Natural convection and entropy generation in a vertically concentric annular space

lable at ScienceDirect

International Journal of Thermal Sciences 49 (2010) 2439e2452

Contents lists avai

International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Natural convection and entropy generation in a vertically concentricannular space

Sheng Chen*, Zhaohui Liu, Sheng Bao, Chuguang ZhengState Key Lab of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 4 December 2009Received in revised form19 August 2010Accepted 19 August 2010Available online 16 September 2010

Keywords:Entropy generationNatural convectionVertically concentric annulusLattice Boltzmann method

* Corresponding author.E-mail address: [email protected] (S. Ch

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.08.011

a b s t r a c t

Natural convection in a vertically concentric annular space is of fundamental interest and practicalimportance. However, available open literature on entropy generation analysis for it is still sparse. In thepresent work we investigate systematically the effects of Rayleigh number, curvature of annulus andPrandtl number on flow pattern, temperature distribution and entropy generation for natural convectioninside a vertically concentric annuli with the aid of the lattice Boltzmann method. The analyzed range iswide, varying from steady laminar convection to unsteady transitional state. Furthermore, we analyszethe non-linear branch of irreversible phenomena from a thermodynamics view point.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Convection heat transfer inside concentric annuli has manysignificant engineering applications. Starting from the micro elec-tronic heat transfer devices, this type of geometry is found in elec-tronic packages, electrical equipment, industrial heat exchangers,petroleum drilling equipment, nuclear reactor, and so on. In practicalsituations, many factors affect the heat transfer from the annularspace. In order to reveal heat transfer performances inside concen-trically cylindrical annuli, a great number of excellentworks have beedone. Davis and Thomas perhaps are the pioneers to investigatenatural convection within concentrically vertical annuli [1]. In theirwork, the flow and heat transfer characteristics with different aspectratioswere studied. Schwab et al. found thatwhenRayleigh numbersgreater than 5�103 for natural convection within concentricallyvertical annuli, therewould appear a fully developed boundary-layerin the cavity [2]. Mixed convection flow pattern and heat transfer forair inside a vertical, cylindrical annular space were investigated byHessami and his cooperators [3]. Yan and Tsay analyzed the fullydeveloped heat transfer performances inside an annular channel [4].EI-Shaarawi and Sarhan [5] and Islam et al. [6] considered the case ofmixed convectional heat transfer at developing flow regime and alsoanalyzed the entrance effect. Ho and Tu studied laminar transition to

en).

son SAS. All rights reserved.

oscillatory convection of cold water in a vertical annulus of aspectratio 8 and radius ratio 2 [7]. Lee et al. investigated fully developedturbulent mixed convection within heated vertical annular pipe bylarge eddy simulation [8]. The study of natural convection in shallowcylindrical annuli was conducted by Leppinen [9]. Usmani et al.reported heat transfer characteristics during natural convectionboiling in an internally heated annulus [10]. Heat transfer in bilater-ally heated vertical narrow annular channels was analyzed by Tianand his partners [11]. Badruddin et al. showed the effect of viscousdissipation and radiation on natural convection in a porous mediumembeddedwithin vertical annulus [12]. Zanchini studied analyticallymixed convection with a temperature-dependent viscosity ina vertical annular duct with uniformwall temperatures [13]. Barlettaet al. investigated heating effects in vertical porous annulus witha radially varying magnetic field [14]. Avci and Aydin analyzedanalytically convectional heat transfer within a micro concentricannulus [15].Weng et al. reported the effect ofwall-surface curvatureon natural convection in an open-ended vertical annular micro-channel [16], to cite only a few. Recently, the present authorsnumerically simulatednatural double-diffusive convection inverticalannuluses with high Rayleigh number [17].

The references mentioned above are based on the first-law ofthermodynamics and avoid complicated analyses for optimumdesign. Recently the entropygenerationanalysismethodology [18,19]which based on the second-law of thermodynamics, is used to opti-mize heat transfer performance inside annuli. The influences of wall

mailto:[email protected]

www.sciencedirect.com/science/journal/12900729

http://www.elsevier.com/locate/ijts

http://dx.doi.org/10.1016/j.ijthermalsci.2010.08.011



Nomenclature

A aspect ratioc fluid particle speedD coefficient in Eq. (12)e!k discrete velocityg! gravityu! fluid velocity vectorgk, fk distribution function in Eqs. (6) and (12)gkeq, fkeq equilibrium distribution function in Eqs. (6) and (12)

H height of simulation domainK curvature ratioS Svanberg vorticityStotal total entropy generation numberSv entropy generation numberBe Bejan numberBr Brinkman numberEc Eckert numberPr Prandtl numberRa Rayleigh numberT temperaturex! phase space

Greek symbolsYo;k;U

0k source terms in Eqs. (6) and (12)

fab strain rate tensorUk collision term in Eq. (12)Dx, Dt grid spacing, time stepDT temperature differencek thermal conductivityn effective kinematic viscosityr densitys relaxation time for Eq. (6)sj relaxation time for Eq. (12)4 irreversibility distribution ratioz dimensionless timezk, xk the weights for equilibrium distribution functiond, c coefficients in Eqs. (9) and (11)u, j vorticity, streamfunction

Subscripts and superscriptsD thermalm viscousk discrete velocity direction0 initial indexh highl lowa, b spatial indexU, total global, total

Fig. 1. The configuration of computational domain.

S. Chen et al. / International Journal of Thermal Sciences 49 (2010) 2439e24522440

conductance ratio, scaling effects and different aspect ratios onconvectional flow are discussed in Refs. [20e25]. Tasnim and Mah-mud analyzed the entropy generation for mixed convection ina vertical annular space [26]. In their work, a correlation wasproposed for calculating optimum radius ratio. The effect of relativerotational motion between the inner and outer cylinders of a cylin-drical annulus on entropy generation was studied by Mahmud andFraser [27]. They found that the entropy generation rate showedasymptotic behavior near the outer cylinder provided that the rota-tion of the inner cylinder was higher than the outer. For theisothermal boundary condition, the Bejan number was maximum atthe inner cylinder. Yilbas and his cooperators investigated entropygeneration characteristics of non-Newtonian fluid flow in annularpipe [28]. It was found that entropy generation due to fluid frictionand heat transfer was high in the region close to the inner wall of theannular pipe due to enhancement of convective heat transfer andincreased fluid friction due to high shear strain in this region.Increasing non-Newtonian parameter reduced the entropy genera-tion number in this region. Allouache and Chikh reported theperformance of a annular heat exchanger with a porous mediumattached over the inner pipe based on second-law analysis [29].Results showed that the minimization of the rate of entropy gener-ation depended on the porous layer thickness, its permeability, theinlet temperature differencebetween the twofluids, and theeffectivethermal conductivity of the porous substrate. An increase in theeffective thermal conductivity of the porous medium seemed to bethermodynamically advantageous. The work to optimize flow andheat transfer performance inside a microannulus was recently con-ducted by Yari [30]. They found that entropy generation decreasedwith an increase in the Knudsen number.

The main originality of the present work is to investigatesystematically the effects of Rayleigh number, curvature of annulusand Prandtl number on flow pattern, temperature distribution andentropy generation for natural convection inside a verticallyconcentric annular space. The analyzed range is wide, varying from

steady laminar convection to unsteady transitional state. The aboveliterature surveyclearly shows that there is no study in the literatureon this topic. Moreover, a lattice Boltzmann (LB) model recentlydeveloped by the present authors [31e35] is used to solve thegoverning equations for heat andfluid flows and entropy generationequation. As our previous work [36] demonstrates, the LB methodpossesses high efficiency for entropy generation analysis task.

2. Governing equations for natural convection in a verticallyconcentric annular space

The configuration of the vertical annulus is illustrated in Fig. 1.The inner wall with the radius Ri and the outer wall with Ro. K¼ Ri/(Ro� Ri) is the curvature ratio. The aspect ratio A¼H/(Ro� Ri). H isthe height of the annular cavity. The temperatures at inner andouter wall are Th and Tl respectively and Th> Tl.

With the Boussinesq assumption, the dimensionless vorticity-streamfunction-based governing equations for natural convectionin a vertically concentric annular space read [1,2,34,35]

Table 1Comparison of average Nusselt number obtained by the present model withprevious work.

Pr Ref. [2] Present (100� 100) Present (200� 200)

0.73 6.13 6.1342 6.13521.0 6.17 6.1690 6.16737.0 6.36 6.3577 6.372125.0 6.31 6.3200 6.3259

S. Chen et al. / International Journal of Thermal Sciences 49 (2010) 2439e2452 2441

vSvt

þ uvSvr

þwvSvz

¼ Pr1rvTvr

þ Pr

Ra1=2

(1rv

vr

�1rv

vr

�r2S

��þ v2Svz2

); (1)

vTvt

þ uvTvr

þwvTvz

¼ 1

Ra1=2

"1rv

vr

�rvTvr

�þ v2T

vz2

#; (2)

v

vr

�1rvj

vr

�þ 1

rv2j

vz2¼ �rS: (3)

where S¼u/r is the Svanberg vorticity and u the vorticity. ThePrandtl number Pr¼ n/k and the Rayleigh number Ra¼ agH3DT/nk.j is the Stokes streamfunction. T is the temperature; n is the kineticviscosity; g is the gravitational acceleration along the negative z-axis; k is the thermal conductivity; DT¼ Th� Tl is the temperaturedifference and a the coefficient of thermal expansion. u and w areradial and axial velocity components:

u ¼ 1rvj

vz; (4)

w ¼ �1rvj

vr: (5)

3. Lattice Boltzmann model

In the last two decades, the LB method has matured as an effi-cient alternative for simulating andmodeling complicated physical,chemical and social systems [37e39]. Parallelization of a LB modelis relative easy since the relaxation is local and the communicationpattern in propagation is one way. Of course in order to get goodperformances, professional techniques must be developed for a LBprocedure [40]. Moreover, the LB moldes have been comparedfavourably with spectral methods [41], artificial compressibilitymethods [42e44], finite volume methods [32,33], finite differencemethods [17,45], projection methods [46,47] and multigridmethods [48,49]. Up to date the LB method has been successfullyused to simulate many problems, from laminar single phase flowsto turbulent multiphase flows [39]. The first LB implementation forthermal convection is proposed by Massaioli et al. [50].

Eq. (1) (governing equation for the flow field) and Eq. (2)(governing equation for the temperature field) both can be solvedby the LB model proposed in Refs. [34,35]. It reads:

gkð x!þ c e!kDt; t þ DtÞ � gkð x!; tÞ ¼ �s�1hgkð x!; tÞ � gðeqÞk ð x!; tÞ

iþDtYo;k (6)

where e!k (k¼ 0.4) are the discrete velocity directions:

-0.8-0.6-0.4

-0.2

0

0.2

0.40.6

0.8

a

Fig. 2. Isotherm (a) and stream

e k ¼ ðcosðk� 1Þp=2; sinðk� 1Þp=2Þ : k ¼ 1;2;3;4
!
� ð0;0Þ : k ¼ 0

c¼Dx/Dt is the fluid particle speed. Dx, Dt and s are the lattice gridspacing, the time step and the dimensionless relaxation timerespectively. Yo;k is the discrete form of the source term Yo [35],where Yo ¼ 3

rPr

Ra1=2vSvr þ Pr1r

vTvr and 1

Ra1=21rvTvr for Eqs. (1) and (2)

respectively. Yo;k satisfies:Xk�0

Yo;k ¼ Yo (7)

The simplest choice satisfying the constraint Eq. (7) is

Yo;k ¼ Yo

5(8)

The equilibrium distribution gk(eq) is defined by

gðeqÞk ¼ d

5

�1þ 2:5

e!k$ u!

c

�(9)

d¼ S, T for Eqs. (1) and (2) respectively and is obtained by

d ¼Xk�0

gk (10)

and the dimensionless relaxation time s is determined by

c ¼ 2c2ðs� 0:5Þ5

(11)

c ¼ PrRa1=2

; 1Ra1=2

for Eqs. (1) and (2), respectively.Eq. (3) is just the Poisson equation, which also can be solved by

the LB method efficiently. In the present study, the D2Q5 modelused in our previous work [34] is employed because this model ismore efficient and more accurate than others to solve the Poissonequation. The evolution equation for Eq. (3) reads

fkð x!þ c e!kDt; t þ DtÞ � fkð x!; tÞ ¼ Uk þU0k (12)

-10-30-50-70-90-110

-130-140

b

function (b) at Ra¼ 104.

a b

Fig. 3. Isotherm (a) and streamfunction (b) at Ra¼ 107.


where Uk ¼ �s�1j ½fkð x!; tÞ � f ðeqÞk ð x!; tÞ�, Uk

0 ¼DtzkQD, Q¼e(r2Sþw) and D ¼ c2

2 ð0:5� sjÞ. sj> 0.5 is the dimensionlessrelaxation time [34]. fk(eq) is the equilibrium distribution function,and defined by

f ðeqÞk ¼� ðx0 � 1:0Þj : k ¼ 0xkj : k ¼ 1;2;3;4

(13)

xk and zk are weight parameters given as x0¼ z0¼ 0, xk¼ zk¼ 1/4(k¼ 1.4). j is obtained by

j ¼Xk�1

fk (14)

Because the vorticityestreamfunction-based governing equationsare used, therefore the continuity equation is satisfied automati-cally. Moreover, the distribution functions used in present work arenot the density distribution functions used in traditional LBmethod, so there is no obvious compressibility effect in the presentscheme, as shown in our previous publication [34].

Because the distribution functions at thewalls are unknown, thenon-equilibrium extrapolation scheme used in our previous studies[34,36] is employed to treat flow and temperature boundaryconditions, namely

gkjw ¼ geqk jw þ gnon�eqk jw ¼ geqk jw þ

�gkjf � geqk jf þ O

�e2��

(15)

a

Fig. 4. Entropy generation number (a)

fjjw ¼ f eqj jwþ f non�eq

j jw

¼ f eqj jwþ fjjf � f eqj j

fþ O e2

� � ��(16)

where the subscript w and f represent the wall boundary grid andneighbouring fluid grid respectively. e is a small quantity [34,36].

In order to validate the present model, we firstly calculate theaverage Nusselt number at the inner wall of free convection betweentwo vertical coaxial cylinders investigated in Ref. [2] with Ra¼ 104

and 0.73� Pr� 25.0. The results are listed in Table 1. The numbers inthe brackets denote the used grid resolutions. It is clear the resultsobtained by the present model agree well with previous data [2].

4. Entropy generation

The entropy generation number is given by [18,51]:

Sv ¼ ðVTÞ2þ4jfj2 (17)

where the irreversibility distribution ratio 4¼ Br/DT [30,51]. Br¼ PrEc is the Brinkman number, where Ec is the Eckert number [18]. jfjis the magnitude of the strain rate tensor [36]

jfj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fabfab

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

"�vuvr

�2

þ�ur

�2þ�vwvz

�2#þ�vwvr

þ vuvz

�2vuut(18)

where fab¼ (vaubþ vbua)/2.

b

and Bejan number (b) at Ra¼ 104.

a b

Fig. 5. Entropy generation number (a) and Bejan number (b) at Ra¼ 105.


Apparently, it is inconvenient to calculate j4j (Eq. (18)) directlyby conventional numerical methods due to its complex form ofspatial derivative [36]. However, in the present LB model, j4j can becalculated easily by [52,53]

jfj ¼ juj ¼ jrSj: (19)

Because the value of Svanberg vorticity S is already known at eachgrid point, therefore no extra computational cost needed for jfj inthe present model.

Recognizing the first term in Eq. (17) as reflecting the entropygeneration due to thermal diffusion and the second due to viscousdissipation, the entropy generation number can be expressed as

Sv ¼ SD þ Sm (20)

where the subscripts D and m are used to indicate the effect ofthermal diffusion and viscous dissipation respectively. Usually SD isreferred to as Heat Transfer Irreversibility HTI and Sm as Fluid Fric-tion Irreversibility FFI [18]. The Bejan number Be is given as [18,36]:

Be ¼ SDSv

¼ HTIHTIþ FFI

(21)

When Be[ 0.5, the irreversibility due to heat transfer dominates,whereas Be� 0.5 the irreversibility due to viscous effect domi-nates. When Be¼ 0.5 heat transfer and fluid friction entropygeneration are equal.

The total entropy generation number is defined as [18]

a b


Stotal ¼ SvvU (22)
ZU
where the subscript U means the global computational domain.

5. Results and discussions

In the present study, A¼ 1; 4¼ 1; 0.09� Pr� 25; 104� Ra� 107

and 0.01� K� 50. The boundary conditions are: j¼ u¼w¼ 0 at allwalls; Th¼ 1.0 at the inner wall; Tl¼�1.0 at the outer wall; vT/vz¼ 0 at the top and bottom walls. The initial conditions arej¼ u¼w¼ 0 and T¼ 0. The grid resolution 100�100 is used. Therelaxation time varies from 1.8 to 1.95 according to the value of Ra.For simplicity, the abscissa is normalized by (r� Ri)/(Ro� Ri), andthe ordinate is normalized by z/(Ro� Ri) for all figures in this paper.

Because theflowfieldbecomesunstablewhenRa� 106, in the restpart of this paper, if not specified, the instant values, such as theentropy generation number, are measured at time step 5�105. Atthat time the unsteady flows with Ra� 106 are fully developed,namely the stable time-averagedmacro-quantities can bemeasured.

The numerical code used here is described and validated indetail in Refs. [35,36].

5.1. Variation in Rayleigh number

We firstly investigate the effect of Rayleigh number on flowpattern, temperature distribution and entropy generation for

and Bejan number (b) at Ra¼ 106.

a b

Fig. 7. Entropy generation number (a) and Bejan number (b) at Ra¼ 107.

a b

Fig. 8. Total entropy generation number (a) and average Bejan number (b) versus dimensionless time at Ra¼ 104.


natural convection in the vertically concentric annulus. K and Pr arefixed at 1 in this subsection.

Figs. 2 and 3 show the isotherm and streamfunction withdifferent Ra. When Ra� 105, the flow is laminar. Whereas Ra� 106,unsteady flow appears. If Ra continues increasing, the convectionwill become transitional flow. The vortex center moves towardsouter wall with Ra increasing. The secondary flow emerges whenRa¼ 107. The shape of the isotherms show how the dominant heattransfer mechanism changes as Ra increases. For low Ra (forexample Ra¼ 104) almost vertical isotherms appear, because heat istransferred by conduction between hot and cold walls. As theisotherms depart from the vertical position, the heat transfer

a b

Fig. 9. Total entropy generation number (a) and average Be

mechanism becomes to change from conduction to convection.When Ra¼ 105 the isotherms at the center of the cavity are nearlyhorizontal and become sharply vertical near the sidewalls. WhileRa¼ 106, the isotherms are distorted in the vicinity of the verticalwalls, especially near the outer wall. The distortedness of theisotherms spreads over the whole domain when Ra¼ 107. FromFig. 3, it is clear that obvious intrusions of isotherms are formed forhigh Ra. The layers of stratification of isotherms near the lateralwalls become narrower and narrower against Ra.

Figs. 4e7 illustrate the entropygeneration andBejannumbers forvariational Ra. The maximum of entropy generation numberincreases with Ra and appears in the vicinity of the corners of the

jan number (b) versus dimensionless time at Ra¼ 105.

a b


a b



annulus. When Ra¼ 104 the maximum emerges at the bottom-leftcorner of the enclosure, whereas for Ra� 105 it jumps to the top-right corner. The convectional flow is clockwise, so the bottom-leftand top-right corners are barriers for the flow. Consequently largervelocity gradients, namely larger irreversibility due to viscous willappear near these two corners. When Ra< 105, the heat conductionis predominant, so the isotherms near the inner wall are morecompact than that near the outer wall. Therefore the largertemperature gradients, namely larger irreversibility due to heattransfer, will emerge near the bottom-left corner. However, sinceRa� 105, the heat convection is predominant, so the isotherms nearthe outer wall are more compact. Accordingly the larger tempera-ture gradients, namely larger irreversibility due to heat transfer, will

a b

Fig. 12. Total entropy generation number (a) and a

emerge near the top-right corner. It is the reasonwhy the location ofthe maximum of entropy generation number will change aroundRa¼ 105. The entropy intensely generates along the lateral walls,especially near the bottom-left and top-right corners. The layerswith significant entropy generation also become narrower with Raincreasing, although their length are lengthened. At Ra¼ 107 thelayers almost overcover the vertical walls. In the center of the cavitythe entropy generation number is very slight for all Ra. However, themaximum of Bejan number appears in the center of the enclosurewhen Ra< 106, as shown in Figs. 4 and 5. When Ra� 105, the irre-versibility due to heat transfer dominates the whole domain excepttwo narrow layers near the sidewalls. The minimum of Bejannumber appears in the vicinity of the outer wall. At higher Ra (i.e.

verage Bejan number (b) vs Rayleigh number.

a b

Fig. 13. Isotherm (a) and streamfunction (b) at K¼ 0.01.

a b

Fig. 14. Isotherm (a) and streamfunction (b) at K¼ 0.1.


Ra¼ 105), the motion of fluid is strengthened than lower Ra (i.e.Ra¼ 104), so the area dominated by irreversibility due to viscouseffect is enlarged. The similar phenomena also can be found in Fig. 6for Ra¼ 106. While Ra¼ 107, the areas dominated by irreversibilitydue to viscous effect and heat transfer are indented and surroundedeach other due to the convectional flow becomes unsteady.

Figs. 8e11 show the variations of total entropy generation andvolume-averaged Bejan numbers versus dimensionless time. WhenRa� 105, the total entropy generation number has a maximum at

a

Fig. 15. Isotherm (a) and strea

the onset of the transient state, then decreases quickly to reacha constant value in the steady state. However for Ra� 106, obviousoscillations of total entropy generation can be observed and theoscillations become regular when the flow is fully developed.Although the amplitude of oscillations for Ra¼ 106 is much biggerthan that for Ra¼ 107, it would cost much more time to achieveregular oscillations for Ra¼ 107 than Ra¼ 106.

Fluctuations of the total entropy generation number at high Ray-leigh numbers indicate that the flow exhibits oscillatory behavior

b

mfunction (b) at K¼ 0.5.

a b

Fig. 16. Isotherm (a) and streamfunction (b) at K¼ 5.

a b

Fig. 17. Isotherm (a) and streamfunction (b) at K¼ 50.


which depends on the boundary conditions [51]. At the very begin-ning of the transient state heat transfer is mainly due to heatconduction. The isotherms are nearly parallel to the active wallsgenerating an horizontal temperature gradient. The streamlines arethose of a single spiral with its center being near the center of thecavity. As time proceeds the isotherms are gradually deformed byconvection generating a vertical temperature gradient while thehorizontal temperature gradient diminishes in the center of the

a b


cavity and becomes locally negative which causes an elongation ofthe central streamline. This transition may induce generation ofinternal waves in the velocity and temperature fields who can be atthe origin of the oscillations of the whole cavity and consequently ofthe entropy generation. Thewhole cavity oscillationswere attributedto the horizontal pressure gradient established by changes in theintrusion temperature field. The two boundary layer oscillations areattributed to travelingwave instabilityon theboundary layer induced

and Bejan number (b) at K¼ 0.01.

a b

Fig. 19. Entropy generation number (a) and Bejan number (b) at K¼ 0.05.

a b

Fig. 20. Entropy generation number (a) and Bejan number (b) at K¼ 0.1.


first by the leading edge effect of the vertical boundary layer andsecond by the impact of the horizontal intrusion from the opposingvertical wall with the boundary layer. The whole cavity oscillationsare said to be caused by the splitting of the horizontal intrusion as itimpacts the opposite wall. From a thermodynamics view point, theoscillatory behavior of the total entropy generation with

a

Fig. 21. Entropy generation number (

dimensionless time at high Rayleigh numbers shows that the systemis in the non-linear branch of the thermodynamics of irreversibleprocesses [51]. The convectional flow for Ra� 106 is relatively farfrom the equilibriumstate, then a rotation around the fully developedstate is possible, and the system is in the case of a spiral approachtowards this state corresponding to an oscillation of the total entropy

b

a) and Bejan number (b) at K¼ 0.5.

a b

Fig. 22. Entropy generation number (a) and Bejan number (b) at K¼ 5.

a b

Fig. 23. Entropy Isotherm (a) and streamfunction (b) at K¼ 10.


generation. Consequently, the system evolves in the non-linearbranch of irreversible phenomena. The same conclusion also can bedraw from the behavior of volume-averaged Bejan number.

Fig. 12 illustrates the variations of time-averaged total entropygeneration number and time-volume-averaged Bejan numberversus Rayleigh number. It can be seen there is an approximativelinear relationship between the logarithm of Stotal and Ra:

a b

Fig. 24. Entropy generation number (a

log10Stotal ¼ 0:268log10Raþ 3:98801 (23)

whereas the time-volume-averaged Bejan number decreasesquickly against the Rayleigh number when Ra> 105. WhenRa< 106, Be> 0.9, namely the irreversibility due to heat transferdominates. While Ra¼ 107, Be z 0.5, which means heat transferand fluid friction entropy generation are approximately equal.

) and Bejan number (b) at K¼ 50.

a b

Fig. 25. Total entropy generation number (a) and average Bejan number (b) vs curvature ratio.

a b

Fig. 26. Entropy generation number (a) and Bejan number (b) at Pr¼ 0.09.


These phenomena result from the motion of working fluid isintensely enhanced with Ra increasing.

5.2. Variation in curvature ratio

To reveal the effect of the curvature on entropy generation ofnatural convection in a vertically concentric annular space, thecases with different value of curvature ratio 0.01� K� 50 but fixedvalues of Rayleigh number Ra¼ 105 and Prandtl number Pr¼ 1 arechosen in this subsection, which can typify all cases in this study.

a b


Figs. 13e17 show the isotherm and streamfunction withdifferent K. It can be seen that there is only one vortex center whenK� 1. And for sufficiently small K (i.e. K� 0.1), the flow patterns arevery similar with that for buoyancy-driven convection in singlecircular vertical cylinder [35]. As K increases, the vortex tends tomove downwards and become elliptic and finally breaks up intotwo vortices at K¼ 5. The vortex near the inner wall is alwaysweaker than that close to the outer wall until K¼ 50. When K is bigenough (for example K� 50), the flow and temperature patternsare nearly same with that for their planar counterparts(namelyK/N) [51]. The isotherms for K� 0.1 are almost vertical except

and Bejan number (b) at Pr¼ 25.

a b

Fig. 28. Total entropy generation number (a) and average Bejan number (b) vs Prandtl number.


near the top wall, which means that for very small curvature ratioheat is mainly transferred by conduction. While K� 0.5, the heattransfer mechanism begins to change from conduction to convec-tion with the isotherms departing from the vertical position andbecoming horizontally parallels in the core of the enclosure.

Figs. 18e24 illustrate the entropy generation and Bejan numbersfor different K. When K� 0.1, the intense entropy generation isrestricted within a layer in the vicinity of the inner wall and themaximum of entropy generation number emerges near the bottom-left corner (except for K¼ 0.01, whose maximum appears at the top-halfpartof the innerwall). AsK increases, the layernear the innerwallbecomes thick and there obviously emerges another layer withintense entropy generation near the outer wall. While K� 0.5, themaximum of entropy generation number appears at the top-rightcorner. However, the differences between these two layers becomebeing erased with K increasing and they are point symmetric withrespect to the geometric center of the cavitywhenK is sufficiently big(for example K� 50) [51]. The similar phenomena also can be foundfor the Bejan number: When K� 0.5, almost the whole domainexcept near the vertical walls is dominated by irreversibility due toheat transfer. Theminimumof Bejan number locates in the vicinity ofthe outer wall. Whereas for K� 0.1, the irreversibilities due to heattransfer and fluid friction are competitive within the core of theenclosure.WhenK¼ 50, thedistributionsof Bejannumberare almostpoint symmetric with respect to the geometric center of the cavity.

Fig. 25 illustrates the variations of time-averaged total entropygeneration and time-volume-averaged Bejan numbers versuscurvature ratio. One can see that the time-averaged total entropygeneration and time-volume-averaged Bejan numbers both arenon-monotonic functions of curvature ratio. Their minima both areobtained at K z 0.08. When K� 0.08 the time-averaged totalentropy generation and time-volume-averaged Bejan numbers aremonotonic decreasing functions of K whereas they become mono-tonic increasing functions when K� 0.08. For K� 1, the variations oftime-averaged total entropy generation and time-volume-averagedBejan numbers versus curvature ratio are slight, and they approachasymptotically to the values of their planar counterparts (i.e.K/N) [51]. These findings are very helpful for optimization designof heat transfer in a vertically concentric annular space. For exampleto set K around 0.08 to improve exergy efficiency.

5.3. Variation in Prandtl number

To reveal the effect of the Prandtl number on entropy generationof natural convection in a vertically concentric annular space, thecases with different value of Prandtl number 0.09� Pr� 25 but

fixed values of Rayleigh number Ra¼ 105 and curvature ratio K¼ 1are chosen in this subsection.

Fig. 26 illustrates the entropy generation and Bejan numberwhen Pr¼ 0.09, which is the representative of the cases Pr< 0.1.Because Pr is very small, therefore heat conduction is predomi-nated. Consequently the map of entropy generation is very similarwith Fig. 4. Fig. 27 shows the entropy generation and Bejan numberwhen Pr¼ 25, which is the representative of the cases Pr [ 1.Together with Fig. 5, which denotes Pr¼ 1, we can find that whenPr> 1, the flow is dominated by heat convection. The maxima ofentropy generation number and Bejan number increase with Prsignificantly. Especially, when Pr¼ 25, the minimum of Bejannumber is bigger than 0.9, which means that the irreversibility dueto viscous effect can be neglected when Pr is very large.

Fig. 28 plots total entropy generation number and average Bejannumber versus Prandtl number. The variable trends of them arevery similar: They both are monotonic increasing functions of Prand they increase sharply when Pr< 5.

6. Conclusion

Natural convection in a vertically concentric annular space is offundamental interest and practical importance. Until now therehave been numerous works on it. However, available open litera-ture on entropy generation analysis for it is still sparse. In thepresent work we investigate systematically the effects of Rayleighnumber, curvature of annulus and Prandtl number on flow pattern,temperature distribution and entropy generation for naturalconvection inside a vertically concentric annular space with the aidof the LB method. The analyzed range is wide, varying from steadylaminar convection to unsteady transitional state. Through thepresent work, four important features of entropy generation ina vertically concentric annular space are revealed:

1. When Pr¼ 1 and K¼ 1, the time-averaged total entropygeneration number is a monotonic increasing function of Ra,and there is an approximative linear relationship between thelogarithm of Stotal and Ra. But the time-volume-averaged Bejannumber has a inverse trend.

2. The entropy intensely generates within two layers along thevertical walls. The differences between these two layersbecome being erased with K increasing and they are pointsymmetric with respect to the geometric center of the cavitywhen K is sufficiently big. The extremumof time-averaged totalentropy generation and time-volume-averaged Bejan numbersare obtained at K z 0.08. When K� 1, the variations of totalentropy generation and time-volume-averaged Bejan numbers


with curvature ratio are slight, and they approach asymptoti-cally to the values of their planar counterparts.

3. The maximum of entropy generation number will jump fromthe inner wall to the outer wall with Ra and K increasing.

4. The total entropy generation number and average Bejannumber both increase monotonously with Pr.

Acknowledgments

This work was supported by the State Key Development Pro-gramme for Basic Research of China (Grant No. 2010CB227004), andthe National Natural Science Foundation of China (Grant No.50936001, 51006043 and 50721005). The present authors wouldgratefully acknowledge the valuable advice from the referees.

References

[1] G. Davis, R. Thomas, Natural convection between concentric vertical cylinders,Phys. Fluids 12 (1969) (II)198e(II)207.

[2] T. Schwab, K. Witt, Numerical investigation of free convection between twovertical coaxial cylinders, AIChE J. 16 (1970) 1005e1010.

[3] M.A. Hessami, G. De Vahl Davis, J.A.E. Reizes, Mixed convection in verticalcylindrical annuli, Int. J. Heat Mass Transfer 30 (1987) 151e164.

[4] W.M. Yan, H.C. Tsay, Mixed convection heat andmass transfer in vertical annuliwith asymmetric heating, Int. J. Heat Mass Transfer 34 (1991) 1309e1313.

[5] M.A.I. EI-Shaarawi, A. Sarhan, Free convection effects on developing laminarflow in vertical concentric annuli, J. Heat Transfer 102 (1980) 617e622.

[6] N. Islam, U.N. Gationde, G.K. Sharma, Mixed convection heat transfer in the entranceregion of horizontal annuli, Int. J. Heat Mass Transfer 44 (2001) 2107e2120.

[7] C.J. Ho, F.J. Tu, Transition to oscillatory natural convection of cold water ina vertical annulus, Int. J. Heat Mass Transfer 41 (1998) 1559e1572.

[8] J.S. Lee, X. Xu, R.H. Pletcher, Large eddy simulation of heated vertical annularpipe flow in fully developed turbulent mixed convection, Int. J. Heat MassTransfer 47 (2004) 437e446.

[9] D.M. Leppinen, Natural convection in a shallow cylindrical annuli, Int. J. HeatMass Transfer 45 (2002) 2967e2981.

[10] M. Khalid Usmani, M. Altamush Siddiqui, S.S. Alam, A.M. Jairajpuri, M. Kamil,Heat transfer studies during natural convection boiling in an internally heatedannulus, Int. J. Heat Mass Transfer 46 (2003) 1085e1095.

[11] W.X. Tian, S.Z. Qiu, D.N. Jia, Investigations on post-dryout heat transfer inbilaterally heated annular channels, Ann. Nucl. Energy 33 (2006) 189e197.

[12] IrfanAnjumBadruddin, Z.A. Zainal, Zahid A. Khan, ZulquernainMallick, Effect ofviscous dissipation and radiation on natural convection in a porous mediumembedded within vertical annulus, Int. J. Therm. Sci. 46 (2007) 221e227.

[13] Enzo Zanchini, Mixed convection with variable viscosity in a vertical annuluswith uniform wall temperatures, Int. J. Heat Mass Transfer 51 (2008) 30e40.

[14] A. Barletta, S. Lazzari, E. Magyari, I. Pop, Mixed convection with heating effectsin a vertical porous annulus with a radially varying magnetic field, Int. J. HeatMass Transfer 51 (2008) 5777e5784.

[15] Mete Avci, Orhan Aydin, Laminar forced convection slip-flow in a micro-annulus between two concentric cylinders, Int. J. Heat Mass Transfer 51(2008) 3460e3467.

[16] H.C. Weng, C. Chen, Drag reduction and heat transfer enhancement overa heated wall of a vertical annular microchannel, Int. J. Heat Mass Transfer 52(2009) 1075e1079.

[17] S. Chen, J. Tolke, M. Krafczyk, Numerical investigation of double-diffusive(natural) convection in vertical annuluses with opposing temperature andconcentration gradients, Int. J. Heat Fluid Flow 31 (2010) 217e226.

[18] A. Bejan, Entropy Generation Through Heat and Fluid Flow, second ed. Wiley,New York, 1994.

[19] A. Bejan, Entropy Generation Minimization. CRC Press, Boca Raton, FL, 1996.[20] K. Hooman, A. Ejlali, Effects of viscous heating, fluid property variation,

velocity slip, and temperature jump on convection through parallel plate andcircular microchannels, Int. Commun. Heat Mass Transfer 37 (2010) 34e38.

[21] K. Hooman, F. Hooman, M. Famouri, Scaling effects for flow in micro-chan-nels: variable property, viscous heating, velocity slip, and temperature jump,Int. Commun. Heat Mass Transfer 36 (2009) 192e196.

[22] K. Hooman, A superposition approach to study slip-flow forced convection instraight microchannels of uniform but arbitrary cross-section, Int. J. Heat MassTransfer 51 (2008) 3753e3762.

[23] M. Famouri, K. Hooman, Entropy generation for natural convection byheated partitions in a cavity, Int. Commun. Heat Mass Transfer 35 (2008)492e502.

[24] K. Hooman, F. Hooman, S.R. Mohebpour, Entropy generation for forcedconvection in a porous channel with isoflux or isothermal walls, Int. J. Exergy5 (2008) 78e96.

[25] K. Hooman, A. Haji-Sheikh, Analysis of heat transfer and entropy generationfor a thermally developing BrinkmaneBrinkman forced convection problemin a rectangular duct with isoflux walls, Int. J. Heat Mass Transfer 50 (2007)4180e4194.

[26] S. Tasnim, S. Mahmud, Mixed convection and entropy generation in a verticalannular space, Exergy Int. J. 2 (2002) 373e379.

[27] S. Mahmud, R. Fraser, Analysis of entropy generation inside concentriccylindrical annuli with relative rotation, Int. J. Therm. Sci. 42 (2003) 513e521.

[28] B. Yilbas, M. Yurusoy, M. Pakdemirli, Entropy analysis for non-newtonian fluidflow in annular pipe: constant viscosity case, Entropy6 (2004) 304e315.

[29] N. Allouache, S. Chikh, Second law analysis in a partly porous double pipe heatexchanger, J. Appl. Mech. 73 (2006) 60e65.

[30] M. Yari, Second-law analysis of flow and heat transfer inside a microannulus,Int. Commun, Heat Mass Transfer 36 (2009) 78e87.

[31] S. Chen, Analysis of entropy generation in counter-flow premixed hydrogen-air combustion, Int. J. Hydrogen Energy 35 (2010) 1401e1411.

[32] S. Chen, J. Li, H.F. Han, Z.H. Liu, C.G. Zheng, Effects of hydrogen addition onentropy generation in ultra-lean counter-flow methane-air premixedcombustion, Int. J. Hydrogen Energy 35 (2010) 3891e3902.

[33] S. Chen, H.F. Han, Z.H. Liu, J. Li, C.G. Zheng, Analysis of entropy generation innon-premixed hydrogen versus heated air counterflow combustion, Int. J.Hydrogen Energy 35 (2010) 4736e4746.

[34] S. Chen, J. Tolke, S. Geller, M. Krafczyk, Lattice Boltzmann model for incom-pressible axisymmetric flows, Phys. Rev. E 78 (2008) 046703.

[35] S. Chen, J. Tolke, M. Krafczyk, Simulation of buoyancy-driven flows ina vertical cylinder using a simple lattice Boltzmann model, Phys. Rev. E 79(2009) 016704.

[36] S. Chen, M. Krafczyk, Entropy generation in turbulent natural convection dueto internal heat generation, Int. J. Thermal Sci. 48 (2009) 1978e1987.

[37] R. Benzi, S. Succi, M. Vergassola, The lattice Boltzmann equation: theory andapplications, Phys. Rep. 222 (1992) 145e197.

[38] S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev.Fluid Mech. 30 (1998) 329e364.

[39] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.Oxford University Press, Oxford, 2001.

[40] G. Amati, S. Succi, R. Piva, Massively parallel LatticeeBoltzmann simulation ofturbulent channel flow, Int. J. Modern Phys. C 8 (1997) 869e877.

[41] D.O. Martinez, W.H. Matthaeus, S. Chen, et al., Comparison of spectral methodand lattice Boltzmann simulations of two-dimensional hydrodynamics, Phys.Fluids 6 (1994) 1285e1298.

[42] X. He, G.D. Doolen, T. Clark, Comparison of the lattice Boltzmann method andthe artificial compressibility method for NaviereStokes equations, J. Comput.Phys. 179 (2002) 439e451.

[43] S. Chen, J. Tolke, M. Krafczyk, Numerical simulation of fluid flow and heattransfer inside a rotating disk-cylinder configuration by a lattice Boltzmannmodel, Phys. Rev. E 80 (2009) 016702.

[44] S. Chen, J. Tolke, M. Krafczyk, Simple lattice Boltzmann subgrid-scale modelfor convectional flows with high Rayleigh numbers within an enclosedcircular annular cavity, Phys. Rev. E 80 (2009) 026702/1e026702/8.

[45] S. Chen, J. Tolke, M. Krafczyk, A new method for the numerical solution ofvorticity-streamfunction formulations, Comput. Methods Appl. Mech. Eng.198 (2008) 367e376.

[46] S. Chen, Z.W. Tian, Simulation of thermal micro-flow using lattice Boltzmannmethod with Langmuir slip model, Int. J. Heat Fluid Flow 31 (2010) 227e235.

[47] S. Chen, Z. Liu, C. Zhang, Z. He, Z. Tian, B. Shi, C. Zheng, A novel coupled latticeBoltzmann model for low Mach number combustion simulation, Appl. Math.Comput. 193 (2007) 266e284.

[48] S. Chen, Z. Liu, Z. Tian, B. Shi, C. Zheng, A simple lattice Boltzmann scheme forcombustion simulation, Comput. Math. Appl. 55 (2008) 1424e1432.

[49] S. Chen, Z. Liu, Z. He, C. Zhang, Z. Tian, C. Zheng, A new numerical approach forfire simulation, Int. J. Modern Phys. C 18 (2007) 187e202.

[50] F. Massaioli, R. Benzi, S. Succi, Exponential tails in two-dimensional Rayleigh-Bnard Convection, Europhys. Lett. 21 (1993) 305.

[51] M. Magherbi, H. Abbassi, A.B. Brahim, Entropy generation at the onset ofnatural convection, Int. J. Heat Mass transfer 46 (2003) 3441e3450.

[52] T.J. Chung, Computational Fluid Dynamics. Cambridge University Press,Cambridge, 2002.

[53] S. Chen, A large-eddy-based lattice Boltzmann model for turbulent flowsimulation, Appl. Math. Comput. 215 (2009) 591e598.

Documents

Natural convection and entropy generation in a vertically concentric annular space