Strojniški vestnik - Journal of Mechanical Engineering 51(2005)7-8, 491-494 UDK-UDC 536.2 Izvirni znanstveni članek - Original scientific paper (1.01) Numerical Simulation on Mixed Convection in a Porous Medium Heated by a Vertical Cylinder Ling LI 1,2, Shigeo KIMURA 1,* 1 Institute of Nature and Environmental Technology, Kanazawa University,2-40-20,Kodatsuno, Kanazawa 920-8667, Japan, * e-mail: skimura@t.kanazawa-u.ac.jp 2 Department of Hydraulic Engineering, Tsinghua University, Beijing, China Abstract Numerical simulation has been performed for mixed convection of heated vertical cylinder with a constant temperature in a saturated porous medium subjected to lateral flow. Four different geometric aspect ratios 5,10,25 and 50 were considered, which give ratios of axial length to diameter. Based on dimensional analysis and nonlinear regression, correlations for the Nusselt number against the Rayleigh and the Peclet numbers have been obtained. It is shown that the average Nusselt number ( Nu ) is a function of the Rayleigh number ( Rad ) and the Peclet number (Pe):Nu=3.1lPefor Rad/Pe<1, and Nu (Rad -3.1 + 0.4 —-1 V Pe for Rad/Pe>1. As compared with Pe1/2 l Pe J Nu = 3.1 Pefor the case of horizontal forced flow around a vertical cylinder, the results of this investigation indicates that heat transfer can be remarkable enhanced by natural convection when forced convection is weak. INTRODUCTION Convective heat transfer and fluid flow around vertical cylinder in porous media has recently received considerable attention in geophysical and engineering applications. Such applications include geothermal systems, chemical catalytic reactor, packed sphere beds, grain storage and thermal insulation engineering. However, most existing fundamental studies have focused on natural convection, or forced convection, while a very few have been reported for mixed convection despite its equal importance in many situations. Mixed convection about a vertical cylinder subjected to horizontal flow has not been investigated extensively [1]. Cheng [2] provided an extensive review of the literature on natural convection heat transfer in fluid saturated porous media He also investigated mixed convection from a horizontal circular cylinder in a saturated porous medium with boundary layer approximation[3]. In connecting with the forced convection by lateral flow, the analysis was made by Kimura [4] to cylinders of elliptic cross sections with integral methods. In his paper, Kimura provided the average Nusselt number for the steady state by integral solution with the formulation of Nu = 3.1 Pe, where the characteristic length in Nu is taken as the circumferential length Kd. In addition, Romero [5] presented solutions for the temperature field due to Darcy flow past a slender body with a prescribed flux distribution embedded in a saturated porous medium. For the natural convection Sano et al. [6] studied the convective flows around a sphere embedded in a porous medium at small Rayleigh number and obtained asymptotic solutions for the transient and steady-state temperature distribution and flow pattern around the sphere. Campos et al. [7] studied the natural convection in the annulus of cylinder filled with porous medium, which has the heated inner wall and the cooled outer wall. However, pure natural or force convection rarely occurs in reality. In low Reynolds number flow conditions, heat transfer mechanisms of both forced convection and natural convection play a vital role and both mechanisms have to be properly accounted for. 491 Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 491-494 Nomenclature Roman symbols specific heat [J/(kg K)] d diameter of cylinder [m] dp diameter of the solid particle [m] g gravitational acceleration [m/s2] K permeability [m2] k thermal conductivity [W/(mK)] L length of cylinder [m] Nu Average Nusselt number p pressure [Pa] Pe Peclet number q" average heat flux [W/m2] ^fl Rayleigh number T d temperature [K] V velocity vector [m/s] u,v,w velocity components in [m/s] the x,y,z system of coordinates x,y,z Cartesian coordinates [m] Greek symbols ¦ ž Thermal diffusivity of porous [m2/s] medium (= km /(pcp)f) ¦ R coefficient of thermal expansion [K1] ^ porosity of porous medium ¦ D density [kg/m3] ¦L dynamic viscosity [Pa s] ¦E kinematic viscosity [m2/s] Subscripts and superscripts f fluid effective s solid wall 0 inlet Recently, numerical results for mixed convection around vertical cylinder by using two-dimensional numerical model have been reported [8,9]. In these research works, fluid flows along the axis of the cylinder are considered so that the problem is axisymmetric. As far as we know, there are very few papers on the problems of mixed convection of a vertical heated cylinder subjected to lateral flow in a saturated porous medium. The work by Ingham and Pop [10] is the only one that dealt with the similar problem. However, they assumed a partially heated infinite vertical cylinder in order to make the mathematical formulations easier. They also relied on the boundary layer approximations, which naturally limit the validity of their results in certain parametric ranges. In the present work, three-dimensional numerical calculation was performed to simulate the mixed convection in a porous medium heated by vertical cylinder. From a parametric study, average Nusselt numbers were obtained and effects of ratio of length to diameter of cylinder, temperature differences, permeability of porous medium and lateral flow velocity on them were investigated. Results obtained from the simulation for two extreme cases, namely pure natural and forced convection, were compared with those in the literatures in order to test our code. Governing equations and numerical methods The analysis is based on solving Darcy’s equations for three-dimensional flow around a vertical cylinder placed in porous medium. The vertical cylinder has a constant surface temperature (TW ). Fluid flows around the cylinder axis with a uniform velocity profile (V0 ) and a constant fluid temperature ( T0 ). It is assumed that the flow is steady, laminar, three-dimensional and incompressible. The radiation and thermal dissipation effect are neglected. In addition, permeability and thermal conductivity of porous medium are homogeneous and isotropic, and it is in local thermal equilibrium with the fluid. It is assumed that the thermophysical properties of the fluid are independent of temperature except for the density in the buoyancy term, that is, the Boussinesq approximation is invoked. The conservation equations for mass, momentum, energy in the fluid region are: V-V = 0 V =-----V(p + pg) jU (V-V)T = aV2T (1) (2) (3) where a is defined by a = km /(pcp)f ,and p and cp denote the density and specific heat of the fluid, respectively. The effective thermal conductivity of the porous medium km is given by km =ks(1-0) + kf0 (4) With jtand k frepresenting the thermal conductivity of solid s and liquid phase, respectively, and f representing the porosity of the porous medium. At the same time, the permeability K is given by K = 180(1 (5) 2 492 Li L. - Kimura S. Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 491-494 With dp representing the sphere’s diameter of porous medium In addition, some nondimensional numbers are defined as follows, Peclect number Pe = V0d / a ; Average Nusselt number Nu=q"nd/[(T -T )k ] ; Rayleigh number Ra^gfiKd(Tw -T0)/(av). Note m that the Nusselt number is defined by nd . The boundary conditions are summarized in Table .1. Table 1 The boundary conditions of numerical mode Boundary type Inflow Top (Bottom) Front (Back) Velocity u^U 0 du/dz = 0 du/dy = 0 Zero dv/dz^0 v = 0 gradient w = 0 dw/dy = 0 Outflow Cylinder Wall slip Tempe- T =T0 dT/dz = 0 dT/dy = 0 Zero T =T rature______________________________________gradient w All the above calculations were performed using the FLUENT code. The governing Eqs. (1)-(3) were discretized using a finite volume method and the SIMPLE algorithm was used to solve the equations. A numerical scheme with non-uniform unstructured grids generated by GAMBIT software was applied to the present physical system. In order to obtain grid independent results, different finest grid spaces adjacent to the wall of size 0.005m, 0.002m and 0.001m are tested. It is proved that the difference between the results predicted based on 0.002m and 0.001m, the finest grid adjacent to the wall, are insignificant (<1%). Thus the finest grid, of size 0.002m and located adjacent to the wall, is chosen to get more detailed situation of the flow field. Results and discussions Numerical results compared with analytical solutions. In order to confirm the validity of the numerical model, the numerical results were compared with analytical solution for forced convection and natural convection in a porous medium heated by a vertical cylinder, respectively. The average Nusselt numbers are in fairly good agreement with the data from References [4,11], as shown in Fig.1 and Fig.2. Effect of Rayleigh number Average Nusselt numbers for L/d=10 and eight different Rayleigh numbers are shown in Fig.3. Also shown on this graph is Nusselt number for the forced. It has shown that Nu is the function of Ra and Pe. Nu will increase with rising of Ra. When Pe is more than 8, Nu variations show a forced convection characteristic. However, just after the onset of the buoyancy drive secondary flow. This is a result of the buoyancy forces that become strong enough to destabilize the lateral flow. Moreover, an increase in the Nusselt number with the Rayleigh number is observed. 100 p 10 1 Numerical Simulation Nu=3.10xPe1/2[Kimura:1988] 1 10 Pe 100 Fig.1. Calculation compared with analytical solution for Forced convection. 5.5 5.0 4.5 4.0 3.5 3.0 2.5 Numerical Simulation Nucylinder/Nuplate=1+0.26^L [Minkowyz and Cheng:1976] 6 8 16 18 10 12 14 LL=4xL/d/Ra1/2 Fig.2. Calculation compared with analytical solution for natural convection( Ra is defined by the height). The region with the secondary flow is described as the mixed convection region. As can be seen from Fig.3, with increase in Ra the point where Nu merges with forced convection occurs is shifted towards large Pe. Therefore, for large Rayleigh number the Nusselt numbers depart from the forced convection values even at large Pe. Even at the smallest Rayleigh number of 0.0119 the secondary flow effects are sufficiently large so that the variation of the Nusselt number is far away from that for forced convection predicted with the formulation of Nu = 3.1yPe. For the largest Rayleigh number the secondary flow effects are largest. Consequently, the larger the Rayleigh number, the larger the mixed convection dominated region, and the larger heat transfer enhancement results. Correlation formulas for mixed convection Heat transfer results for mixed convection can be best presented in terms of two governing parameters, Nu/Pe1/2 and Ra/Pe(Fig.4). As observed in all cases studied, the good collapse of the data points suggests that Numerical simulation on mixed convection in a porous medium heated by a vertical cylinder 493 Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 491-494 the transition of heat transfer processes occurring around vertical cylinder are fairly gradual. 10 1 L/d=0.2m/0.02m=10 Rad=0.0119 Rad=0.0358 Rad=0.119 Rad=0.298 Rad=0.894 Rad=2.98 Rad=8.05 Rad=26.8 forced convection 0.01 0.1 1 10 100 1000 Pe Fig.3. Average Nusselt number with Pe. (L/d=10) The ratio Rad / Pe can be used to indicate the relative strengths of the two modes of convection in a mixed convection environment. It has shown the buoyancy effects become noticeable when Rad / Pe approaches unity. For Rad/Pe <<1, the forced convection component controls the heat transfer processes, while Rad /Pe>>1, buoyancy effects predominate. For forced convection, the correlation of heat transfer results based on numerical data is given by Nu = 3.1 Pe , which is in very good agreement with the results of Ref [4]. For mixed convection, a least-squares fit of the data provided the following correlation: Nu -----= 3 1 1/2 . Pe 0.4 d-1 Pe (6) That is also indicated as a solid line in Fig.4. When the Rayleigh number is increased to large values, one would expect that the effect of forced convection would become negligible in comparison with the effect of natural convection, and consequently the Nusselt number should become nearly independent of the Peclet number. Accordingly, the power of correlation has been used as 1/2, which is consistent with the natural convection results. Eq.(6) reflects correctly the above nature of mixed convection heat transfer. 100 1 0.1 aa a a* * * •«* a****a«ta*>#* #*«*••»* L/d=10 L/d=25 L/d=50 1E-4 1E-3 0.01 0.1 1 10 100 1,000 Rad/Pe Fig.4. Heat transfer rate in the mixed convection Conclusions Mixed convection from a heated vertical cylinder in a saturated porous medium subjected to lateral flow has been examined numerically. It has been shown that the average heat transfer rate from a vertical cylinder is described by the following equations: and Nu=3.1 Pe for Rad /Pe<1 Nu Pe1/ 2 3.1 + 0.4 Ra V Pe for Rad / Pe>1 References [I] D.A. Nield and A. Bejan, Convection in porous media, 2nd edition, Springer, 1999. [2] P. Cheng, Heat transfer in geothermal systems, Advances in Heat Transfer, Vol.14, Academic Press, New York, 1978, pp.1-105. [3] P. Cheng, Mixed convection about a horizontal cylinder and a sphere in a fluid-saturated porous medium, Int. J. Heat Mass Transfer 25 (1982) 1245-1246. [4] S. Kimura, Forced convection heat transfer about an elliptic cylinder in a saturated porous medium, Int. J. Heat Mass Transfer 31 (1988) 197-199. [5] L. A. Romero, Forced convection past a slender body in a saturated porous medium, SIAM Journal of Applied Mathematics 55 (4) (1995) 975-985. [6] T. Sano and R. Okihara, Natural convection around a sphere immersed in a porous medium at small Rayleigh numbers, Fluid Dynamics Research 13, (1994) 39-44. [7] H. Campos, J.C. Morales and V. Lacia, Thermal aspect of a vertical annular enclosure divided into fluid region and a porous region, Int. Commun. Heat Mass transfer 17 (1990) 343-353. [8] T.K.Aldoss, M.A.Jarrah and B.J.Al-shaer, Mixed convection from a vertical cylinder embedded in porous medium: non-Darcy model, Int.J.Heat Mass Transfer 39 (6) (1996) 1141-1148. [9] K.A.Yih, Coupled heat and mass transfer in mixed convection about a vertical cylinder in a porous medium: The entire regime, Mechanics Research Communications 25 (6) (1998) 623-630. [10] D.B. Ingham and I. Pop, A horizontal flow past a partially heated infinite vertical cylinder embedded in porous medium, Int. J. Engng. Sci. 24 (8) (1986) 1351-1363. [II] W.J.Minkowycz and P.Cheng, Free convection about a vertical cylinder embedded in a porous medium, Int. J. Heat Transfer 19 (1976) 805-813. 494 Li L. - Kimura S.