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NUMERICAL STUDY OF THERMALWALL BOUNDARY EFFECTS FORTRANSIENT NATURAL CONVECTIONBETWEEN CONCENTRIC AND VERTICALLYECCENTRIC SPHERESWen Ruey Chen aa Department of Mechanical Engineering, Far East College,Tainan,Taiwan, Republic of ChinaPublished online: 02 Feb 2011.

To cite this article: Wen Ruey Chen (2003) NUMERICAL STUDY OF THERMAL WALL BOUNDARYEFFECTS FOR TRANSIENT NATURAL CONVECTION BETWEEN CONCENTRIC AND VERTICALLY ECCENTRICSPHERES, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation andMethodology, 44:4, 433-449, DOI: 10.1080/713838229

To link to this article: http://dx.doi.org/10.1080/713838229

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NUMERICAL STUDY OF THERMAL WALL BOUNDARYEFFECTS FOR TRANSIENT NATURAL CONVECTIONBETWEEN CONCENTRIC AND VERTICALLY ECCENTRICSPHERES

Wen Ruey ChenDepartment of Mechanical Engineering, Far East College,Tainan, Taiwan, Republic of China

Transient analysis has been investigated numerically to determine heat transfer by natural

convection between concentric and vertically eccentric spheres with specified constant heat

flux wall boundary conditions. The governing equations, in terms of vorticity, stream

function and temperature are expressed in a spherical polar coordinate system. The alter-

nating direction implicit method and the successive over-relaxation techniques are applied

to solve the finite difference form of governing equations. Results were obtained for steady

and transient heat-transfer in vertically eccentric spheres at a Prandtl number of 0.7, with

the modified Rayleigh number ranging from 103 to 56105, for a radius ratio of 2.0 and

eccentricities varying from 7 0.625 to þ 0.625. Comparisons are attempted between the

isothermal boundary condition and the present computations for constant heat flux

boundary condition.

INTRODUCTION

The problem of natural-convection heat transfer in the annulus between twoconcentric and eccentric spheres has received considerable attention from researchersin many diverse fields of application. Such problems commonly occur in the geo-physical fields, solar energy collectors, thermal storage systems, and many otherpractical situations. For some engineering applications, such as gyroscopes, thepredication of transient temperature distribution and heat transfer rate from initialstate to steady state is very important. As a result, extensive experimental and the-oretical work dealing with flow and associated heat transfer characteristics of naturalconvection in annuli between two isothermal concentric spheres has been reported inthe literature. Experimental heat transfer results for isothermal concentric spheres,the inner being hotter, have been obtained by Bishop et al. [1] for air, and by Scanlanet al. [2] for water and silicone oils. Semianalytical studies of the problem have beendone by Mack and Hardee [3] and by Singh and Chen [4]. Numerical results havebeen reported by Astill et al. [5] and by Ingham [6] for radius ratios up to 2, whereas

Received 7 March 2002; accepted 9 January 2003.

Address correspondence to Wen Ruey Chen, Far East College, Department of Mechanical Engi-

neering, 49 Chung Hua Road, Hsin-Shih, Tainan Prefecture, Taiwan 744, Republic of China. E-mail:

ruey666@yahoo.com.tw

Numerical Heat Transfer, Part A, 44: 433–449, 2003

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/110407780390210620

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Geoola and Cornish [7] and Fujii et al. [8] considered very high radius ratios in theirinvestigations. The transition toward a multicellular flow has been studied numeri-cally by Caltagirone et al. [9]. Chu and Lee [10] computed a numerical solution fortransient natural convection with a large range of Rayleigh number. In all the aboveanalyses, the process is taken to be two-dimensional. Recently, Ozoe et al. [11] did a3-D numerical study under asymmetrical thermal boundary conditions for a lowRayleigh number. A little attention has been paid to natural convection in eccentricspheres of engineering interest. Weber et al. [12] obtained heat transfer correlationsfor water and silicone and eccentricities from 70.75 to þ0.75. Powe et al. [13, 14]have studied air and silicone oils for radius rations of 1.4 and 2.17. Numerical resultsfor transient natural convection between eccentric sphere has been reported by theauthors (Chiu and Chen [15]). However, most of the previous studies were concernedwith the spherical annulus enclosed by two isothermal boundaries; little work hasbeen reported on a spherical annulus with other types of thermal boundary condi-tions of engineering interest, even though Singh and Elliott [16] presented an ana-lytical solution of constant heat flux boundary condition on the inner surface and avariable temperature on the outer sphere surface, using a perturbation method.

To further extend the existing knowledge on natural-convection heat transferin spherical enclosures, the consideration in the present study is given to laminartransient natural convection in concentric and eccentric spherical annuli with

NOMENCLATURE

Cp specific heat at constant pressure

e vertical eccentricity

g local gravitational acceleration

h heat transfer coefficient

k thermal conductivity

L annular gap ð¼ �rro � �rriÞNu local Nusselt number (=hL=k)

Nu average Nusselt number (=hL=k)

Pr Prandtl number (¼ n=a)q heat flux

r dimensionless coordinate ð¼ �rr= �LLÞ�rr radial coordinate

R dimensionless radial profile of outer

sphere ð¼ �RR=LÞ�RR radial profile of outer sphere

R* radius ratio ð¼ ro=riÞRa Rayleigh number ð¼ gbD �TTL3=naÞRa* modified Rayleigh number

ð¼ gbqiL4=knaÞt time

T dimensionless temperature

½¼ ð �TT� �TToÞk=qiL��TT temperature

v velocity

V dimensionless velocity

(¼ vL=a)

Greek symbols

a thermal diffusivity

b thermal expansion coefficient

DT mean boundary temperature difference

between spheres, defined by Eq. (16)

e dimensionless vertical eccentricity

ð¼ e=LÞZ radial coordinate in transformed plane

½¼ ðr� riÞ=ðro � riÞ�y dimensionless angular coordinate

ð¼ �yy=pÞ�yy angular coordinate

y� angular position at vortex center

n kinematic viscosity

r fluid density

t dimensionless time ð¼ ta=L2Þc dimensionless stream function

ð¼ �cc=aLÞ�cc stream function in spherical

coordinates

o dimensionless vorticity ð¼ �ooL2=aÞ�oo vorticity

Subscripts

i,o inner, outer

max maximum

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constant heat flux boundary conditions. In practice, iso-heat flux boundary condi-tions are not usually obtained in most practical applications. However, since it ismathematically amenable, the emphasis of the present study is on the effect or thespecified flux boundary conditions and eccentricity on the transient behavior of theflow and temperature fields.

MATHEMATICAL FORMULATION AND NUMERICAL COMPUTATION

The geometric configuration of the physical system is a concentric or verticallyeccentric arrangement of two circular spheres of radii ro and ri located at O0 and O,respectively. The eccentricity of the outer sphere is measured by the distance e. If theouter sphere is placed above the central position, e has a positive value; otherwise e isnegative. For a natural-convective heat transfer problem, the local heat transfervariables are changed by the local circulation along the inner and outer spheres whenthe eccentricity is aligned with the gravitational direction. Therefore, this studyfocuses on the problem in which e is shifted vertically.

The space between the inner and outer spheres is filled with a viscous andincompressible Newtonian fluid. Initially, the annulus is at a uniform temperature,and a quiescent state is assumed. At t¼ 0þ , the inner sphere is heated uniformlywith a constant heat flux qi, while the outer sphere is cooled uniformly with aconstant heat uniformly flux qo, such that

Aiqi ¼ Aoqo

The heat transfer takes place between the spheres by natural convection. Amodel to describe the process has been derived making the following assumptions:(1) the flow within the annulus is laminar; (2) all fluid properties are taken to beconstant, except for the density variation with temperature in the buoyancy term,i.e., the Boussinesq approximation is valid; (3) the flow is symmetrical about thevertical axis, which is parallel to the line of gravity acceleration; (4) viscous dis-sipation and radiation effects are neglected.

A spherical polar coordinate system (r, y, j) was chosen as shown in Figure 1.To deal with the numerical formulation associated with the complex physical do-main of the vertically eccentric annulus, a radial coordinate transformation isadopted to map the eccentric annular gap into a unit sphere. The outer-sphere radiusr¼R(y) is transformed into the unit sphere Z¼ 1, while the inner-sphere radius r¼ riis transformed into the pole Z¼ 0.

This transformation is obtained by defining a new radial coordinate as

Z ¼ r� riRðyÞ � ri

ð1Þ

where R(y) denotes the variable dimensionless profile of the outer spheremeasured from the center of the inner sphere, which is symmetric with respect to thevertical axis in any angular position of j direction and is expressed by

RðyÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2o � e2 sin y

q� e cos y ð2Þ

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The governing equations for the two-dimensional problem in dimensionlessterms can be written as follows.

Vorticity transport equation:

qoqt

þ 1

pr2 sin pyð ÞqZqr

� �qcqZ

qoqy

�qcqy

qoqZ

� �þ 1

r

qZqy

�p cot pyð Þ qZqr

� �qcqZ

þ1

r

qcqy

� �o

� �

¼ Pr H21�

1

r2 sin2 pyð Þ

� �o þ PrRa� sin pyð Þ qZ

qrþ cos pyð Þ

prqZqy

� �qTqZ

þ cosðpyÞpr

qTqy

� �ð3Þ

Stream function equation:

D21c ¼ or sin pyð Þ ð4Þ

Energy equation:

qTqt

þ 1

pr2 sin pyð ÞqZqr

qcqZ

qTqy

� qcqy

qTqZ

� �¼ H2

1T ð5Þ

where

qZqr

¼ 1

R� rið6Þ

Figure 1. Coordinate system for the spherical annulus.

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qZqy

¼ �ZR� ri

qRqy

ð7Þ

q2Z

qy2¼ �1

R� riZ

q2R

qy2þ 2

qZqy

qRqy

� �ð8Þ

H21 ¼

qZqr

� �2

þ 1

p2r2qZqy

� �2" #

q2

qZ2þ 2

p2r2qZqy

� �� �q2

qZ qyþ 1

p2r2

� �q2

qy2

þ 2

r

qZqr

� �þ 1

p2r2q2Z

qy2

� �þ cot pyð Þ

pr2qZqy

� �� �qqZ

þ cot pyð Þpr2

� �qqy

ð9Þ

D21 ¼

qZqr

� �2

þ 1

p2r2qZqy

� �2" #

q2

qZ2þ 2

p2r2qZqy

� �� �q2

qZ qyþ 1

p2r2

� �q2

qy2

þ 1

p2r2q2Z

qy2

� �� cot pyð Þ

pr2qZqy

� �� �qqZ

� cot pyð Þpr2

� �qqy

ð10Þ

The fluid flows within the annulus between the stationary inner (Z¼ 0) andouter (Z¼ 1) sphere boundaries, assuming no slip at the boundaries and no flowacross the boundaries. The associated initial and boundary conditions for the pro-blem considered are as follows.

For t ¼ 0 :

o ¼ c ¼ qcqZ

¼ qcqy

¼ 0 T ¼ 0 everywhere ð11Þ

For t > 0 :

At Z ¼ 0;

c ¼ qcqZ

¼ 0qTqZ

¼ �ðR� riÞ o ¼ 1

ri sin pyð ÞqZqr

� �2 q2cqZ2

ð12Þ

At Z ¼ 1;

c ¼ qcqZ

¼ 0qTqZ

¼ ðR� riÞðcos d�1

prqRqy

sin dÞ r2ir2o

� �

where d ¼ sin�1 erosinðpyÞ

� �

o ¼ 1

ro sin pyð ÞqZqr

� �2

þ 1

p2r2qZqy

� �2" #

q2cqZ2

ð13Þ

CONVECTION BETWEEN CONCENTRIC AND ECCENTRIC SPHERES 437

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At y ¼ 0; 1;

c ¼ o ¼ qTqy

¼ 0 ð14Þ

From the above formulation, the governing parameters for the present problem arethe modified Rayleigh number Ra*, the Prandtl number Pr, the radius ratio R*, andthe eccentricity e.

Based on the difference in mean boundary temperatures of the inner and outerboundaries, an average Nusselt number can be defined as

Nu ¼ qiL

kD �TT¼ 1

DTð15Þ

in which

DT ¼ Tmð Þi� Tmð Þo ð16Þ

Tm ¼RR

A T dARRA dA

¼ p2

Z 1

0

T sin y dy ð17Þ

In the evaluation of Tm, the area integration over the circumference is done nu-merically.

Equations (3)–(5) expressing the vorticity transport, energy transport, andstream function equations, together with initial and boundary conditions in Eqs.(11)–(14), provide a complete description of the problem. Because the flow is knownto be parabolic in time but elliptic in space, the solution for the problem can only bemarched in time. In this study, the time-dependent verticity transport and energytransport equations were solved by employing the alternating direction implicit(ADI) finite technique, while the stream function equation was solved by employingthe successive overrelaxation (SOR) technique. The first and second derivatives inspace were approximated by central difference, while the time derivatives were ap-proximated by forward difference. Derivatives at the boundaries were approximatedby a three-point forward or backward difference. The results presented in this articlewere all obtained using a mesh of 416 41 cells. Numerical test calculations were alsoperformed for different time step sizes. Two different time step sizes, depending onthe geometry, have been used for the calculations: 16 1074 for e¼ 0.0 and 56 1075

for e¼�0.625.The solution was considered convergent when the relative error between the

new and old values of the field variables F during every time step was less than aprescribed criterion (107 4), where F represents o, c, and T.

Fnew � Foldj jmax

Fnewj jmax

� 10�4 ð18Þ

Further, the steady-state solution was determined by requiring the relativeerror between the present and next time step values of all field variables for the innerand outer spheres to be

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Fnþ1 � Fn�� ��

max

Fnþ1�� ��

max

� 10�4 ð19Þ

where the superscripts n and (nþ 1) indicate the nth and (nþ 1)st time steps,respectively.

RESULTS AND DISCUSSION

Numerical calculations have been performed systematically for a spherical annu-lus of radius ratio fixed at 2.0 and a Prandtl number of 0.7, with modified Rayleighnumberranging from103to5.06105andeccentricitiesvaryingfrom 70.625 toþ0.625.

Figure 2a shows an evolution of streamlines and isotherms for a modifiedRayleigh number of 103. Figure 2b presents a corresponding evolution for a modifiedRayleigh number of 105. This evolution of results is designed to demonstrate theeffect of modified Rayleigh number on the heat and fluid flow patterns in the con-centric annulus. Because the problem is symmetric about the axis, each annuluscontains the isotherms on the left and streamlines on the right. To facilitate com-parison of the different configurations, the number of contours within each geometryis kept constant at 11 for temperature and at 6 for stream function. Since the innersphere is kept hotter, the hot fluid near the inner sphere rises upward due to thermalexpansion. The rising plume is then cooled by the colder fluid near the upper part ofthe outer sphere. The colder and denser fluid will eventually flow downward alongthe surface of the outer sphere. At the first time step, corresponding to (I), the fluidflow in the spheres is weak and forms a recirculation zone in the clockwise direction.The isotherms are nearly circular and distribute near the sphere wall, which impliesthat the local heat transfer near both the inner and outer sphere wall is almost thesame as in the pure conduction case, further indicating that there is little influenceof the convective flow on heat transfer. When the time step proceeds from (II) to(III), the fluid motion in the annulus becomes gradually strong, which demonstratesthat the buoyant effect is increasing along the inner sphere wall and decreasing alongthe outer sphere wall, the maximum value of the stream function is increasing, andthe vortex center of the eddy is shafting upward above the horizontal axis. Finally,during a later time step, at (III)–(IV), correspondingly the heat and flow will tend toa quasi-steady state.

For the low value of modified Rayleigh number of Ra*¼ 103 in Figure 2a, astime proceeds, it is found that the position of the vortex center of the eddy firstmoves upward and then slowly moves downward along the annular space, while themaximum value of the stream function first increases and then slowly decreases, untilat steady state the maximum value of stream function cmax¼ 1.350 and the angularposition of the crescent-shaped vortex center lies at y*¼ 76.5� from the upper ver-tical line symmetry about the midgap position. Simultaneously, the transient tem-perature contours exhibit features with specified heat flux conditions quite differentfrom the cases of isothermal boundaries available in the literature. In general, iso-therms start from the inner sphere wall and end at the outer sphere wall. This featureincreases with increasing time step until the system reaches steady state. However,for the positive eccentricity geometry, some of the isotherms begin from and end atthe outer sphere boundary (Figure 3a).

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At Ra*¼ 105, as shown in Figure 2b, the fluid motion becomes stronger, asindicated by the increase in value of the stream function, and the vortex center of theeddy shifts upward. Examination of the isothermal patterns also reveals that laminarconvection is the dominant mode of heat transfer, as opposed to, the pseudo-conduction heat transfer regime which appeared in Figure 2a.

Next, the isotherms and streamlines for the eccentric configurations consideredin this study will be examined. Figures 3 and 4 illustrate the transient streamlines andisotherms at different Ra* for positive and negative eccentricity, respectively.Common to all three geometries considered is the existence of higher temperaturegradients as seen at the outer wall at y¼ 0 for higher values of the modified Rayleighnumber when a quasi-steady state is approached. The presence of large convection atthe higher values of the modified Rayleigh numbers is demonstrated by the

Figure 2. Isotherms (left) and streamlines (right), for R*¼ 2.0, Pr¼ 0.7, and e¼ 0.0 at different time steps:

(a) Ra*¼ 1.06103; (b) Ra*¼ 1.06105.

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appearance of temperature inversions mainly in the upper half of each annulus. Theeffect of eccentricity on the flow is revealed by comparing, for different eccentricities,the isotherms and streamlines at fixed values of the modified Rayleigh number. Forthe positive eccentric geometry (Figure 3), it is evident that the convective flows areboth larger and stronger than for the concentric spheres. In such a favorable con-figuration for convective motion, the two spheres are very close to each other alongy¼ p. In this region, the effects of convection are diminished, and the shape of theisotherms is determined largely by the conductive mode of heat transfer.

The effect of negative eccentric geometry is shown in Figure 4. Here the con-vective cell center has moved toward y¼ p. This cell is also less powerful than in theconcentric case. Clearly, this geometric configuration of the two spheres inhibitsconvective motion in the fluid. Thus the isotherms show smaller temperature

Figure 3. Isotherms (left) and streamlines (right), for R*¼ 2.0, Pr¼ 0.7, and e¼ 0.625 at different time

steps: (a) Ra*¼ 1.06103; (b) Ra*¼ 1.06105.

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inversions when compared to the concentric case, Figure 2. The bottom of theannulus (near y¼ p) exhibits an enlarged stagnant region of fluid. In each case, themotion of the fluid is clockwise. A region of almost stagnant fluids exists near y¼ p,at the lower region; this zone increases in size for the negative eccentricities becauseof the increased resistance to flow.

Compared to the cases of isothermal boundaries available in the literature(Chu and Lee [10] and Chiu and Chen [15]), the flow pattern obtained with specifiedheat flux boundary conditions is essentially similar to results obtained under iso-thermal boundary conditions with fluid rising along the inner boundary and re-circulating by falling down along the outer colder boundary. However, thetemperature field obtained with specified heat flux boundary conditions is differentfrom results obtained under isothermal boundary conditions. In general, isotherms

Figure 4. Isotherms (left) and streamlines (right), for R*¼ 2.0, Pr¼ 0.7, and e¼70.625 at different time

steps: (a) Ra*¼ 1.06103; (b) Ra*¼ 1.06105.

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start from the inner wall and end up on the outer wall due to constant heat fluxcondition on the inner and outer boundaries. Common to all three geometriesconsidered is the existence of a stagnant region at the bottom of the annulus, becausethe convection cells are weak for thermally stable fluid in these regions. The presenceof heat transfer begins with the mode of conduction dominated in the middle of theannulus at the initial stage. This initial stage of conduction corresponds to a regionof static fluid between two flat plates.

For the specified uniform heat flux condition on the inner and the outerspheres, surface temperature distribution is one of the important parameters in thecalculations. The transient variation of circumferential temperature distributionsalong the inner and the outer boundaries at Ra* of 103 and 105 are presented inFigures 5–7 for the three configurations under study. Common to the results dis-played for the three geometries is that, as the modified Rayleigh number increases,the local dimensionless surface temperature of the inner boundary decreases, in-dicative of a higher rate of heat transfer due to the stronger buoyancy flow in theannulus. For the concentric annulus, as expected, an isothermal surface temperatureprofile occurs for pure conduction, Ra*¼ 0, as illustrated in Figure 5. The enhancedconvective fluid flow in the annulus with increasing Ra* causes a significant variationof the surface temperature along the circumference of both spheres at Ra*¼ 103–105.The local surface temperature decreases from the top to the bottom of the inner andthe outer boundaries, as expected from physical considerations. Similar observationscan be made for the positive eccentric geometry considered for Ra*¼ 1.06105, asshown in Figure 6b. However, in Figure 6a, for the case of low Rayleigh number(Ra*¼ 1.06103) of the positive eccentricity, the minimum local surface temperatureof the inner boundary is located at y¼ 65� and the outer boundary at y¼ 72�.However, the location of the maximum surface temperature shifts to the bottom ofthe inner and outer boundaries because the convection cells are not strong enough atthe bottom of both spheres. Similar observation concerning the distribution ofconvection cells in the positive eccentric annulus for the case of Ra*¼ 1.06103 canbe made in the isotherm patterns (Figure 3a). As for the negative eccentric annulus,Figure 7, the temperature profile along the inner boundary for pure conduction isopposite that for the positive eccentricity geometry shown in Figure 6. The surfacetemperature decreases from the top to the bottom of each sphere. For Ra � 105, atemperature profile similar to the cases described above can be clearly detected in thefigure. Finally, the circumferentially averaged Nusselt numbers at steady state ob-tained in the present study are given in Table 1 for various modified Rayleighnumbers in the three annular geometries under consideration. As the modifiedRayleigh number increases beyond the conduction regime, the curves beyond thepseudo-conduction region are straight lines on log-log coordinates. Nu for a dia-meter ratio of 2.0 may be represented by an equation of the form

Nu ¼ CðRa�Þm ð20Þ

The values of C and m are listed in Table 2 for the three configurations consideredhere.

In order to compare the heat transfer results of constant heat flux and iso-thermal boundary conditions, Eq. (20) can be rewritten as Eq. (21), where the cor-responding coefficients C1 and m1 are listed in Table 3:

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Nu ¼ C1ðRaÞm1 ð21Þ

Noting that the modified Rayleigh number Ra*¼NuRa, in Figure 8 theaverage Nusselt number at steady state is also plotted versus the Rayleigh number

Figure 5. Transient variation of local surface temperature distribution on the inner and outer spheres for

R*¼ 2.0, Pr¼ 0.7, and e¼ 0.0: (a) Ra*¼ 1.06103; (b) Ra*¼ 1.06105.

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for a diameter ratio R* of 2.0. For comparison, data for similar configurations withthe isothermal boundary conditions of Bishop et al. [1], Chiu and Chen [15], Chu andLee [10], and Scanlan et al. [2] are also plotted in the figure. For all three geometries

Figure 6. Transient variation of local surface temperature distribution on the inner and outer spheres for

R*¼ 2.0, Pr¼ 0.7, and e¼ 0.625: (a) Ra*¼ 1.06103; (b) Ra*¼ 1.06105.

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considered here, the average Nusselt number for the heat flux boundary conditions islarger than for the isothermal boundaries due to the definition of Ra*¼Nu Raemployed. Closer examination of Figure 8 reveals that the increase in the average

Figure 7. Transient variation of local surface temperature distribution on the inner and outer spheres for

R* ¼ 2.0, Pr ¼ 0.7, and e¼7 0.625: (a) Ra*¼ 1.06103; (b) Ra*¼ 1.06105.

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Table 2. Empirical constants and deviations for Eq. (20)

e C m Ra* Max. deviation (%)

0.625 0.6714 0.1814 103–105 3.61

0.000 0.8033 0.1588 103–56105 6.18

70.625 0.9496 0.1309 103–105 4.30

Table 1. Average Nusselt number (R*¼ 2.0, Pr¼ 0.7)

Nu for Ra*

e 103 56103 104 56104 105 56105

0.625 2.3028 3.3075 3.4938 4.7577 5.4074 —

0.000 2.2655 3.1816 3.6278 4.6116 5.0451 6.1442

70.625 2.2782 2.9476 3.2520 4.0186 4.1070 —

Table 3. Empirical constants and deviations for Eq. (21)

e C1 m1 Ra Max. deviation (%)

0.625 0.7325 0.2005 4.346102–1.856104 3.65

0.000 0.8716 0.1636 4.416102–8.146104 6.25

70.625 1.1953 0.1064 3.396102–2.436104 4.33

Figure 8. Average Nusselt number as a function of Rayleigh number for R*¼ 2.0 and Pr¼ 0.7.

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Nusselt number for the heat flux boundary conditions is less prominent at highmodified Rayleigh number (Ra*> 3.56103) for the negative eccentric geometry, dueto its less favorable circumstances for natural convection.

CONCLUSIONS

The transient natural convection in concentric and vertically eccentric sphereswith constant heat flux boundary conditions has been analyzed numerically by afinite-difference method. The transient behavior of the heat and fluid flow in theannuli have been visualized by means of contour maps of isotherms and streamlines.Although the streamline patterns look similar to those obtained in previous in-vestigations with isothermal boundary conditions, the isotherm plots show differentheat flow patterns. The numerical results obtained further indicate that heat andfluid flow patterns in the annuli are primarily dependent on the modified Rayleighnumber and the eccentricity. The average Nusselt number increases with Rayleighnumber in each eccentricity displacement. The positive eccentric geometry can en-hance convective heat transfer rates, but the negative eccentric geometry provides theleast favored circumstance for the development of natural convection between twospheres in the annulus. Above all, the heat transfer rate is higher for the constantheat flux case than for isothermal heating when the equivalent temperature differenceis the same in both cases, due to the definition of Ra*¼NuRa employed.

REFERENCES

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13. E. Powe, R. C. Baughman, J. A. Scanlan, and J. T. Teng, Free Convection Flow Patternsbetween a Body and Its Spherical Enclosure, J. Heat Transfer, vol. 97, pp. 296–298, 1975.

14. E. Powe, R. O. Warrington, and J. A. Scanlan, Natural Convection between a Body andIts Spherical Enclosure, Int. J. Heat Mass Transfer, vol. 23, pp. 1337–1350, 1980.

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Concentric and Vertically Eccentric Spheres, Int. J. Heat Mass Transfer, vol. 39, pp. 1439–1452, 1996.

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Slightly-Thermally Stratified Medium, Int. J. Heat Mass Transfer, vol. 24, pp. 395–406,1981.

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