•• EG0600070 Proceedings of the Environmental Physics Conference, 24-28 i _„. .
HEAT AND MASS TRANSFER EFFECT ON HYDROMAGNETIC FLOW OF A MOVING PERMEABLE VERTICAL SURFACE
M.M. Abdelkhaiek Atomic Energy Authority, Nuclear Research Centre, Nuclear Physics Department, Cairo, Egypt, P.O. Box 13759
ABSTRACT
Numerical results are presented for the effects of heat and mass transfer on hydromagnetic flow of a moving permeable vertical surface. The surface is maintained at linear temperature and concentration variations. The nonlinear- coupled boundary layer equations were transformed and the resulting ordinary differential equations were solved by perturbation technique. Numerical results for the dimensionless velocity profiles, the temperature profiles, the local friction coefficient and the local Nusselt number are presented for various values of Prandtl number, suction/blowing parameter, Schmidt number, buoyancy ratio and Hartmann number. The effects of the different parameters on the velocity and temperature profiles as well as skin friction and wall heat transfer are evaluated. Favorable comparisons with previously published work confirm the correctness of numerical results. Key Words: Heat and Mass Transfer, Hydromagnetic Flow, Perturbation Technique/
INTRODUCTION
Hydromagnetic incompressible viscous flow has many important engineering applications such as magnetohydrodynamic power generator and the cooling of reactors. Heat transfer and its related topics are heavily studied fields. These studies range from over simplified problems to highly complex types of interactions and configurations, which require sophisticated numerical schemes and high-speed computers to obtain reasonably accurate results. However, the common ground for most of these .-jtudies is that they are solved and analyzed by assuming a pure fluid with no contaminants. While this assumption approximates the reality in some cases quite well, especially for low particle contamination levels, it. is not, however, valid in a lot of other cases in which the contaminants in the fluid play a major role in altering the resultant flow and heat transfer characteristics. In many world environments, such as in Egypt, for example, dust storms or fine dust Suspension in the air are encountered for many months during the year. These fine particles of dust penetrate the enclosures and the various devices, and have a serious impact on the performance of many equipments. This example represents a situation of participate suspension where a pure fluid assumption does not accurately represent the reality. The advent of technology in the field of magnetohydrodynamic power generators and magnetohydrodynamic devices, nuclear engineering and thermonuclear power has created a great practical need for understanding the dynamics of conducting fluids. The use of liquid metals as heat transfer agents and as working fluids in magnetohydrodynamic power generators has created a growing interest in the study of liquid metal flows and nature of interaction with magnetic field. The interaction between
185 Proceedings of the Environmental Pliysics Conference, 24-28 Feb. 2004, Mi/tya, Egypt
the conducting fluid and magnetic field radically modifies the flow and heat transfer characteristics. The outlook for a direct coal-fired magneiohydrodynamic power generator as potentially significant source of energy seems promising in view of its efficiency, its effect on environment and the availability of needed natural resources. These studies are useful in understanding the effect of the presence of a slag layer on heat transfer characteristics of a coal-fired magnetohydrodynamic generator. Many oilier similar situations are faced in such industries as food processing, various grinding operations, settling of various liquid contaminants in large storage tanks, waslc processing and recycling and many others. There has been considerable work done on various aspects of free convection flow of a clean fluid over vertical and inclined surfaces. There has been considerable interest in studying flow and heat transfer characteristics of electrically conducting and heat generating/ absorbing fluids (see for instance, Moalem(l), Chakrabarti and Gupta(2); Vajravelu and Nayfeh(3); Chiam(4); Chamkha(5); Chandran et al(6); Vajravelu; Hadjinicalaou(7) and Abdelkhalek(8"15). This interest stems from the fact that hydromagnetic flows and heal '/ansfer have been applied in many industries. For example, in many metallurgical processes such as drawing of continuous filaments through quiescent fluids, and annealing and 'inning of copper wires, the properties of the end product depend greatly on the rate of cooling involved in these processes. The problem of magnetohydrodynamic natural convection about a vertical impermeable flat plate is presented by Sparrow and Cess(16) and Wilks and Hunt(l7). Watanabe and Pop(18) studied the case of a wedge. The laminar plume above a line heat source in a transverse magnetic field was studied by Gray(l9). Kafoussias^20-1 investigated the MI-ID free convective flow through a porous medium over an isothermal cone surface. Balder et al(2l) presented non similar solutions for free convection from a vertical permeable plate in porous media. Similarity solutions of natural convection boundary layers adjacent ic vertical and horizontal surfaces in porous media with internal heat generation were studied by Pop(22). Grubka and Bobba(23) investigated heat transfer characteristics of a continuous, stretching surface with variable wall temperature. Ali(24) numerically presented the heat transfer characteristics of a power law continuous stretched surface without and with suction injection. Takhar et al(25) studied a MHD asymmetric flow past a semi-infinite moving plate and numerically obtained the solutions. Yih(26) studied a free convection effect on MHD coupled heat and mass transfer of a moving permeable vertical surface. In the present paper we studied analytically and numerically the heat and mass transfer in a hydromagnetic flow of a moving permeable vertical surface. The nonlinear-coupled boundary layer equations were transformed and the recaUir.g ordinary differential equations were solved by perturbation technique. The effect of different values of parameters, Prandtl number, suction/blowing parameter, Schmidt number, buoyancy ratio and Hartmann number, on the velocity and temperature profiles as well as skin friction and wall heat transfer are evaluated.
FORMULATION OF THE PROBLEM
Let us consider two-dimensional free convection effect on the steady incompressible laminar magnetohydrodynamic heat and mass transfer characteristics of a linearly moving permeable vertical surface when the velocity of the fluid iar away from the plate is equal to zero. The variations of surface temperature and concentration are linear. All the fluid properties are assumed to be constant except for the density variations in the buoyancy force term: Introducing the boundary layer and Boussinesq approximations, the governing equations can be written as follows:
186 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004, Minya, Egypt
+ 0 (1) d x d y du dv d2u = v P,\c~C,) » (2) ox o y o y~ p p dT dT d2r ... u + v = a (3) dx dy d y" 8C dC d2C u + v = D ^ (4) d x d y d y~ where, u,v is the velocity components in x,y directions, respectively, x,y are the tangential (or radial) distance along the wall and the normal distance, respectively, T is the temperature,
y-+co, u = 0, T = Ta, C = Ca (6) where v(1. is the uniform surface mass flux, B, a,b are prescribed constants. The stream function is defined by u = and v = , d y d x Put the non dimensional quantities defined by
and substituting Eq. (7) into Eqs. (l)-(6), we obtain
/'" + ff" - (M + /')/' - -%•(* + JVC) (8)
K t 0 + Pr{f6 -f'6) = O (9)
C' + Se(fC'-fC) = 0 (10) The boundary conditions are written as follows, 7 = 0: / = /„., /' = 1, 0 = 1, C = \ 7->oo: /' = 0, 0 = 0, C = 0
The velocity components are: it •= Bxf, v - -(B V)°'* f
187 Proceedings of tins Environmental Physics Conference, 24-28 Feb. 2004, Minva, Eg)'pi
where, the primes denote the differentiation with respec!. to //, M = — is the magnetic Bp G T gfiTa G T parameter, - •*•• = —-— is the buoyancy parameter. When —-—• = 0 the governing equations are reduced to forced convection limit. However, as -A—> co free convection is dominated. K The buoyancy ratio N = —— measures the relative importance of mass and thermal diffusion fiTa 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, V V V Pr = — is the Prandtl number, and Sc -— is the Schmidt number, fw = "^-^j is the suction / blowing parameter. For the case of suction, vw > 0 and hence fw > 0 . For the case of blowing, vH, <0 and hence /\<0. The resulting differential equations contain arbitrary parameters, the Prandtl number/^., the magnetic field strength (M) and the buoyancy parameter GrTlR], the
Schmidt number (Sc )and buoyancy ratio (N). Solution of the resulting semi-infinite domain, nonlinear equations is accomplished with a three part seder. method(27l The employed power series contains a term K that satisfies the boundary conditions and differential equations at infinity, a second term that satisfies the boundary corHitions at zero and is the solution to the initial homogeneous differential equation, and additional terms that are utilized to obtain increased numerical accuracy. This accuracy is limited by number of terms that will not initiate divergence of the numerical results: 2 3 f = K + efl+e f2+£ f3+- (12) 2 i # = £#, +£ 02+£ 9,+--- (13) 2 C = sCx +£ C2 +s'C3 +••• (14)
7 = 0, //(0) = l, /2'(0) = /3'(0) = 0, /,(0) = /n., /,(0) = /J(0) = 0,
,0,(O) = 1, 02(O) = 0,(O) = O, C,=l, C,=C3=0, (15)
7-**>> /„'(«) = 0, ^H(oo) = 0, Cfl(oo) = 0, «= 1,2,3
Equation (13), the temperature representation, along with equation (12) and the associated boundary conditions equation (15), contain an undetermined parameter s which aids in the collection of terms for each set of the resulting linear differential equations. In some problems, it will have a physical meaning which results in a power series in that parameter. However, the present case s equals unity. Substitution of the series representation into the differential equations and collection of terms by WVd powers of s result in a family of linear differential equations, and the first three sets are:
^(^ X) (16) Re 6'; (17) Pr
188 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004, Minya, Egypt
— C';+KC:=O (18) sc
2 2f[f;x (20) "r ± (21)
;- fjx'-2j£.(o3 + NC,) (22) R e
ye; + KO[ = /,», + /# - /,# - f2e; (23)
~ c;+KC3 = f;c2 + /2'c, - /, c; - /2 c; (24)
The solutions to the first two sets, Eqs. (25)-(30), when substituted into Eqs. (12)-(14), provide the required representations for /,<9and C. The constant K is determined by satisfying the boundary conditions /(0) and is a function of Pr and M : (25)
KSc?? C] =e~ (26) -an -KP TJ -KS ? l r c ' ^ae +ae (27)
KPr11 V| r -PraAJj)r +a6e '-va^e '' '" *'' (28)
= a + a lS a 7 2 "l 8 9 ~ c 4' ^ 8 a +KS -( i1 ch c
-2a i} -2KPij -IKS n ( \-KPn
^ \ - ^ /; - (K/J -I- KS }/ 28 " "33 • "32"'^ ' "29
where, a]}a2,a3, a35 are in appendix. The series for G, its first derivative #'(0) (the wall temperature gradient), /' (the velocity profile) and f"(Q) (the wall velocity gradient) were evaluated. The general results of the investigations are that the imposed magnetic field diminished the velocity field, wall shear, flow rate, and wall heat transfer, also the onset of free convection was retarded while the fluid temperatures and the time required for the flow to each steady state are increased. In addition, sizable influences on the flow and thermal fields can be produced with moderate magnetic field strengths only for liquid metal flows while the effects of induced magnetic
189 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004, Minya, Egypt fields and Joule heating are very small. The magnetic field strength is to reduce the values of wall shear stress regardless of blowing and suction strength. Magnetic force is known to have a retardation effect, thus decreases the wall shear stress. Blowing has a similar effect thus aiding the magnetic field in reducing the wall shear stress. However suction has an opposite effect and increases the shear stress at the wall opposing the magnetic effect. In case of free convection, suction decreases wall shear stress, this may be because increases suction will decrease the flow velocity and in turn diminishing the retardation effect of the magnetic field force. Knowing the velocity, we can calculate the skin friction and from the temperature field the rate of heat transfer in terms of the Nusselt number.
RESULTS AND DISCUSSIONS
In order to get the physical insight into the problem, numerical calculations are carried out for different values of Prandtl number, suction/blowing parameter, Schmidt number, buoyancy ratio and Hartmann number. In order to verify the accuracy of our present, we have compared our results with those of Grubka et al(23), Ali(24), Takhar et al(25), and Yih(26), Table [1] compares the values of (/"(0)) for various values of magnetic parameter (M). The comparison in all above cases is found to be in excellent -agreement. Table (2) shows the selected values of non-dimensional wall temperature gradient (-O'(0)) for the various values of Prandtl number Pr. The comparison is found to be in good agreement.
Table 1. Comparison of non-dimensional wall velocity gradient /"(0) for various values 2 of Hartmann number (M) andG,.r /Re = fsr = 0 . M = 0.0 M = 0.5 M= 1.0 M = 1.5 M -2.0 Takhar et al -1. 0 -1 22 -1.41 -1.58 -1 73 [25] Yih [26] -1. 00 -1 2247 -1.4142 -1.5811 -1 7321 Present results -1. 000 -1 2356 -1.4156032 -1.58212 -1 73342
Table 2. Comparison of the values of non dimensional wall temperature gradient -0'(Q) 2 for various values of Pr andM =GrT /Re = fw = 0. 0.72 1.0 3.0 10.0 100.0 Grubka et al 0.8086 1.000 1.9237 3.7207 12.294 [23] Ali [24] 0.8058 0.9961 1.9144 3.7006 _„ Yih [26] 0.8086 1.000 1.9237 3.7207 12.2940 Present work 0.80865 1.000 1.9246 3.7216 12.29453
The effect of various values of suction / blowing parameter fw on Sherwood number (-c'(0)) is illustrated in Fig. (1). The Sherwood number increases as suction / blowing parameter decreases. It is seen from the figure that the Sherwood number profiles increase monotonically with increasing buoyancy parameter. When buoyancy parameter G -»0 , Nusselt number and Sherwood number approach the values of pure forced convection for each/H.. Moreover as buoyancy parameter increases both the heat transfer rate and the mass transfer rate increase.
190 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004. Minya, Egypt
- O(O)
s.oo —
Fig. 1. Effects of suction / blowing parameter on Sherwood number with, Re = 50.0, M = 0.0, N = 10.0, /?= 1.0, P...=0.7, Sr =0.2
The effects of both buoyancy parameter and various values of Schmidt number on the Sherwood number is illustrated in Fig. (2). The Sherwood number increases as the Schmidt number increases. The Schmidt number has a more significant effect on the Sherwood number than it does on the Nusselt number.
O.2
O.T2
—rf-CB 1 O.OO
Fig. 2. Effects of Schmidt number on Sherwood number with, Re=50.0,M = 1.0, N = 5.0, 77= 1.0, Pr =0.7, fw =1.0
The effects of both buoyancy number G and various values of buoyancy ratio N on the non dimensional wall temperature gradient and Sherwood number, respectively, are depicted in Fig. (3) . The non dimensional wall temperature gradient and Sherwood number are increasing for the increase of buoyancy ratio N. This is because that increasing the buoyancy ratio N tends to reduce the thermal and concentration boundary layer thickness.
191 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004, Minya, Egypt
Fig. 3. Effects of buoyancy ratio on non dimensional wall temperature gradient with,
Re= 50.0, M = 5.0, 77=1.0, Pr=0.7, /„, = -0.5,5"f =1.0
Figure (4) presents the effects of both buoyancy number and various values of Hartmann number on the Sherwood number. It is observed that the Sherwood number decreases with increasing Hartmann number. It is seen from the figure that the Sherwood number profiles increase monotonically with increasing buoyancy parameter.
Fig. 4. Effects of Hartmann number on Sherwood number with,
tf,= 50.0,N = -0.5, ?7=l-0, P,=0.7t /„. =0.5,Sc =1.5
CONCLUSION
The problem of free convection effect on the steady, laminar, incompressible, two dimensional magnetohydrodynamic heat and mass transfer of a linearly moving permeable surface when the velocity of the fluid far away from the plate is equal to zero. The coupled nonlinear boundary layer equations were transformed and the resulting ordinary differential equations were solved by a perturbation technique. Numerical results for various values of magnetic parameter, buoyancy parameter, Prandtl number, Schmidt number, buoyancy ratio and suction/blowing parameter are given for the skin friction, the Nusselt number and the Sherwood number. As the Prandtl number increases, the non dimensional wall temperature gradient increases while the non dimensional wall velocity gradient decreases. The Sherwood number increases as the suction/blowing parameter decreases, as the buoyancy parameter
192 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004, Minya, Egypt increases both the heat transfer rate and the mass transfer rale increase. Increasing the values of Schmidt number tends to reduce the non dimensional wall temperature gradient. The Sherwood number increases as the Schmidt number increases. The non dimensional wall temperature gradient and the Sherwood number are increases for the increase of buoyancy ratio. The Sherwood number increases as the Schmidt number increases. The Sherwood number decreases with increasing Hartmann number.
REFERENCES
[I] D. Moalem; Int. J. Heat Mass Transfer; 19, 529, (1976). [2] A. Chakrabarti and A.S. Gupta, Q. Appl. Math., 37, 73, (1979). [3] K. Vajravelu and J. Nayfeh; Int. Commun. Heat Mass Transfer 19, 701, (1992). [4] T. C. Chiam; Int. J. Engng. Sci., 33, 429, (1995). [5] A. J.. Chamkha; Int. Commun. Heat Mass Transfer 23, 875, (1996). [6] P. Chandran; N.C. Sacheti and A.K. Singh; Int. Commun. Heat Mass Transfer 23, 889 (1996). [7] K. Vajravelu and A. Hadjinicalaou; Int. J. Engrg. Sci. 35, 1237, (1997). [8] M.M. Abdelkhalek; Al-Azhar Engineering Fifth International Conference; Vol. 8, P.31- 42. Dec. 19-22(1997), [9] M.M. Abdelkhalek; Egypt. J. Phys.; 29,No.l, .39 (1998). [10]. M.M. Abdelkhalek and M.N.H. Comsan; Egypt. J. Phy;.., 30, No.l, 107 (1999). [II] M.M. Abdelkhalek; Arab J. Nucl. Sci. Appl.; 36(2), 179 (2003). [12] M.M. Abdelkhalek; Arab J. Nucl. Sci. Appl; 36(2), 189 (2003). [13] M.M. Abdelkhalek; Egypt. J. Phys; 34(2), 267 (2003). [14] M.M. Abdelkhalek; Arab J. Nucl. Sci. Appl; 37(1), 257-267 (2004). [15] M.M. Abdelkhalek; Proceedings of the International Conference on Mathematics Nuclear Physics and Applications in the 21 Century; 479 ( Cairo 8-13, March 2003) [16] E.M. Sparrow, and R.D. Cess ; Int. J. Heat Mass Transfer, 4, 267 (1961). [17] G. Wilks and R. Hunt., J. Appl. Math. Phys., 35, 34 (1984). [18] T. Watanabe and I. Pop ; Int. Commun. Heat Mass Transfer 20, 871, (1993). [19] D.D. Gray ; Appl. Sci. Res., 33, 437 (1977). [20] N.G. Kafoussias ; Mech. Res. Commun. , 19, 89, (1992). [21] A.Y. Bakier, M.A. Mansour , R.S.R. Gorla and A.B. Ebiana , Heat Mass Transfer, 33, 145,(1997). [22] I. Pop; Int. Commun. Heat Mass Transfer 26, 1183 (1999). [23] L.J. Grubka and K.M. Bobba ., ASME J. Heat Transfer, 107, 248 (1985). [24] M.E. Ali ; Warme und Stoffubertragung 29, 227 (1994). [25] U.S. Takhar, A.A. Raptis and C.P. Perdikis ; Acta Mech. 65, 287 (1986). [26] K.A. Yih ; Int. Coram. Heat Mass Transfer, Vol. 26, No.l, 95, (1999). [27] R. Kenneth Cramer and Shih-I Pai; " Magneto fluid Dynamics for Engineers and Applied Physicists" , P. 166-167,(1973),
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APPENDIX
+4M GrT GrT a2 2 Ch = 2 ' ~KPrR;(K P,.(P,.-l)~My KScr-(K S,(S
KP,a2+KSca3+l />,(fl,as - KasPr) aa a = a = ~ ' ' "> (al+KPj(al+KP,)-KY -a5Sc _ a2Sc(KSc-KPr) 7 ,r _ 7 9 {{KPr+KSc)-KY \ ((a,+^c)-^)' (KSc+KPj(KSc+KPr)-K)
8 n ^ Re Re fl1177 = = "V^C"V^+-+7ff (fl8+fl
7f(7%) 19Tf^4 20Tf e e e
21
22 24 -(al+KPrX(al+KPrX(al+KPf)-K)-M) ' -2KPr(2KPr{2KPr-K)-
^25 ~ 21 -2KSC{2KSC(2KSC-K)-M)' -KPr(KPr{KPr-K)-M)'
"is
= 29 2 {K Pr(3Pr-2)-M)a 30 a,
2l +aa +a23 +a24 +a25
194