heat and mass transfer by natural convection around a hot

Scientia Iranica B (2013) 20(5), 1474{1484

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Heat and mass transfer by natural convection around ahot body in a rectangular cavity

G.A. Sheikhzadeha, R. Heydaria, N. Hajialigola;�, A. Fattahiaand M.A. Mehrabianb

a. Department of Mechanical Engineering, University of Kashan, Kashan, P.O. Box 87317-51167, Iran.b. Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Iran.

Received 1 February 2012; received in revised form 7 December 2012; accepted 6 May 2013

KEYWORDSHeat and masstransfer;Natural convection;Cavity.

Abstract. The simultaneous heat and mass transfer by natural convection around ahot body in a rectangular cavity is investigated numerically. The cavity is �lled withair and the ratio of body's length to enclosure's length is assumed to be constant at1/3. The di�erential equations for continuity, momentum, energy and mass transfer aresolved using the Patankar technique. The results are displayed in the form of isotherms,isoconcentrations and streamlines, and the e�ects of Rayleigh number, Lewis number andbuoyancy ratio on average Sherwood and Nusselt numbers are investigated. The studycovers a wide range of Rayleigh numbers, Lewis numbers, and buoyancy ratios. It isobserved that by increasing Lewis number, the average Sherwood number increases andthe average Nusselt number decreases. Additionally, by increasing the absolute value ofbuoyancy ratio, the average Sherwood and Nusselt numbers enhance. This work presents anovel approach in this �eld in the light of geometry and the range of dimensionless numbers.

© 2013 Sharif University of Technology. All rights reserved.

1. Introduction

Natural convection has become the subject of manyexperimental and numerical studies in recent yearsdue to its widespread applications in engineering andindustry, such as cooling of electronic systems, oatglass manufacturing, solar ponds and metal solidi�-cation processes. Davis [1] investigated a benchmarkfor numerical solution on natural convection of airin a square enclosure. Ostrach [2] traced out therich diversity of natural convection problems in scienceand technology. The numerous studies have focusedextensively on convective ows driven by the densityinversion e�ect [3-10].

The steady-state ow structure, temperature and

*. Corresponding author.E-mail address: [email protected] (N.Hajialigol)

heat transfer in a square enclosure, heated and cooledon opposite vertical walls and containing cold water,are numerically investigated by Lin and Nansteel [3].Nansteel et al. [4] studied the natural convection ofwater in the vicinity of its maximum density in arectangular enclosure in the limit of small Rayleighnumber. They observed that the strength of thecounter rotating ow decreases with decreasing aspectratio. Hossain and Rees [8] studied the naturalconvection in an enclosure with heat generation. Inthis analysis, when the heat generation parameteris su�ciently strong, the circulation of the ow isreversed.

Lee and Ha [11] numerically investigated naturalconvection in a horizontal enclosure with a conductingbody. They compared the results of the case ofconducting body with those of adiabatic and neutralisothermal bodies. They showed that when the dimen-sionless thermal conductivity is 0.1, a pattern of uid

G.A. Sheikhzadeh et al./Scientia Iranica, Transactions B: Mechanical Engineering 20 (2013) 1474{1484 1475

ow and isotherms and the corresponding surface- andtime-averaged Nusselt numbers are almost the same asthe case of an adiabatic body.

Das and Reddy [12] studied natural convection ow in a square enclosure with a centered internalconducting square block both having an inclinationangle using SIMPLE algorithm. They considered anangle of inclination in the range of 15-90 degrees andratio of solid to uid thermal conductivities of 0.2and 5. Sheikhzadeh et al. [13,14] studied the steadymagneto-convection around an adiabatic body inside asquare enclosure.

The simultaneous heat and mass transfer occursin many processes in industry and engineering equip-ment and environmental applications, involving thetransport of water vapor and other chemical contami-nants across enclosed spaces. Comparison of the scalesof the two buoyancy terms in the momentum equationsuggests three classes of ows:

(i) Heat transfer driven ows when the buoyancye�ect, due to heating from the sides, is dominant;

(ii) Combined heat and mass transfer driven owswhen both buoyancy terms are important;

(iii) Mass transfer driven ows, when the buoyancy,due to heating from the sides, is negligible.

The distinction between these ows can be made usingthe buoyancy ratio (N); jN jhO(1) for class (i), jN j �O(1) for class, (ii), and jN jiO(1) for class (iii).

Gebhart and Pera [15], using the similaritymethod, investigated the laminar ows which arise in uids due to the interaction of the force of gravityand density di�erences caused by the simultaneousdi�usion of thermal energy and chemical species. Bejan[16] presented a fundamental study of laminar naturalconvection in a rectangular enclosure with heat andmass transfer from the side. He used scale analysisto determine the scales of the ow, temperature andconcentration �elds in boundary layer ow for all valuesof Prandtl and Lewis numbers. He investigated the caseof N = 0 to study the heat-transfer-driven ows. Weeet al. [17] investigated numerically and experimentallythe same problem for both horizontal and verticalcavity. The experimental technique employed twoporous plastic plates as two cavity walls allowing theimposition of simultaneous moisture and temperaturegradients.

Nishimura et al. [18] studied numerically theoscillatory double-di�usive convection in a rectangularenclosure with combined horizontal temperature andconcentration gradients. Natural convection ow, re-sulting from the combined buoyancy e�ects of thermaland mass di�usion in a cavity with di�erentially heatedsidewalls, was numerically studied by Snoussi et al. [19].They considered a wide range of Rayleigh numbers

and used �nite-element method. They showed thatmass and temperature �elds are strongly dependenton thermal Rayleigh number and the aspect ratio ofthe cavity. Chouikh et al. [20] studied numerically thenatural convection ow resulting from the combinedbuoyancy e�ects of thermal and mass di�usion inan inclined glazing cavity with di�erentially heatedside walls. The double-di�usive natural convection ofwater in a partially heated enclosure with Soret andDufour e�ects is numerically studied by Nithyadeviand Yang [21]. The e�ect of various parameters suchas thermal Rayleigh number, buoyancy ratio number,Schmidt number and Soret and Dufour coe�cientson the ow pattern, heat and mass transfer wasdepicted. Nikbakhti and Rahimi [22] studied double-di�usive natural convection in a rectangular cavity withpartially thermal active side walls. They found thatin aiding ow, heat transfer increases by increasingthe buoyancy ratio. Their results also showed thatin opposing ow, with increasing buoyancy ratio untilunity, heat transfer decreases.

Al-Amiri et al. [23] and Teamah and Magh-lany [24] studied the heat and mass transfer in a lid-driven cavity. More recently, Mahapatra et al. [25] an-alyzed the e�ects of uniform and non-uniform heatingof wall(s) on double-di�usive natural convection in alid-driven square enclosure.

In the present study, natural convection arounda hot body in a square cavity �lled with air isstudied numerically, using the �nite volume method.Simultaneous heat and mass transfer and the e�ects ofpertinent parameters such as buoyancy ratio, Rayleighand Lewis numbers on the ow are studied. Despitenumerous works in this �eld, to the best of authors'knowledge, no similar study has been carried out tosimulate simultaneous heat and mass transfer in thecurrent geometry with the range of buoyancy ratio,Rayleigh and Lewis numbers used in this study.

2. Mathematical model

A schematic diagram of the enclosure with coordinatesystem and boundary conditions is shown in Figure 1.Both body and enclosure walls are held at constantbut di�erent temperatures and concentrations; hightemperature and concentration (Th, Ch) for the bodyand low temperature and concentration (Tc, Cc) for thecavity are considered. In this study, the ratio of W=Lis maintained constant at 1/3, and the body is locatedin the center of the cavity.

The Boussinesq approximation holds, meaningthat density is linearly proportional to both temper-ature and concentration.

� = �0 (1� �T (T � Tc)� �M (c� cc)) : (1)

With these assumptions, two dimensional laminar

1476 G.A. Sheikhzadeh et al./Scientia Iranica, Transactions B: Mechanical Engineering 20 (2013) 1474{1484

Figure 1. Geometry and boundary conditions.

governing equations including continuity, momentum,concentration and energy in steady state conditions canbe written as:

Continuity equation:

@u@x

+@v@y

= 0: (2)

x-momentum equation:

u@u@x

+ v@u@y

= �1�@p@x

+ ��@2u@x2 +

@2u@y2

�: (3)

y-momentum equation:

u@v@x

+ v@v@y

= �1�@p@y

+ ��@2v@x2 +

@2v@y2

�+ g[�T (T � Tc) + �M (c� cc)]: (4)

Energy equation:

u@T@x

+ v@T@y

= ��@2T@x2 +

@2T@y2

�: (5)

Concentration equation:

u@c@x

+ v@c@y

= D�@2c@x2 +

@2c@y2

�: (6)

Using the dimensionless variables:

X =xL; Y =

yL; U =

uL�;

V =vL�; P =

p�L2

�2 ; � =T � TcTh � Tc ;

C =c� ccch � cc ; (7)

the dimensionless form of the equations become:

@U@X

+@V@Y

= 0; (8)

U@U@X

+ V@U@Y

= � @P@X

+ Pr�@2U@X2 +

@2U@Y 2

�; (9)

U@V@X

+ V@V@Y

= �@P@Y

+ Pr�@2V@X2 +

@2V@Y 2

�+ RaT Pr(� +NC); (10)

U@�@X

+ V@�@Y

=@2�@X2 +

@2�@Y 2 ; (11)

U@C@X

+ V@C@Y

=1

Le

�@2C@X2 +

@2C@Y 2

�: (12)

The thermal and mass Rayleigh numbers, Lewis num-ber, Prandtl number, and buoyancy ratio in the aboveequations are de�ned as:

RaT =g�TL3�T

��;

RaM =g�ML3�c

D�= RaTLeN;

Le =�D;

Pr =��;

N =�M�c�T�T

; (13)

where �T and �M are thermal and concentrationexpansion coe�cients. �T is positive but �M can benegative or positive [16]:

�T = �1�

�@�@T

�P;

�M = �1�

�@�@c

�P: (14)

The values of �M for some species are presented byGebhart and Pera [15].

Dimensionless boundary conditions are as follows:

0 � X � 1!U(X; 0) = U(X; 1) = V (X; 0) = V (X; 1) = 0;

�(X; 0) = �(X; 1) = C(X; 0) = C(X; 1) = 0:

0 � Y � 1!U(0; Y ) = U(1; Y ) = V (0; Y ) = V (1; Y ) = 0;

�(0; Y ) = �(1; Y ) = C(0; Y ) = C(1; Y ) = 0:


L2� W

2� X � L

2+W2!

U�X;

L2� W

2

�= U

�X;

L2

+W2

�= V

�X;

L2� W

2

�= V

�X;

L2

+W2

�= 0;

��X;

L2� W

2

�= �

�X;

L2

+W2

�= C

�X;

L2� W

2

�= C

�X;

L2

+W2

�= 1:

L2� W

2� Y � L

2+W2!

U�L2� W

2; Y�

= U�L2

+W2; Y�

= V�L2� W

2; Y�

= V�L2

+W2; Y�

= 0;

��L2� W

2; Y�

= ��L2

+W2; Y�

= C�L2� W

2; Y�

= C�L2

+W2; Y�

= 1: (15)

Local and average Nusselt and Sherwood numbers onthe vertical left side hot wall are de�ned as follows:

Nu = ��@�@X

��X=L

2 �W2; (16)

Nua = � 1A

AZ0

�@�@X

�X=L

2 �W2dY; (17)

Sh = ��@C@X

��X=L

2 �W2; (18)

Sha = � 1A

AZ0

�@C@X

�X=L

2 �W2dY: (19)

In general, Sherwood number represents the masstransfer strength like Nusselt number that symbolizesthe heat transfer strength.

3. Numerical method

The governing non-linear equations with appropriateboundary conditions were solved by iterative numericalmethod, using �nite volume technique. In order tocouple the velocity �eld and pressure in momentum

equations, the well-known SIMPLER (Semi-ImplicitMethod for Pressure-Linked Equations Revised) algo-rithm was adopted. Uniform grid is used for this prob-lem and grid independency test was performed. Thediscretized equations were solved by the Gausse-Seidelmethod. The iteration method used in this programis a line-by-line procedure which is a combination ofthe direct method and the resulting Three DiagonalMatrix Algorithm (TDMA). The di�usion terms inthe equations are discretized by a second order centraldi�erence scheme, while a hybrid scheme (a combina-tion of the central di�erence scheme and the upwindscheme) is employed to approximate the convectionterms. The under-relaxation factors for U-velocity,V-velocity, energy and concentration equations were0.6, 0.6, 0.4 and 0.4, respectively. The solution isterminated until the following convergence criterion ismet:

Error =

mPj=1

nPi=1

��t+1 � �t��mPj=1

nPi=1j�t+1j

� 10�10; (20)

where m and n are the number of cells in X and Ydirections, respectively, � is a transport quantity, andt is the number of iteration.

3.1. Code validationTo reach the grid independent solution, various grids(21 � 21, 61 � 61, 81 � 81 and 101 � 101) weretested for calculating the average Nusselt number onthe hot walls (Table 1). The grid size of 81 � 81 isadequately appropriate to ensure a grid-independentsolution.

Figures 2 and 3 present a comparison between thepresent work and the results of Lee and Ha [11] and

Table 1. Grid independency test for RaT = 104, N = 1and Le = 1.

Grid 21� 21 61� 61 81� 81 101� 101Nua 5.310 6.056 6.162 6.167

Figure 2. The comparison between Isotherms foradiabatic body at Ra = 103 (left) and Ra = 104 (right);Lee and Ha [11] (solid lines), present work (dashed line).


Figure 3. Isotherm (right), and stream line (left) forLe = 1, N = 10, Ri = 1 and Uup = �1; Teamah [24] (solidlines), present work (dashed lines).

Teamah and Maghlany [24], respectively. It is obviousthat the two sets of results are in good agreement.

4. Results and discussion

The present study considers the e�ects of buoyancyratio (�10 � N � 10), Rayleigh number (103 �RaT � 105) and Lewis number (0:1 � Le � 50) onthe isotherms, isoconcentrations, streamlines, averageNusselt number and average Sherwood number.

4.1. E�ect of buoyancy ratioDependence of streamlines, isoconcentrations andisotherms on buoyancy ratio is shown in Figures 4, 5,and 6. Figure 7 shows the e�ect of N and Le number onaverage Sherwood and Nusselt numbers. To investigate

the e�ect of buoyancy ratio, the thermal Rayleighnumber is kept constant at 104, and Lewis numbersare chosen 0.1, 1 and 10. At all Lewis numbers andbuoyancy ratios, two primary vortices are formed at theright and left sides of the hot body, respectively. Byincreasing the absolute value of N and Le number, themaximum absolute values of stream function for theseprimary vortices increase and decrease, respectively.Discussion about the e�ect of N on uid ow, heat andmass transfer is done in three cases: N > 0, N < 0 andN = 0.

N > 0: At Le=1 and 10, by increasing the value of Nfrom 0 to 10, two secondary vortices are appeared atthe top of the hot body; the strength of these secondaryvortices are lower than that of the primary vortices.At Le = 0.1, the mass di�usivity is higher than thethermal di�usivity and for all N , isoconcentration linesare approximately parallel to the walls (Figure 4).As shown in Figure 4, by increasing the buoyancyratio, isotherms become denser near the bottom wallof the hot body, and two weak isothermal plumes areappeared at the top corners of the body. Accordingto Figure 7, it is observed that by increasing thebuoyancy ratio, the average Nusselt number increasesbut the average Sherwood number is approximatelyconstant. Figure 7 shows also that the values of averageNusselt number for Le=0.1 are higher than those ofSherwood number. At Le=1, plots of isoconcentrationsand isotherms are the same, and the average Nusseltnumber and Sherwood number are equal [11] and theyincrease with increasing N . According to Figure 5, twoisothermal and isoconcentration plumes are appeared

Figure 4. Isotherms, streamlines and isoconcentrations for Ra = 104 and Le = 0:1.


Figure 5. Isotherms, isoconcentrations and streamlines for Ra = 104 and Le = 1.

Figure 6. Isotherms, streamlines and isoconcentrations for Ra = 104 and Le = 10.

Figure 7. E�ect of Lewis number and buoyancy ratio on average Sherwood and Nusselt numbers.


at the top corners of the body. At Le=10, theisoconcentration plumes are more obvious than theisothermal plumes.

N < 0: At Le=1 and 10, by decreasing N from0 to -10, two secondary vortices are appeared at thebottom of the hot body. At Le=10, the strength ofthe secondary vortices bocomes more than that of theprimary vortices. At Le=0.1, two weak isothermalplumes are appeared at the bottom corners of thehot body. It is also observed from Figure 7 thatby decreasing N from -1 to -10, the average Nusseltnumber increases, but the average Sherwood number isapproximately constant. At Le=1, two isothermal andisoconcentration plumes are appeared at the bottomcorners of the hot body at N = �10 (Figure 5). Bydecreasing N from -1 to -10, the average Sherwood andNusselt number increase (Figure 7). By decreasing N ,both heat and mass transfer are increased. At Le=10,the formed isoconcentration plumes, at the bottomcorners of the hot body, are more obvious than theisothermal plumes. At N = �1 and Le=1 there isno convection ow, and therefore the average Nusseltnumber and Sherwood number have their minimumvalues at this condition. As Figure 7 shows, at lowLe, Sha and Nua are not a function of N .

N = 0: At N = 0, convection ow is formed only dueto temperature gradient. One isoconcentration plumeat the top of the hot body is appeared at Le=10.

Flow direction in the vortices for negative andpositive values of N are shown in Figure 5; forinstance at N = 10 (aiding e�ect of temperature andconcentration gradients) and N = �10 (opposing e�ectof temperature and concentration gradients). It isobserved that at N = �10, the primary vortex is CCWand the secondary vortex is CW at the right side of thehot body (in the direction caused by the predominantmass gradient), and at N = 10, the primary vortex isCW and the secondary vortex is CCW at the right sideof the hot body.

4.2. E�ect of Lewis numberThe e�ect of Lewis number at Ra = 104 on plotsof isotherms, isoconcentrations and streamlines forN = 1, 10 and -1 are shown in Figures 8, 9 and10, respectively. When Le=1, the isotherms and iso-concentrations are similar to each other due to thesimilarity between the energy and concentration equa-tions. It is observed that by increasing Le, convection ow suppresses and so the maximum absolute valueof stream function decreases. At N = 1 for Le > 10there is an isoconcentration plume at the top of thehot body. At N = 10, two secondary vortices and twoplumes are appeared at the top of the hot body at Le=1and 10. The secondary vortices vanish at Le=30 and50, and two plumes become one plume. It is observedfrom Figure 9 that isoconcentration plumes are moreobvious than that of isothermal plume. It is foundfrom Figure 7 that mass transfer increases when Le

Figure 8. Isotherms, Isoconcenterations and streamlines for N = 1 and Ra = 104.


Figure 9. Isotherms, isoconcenterations and streamlines for N = 10 and Ra = 104.

Figure 10. Isotherms, isoconcenterations and streamlinesfor N = �1 and Ra = 104.

enhances as a result of densing solutal boundary layer(it is seen in Figure 8). On the other hand, Nua reduceswhen Le increases. As Le increases, thermal di�usivityincreases (� = k=�cp). It causes conduction heattransfer to be stronger at high Le compared to the lowLe cases where convection is stronger and dominant.Because conduction has a minor e�ect on heat transferin comparison with convection, Nu decreases as Leincreases.

The e�ect of Le on isotherms, isoconcentrations

Table 2. Correlations of Sha and Nua in terms of Le andN .

Range of N Correlation

N � 0 Sha = 7:3167Le0:2236N0:1103

Nua = 7:0775Le�0:0764N0:0818

N � �1 Sha = 5:2186Le0:2379jN j0:2418

Nua = 6:4143Le�0:0525jN j0:0937

and streamlines at N = �1 is shown in Figure 10. Theisotherms are approximately the same. It is observedthat by increasing Le, the maximum absolute value ofstream function increases.

The predicted values of Sha and Nua in theranges investigated for N and Le and at Ra = 104

are correlated and presented in Table 2. The validityranges for these correlations are 0:1 � Le � 50 and�10 � N � 10.

4.3. E�ect of Rayleigh numberE�ect of Rayleigh number on plots of isotherms, iso-concentrations and streamlines, for instance at N = 1and Le=1, are shown in Figure 11. It is observed,as expected, that convection ow becomes stronger byincreasing Ra, and the secondary vortices are appearedat Ra = 105. Variations of Sha and Nua with respect toN at various Ra are shown in Figure 12. In this �gureit is shown that Nu and Sh increase with Rayleighnumber.

The predicted values of Sha and Nua over theranges investigated for N and Ra and at Le=1 arecorrelated and presented in Table 3. The validityranges for these correlations are: 0:1 � Le � 50 and


Figure 11. E�ect of Rayleigh number on isotherms (and also isoconcentrations) and streamlines, for Le = 1 and N = 1.

Figure 12. E�ect of Rayleigh number on average Nusseltand Sherwood numbers on hot wall at Le = 1 and variousN .

Table 3. Correlations of Sha and Nua in terms of Ra andN .

Range of N CorrelationN � 0 Nua = Sha = 0:91526Ra0:2198N0:1557

N � �1 Nua = Sha = 0:66599Ra0:2002N0:4114

�10 � N � 10. Figure 13 shows the comparison be-tween numerical results and the correlations in Table 3.

5. Conclusion

In this article, simultaneous natural convection heatand mass transfer around a hot body in a rectangularair-�lled cavity was studied numerically. According tothe present numerical results, the following conclusionsare drawn:

1. By increasing the absolute value of Buoyancy

Figure 13. Comparison between numerical results andthe correlation for Ra = 105 and Le = 1.

ratio, the average Nusselt and Sherwood numbersincrease.

2. By incresing Lewis number, the average Sherwoodnumber increases but the average Nusselt numberdecreases.

3. By increasing Rayleigh number, the averageNusselt and Sherwood numbers increase at Le=1and N = 1.

4. The predicted values of average Sherwood andNusselt numbers in the range investigated forbuoyancy ratio, Rayleigh and Lewis numbers arecorrelated.

Nomenclature

C ConcentrationC Dimensionless concentration


CCW CounterclockwiseCW ClockwiseD Mass di�usivityH Enclosure heightL Enclosure lengthLe Lewis number (Sc/Pr)N Buoyancy ratioNu Nusselt numberp PressureP Dimensionless pressurePr Prandtl numberRaM Mass transfer Rayleigh numberRaT Heat transfer Rayleigh numberSc Schmidt numberSh Sherwood numberT Temperatureu; v Components of velocityU; V Dimensionless velocity componentsW Body lengthx; y Cartesian coordinatesX;Y Dimensionless Cartesian coordinates

Greek symbols

� Thermal di�usivity Streamfunction� Dimensionless temperature� Kinematic viscosity�T Volumetric coe�cient of thermal

expansion�M Volumetric coe�cient of expansion

with concentration

Subscript

a Averagec Cold wallh Hot wallT ThermalM Mass

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Biographies

Ghanbar Ali Sheikhzadeh is an Associate Professorof Mechanical Engineering Department at Universityof Kashan, Kashan, Iran. He received his PhD in Me-chanical Engineering from Shahid Bahonar Universityof Kerman. His research works concern numerical anal-ysis and application of heat transfer in nano-systemsand other areas of thermal and uid sciences. Dr.Sheikhzadeh has published many papers in journalsand conferences in his research �elds.

Roghayeh Heydari received her MS degree in Me-chanical Engineering at University of Kashan, Kashan,Iran. Her research interests focus on numerical analysisof heat and mass transfer in nano-systems and otherareas of thermal and uid applications. Her researchworks are also paid to combustion systems such asnumerical investigations of ames and furnaces.

Najmeh Hajialigol is a PhD student at TarbiatModares University of Tehran, Tehran, Iran. Shereceived her MSc degree from University of Kashan,Kashan, Iran. Her research activities are paid tocombustion, computational uid dynamics and heattransfer.

Abolfazl Fattahi is a PhD student at Iran Universityof Science and Technology, Tehran, Iran. He receivedhis MSc degree from University of Kashan, Kashan,Iran. His research activities are paid to combustion,numerically and empirically, computational uid dy-namics and heat transfer.

Mosa�ar Ali Mehrabian is a full professor at ShahidBahonar University of Kerman, Kerman, Iran. Hereceived his PhD degree from University of Bristol,England. His researches are focused on uid mechanics,thermodynamics and heat transfer.

heat and mass transfer by natural convection around a hot

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