Int. Comm. Heat Mass Tmnsfc Vol. 28, No. 2. pp. 289-291, 2001 Copyright 0 2001 Elsevier Science Ltd Frinted in the USA. All rights reserved

07351933/016-see front matter

PII: So7351933(01)002354

ON FORCED CONVECTIVE HEAT TRANSFER FOR A STORES FLOW IN A WAVY CHANNEL

M. Vasudeviah and K. Balamurugan Department of Mathematics

Anna University Chennai 600 025, India.

(Communicated by A.R. Balakrishnan)

ABSTRACT Study of heat transfer in a two-dimensional wavy channel due to a pressure driven Stokes flow, normal to the wall corrugations, is made. Expression for the mean Nusselt number describing the average rate of heat transfer from the warmer surface, obtained analytically, predicts a decreased heat transfer rate due to corrugations. This observation is found to be significantly opposite to the prevailing results of moderate or large Reynolds number flows. 43 2001 Elsevier Science Ltd

introduction

Flow past wavy boundaries are encountered in many situations such as the rippling of melting surfaces

and in physiological applications. Laminar flow of a viscous incompressible fluid in a wavy/corrugated

channel bounded by the sinusoidal surfaces z = b sin(27ty/L) and z = a+b sin(2ay/L+p), which are

otherwise parallel except for a phase shift p , Fig.1, was studied by Wang [ 1,2] when the pressure driven

flow is (i) parallel to the corrugations and (ii) transverse to the corrugations. Wang [2] has shown

analytically that for a given mean pressure drop, the flow transverse to the corrugations always decreases

and that the phase shift p could be utilized to control the flow through corrugated surfaces. Nishimura,

Yoshino and Kawamura [3], adopting a finite element analysis, studied two-dimensional pulsatile flow in

a wavy channel with periodically converging - diverging cross-sections.

The related study of heat transfer has been made by several investigators from energy

considerations, such as to simulate forced convective cooling of electronic packages etc. Straight ducts of

constant cross-section with sinusoidally curved walls may be found in plate heat exchangers as well as

in packing structures used in mass transport operations or in catalytic reactors. Enhancement of heat

transfer by the wavy walls has been uniformly predicted by almost all workers in the field. Experimental

289

290 M. Vasudeviah and K. Balamurugan Vol. 28, No. 2

a Flow

FIG.1 Flow geometry : vertical section of the channel

evidence in this context can be traced to the work of Goldstein and Sparrow [4]. Sparrow and Hossfeld [5]

showed that the rounding of the wavy peaks brought about a decrease in the Nusseh number. Numerical

predictions of laminar flow and heat transfer in wavy channels of uniform cross-sectional area, for equal

wall temperatures, was made by Xin and Tao [6] for moderately large Reynolds number Re in the range

100 c Re < 1000. Using Greens function technique, Rutledge and Sleicher [7] numerically studied the

possibility of enhancement of heat transfer rates by intentional roughening of surfaces. Recently Snyder.

Li and Wartz [8] experimentally investigated forced convective heat transfer rates and pressure drops in

the thermally developed region of a two-dimensional serpentine channel in the regime 250 < Re c 10000.

and observed that the heated surfaces of the channel outperformed the base-line parallel plate channel.

Fully developed, laminar, natural convection flow in a vertical channel with isothermal corrugated walls

was studied by Faghri and Asako [9]. Using an algebraic transformation to map the wavy channel into

a rectangular channel with plane surfaces, they presented averaged heat transfer results for Grashoff s

number Gr upto O(10) for different aspect ratios. The significant feature of their finding is that the

average Nusselt number and consequently mass flow rates were found to be always less than those for a

channel with plane walls. However the work of Bhavnani and Bergles [lo] show enhanced heat transfer

results.

Observing that all those investigations are concerned with large or atleast moderately large

Reynolds number flows, it appears interesting to study how the results would shape in low Reynolds

number flows. With this motivation. we wish to study the effect of corrugations and phase difference on

heat transfer in the laminar regime of flow transverse to the corrugations of sinusoidal wavy channels.

Vol. 28, No. 2 STOKES FLOW IN A WAVY CHANNEL 291

Such a study. coupled with the ease of controlling flow distribution in such wavy channels as evidenced

in the literature, can be effectively made use of in food processing industry where frequent cleaning and

sterilisation is necessary. Wangs [2] velocity analysis forms a basis for this study. The analogous heat

transfer problem in respect of flow parallel to the corrugations, Wang [l] has been studied recently by

Vasudeviah and Patturaj [ 111.

Mathematical Analvsis

Velocitv Field

The corrugated walls constitute stream surfaces for the flow, hence volumetric flow discharge per

unit time, per unit thickness of the channel is a constant Q for a given pressure drop. The appropriate

velocity field is (0, av//&, -&p/Jy), where I# is the stream function. We shall assume low Reynolds

flow such that the Stokes equation holds. Normalising the stream function by Q and all lengths by a

the mean gap width between the walls, the Stokes flow is obtainable as a solution of the boundary value

problem:

v4w=o (1)

w = 0, awlan = 0 on 2 = E sin hy (2)

y = 1, *ian = 0 on z = 1 + E sin (hy + p) (3)

a2 a2 where V2 = - + - and 1= 2xafL.

ay2 az2 (4)

Here II is the normal direction to the boundary and L is the wave length of the sinusoidal surfaces. A

regular perturbation solution for the flow field is sought in the form

I+l = vo+&W, +EZ~2+..., (5)

where V4w, = 0, i = 0,1,2,... (6)

The respective boundary conditions for vo, w,. v2 etc. are obtainable from the Eqns.(2-3) subjecting them

to Taylor expansions around z=O and z=l respectively. The corresponding solutions, Wang [21 are as

follows :

WIJ = 322 - 223 (7)

y, = eiaY (C, eL + C, emb + C, z eh + Cd Z e-9 (8)

IJI~ .= (312 + B,z + B2z2 - 2B3z3) + eIzky w2,(z) + e-i2)iy Wz,(z) (9)

292 M. Vasudeviah and K. Balamurugan Vol. 28, No. 2

where the real parts of w, and v2 only have any physical significance. The values of the constants

appearing in the above solutions are listed in Wang [2]. The key result of Wangs work is the expression

for the total flow Q interns of mean pressure gradient, given as follows :

Q = ;= 1 - E*B, + 0 (a4) .

12 P

(10)

Temperature Field

We now study the corresponding heat transfer problem. The wavy section of the upper wall binded

by heaters on its outer surface is maintained at an elevated uniform temperature T,. Attaching a cooling

jacket to the outer surface of the lower wall, its surface temperature may be regulated to remain at the

entrance flow temperature Tt of the fluid. Dropping convective terms in the energy equation in

comparison to viscous dissipation terms, in the context of Stokes approximation adopted in the velocity

analysis, the governing equation for the temperature distribution is given as

V2T=-Br[ 4 ( 2 p + ( $- $ 1 ] , (11)

T - Tp where T = - and

Br = MQW* is the Brinkman number. (12)

T - Tl k&-T1 )

Ensuing analysis is relevant for flow environment characterised by Brinkmans number of order

unity. The appropriate boundary conditions are :

2 = & sin hy on T=O

I

(13)

z = 1 + E sin(hy + /3) on T = 1 (14)

As in the velocity analysis, we proceed to obtain T in the form

T = T,,+ET~+E*T~+... (15)

The Taylor expansion of the boundary conditions (13,14) give the corresponding boundary conditions for

To, T,. T2 as follows :

2= 0 z=l

To = 0 To= 1

T, + sin(hy) aTdaz = 0 T, + sin(Ly+P) BTo/az = 0

T, + sin(hy) dT,/az + $5 sir&y) a2T&iz2 = 0 T2 + sin(Xy+P) dT,/az + w sin2(1y+8) $Tt-,Gz2 = 0

(16)

Vol. 28, No. 2 STOKES FLOW IN A WAVY CHANNEL 293

Substituting for v from Wangs analysis, T, solvable from

V2To = -36 Br (4~ - 42 + 1)

is obtained as

T, = z - 6 Br (2z4 - 4z3 + 3z2 - z) .

T, is obtainable as a real part of the solution of

V?, = -24 Br eY (l-22) [(C,h2 + C,I. + C, hz) et + (C2h2 - C,X + C4h2z) e-1 ,

where C,, C,. C,, C4 are listed in Wang [2].

The appropriate boundary conditions are :

T,=(l+6Br)ieyon z=O and T, = (1 - 6 Br) ieiP eY on z = 1

The corresponding solution is

T, = ey f(z),

where f(z) = M, eAz + M, e-h + Br [(8hC3z3 + 6h(2C,-C,)z* - 6(2C,h + 2C,+C3)z) exz

+ (-8hC4z3 + 6h(C4-2C2)z2 + 6(2C& - 2C2 - C~)Z) e-*z] ,

(e - e-3 M, = - (1 + 6Br)ie- + (1 - 6Br)ieib - Br g(k) ,

(e - e-3 M, = (1 + 6Br)ie* - (1 - 6Br)ie iP + Br g(A)

g(k) = (2C3X - 6C3 - 12C,) e - (2C& + 6C4 + 12C2)e- .

(17)

(18)

(19)

(20)

(21)

(22)

We see that this first order solution T, is periodic in y and therefore cannot contribute to the mean rate

of heat transfer. The second order solution T,, however may contain an aperiodic term which therefore

can contribute to the mean rate of heat transfer. The most general solution for T, is of the form

T2(y,z) = TZO(z) + ei2y T2,(z) + eSi2*y T22(~) (23)

Our interest is however on T20 (z) only, which is solvable from

d?, I dz2 = -24 Br (l-22) (B2 - 6B,z) (24)

under the boundary conditions

Tzo(0) = 9 Br - i/2 f (0) and Tzo(l) = 9 Br - i/2 e-B f (1) . (25)

The appropriate solution is

T2,$z) = d,z + d, - 4Br [3B,z2 - (2B, + 6B3)z3 + 6B3z4], (26)

where d, = 4Br B2 + i/2 f(0) - i/2 e-P f (1) and $ = 9Br - i/2 f (0). (27)

294 M. Vasudeviah and K. Balamurugan Vol. 28, No. 2

Results and Discussion

The point of physical interest here is to understand whether heat transfer enhancement is possible

due to corrugated walls; if so for what values of the phase difference p its optimal value is real&d. This

is important from design considerations directed towards effective transmission of heat from one medium

to another and consequential energy savings. Accordingly we compute the mean Nusselt number

describing the average rate of heat transfer from the warmer surface, per unit width along the corrugations,

for one full wave length of the wall, using the formula

Nu = (l/A) I (- JTIW, = l+s sin(ky+p) a p (28) where A = (2Ra/h) (1 + ~*h*/4) is the corresponding surface area of the wall.

Making use of the available solution T upto O(&*), and subjecting @T/&I)~ = r+r sin(Iy+P) to a Taylor

expansions about z = 1, we finally obtain

Nu = (6 Br-1) + a* (N, + N2) + O(E~), (29)

where N, = X2/4 - h coth h + h cosech 1 co@ (30)

and N, = Br(-36- 1.5X2). (31)

The flow rate in the channel, Wang 121, quantified by the equation (10) is G= 1 - E* B, + O(E).

The expression for the real part of B,, computed and plotted by Wang for increasing 1 is found to be

always positive. Hence the flow rate decreases due to corrugations. Effectively this means that corrugations

offer flow resistance, consequently causing viscous heat generation. This generation of heat is represented

by the expression E* N2, which is readily seen to be negative for all 1, and hence accounts for heat

transfer from the plate to the fluid. It may also be noted that N2 is not influenced by the phase difference

factor /3. Table 1 gives the numerical values of N2/Br for various values of 1 ranging from 0 to 10.

TABLE 1 Effect of corrugations on viscous heating of the fluid : N-/BR

J.- 0 1 2 3 4 5 6 7 8 9 10

N$Br -36 -37.5 -42 -49.5 -60 -73.5 -90 -109.5 -132 -157.5 -186

The expression a*N, appearing in equation (31) may hence be understood as measuring the effect of

corrugations on heat transfer due to the impressed temperature difference T, - TI. Table 2 gives the

numerical values of N, for various values of A ranging from 0 to 10 for the phase difference @O. 90

and 180.

Vol. 28, No. 2 STORES FLOW IN A WAVY CHANNEL

TABLE 2

Effect of corruaations on heat transfer due to the impressed temperature difference T,,-Tr : N, L

b 0 1 2 3 4 5 6 7 8 9

P

O0 0.0000 -0.2121 -0.5232 -0.4654 0.1439 1.3169 3.0297 5.2628 8.0054 11.2522

90 -1.0000 -1.0630 -1.0746 -0.7649 -0.0027 1.2495 2.9999 5.2500 8.0000 11.2500

180 -2.0000 -1.9140 -1.6261 -1.0644 -0.1493 1.1822 2.9702 5.2372 7.9946 11.2478

295

For B=O, N, starts at zero and goes upto -0.5582 corresponding to k2.5, thus causing an increased

heating of the fluid. This increase reduces in size as h goes upto 3.5. Between k3.5 and 4, Nl changes

in sign thus accounting for heat reversal. Such a kind of heat transfer persists and increases with the

frequency of the corrugations. Similar phenomena is observed upto B=90 . Beyond p=90 , a little

difference is seen; heat transfer from the surface to the fluid always decreases steadily upto h=4, beyond

which N, becomes positive and keeps increasing with h, reflecting the earlier character.

It may be seen that for a given I, 1 N, 1 is maximum at B=I? for a I 4 and at S=O for b4;

also the effect of p appears more pronounced for hc4. This intricate interaction of the frequency of

corrugations and the phase difference is interesting; for some wave length inducing thermally driven

convection from the surface to the fluid and for some other wave lengths absorbing heat from the fluid.

From Tables 1 and 2 it is seen that the contribution of N, is substantial compared to N, and hence

&2 (Nl+N2) remains negative for all A. The results of the entire analysis thus leads to the following key

observation : Corrugations not only retard the flow in the channel but also diminish the rate of heat

transfer from the warmer wall. Alternatively one can say that the heat transfer performance of a wavy wall

is inferior to that of a plane wall. Fig.2 shows Nu verses h for selected values of Br = 2.4.6,8 and E =

0.1.

This prediction of heat transfer reduction in respect of low Reynolds number flows, is significantly

different from the successful attempts of many workers in simulating heat transfer enhancement in the case

of moderate or large Reynolds number flows, by incorporating small corrugations into the heat exchange

surface. However our findings qualitatively agree with the reports of Faghri and Asako [9], though their

results are applicable to free convection flows only. It may also be observed mat the effect of phase

difference on mean Nusselt number is very very marginal; maximum heat transfer occurring when the

walls are in phase and minimum when they are out of phase, for a given 1. However, in the case of flow

parallel to the corrugations, the effect of /3 appears more pronounced as can be realised from the results

of Vasudeviah and Patturaj [ 1 I]. It is thus seen that orientation of pressure drive, coupled with factors like

wave length, phase difference, mean distance between the walls etc. weave a complex mechanism to

simulate heat transfer in the channel and each configuration offers a study of its own.

296 M. Vasudeviah and K. Balamurugan Vol. 28, No. 2

FIG.2 Mean Nusselt number : Nu vs 1,

Nomenclature

a

b

Br

Gf

k

L

n

Nu

P

Q

6

Re

TR IT

T

T

X,Y,Z

x.y,z

mean gap-width of the channel, m

amplitude of the corrugation, m

Brinkman number

Grashoff number

thermal conductivity, wl(m.K)

wave-length of the corrugations, m

outward normal direction to the wall. m

mean Nusselt number

dimensional pressure, N/m

volumetric flow discharge per unit time, per unit thickness of the channel, m*/s

non-dimensional volumetric flow

flow Reynolds number

temperatures on the lower and upper walls, respectively, K

temperature of fluid, K

non-dimensional temperatures, respectively

dimensional spatial coordinates, m

non-dimensional spatial coordinates

Vol. 28, No. 2 STOKES FLOW IN A WAVY CHANNEL 291

P phase shift of the corrugated surfaces 1 frequency of the corrugation

p dynamic viscosity, N.s/m2

E dimensionless amplitude of the corrugations (b/a)

d stream function, m2/s

V non-dimensional stream functions

References

1. C.-Y. Wang, ASCE, 1. Engg Mech, 102, 1088 (1976).

2. C.-Y. Wang, ASME, J. Appl. Mech, 46, 462 (1979).

3. T. Nishimura, T. Yoshino and Y. Kawamura, J. Chem. Engg. Japan, 20(S), 479 (1987).

4. L. Goldstein and J.M.E. Sparrow, J. Heat Transfer, 99, 187 (1977).

5. E.M. Sparrow and L.M. Hossfeld, Zt~r. J. Hear Mass Transfer, 27(10), 1715 (1984).

6. R.C.Xin and W.Q. Tao, Nr. Heat Transfer, 14,465. (1988).

7. J. Rutledge and CA. Sleicher, Comm. Nr. Methods in Engg, 10, 489 (1994).

8. B. Snyder, K.T. Li and R.A. Wartz, In?. J. Heat Mass Transfer, 36(12), 2965 (1993).

9. M. Faghri and Y. Asako, Int. J. Heat Mass Transfer, 29, 1931 (1986).

10. S.H. Bhavnani and AI. Bergles, Wiierme-und stofiibertragung, 26, 341 (1991).

11. M. Vasudeviah and R. Patturaj, Proc. First Annual Conf. ISIAM, Roorkee. India. 55. (1993).

Received October 6, 2000