Effect of Obstacle Presence for Heat Transfer in Porous Channel ...

Effect of Obstacle Presence for Heat Transfer in Porous Channel Saida Chatti*, Chekib Ghabi*, and Abdallah Mhimid* Energetic department, National Engineering School of Monastir, Tunisia [email protected], {ghabichekib,abdmhimid}@yahoo.fr

Abstract. In this study, heat transfer enhancement in porous medium was inspected. For this aims the lattice Boltzmann method was adopted to simulate mixed convection in porous domain. The Brinkman Forchheimer model was implemented to simulate porous channel including obstacles maintained at constant temperature. The velocity and the temperature are plotted at various parameters. The simulation was carried out for different porosity, Darcy and Reynolds. The Results show that by decreasing porosity, the heat transfer enhances. Also the result points that the obstacle position has effect on heat transfer. Keywords: incompressible flow, lattice Boltzmann method, heat transfe, obstacle, porous media.

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Introduction

During the past several decades, convective heat transfer in porous media has attracted many scientific researchers because of its various applications. The porous medium are usually used for improving heat transfer in industry such as nuclear Reactors cooling, heat pipes, packed bed reactors, fuel cells and heat exchangers. Huang and Vafai [1] simulated forced convection in a channel containing blocks arranged on the bottom wall. Kaviany [2] studied fluid flow and heat transfer in porous media with two isothermal parallel plates. Based on forced convection in a porous channel containing discrete heated blocks Rizk and Kleinstreuer [3] showed that an increase in heat transfer can be obtained by using porous channel. The lattice Boltzmann method (LBM) is an efficient and powerful numerical tool founded on kinetic theory for simulation of fluid flows and modeling the physics proprieties. The incompressible flows through porous media by using lattice Boltzmann method was studied by many researchers such as Seta and al. and Zhao and Guo [4]. In this study LBM is used to simulate flow behaviors and heat transfer in a channel with solid block located inside a porous media. *

Corresponding authors.

© Springer International Publishing Switzerland 2015 M. Chouchane et al. (eds.), Design and Modeling of Mechanical Systems - II, Lecture Notes in Mechanical Engineering, DOI: 10.1007/978-3-319-17527-0_82

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S. Chatti, C. Ghabi, and A. Mhimid

Lattice Boltzmann Method for Incrompressible Flow in Porous Media

The LBM is defined as one of the computational fluid dynamics (CFD) methods. Counter to the others a macroscopic Navier Stokes (NS) method; the Lattice Boltzmann Method (LBM) is based on a mesoscopic approach to simulate fluid flows [5] [6]. The general form of Lattice Boltzmann equation with external force can be written as [7]:

f ( x+ δ tci , t+ dt ) i − f ( x, t ) i =

f ( x, t ) i − f ( x, t )

Where δ t denotes lattice time step, direction,

eq

Γν

c i is

+ δ t Fi ⋅

(1)

the discrete lattice velocity in

Fi is the external force in direction of lattice velocity c i , Gn denotes

the lattice relaxation time, f

eq i

is the equilibrium distribution function. The local

equilibrium distribution function determines the type of problem that needs to be solved. Equation (1) can be interpreted as two successive processes collision and streaming steps. The collision expresses various fluid particle interactions such as collisions and calculates new distribution functions [8]. Many models are advanced for the simulation of the fluid flow in the porous medium. The Brinkman-Forchheimer approach has been used successfully in simulation porous media in large values of porosities, Darcy, Rayleigh and Reynolds numbers [9]. This model includes the viscous and inertial terms by the local volume averaging technique. The Brinkman-Forchheimer equation is written as:

æu ö ¶u 1 + (u⋅)çç ÷÷ = -  (e p) + ue2 u+ F . ç ÷ èeø ¶t r

(2)

e is the porosity, ue the effective viscosity, F is the total body force which contains the viscous diffusion, the inertia due to the porous medium, and an external force given by the Ergun’s relation. The forcing term model is written as [10] [11]:

æ 1 ÷ö éê 3ci F 9 (u F : ci ci ) 3u⋅ F ùú Fi = wi r çç1 ⋅ ÷ 2 + çè 2Gu ÷÷ê e c4 e c 2 úû øë c

(3)

The equilibrium distribution functions are calculated by [12]: fi

eq

2 ⎡ 3 ci u 9 (ci u ) 3 u2 ⎤ ⎥⋅ = ω i ρ ⎢1 + + − c2 2ε c 4 2ε c 2 ⎥ ⎢⎣ ⎦

(4)

Effect of Obstacle Presence for Heat Transfer in Porous Channel

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are given by:

For D2Q9 model, the discrete velocities

⎧ ⎪c = 0. ⎪0 ⎪⎪ π ⎞ ⎛⎛ π ⎞⎞ ⎛ ⋅ ⎨ci = c cos ⎜ ( i − 1) ⎟ ;sin ⎜ ⎜ ( i − 1) ⎟ ⎟ i = 1,4 2⎠ 2 ⎠⎠ ⎝ ⎝⎝ ⎪ ⎪ π π⎞ ⎛ π π⎞ ⎛ ⎪ci = c 2cos ⎜ ( i − 5) + ⎟ ;sin ⎜ ( i − 5) + ⎟ i = 5,8 2 4⎠ 2 4⎠ ⎝ ⎝ ⎩⎪

(5)

Fig. 1 The velocity distribution in the D2Q9 models

The weights are expressed as follows:

4 9

ω0 = , ωi =

1 1 i = 1, 4, ω i = i = 5, 8 9 36

(6)

The macroscopic quantities are accessed through the distribution functions. Indeed the density and the fluid velocity: 8

ρ ( x, t ) = ∑ f ( x, t ) i ⋅

(7)

i =0

8

u ( x, t ) = ∑ i =0

c i f (x, t)i

ρ

+

Fδ t ⋅ 2

(8)

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In LBM the fluid viscosity is determined using the following relation:

d tc 2 æç 1ö u= ççGu - ÷÷÷ ⋅ 3 è 2ø

(9)

The temperature is carried out by a second distribution function denoted

g ( x , c , t ) . It is governed in direction by the following equation: g ( x+ci δ t,t+ δ t ) i − g ( x,t ) i = −

g ( x,t ) i − g ( x,t ) i Γc

eq

⋅

(10)

The equilibrium distribution functions are given by the following expressions:

ìï ïï 2 ïïg0eq =- 2re u ïï 3c2 ïï 2 é 2ù ïïg eq = re ê 3 + 3 ci u + 9 (ci u) - 3u ú i = 1,4 . í i ïï 9 êê 2 2 c2 2 c4 2c2 úú ë û ïï 2 ïï é 2ù ïïg eq = re ê 3 + 3 ci u + 9 (ci u) - 3u ú i = 5,8 i ê ïï 9 ê 2 2 c2 2 c4 2c2 úú ë û ïî

(11)

The fluid temperature is obtained from the distribution function by:

T ( x, t ) =

1 r

8

å c g (x, t ) ⋅ i

i=0

i

(12)

The thermal diffusivity is:

a = s c2 s (Gc - 0.5)⋅ 3

(13)

Problem Description

In the present study we consider a fluid flow in porous channel of width . The walls are fixe. The upper plate is hot and the bottom one is cold. The computational domain contains hot solid blocks at different positions as shown in figure 2. In simulation we adopt that the laminar and incompressible flow is viscous Newtonian, and the buoyancy effects are assumed negligible. All physical properties of the fluid and solid are constant.


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0 Hot solid blocks

0 0

y x

Fig. 2 Schematic of flow in the channel

For this flow in porous media the fluid behavior can be described by the following equations: The continuity equation:

⋅ u = 0 ⋅

(14)

The momentum equation: æu ö ¶u eu 1 + (u⋅)çç ÷÷ = -  (e p ) + e Gu+ ue  2 u⋅ ç ÷ è ø ¶t e r K

(15)

Energy equation:

s

¶T + ⋅ (u T ) = a2T ⋅ ¶t

(16)

From physical limits (the velocity in the inlet of the flow is defined) to numerical ones we occur to the distribution functions. The distribution functions out of the domain are known from the streaming process. The unknown distribution functions are those toward the domain. The solid walls are assumed to be no slip, for this reason the bounce-back scheme is applied. For example in the north boundary the following conditions are used [13].

f i = f-i ⋅

(17)

i = 4, 7,8 For the inlet the Zou and He boundary conditions are applied and an extrapolation in the outlet boundary is used [10] [11]. For the thermal boundary the Direchlet boundary are necessary.

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Resultats and Discussion

In this simulation the Reynolds number changes from 150 to 50, the porosity is equal to 0.7 and the Darcy number set to be 0.1.

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Fig. 3 Temperature (left) and velocity (right) profile at different Reynolds number (respectively 150, 80 and 50)

Fig. 4 temperature (left) and velocity (right) profile at different porosity (respectively 0.99, 0.5 and 0.3)


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The figure shows that by increasing the Reynolds number the heat transfer in the channel raises. At high value of Reynolds the fluid velocity increases. Then the porosity change from 0.99 to 0.3, the Reynolds number is equals to 50 and the Darcy one is 0.1. The figure shows that for low value of porosity the heat transfer and the velocity of the flow are more important. Indeed With increasing the porosity the fluid temperature reduces due to lower values of effective thermal conductivity in blocks which can causes the heat transfer decreases. An Increase in the porosity value, the velocity rises. With increasing the porosity, it easier for fluid to change its path. In this study the Darcy number changes from 10-3 to 1, porosity is 0.7 and the Reynolds number is taken 50.

Fig. 5 temperature (left) and velocity (right) profile at different Darcy number (respectively 10-3, 10-2 and 1)

The heat transfer is more important at high value of Darcy number. Finally the porosity is equal to 0.7, the Reynolds number is 50 and the Darcy one is 0.1. The thermal conductivity ratio Rk changes from 29 to 69.

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Fig. 6 temperature (left) and velocity (right) profile at different thermal conductivity ratio (respectively 29, 59 and 69)

The heat transfer and the fluid velocity increase by decreasing the thermal conductivity ratio.

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Conclusion

The heat transfer phenomena is abundant in many scientifique and engeeniring field. In this study a numerical simulation was carried out for Fluid flow and heat transfer in a porous channel containing hot solid blocks. This paper interested on the effect of parameters such as Reynolds number, thermal conductivity ratio and porosity on the flow field and thermal behavior is simulated by using thermal lattice Boltzmann method. Indeed the Brinkman-Forchheimer approach was adopted for simulation. The temperature of fluid reduces by increasing the porosity due to lower values of thermal conductivity. So that the heat transfer decreases with blocks. Increasing the thermal conductivity ratio the fluid temperature will be reduced. The results indicate that increasing the Reynolds and the Darcy number raises the heat transfer. The lattice Boltzmann method is a powerful tool for simulation of fluid flow and heat transfer in porous media and many phenomena.


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Nomenclature

c ci Da f f

eq

Lattice spacing U Fluid velocity Discrete velocity for D2Q9 model Greek letters Darcy number Density distribution function Density equilibrium distribution function Total body force Geometric factor

a Thermal diffusivity

Gc Thermal time Gn

relaxation Dynamic time relaxation Time step Porosity

δt F e Fe g Thermal distribution function u Viscosity eq Thermal equilibrium distribution ue Effective viscosity g H i j

function Channel width Lattice index in the Lattice index in the

Permeability Thermal conductivity p Pression

K k

Ra Re T Tc

Rayleigh number Reynolds number Fluid temperature Cold temperature

Th

Hot temperature

r

direction direction

Density W Collision operator wi The weights coefficient in the direction Subscripts e effective f Fluid Discrete velocity direction Superscript eq equilibrium

i

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[6] Azmi, M.: Numerical study of convective heat transfer and fluid flow through porous media, Thèse de doctorat, Université de Technologies de Malaysia (2010) [7] Seta, T., Takegoshi, E., Kitano, K., Okui, K.: Thermal lattice Boltzmann model for incompressible flows through porous media. Jounal of Thermal Science and Technology, 90–100 (2006) [8] Arab, M.: Reconstruction stochastique 3D d’un matériau céramique poreux à partir d’images expérimentales et évaluation de sa conductivité thermique et de sa perméabilité, Thèse de doctorat, Université de Limoges (2010) [9] Jie, X.: A generalized Lattice-Boltzmann Model of fluid flow and heat transfer with porous media, National University of Singapore, Master of engineering (2007) [10] Mohamad, A.A.: lattice Boltzmann method fundamentals and engineering applications with computer codes, Springer Verlag, London, 195 (2011) [11] Janzadeh, N., Delavar, M.A.: Using Lattice Boltzmann Method to Investigate the Effects of Porous Media on Heat Transfer from Solid Block inside a Channel. Transport Phenomena in Nano and Micro Scales 1, 117–123 (2013) [12] Farhadi, M., Mehrizi, A.A., Sedighi, K., Afrouzi, H.H.: Effect of Obstacle Position and Porous Medium for Heat Transfer in an Obstructed Ventilated Cavit, Jurnal Teknologi, pp. 59–64 (2012) [13] Che Sidik, N.A., Khakbaz, M., Jahanshaloo, L., Samion, S., Darus, A.N.: Simulation of forced convection in a channel with nanofluid by the lattice Boltzmann method. Nanoscale Research Letters, 1–8 (2013)