International Communications in Heat and Mass Transfer 36 (2009) 269–273

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International Communications in Heat and Mass Transfer

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Integral treatment of coupled heat and mass transfer by natural convection from acylinder in porous media☆

B.B. Singh ⁎, I.M. ChandarkiDepartment of Mathematics, Dr. Babasaheb Ambedkar Technological University, Lonere - 402 103, Dist. Raigad (M.S.), India

☆ Communicated by A.R. Balakrishnan and S. Jayanti.⁎ Corresponding author.

E-mail address: brijbhansingh@yahoo.com (B.B. Sing

0735-1933/$ – see front matter © 2008 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2008.11.007

a b s t r a c t

a r t i c l e i n f o

Available online 9 January 2009

Keywords:

This paper deals with therevolution embedded in a swith linear temperature an

BuoyancyHeat transferNatural convectionBoundary layer thicknessBoundary layer thickness ratio

d concentration distributions. The governing parameters for the problem understudy are buoyancy ratio (N) and Lewis number (Le). The numerical values of local Nusselt and localSherwood numbers have also been computed for a wide range of N and Le. The results pertaining to thevariations of local Nusselt number, local Sherwood number, N and Le with one another have been studiedgraphically, and it has been concluded that the local Nusselt number decreases while the local Sherwood

study of the buoyancy induced heat and mass transfer from a slender body ofaturated porous medium. The study has reported the important case of a cylinder

number increases along with NN0 for increasing Lewis number. The local Nusselt number decreases whilethe local Sherwood number increases along with Le for positive values of N. Also the boundary layerthickness ratio decreases along with Le for NN=0. In this study, an integral method of Von-Karman type hasbeen used in order to obtain mathematical expressions for local Nusselt and local Sherwood numbers.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The study of dynamics of hot and salty springs of a sea, where thecombined convection of heat and mass transfer is involved, has beenanalyzed by Dagan [1]. The analysis of natural convection flows arisingfrom a heated impermeable surface, embedded in fluid saturatedporous media, has been used by Cheng and Minkowcyz [2] to modelthe heating of ground water in an aquifer by a dike. Mahajan andGebhart [3] used higher order approximations for Newtonian flows.Bejan and Khair [4] studied buoyancy induced heat and mass transferfrom a vertical plate embedded in a saturated porous medium.Nakayama and Koyama [5] used Karman–Pohlhausen integral relationfor a Darcian fluid flow over a non-isothermal body of arbitrary shape.A review of combined heat and mass transfer by free convection in aporous medium was given by Trevisan and Bejan [6]. Lie et al. [7]studied the coupled heat and mass transfer of slender bodies ofrevolution by natural convection in porous media. Singh and Sharma[8] used integral method of Von-Karman type of free convection inthermally stratified porous medium. Similar studies were also carriedout by researchers [9–18].

The objective of the present paper is to investigate the problem ofnatural convection induced by the combined action of temperatureand concentration gradients from slender bodies of revolution; e.g. acylinder, in saturated porous media. Emphases have been placed on a

h).

l rights reserved.

fundamental examination of these effects on the flow, temperatureand concentration fields. It is expected that the results thus obtainedwill not only provide useful information for applications but also serveas a complement to the previous studies [4,7]. In the present study, anintegral method of Von-Karman type has been used to derivemathematical expressions for local Nusselt and local Sherwoodnumbers. Based on these expressions, numerical values have beencomputed for a wide range of buoyancy ratio and Lewis number. It hasbeen found that these results are in complete agreement with thoseobtained by Lai et al. [7] who investigated the same problem withRunge-Kutta fourth order method along with Shooting technique.

2. Formulation of the problem

The governing equations for a slender body of revolution placed ina saturated porous medium (Fig. 1) in terms of stream-functions, bytaking into account the boundary layer and Boussinesq approxima-tions, are:

1r@ψ@r

=gKυ

βT T − T∞ð Þ + βc C −C∞ð Þ½ � ð1Þ

@ψ@r

@T@x

−@ψ@x

@T@r

= α@

@rr@T@r

� �ð2Þ

@ψ@r

@c@x

−@ψ@x

@c@r

=D@

@rr@c@r

� �ð3Þ

where u = 1r@ψ@r , v = −

1r@ψ@x.

Nomenclature

A constant defined in Eq. (5)a constant defined in Eq. (5)B constant defined in Eq. (5)b constant defined in Eq. (5)C dimensionless concentrationc concentrationD mass diffusivityf dimensionless stream function defined in Eq. (7)g acceleration due to gravityh local heat transfer coefficientK permeabilityk effective thermal conductivityLe Lewis numberm local mass flux at the wallN buoyancy ratioNu local Nusselt numberRa modified local Rayleigh numberSh Sherwood numberT Temperatureu Darcy's velocity in the x-directionv Darcy's velocity in the r-direction

Greek symbolsα thermal diffusivity of the porous mediumβT coefficient of thermal expansion, − 1

ρ@ρ@T

� �P

βc coefficient of concentration expansion, − 1ρ

@ρ@c

� �P

η independent similarity variableη0 dimensionless radius of the slender bodyθ dimensionless temperatureυ kinematic viscosity of convective fluidρ density of convective fluidψ stream function

Subscriptw condition at the wall∞ condition at infinity

270 B.B. Singh, I.M. Chandarki / International Communications in Heat and Mass Transfer 36 (2009) 269–273

The density, in accordance with the linear Boussinesq approxima-tion, is given by

ρ = ρ∞ 1 −βT T −T∞ð Þ−βc c −c∞ð Þ½ � ð4Þwhere βT and βc are the coefficients for thermal and concentrationexpansion.

Fig. 1. A slender body of revolution embedded in saturated porous medium.

If the transfer process occurs at low concentration difference suchthat the interfacial velocity due to mass diffusion is negligible, theboundary conditions will be as given below:

At the surface, r=R(x),

v = 0 : T = Tw xð Þ = T∞ + Axa

c = cw xð Þ = c∞ + Bxb ð5Þ

At the infinity, r→∞,

u = 0 : T = T∞; c = c∞ ð6Þ

where R xð Þ = υαη0KgβT A

n o1=2x1 − a2 prescribes the surface shape of the axi-

symmetric body.To solve the set of simultaneous differential Eqs. (1)–(6), the

following dimensionless variables are introduced:

η = Rarx

� �2;ψ = αxf ηð Þ ð7Þ

θ =T −T∞Tw−T∞

;C =c −c∞cw−c∞

ð8Þ

where Ra =KgβT Tw−T∞ð Þx

αυ (modified local Rayleigh number).After transformation, the resulting equations are:

f V=12

θ +NCð Þ ð9Þ

θW =12η

af Vθ − 2 + fð ÞθV½ � ð10Þ

CW =12η

bLef VC − 2 + Lefð ÞCV½ � ð11Þ

with boundary conditions given by

atη = η0 : f + a − 1ð Þnf V= 0; θ = 1; C = 1 ð12Þ

atη =∞ : f V= 0; θ = 0; C = 0 ð13Þ

where η0 is a constant and is numerically small for slender bodies.Here the parameter

N =βC cw−c∞ð ÞβT Tw−T∞ð Þ bouyancy ratioð Þ ð14Þ

measures the relative importance of mass and thermal diffusion in thebuoyancy driven flow. N will be zero for purely thermal buoyancydriven flow, infinite for mass driven flow, positive for aiding the flow,and negative for opposing the flow.

Here the primes denote differentiation with respect to η, η ∊ [η0,∞], and f′(η) is a non-dimensional velocity related to the stream-function ψ (x,y). Le is Lewis number defined by

Le =αD: ð15Þ

3. Integral method

The transformed energy Eq. (10) and the constituent massconservation Eq. (11) can be integrated with respect to η from η=η0to η=∞, to obtain

−2η0θV η0� �

= a + 1ð Þ ∫∞

η0f Vθdη − a − 1ð Þη0 1 +Nð Þ ð16Þ

−2η0CV η0� �

= b + 1ð ÞLe ∫∞

η0f VCdη − Le a − 1ð Þη0 1 +Nð Þ: ð17Þ

Fig. 2. Heat transfer coefficient as a function of buoyancy ratio.

271B.B. Singh, I.M. Chandarki / International Communications in Heat and Mass Transfer 36 (2009) 269–273

The infinity is boundary layer thickness for temperature as well asfor concentration.

The temperature and concentration profiles are now expressed interms of the following exponential functions:

θ ηð Þ = exp −η −η0δT

� �� ð18Þ

C ηð Þ = exp −nη −η0δT

� �� ð19Þ

which satisfy the boundary conditions (12) and (13).Here δT is the arbitrary scale for the thermal boundary layer

thickness, whereas ξ is its ratio to the concentration boundary layerthickness.

With the help of above profiles and using Eq. (9), the Eqs. (16) and(17) can be combined into two distinct expressions for δT2 as

a + 12

� �12+

Nn + 1

�δ2T−

a − 1ð Þη0 1 +Nð Þ2

δT = 2η0 ð20Þ

and

Le b + 1ð Þ2n

1n + 1

+N2n

�δ2T−

Le a − 1ð Þη0 1 +Nð Þ2n

δT = 2η0 ð21Þ

The above equations govern the thermal boundary layerthicknesses for the coupled heat and mass transfer by naturalconvection from slender bodies of revolution in porous media.Here, the particular case a=b=1, which corresponds to a cylinderwith linear temperature and concentration distributions, has beenconsidered. For this case, the above two equations can be combinedinto one cubic equation for determining the boundary layerthickness ξ, as

n3 + 1 + 2Nð Þn2− 2 +Nð ÞLen −NLe = 0: ð22Þ

As ξ is determined from Eq. (22), the local Nusselt and Sherwoodnumbers, which are of main interest in terms of heat and masstransfer respectively, are given as

Nu =hxk

= −@T@r jr = R xð ÞTw−T∞

= − 2η0θ0 η0� �

Rað Þ1=2 = 2η0δT

Rað Þ1=2 from Eq: 18ð Þ½ �

Table 1Local Nusselt and Sherwood numbers

N Le Nu/(Rax)1/2 Sh/(Rax)1/2

0 1 0.95 0.954 0.95 2.2537

10 0.95 3.850 0.95 9.0369

100 0.95 12.96841 1 1.3435 1.3435

4 1.1920 2.959310 1.1128 4.870350 1.0260 11.3080

100 1.0041 16.12945 1 2.3270 2.3270

4 1.8432 4.824210 1.5513 7.756350 1.2195 17.6084

100 1.1384 24.978210 1 3.1508 3.1508

4 2.4115 6.430310 1.9428 10.267950 1.3863 23.1598

100 1.2512 32.8070

= η1=20n + 1 + 2N

n + 1

�1=2Rað Þ1=2 from Eq: 20ð Þ½ �

= 0:1n + 1 + 2N

n + 1

�1=2Rað Þ1=2 ð23Þ

Sh =mx

D cw−c∞ð Þ = − 2η0C0 η0� �

Rað Þ1=2 = 2η0n Rað Þ1=2δT

from Eq: 19ð Þ½ �

= η1=20 nn + 1 + 2N

n + 1

�1=2Rað Þ1=2 from Eqs: 20ð Þ and 21ð Þ½ �

= 0:1nn + 1 + 2N

n + 1

�1=2Rað Þ1=2: ð24Þ

Since our concern in this study is on combined heat and masstransfer effects by natural convection from a slender body ofrevolution, our calculations are restricted to specific case ofη0=0.01 only. Attention has been placed to the important roleplayed by the governing parameters; i.e., Lewis number (Le) andbuoyancy ratio (N).

Also, the accuracy acquired in the above approximate expressionsmay be examined by comparing the heat and mass transfer resultsagainst the similarity solution for the two limiting cases of purethermal driven flow (i.e. N=0) and pure mass driven flow (i.e. N→∞).Our approximate formulae given by Eqs. (23) and (24) tend tooverestimate heat andmass transfer rate under these physical limitingconditions. It is not unusual to have an error of 5% or more, depending

Fig. 3. Mass transfer coefficient as a function of buoyancy ratio.

Fig. 6. Boundary layer thickness ratio.

Fig. 4. Heat transfer coefficient as a function of Lewis number.

272 B.B. Singh, I.M. Chandarki / International Communications in Heat and Mass Transfer 36 (2009) 269–273

on the assumed profile. However, the situation can be remedied byadjusting the multiplicative constant, namely, replacing 0.1 by 0.95.Thus, the following approximate formulae are proposed:

Nu = 0:95n + 1 + 2N

n + 1

�1=2Rað Þ1=2 ð25Þ

Sh = 0:95nn + 1 + 2N

n + 1

�1=2Rað Þ1=2: ð26Þ

4. Results and discussions

Wehavedone calculations for awide rangeof parametersN (buoyancyratio) and Le (Lewis number) in order to study their effects on thecombined heat and mass transfer along a vertical wall due to freeconvection. Thesevalues are given inTable1. Fromthis Table, it is observedthat thenumerical values of localNusselt andSherwoodnumbers turnoutto be the same and are equal to 0.95 forN=0 and Le 1. From the Table, it isalso obvious that the Lewis number has more pronounced effect on theconcentration field than it has on the temperature field.

The local Nusselt number has been plotted against buoyancy ratioin Fig. 2 for different values of Lewis number (Le=1,10,50,100). It isobserved from the nature of the graphs that the local Nusselt numberi.e. rate of heat transfer decreases with increasing Lewis number forNN0. Similarly, the local Sherwood number has been plotted in

Fig. 5. Mass transfer coefficient as a function of Lewis number.

Fig. 3 against the buoyancy ratio N for various values of Lewis number(Le=1,10,50,100), and increases with increasing Le for all NN0.

The local Nusselt number in Fig. 4 has been plotted against Lewisnumber for various values of buoyancy ratio N=0, 1, 5 and 10.

It is found that the local Nusselt number decreases with increasingLewis number for NN0. Similarly, the local Sherwood number in Fig. 5has been plotted against Lewis number for various values of buoyancyratio (N=0, 1, 5, 10). It has been found from the graphs that the localSherwood number increases with increasing Lewis number for all NN0.

In Fig. 6, the ratio of local Nusselt number to local Sherwoodnumber Nux

Shx

� �has been plotted against the Lewis number for N=0, 10.

It has been found from the graphs that the value of this ratio, i.e.boundary layer thickness ratio (ξ) decreases with increasing values ofLewis number for N≥0.

5. Concluding remarks

For coupled heat andmass transfer by natural convection in porousmedia, solutions have been presented for the important case ofcylinder with linear temperature and concentration distribution. Thegoverning parameters for the problem under consideration are theLewis number (Le) and buoyancy ratio (N). For NN0, the inclusion of aconcentration gradient may assist the flow induced by thermalbuoyancy. The Lewis number is found to have a more pronouncedeffect on the concentration field than it does on the flow andtemperature fields.

The result obtained in this study are in complete agreement withthose obtained by Lai et al. [7] who solved the same problem by thenumerical integration using a fourth order Runge-Kutta method andShooting technique.

Acknowledgement

The authors are highly obliged to Professor A. K. Singh of theDepartment of Mathematics, Banaras Hindu University, Varanasi for hisable guidance given at the time of the preparation of the manuscript.

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