Baytas - Thermal Non-equilibrium

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INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2003; 27:975988 (DOI: 10.1002/er.929)

Thermal non-equilibrium natural convection in a square

enclosure filled with a heat-generating solid phase,non-Darcy porous medium

A.C. Bayta,sn,y

Istanbul Technical University Institute For Nuclear Energy, 80626 Maslak, Istanbul, Turkey

SUMMARY

The aim of the present paper is to study the steady state natural convection in a square porous enclosureusing a thermal non-equilibrium model for the heat transfer between the fluid and the solid phases. Theanalysis assumes that the porous medium is homogeneous and isotropic. The present study also assumes

the non-Darcy model of natural convection in porous media. It is assumed that the heat generation is onlyin solid phase. Two dimensional steady convection in a cavity bounded by isothermal walls at constanttemperatures has been studied numerically by adopting a two-temperature model of microscopic heattransfer. Such a model, which allows the fluid and solid phases not to be in local thermal equilibrium, isfound to modify the flow behaviour and heat transfer rates. Knowledge of this behaviour is very importantfor the design of the many engineering applications. Copyright # 2003 John Wiley & Sons, Ltd.

KEY WORDS: natural convection; porous media; local thermal non-equilibrium; non-Darcy; numericalstudy; microscopic heat transfer; heat generation

1. INTRODUCTION

Free convection within enclosures filled with porous media has been studied widely during thepast two decades due to their widespread engineering applications. Some of these applications

are geothermal systems, underground spread of pollutants, storage of nuclear waste materials,

pebble-bed nuclear reactor, solidification of casting, thermal insulation, electronic cooling,

petroleum reservoir modelling, design of chemical catalytic reactors, powder metallurgy,

ceramic engineering, food and medical industries and so on. Most of the early theoretical studies

were based on Darcys law with the assumption of thermal equilibrium between solid and fluid

phases. Based on this model, the fluid and solid structure are assumed to be in local thermal

equilibrium. This assumption is adequate for small-pore media such as geothermal reservoirs

and fibrous insulations.

As mentioned previously, natural convection in porous cavities is widely applied in

engineering, therefore the acquisition of accurate results for a system is very important for

the design of such a system. For many applications involving high-speed flows, it is important totake account of thermal non-equilibrium effects. In addition to this, there is often a large

Received 31 January 2002Accepted 6 January 2003Copyright# 2003 John Wiley & Sons, Ltd.

nCorrespondence to: A.C. Bayta,s, Istanbul Technical University Institute for Nuclear Energy, 80626 Maslak, Istanbul,Turkey.yE-mail: [email protected]


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temperature difference between fluid and solid phase in the study of nuclear reactor cores and

especially in the study on hypothetical nuclear reactor accidents. If the temperature between

phases is a very important safety parameter, the thermal non-equilibrium convection model in

porous matrix is an indispensable model. For example, the thermal non-equilibrium model is

important for post-accident heat removal from pebble-bed nuclear reactors. As aforesaid, theassumption of local thermal equilibrium between the fluid and the solid phases is inadequate for

a number of problems. Although Schumann (1929) proposed the classical thermal non-

equilibrium model 70 years ago, in recent years more attention has been paid to the local

thermal non-equilibrium model and its use has increased in theoretical and numerical

investigation of the convection heat transfer in porous media. Much of this activity has been

summarised in recent books by Nield and Bejan (1999) and Ingham and Pop (1998). The reviews

by Kuznetsov (1998) and Kuznetsov (1996) give detailed information about the thermal non-

equilibrium effects of the fluid flow through a porous packed bed. Jiang and Ren (2001) carried

a numerical investigation of forced convection heat transfer in porous media using a thermal

non-equilibrium model. Rees and Pop (1999, 2000) performed a numerical and analytical study

on the free convective boundary-layer flow and stagnation-point flow using a thermal non-

equilibrium model. Bayta,s and Pop (2002) studied free convection in a differentially heatedsquare cavity using a thermal non-equilibrium model.

The aim of present paper is to study the steady state non-Darcy natural convection in a

square porous enclosure using a thermal non-equilibrium of microscopic heat transfer between

the solid phase with the heat generation and the fluid phase.

2. MATHEMATICAL FORMULATION

A schematic diagram of a two-dimensional rectangular cavity with isothermally cooled walls is

shown in Figure 1. The cavity is filled with a fluid-saturated porous medium, which generates

heat in solid phase at a uniform rate. All walls of the cavity are assumed to be impermeable.The analysis assumes that the porous medium is homogenous and isotropic. The fluid flow is

Figure 1. Schematic diagram of the physical situation.

Copyright # 2003 John Wiley & Sons, Ltd. Int. J. Energy Res. 2003; 27:975988

A.C. BAYTAS,976


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For momentum balance equation in y direction

@V

@t U

@V

@X V

@V

@Y jrP JPrr2V

fPr

DaV j2FoU

2 V21=2V jPrRayf 9

The energy balance equation for the solid phase is

G@ys@t

@2ys@X2

@2ys@Y2

gH yf ys 1 10

and the energy balance equation for the fluid phase is

@yf@t

U@yf@X

V@yf@Y

@2yf@X2

@2yf@Y2

H ys yf 11

are obtained by employing the following non-dimensional variables:

X; Y x; y=D; U; V u; vD=af; t kft=rcpfD2 12

P pD2=ra2f; y T Tw=DT; DT%q000s D

2=ks

and the parameters,

J me=m; Pr n=a; Ra gbD3

naf

q000D2

ks

; Da

K

D213

F0 DC; G af=as; H hD2

jkf; g

jkf

1 jks

The non-dimensional initial and boundary conditions of Equations (7)(11) are

t 0 : U V yf ys 0 at X 0; 1 and Y 0; 1

t > 0 : U V yf ys 0 on X 0; 1 and Y 0; 1 14

The fluid and solid phase average Nusselt numbers are defined as

Nuf Dkf

RD0

@Tf@n wj dn

kfDT; Nus

DksR

D0

@Ts@n wj dn

ksDT15

After using non-dimensional variables (in Equation (12)), the average Nusselt numbers are

obtained for fluid and solid phase, respectively,

Nuf

Z10

@yf@n

w

dn; Nus

Z10

@ys@n

w

dn 16

where n denotes the x- or y-directions.

3. NUMERICAL SOLUTION

The dimensionless governing Equations (7)(11), in primitive variables, were discretized by the

two dimensional Finite Volume method of Patankar (1980), along with boundary conditions

given in Equation (14). The grid layout was arranged by utilizing non-staggered (collocated)

grid procedure, while the power law differencing scheme was adopted for convectiondiffusion

terms for calculations in the fluid domain. Computing on a non-staggered grid requires the cell


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face velocities to be computed. The necessary coupling between velocity and pressure can be

established by using momentum equation at the cell face to find the interface velocity. There are

different approximations for face velocity values. In this study, the linear interpolation is used to

find the cell face values.

To handle the pressure, temperature and velocity coupling of Equations (7)(11) the SIMPLE

algorithm was utilized. The discretized energy and momentum equations were solved using

alternating direction implicit (ADI) method. ADI leads to a tridiagonal matrix which was solved

with the tridiagonal matrix algorithm (TDMA) as described by Versteeg and Malalasekera

(1995). The pressure equations used in the SIMPLE algorithm were solved by point successive

over relaxation; the relaxation parameter for the pressure correction equation was found to be

1.925 for a non-uniform grid of 42 42. A high grid density was utilized near the solid walls of

the enclosure where sharp gradients of velocity and temperature are expected. The numerical

iterations were advanced in time untilXl1i;j l

i;j

=X l1i;j 4e 17

where l stands for y, P; is the iteration level and e is a prescribed error, e is 105.Accuracy tests were performed at steady state using three sets of grids as shown in Table I.

The results show insignificant difference between the 42 42 and 82 82. The grid was selected

as a trade off between numerical accuracy, stability, and computational time. A non-uniformgrid (42 42) was used in all calculations. The chosen grid generation used by Baytas (1996) is

the similar in this study.

4. RESULTS AND DISCUSSION

In this investigation, the possible values of each dimensionless parameter affecting the problem

are studied first. The Rayleigh, Prandtl and Darcy numbers, aspect ratio of cavity, dynamic

viscosity ratio (J), non-dimensional form coefficient (F0) and thermal diffusivity ratio (G) were

all held fixed throughout this investigation in order to focus our attention on the parameters

that directly pertain to the solid/fluid-scaled heat transfer coefficient (H) and the porosity- scaled

conductivity ratio (g). All analyses were carried out assuming Ra=107, Pr=7, Da=102,J=1.0, G=1.0, F0=5.648 from Equation (6). To reduce the numerical effort and benchmarking,

the problem was studied for a square enclosure. Numerical results were obtained for some

values of the parameters Hand g varying in the range 05H52000 and 05g5100, respectively.

Isotherms and streamlines are presented in Figures 26 while the temperature profiles at

midplane and average Nusselt number for different H values are illustrated in Figures 7 and 8,

Table I. Accuracy test with Pr=7, Ra=107, Da=102, Fo=5.648, j=0.4 at steady state.

Nodes Grid streacing The step size Nutop Nubottom Nuright and leftparameter at the walls

22 22 1.20 0.01926 0.3257 0.1506 0.2595

42 42 1.10 0.00873 0.3315 0.1508 0.259882 82 1.05 0.00413 0.3403 0.1501 0.2575


THERMAL NON-EQUILIBRIUM NATURAL CONVECTION 979


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respectively. Figure 2 illustrates the isotherms and streamlines for the case of classical heatgenerating square porous cavity filled with a fluid saturated porous medium, which is in

thermodynamical equilibrium for Ra=106, 107, and 108. Figures 36 show for the case of non-

equilibrium model for Ra=107 and Pr=7. In Figure 3, the isotherms for fluid phase change but

the isotherms for solid phase do not change considerably when His increased between 1 to 1000

for g=103 because there is little transfer of heat from the fluid phase to the solid phase for the

0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

Figure 2. Isotherms and streamlines at steady state for the case of local thermal equilibrium model forRa=106, 107 and 108, respectively.


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0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

Figure 3. Isotherms for fluid and solid and streamlines for the case of non-equilibrium model for H=1, 50,200, 500, 1000 and g=103.




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1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

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0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

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0.20

0.00

0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00

Figure 4. Isotherms for fluid and solid and streamlines for the case of non-equilibrium model for H=1, 50,200, 500, 1000 and g=101.


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1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00

Figure 5. Isotherms for fluid and solid and streamlines for the case of non-equilibrium model for H=1, 50,200, 500, 1000 and g=1.0.




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1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00

0.80

0.60

0.40

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0.00

1.00

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0.00

1.00

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0.20

0.00

0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00 0.20 0.40 0.60 0.80 1.000.00

Figure 6. Isotherms for fluid and solid and streamlines for the case of non-equilibrium model for H=1, 50,200, 500, 1000 and g=100.0.


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small value of g. As H increases, the isotherms for fluid and solid phases change in Figures. 4

and 5 for g=0.1 and 1, respectively. The isotherms for fluid and solid phases are approximately

similar in Figure 6. The temperature profiles at vertical midplane of cavity are shown for solid

and fluid phases in Figure 7. As H increases, the maximum temperature of fluid increases while

the temperature of the solid phase does not change with H for g=103 in Figure 7(a). As H

increases, the maximum temperature of the fluid increases and that of the solid decreases for

g=0.1 in Figure 7(b). The temperature of the fluid and solid phases close each other as H

increases in Figure 7(c) and 7(d). It is also seen from Figure 8 that when His small there is a verysubstantial difference between the heat transfer rates of the fluid and solid phases, indicating

that nonequilibrium effects are stronger when H is small. As H increases the heat transfer

between the phases occurs more readily and this is reflected by the increasing similarity between

the heat transfer rates of the phase. For very large values of H the thermal equilibrium case

where the fluid and solid temperature fields are identical have been almost recovered.

H=1, H=50, H=200, , H=500, H=1000

Fluid phase

Solid phase

H

H

0.0 0.2 0.4 0.6 0.8 1.0

Y

0.00

0.02

0.04

0.06

0.08

Temp.profilea

tmidplane

0.00 0.20 0.40 0.60 0.80 1.00

Y

0.00

0.02

0.04

0.06

0.08

0.00 0.20 0.40 0.60 0.80 1.00

Y

0.00

0.02

0.04

0.06

0.08

Temp.profileatm

idplane

0.00 0.20 0.40 0.60 0.80 1.00

Y

0.000

0.004

0.008

0.012

Temp.profileatmidplane

Temp.profilea

tmidplane

(a) (b)

(c) (d)

Figure 7. Temperature profile at vertical midplane; (a) g=103, (b) g=0.1, (c) g=1.0, (d) g=100.




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5. CONCLUSION

The steady state flow and heat transfer characteristics have been numerically investigated for the

nonequilibrium model of thermal energy transport in a square cavity filled with a porous

medium. The heat generation in the porous cavity takes place within the solid phase. For large

values of H and g the thermal equilibrium case has been recovered where the fluid and solid

temperatures fields are identical. In this case, the local thermal equilibrium assumption may be

valid. For small values ofg the temperature of the fluid phase is larger than the temperature of

the solid phase in the top region of cavity. Namely, the temperature differences between solidand fluid phases are negative at the upper part of the cavity and positive at the lower part of the

cavity. When H is small, the heat transfer between phases is evidently poor, therefore, the

temperature of the solid phase is greater than the temperature of the fluid phase. In these cases,

the nonequilibrium assumption is very important and indispensable.

=0.1 =1, =50

0 400 800

H1200 1600 2000

0.0

0.1

0.2

0.3

0.4

Nu

Fluid phase

solid phase

Figure 8. Variation of the average top wall Nusselt number with H for different values of g.

NOMENCLATURE

CE =Forchheimer coefficient

cp =fluid specific heat at constant pressure

C =form coefficient

D =height of porous cavityDa =Darcy number

F0 =non-dimensional form drag coefficient

g =acceleration due to gravity

h =volumetric heat transfer coefficient between solid and fluid

H =dimensionless scaled value of h


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REFERENCES

Baytas AC. 1996. Buoyancy-driven flow in an enclosure containing time periodic internal sources. Heat Mass Transfer31:113119.

Baytas AC, Pop I. 2002. Free convection in a square porous cavity using a thermal nonequilibrium model. InternationalJournal of Thermal Science 41:861870.

Erg .uun S. 1952. Fluid flow through packed columns. Chemical Engineering and Progress 8:8994.Ingham DB, Pop I (eds). 1998. Transport Phenomena in Porous Media. Pergamon: Oxford.

J =dynamic viscosity ratio

k =thermal conductivity

K =permeability of the porous medium

Nu =average Nusselt number

p =difference between static and hydrostatic pressureP =dimensionless pressure

Pr =Prandtl number

q000s =volumetric heat source strength in solid phase

Ra =modified Rayleigh number for a porous medium

t =time

T =temperature

u,v =velocity components along the x and y axes

U,V =non-dimensional velocity components

x,y =Cartesian coordinates

X,Y =non-dimensional coordinates

Greek symbols

a =thermal diffusivity

b =coefficient of thermal expansion

e =prescribed error

g =modified conductivity ratio

G =thermal diffusivity ratio

j =porosity

m =fluid dynamic viscosity

me =effective viscosity (Brinkman)

n =fluid kinematic viscosity

y =dimensionless temperature

r =fluid densityt =non-dimensional time

Subscripts

f =fluid

s =solid

w =wall




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Jiang PX, Ren ZP. 2001. Numerical investigation of forced convection heat transfer in porous media a thermal non-equilibrium model. International Journal of Heat and Fluid Flow 22:102110.

Kuznetsov AV. 1998. Thermal nonequilibrium forced convection in porous media. In Transport Phenomena in PorousMedia, Ingham, DB, Pop I (eds). Pergamon: Oxford, 103129.

Kuznetsov AV. 1996. A perturbation solution for a non thermal equilibrium fluid flow through a three dimensionalsensible storage pached bed. Trans. ASME. Journal of Heat Transfer 118:508510.

Lage JL. 1992. Effect of the convection inertia term on Benard convection in a porous medium. International Journal ofHeat and Mass Transfer 22:469485.

Nield DA, Bejan A. 1999. Convection in Porous Media. (2nd edn.). Springer: New York.Patankar SV. 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere: Washington.Rees DAS, Pop I. 1999. Free convective stagnation-point flow in a porous medium using a thermal nonequilibrium

model. International Communication on Heat Mass Transfer 26:945954.Rees DAS, Pop I. 2000. Vertical free convective boundary-layer flow in a porous medium using a thermal

nonequilibrium model. Journal on Porous Media 3:3144.Schumann TEW. 1929. Heat transfer: liquid flowing through a porous prism. Journal of the Franklin Institute 208:

405416.Vesteeg HK, Malalasekera W. 1995. An Introduction to Computational Fluid Dynamics. The finite Volume Method.

Longman, London.


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