International Renewable Energy CongressNovember 5-7, 2009 - Sousse Tunisia

Application of the lattice Boltzmann method for solving conduction problems with heat flux boundary condition.

Raoudha CHAABANE, Faouzi ASKRI, Sassi Ben NASRALLAH

Laboratoire d’Etudes des Systèmes Thermiques et EnergétiquesEcole Nationale des Ingénieurs de Monastir

Av. Ibn ElJazzar 5019 Monastir- TunisieRaoudha.Chaabane@issatgb.rnu.tn

ABSTRACT

The lattice Boltzmann method (LBM) has been developedover the last decade as an alternative promising tool forfluid flows. It has been widely used in many kinds ofcomplex flows such as turbulent flow, solar collectors,multiphase flow and micro flow. This article deals withthe implementation of the lattice Boltzmann method(LBM) for the solution of conduction problems with heatflux and temperature boundary conditions. Problems intwo dimensional rectangular geometries have beenconsidered. In the 2-D geometry, the south and the northboundary are subjected to constant heat flux condition.The remaining boundaries are at prescribed temperatures.The energy equation is solved using the LBM. The resultsof the LBM have been found to compare very well withthose available in the literature.

Index Terms: Lattice Boltzmann method, heatconduction, uniform lattices, heat flux, 2D heat transfer.

1. INTRODUCTION

Nowadays, the lattice Boltzmann method (LBM) is beingviewed as a potential computational tool to analyze a largeclass of problems in science and engineering [1–11].Recently, thermal lattice Boltzmann method has attractedmuch attention because of its potential applications as wellas practical importance in engineering designs and energyrelated problems, such as solar collectors, thermalinsulation, cooling of electronic components, heatexchangers, air heating systems for solar dryers, passivesolar heating and storage technology to name just a few.Various configurations may be considered for thisproblem. In comparison with the conventionalcomputational techniques based on the finite differencemethod, the finite element method and the finite volumemethod, this surge in interest is attributed to the bottom-upapproach inherent with the LBM. A simple calculationprocedure, simple and more efficient implementation forparallel computation, straightforward and efficienthandling of complex geometries and boundary conditionsand high computational performance with regard tostability and precision are some of the main advantages ofthe LBM [1–8]. Owing to the above attributes, these days,

the LBM is increasingly being applied in the analysis of alarge class of fluid flow and heat transfer problems [1–11].

Analysis of conduction problems with flux boundaryconditions finds applications in furnace design, fireprotection systems, foam insulations, solidification/meltingof semitransparent materials, high-temperature porousinsulating materials, glass-fluidized bed, electronics chip andpower plants, etc. [12-13]. A few papers have discussed theproblems with flux boundary conditions [14-15].

Thus, the present work aims at further extending theapplication of the LBM to solve heat conduction problemsdealing with temperature as well as heat flux boundaryconditions.

We consider two-dimensional rectangular geometry whereone or two boundaries can be at prescribed heat fluxconditions. The energy equation is solved using the LBM andobtained results are compared with those available in theliterature.

2. FORMULATION AND KINETIC EQUATION

We have considered transient heat conduction heat in a 2-Drectangular geometry. Thermo-physical properties of themedium are assumed constant. The system is initially attemperature ET . For time t>0, the south and the north

boundaries are subjected to heat fluxes STq , and NTq , ,

respectively. The east and the west boundaries are kept attemperatures ET and WT , respectively. For the problem under

consideration, and in the absence of convection andradiation, the energy equation is given by

2TT Q

t

(1)

Where is the thermal diffusivity.

The starting point of the LBM is the kinetic equation satisfiesa discretized evolution equation of the form [16]

( , ). ( , ) , 1, 2,3,..,i

i i i

f r te f r t i b

t

(2)

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The collision operator i represents the rate of change

of if due to collisions. It incorporates all the physics and

modelling of any particular problem at hand. The simplestmodel for i is the Bhatnagar–Gross–Krook (BGK) model

[16]

)],(),([1 )0( trftrf iii

(3)

if is the particle distribution function denoting the number

of particles at the lattice node r at time t moving in

direction i with velocity ie along the lattice link

ter i connecting the nearest neighbours.

b is the number of directions in a lattice through which theinformation propagates.

The basis of the discrete velocity model is a finite set of

virtual velocities ie or equivalently, of virtual fluxes of the

considered scalar field ),( trT which given by

b

i i trftrT0

),(),( (4)

The observed flux is expressed by

ib

i i etrf 0),( (5)

Fig.1: Schematic diagram of the D2Q9 lattice.

The well-known D2Q9 lattice model (Fig.1) will beconsidered here. In that model, the set of ie

‘s is such that

they connect the point, on which the lattice stencil iscentred, to its nearest neighbours on a spatial grid withuniform spacing in both coordinate directions.

Any LBM advances the probability densities ),( trf i in

time and thereby computes the evolution of the consideredscalar. In the absence of external sources or fluxes for thescalar, the corresponding discrete evolution equation canbe written in the following general form:

(0)( , ) 1. ( , ) [ ( , ) ( , )]i

i i i i

f r te f r t f r t f r t

t

(6)

It is a single-relaxation-time model with relaxationconstant that can be related, via Chapmann–Enskog

analysis, to the diffusivity of the medium. (0)if is the

equilibrium distribution function.

The relaxation time can be related with the thermaldiffusivity, the lattice velocity C and the time step [17] bythe following relation

2

32

t

C

(7)

For the D2Q9 model in particular, the 9 velocities ie

and

their corresponding weights iw are the following

)0,0(0 e (8)

Cii

ei )).2

1sin(),

2

1(cos(

for 4,3,2,1i (9)

Cii

ei )).4

12sin(),

4

12(cos(2

for 8,7,6,5i (10)

9

4iw (11)

9

1iw for 4,3,2,1i (12)

36

1iw for 8,7,6,5i (13)

It is to be noted that in the above equations, tytxC //

and the weights satisfy the relation

b

i iw0

1.

After discretization, and considering heat generation,equation (6) can be written as

*)0( )],(),([),(),( tQwtrftrft

trfttterf iiiiii

(14)

where *Q is the non dimensional heat generation and iw is the

weight in corresponding direction.

To process equation (8), an equilibrium distribution functionis required. For heat conduction problems, this is given by

),(),()0( trTwtrf ii (15)

3. RESULTS AND DISCUSSION

We consider transient heat conduction problems in 2-DCartesian geometry with the following conditions:

Case1: the four boundaries are at known temperatures

The initial and the boundary conditions for cases 1 are thefollowing

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Initial condition ( , ,0) refT x y T(16)

Boundary conditions

e( ,0, ) 0.25 r fT x t T (17)

( , , ) (0, , ) ( , , ) refT x Y t T y t T X y t T (18)

Steady state conditions were assumed to have beenachieved when the temperature difference between twoconsecutive time levels at each lattice centre did notexceed 610 . Non dimensional time was defined

as 2/t L where L is the characteristic length. was

taken as 410 .

To check the accuracy of the present LBM algorithm, the

same problem was solved using the finite volume method

and the results given by the two algorithms are compared

with those available in the literature.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

distance y/Y for x/X=1/2

Tem

pera

ture

=T

/Tre

f

=SS=0.05

=0.01=0.001

FVMLBMReference[18]

Fig. 2. Centerline(x/X=0.5) temperature evolution fordifferent instants (case 1).

In fig.2, the non dimensional centreline ( x/X=0.5 )

temperature has been compared at different instants for

the case 1.

Case2: Effects of heat generation and the four boundariesare at specified temperatures

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

distance y/Y for x/X=1/2

Tem

pera

ture

=T

/Tre

f

=SS=0.05

=0.01=0.001

without Heat Generationwith Heat Generation

Fig. 3. Comparison of centreline (x/X=0.5) temperature inthe presence and the absence of heat generation.

In fig. 3, the effects of volumetric heat generation are shown.

The non dimensional volumetric heat generation is taken as

unity. Effect of heat generation is very less in the beginning

compared to steady state because it takes some time to

influence the temperature profile. For the 2-D geometry, the

number of iterations for a 50x50 grid is 3719 (135.95

seconds) compared to that cited at the literature 3257 [18].

Case3: The bottom and top boundaries are at prescribedfluxes and remaining two boundaries at knowntemperatures

Initial condition 0( , ,0)T x y T (19)

Boundary conditions

( ,0, ) sq x t q ; ( , , ) Nq x Y t q ;

0(0, , ) ( , , )T y t T X y t T (20)

It is seen from the figure 4 that the steady state results match

exactly which each other.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

distance y/Y for x/X=1/2

Tem

pera

ture

=T

/Tre

f

=SS

=0.05=0.01=0.001

LBMReference [18]

Fig. 4. Centreline(x/X=0.5) temperature evolution fordifferent instants (case3).

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00x/X

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

y/Y

0.001

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00x/X

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

y/Y

0.01

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00x/X

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

y/Y

0.05

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00x/X

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

y/Y

at steady state

Fig.5. Isotherms when the bottom and the top boundaries areat prescribed fluxes and remaining two boundaries atknown temperatures for different .

In fig. 5, we present the time-space evolution of theisotherms when the bottom and the top boundaries are atprescribed fluxes and remaining two boundaries at knowntemperatures.

Table 1:

CPU times (second) and number of iterations of the LBMcode (case3)

sizeLattices

iterations CPUtime

(seconds)

Temperatureat steady

state(x/X=0.5)8x8 6251 12.42 0.44170

12x12 6076 24.33 0.3772220x20 6051 53.026 0.3441350x50 6199 286.011 0.34493

Table 2:Effect of heat generation on CPU times (second) and numberof iterations of the LBM code (case3)

Lattice size iterations CPU time

In the absence of heat generation

50x50 6199 549.69

In the presence of heat generation

50x50 6317 555.13

To have an idea of the number of iterations for the converged

solutions and the CPU time, tests were performed with

different lattices. The LBM code was found to take slightly

less number of iterations for the little lattices (Table 1).

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The effect of heat generation on CPU times (second) and

number of iterations when all boundaries at known

temperatures, was highlighted in table 2.

5. CONCLUSIONS

The LBM is used to solve transient heat conductionproblems in two dimensional geometries with uniformlattices having constant temperature and/or flux boundaryconditions.Effect of heat generation is also studied. The sameproblems are solved using the finite volume method. Theresults given by the two numerical approaches arecompared with those available in the literature and goodagreement is obtained. On the other hand, the effect oflattice size is highlighted via the number of iterations andthe CPU time. The considered 2D geometry is a simpleone, to allow simple validation. Advection and radiationare omitted. Thus, it remains to demonstrate the viabilityof the LBM as heat diffusion-advection solver.

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