Indian Journal of Engineering & Materials SciencesYol. I, August 1994,pp.181-188

Finite element analysis of laminar convection heat transfer and flow of thefluid bounded by V-corrugated vertical plates of different corrugation

frequencies

Mohammad Ali & M Nawsher Ali

Department of Mechanical Engineering, Bangladesh Institute of Technology, Khulna 9203, Bangladesh

Received 27 December 1993; accepted 5 May 1994

A study on steady-state natural convection heat transfer and flow characteristics of the fluid in ahorizontal duct with V-corrugated vertical walls has been tarried out using control volume based Fi­nite Element Method. The effects of corrugation frequency and Grashof numbers on local Nusseltnumber, total heat flux, vertical velocity and temperature distributions at the horizontal mid-planehave been investigated. Corrugation frequency from 1 to 3 and Grashof number from 103 to 105have been considered in this study. The results are compared with the available information onnatural convection and fluid flow.

Natural convection is one of the important phe­nomena in everyday engineering problem. It is ofparticular importance in designing nuclear reac­tors, solar collectors and in many other designproblems where heat transfer is occurring by natu­ral convection. Heat transfer by natural convec­tion depends on the convection currents deve­loped by thermal expansion of the fluid particles.Further, the development of the flow is influencedby the shape of the heat transfer surfaces. There­fore, the investigation for different shapes of theheat transfer surfaces is necessary to ensure theefficient performance of the various heat transferequipments ..

There are several studies on convective heattransfer with corrugated walls. In these studies,however, the heat transfer has been consideredonly from a lower hot corrugated plate to an up­per cold flat plate, both being horizontally placed.No research has, so far, been carried out on con­vective heat transfer with vertical hot and coldcorrugated plates. Chinnapa 1 carried out an ex­perimental investigation on natural convectionheat transfer from a horizontal lower hot V-corru­

gated plate to an upper cold flat plate. He collect­ed data for a range of Grashof numbers from 104to 106• The author noticed a change in the flowpattern at Gr= 8 x 104, which he concluded as atransition point from laminar to turbulent flow.Randall et aU studied local and average heattransfer coefficients for natUral convectionbetween a V-corrugated plate (600 V-angle) and aparallel flat plate to find the temperature distribu-

tion in the enclosed air space. From this tempera­ture distribution they used the wall temperaturegradient to estimate the local heat transfer coeffi­cient. Local values of heat transfer coefficient

were investigated over the entire V-corrugatedsurface area. The author recommended a correla­tion of the average heat flux which presents 10%higher value than that for parallel flat plates.

Zhong et af.3 carried out a finite-differencestudy to determine the effects of variable propert­ies on the temperature and velocity fields and theheat transfer rate in a differentia,lly heated, two­dimensional square enclosure. Nayak and Cheny4considered the problem of free and forced con­vection in a fully developed laminar steady flowthrough vertical ducts under the conditions ofconstant heat flux and uniform peripheral walltemperature. Chenoweth and Paoluccis" obtainedsteady-state, two-dimensional results from thetransient Navier-Stokes equations given for lami­nar cunvective motion of a gas in an enclosedvertical slot with 'large horizontal temperature dif­ferences.

System DescriptionThe problem schematic is shown in Fig. 1. The

top and bottom walls of the enclosure are insulat­ed and the left and right vertical walls are V-cor­rugated. The left and right walls are kept at con­stant temperature. The temperature of the leftwall is Th and that of the right wall is ~, whereTh > ~. The characteristic length of the squareenclosure is L. The origin of the X-Y coordinate

\

INDIANJ. ENG. MATER. SCL AUGUST 1994182

Insulat~

InsulatedL

Corrugation frequency = 1

Insulated

~

'II Insulated

Insulated

Corrugatio frequency = 2

~

Insulat~l

V Corr~~

Fig. l-Schematic of the calculation domain

system is located at the left bottom corner of thecavity.

u au + v au = _ ~ ap + v (a2 ~ + a2 ~) + SUax ay p ax ax ay... (2)

Basic EquationsThe Navier-Stokes equations6 for two-dimen­

sional, incompressible flow with constant propert­ies in cartesian coordinates can be written as fol­lows:

Continuity equation, In the above equations, u and v represent thevelocity components in the x and y directions re­spectively and p is the pressure. The source termsSU and SV consider the other body and surface for­ces in the x and y directions respectively and v isthe kinematic viscosity.

au av-+-=0ax iJy

x-momentum equation,

... (1)

.y-momentum equation,

u av + v av = _ ~ ap + v (a2v + a2v) + SVax ay p ay ax2 a/... (3)

ALl & ALl: NATURAL CONVECTION HEAT TRANSFER 183

where w is the vorticity, defined as,

aw aw (a2w ~w) (asV as")u-+v-=v -2+-2 + --- ... (4)

ax ay .ax ay ax ay

By differentiating Eqs (2) and (3) with respectto y and x respectively and then subtracting theresults of the former from the later, a single vor­ticity transport equation can be obtained,

. " (11)

" . (12)

to yield,

c3Q c3Q a2Q a2Q aeu-+ v-=-+-+ Gr-ax ay ax2 ay2 ax

Method of SolutionThe calculation domain is first discretized9 and

an example of discretization of the domain isshown in Fig. 2. Following the domain discretiza­tion, the integral formulation of the relevant trans­port equation 7 is impossed on each control

where Gr and Pr are the Grashof and Prandtl

numbers 10 respectively and defined as:Gr=gf3(Th- TJVlv2

Pr= via

Here the parameters g, f3 and a represent the ac­celeration due to gravity, the coefficient of ther­mal expansion and the thermal diffusivity of thefluid respectively.

The boundary conditions of the problem are asfollows:- .

(i) U= V= 0 at all walls(ii) lJI= 0 at all walls

(iii) e= 1 at left walle= 0 at right wall

C ) (ae) - (ae) - 0IV ay y=o - ay Y= I -

" . (5)

... (6)

... (7)

av _ auw= ax iJy

a1jJ = uay

_ a1jJ = vax

Upon defining the stream function, 1jJ as

The details for implementing wand 1jJ conditionsare available in the reference 7.

Assuming the properties to be constant· otherthan the density variation in the buoyant forces,the Boussinesq approximation8 which consists ofretaining .only the variations of density in thebuoyancy terms, may be used on Eq. (4) whichresults in

the Poisson equation relating w to 1jJ may be ob­tained by substituting Eqs (6) and (7) into Eq. (5)~

a21jJ a21jJ-2+~+W=O ... (8)ax ay

The energy transport equation for two-dimen­sional incompressible flow with constant propert­ies6,9 can be written as,

Fig. 2-Domain discretization

... (9)

... (10)

v= vLv

T- T.e=-_cTt,-T::

u= uLv '

lJI=!!!v'

y=lL'

aw aw (a2w a2w) aTu-+v-=v -2+-2 +gf3-

ax ay ax ay ax

wL2Q=

V '

xX=-,L

u aT + v aT = a (a2T + a2~ax ay ax2 al)

where a is the thermal diffusivity of the fluid.Eqs (4 )-( 10) can be normalised by introducing

the following nondimensional quantities,

184 INDIAN J. ENG. MATER. SCL, AUGUST 1994

Table I-The variation of the total heat flux for corrugatedwalls with different Grashof numbers

The dimensionless total heat flux at the hotwall,

the horizontal distance measured from the left ex­

tremity of the left wall to its right extremity (seeFig. 1). Henceforth, the left and right extremitie~of the hot wall will be referred to as the "trough"and "peak", respectively. A numerical value 10-5was considered as the convergence criterion forreceiving better accuracy of the results. It may bementioned that the convergence checking wasperformed by calculating the residue from thevorticities and coefficients of the relevant nodal

equations and comparing it with the convergencecriterion. It was found that there was no appreci­able change in results for lower value of converg­ence criterion than the above.

Effects of corrugation frequency (CF) on totaiheat flux-Figs 4 and 5 show the streamline pat­tern of the fluid flow and isotherm plot into theenclosure for Grashof number 105, respectivelyand Table 1 shows the effects of CF on total heatflux (Q) with different Grashof numbers. Thistable shows that Q increases continuously withthe increase in CF for Gr= 103 but for Gr= 104

and 1OS, Q decreases with increase in CF from 1to 3. The increase of CF from 1 to 3 leads to a

higher value of Q for Gr= lO3, which may be at­tributed to the enhancement of surface area, butthe decrease in Q for higher Gr may be explainedas the retardation of flow due to increased wavi­ness of the corrugation. This behaviour may alsobe explained by asserting that at high Gr the fluidvelocity increases near the peaks but drops nearthe troughs as the boundary layer tends to separ­ate. Thus the fluid fails to maintain close contact

near the troughs of the corrugation, resulting indecreased convection heat transfer, whereas for

... (14)

... (13)of)

oN

Grashof number QforCF= 1QforCF=2Qfor CF= 3

103

1.1261.1321.135104

2,2952.2712.238

105

4.8374.7534.573

qNL =Nuy= k(Th - '4)

Results and DiscussionIn this investigation local Nusselt number along

the hot wall, the total heat flux through the enclo­sure {ll1dvertical velocity and temperature distrib­utions at the horizontal mid-plane have been ex­amined and discussed for corrugation frequency(CF) 1, 2, 3, and Grashof number (Gr) 103, 10\105• The discussion of the results that follows arethose obtained by using a 31 x 31 mesh. All theresults are presented in dimensionless form.

In this investigation the corrugation amplitudewas fixed at 5% of the enclosure height for allruns, where the amplitude ''N.' is defined as half

Jy~ I of)Q=- -dS(Y)

Y~ooN

Where S is the dimensionless distance measuredalong the corrugation of the wall and N is the di­mensionless distance measured normal to same.In Eq. (13) qN is the heat flux rate per unit length,at the hot wall and kis the thermal conductivity.

volumell (Fig. 3) of the overall region. The solu­tion of the governing equations is obtained iterat­ively. and once convergence of the equations hasbeen achieved the following quantities are calcu­lated.

The local Nusselt number along the hot wall,Nuy,

Fig. 3-A typical control volume shown shaded Fig. 4-Streamline plot (Gr= 105 CF'= 3)

ALl & AU: NATURAL CONVECTION HEAT TRANSFER 185

Gr= 103, the low vertical velocities thus generatedenable the fluid to maintain better contact withthe corrugated wall. Thus with increasing CF thecorresponding enhancement of heat transfer sur­face leads to an increase in total heat flux at lowGr, but for the case of high Gr, the lower velocit­ies and consequent decrease in convective heattransfer at the troughs offsets the effect of in­creased surface area.

Effects of corrugation frequency on local Nu~selt number-The decreasing nature in convectionheat transfer is evident from Figs 6-8 where itmay be observed that the local Nusse\t numberwhich is identical to the dimensionless local heatflux attains minimum values at the troughs of thecorrugation. It can be noted from these figuresthat there is a significant increase in local Nusseltnumber at the peaks of the corrugation. The

Shope of the vertical wall

-<>-<>- G r = 10 5~

-A--A- Gr = 104 C F=2~ Gr = 103

Grid size =(31x 31)12

>-

10OJ z~•..t).0E;Jc

••

••••OJZ0ua-.J

Fig. 7- Local Nusselt n.umber distribution on the hot wall(CF=2)

02 0'" 06 0'8 1·0

Oimt'nsionless vt'rtical distance ( Y)

Grid size = (31 x 31 )

Fig. 5-Isotherm plot (Gr= 105 CF= 3)

14

GrId size =(31)(31)

Gr = !OS}Gr = 104 CG r = T03

0·2 0·" tl6 (J·ll 1·0Oimt'nsionless vt'rtlcal dlslonc e (Y)

••••••OJ

Z 4

0_00

12

t:lua

-.J

~ 8.0E;Jc

>- 10OJ

Z

Fig. 8-Local Nusselt number distribution on the hot wall(CF=l)

Gr = to"}Gr ': trf C.F = 3

Gr = t 03

o0'0 0·2 0'4 0'6 0·8 1·0

Dimens ion less vertic~1 dlstanc e (V)

Fig. 6- Local Nusselt number distribution on the hot wall(CF=3)

>:':JZ~ 10L."DE:JC 8

2

12

IIIIIIII

~ 6Iiiu.3 "

186 JNDIAN J. ENG. MATER. seL, AUGUST 1994

10

•

t , I~ I. ,.20-

.0 I IGr = 103

Grief size = (31 x31)

° Present lor CF = 1

,. Present lor C F = 2

l] Present tor C F = 3

30

> 0

-10-,a l

1.

.-3 o~

,..t,,,,

_I. 0 I ,,,,

00·2040'60·8'·0

Effects of Gr on total heat flux-Fig. 12 showstJ:le variation of Q as a function of Gr for differ­ent CF and portrays the comparison between theresults of the present study and Hussain7• Here itcan be pointed out that for higher values of Grthe rate of increase of Q increases with decreas­ing CF and the different curves "cross" at aroundGr= 103 indicating a trend reversal which havebeen discussed earlier.

Effects of corrugation frequency (CF) on verti­cal velocity and temperature distribution-Figs 9and 10 exhibit the effects of CF on vertical veloc­ity distribution at the horizontal mid-plane forGr= 105 and 103, respectively. Fig. 9 indicatesthat the peak value of the vertical velocity dec­reases with increase in CF. This trend can be ex­plained by examining Fig. 11 which indicates thatthe temperature gradient is lower for higher CF,causing a lower buoyant force and hence a lowervertical velocity. Because of this lower velocity,the strength of convection heat transfer decreaseswith increasing CF. But in Fig. 10 the vertical ve­locity increases with increase in CF, which leadsto an increase in overall heat transfer.

reason for this is that the peaks cause the fluid tocome in contact more intimately with the surface,resulting in large convection heat transfer andconsequently the local Nusselt number increases.These figures also show that the peak values ofNuy decrease with increasing vertical distancealong the corrugated wall. This may be explainedby the fact that the colder fluid collects at thebottom-left corner of the enclosure creating alarge temperature gradient with thc hot wall,which is the main driving force for heat transferand as it moves up and receives heat, the temper­ature gradient decreases, causing the decrease inlocal Nusselt number.

Gr = 105

Grid size = (31 x31)

-e-<>- Present lor CF = 1

~ Presenl tor C F = 2

~ Present tor CF = 3

0,1

o I L I I I d

o 0·2 0'" 0,6 0,8 1·0

.e

.6

•o a.o~XiJBBi8i!~!i:!i!fiio ..

G r = 105

Grid size =(31 x 31)° Present tor CF = ~

,. Present tor CF = 2

: D Present tor CF = 3.0·9. a.

o

1-0

0·8

CD 0·7

~ 0·6o\.. 0·5<1>

\..;:J

a 0'"\..C!J

0. 0·3E<1>

l- 0·2

x

Fig. 10-Vertical velocity distribution at the horizontal mid­plane (Gr= 103)

\"0060'402

80

604020>

-

-20_40-6 0- 80

0

x

Fig. 9-Vertical velocity distribution at the horizontal mid­'plane for different corrugation frequencies (Gr= 105)

X

Fig. 11-Temperature distribution at the horizontal mid-plane(Gr= 105)

ALl & ALl: NATURAL CONVECTION HEAT TRANSFER 187

36

Slope = 0'132

C = 2'OOh 105

16 20 24 28

lC r]0'311 It rf'019

12

I8

4

5

a 3

AcknowledgementThe authors are highly grateful to Computer

Centre of Bangladesh University of Engineeringand Technology, Bangladesh for providing facilit­ies of the numerical computation.

Fig. 13-Graphical representation of the correlation of V-cor­rugation

1 The overall heat transfer decreases with the in­crease of corrugation frequency for higherGrashof number but the trend is reversed forlower Grashof number.

2 At low Grashof number the conduction modeprevails and as corrugation frequency in­creases, total heat flux is enhanced for the en­hancement of surface area whereas for highGrashof number the fluid fails to collect heatby transport due to increasing corrugation.

3 The peak values of local Nusselt number dec­rease with the increase of vertical distancealong the corrugated surface and local Nusseltnumber increases significantly at the peaks andattains minimum values at the troughs of thecorrugation.

4 The vertical velocity of the fluid decreases withthe increase of corrugation frequency for highGrashof number because of the lower tempera-ture gradient as the boundary layer tends to se­parat~ but the trend is reversed for lowGrashof number.

5 The decreasing trend of total heat flux mayfind application in practical situations whereheat transfer reduction is desired across largetemperature diffcrences by increasing the cor­rugation of the vertical wall.

105

Qfor CF= 3

4.5732

4.5734

study)

QforCF=2

4.7529

4.7531

QforCF= I

4.8371

4.8372

104Gr

Fig. 12- Total heat flux vs Grashof number

CF : 3}---'-- C F: 2 ( Pr~s~nt---- -- CF: 1----- CF: O(R~f. 7)

1

103

2

4

5

Table 2 - Total heat flux for grids 31 x 31 and 49 x 49Grashof number = 105

Grid size

31 x 31

49 x 49

Effects of grid refinement on total heat flux-Toobserve the effect of mesh size a grid refinementstudy has been made using 49 x 49 mesh forGrashof number 105. The results of the runs that ,follows the Table 2, show that the total heat fluxfor the mesh 31 x 31 agrees quite well with that of49 x 49 mesh.

Correlation-On the basis of the above results

a correlation equation can be developed in termsof Q, Gr and CF which is found to be capable ofcorrelating the computed data within ± 1.4%. Thecorrelation is:

Q= 0.132 [Gr]0311·[CF]-00l9

where,1000 ~ Gr~ 100000

1 ~CF~3

The graphical representation of the abovecorrelating equation is shown in Fig. 13.

d J

ConclusionsThe results obtained in this study indicate the

followings:

Nomenclatureu = horizontal velocity, m/sv = vertical velocity, m/sU - dimensionless horizontal velocity

188 INDIAN J. ENG. MATER. SCI., AUGUST 1994

v = dimensionless vertical velocityx = horizontal cartesian coordinate, my = vertical cartesian coordinate, mX = dimensionless horizontal cartesian coordinateY = dimensionless vertical cartesian coordinate

J1. = absolute viscosity, kg/msv = kinematic viscosity, m2/s

p = density, kg/m31/1 = stream function, m2/sljI = dimensionless stream function

w .= vorticity, S-IQ = dimensionless vorticitya = thermal diffusivity, m2/s

fJ = coefficient of thermal expansion, 11Kg = acceleration due to gravity, mls2h = convective heat transfer coefficient, W/m2 Kk = thermal conductivity, W/m KGr= Grashof numberPr = Prandtl number

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117-123.

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6 Yuan S W, Foundation of fluid mechanics (Prentice-HallofIndia, New Delhi), 1969, 103-139.

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