The velocity gradient and temperaturegradient near the hot and cold walls are steeper for φ = -30o. When φ > φc, the effect ofviscous forces becomes predominating and Nu values decrease. At very large φ, the effectbecomes significant and the mode of heat transfer becomes primarily conductive leading to aconstant Nu for all values of Fc/Pr [4].Fig. 2: Iso-vector-potential ( xΨ ) for Ra = 1000, ARY =ARZ =1.0, Fc/Pr = 10-2and φ = -30o-60 -40 -20 0 20 40 600246810121416182022NuφFc/Pr=10-2Fc/Pr=10-3Fc/Pr=10-5Fig.3: Variation of Nu with ϕ for Fig. 4 : Variation of Nu with ϕ forRa=2000, ARy=1.0 and ARz=1.0 Ra=1000, ARy=0.5 and ARz=1.0
  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) © IAEME2647.3 CONCLUSIONSNumerical solutions to the equations governing natural convection heat transfer in aninclined porous box have been obtained employing SAR scheme. The flow description iswithin the framework of Darcy-Forchheimer flow model. The flow and temperature fields forDarcy-Forchheimer flow description are similar to that of two-dimensional Darcy flowmodel. The strength of natural convection is lower than the Darcy flow model due to inertiaeffect. The average Nusselt number values are independent of horizontal aspect ratio (ARZ)i.e. the case of two-dimensional flow. For Fc/Pr < 10-5, the average Nusselt number valuesare close that of Darcy flow model. The critical angle of inclination is –30othat is same asDarcy-Brinkman flow modelREFERENCES[1] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress, pp.89-94, 1952.[2] Poulikakos and A. Bejan, The departure from Darcy flow in natural convection in avertical porous layer, Physics of Fluids, Vol. 28 pp.3477-3484, 1985.[3] T.W. Tong and E. Subramanian , A boundary layer analysis for naural convection inporous enclosure use of Brinkman extended Darcy flow model, International Journalof Heat and Mass Transfer, Vol. 28 pp.563-571, 1985.[4] V. Prasad and A. Tuntomo, Inertia effects on natural convection in a vertical porouscavity, Numerical Heat Transfer, Vol. 11 pp.295-320, 1987.[5] V.V.Satyamurthy and M.D. Rao, Relative effects of variable fluid properties andnon-Darcy flow on convection in porous media, ASME trans. HTD-96 ,pp.618-627,1988.[6] A Zebib and D.R. Kassoy, Three-dimensional natural convection motion in a confinedporous medium, Physics of Fluids, Vol.21 pp.1-3, 1978.[7] P.H. Holst, Transient three-dimensional natural convection in confined porous media,International Journal of Heat and Mass Transfer, pp. 73-90, 1972.[8] R.N. Horne, 3-D natural convection in a confined porous medium heated from below,Journal of Fluid Mechanics, Vol. 92, pp. 751-766, 1979.[9] Y.T. Chan and S. Bannerjee, Analysis of transient three-dimensional naturalconvection in a porous media, ASME Trans. Journal of Heat Transfer, Vol.103pp.242-248, 1981.[10] Amir S. Dawood and P.J. Burns, Steady three-dimensional convective heat transfer inporous box via multigrid, Numerical Heat Transfer, Part A, Vol. 22 pp.167-198,1992.[11] P.Vasseur, M.G. Satish and L. Robillard, Natural Convection in a thin, InclinedPorous Layer exposed to a constant Heat Flux , International Journal of Heat andMass Transfer , Vol. 3 pp.537-549, 1987.[12] G.J. Hirasaki and J.D. Hellums, A general formulation of the boundary conditions onvector potential in three-dimensional hydrodynamics. Q. Appl. Math., Vol. 16, pp.331-342, 1968.[13] K. Aziz and J.D. Hellums, Numerical solution of the three-dimensional equations ofmotion for laminar natural convection. Physics of Fluids, Vol. 10, pp. 314-324, 1967.
  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) © IAEME265[14] 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.[15] 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.

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