Free convection effect on mhd coupled heat and mass transfer of a moving permeable vertical surface

Pergamon

Int. Comm. Heat Mass Transfer, Vol. 26, No. 1, pp. 95-104, 1999 Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/99/S-see front matter

PII S0735-1933(98)00125-0

FREE CONVECTION EFFECT ON MHD COUPLED HEAT AND MASS TRANSFER OF A MOVING PERMEABLE VERTICAL SURFACE

K.A. Yih Air Force Aeronautical and Technical School Department of General Course Kangshan, Kaohsiung, Taiwan 90395-2, R.O.C.

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT The purpose of this paper is to consider numerically the free convection effect on magnetohydrodynamic heat and mass transfer of a continuously moving permeable vertical surface. The surface is maintained at linear temperature and concentration variations. The similar equations were solved by using a suitable variable transformation and employing an implicit finite difference method. Numerical results were graphically given for the Nusselt number and the Sherwood number for various parameters. Generally, it is found that the Nusselt number and the Sherwood number increase for the suction case, increasing the buoyancy ratio N and the buoyancy parameter GrT/Re 2, and for the decrease of magnetic parameter M. Furthermore, the Nusselt (Sherwood) number increases for the decrease (increase) of Schmidt number Sc. © 1999ElsevierScienceLtd

Introduction The research of incompressible viscous boundary layer flow over a moving continuous surface has many important applications such as polymer technology and metallurgy. In the aspect of moving surface with a velocity proportional to x (x being the distance along the wall), Grubka and Bobba [1] investigated heat transfer characteristics of a continuous, stretching surface with variable wall temperature. Chen and Char [2] analyzed the heat transfer of a continuous, stretching surface with suction or blowing. The results in [1,2] are expressed in terms of Kummer's functions [3]. Boundary layer flow and heat transfer for the stretching plate with suction was studied by Ahmad and Mubeen [4]. For the effect of power-law stretched surface on heat transfer was reported by Afzal [5] using 95

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asymptotic method. All [6,7] numerically presented the heat transfer characteristics of a power-law continuous stretched surface without and with suction/injection. However, the above researches [1-7] are restricted to hydrodynamic flow and heat transfer problem. Recently, the problems of hydromagnetic flow and heat transfer have become more important industrially. Chakrabarti and Gupta [8] studied the hydromagnetic flow and heat transfer over a linearly stretching isothermal sheet with suction or blowing. Takhar et al. [9] studied a MHD asymmetric flow past a semi-infinite moving plate and numerically obtained the solutions. An analysis of heat transfer characteristics in an electrically conducting fluid over a linearly stretching surface with variable wall temperature was investigated by Vajravelu and Rollins [10]. Char [11] presented the solutions for the heat transfer in a hydromagnetic flow over a stretching sheet subjected to the thermal boundary condition with either a prescribed surface temperature (PST) or a prescribed heat flux (PHF). Chiam [12] obtained the solutions of magnetohydrodynamic heat transfer over a non-isothermal stretching sheet. The prescribed wall temperature is a quadratic function of x. Previous researches [1-12], however, have only concentrated upon the problem of forced convection. As the difference between the wall temperature and the surrounding temperature is large, it is interesting to study the effect of free convection on the flow and heat transfer. Convective heat transfer at a stretching vertical sheet with the effect of free convection was numerically studied by Vajravelu and Nayfeh [13] for local similarity solution. Chen [14] numerically investigated the laminar mixed convection adjacent to vertical, continuously stretching sheets by using Keller's box method. The problem of heat and mass transfer of forced convection on a stretching sheet with suction or blowing for the case of uniform wall temperature and concentration was analyzed by G u p t a and Gupta [15]. In the present paper, we studied numerically the free convection effect on the MHD laminar boundary layer for the heat and mass transfer of a linearly stretching permeable vertical surface.

Analysis Consider the problem of free convection effect on the steady incompressible twodimensional laminar magnetohydrodynamic heat and mass transfer characteristics of a

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linearly moving permeable vertical surface when the velocity of the fluid far away from the plate is equal to zero. The variations of surface temperature and concentration are linear. All the fluid properties axe assumed to be constant except for density variations in the buoyancy force term. Introducing the boundary layer and Boussinesq approximations, the governing equations can be written as follows: Ou

Ou

~+v

Ou Ny

=u

02u Oy 2

Ov

o-7 + ~ = o,

(1)

~,B2o -- u

(2)

+ g / 3 T ( T -- Too) + g/3c(c - coo),

P OT

~,~

OT

02T

Oc

02c

+ v Ny = ~-b-jv~ ,

Oc

~ N + v N = D--oy2,

(3)

(4)

where a is the electrical conductivity; Bo is the externally imposed magnetic field in the y-direction. We assumed the magnetic Reynolds number is small. Therefore, the induced magnetic field effect is negligible compared with the applied magnetic field. The Hall effect, the viscous dissipation and the Joule heating (electrical dissipation) terms are also neglected. The boundary conditions are defined as follows: y = O : v = -vw,

u = Bx,

T = T ~ + a x , c = coo + bx,

y--,ec : u = O , T = T o o ,

c=coo,

(5) (6)

where Vw is the uniform surface mass flux; B, a, b are prescribed constants. The stream function ¢ is defined by u = O ¢ / O y and v = - O ¢ / O x ,

therefore, the

continuity equation is automatically satisfied. Invoking the following similarity variables

(7a) ¢ f(~) = ,/N-;.~'

(7b)

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T-Too 0(7) -

T~(x)

C(,7) =

(7c)

- T~'

c cw(x)

(Td)

c~ - c~

and substituting Eq. (7) into Eqs. (1)-(6), we obtain /

GrT

\

f ' " + f f " _ { f, )2 _ M I ' + --~-~e2(O + N C ) = O,

(s)

1--~-0" + fO' -- f'O = O, Pr

(9)

!C" Sc + fC'

(lO)

- f ' C = O.

The boundary conditions are rewritten as follows: rl=O: f=fw,

f'=l,

r?-*cc : i f = 0 ,

0=1,

0=0,

C=1,

C=0.

(11) (L2)

In addition, the velocity components are

u = Bxf',

v =- - ~ - u u f .

(13)

In the foregoing equations, the primes denote the differentiation with respect to rI. M

= a B 2 / ( B p ) is the magnetic parameter. G r T / R e 2 = gt3Ta/B 2 is the buoyancy parameter. When G r T / R e 2 = 0 the governing equations are reduced to forced convection limit. However, as GrT/Re2--+oc free convection is dominated. The buoyancy ratio N = /3cb/(/3Ta) measures the relative importance of mass and thermal diffusion in the buoyancy-driven flow. It is apparent that N is zero for thermal-driven flow, infinite for mass-driven flow, positive for thermally assisting flow, negative for thermally opposing flow. Pr = u / a and Sc = u/D are the Prandtl number and Schmidt number, fw = v~/x/-ff~ is the suction/blowing parameter. For the case of suction, v~ > 0 and hence f~ > 0. On the other hand, for the case of blowing, v~ < 0 and hence fw < O.

Numerical

Method

The system of the governing equations (8)-(10) together with the boundary conditions (11)-(12) is nonlinear ordinary differential equations depending on the various values of the

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99

magnetic parameter M, buoyancy parameter GrT/Re 2, buoyancy ratio N, P r a n d t l number Pr, Schmidt n u m b e r Sc, and suction/blowing parameter fw. The system of Eqs. (8)-(10) is solved numerically by the well-known Keller's box method. The effective finite difference scheme has been described in details [16].

T h e computations were carried out on an

AcerPower 590hd computer. The edge of the boundary-layer r/o¢ was adjusted for different range of Pr and Sc.

T h e requirement that the variation of the velocity, temperature,

and concentration distributions is less than 10 - s between any two successive iterations is employed as the criterion of convergence.

The integrated values of the velocity, the

temperature, and the concentration fields from Eqs. (8)-(10) thus obtained are used to calculate the corresponding values of the skin-friction coefficient, the Nusselt number, and the Sherwood number from the following relations:

0.5Cfne 1/2 =

N u / R e 1/2 = --8'(0),

f't(0),

S h / R e 1/2 = - C ' ( 0 ) ,

(14)

respectively. In the Eq. (14), the Reynolds n u m b e r Re = Bx2/~.

Results

and Discussion

In order to verify the accuracy of our present method, we have compared our solutions with d a t a of Chen [14] where no magnetic force effect (M = 0), no buoyancy force effect from mass diffusion (N --- 0), and no blowing/suction effect (fw = 0). The comparison is in very good agreement, as shown in Figs. 1 and 2. 5 Pr = 0.7//° [ 1 4 ] ~

2 ~Chen

4

~Chen [14] O Present -------43

o

2 ~

Z

0

0

0

Pr = 10

3 L

O

O-----~

3

o

0.7 M = N =

f.

=

M = N = f,,= 0

0

0 0

2

4

6

8

2

10

GrT/Re z

FIG. 1 Comparison of the skin-friction coefficient

4 6 GrT/Re z

8

10

FIG. 2 Comparison of the Nusselt number

We also have compared our numerical results with those of Takhar et al. [9], Grubka and Bobba [1], Afzal [4], and All [5,6]. The comparisons in all the above cases axe found to be in good agreement, as shown in Tables 1 to 3.

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Vol. 26, No. 1

TABLE

I

C o m p a r i s o n o f t h e V a l u e s o f f " ( 0 ) ~ r V a r i o u s Valuesof M and G r T / R e 2 = f ~ = 0 0.~

Present results -1.0000 -1.2247 -1.4142 -1.5811 -1.7321

Takhar et al. [9] -1.00 -1.22 -1.41

0.5 1.0 1.5 I 2.0

-1.58 -1.73

TABLE 2 Comparison of the Values of - 8 ' ( 0 ) for Various Values of Pr and M = G r T / R e 2 = f w = 0 Grubka and Bobba 0.001

All [6]

[1]

Pr .

- -

,

Present results 0 . 0 0 2 0

(0.0020) 0.0197 0.8086

0.0197 0.8086 1 1.0000 1.0000 3 1.9237 1.9237 10 3.7207 3.7207 100 12.2940 12.2940 108 1253.0668 (1253.3141) Results in parentheses are those of Afzal [5]

0.01

0.72

•

0.8058 0.9961 1.9144 3.7006

- -

TABLE 3 Comparisonofthe VMuesof-0'(0)~rVariousValuesofPrandf~and

Aft [71

fW 0.6 0.4 0.2 -0.2 -0.4 -0.6 -1.0 -1.5

Pr=0.72 1.0365 0.9529 0.8758 0.7418 0.6679 0.6323 0.5437 0.4559

Pr= 1 1.3364 1.2135 1.0999 0.9018 0.8077 0.7421 0.6167 0.4992

Pr=3 3.0753 2.6540 2.2646 1.6072 1:3426 1.1314 0.8242 0.5990

I

Pr = 10 I Pr = 0.72 7.9392 1.0419 6.3735 0.9574 4.9437 0.8796 2.6984 O.7443 1.9593 0.6863 1.4674 0.6341 0.9405 0.5451 0.6446 0.4570

M = GrT/Re2=

Present results Pr= 1 Pr=3 1.3440 3.0978 1.2198 2.6708 1.1050 2.2775 0.9050 1.6139 0.8198 1.3510 0.7440 1.1348 0.6180 0.8258 0.5000 0.5999

Pr=10 8.0177 6.4258 4.9764 2.7096 1.968i 1.4709 0.9418 0.6452

Numerical results are graphically given for Pr = 0.72, Sc ranging from 0.2 to 2.0, Grr/Re 2

ranging from 0 to 100, M ranging from 0 to 100, N ranging from - 0 . 5 to 100,

and fw ranging from - 1 . 0 to 1.0.

0

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MHD COUPLED HEAT AND MASS TRANSFER

101

T h e N u s s e l t n u m b e r Nu/l:te 1/2 a n d t h e S h e r w o o d n u m b e r S h / R e 1/2 for various values of s u c t i o n / b l o w i n g p a r a m e t e r fw w i t h M = 0, N = 10, P r = 0.72 a n d Sc = 0.5 are i l l u s t r a t e d in Figs. 3 a n d 4, respectively. N u / R e 1/2 and S h / R e 1/2 i n c r e a s e ( d e c r e a s e ) for t h e ease of s u c t i o n ( b l o w i n g ) . W h e n b u o y a n c y p a r a m e t e r GrT / Re2--+O, N u / Re 1/2 and S h / Re ~/2 a p p r o a c h t h e v a l u e s of p u r e forced c o n v e c t i o n for e a c h fw. M o r e o v e r , as G r T / R e 2 increases b o t h t h e h e a t t r a n s f e r r a t e a n d the m a s s t r a n s f e r r a t e increase. 10

"~1

I0"

M = 0, N = 10 Pr --- 0.72, Sc = 0.5

M = 0, N = i0 Pr = 0.72, Sc = 0.5

fw

z

_______=_1-11

10

-310-=10-,

1 Grr/Re 2

10

10

..........................................

0.1

............................................

0.1

10 "~10

z

FIG. 3 N u s s e l t n u m b e r for v a r i o u s values of fw

-' i GrT/Re z

-210

i0

10

~

FIG. 4 S h e r w o o d n u m b e r for v a r i o u s values of ft,

F i g u r e s 5 a n d 6 s h o w t h e Nusselt n u m b e r a n d t h e S h e r w o o d n u m b e r for various values of Sc w i t h M = 1, N = 5, P r = 0.72 a n d fw = 1, r e s p e c t i v e l y . i n c r e a s e s as S c h m i d t n u m b e r Sc decreases (increases).

N u / R e I/2 ( S h / R e 1/2)

T h i s is d u e to t h e fact t h a t a

s m a l l e r Sc (or, e q u i v a l e n t l y , Le for a fixed value of P r ) is a s s o c i a t e d w i t h a t h i n n e r t h e r m a l b o u n d a r y layer. T h e t h i n n e r t h e t h e r m a l b o u n d a r y layer thickness, t h e g r e a t e r t h e Nusselt n u m b e r . T h e S e h m i d t n u m b e r has a m o r e significant effect on t h e S h e r w o o d n u m b e r t h a n it does o n t h e N u s s e l t n u m b e r , as c o m p a r e d Figs. 5 a n d 6. 4.0

10

M = 1, N = 5 Pr = 0.72, f. = 1

3.5

M = 1, N = 5 Pr = 0.72, f,, = 1

/

"--- 3.o

Sc = 0 . 2 ~

/~

2.5

co

Z 2.0

1.5 1.0

. . . . . . . .

10

-a 1 0

,

. . . . . . . .

-7 1 0

,

. . . . . . . .

-t

,

1

. . . . . . . .

,

10

. . . . . . . .

10

O.l

,

z

Grz/Re 2 FIG. 5 N u s s e l t n u m b e r for v a r i o u s values of Sc

.......................................... 10 -310

-2

10 -~ 1 GrT/Re 2

10

10 ~

FIG. 6 S h e r w o o d n u m b e r for v a r i o u s values of Sc

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K.A. Yih

Vol. 26, No. 1

T h e N u s s e l t n u m b e r a n d t h e S h e r w o o d n u m b e r for various values of b u o y a n c y r a t i o N w i t h M = 5, f~, = - 0 . 5 , P r = 0.72 a n d Sc = t, are d i s p l a y e d in Figs. 7 a n d 8, r e s p e c t i v e l y . T h e N u s s e l t n u m b e r a n d t h e S h e r w o o d n u m b e r are i n c r e a s i n g for t h e i n c r e a s e of N. T h i s is b e c a u s e t h a t i n c r e a s i n g t h e b u o y a n c y r a t i o N r e d u c e s t h e t h e r m a l a n d c o n c e n t r a t i o n b o u n d a r y layer thicknesses. 10:

M = 5, f,, = - 0 . 5 Pr = 0.72, Sc = 1

10

M = 5, f, = - 0 . 5 Pr = 0.72, Sc -- 1 /

N = 1010~/ ,= Z

O3

0.1

" "~""'

I0

' ........

0.1

' ..........................

i0-' I GrT/IRe 2

-3 I 0 - '

I0

I0

FIG. 7 N u s s e l t n u m b e r for v a r i o u s values of N

.............................................

10 -~ 10 -~ 1 0 - ' 1 Gr~/Re 2

2

10

10'

FIG. 8 S h e r w o o d n u m b e r for v a r i o u s values of N

F i g u r e s 9 a n d 10 p r e s e n t t h e N u s s e l t n u m b e r a n d t h e S h e r w o o d n u m b e r for v a r i o u s values of m a g n e t i c p a r a m e t e r M w i t h N = - 0 . 5 ,

f ~ = 0.5, P r = 0.72 a n d Sc = 1.5,

respectively. It is o b s e r v e d t h a t t h e Nusselt n u m b e r a n d t h e S h e r w o o d n u m b e r d e c r e a s e w i t h i n c r e a s i n g m a g n e t i c p a r a m e t e r M. T h e p h e n o m e n o n of t h e N u s s e l t n u m b e r is s i m i l a r to w h a t has b e e n r e p o r t e d by C h a k r a b a r t i a n d G u p t a [8] for p u r e f o r c e d c o n v e c t i o n . 2.0

3.0

N = - 0 . 5 , f, = 0.5 Pr = 0.72, Sc = 1 . 5 /

2.5

1.5 M

=

N = - 0 . 5 , f, = 0.5 Pr = 0.72, Sc = l . h j

2.0

0

c~ 1.5 .c= m 1.0

1.0 z

I00

0.5

0.5

0.0

..................................

10

-3 1 0

Nusselt n u m b e r

-2

I0-' I GrT/Re z

•

I0

.....

10

0.0 2

FIG. 9 for various values of M

............................................

i0 -~ I0 -~ 10 -~ 1 GrT/Re 2

10

10

~

FIG. I0 S h e r w o o d n u m b e r for various va/ues of M

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103

Conclusions Free convection effect on magnetohydrodynamic laminar boundary layer coupled heat and mass transfer is numerically analyzed when a vertical moving surface with suction or blowing and linear temperature and concentration variations moves with a linear surface velocity. The nonlinear boundary-layer equations were transformed and the resulting ordinary differential equations were solved by Keller's box method. Numerical results for magnetic parameter M, buoyancy parameter G r T / R e 2, buoyancy ratio N, Prandtl number Pr, Schmidt number Sc, and suction/blowing parameter fw are given for the Nusselt number and the Sherwood number. As N, G r T / R e 2, fw are increased and M is decreased, the Nusselt number and the Sherwood number increase. Increasing (Decreasing) the Schmidt number Sc gives rise to an increase in the mass (heat) transfer rate. Furthermore, the Schmidt number has a more significant effect on the Sherwood number than it does on the Nusselt number.

References

1.

L. J. Grubka and K. M. Bobba, A S M E J. Heat Transfer 107, 248 (1985).

2.

C. K. Chen and M. I. Char, J. Math. Anal. Appl. 135, 568 (1988).

3.

M. Abramowitz and L. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, AMS 55, Dec. (1972).

4.

N. Ahmad and A. Mubeen, Int. Comm. Heat Mas~ Transfer 22,895 (1995).

5.

N. Afzal, Int. J. Heat Mass Transfer 36, 1128 (1993).

6,

M. E. Ali, W'drme- und Stoffi~bertragung 29, 227 (1994).

7.

5/I. E. Ali, Int. ]. Heat and Fluid Flow 16, 280 (1995).

8.

A. Chakrabarti and A. S. Gupta, Q. Appl. Math. 37, 73 (1979).

9.

H. S. Takhar, A. A. Raptis and C. P. Perdikis, Acta Mech. 65,287 (1986).

10.

K. Vajravelu and D. Rollins, Int. J. Non-Linear Mech. 27, 265 (1992).

11.

M. I. Char, W'drme- und Stoffdbertragung 29~ 495 (1994).

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K.A. Yih

Vol. 26, No. 1

12.

T. C. Chiam, Acts Mech. 122,169 (1997).

13.

K. Vajravelu and J. Nayfeh, Acta Mech. 96, 47 (1993).

14.

C. H. Chen, Heat and Mass Transfer 33~ 471 (1998).

15.

P. S. Gupta and A. S. Gupta, Can. J. Chem. Engn9. 55,744 (1977).

16.

T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, 1st ed., p. 385. Springer-Verlag, New York (1984). Received June 17, 1998