W/irlneundStoffiibertragung

W/irme- und Stofffibertragung 23, 213-217 (1988)

9 Springer-Verlag 1988

Free convection flow over an uniform-heat-flux surface with temperature-dependent viscosity J.-Y. Jang and C.-N. Lin, Tainan, Taiwan, Republic of China

Abstract. The role of temperature-dependent viscosity is studied in laminar free convection flow adjacent to a vertical surface with uniform heat flux. The resulting non-similar equations are solved by using a suitable variable transformation and employing an implicit finite difference method. It is shown that the constant viscosity results evaluated at the ambient fluid temperature underestimate the Nusselt number and overestimate the drag coefficient. The heat transfer predictions for large values of the viscosity parameter may be two times the constant viscosity parameter prediction. The present analysis is in good agreement with the corresponding correlation of previous experimental investigation. Striimung bei freier Konvektion fiber eine Oberflfiche mit einheitlichem Wiirmestrom und mit temperaturabhiingiger Viskositiit Zusammenfassung. Die Rolle der Temperaturabh/ingigkeit der Viskosit/it wird in laminarer Str6mung mit freier Konvektion, angrenzend an eine senkrechte Oberflfiche mit einheitlichem W/irmestrom, untersucht. Die restlichen Gleichungen werden durch Anwendung einer geeigneten Variablentransformation und einer impliziten Finiten-Differenzen-Methoden gel6st. Es wird gezeigt, dab die Ergebnisse bei konstanter Viskosit/it, berechnet mit der Temperatur der umgebenden Flfissigkeit, die Nusseltzahl unterschreiten und den Widerstandsbeiwert fiberschreiten. Die Nusseltzahl bei W/irmefibertragung mit grogen Viskosit/itsparametern ist ungeffihr zweimal h6her als die bei konstanter Viskositfit. Die vorliegende Untersuchung stimmt gut mit vorhergehenden experimentellen Ergebnissen iiberein. Nomenclature

Cd, x

Cd, i~

c. f

Gr* g K* k L Nu x Nu L Pr q" T H~ 1) x, y

local drag coefficient average drag coefficient specific heat of fluid dimensionless stream function local flux Grashof number acceleration due to gravity viscosity parameter thermal conductivity of fluid length of surface in flow direction local Nusselt number average Nusselt number Prandtl number heat flux at the wall temperature convection velocity streamwise and transverse velocity component streamwise and transverse coordinate

Greek symbols

/~ r/ #, v v* ~ ~o ql ~b

coefficient of thermal expansion dimensionless transverse coordinate absolute and kinematic viscosity of fluid viscosity ratio parameter dimensionless streamwise coordinate density dimensionless temperature stream function wall shear stress

Subscripts

F P w oe

referred referred referred referred

to to to to

results from Fujii et al. [8] results from the present analysis condition at wall temperature condition at ambient temperature

1 Introduction

W h e n free convection heat transfer takes place under conditions when there are large temperature differences within the fluid, it is necessary (for accuracy) to consider the effects of variable fluid properties. Two approaches, c o m m o n l y used during the past several decades, to account for such effects are the reference temperature m e t h o d and the p r o p e r t y ratio method. M a n y investigations have been carried out on laminar free convection flow adjacent to an isothermal vertical surface in gases and liquids whose properties are temperature dependent [ 1 - 7 ] . The results showed that variable properties have a significant effect on the thermal and momentum transport predictions. Some studies [ 2 - 4 ] indicate that variable property influence can be adequately accounted for by using the reference temperature m e t h o d with the reference temperature simply set equal to the film temperature, an arithmetic average of the wall temperature (T~) and ambient fluid temperature (To). On the other hand, considerably less work has been done concerning the variable property effects on the vertical laminar free convection flow over an uniform-heat-flux surface, where the wall temperature is not a constant and increases along the streamwise direction. Two methods of correlating

W~irme- und Stoffiibertragung 23 (1988)

214 the effects of variable properties on heat transfer for free convection flow from a vertical surface with uniform heat flux boundary condition to liquids (water, spindle oil and Mobiltherm 600 oil) were examined experimentally by Fujii et al. [8]. The first method of correlating data consisted of using constant property correlation for the Nusselt number and evaluating all properties at the reference temperature, T~= T~ - 0.25 (T~ -- T~). The second method is the property ratio method which applies only to liquids for which viscosity variation is dominant. This amount to a Nusselt number correction factor which consists of the ratio of kinematic viscosities (v) at the surface and ambient temperature raised to some power. The resulting correction has the form, NUx(Vw/V~) ~ oc (GG Pr)~4, with all other properties evaluated at the ambient fluid temperature. Carey and Mollendorf [9] developed a regular perturbation analysis for the three laminar free convection flows in liquids with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source and the flow adjacent to a vertical uniform heat flux surface. While these flows have well-known power-law similarity solutions when the liquid viscosity is taken to be constant, they are non-similar when the viscosity is considered to be a function of temperature. In [9], the linear variation of viscosity with temperature is assumed. Their solutions are only applicable for small values of the perturbation parameter. Therefore, they can't satisfy most of the engineering applications, except for very small wall and ambient temperature differences. The present study examines in detail the effects of temperature-dependent viscosity on the laminar free convection in liquids from a vertical uniform heat flux surface when other property variations can be neglected. The variation of viscosity with temperature is represented by an exponential function. The non-similar equations are solved by using a suitable variable transformation and employing an implicit finite difference method similar to that described in Cebeci and Bradshaw [11]. Average heat transfer rate,drag coefficient, local velocity and temperature profiles are given for different values of Prandtl number and visocity parameter. These results are also compared with available experimental results [8]. Solutions would be useful in problems of heat transfer and in problems of stability of liquid boundary layers.

8T 0T k ~2T u ~ - + ~ a~- - ~ c . ay2 9

(3)

The boundary conditions appropriate to the problem are y=0, u=v=0, y-+ oo, u-+0,

-kOT/Sy=q'=constant T-+Too

(4)

where x represents the distance along the plate from its leading edge and y represents the distance normal to the surface; and 9 is the gravitational acceleration and is taken to be in the negative x direction. The various symbols are defined in the Nomenclature. It is necessary, bvefore proceeding, to choose a particular viscosity-temperature relation. For liquids, the absolute viscosity # is appropriately assumed to vary exponentially with temperature according to # = B exp ( - A T), where A and B are positive constants adopted from the least square fitting. Equations (1)-(4) can be transformed from the (x, y) coordinates to the dimensionless coordinates (4 (x), t/(x, y)) by introducing a pseudo-similarity variable t/and an x-dependent non-similar parameter ~ as

along with the reduced stream function f(~, r/), the dimensionless temperature ~(~, q), respectively, by

f (4, ~) = 4J(x, y)/[5 v~ (Gr* / 5)1/51,

T-- T~ ( Gr* ~ 1/5 P(r rl) = q" x / ~ \ ~ - J '

(6)

where Gr* = g fl q" x4/k v2 is the flux Grashof number in which the kinematic viscosity is evaluated at Too, and the stream function r y) satisfies the continuity equation (I) with u -- ~ / S y , v = -5~/5x. Upon transformation, one can arrive at the following system of equations: E x p ( - - K * ~l/~b) 9f " ' + r + 4 i f " - - 3 f '2 - K * 4 u4

9exp(--K*~l/4r

~f",

r (8r , ~f ) +4fO'-f'O=4{ ~ f -~O' Pr~,

)

(7) (8)

with the boundary conditions f(~, 0) = f ' ( ~ , 0) = 0,

r

0) = - 1 ,

2 Mathematical analysis

f'(~, oo)--, 0

The present formulation assumes steady, two dimensional vertical free convection flow and incorporates the boundary layer and Boussinesq approximations. The absolute viscosity, #, is taken to be variable. The equations expressing conservation of mass, momentum and energy are as follows:

where the primes indicate partial derivatives with respect to t/, Pr~ =#~cp/k is the ambient temperature Prandtl number and K* is the viscosity parameter, which is defined as

5u

8v

~x F ~ = 0 , eu

0u

u~+V-ay-~

ay\

By/

r

oo)-~ 0

(9)

(1)

For a given fluid, the values of K* can usually be increased by increasing the heat flux from the wall. It is noted that the

(2)

higher the Prandtl fluid is, the larger the values of A -

J.-Y. Jang and C.-N. Lin: Free convection flow over a uniform-heat-flux surface are. Consequently, for a given heat flux from the wall, K* is larger for the high Prandtl number fluid than for the low one. It is also noted that K * = ~ = 0 corresponds to the traditional constant viscosity case evaluated at some reference temperature, in which the similarity solutions exist. The physical quantities of interest are the local Nusselt number, Nu:~, the average Nusselt number N u based on the wall and ambient temperature difference at x = L / 2 [10], the local drag coefficient Cd. ~ based on a convection velocity, U~=5v~ ~ fm~x/X ' [9], and the average drag coefficient Ca, L, they are defined, respectively, by Nu~-

q" Tw(x) _ T

x k '

q" L NUL = k(T~ -- T~)L/2 ' (11)

L

G~'

~w(~)

GL-

0 U~2 '

,

o oLUZ

the boundary layer for a given 4 and % ( x ) = # ~yy 8u ~=o is the local wall shear stress. In term of the dimensionless variables, Eqs. (11) can be reduced to ~(4, 0 ) '

NuL = 2

[~(~, 0)k/2'

exp [ - - K * ~1/4 ~b(~, 0)] f " ( 4 , 0) G ~:

5

'

,

~

9~ ( f max)

,

(12)

L

1 ! 43/4 e x p [ - K * ~1/4 ~b(~, 0)] 'f"(4, 0) d~ ca, L - 4

integration scheme. After obtaining a converged solution along ~ = 0, this solution is then employed in a Keller box scheme with second-order accuracy to march step-by-step along the boundary layer. For a given 4, the iterative procedure is stopped to give the final velocity and temperature distributions when the difference in computing the velocity and the temperature in the next procedure becomes less than 10 -6, i.e., I f i - f i - l [ , [f,lf, i 11 and [ o i - - ~ i 1 [ ~ 1 0 - - 6 , where the superscript i denotes the number of iteration. In the calculations, the value of t/~ = 9 was found to be sufficiently accurate for I f~[ and [~b~o]<10 -3. A uniform step size of At/= 0.06 in the t/-direction was used in the calculations. On the other hand, the variable step sizes were used in the ~-direction. The step size was increased gradually with increasing 4 and ranged from A~ = 0.05 for 0 < 4 < 1 to ~ = 1 for 4>7.

'

where fm.x is the m a x i m u m of axial velocity f ' ( ~ , ~l) across

Nu: - - -

215

r 11/4 (f.~.0~

3 Numerical analysis The system of Eqs. (7)-(9) was solved by an efficient and accurate implicit finite difference method similar to that described in Cebeci and Bradshaw [/1]. To begin with, the partial differential equations (7)-(8) are first converted into the system of first order equations. Then these first order equations are expressed in finite difference forms by approximating the functions and their first derivatives in terms of center difference. Denoting the mesh points in the (4, t/)plane by ~i and t/j, where i = 0, 1. . . . . M and j = 0, 1. . . . . N, central difference approximations are made, such that those equations involving ~ explicitly are centered at (~_ 1/2, q j- 1/2) and the remainder at (4i, t/j_ 1/2), where r/j_ 1/2 = ~1 (qj + tlj_ 1), ect. This results in a set for non-linear difference equations for the unknowns at ~i in terms of their values at ~ 1. The resulting non-linear finite difference equations are then solved by Newton's iterative method, taking the initial iteration to be given by the converged solution at ~ = ~i-1. The boundary layer equations are thus solved step-by-step. To initiate the process, Eqs. (7)-(9) with 4 = 0 are first solved by using a sixth order variable step-size Runge-Kutta

4 Results and discussion The numerical calculations are performed on moderate and high Prandtl fluids, whose viscosities are strongly temperature dependent and can be approximately represented by an exponential function. Figures 1 - 3 show simultaneously the variations of dimensionless velocity and temperature profiles across the boundary layer at various streamwise locations (Gr*) for Pr o = 100, 1000 and/0,000, respectively. The volocity profiles are referred to the left and lower axes, while the temperature profiles are referred to the right upper axes. Dashed lines represent the traditional constant viscosity case (K* = 0) evaluated at the ambient fluid temperature, in which the similarity solution exist and the solution is independent of Gr*. Because the wall temperature increases along the streamwise direction, the liquid viscosity near the wall decreases as Grashof number increases. Therefore a large velocity (i.e. large velocity gradient) near the wall is expected with increasing Gr*. This agrees with the theoretical results shown in Figs. 1-3. In addition, the increased velocity along the wall creates higher heat transfer rate which, in turn, decreases the wall temperature as seen in the figures. It is also shown that the effect of temperature-dependent viscosity on the velocity profiles is more pronounced than the effect on temperature profiles. Fujii et al. [8] measurements of temperature profile for spindle oils are also shown in Fig. 1. The measured data have been transformed into our dimensionless coordinates. It is seen that the corresponding measurements of the temperature field are in good agreement with the theoretical predictions. Figure 4 a - c show simultaneously the average Nusselt number N u L and average drag coefficient Cd, L VS. Gr* for various values of viscosity parameters K*. Dashed lines in each figure indicate the constant viscosity results. With increasing K* or Gr*, the deviations of the variable viscosity results against the constant viscosity results are seen to be larger. One can see that the constant viscosity results underestimate the Nusselt number for liquid heating. The N u L at

216

W a r m e - u n d Stofft~bertragung 23 (1988)

i"1 1.0 0.0354

0.8 I

0.6

I

I

0.4

I

I

103

0.2

I

I

r

102

0.5

I

-.~...}

oFujii et. [8] [,,./ I 0.030 /Gr*=l.07* 101~ \ // !/,,,~~~.0r*=1.54 • 104 ~ / 1 / I I o -0.4 0.0251 0.3 l o.o2oI :,_ 0.015-

Proo =I00

101

i 102.

0.2 !

0.010 -

K*:1.4

"

IJ

~

100

-

0.1

0,005-

101.=

O

i

i

i

i

1

i

7_

i

i

f

3

i

4

TI --.I-..

F

0.6

0.4,

l

5

0.012 / ~

/

i

I

100

I

I

102

I

I

104

I

1

106

Gr~=l.O7x101~

l

sX Grx %7.49* 103

%//

I

108

10-2 10~o

I

=

10~

/

"""-,.,. 1~"

////

I 0.008

1

0.3

I

Gr*=9.33•

0.010 ~-,,,,,~//

a,

Gr~

02

I

104

i

Fig. l i Variations of velocity and temperature profiles across the boundary layer at various Gr*. Dashed lines denoted the constant viscosity results. Pro~ = 100

0.014

I!

Proo =1000

, =102

/

:,- 0.006

-~

0"004[

....

Proo= 1000

/-'///

0.002;- r=4 o

1

2

,q

- § -0.1

~: - ~ - - ~

3

_.-_

.10~

o

6

Fig. 2. Variations of velocity and temperature profiles across the boundary layer at various Gr*. Dashed lines denoted the constant viscosity results. Proo = 1000 -,-----------~1

0.1 0 f I 0.20 ~ er* =1.07,10 io l[... _ - ~.----....~ / 6rx%9.33• s ~ //;' ~,~ O.OOZO- ~--_,,,,.,,_~~'----,,~/;'r*= 7.49,103 X//"~/-u''~

0.0025

0.4

I

0.3

I

0.2

I

I

0"0015-I

I

10~

-7 40-1

b

100

I

I

102

I

l

104 Br*

I

I

10~ =

I

J

10a

I

........ / .--///---_

a- 0.0010_

,'//~

K*=2.6

0.0005-_ o

jj

W

. . . .

. . . . 4 5

"101t

~ _ S

,- - , - - 7 = ~ 1 2 3

'r]

i

o no u.uo

I 0"04 . . . . . 4 5

IJ

1o 6

Fig. 3. Variations of velocity and temperature profiles across the boundary layer at various Gr*. Dashed lines denoted the constant viscosity results. Pro~ = 10,000 Fig. 4. a Variations of average Nusselt number N u L and drag coefficient Cd, L VS. Gr* for various K*. Pr~o = 100; b Variations of average Nusselt number NUL and drag coefficient Ca, L VS. Gr* for various K*. Proo = 1000; e variations of average Nusselt number N u L and drag coefficient Ca, s vs. Gr* for various K*. Proo = 10,000

!i 10~

GC

-

i!

217

J.-Y. Jang and C.-N. Lin: Free convection flow over a uniform-heat-flux surface

/

1.10t 1.081 1.061 1.04 i 1.02-

5 Conclusions

AviscosityparameterK*=(

0.98 0.960.94

i

0

I

1

I

-1

I

-2

-3

tn (v*)

Fig. 5. Hp/HF vs. In (v*) for selected values of Pr~

Gr* = 101~ for large values of K* can be a b o u t two times that for constant viscosity predictions. It is also shown that heating ( K * > 0) always decreases the drag coefficient (wall shear) even though the velocity gradient is higher. This is a result of the competition between viscosity and velocity gradient (at the wall) in the determination of wall shear. In this instance, the decrease in viscosity is larger than the increase in velocity gradient, which results in lower drag coefficient. It should be noted that the variable viscosity effect on the drag coefficient is more p r o n o u n c e d than on the Nusselt number. The heat transfer correlation of Fujii et al. [8] for laminar free convection from a vertical uniform heat flux surface in moderate and high P r a n d t l number liquids with temperature dependent viscosity is ( V w ~ 0"17

(Nux) ~

= 0.62(Gr*

Vr)~ 5 .

2 Vo~/ The above correlation is rewritten for convencience as

HE --

(Nux)~ , 1/5 ( Gr x Pr)~

0.62(v*) 0.a7

where v* = v~/v~, is the ratio of viscosities at the wall and ambient temperatures. HF m a y be c o m p a r e d to the transformed expression of the present analysis

(Nux)~176 - (0'2)1/5 (Pr j - 115. Hp - (Cw* p~1/5 r O) \-- x

has

been successfully employed in the present analysis to include temperature dependent of viscosity in the free convection b o u n d a r y layer adjacent to a vertical surface with uniform heat flux. The present results predict that variable viscosity has a significant effect on the temperature and velocity profiles as well as Nusselt number and D r a g coefficient. It is shown that variable viscosity effect causes moderate increases in heat transfer rate anbd dramatic decreases in wall shear stresses. Excellent agreement are found between the present numerical analysis and the heat transfer data and correlation of Fujii et al. [8].

//~Proo=lO000

/

#1 d# ) { \ 5v~ ~ Jq'3

References

1. Schuh, H.: The solution of laminar boundary layer equation for the flare plat for velocity and temperature field for variable physical properties and for diffusion field at high concentration. NACA TM 1275 2. Sparrow, E. M.; Greg, J. L.: The variable fluid property problem in free convection. ASME 80 (1958) 869-876 3, Minkowycz, W. J.; Sparrow, E. M.: Free convection heat transfer to stream under variable property condition. Int. J. Heat Mass Transfer 9 (1967) 1145-1147 4. Carey, V. P.; Mollendorf, J. C.: Natural convection in liquids with temperature-dependent viscosity. Proceedings of the sixth Int. Heat Transfer Conference, Toronto, V.2. Washington: Hemisphere 1978, pp. 211-217 5. Clausing, A. M.; Kempa, S. N.: The influence of property variation on natural convection from vertical surface. ASME J. Heat Transfer 103 (1981) 609 612 6. Clausing, A. M.: Natural convection correlation for vertical surface including influence of variable property. ASME J. Heat Transfer 105 (1983) 138-143 7. Cairnie, L. R.; Harrison, A. J.: Natural convection adjacent to a vertical isothermal hot plane with a high surface-ambient temperature difference. Int. J. Heat Mass Transfer 25 (1982) 925934 8. Fujii, T.; Takeuchi, M.; Fujii, M.; Suzaki, K.; Uehara, H.: Experiments on natural convection heat transfer from the outer surface of a vertical cylinder to liquids. Int. J. Heat Mass Transfer 13 (1970) 753-787 9. Carey, V. P.; Mollendorf, J. C.: Natural convection effect in several natural convection flows. Int. J. Heat Mass Transfer 23 (1980) 95-109 10. Sparrow, E. M.; Gregg, J. L.: Laminar free convection from a vertical plate with uniform surface heat flux. ASME J. Heat Transfer 78 (1956) 435-440 11. Cebeci, T.; Bradshaw, P.: Momentum Transfer in Boundary Layers. Washington: Hemisphere 1977

-/ao

Finally, Fig. 5 shows the comparison of H e and Hv for various values of v* and P r o. F o r Pr o = 1000, the present analysis are seen to agree with the correlation of Fujii et al. within 2% up to v* = 1/20. F o r P r ~ = 100 and 10,000, the theoretical results lie within 10% of the correlation of Fujii et al. up to v* = 1/10.

Jiin-Yuh Jang, Associate Professor Chien-Nan Lin, Graduate Student Department of Mechanical Engineering National Cheng-Kung University Tainan, Taiwan, 70101, R.O.C. Received December 14, 1987