Entropy generation inside disk driven rotating convectional flow

lable at ScienceDirect

International Journal of Thermal Sciences 50 (2011) 626e638

Contents lists avai

International Journal of Thermal Sciences

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

Entropy generation inside disk driven rotating convectional flow

Sheng Chen a,b,*

a State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, ChinabNational Technology Center, WISCO, Wuhan 430083, China

a r t i c l e i n f o

Article history:Received 17 March 2010Received in revised form1 September 2010Accepted 23 November 2010Available online 24 December 2010

Keywords:Entropy generationDisk driven rotating flowConvectional flow

* R&D Center, WISCO, Wuhan 430083, China.E-mail address: [email protected].

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.11.012

a b s t r a c t

In the present study, entropy generation inside disk driven rotating convectional flow is investigated forthe first time. In order to reveal the effects of rotating of the disk (described by the Reynolds number Re)and the buoyancy force (described by the Grashof number Gr) on entropy generation systematically, wedivide the cases investigated in the present study into three groups: (1) Res0 but Gr¼ 0 (correspondingto forced convection); (2) Re¼ 0 but Grs0 (corresponding to natural convection); (3) both Res0 andGrs0 (corresponding to mixed convection). We find the situation becomes quite complex when theinfluences of rotating of the disk and the buoyancy force both should be considered. The conclusionsdrew from the first two groups can not be applied to the last one straightforwardly. Especially, somephenomena observed in Group (3) are fully contrary to that in the first two groups.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

The complex hydrodynamic behavior of convectional flowinside a coaxial rotating disk-cylinder configuration has for a longtime been a subject of considerable attention because of its widevariety of immediate technological applications, for example, Czo-chralski crystal-growth in material engineering [1] and electrodeposition process in electrochemistry [2]. Due to importance ofsuch flows, considerable research efforts, experimental works aswell as analytical and numerical studies, have been performed inorder to realize their characteristics [3e8]. Inamuro et al. [3]investigated the effects of disk rotation and Prandtl numbers onthe fluid flow and heat transfer in the container. In their workNusselt numbers on the walls were calculated for the temperaturefields and were compared with available experimental data.Numerical and experimental study of the forced convection insidea rotating disk-cylinder configuration were conducted by Ruiz andhis cooperators [4]. The authors discussed the influences ofdifferent boundary conditions on flow patterns. Lee and Hyun [5]reported the effect of the Prandtl number on global flow patternsfor both gravitationally unstable and gravitationally stable config-urations. They also described the behavior of the vortex breakdownstagnation bubble under the influence of buoyancy. Influence of themagnetic field and the conductance ratio on the mass transferrotating lid driven flow was reported by Kharicha’s group [6] using

son SAS. All rights reserved.

a finite volume computational fluid dynamics (CFD) method as wellas experimental tools. Bessaih and his cooperators [7] numericallyinvestigated the thermal instability of magnetohydrodynamicflowswhich was created in a cylindrical enclosurewith rotating topdisk, having an aspect ratio equal to 2. Recently, Chen et al. [8]proposed a lattice Boltzmann (LB) model for convectional flowinside a coaxial rotating disk-cylinder configuration to overcomethe disadvantages of traditional CFD methods in this field. Onlya few of them can be cited here since the amount of the publica-tions is too large. However, almost all of them are based on the first-law of thermodynamics as well as the fluid mechanics, and ignoredcomplicated analysis for entropy generation and optimum design.

After Bejan [9] showed that entropy generation in convectivefluid flow is due to heat transfer and viscous shear stresses, a lot ofnumerical investigations on the entropy generation in convectionalflow were carried out. Tasnim and Mahmud [10] analyzed theentropy generation for mixed convection in a vertical annularspace. The effect of relative rotational motion between the innerand outer cylinders of a cylindrical annulus on entropy generationwas studied by Mahmud et al. [11]. Yilbas and his cooperators [12]investigated entropy generation characteristics of non-Newtonianfluid flow in annular pipe. Allouache and Chikh [13] reported theperformance of a annular heat exchanger with a porous mediumattached over the inner pipe based on second-law analysis. Chenet al. [14] investigated numerically entropy generation in turbulentnatural convection due to internal heat generation. Recently, theentropy generation analysis has been extended widely for reactiveflows [15e17]. But surprisingly, to date there is no study on entropy

mailto:[email protected]

www.sciencedirect.com/science/journal/12900729

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

http://dx.doi.org/10.1016/j.ijthermalsci.2010.11.012



Nomenclature

A aspect ratioBe Bejan numberc fluid particle speedcs speed of sounde!k discrete velocity vectorEc Eckert numberFFI Fluid Friction Irreversibilityfj, gk distribution function for flow and temperature fieldf eqj ; geqk equilibrium distribution function for flow and

temperature fieldGr Grashof numberH height of cylinderHTI Heat Transfer IrreversibilityP pressurePr Prandtl numberRe Reynolds numberRc radius of cylinderRx radius of diskbS Svanberg vorticityStotal total entropy generation numberS local entropy generation numberT temperatureu! fluid velocity vectorx! phase space

Greek symbolsQ source term in Eq. (26)L;LR aspect ratio, radius ratioYo source term in Eq. (28)F swirlUk;U

0k source terms in Eq. (33)

k thermal diffusivitym dynamics viscosityn kinematic viscosityxk the weights for equilibrium distribution functionr densitysj relaxation time for Eq. (33)s relaxation time for Eq. (27)fab strain rate tensorjfj magnitude of the strain rate tensor4 irreversibility distribution ratioc parameter in Eq. (32)j stream functionDx, Dt grid spacing, time stepDT dimensionless temperature difference

Subscripts and superscriptsk discrete velocity direction- averagea, b spatial indexU globalm viscousD thermal

zRx

(Rotational axis)

S. Chen / International Journal of Thermal Sciences 50 (2011) 626e638 627

generation inside disk driven rotating convectional flow, whichinspires the present study.

The objective of the present work is to perform a detailed studyon the effects of Reynolds number (i.e. rotating of the disk) andGrashof number (the buoyancy force) on entropy generation insidedisk driven rotating convectional flow for the first time. The aboveliterature survey clearly shows that there is no open study on thistopic. The governing equations for flow and scalar fields as well aspart of the entropy generation equation are solved by the latticeBoltzmann (LB) method recently developed by the present authors[8,18,19] instead of traditional numerical methods [9]. As shown inour previous study, the LBmethod possesses significant advantagesover traditional CFD methods for entropy generation analysis[14,20]: for example, the LB method is more suitable for massiveparallel computing of engineering flows and more efficient toevaluate entropy generation due to fluid friction.

r

H

0 Rc

stationary partof upper surface

sidewall

cylinder bottom

rotating disk

working fluid

Fig. 1. The configuration of coaxial rotating disk-cylinder.

2. Governing equations

The configuration of a coaxial rotating disk-cylinder is shown inFig. 1, which is a benchmark research prototype popularly used inprevious works [1,3,18]. It consists of a vertical cylindrical crucibleof radius Rc filled with working fluid to a height H. The workingfluid is bounded above by a coaxial disk of radius Rx � Rc rotatingwith angular velocity Ux.

The primitive-variables-based formulae of the governing equa-tions in the cylindrical coordinate system can be written as [1,18]

1rvðruÞvr

þ vwvz

¼ 0; (1)

vuvt

þ uvuvr

þwvuvz

� v2

r¼ �1

r

vpvr

þ n

�V2u� u

r2

�; (2)

vv

vtþ u

vv

vrþw

vv

vzþ uv

r¼ n

�V2v� v

r2

�; (3)

vwvt

þ uvwvr

þwvwvz

¼ �1r

vpvz

þ nV2wþ gaðT � TcÞ; (4)

vTvt

þ uvTvr

þwvTvz

¼ kV2T: (5)

where

V2 ¼ 1rv

vr

�rv

vr

�þ v2

vz2:

Table 1Some results for the test cases by the present model, the hybrid LB schemes [36] and QUICK [37].

Gr Rex Rec jðaÞmin jðaÞ

max jðbÞmin jðbÞ

max jðcÞmin jðcÞ

max

0 102 0 �2:381� 10�1 7:891� 10�6 �2:272� 10�1 7:921� 10�6 �2:172� 10�1 4:063� 10�6

0 103 0 �3:286� 100 7:403� 10�5 �4:979� 100 7:311� 10�5 �4:994� 100 1:826� 10�5

105 0 0 �2:337� 10�3 2:835� 101 �1:213� 10�3 2:863� 101 �5:798� 10�4 2:841� 101

106 0 0 �2:715� 10�1 9:270� 101 �3:805� 10�1 9:320� 101 �1:200� 10�1 9:251� 101

105 101 0 �7:810� 10�4 2:815� 101 �1:178� 10�3 2:863� 101 �5:785� 10�4 2:841� 101

105 102 0 �1:618� 10�4 2:837� 101 �1:564� 10�4 2:860� 101 �4:517� 10�4 2:838� 101

105 103 0 �4:284� 10�1 2:511� 101 �5:562� 10�1 2:528� 101 �5:677� 10�1 2:517� 101

Note. (a), present work; (b), Ref. [36]; (c), Ref. [37].

S. Chen / International Journal of Thermal Sciences 50 (2011) 626e638628

u, v and w are radial, azimuthal and axial velocity components; p isthe pressure; T is the temperature; n is the kinetic viscosity; g is thegravitational acceleration along the negative z-axis,; k is thethermal conductivity; r is the density of the working fluid; anda the coefficient of thermal expansion. The subscripts c and x standfor the cylinder wall and the disk.

Because the coupling term uv/r in the discretized governingequation leads to a false production of angular momentum that isdifficult to eliminate, usually a physically conserved quantity theswirlF ¼ rv, viz the angularmomentumperunitmass, is introducedto overcome this difficulty. Then Eqs. (2) and (3) become [21,22]

vuvt

þ uvuvr

þwvuvz

� F2

r3¼ �1

r

vpvr

þ n

�V2u� u

r2

�; (6)

vF

vtþ u

vF

vrþw

vF

vz¼ n

rv

vr

�r3

v

vr

�F

r2

��þ n

v2F

vz2: (7)

However, in most practical simulations the more frugal vorti-cityestreamfunction method are employed instead of the primitiveequations [21,22]. Because for rotationally symmetric flow,computation time can be reduced if the problem is reformulated sothat the three variables u, v, p are eliminated in favor of the vorticityu and Stokes streamfunction j, which are defined as [18]

u ¼ vwvr

� vuvz

; (8)

u ¼ 1rvj

vz; (9)

w ¼ �1rvj

vr: (10)

The dimensionless vorticityestreamfunction-based governingequations read [8,18]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a b

Fig. 2. Streamlines (a) and is

vbS þ uvbS þw

vbS þ v F2

¼ Gr 1 vT
vt vr vz vz
r4

!L3RRe

2 r vr( � � �� ) (11)

þ 1Re

1rv

vr1rv

vrr2bS þ v2bS

vz2;

vF

vtþ u

vF

vrþw

vF

vz¼ 1

Re

(1rv

vr

�r3

v

vr

�F

r2

��þ v2F

vz2

)(12)

vTvt

þ uvTvr

þwvTvz

¼ 1Re Pr

"1rv

vr

�rvTvr

�þ v2T

vz2

#; (13)

v

vr

�1rvj

vr

�þ 1

rv2j

vz2¼ �rbS: (14)

with following boundary conditions [8]:

bS ¼ � 2r2Dz2

jðr;LRL�Dz; tÞ; U ¼ r2; T ¼ Tx ¼ �0:5; j ¼ 0

(15)

on the disk,

bS ¼ � 2L2RDr

2jðLR � Dr; z; tÞ; U ¼ 0; T ¼ Tc ¼ 0:5; j ¼ 0

(16)

on the cylinder wall,

bS ¼ � 2r2Dr

jðr;Dz; tÞ ; U ¼ 0;vTvz

¼ 0; j ¼ 0 (17)

on the cylinder bottom,

0.45

0.40.35

0.3

0.25

0.2

0.150.10.050

-0.1-0.2

-0.35-0.45

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

otherms (b) at Re¼ 100.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.45

0.40.35

0.3

0.25

0.2

0.150.10.05

-0.05-0.15-0.25

-0.45-0.35

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 3. Streamlines (a) and isotherms (b) at Re¼ 500.


bS ¼ 0;vU

vz¼ 0; T ¼ Tx þ r � Rx

Rc � RxðTc � TxÞ; j ¼ 0 (18)

on the gap surface, and the axisymmetric boundary condition [8]for the rotational axis r¼ 0. In the above equations the vorticity uis replaced by the Svanberg vorticity bS ¼ u=r for numerically stablemodeling of physically unstable flows [18,21]. Other independentscaling parameters are:

(1) Aspect ratio L ¼ H=Rc;(2) Ratio of cylinder radius to disk radius LR ¼ Rc=Rx;(3) Prandtl number Pr ¼ n=k;(4) Reynolds number Re ¼ R2cUx=n;(5) Grashof number Gr ¼ agR3c ðTc � TxÞ=n2;(6) Dimensionless temperature T ¼ ðT � TxÞ=ðTc � TxÞ, for simpli-

fication the overline is omitted in the rest part.

Eqs. (11)e(14) can be further transformed as [8]:

vbSvt

þ uvbSvr

þwvbSvz

¼ 1Re

v2bSvr2

þ v2bSvz2

!þ bSo; (19)

vF

vtþ u

vF

vrþw

vF

vz¼ 1

Re

v2F

vr2þ v2F

vz2

!þ Fo; (20)

vTvt

þ uvTvr

þwvTvr

¼ 1Re Pr

v2Tvr2

þ v2Tvz2

!þ To; (21)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

z

Fig. 4. Streamlines (a) and iso

v2j

vz2þ v2j

vr2¼ Q: (22)

In Eqs. (19)e(22), the source terms caused by the transformationread

bSo ¼ 3rRe

vbSvr

þ Gr

L3RRe

2

1xvTvr

� 2Fr4

vF

vz; (23)

Fo ¼ � 1rRe

vF

vr; (24)

To ¼ 1rRe Pr

vTvr

; (25)

Q ¼ ��r2bS þw

�: (26)

3. Lattice Boltzmann model for disk driven rotatingconvectional flow

In the last two decades, the LB method has matured as an effi-cient alternative for simulating andmodeling complicated physical,chemical and social systems [23,24]. The implementation of a LBprocedure is quite easy. Parallelization of a LBmodel is natural sincethe relaxation is local and the communication pattern in

0.450.4

0.35

0.3

0.25

0.2

0.150.10.05-0.05-0.15

-0.45-0.35-0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

b

therms (b) at Re¼ 1000.

0.0550993

0.107974

0.20873

0.538655

1

2

34549

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.999104

0.998

0.996

0.994

0.9920.990.9880.984

0.998

0.9940.998

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 5. Distributions of entropy generation number (a) and Bejan number (b) at Re¼ 100.


propagation is one way, and the performance increases nearlylinearly with the number of CPUs. Moreover, the LB moldes havebeen compared favourably with spectral methods [25], artificialcompressibility methods [21,26,27], finite volume methods[28e30], finite difference methods [31e33], projection methods[22] and multigrid methods [34,35], all quantitative results furthervalidate excellent performance of the LB method not only incomputational efficiency but also in numerical accuracy. Dueto these advantages, the LB method has been successfully used tosimulate many problems, from laminar single phase flows toturbulent multiphase flows [24].

Eqs. (19)e(21) can be solved using the LB model developed inour previous work [8]. It reads:

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

�gðeqÞk ð x!; tÞiþ DtYo;k (27)

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

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

c ¼ Dx=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 first-order term of the expansion of thesource term Yo, Yo ¼ bSo;Fo; To for Eqs. (19)e(21) respectively. Yo;ksatisfies:

0.052039

0.0765937 0.0765937

0.104601

0.149347

0.23042

0.376913

0.616787

1

23.339

72456

10

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

z

b

Fig. 6. Distributions of entropy generation num

Xk�0

Yo;k ¼ Yo (28)

In this study we simply choose

Yo;k ¼ Yo5

(29)

The equilibrium distribution gðeqÞk is defined by

gðeqÞk ¼bS5

�1þ 2:5

e!k$ u!

c

�(30)

The Svanberg vorticity bS is obtained by

bS ¼Xk�0

gk (31)

and the dimensionless relaxation time s is determined by

c ¼ 52c2ðs� 0:5Þ (32)

c ¼ Re; Re; Re Pr for Eqs. (19)e(21) respectively.The evolution equation for Eq. (22) reads [8]

fkð x!þ c e!kDt; t þ DtÞ � fkð x!; tÞ ¼ Uk þ U0k (33)

where Uk ¼ �s�1j ½ fkð x!; tÞ � f ðeqÞk ð x!; tÞ�, U0

k ¼ DtzkQD andD ¼ c2

2 ð0:5� sjÞ. sj > 0:5 is the dimensionless relaxation time. f ðeqÞkis the equilibrium distribution function, and defined by

0.99

0.99274

0.995

0.995888

0.996637

0.996637

0.997235

0.998084

0.999417

0.998084

0.9988120.999417

0.999931

0.992740.99274

0.97

0.99

0.99274

0.996637

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

ber (a) and Bejan number (b) at Re¼ 500.

0.0460491

0.0806436 0.0806436

0.126373

0.193823

0.418959

1

23

457

11

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.99

0.9918370.994216

0.996116 0.997161

0.9985040.998815

0.999693

0.994216

0.9918370.998815

0.9998740.999693

0.990.9985040.9985040.97

0.996116

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 7. Distributions of entropy generation number (a) and Bejan number (b) at Re¼ 1000.


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

(34)

xk and zk are weight parameters given as x0 ¼ z0 ¼ 0,xk ¼ zk ¼ 1=4ðk ¼ 1/4Þ. j is defined by

j ¼Xk�1

fk (35)

4. Entropy generation

The entropy generation number is given by [9,14]:

S ¼ ðVTÞ2þ4jfj2 (36)

where the irreversibility distribution ratio 4 ¼ Br=DT [14].Br¼ PrEc is the Brinkman number, where Ec is the Eckert number[9]. jfj is the magnitude of the strain rate tensor [19,27]

jfj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fabfab

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

"�vuvr

�2

þ�ur

�2þ�vwvz

�2#þ�vwvr

þ vuvz

�2vuut(37)

where fab ¼ ðvaub þ vbuaÞ=2.Apparently, it is inconvenient to calculate jfj (Eq. (37)) directly

by conventional numerical methods due to its complex form ofspatial derivative [9]. However, in the present LB model, jfj can becalculated easily by [14,27]

a b

Fig. 8. Total entropy generation number (a) a

jfj ¼ juj ¼rbS: (38)

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

Recognizing the first term in Eq. (36) as reflecting the entropygeneration due to thermal diffusion and the second due to viscousdissipation, the entropy generation number can be expressed as

S ¼ SD þ Sm (39)

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 FluidFriction Irreversibility FFI [9]. The Bejan number Be is given as[9,14]:

Be ¼ SDS

¼ HTIHTIþ FFI

(40)

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 [9]

Stotal ¼ZU

SvU (41)

where the subscript U means the global computational domain.Similar expression can be written for the Bejan number [17].

nd Bejan number (b) versus variable Re.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.45

0.40.35

0.3

0.25

0.20.150.1

0.050

-0.05

-0.1

-0.15

-0.2-0.25

-0.45-0.35

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 9. Streamlines (a) and isotherms (b) at Gr¼ 105.


5. Numerical validation

Firstly, the numerical simulation was validated. Table 1 lists thecomparison of computed minimum and maximum stream functionfor different cases with grid resolution 100�100. For comparison,we also present the results of Xu et al. [37] using the second-orderdifference scheme and those in Ref. [36] obtained by a hybrid LBscheme. In all cases, the maximum absolute values of stream

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a b

Fig. 10. Streamlines (a) and i

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

z

Fig. 11. Streamlines (a) and i

function computed by the present model agree very well with theprevious data [36,37]. Some very small deviations between thecomputed minimum absolute values of stream function can beneglected since the minimum absolute values of stream functionare so small compared with the maximum absolute values.

The computer code for entropy generation analysis developed inour previous works [14,17,20] is used in the present study. Thevalidation and description of the code can be found in Refs. [14,17]

0.45

0.4

0.35

0.3

0.250.20.15

0.1

0.05

0-0.05-0.

1-0.15

-0.2-0.25

-0.45-0.35

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

sotherms (b) at Gr¼ 106.

0.450.4

0.35

0.3

0.250.2

0.150.1

0.05

0

-0.05

-0.1

-0.15

-0.2-0.25

-0.45-0.35

-0.15-0.1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

b

sotherms (b) at Gr¼ 107.

3.7044521.07123

0.341341

0.341341

0.1757590.175759

0.175759

0.615333

1.07123

20 128

3.70445

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.95

0.950.9

89328

0.989328

0.85

0.80.35

0.15

0.1 0.30.6 0.8

0.989328

0.95

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 12. Distributions of entropy generation number (a) and Bejan number (b) at Gr¼ 105.


6. Results and discussions

In the present work, Pr¼ 0.05, LR ¼ 2:5, L ¼ 1 and Br¼ 1.0,which are popularly used in previous publications [8,36,37]. Theprototype of this configuration is Czochralski crystal-growth inmaterial engineering [1,8]. In order to reveal the effects of Re and Gron entropy generation systematically, we divide the cases investi-gated in the present study into three groups: (1) Res0 but Gr¼ 0(corresponding to forced convection); (2) Re¼ 0 but Grs0

0.286666

35

55

30

0.286666

1.25333

20

1551.25333

1.25333

0.877726

1.25333

101.25333

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a


20

8

0.555672

0.2800210.2800

210.555672

2084

22

1.415 0.806839

6

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a


(corresponding to natural convection); (3) both Res0 and Grs0(corresponding to mixed convection).

6.1. The sole effect of Re

In order to investigate the sole effect of Re on entropy generationinside disk driven rotating convectional flow, we set Gr¼ 0 while100 � Re � 1000, namely no buoyancy force in this group.

0.9

0.90.60.2

0.10.9

0.995909

0.9

0.7

0.7

0.1

0.7

0.6

0.9

0.4

0.2

0.50.7

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

ber (a) and Bejan number (b) at Gr¼ 107.

0.98478

0.9

0.1

0.1

0.98478

0.50.4

0.984780.8 0.6

0.4

0.20.50.7

0.98478

0.4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

ber (a) and Bejan number (b) at Gr¼ 106.

a b

Fig. 15. Total entropy generation number (a) and Bejan number (b) versus variable Gr.


Figs. 2e4 illustrate the streamlines and isotherms for differentRe. It can be observed that when the disk rotation speed isincreased, the intensity of vortex increases, and the center of thevortex induced by the rotation moves towards the side wall of thecylinder and is deformed increasingly. Meanwhile, the highestvelocity region moves from the upper left corner to the upper rightcorner, while the lowest velocity region is enlarged in the lower leftcorner. The temperature fields for the cases of forced convectionGr¼ 0 are very similar.

Figs. 5e7 show the corresponding distributions of entropygeneration number and Bejan number for the cases Gr¼ 0. The

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

Fig. 17. Streamlines (a) and isotherm

a

Fig. 16. Total entropy generation number (a) and Bejan

features of them are also similar: entropy generates intensivelynear the rotating disk. Especially the maximum of S appears in thevicinity of the edge of the rotating disk. Far away from the disk,entropy generation decreases quickly. Although the movement ofthe working fluid is driven by the rotation of the disk instead ofbuoyancy force, the irreversibility due to heat transfer dominatesthe whole cavity and at every point of the domain Be approachesunity. It can be confirmed by Fig. 8(b), in which the total Bejannumber Be> 0.99 for all the cases in this group. According to thisfigure, it is clear that the total Bejan number is a monotonicdecreasing function of Re. However, the total entropy generation

0.450.40.350.30.250.20.150.10.0

50-0.05

-0.45-0.35

-0.25

-0.15

-0.1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

s (b) at Gr¼ 105 and Re¼ 100.

b

number (b) versus dimensionless time for Gr¼ 107.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.45

0.40.40.350.3

0.250.20.150.1

0.05

0-0.05-0.1

-0.45-0.35

-0.25

-0.15

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 18. Streamlines (a) and isotherms (b) at Gr¼ 105 and Re¼ 1000.


number Stotal is a monotonic increasing function of Re. WhenRe< 600, the increment of Stotal is sharp but it becomes slow andslow since Re> 600, as shown in Fig. 8(a).

6.2. The sole effect of Gr

In order to investigate the sole effect of Gr on entropy generationinside this kind of convectional flow, we set Re¼ 0while 105 � Gr �

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a b


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

a

z

b


107. Namely, the disk is at rest and the motion of the working fluidis induced only by buoyancy force.

The streamlines and isotherms for different Gr are plotted inFigs. 9e11. When Gr increases, the vortex gets enlarged and itshighest velocity layer moves to the side wall of the cylinder.Different from that in Figs. 2e4, the isothermal lines in thisgroup become vertical in the vicinity of the side wall of thecylinder due to the existence of buoyancy force. The flow field

0.450.40.350.30.250.2

0.150.10.05

-0.45-0.35

-0.25

0-0.05

-0.1

-0.15-0.2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

s (b) at Gr¼ 106 and Re¼ 1000.

0.45

0.40.350.30.25

0.20.150.1

0.05

0-0.05

-0.1

-0.45-0.35

-0.25

-0.2

-0.15

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

s (b) at Gr¼ 106 and Re¼ 100.

2.51695

2.51695

12

4

2.51695

1.40173

1.02538

0.675853

0.582143

0.557935 0.261635

0.457203

0.457203

1.02538

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a 0.95

0.950.9

0.85

0.9

0.80.60.40.1 0.05 0.05

0.55

0.975141

0.40.55

0.95

0.975141

0.975141

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 21. Distributions of entropy generation number (a) and Bejan number (b) at Gr¼ 105 and Re¼ 100.


becomes unsteady as Gr¼ 107 and the isotherms are distortedsignificantly.

Figs. 12e14 illustrate the corresponding distributions of entropygeneration number and Bejan number for the cases of naturalconvection Re¼ 0. From these figures it is clear that when Gr> 105

entropy generates intensively not only near the edge of the rest diskbut also in the vicinity of the side and bottomwalls of the cylinder.The motion of working fluid is enhanced significantly with Gr, sothe area where the irreversibility due to viscous effect dominates

16

12

124

0.4255540.683605

0.425554

0.683605

1.16616

2

46 8

4

2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

a

Fig. 23. Distributions of entropy generation number (a

2

0.21098

0.46616

0.615393

0.975879

1.42106

22.68123

46810

2

0.46616

0.21098

0.46616

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

Fig. 22. Distributions of entropy generation number (a

becomes large and large, even though the whole domain is stillpredominated by the irreversibility due to heat transfer. It can beconfirmed by Fig. 15(b): the total Bejan number decreases sharplyagainst Gr and is always less than 0.8. According to Fig. 15(a) it isclear that the total entropy generation number Stotal increases withGr.

Fig. 16 shows the oscillations of the total entropy generationnumber and Bejan number versus dimensionless time whenGr¼ 107. One can observe that there is a obvious peak value for the

0.950.75

0.95

0.9

0.950.95

0.75

0.60.55

0.80.950.85

0.05 0.10.95

0.05

0.2

0.95

0.350.50.55

0.95

0.95

0.6

0.55

0.45

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

b

) and Bejan number (b) at Gr¼ 106 and Re¼ 100.

0.95 0.9

0.80.8

0.85

0.95

0.9

0.85

0.95

0.95

0.9

0.8 0.75

0.95

0.45

0.15

0.05

0.15

0.10.55

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

) and Bejan number (b) at Gr¼ 105 and Re¼ 1000.

a b

Fig. 25. Total entropy generation number (a) and Bejan number (b) versus variable Gr and Re.

0.26028

16

10 1.3240240.

468743

0.26028

0.26028

0.812648

0.687773

0.687773

0.812648

1.32402

22.7586

4610

16

4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

a

0.9

0.9

0.95

0.750.55

0.05 0.2 0.1 0.95

0.950.9

0.95

0.05

0.10.2

0.50.650.850.95

0.95

0.95

0.95 0.95

0.95

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

b

Fig. 24. Distributions of entropy generation number (a) and Bejan number (b) at Gr¼ 106 and Re¼ 1000.


total entropy generation number and Bejan number respectively atthe onset of the transient state and the oscillations become regularwhen the flow is fully developed. From a thermodynamics viewpoint, the oscillatory behavior of the total entropy generation withdimensionless time at high Grashof numbers shows that the systemis in the non-linear branch of the thermodynamics of irreversibleprocesses [38]. The convectional flow for Gr � 107 is relatively farfrom the equilibrium state, then a rotation around the fullydeveloped state is possible, and the system is in the case of a spiralapproach towards this state corresponding to an oscillation of thetotal entropy generation. Consequently, the system evolves in thenon-linear branch of irreversible phenomena.

6.3. The combined effects of Re and Gr

In this subsection we discuss the combined effects of Re and Gron entropy generation inside disk driven rotating convectionalflow. The motion of working fluid in mixed convection is driven byboth the rotation of the disk and the buoyancy force.

Figs. 17e20 plots the streamlines and isotherms for variable Reand Gr. It can be observed that when Gr is small (for exampleGr¼ 105), whatever Re is the forced convective flow dominates theflow. However, as Gr is big enough (for example Gr¼ 106), whateverRe is the natural convective flow dominates the flow while theforced convective flow induced by the disk only has minor effect.

It is very interesting that whatever predominates the flow, thedistributions of entropy generation number and Bejan number forthe mixed convection have similar characteristics, as shown in Figs.21e24: entropy generates intensively not only near the edge of therotating disk but also in the vicinity of the side and bottomwalls of

the cylinder. Near the right-bottom corner of the cylinder, there isa obvious zone where the irreversibility due to viscous effectdominates, which are similar with that in their pure naturalconvection counterparts. However, the features in their pure forcedconvection counterparts can not be observed even though for thecases in which the forced convective flow dominates the flow (forexample Figs. 17 and 18). It implies that Gr has greater influencethan Re on entropy generation inside disk driven rotating convec-tional flow, which also can be confirmed by Fig. 25. From this figure,it is clear that, compared with Gr, the variations of the total entropygeneration number and Bejan number are nearly insensitive to Re.The total entropy generation number increases significantlywith Grbut the total Bejan number decreases apparently against Gr, whichagreewith the conclusions drew from their pure natural convectioncounterparts (c.f. Fig.15). More important, it can be observed that inmixed convection, the total entropy generation number decreasesagainst Re, which is fully contrary to the conclusion drew from theirpure forced convection counterparts (c.f. Fig. 8). Perhaps it resultsfrom that high speed rotation of the disk reduces the temperaturegradients (namely the irreversibility due to heat transfer) inducedby the buoyancy force. The variation trend of the total Bejannumber versus Re depends on Gr. For Gr¼ 105, the total Bejannumber increases slightly with Gr while it decreases tinily againstGr when Gr¼ 106.

7. Conclusion

In the present study, entropy generation inside disk drivenrotating convectional flow is investigated for the first time. In orderto reveal the effects of rotating of the disk (Re) and the buoyancy


force (Gr) on entropy generation systematically, we divide the casesinvestigated in the present study into three groups: (1) Res0 butGr¼ 0 (corresponding to forced convection); (2) Re¼ 0 but Grs0(corresponding to natural convection); (3) both Res0 and Grs0(corresponding to mixed convection). For the cases in Group (1),the total entropy generation number increases with Re while thetotal Bejan number decreases against Re. For the cases in Group (2),the total entropy generation number increases with Gr while thetotal Bejan number decreases against Gr. The situation becomescomplex when the influences of rotating of the disk andthe buoyancy force both should be considered. The variations of thetotal entropy generation number and Bejan number depend on thedetailed combination of Re and Gr. The conclusions drew fromGroup (1) and Group (2) can not be applied to this groupstraightforwardly. For example, The variation trend of the totalBejan number versus Re depends on the value of Gr.

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (Grant Nos. 51006043 and 50936001)and the Research Foundation for Outstanding Young Teachers,HUST (Grant No. 2010QN027).

References

[1] W.E. Langlois, Buoyancy-driven flows in crystal-growth melts, Annu. Rev.Fluid Mech. 17 (1985) 191e215.

[2] Ph. Mandina, C. Fabian, D. Lincot, Importance of the density gradient effects inmodelling electro deposition process at a rotating cylinder electrode, Elec-trochim. Acta 51 (2006) 4067e4079.

[3] T. Inamuro, A. Yamaguchi, F. Ogino, Numerical simulation of fluid flow andheat transfer in a rotating cylindrical container with a counter-rotating disk atthe fluid surface, Heat Transfer Asian Res. 28 (1999) 172e182.

[4] X. Ruiz, M. Aguilo, J. Massons, F. Diaz, Numerical and experimental study ofthe forced convection inside a rotating disk-cylinder configuration, Exp. Fluids14 (1993) 333e340.

[5] C.H. Lee, J.M. Hyun, Flow of a stratified fluid in a cylinder with a rotating lid,Int. J. Heat Fluid Flow 20 (1999) 26e33.

[6] A. Kharicha, A. Alemany, D. Bornas, Hydrodynamic study of a rotating MHDflow in a cylindrical cavity by ultrasound Doppler shift method, Int. J. Eng. Sci.43 (2005) 589e615.

[7] R. Bessaih, A. Boukhari, P. Marty, Magnetohydrodynamics stability of a rotatingflow with heat transfer, Int. Commun. Heat Mass Transfer 36 (2009) 893e901.

[8] 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) 016702/1e016702/9.

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

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

[11] S. Mahmud, R. Fraser, Analysis of entropy generation inside concentric cylin-drical annuli with relative rotation, Int. J. Thermal Sci. 42 (2003) 513e521.

[12] B. Yilbas, M. Yurusoy, M. Pakdemirli, Entropy analysis for non-Newtonianfluid flow in annular pipe: constant viscosity case, Entropy 6 (2004) 304e315.

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

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

[15] K. Nishida, T. Takagi, S. Kinoshita, Analysis of entropy generation and exergyloss during combustion, Proc. Combust. Inst. 29 (2002) 869e874.

[16] S.K. Soma, A. Datta, Thermodynamic irreversibilities and exergy balance incombustion processes, Prog. Energy Combust. Sci. 34 (2008) 351e376.

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

[18] 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) 016704/1e016704/8.

[19] 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.

[20] 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.

[21] S. Chen, J. Tolke, S. Geller, M. Krafczyk, Lattice Boltzmann model for incom-pressible axisymmetric flows, Phys. Rev. E 78 (2008) 046703/1e046703/8.

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

[23] C.K. Aidun, J.R. Clausen, Lattice-Boltzmann method for complex flows, Annu.Rev. Fluid Mech. 42 (2010) 439e472.

[24] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.Oxford University Press, Oxford, 2001, pp. 87.

[25] D.O. Martinez, W.H. Matthaeus, S. Chen, D.C. Montgomery, Comparison ofspectral method and lattice Boltzmann simulations of two-dimensionalhydrodynamics, Phys. Fluids 6 (1994) 1285e1298.

[26] 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.

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

[28] 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.

[29] S. Chen, Z. Liu, B. Shi, C. Zheng, Computation of gas-solid flows by finitedifference Boltzmann equation, Appl. Math. Comput. 173 (2006) 33e49.

[30] A. Al-Zoubi, G. Brenner, Comparative study of thermal flows with differentfinite volume and lattice Boltzmann schemes, Int. J. Mod. Phys. C 15 (2004)307e319.

[31] S. Chen, Z.W. Tian, Simulation of microchannel flow using the lattice Boltz-mann method, Phys. A 388 (2009) 4803e4810.

[32] 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.

[33] S. Chen, Z. Liu, B. Shi, Z. He, C. Zheng, A novel incompressible finite-differencelattice Boltzmann equation for particle-laden flow, Acta Mech. Sinica 21(2005) 574e581.

[34] S. Chen, Z. Liu, Z. He, C. Zhang, Z. Tian, B. Shi, C. Zheng, A novel latticeBoltzmann model for reactive flows with fast chemistry, Chin. Phys. Lett. 23(2006) 656e659.

[35] 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.

[36] H. Huang, T.S. Lee, C. Shu, Hybrid lattice Boltzmann finite-difference simula-tion of axisymmetric swirling and rotating flows, Int. J. Numer. Meth. Fluids 53(2007) 1707e1726.

[37] D. Xu, C. Shu, B.C. Khoo, Numerical simulation of flows in Czochralski QUICKscheme, J. Crystal Growth 173 (1997) 123e131.

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

Entropy generation inside disk driven rotating convectional flow

Documents

Transcript of Entropy generation inside disk driven rotating convectional flow