Viscosity Variation Effects on Heat Transfer and Fluid Flow Through Two-Layered Porous Media

Journal of ChemicalBahadori Technology Fatemeh, and Sima Metallurgy, Rezvantalab 50, 1, 2015, 35-38

VISCOSITY VARIATION EFFECTS ON HEAT TRANSFER AND FLUID FLOW THROUGH TWO-LAYERED POROUS MEDIA

Bahadori Fatemeh, Sima Rezvantalab

Chemical Engineering Department, Received 10 June 2014 Urmia University of Technology, Urmia, Iran Accepted 24 November 2014 E-mail: [email protected]

ABSTRACT

Temperature dependent viscosity effects on natural convection in a composite cavity, isothermally heated from below and cooled from the opposing surface, are analyzed. The two-layered porous media was composed of two regions with different permeability. It is observed that viscosity reduction, due to temperature rising, affects heat transfer rate and fluid motion at the interface. Keywords: two layered porous media, viscosity, heat transfer, permeability ratio (Kr), streamline.

INTRODUCTION tion for calculation of the overall Nusselt number for N horizontal sub layers. Lai and Kulacki [13] discussed Heat transfer in porous media has been extensively convection in a vertically divided rectangular cavity, of studied due to its numerous engineering and environ- the right hand side wall which was heated with constant mental applications. Oil recovery, underground spread heat flux. of pollutants, grain storage, geothermal systems, solar Egorov and Polezhaev [14] investigated a theoretical power collectors, optimal design of furnaces, compact model and showed that their numerical results are in the heat exchangers, packed-bed catalytic reactors and pas- line of their experimental data for multilayer insulation. sive thermal control devices are examples of applications In addition, Merrikh and Mohamad [15], Belghazi et al. of heat transfer in porous media. [16] and Ordo´n˜ez-Miranda and Alvarado-Gil [17] and Studies concerning natural convection in porous others, have investigated multilayered porous media. media can be found in the monographs of Ingham It can be seen that composite cavity under natural and Pop [1], and Nield and Bejan [2]. The case of free convection is evaluated extensively in literature, how- convection in a rectangular porous medium, heated on ever, the viscosity variation due to temperature rising is one side and cooled at the opposing side, is also an im- considered negligible. portant problem in thermal convection in porous media. The aim of this paper is the investigation of viscosity The work of Walker and Homsy [3], Bejan [4], Prasad variation effects on the natural convection in a saturated and Kulacki [5], Manole and Lage [6], Saeid and Pop enclosure, divided into two different porous layers, with [7] and others have further contributed some classical different permeabilities and subjected to a heating from results to this problem. one side and cooling from the other side. The presence of a free-flow domain adjacent to a Governing equations porous media, has been studied in literature by Cieszko, An incompressible, 2-D, natural convection in a Kubik [8], Mharzi et al. [9], Edimilson and de Lemos two-layered cavity, filled with a saturated, isotropic Marcelo [10] and Baytas et al. [11], and others. Fur- porous medium with temperature dependent viscosity thermore, multilayered cavity has been investigated. and density variation in the buoyancy term, was studied Poulikakos and Bejan [12] studied horizontally and numerically. It was assumed that the solid matrix and vertically layered cavity and approximated a correla- the fluid are in local thermal equilibrium. As suggested 35 Journal of Chemical Technology and Metallurgy, 50, 1, 2015 by Nield and Bejan [2], the equations that govern the conservation of mass, momentum, and energy can be written as follows: ∇=u 0 (1)

ρ f  ∂∂∂uuP µ  uv +=−  u (2) ε 2  ∂∂∂x y xK ρ  ∂∂∂vvP µ f +=−+ρβ − (3) 2 u v [v g() TT∞ ] ε  ∂∂∂x y xK a) b)

∂∂TT 2 uv+=∇α T (4) ∂∂xy The average Nusselt number is defined by: 2L Nu= Nu dx ∫ x (5) c) 0 It was assumed that viscosity is varying with tem- Fig. 2. Streamlines for horizontally divided porous media, Ra = 104, Da = 0.001, Kr = 0.001:(a) constant viscosity, perature in the following models: 1 µ = C (6a) (b) linear variation of viscosity with temperature and (c) exponential variation of viscosity with temperature.

µλ=−−T00() TT (6b)

µγ= TT0 exp( ) (6c)

Numerical procedure and grid independence The model equations were solved using the commer- cial flow simulation software FLUENT (version 6.3.26). Fig. 1 shows the geometry of the simulated media. As a) b) it can be seen, the rectangular enclosures with different permeability are separated at Y=L/2. The results for

Ram=1000 are in line with those in the literature.

c) Fig. 3. Streamlines for horizontally divided porous media, 4 Ra1 = 10 , Da = 0.001, Kr = 1000; (a) constant viscosity, (b) linear variation of viscosity with temperature and (c) exponential variation of viscosity with temperature.

RESULTS AND DISCUSSION

The results of simulation are presented in the next sections. Fig. 1. Scheme of the layered porous media and the coor- Effect of viscosity variation dinate system. Fig. 2 and Fig. 3 illustrate the streamlines for 36 Bahadori Fatemeh, Sima Rezvantalab

a) b)

Fig. 4. Variation of cavity average Nusselt number for a horizontally divided enclosure (a) Da = 0.01, Kr = 0.1 and (b) Da = 0.01, Kr = 10. a) b)

Fig. 5. Variation of v-velocity at the mid-height of the horizontally divided enclosure for (a) Da = 0.01, Kr = 0.1 and (b) Da = 0.01, Kr = 10.

4 Ra1=10 , Da=0.001, Kr= 1000 for a horizontally divided Velocity profile porous media. As it can be seen, viscosity reduction due Fig. 5 illustrates the velocity profile in the horizon- to temperature raising causes increase of the flow pen- tally divided porous media. As it can be seen, decreas- etration rate from one layer to another. In addition, the ing of the velocity due to temperature rising, causes an flow penetration rate for the case, which is modeled by increase of fluid movement. In addition, comparison of considering exponential variation of viscosity, is higher Fig. 5a and Fig. 5b shows that the velocity magnitude than the case with linear variation of viscosity. in two layered porous media with Kr < 1 is higher than Heat transfer Kr > 1. Fig. 4 shows Nu numbers obtained by assuming: of (a) exponential variation of viscosity with temperature, CONCLUSIONS (b) linear variation of viscosity with temperature and (c) constant viscosity. Fig. 4 shows that the mean Nu number The viscosity variation effect on natural convection obtained when considering exponential variation with in a horizontally layered porous media was investigated temperature (Nuexp) is greater than the Nu number ob- numerically using the finite volume method. Results tained with linear variation of viscosity with temperature were presented for flow field and heat transfer of enclo-

(Nulin). As it is seen, Nucon is in line with the findings of sures subjected to heated from one side, and cooled from Poulikakos and Bejan [12]. the opposite wall, while the other walls were isolated. 37 Journal of Chemical Technology and Metallurgy, 50, 1, 2015

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