Numerical study of natural convection in porous media (metals) using Lattice Boltzmann Method (LBM)

International Journal of Heat and Fluid Flow 31 (2010) 925–934

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Numerical study of natural convection in porous media (metals) using Lattice Boltzmann Method (LBM) C.Y. Zhao a,b,*, L.N. Dai b, G.H. Tang b, Z.G. Qu b, Z.Y. Li b a b

School of Engineering, University of Warwick, Coventry CV4 7AL, UK School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

a r t i c l e

i n f o

Article history: Received 4 June 2009 Received in revised form 1 June 2010 Accepted 3 June 2010 Available online 2 July 2010 Keywords: Lattice Boltzmann thermal model Doubled populations BGK Natural convection Porous media Porosity Pore density

a b s t r a c t A thermal lattice BGK model with doubled populations is proposed to simulate the two-dimensional natural convection flow in porous media (porous metals). The accuracy of this method is validated by the benchmark solutions. The detailed flow and heat transfer at the pore level are revealed. The effects of pore density (cell size) and porosity on the natural convection are examined. Also the effect of porous media configuration (shape) on natural convection is investigated. The results showed that the overall heat transfer will be enhanced by lowering the porosity and cell size. The square porous medium can have a higher heat transfer performance than spheres due to the strong flow mixing and more surface area. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Porous metals, as a new type of materials, possess a number of advantages, such as low density, high porosity and large surface area, and they have been widely used in many applications including traffic, microelectronics, oil extraction, aerospace industry, organism, medical treatment and buildings. Zhao et al. (2005) presents a combined experimental and numerical study on natural convection in open-celled metal-foams. In that study the Representative Effective Volume (REV) method is used in the numerical study, and fairly good agreement between the model predictions and experimental measurements is obtained. An analytical study of the forced convection heat transfer characteristics in the high porosity open-cell metal-foam filled tubular heat exchangers has been carried out by Lu et al. (2006) and Zhao et al. (2006). The results showed that the use of metal-foam can dramatically enhance the heat transfer. The flow boiling heat transfer and thermal radiation in metal-foams have been carried out by Zhao et al. (2004a,b, 2008, 2009). Lu et al. (1998) developed an analytical model for metal-foams with simple cubic unit cells consisting of heated slender cylinders. Due to the complex structure of porous metals, it is unlikely to capture the flow and heat transfer at the pore level by using the REV method.

* Corresponding author at: School of Engineering, University of Warwick, Coventry CV4 7AL, UK. E-mail address: [email protected] (C.Y. Zhao). 0142-727X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2010.06.001

Recently Lattice Boltzmann Method (LBM) has been developed as a new tool for simulating the fluid flow, heat transfer and other complicated physical phenomena. Compared with the traditional computational fluid dynamics methods, the Lattice Boltzmann Method is a micro and meso scale modeling method based on the particle kinematics. It has many advantages, such as simple coding, easy implementation of boundary conditions and fully parallelism. At present the applications of LBM have achieved great success in multiphase flow, chemical reaction flow, flow in porous medium, thermal hydrodynamics, suspension particle flow and magneto hydrodynamics. Tang et al. (2005) simulates the gaseous slip flow at the pore scale in micro scale porous geometries using LBM. Guo and Zhao (2002) and Peng et al. (2003) conducted the simulation for incompressible flow in porous media by using LBM. D’Orazio et al. (2004) and Shu et al. (2002) performed the numerical calculations for the natural convection in a cavity. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium. The natural convection in porous media has been extensively investigated by Hossain et al. (1999), Kramer et al. (2007), Cheng (2009), Kumari and Nath (2008), Varol et al. (2009), Abbas et al. (2009). However the fluid flow and heat transfer in the pore level are not carefully examined and their effects on the bulk heat transfer need to be further investigated. In this paper the porous metals are idealized as porous regular structures. The natural convection among the pores is investigated by using the LBM, and the pore-level fluid flow and heat transfer are revealed.

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Nomenclature a e ~f i fie g e gi g ei H L Nu Pr q r R

length of the square metal blocks thermal energy density distribution function equilibrium distribution function gravitational acceleration, m/s2 distribution function equilibrium distribution function cavity height, m the height and width of each metal block Nusselt number Prandtl number = m/a heat flux radius of circular metal blocks the total contact surface area ratio of the square and circular metal blocks Rayleigh number = gbDTH3/(m/a) temperature, K average temperature, K x-component velocity, m/s

Ra T T u

v

y-component velocity, m/s

Greek symbols thermal diffusivity, m2/s b thermal expansion coefficient, K1 e porosity q average density, kg/m3 sf relaxation time sg relaxation time t kinematic viscosity, m2/s

a

Subscripts E East N North S South W West

viscous heating. This new distribution functions obey thus a set of lattice BGK equations in the form (add the buoyancy effect):

2. Mathematical formulations and numerical methods 2.1. Physical problem

~f ðx þ c dt; t þ dtÞ  ~f ðx; tÞ ¼  i i i

Lu et al. (1998) developed an analytical heat transfer model by treating metal-foams as simple cubic unit cells consisting of heated slender cylinders. In an attempt to reveal the interfacial heat transfer among the high porosity metal-foam structures, the regular square (Fig. 1) or circular metal blocks with different porosity (e) and pore density (ppi, Pore number Per Inch) are assumed in the present paper. With the Boussinesq approximation, all the fluid properties are considered as constant, except in the body force term in the Navier–Stokes equations, where the fluid density is assumed  ½1  bðT  TÞ; b is the thermal expansion coefficient, q ; T q¼q are the average fluid density and temperature and g is the gravitational acceleration vector. The major control parameter is the Rayleigh number Ra = bgDTH3Pr/m2, where DT is the temperature difference between the hot (the left wall) and cold wall (the right wall); Pr = m/a is the Prandtl number, an indicator of the momentum to heat diffusivity ratio, and H is the height or width of the cavity. To simulate the real heat transfer mechanism of metalfoams, the temperature of the metal blocks is assumed the linear temperature distribution from the hot temperature (the left wall) to the cold temperature (the right wall).

g~i ðx þ ci dt; t þ dtÞ  g~i ðx; tÞ ¼ 

~f ¼ f þ 0:5dt ðf  f e Þ i i i i

sf

g~i ¼ g i þ

0:5dt

sg

ðg i  g ei Þ þ

dt Z i fi 2

ð~f  f e Þ

dt

ð3Þ

ðg~  g ei Þ

sg þ 0:5dt i dt sg Zf  sg þ 0:5dt i i

ð4Þ

where sf, sg are the relaxation times and fie , g ei are the equilibrium distribution functions. G1 ¼ bgðT  TÞ represents the buoyancy effect. In the sequel, the reader is referred to the two-dimensional square lattice with the nine speeds, which is sufficient to guarantee the recovering of the Navier–Stokes equations after a Chapman– Enskog expansion:

   i1 ci ¼ cos p ; 4

sin

  i1 p c; 4

i ¼ 1  8;

~ c0 ¼ 0

ð5Þ

where c2 = 3RT and T is the temperature. The equilibrium density distributions are chosen in the form of a quadratic expansion of a Maxwellian as follows:

(

2.2. The thermal LBE model The double-populations LBE model proposed by D’Orazio et al. (2004) is employed in this study. The main principle is to view the thermal flow as a mixture of material particles and thermal excitations described by two separate distribution functions f and g, respectively. In order to avoid the implicitness of the scheme, two new discrete distribution functions ~f i and g~i were proposed as

dt

sf þ 0:5dt i i dt sf 3G1 ðcix  uÞ e fi þ c2 sf þ 0:5dt

fie

¼ wi

"

#) 3ci  u 9ðci  uÞ2 3ðu2 þ v 2 Þ q þ q0 þ  c2 2c2 2c4

ð6Þ

3qe ðu2 þ v 2 Þ 2 " c2 # 1:5ci  u 4:5ðci  uÞ2 1:5ðu2 þ v 2 Þ ¼ w1 qe 1:5 þ þ  c2 c2 c4 " # 6ci  u 4:5ðci  uÞ2 1:5ðu2 þ v 2 Þ ¼ w2 qe 3 þ þ  c2 c2 c4

g e0 ¼ w0

ð7Þ

g e1:2:3:4

ð8Þ

ð1Þ

g e5:6:7:8

ð2Þ

In the above equations, ~ u  ðu; v Þ, qe = qRT (in 2D), the weights of the different populations are

where fi and gi are the discrete populations which evolve when a standard first order integration strategy is adopted. Z i ¼ ½ci  uðx; tÞ  ½uðx þ ci dt; t þ dtÞ  uðx; tÞ=dt represents the effects of

w0 ¼ 4=9;

w1 ¼ 1=9;

i ¼ 1; 2; 3; 4;

w2 ¼ 1=36;

ð9Þ

i ¼ 5; 6; 7; 8 ð10Þ

C.Y. Zhao et al. / International Journal of Heat and Fluid Flow 31 (2010) 925–934

(a) ppi=2, ε=0.8

(b) ppi=5, ε=0.8

(c) ppi=2, ε=0.9

(d) ppi=5, ε=0.9

(e) ppi=2, ε=0.95

(f) ppi=5, ε=0.95

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Fig. 1. A square cavity filled with porous square metal blocks: (a) ppi = 2, e = 0.8; (b) ppi = 5, e = 0.8; (c) ppi = 2, e = 0.9; (d) ppi = 5, e = 0.9 (e) ppi = 2, e = 0.95; (f) ppi = 5, e = 0.95.

Finally, the hydrodynamic variables, density and momentum can be calculated as Eq. (11), and thermal energy density and corresponding heat flux can be calculated as Eqs. (12) and (13).

X

~f ¼ q; i

i

qe ¼

~f c ¼ qu; i ix

i

X

g~i 

i

q¼

X

X i

X i

~f c ¼ qv  dt F i iy b 2

dt X Z i fi 2 i

dt X ci g~i  qeu  ci Z i f i 2 i

where Fb = qG1.

t ¼ sf RT; a ¼ 2sg RT

sg sg þ 0:5dt

ð14Þ

ð11Þ ð12Þ

!

The kinematic viscosity and the thermal diffusivity are given by:

ð13Þ

2.3. Boundary conditions With regard to the velocity field, the non-slip boundary conditions are applied to the four walls of the cavity and metal blocks. These are obtained by means of the non-equilibrium bounce back rule (Zou and He, 1997), which can be described as follows. If the East wall of metal block is considered, the unknown particle distri-

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bution functions are ~f 1 , ~f 5 , ~f 8 . The wall speed (U, V) = (0, 0) has been imposed by means of the constraints:

X

X

~f ¼ q ; i E

i

~f c ¼ q u; i ix E

i

X i

~f c ¼ q v  dt F i iy b E 2

ð15Þ

plus a ‘‘bounce-back” condition on the non-equilibrium part of the particle distribution perpendicular to the boundary:

~f 1  f e ¼ ~f 3  f e 1 3

~f 1  f e ¼ ~f 3  f e 1 3 ~f 4  f e ¼ ~f 2  f e 4 2

where eS denotes the thermal energy density current value at the South wall of the cavity. The unknown populations g~2 , g~5 , g~6 , as

g~i ¼ qðeS þ e0 Þ ½corresponding form for equilibrium Become

qeE Vc þ dt2

P

i c ix Z i fi

1 3

þ ðg~4 þ g~7 þ g~8 Þ

þ 12 Vc þ 12

V2 c2

ð32Þ

2.4. Code validation: natural convection in a square cavity

ð19Þ

ð21Þ

In order to validate the thermal LBE model, the natural convection in a square cavity is simulated and compared with the benchmark solutions. Results for the Rayleigh number ranging from 103 to 106, and the Prandtl number Pr = 0.71 are reported. The average temperature used in particle equilibrium is T ¼ ðT h þ T c Þ=2; the relaxation times are chosen sf = 0.1 and sg = 0.0704, respectively. The mesh of 201  201 (in X–Y) is used and the grid independent solution is obtained. The average Nusselt number is calculated as

g~i ¼ qðeE þ e0 Þ ½corresponding form for equilibrium

ð23Þ

þ

1 U 2 c

ð24Þ Nu ¼  ð25Þ ð26Þ

ð27Þ

ð28Þ

i Z i fi

þ

K

1 U2 2 c2

RH   dy k 0 @T q @x x¼0 ¼ H DT DT

ð33bÞ

The average Nusselt number can be obtained as

which yields

P

ð33aÞ

where h is the average heat transfer coefficient, k is the thermal conductivity of the fluid.

ð29Þ

where K is the sum of the six known populations coming from nearest wall and fluid nodes, eE denotes the imposed thermal energy density at the East wall. For the insulated walls, the constraint on the heat flux is obtained by imposing qx = 0 in Eq. (13), so that

1 DT

Z

H 0

 @T  dy @x x¼0

ð33cÞ

The predicted streamlines and temperature profile are shown in Figs. 2 and 3, respectively. Table 1 reports the comparison of average Nusselt number with the benchmark solutions (De Vahl Davis, 1983) at different Rayleigh numbers. The results show that a very good agreement has been obtained. 3. Results and discussions Natural convection in a square cavity with square metal porous media is first simulated at different Rayleigh numbers, different porosities (0.8, 0.9 and 0.95) and different pore densities (2 ppi and 5 ppi). To facilitate the description, the metal blocks’ South, East, North and West walls are denoted by 1–4, correspondingly, and the height and width of each metal block are equal to L. Since heat is transferred through the side wall and the four surfaces of all metal blocks, therefore, similar to the definition in Eq. (33), the average Nusselt number can be written as:

Nu ¼

qeE þ dt2

hH k

Since h ¼

By definition:

dt X Z i fi 2 i

Nu ¼

ð22Þ

where qSE can be taken equal to the density value of the nearest neighbor node. With regard to the thermal field, a thermal counter-slip approach is used here (D’Orazio et al., 2004). The incoming unknown thermal populations are assumed to be equilibrium distribution functions with a counter-slip thermal energy density e0 , which is determined so that suitable constraints are verified. If the East wall of metal block is considered, where T = TE, the unknown g~1 , g~5 , g~8 are chosen

1 3

ð31Þ

ð20Þ

~f 1 ¼ ~f 3 þ 2 q U 3 SE c ~f 4 ¼ ~f 2  2 q V 3 SE c ~ ~f 5 ¼ qSE  f 0  ð~f 2 þ ~f 3 þ ~f 6 Þ  1 q U þ 1 q V 3 SE c 2 SE c 2 ~ ~f 7 ¼ qSE  f 0  ð~f 2 þ ~f 3 þ ~f 6 Þ  1 q U þ 1 q V þ dt F b 2 SE c 3 SE c 4 2 1 U 1 V dt ~f ¼ ~f þ q  q  Fb 8 6 6 SE c 6 SE c 4

qðeE þ e0 Þ ¼

ð30Þ

ð18Þ

The five unknown distributions at the corner of metal block are thus found as:

i

dt X cix Z i fi 2 i

ð17Þ

The constrains for a corner node (the South-East corner of metal block as an example) are the same in Eq. (15) plus the two nonequilibrium bounce back conditions on the particle distributions perpendicular to the boundaries

g~i 

i

qðeE þ e0 Þ ¼

~f 1 ¼ ~f 3 þ 2 q U 3 Ec ~f 4  ~f 2 1 U 1 V dt ~f 5 ¼ ~f 7 þ þ qE þ qE  F b 6 c 2 c 4 2 ~f  ~f 1 U 1 V dt 2 ~f ¼ ~f  4 þ qE  qE  F b 8 6 6 c 2 c 4 2

X

cix g~i ¼ qeS U þ

ð16Þ

It follows that

qeE ¼

X

  Z 1 @T  H L 1 @T  H dy þ  dx þ L 0 L DT @x x¼0 DT @y 1 0   Z L Z 1 @T  H L 1 @T  H   dy þ  dx þ L 0 L DT @x 2 DT @y 3 0  Z L  1 @T   dy  DT @x 4 0

Z

H



ð34Þ

The average Nusselt number in the above Eq. (34) includes two parts: one part is the heat through the hot side wall and the other part is through the surfaces of all metal blocks.

C.Y. Zhao et al. / International Journal of Heat and Fluid Flow 31 (2010) 925–934

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Fig. 2. Predicted streamlines for different Rayleigh numbers: (a) Ra = 103; (b) Ra = 104; (c) Ra = 105; (d) Ra = 106.

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Fig. 3. Predicted temperature profile for different Rayleigh numbers: (a) Ra = 103; (b) Ra = 104; (c) Ra = 105; (d) Ra = 106.

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C.Y. Zhao et al. / International Journal of Heat and Fluid Flow 31 (2010) 925–934

3.3. Effect of porosity, e

Table 1 Comparison of average Nusselt number with previous works. Nu

Ra = 103

Ra = 104

Ra = 105

Ra = 106

This work G. Barakos et al. (1994) De G et al. (1983) He X et al. (1998)

1.115 1.114 1.118 1.117

2.242 2.245 2.243 2.235

4.503 4.510 4.519 4.504

8.698 8.806 8.880 8.767

3.1. Streamlines and isotherms Figs. 4 and 5 show the flow and temperature fields for two different cell sizes ppi = 2 and 5 for a fixed porosity. Under the buoyancy and gravity effect, the global vortex is produced in the center of the cavity. The streamlines go around the metal blocks. As the Rayleigh number increases, the fluid flow intensifies, and this tends to flatten the streamlines. Due to the presence of the metal blocks, the deformation of the streamlines near cavity walls is small and the two vortexes observed in Fig. 2 disappear for this case. For Ra = 103, the temperature is linear along the x-axis, and the isotherms are almost vertical. Heat transfer in the cavity is mainly through the heat conduction. With the increase of the Rayleigh number, the isotherms become more and more distorted, and the main heat transfer mode in the cavity changes from the heat conduction to convection dominated mechanism. It can be seen from Fig. 5 that both the streamlines and isotherms exhibit the periodic distributions, for each metal block, the local heat transfer at the top surface is quite different from that at the bottom surface due to the buoyancy effect of natural convection. Also it should be noted that the local heat transfer at the bottom surface is affected by the flow field that originates from the lower blocks, but the interaction between the lower and upper blocks is not very strong due to the relatively large distance between them for the case of 5 ppi and 90% porosity. It is expected that the local heat transfer interaction between the lower and upper blocks will become stronger for the case with a bigger pore density (ppi) and lower porosity.

3.2. Effect of Rayleigh number (Ra) and pore density, ppi To examine the effect of Rayleigh number and pore density on flow and heat transfer, the numerical simulations of natural convection in a metal block filled square cavity are conducted for two pore densities ppi = 2 and ppi = 5 for a fixed porosity e = 0.9. Fig. 6 shows the velocity-v profile at y/H = 0.4 of the cavity (a) and the velocity-u distribution at x/H = 0.4 (b) at different Rayleigh numbers for a fixed porosity (0.9) and pore density (5 ppi). As expected, the velocity at Ra = 104 is higher than that at Ra = 103 and it exhibits quite different flow behavior due to the stronger buoyancy effect for a higher Rayleigh number. Fig. 7 presents the effect of pore density on the velocity-v distribution at y/H = 0.4 of the cavity (a) and velocity-u distribution at x/ H = 0.4 of the cavity (b) for a given porosity and Rayleigh number. The velocity at 2 ppi is much larger than the velocity at 5 ppi, since the increase of the number of metal blocks (5 ppi) leads to the higher viscosity resistance to the fluid flow. Due to the viscosity, the fluid velocity near metal blocks is lower. The comparison of average Nusselt number with and without metal blocks is shown in Fig. 8 for different Rayleigh numbers. The average Nusselt number in the metal block filled square cavity is much larger than that of empty cavity. Since the metal blocks are assumed a linear temperature distribution (from the hot temperature to the cold temperature) in this study, the metal blocks can enhance the heat transfer. With the increase of pore density (more numbers of metal blocks) the total contact surface area with the fluid will be higher, and thereby leading to a higher Nusselt number.

The effect of porosity (e) on natural convection is presented in Fig. 9 for a fixed Rayleigh number and pore density. The predicted results show that the heat transfer is higher for the case with the lower porosity. For a given pore density (5 ppi), a lower porosity leads to a higher contact surface area, and thereby intensifying the heat transfer process. From Fig. 9, it can be seen that the Nusselt number increases if the porosity is reduced from 0.95 to 0.9 (only 0.05 difference), but if the porosity is further reduced by 0.1 to 0.8 from 0.9 (porosity change is doubled), the increase of Nusselt number is roughly only half of the increase for the porosity change from 0.95 to 0.9. Therefore, it can be expected that the Nusselt number will approach to a maximum value as the porosity continuously decreases, since the distance among the square blocks decreases significantly as the porosity decreases, this will lead to a considerable increase in the fluid flow resistance. 3.4. Effect of the porous media shape/configuration In order to examine the porous media structure/configuration effect on the flow and heat transfer at the pore level, the square metal blocks shown in Fig. 1 are replaced by the metal spheres. The Lattice Boltzmann Model is the same as the model in 2.2, but the boundary conditions are different. Fig. 10 gives the boundary distribution mark for the metal spheres. Table 2 shows the unknown particle distribution functions, and the numerical procedures are as same as before. The effect of porous media shape on the heat transfer is presented in Fig. 11 for a given porosity (90%) and pore density (2 ppi). The Nusselt number for the square metal blocks is much higher than that of the circular one due to the larger total contact surface area and strong flow mixing. It should be noticed that the size of the blocks decreases as the porosity increases, the effect of the shape on the local fluid flow and heat transfer coefficient tends to vanish. However, the ratio of the total contact surface areas between the circular and square metal blocks never change, the derivation of surface area ratio is given below: The pore density and porosity of the metal blocks are assumed as N ppi (pore number per inch), and e, respectively. The length of the square metal blocks is a, while the radius of the circular metal blocks is r. From the definitions of ppi and porosity, the following equation can be obtained as: The surface area occupied by metal blocks in one inch square area is

N2 a2 ¼ 0:02542  0:02542 e for square metal blocks N

2

2

2

2

pr ¼ 0:0254  0:0254 e for circular metal blocks

ð35Þ ð36Þ

From Eq. (35), the metal block length a can be obtained as

a¼

0:254 pffiffiffiffiffiffiffiffiffiffiffi 1e N

ð37Þ

From Eq. (36), the circular metal block radius can be obtained as

r¼

0:254 N

rffiffiffiffiffiffiffiffiffiffiffi 1e

ð38Þ

p

Then the total contact surface area of square metal blocks for heat transfer in one inch square is

pffiffiffiffiffiffiffiffiffiffiffi As ¼ N2 4a ¼ 4  0:254N 1  e

ð39Þ

The total contact surface area of circular metal blocks for heat transfer in one inch square is

Ac ¼ N2 2pr ¼ 2  0:254N

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  eÞ

ð40Þ

Thus, the total contact surface area ratio of the square and circular metal blocks is

R¼

Ac 1 pffiffiffiffi ¼ p ¼ 0:887 As 2

ð41Þ

C.Y. Zhao et al. / International Journal of Heat and Fluid Flow 31 (2010) 925–934

931

a

(1) Ra=103

(2) Ra=104

(3) Ra=105

(4) Ra=106

(1) Ra=103

(2) Ra=104

(3) Ra=105

(4) Ra=106

b

Fig. 4. Predicted results for the case of ppi = 2, e = 0.9 (square metal blocks). (a) Streamlines: (1) Ra = 103; (2) Ra = 104; (3) Ra = 105; (4) Ra = 106. (b) Isotherms: (1) Ra = 103; (2) Ra = 104; (3) Ra = 105; (4) Ra = 106.

For a given pore density N ppi, when the porosity increases, the size of the blocks decreases, the effect of the shape on the local fluid flow and heat transfer coefficient tends to vanish. However the me-

tal blocks act as the heat source, the ratio of the overall heat transfer performance should be almost the same as the ratio (0.887) of the contact surface area between the circular and square metal blocks.

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C.Y. Zhao et al. / International Journal of Heat and Fluid Flow 31 (2010) 925–934

a

(1) Ra=103

(2) Ra=104

(3) Ra=105

(4) Ra=106

(1) Ra=103

(2) Ra=104

(3) Ra=105

(4) Ra=106

b

Fig. 5. Predicted results for the case of ppi = 5, e = 0.9 (square metal blocks). (a) Streamlines: (1) Ra = 103; (2) Ra = 104; (3) Ra = 105; (4) Ra = 106. (b) Isotherms: (1) Ra = 103; (2) Ra = 104; (3) Ra = 105; (4) Ra = 106.

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0.0010 0.0008

1.0

0.9,ppi=5,Ra=1000 0.9,ppi=5,Ra=10000

0.0006

0.8

0.0004

y/H

v

0.0002 0.0000 -0.0002

0.9,ppi=5,Ra=1000 0.9,ppi=5,Ra=10000

0.6 0.4

-0.0004

y/H=0.4

-0.0006

0.2

-0.0008

x/H=0.4

0.0

-0.0010 0.0

0.2

0.4

0.6

0.8

-0.0008

1.0

-0.0004

0.0000

0.0004

0.0008

x/H

u

(a) the velocity-v distribution at y/H=0.4

(b) the velocity-u distribution at x/H=0.4

Fig. 6. Effect of Rayleigh number on fluid flow. (a) The velocity-v distribution at y/H = 0.4 (b) the velocity-u distribution at x/H = 0.4.

0.0020

1.0

0.9,ppi=2,Ra=10000 0.9,ppi=5,Ra=10000

0.0015

y/H

v

0.0005 0.0000 -0.0005 y/H=0.4

-0.0010

0.6 0.4 x/H=0.4

0.2

-0.0015 -0.0020

0.9,Ra=10000,2ppi 0.9,Ra=10000,5ppi

0.8

0.0010

0.0 0.0

0.2

0.4

0.6

0.8

1.0

-0.002

-0.001

0.000

0.001

0.002

x/H

u

(a) the velocity-v distribution at y/H=0.4

(b) the velocity-u distribution at x/H=0.4

Fig. 7. Effect of pore density on fluid flow. (a) The velocity-v distribution at y/H = 0.4 (b) the velocity-u distribution at x/H = 0.4.

80

53.0 Empty

70

0.9,2ppi

60

0.95,5ppi

51.5 51.0

Nu

40

50.5 50.0

30

49.5

20

49.0

10 0

0.9,5ppi

52.0

0.9,5ppi

50

Nu

0.8,5ppi

52.5

48.5 103

104

105

106

Ra Fig. 8. Effect of Rayleigh number and pore density on heat transfer.

48.0

103

104

105

Ra Fig. 9. Effect of porosity on heat transfer.

106

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C.Y. Zhao et al. / International Journal of Heat and Fluid Flow 31 (2010) 925–934

square metal blocks is much stronger than that of circular one due to the larger total contact surface area and strong flow mixing. Acknowledgements The present study is supported by the UK Engineering and Physical Science Research Council (EPSRC, Grant Number: EP/F061439/ 1), Warwick Research Development Fund (RDF) Strategic Award (RD07110), National Natural Science Foundation of China (Grant Number 50576069). References

Fig. 10. The boundary distribution mark of circular metal block.

Table 2 The unknown particle distribution functions. Mark

The unknown distribution

Mark

The unknown distribution

1 2 3 4

f3, f1, f1, f1,

5 6 7 8

f1, f2, f2, f3,

f4, f4, f4, f2,

f7, f7, f5, f5,

f8 f8 f8 f8

f2, f3, f3, f4,

f5, f5, f6, f6,

f6 f6 f7 f7

20 18

square

16

circle

14

Nu

12 10 8 6 4 2

103

104

Ra

105

106

Fig. 11. Effect of the shape on heat transfer.

4. Conclusions The double-populations LBE model is employed in this paper to simulate the natural convection in a square cavity filled with square and circular metal blocks. The detailed fluid flow and local heat transfer behavior at the pore level are examined. The parametric study has been carried out to reveal the effect of Rayleigh number, porosity, pore density and metal block’s shape on the flow and heat transfer. The results show that the metal porous medium enhances the natural convection in a square cavity. The heat transfer is enhanced with the increase of the pore density, however it is weakened with the increase of the porosity. The heat transfer with

Abbas, I.A., El-Amin, M.F., Salama, A., 2009. Effect of thermal dispersion on free convection in a fluid saturated porous medium. International Journal of Heat and Fluid Flow 30, 229–236. Barakos, G., Mitsoulis, E., Assimacopoulos, D., 1994. Natural convection flow in a square cavity revisited: Laminar and turbulent models with wall functions. International of Journal Numerical Methods in Fluids 18, 695–719. Cheng, C.Y., 2009. Natural convection heat transfer of non-Newtonian fluids in porous media from a vertical cone under mixed thermal boundary conditions. International Communications in Heat and Mass Transfer 36, 693–697. De Vahl Davis, G., 1983. Natural convection of air in a square cavity: a bench mark numerical solution. International Journal for Numerical Methods in Fluids 3, 249–264. D’Orazio, A., Corcione, M., Celata, G.P., 2004. Application to natural convection enclosed flows of a lattice Boltzmann BGK model coupled with a general purpose thermal boundary condition. International Journal of Thermal Sciences 43, 575–586. Guo, Z., Zhao, T.S., 2002. Lattice Boltzmann model for incompressible flows through porous media. Physical Review E 66, 036304–036312. He, X., Chen, S., Doolen, G.D., 1998. A novel thermal model for the lattice boltzmann method in incompressible limit. Journal of Computational Physics 146, 282– 300. Hossain, M.A., Vafai, K., Khanafer, K.M.N., 1999. Non-Darcy natural convection heat and mass transfer along a vertical permeable cylinder embedded in a porous medium. International Journal of Thermal Sciences 38, 854–862. Kramer, J., Jecl, R., Škerget, L., 2007. Boundary domain integral method for the study of double diffusive natural convection in porous media. Engineering Analysis with Boundary Elements 31, 897–905. Kumari, M., Nath, G., 2008. Unsteady natural convection from a horizontal annulus filled with a porous medium. International Journal of Heat and Mass Transfer 51, 5001–5007. Lu, T.J., Stone, H.A., Ashby, M.F., 1998. Heat transfer in open-cell metal foams. Acta Materialia 46, 3619–3635. Lu, W., Zhao, C.Y., Tassou, S.A., 2006. Thermal analysis on metal-foam filled heat exchangers. Part I: metal-foam filled pipes. International Journal of Heat Mass Transfer 49, 2751–2761. Peng, Y., Shu, C., Chew, Y.T., 2003. Simplified thermal lattice Boltzmann model for incompressible thermal flows. Physical Review E 68, 026701. Shu, C., Peng, Y., Chew, Y.T., 2002. Simulation of natural convection in a square cavity by Taylor series expansion and least squares-based lattice Boltzmann method. International Journal of Modern Physics C 13, 1399–1414. Tang, G.H., Tao, W.Q., He, Y.L., 2005. Gas slippage effect on microscale porous flow using the lattice Boltzmann method. Physical Review E 72, 056301. Varol, Y., Oztop, H.F., Pop, I., 2009. Natural convection in a diagonally divided square cavity filled with a porous medium. International Journal of Thermal Sciences 48, 1405–1415. Zhao, C.Y., Lu, T.J., Hodson, H.P., 2004a. Thermal radiation in metal foams with open cells. International Journal of Heat and Mass Transfer 47, 2927–2939. Zhao, C.Y., Lu, T.J., Hodson, H.P., Jackson, J.D., 2004b. The temperature dependence of effective thermal conductivity of open-celled steel alloy foams. Materials Science and Engineering: A 367, 123–131. Zhao, C.Y., Lu, T.J., Hodson, H.P., 2005. Natural convection in metal foams with open cells. International Journal of Heat Mass Transfer 48, 2452–2463. Zhao, C.Y., Lu, W., Tassou, S.A., 2006. Thermal analysis on metal-foam filled heat exchangers. Part II: tube heat exchangers. International Journal of Heat Mass Transfer 49, 2762–2770. Zhao, C.Y., Lu, T.J., Tassou, S.A., 2008. Analytical considerations of thermal radiation in cellular metal foams with open cells. International Journal of Heat and Mass Transfer 51, 929–940. Zhao, C.Y., Lu, W., Tassou, S.A., 2009. Flow boiling heat transfer in horizontal metal foam tubes. ASME Journal of Heat Transfer 131. Zou, Q., He, X., 1997. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Physics of Fluids 9, 1591–1598.