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International Journal of Heat and Mass Transfer 55 (2012) 2122–2128
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat and mass transfer of thermophoretic MHD flow over an inclined radiate isothermal permeable surface in the presence of heat source/sink N.F.M. Noor a, S. Abbasbandy b, I. Hashim c,d,⇑ a
Department of Mechatronic, Faculty of Engineering, Universiti Selangor, 45600 Berjuntai Bestari, Selangor, Malaysia Department of Mathematics, Imam Khomeini International University, Ghazvin 34149-16818, Iran c School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia d Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia b
a r t i c l e
i n f o
Article history: Received 9 June 2011 Accepted 20 November 2011 Available online 5 January 2012 Keywords: Boundary layer MHD Thermophoresis Heat source
a b s t r a c t In this paper, a free convection thermophoretic hydromagnetic flow over a radiate isothermal inclined plate with heat source/sink effect is considered. The shooting method is employed to yield the numerical solutions for the model. The effects of thermophoretic parameter and internal heat generation/absorption for both suction and injection cases are discussed and presented graphically. The values of skin friction, wall heat flux and wall deposition flux are also tabulated with the variation of thermophoresis, heat source/sink and suction/injection parameters. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Thermophoresis has been the subject of abundant studies for many years [1–3]. It describes the migration of suspended small micron sized particles in a non-isothermal gas to the direction with decreasing thermal gradient. The velocity acquired by the particle is known as thermophoretic velocity while the force experienced by the suspended particles due to the temperature differences is known as thermophoretic force [4,5]. Being a potential mechanism for particles collection on cool surfaces, thermophoresis has numerous practical applications such as in aerosol particles sampling, air cleaning, scale formation on surfaces of heat exchangers, modified chemical vapor deposition, particulate material deposition on turbine blades, microelectronics manufacturing, nuclear reactor safety and removal of soot particles for combustion exhaust gas systems. One of the earliest studies on the role of thermophoresis in laminar flow over a horizontal plate with analysis on cold and hot plate conditions is given by Goren [6]. Some recent studies on the effect of thermophoresis in laminar flow are reported in [7–10]. The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal-working processes. The study of MHD flow and heat transfer are deemed as of great interest due to the effect of ⇑ Corresponding author at: School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia. Tel.: +60 3 8921 5758; fax: +60 3 8925 4519. E-mail address: [email protected] (I. Hashim). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.12.015
magnetic field on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. Some of the engineering applications are in MHD generators, plasma studies, nuclear reactor, geothermal energy extractions, purifications of metal from non-metal enclosures, polymer technology and metallurgy. Thermophoresis is also a key mechanism of study in semi-conductor technology especially controlled high-quality wafer production as well as MHD energy generation system operations. Since various industrial heat transfer process involved both the hydromagnetic flows and thermophoresis such as in MHD energy systems, many numerical studies on magnetohydrodynamic heat and mass transfer have been reported with buoyancy, Joule heating effects and heat source/sink parameters. The hydromagnetic free convection boundary layers in non-Darcian regimes via Keller-Box method was considered by Bég and Takhar [11]. Other hydromagnetic flow studies by Bég et al. include [12] on non-Newtonian wedge flow hydromagnetics and [13] on dissipative hydromagnetic boundary layers with wall suction/injection in porous media. The effects of suction and injection along a flat plate for free convection flow have attract interest of many researchers [9,14] due to the double impacts projected with respect to heat transfer. The temperature profile is influenced by the changed velocity field in the boundary layer leading to a change in the heat conduction at the wall. On the other hand, convective heat transfer occurs at the wall along with heat conduction for various values of suction/injection parameters. A computational analysis of heat generation/absorption and thermophoresis on hydromagnetic flow over a flat surface was presented by Chamkha and Camille [15].
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Using a perturbation approach, the transient oscillatory MHD convection past a flat plate adjacent to a porous medium with heat generation effects was solved in [16] and transient MHD natural convection with viscous heating was considered by Zueco [17]. Recently the effects of variable suction and thermophoresis on steady MHD flow over a permeable inclined plate was analyzed by Alam et al. [18] while the heat and mass transfer of thermophoretic hydromagnetic flow with lateral mass flux, heat source, Ohmic heating and thermal conductivity effects using the network simulation method was studied by Zueco et al. [19]. To our best knowledge, study on MHD flow over an inclined plate with thermophoresis and heat source has never been considered before. Therefore, in this paper, we will extend the previous work on a steady MHD flow with thermophoresis over a permeable radiate inclined plate by Alam et al. [20] to include the heat source/sink parameter. The present objective is to investigate the effects of thermophoretic parameter with the impact of heat source/sink in the model for both suction and injection cases. The values of skin friction, wall heat transfer and wall deposition flux are also tabulated. 2. Problem formulation 2.1. Governing equations and boundary conditions Consider a two-dimensional steady laminar flow of an incompressible electrically conducting fluid over a continuously moving semi-infinite inclined permeable flat plate with an acute angle a to the vertical. With the x-axis measured along the plate, a magnetic field B(x) is applied in the y-direction that is normal to the flow direction. Suction or injection is imposed on the permeable plate. The temperature of the surface is held uniform at Tw which is higher than the ambient temperature T1. The species concentration at the surface is maintained uniform at Cw = 0 while the ambient fluid concentration is assumed to be C1. The presence of uniform internal heat source/sink and thermophoresis are considered to study the variation of velocity, heat transfer and concentration deposition on the inclined surface. The following assumptions are made: (a) the concentration flux of particles is sufficiently small so that the main stream velocity and temperature fields are not affected by the thermophysical processes experienced by the relatively small number of particles, (b) the temperature gradient in the y-direction is larger than that in the x-direction due to the boundary layer behaviour. Hence only the thermophoretic velocity component normal to the surface is of importance, (c) the fluid has constant kinematic viscosity and thermal diffusivity and that the Boussinesq approximation may be adopted for steady laminar flow, (d) the particle diffusivity is assumed to be constant and the particles concentration is sufficiently dilute to assume the particle coagulation in the boundary layer is negligible, (e) the magnetic Reynolds is assumed to be sufficiently small so that the induced magnetic field is negligible in comparison to the applied magnetic field, and (f) the fluid is considered to be gray; absorbing-emitting radiation but non-scattering medium and the Rosseland approximation is used to describe the radioactive heat flux in the x-direction which is considered negligible in comparison to the y-direction. Under the above assumptions, the governing equations for this problem are written as @u @ v þ ¼ 0; @x @y
ð1Þ
u
@u @u @2u rB2 ðxÞ þv ¼ m 2 þ gbðT T 1 Þ cos a u; @x @y @y q
u
@T @T kg @ 2 T 1 @qr QðxÞ l @u 2 rB2 ðxÞ 2 þv ¼ ðTT 1 Þþ þ u ; ð3Þ þ 2 @x @y qcp @y qcp @y qcp qcp @y qc p
ð2Þ
u
@C @C @ 2 C @ðV T CÞ þv ¼D 2 ; @x @y @y @y
ð4Þ
where u and v are the velocity components in the x, y directions, m is the kinematic viscosity, g is the acceleration due to gravity, b is the volumetric coefficient of thermal expansion, T, Tw and T1 are the temperatures of thermal boundary layer fluid, the inclined plate and the free respectively, r is the electrical conductivity, pffiffiffiffiffistream ffi BðxÞ ¼ B0 = 2x is the magnetic induction, kg is the fluid thermal conductivity, q is the fluid density, cp is the specific heat at constant pressure, qr is the radiative heat flux in the y-direction, Q(x) = Q0/x is the internal heating, l is the dynamic viscosity, D is the molecular diffusivity of the species concentration and VT is the thermophoretic velocity. The boundary conditions for the model are as follows [20]:
v ¼ v w ðxÞ;
u ¼ U0 ; u ¼ 0;
T ¼ T1;
T ¼ Tw;
C ¼ C1;
C ¼ C w ¼ 0 at y ¼ 0;
as y ! 1;
ð5Þ ð6Þ
where U0 is the uniform plate velocity and vw(x) represents fluid suction/injection on the porous surface. The transpiration function variable vw(x) of order x1/2 is considered [20]. The radiative heat flux qr under Rosseland approximation has the form
qr ¼
4r1 @T 4 ; 3v @y
ð7Þ
where r1 is the Stefan–Boltzmann constant and v is the mean absorption coefficient. The temperature differences within the flow are assumed to be sufficiently small such that T4 may be expressed as a linear function of temperature. Expanding T4 using Taylor series and neglecting higher order terms yields
T 4 ffi 4T 31 T 3T 41 :
ð8Þ
Using Eqs. (7) and (8), we have
u
@T @T kg @ 2 T 16r1 T 31 @ 2 T Q ðxÞ þ þ ðT T 1 Þ þv ¼ @x @y qcp @y2 3qcp v @y2 qc p l @u 2 rB2 ðxÞ 2 þ u : þ qcp @y qcp
ð9Þ
The second, third, fourth and fifth terms on the RHS of the Eq. (9) denote the thermal radiation, internal heating, viscous and magnetic heating terms respectively. Now the thermophoretic velocity VT which appears in the Eq. (4) can be written as [22]:
V T ¼ km
rT Tr
¼
km @T ; T r @y
ð10Þ
where Tr is a reference temperature and k is the termophoretic coefficient with range of value from 0.2 to 1.2 as indicated by Batchelor and Chen [21] and is defined from the theory of [22] by
k¼
2C s ðkg =kp þ C t KnÞ½C 1 þ C 2 eC3 =Kn ; ð1 þ 3C m KnÞð1 þ 2kg =kp þ 2C t KnÞ
ð11Þ
where C1, C2, C3, Cm, Cs, Ct are constants, kg and kp are the thermal conductivities of the fluid and diffused particles respectively and Kn is the Knudsen number. A thermophoretic parameter s can be defined [7,23] as follows:
s¼
kðT w T 1 Þ : Tr
ð12Þ
Typical values of s are 0.01, 0.05 and 0.1 corresponding to approximate values of k(Tw T1) equal to 3 K, 15 K and 30 K for a reference temperature of Tr = 300 K.
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2.2. Similarity transformation The governing Eqs. (2)–(4) can be transformed to a set of nonlinear ordinary differential equations by introducing the following non-dimensional variables [20]:
g¼y
rffiffiffiffiffiffiffiffi U0 ; 2mx
w¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2U 0 mxf ðgÞ;
hðgÞ ¼
T T1 ; Tw T1
/ðgÞ ¼
C ; C1 ð13Þ
where w is the stream function that satisfies the continuity Eq. (1) with
u¼
@w ¼ U 0 f 0 ðgÞ and @y
v¼
rffiffiffiffiffiffiffiffiffi @w U0 m ½f ðgÞ gf 0 ðgÞ: ¼ @x 2x
ð14Þ
Using (13) and (14), the following similarity equations with the corresponding boundary conditions are obtained: 00
0
f 000 þ ff þ ch cos a Mf ¼ 0; h i ð3R þ 4Þh00 þ 3RPr f h0 þ Ecðf 00 Þ2 þ EcMðf 0 Þ2 þ 2dh ¼ 0; 00
0
0
00
/ þ Scðf sh Þ/ Scsh / ¼ 0;
ð15Þ ð16Þ ð17Þ
subject to
f ¼ fw ;
f 0 ¼ 1;
h ¼ 1;
f 0 ¼ 0;
; h ¼ 0;
/ ¼ 1;
/ ¼ 0 at g ¼ 0; as g ! 1:
ð18Þ ð19Þ
Here c ¼ Grx =Re2x is the local buoyancy parameter, Grx = gb (Tw T1)(2x)3/m2 is the local Grashof number, Rex = 2xU0/m is the local Reynolds number, M ¼ rB20 =ðqU 0 Þ is the Hartmann number, R ¼ kg v=ð4r1 T 31 Þ is the conduction-radiation parameter, Pr = m qcp/ kg is the Prandtl number, Ec ¼ U 20 =ðcp ðT w T 1 ÞÞ is the Eckert number, d = Q0/ (qcpU0) is the internal heat source/sink, Sc = m/D is the Schmidt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number and fw ¼ v w ðxÞ 2x=mU 0 is the permeability of the porous surface with positive value indicates suction while negative value indicates injection. The physical quantities of interest are the local skin friction coefficient, the wall heat transfer coefficient (or the local Nusselt number) and the wall deposition flux (or the local Stanton number) which are defined as
Cfx ¼
sw qw x J ; Nux ¼ ; Stx ¼ s ; kg ðT w T 1 Þ U0 C1 qU 20 =2
ð20Þ
respectively where the skin friction sw, the heat transfer from the wall qw and the deposition flux from wall Js are given by
! @u @T 4r1 @T 4 sw ¼ l ; qw ¼ kg ; @y y¼0 @y y¼0 3v @y y¼0 @C J s ¼ D : @y y¼0
ð21Þ
Hence the expressions for the skin friction, the rate of heat transfer and the deposition flux for general flow over a radiative surface are written as
Cfx Rex1=2 ¼ 2f 00 ð0Þ;
Nux Rex1=2
Stx ScRe1=2 ¼ /0 ð0Þ: x
3R 1 ¼ h0 ð0Þ; 3R þ 4 2
ð22Þ
3. Method of solution The system of ordinary differential Eqs. (15)–(17) subject to the boundary conditions (18), (19) are solved numerically using shooting method [18,24]. A step size of Dg = 0.01 was selected to be satisfactory for a convergence criterion of 106 in all cases. The value of g1 was found to each iteration loop by the statement g1 = g1 + Dg. The maximum value of g1 to each group of param-
eters c, Pr, M, R, a, Ec, Sc, s, fw and d determined when the value of the unknown boundary conditions at g = 0 not change to successful loop with error less than 106. Our computations show that step size Dg = 0.01 is sufficient to getting a convergent solution. The value of c = 10 which corresponds to pure free convection and the value of Pr = 0.7 which represents air at 293 K and 1 atmosphere of pressure are applied in all calculations done in this paper. The default values of other parameters are M = 0.5, R = 1, Ec = 0.1 and Sc = 0.6. On the other hand, the values of the parameters a, fw, s and d will be specified later in the discussion.
4. Results and discussion The skin friction jf00 (0)j value and the local Stanton numbers obtained in this study are compared with the results of other researchers in Tables 1 and 2. In order to show the accuracy of the computations done, we plotted some figures of the residual error for the Eqs. (15)–(17). Fig. 1 shows the residual errors for the suction case while Fig. 2 is for the injection case. For simplicity, only a few set of data considered for showing the residual errors although the same figures for other parameters’ values can be obtained. The values of the skin friction f00 (0), the wall heat transfer h0 (0) and the wall deposition flux /0 (0) with the variations of the suction/injection parameter fw, the heat source/sink d and the thermophoretic parameter s are listed in Tables 3–5 respectively. Based on Table 3, the skin friction is higher when there is an injection fw = 0.5 on the inclined permeable surface. However, the heat transfer from the surface and the deposition flux are greater for the suction case fw = 0.5. According to mass conservation theory in three dimensions, change of force in the flow direction causes change of forces in other directions including in the direction of normal to the surface. When fw = 0.5, suction caused change of force from the bottom of the inclined plate. This downward force combined with force in the direction of the flow is greater than the one in the opposite direction of the flow. Hence the skin friction decreases. With greater forces in the direction of the flow and downward the surface, greater amount of mass can be deposed and greater amount of heat can be transferred to the surroundings. Thus h0 (0) and /0 (0) increase when fw = 0.5. The influence of heat source/sink parameter d when suction parameter fw = 0.5 is observed from Table 4. Here d = 1 represents heat sink, d = 0 is without heat source/sink parameter whereas d = 1 represents heat source. With heat source in the model, greater amount of heat is generated by the flow which reflects greater amount of heat can be transferred from the wall. However, when there is a heat sink in the flow, more heat is absorbed and more fluid concentration is deposed on the wall which increases the skin friction. Table 5 shows the evidence that the thermophoretic parameter s does not has any impact in the velocity and temperature profiles. This is due to the unique appearance of the parameter in the Eq. (17). It can be assumed that higher value of the thermophoretic parameter contributes higher deposition flux on the surface. This phenomenon is caused by the characteristics of thermophoresis where small or nano particles tend to be driven away from hot surface to a cooler one. Since the wall temperature Tw is higher than the ambient temperature T1, fluid mass is deposed away from the wall to the surroundings.
Table 1 Comparison of jf00 (0)j value for c = M = fw = 0. Cortell [25]
Javed et al. [26]
Yazdi et al. [27]
Present work
0.6275
0.6275
0.6275
0.6276
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a
Table 2 Comparison of local Stanton number for Sc = 1000, Pr = 0.7, a = 90°, s = 1 and c = M = 1/R = Ec = d = 0. fw
Alam et al. [20]
Present work
1.000 0.500 0.000 0.004 0.005 0.250
0.8691 0.5359 0.2076 0.2070 0.2065 0.0349
0.9532 0.5997 0.2461 0.2433 0.2426 0.0693
a
b
c
b
c
Fig. 2. Residual error: c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1, fw = 0.5, d = 1 for (15)–(17).
Table 3 Values of f00 (0), h0 (0) and /0 (0) for c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1 and d = 1. fw
f00 (0)
h0 (0)
/0 (0)
0.5 0 0.5
3.39924 3.40786 3.31519
0.739454 0.823000 0.912593
0.643325 0.799261 0.972771
Table 4 Values of f00 (0), h0 (0) and /0 (0) for c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1 and fw = 0.5.
Fig. 1. Residual error: c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1, fw = 0.5, d = 1 for (15)–(17).
d
f00 (0)
h0 (0)
/0 (0)
1 0 1
6.51396 4.28480 3.31519
0.47729 0.468335 0.912593
1.163990 1.050370 0.972771
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Table 5 Values of f00 (0), h0 (0) and /0 (0) for c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, d = 1 and fw = 0.5.
s
f00 (0)
h0 (0)
/0 (0)
0.1 0.5 1.0
3.31519 3.31519 3.31519
0.912593 0.912593 0.912593
0.924569 0.946524 0.972771
a
a
b
b
c
c Fig. 4. (a) Velocity, (b) temperature and (c) concentration profiles for both suction and injection cases: c = 10, Pr = 0.7, M = 0.5, R = 1, d = 1, Ec = 0.1, Sc = 0.6, a = p/6 and s = 0.1, 0.5, 1.
Fig. 3. (a) Velocity, (b) temperature and (c) concentration profiles for both suction and injection cases: c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1 and d = 1, 0, 1.
The effects of suction/injection parameter fw when heat source/ sink d, the inclined plate angle a and the thermophoretic parameter s are varied can be observed from Figs. 3–5 respectively. Lower skin friction indicates lower flow velocity. Since the skin friction value is
smaller for suction compared to injection case as enlisted in Table 3, then the flow velocity with suction (fw = 0.5) is less than the velocity with injection (fw = 0.5) as depicted in Figs. 3(a), 4(a) and 5(a). Furthermore, more heat is transferred to surroundings when suction is imposed on the surface. As a result, the flow becomes cooler than the one with injection. These consistent occurrences are revealed in Figs. 3(b), 4(b) and 5(b), respectively. Oppositely, as the wall deposition flux /0 (0) is greater for suction than injection as indicated in the Table 3, the concentration profile for suction are always greater than injection as shown in Fig. 3(c), 4(c) and 5(c). The profiles of velocity, temperature and concentration in Fig. 3 are consistently lower for heat source impact when d = 1 compared to heat sink when d = 1. These correspond to smaller values of the skin friction f00 (0), the heat transfer h0 (0) and the deposition flux / 0 (0) as given in the Table 4. As discussed previously in the Table 5, the thermophoretic parameter s can only affect the wall deposition
N.F.M. Noor et al. / International Journal of Heat and Mass Transfer 55 (2012) 2122–2128
a
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values of skin friction, wall heat flux and wall deposition flux with variation of the suction, thermophoretic and heat source parameters are tabulated. Furthermore the effects of these parameters to the velocity, temperature and concentration profiles are presented graphically. The distributions of velocity and temperature for the flow with suction are smaller than the flow with injection. Opposite occurrence is observed for the concentration distribution. On the other hand, with the effect of heat source, the velocity, temperature and concentration profiles for the flow are lesser compared to the flow with heat sink. Finally, higher value of thermophoretic parameter contributes to lower fluid flow concentration on the inclined surface. Acknowledgments
b
We gratefully acknowledge the comments of the referee which lead to an improvement in the paper. Financial support received under the grant FRGS/2/2010/SG/UNISEL/03/1 is greatly appreciated. References
c
Fig. 5. (a) Velocity, (b) temperature and (c) concentration profiles for both suction and injection cases: c = 10, Pr = 0.7, M = 0.5, R = 1, d = 1, Ec = 0.1, Sc = 0.6, s = 1 and a = 0, p/9, p/6.
flux. From Fig. 4(c), the distribution of fluid concentration obviously declines as the value of s raises. This is influenced by the nature of thermophoresis where more fluid particles are driven away from the hot inclined surface to cooler surroundings. Due to the first appearance of a in the Eq. (15), the inclined plate angle contributes greater impact to the velocity profile than the temperature and concentration profiles. Based on Fig. 5, it can be concluded that higher angle of the inclined surface denotes greater gravity force acting on the surface. Thus the velocity, the temperature and the concentration of the flow decrease as the angle a increases from 0 to p/6. 5. Concluding remarks The effects of thermophoretic and heat source/sink parameters for both suction and injection cases on MHD flow over an inclined radiate isothermal permeable surface have been discussed. The
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[24] P.R. Nachtsheim, P. Swigert, Satisfaction of the asymptotic boundary conditions in numerical solution of the system of non-linear equations of boundary layer type, NASA TND-3004; 1965. [25] R. Cortell, Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math. Comput. 184 (2007) 864–873. [26] T. Javed, Z. Abbas, M. Sajid, N. Ali, Heat transfer analysis for a hydromagnetic viscous fluid over a non-linear shrinking sheet, Int. J. Heat Mass Transfer 54 (2011) 2034–2042. [27] M.H. Yazdi, S. Abdullah, I. Hashim, K. Sopian, Slip MHD liquid flow and heat transfer over non-linear permeable stretching surface with chemical reaction, Int. J. Heat Mass Transfer 54 (2011) 3214–3225.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat and mass transfer of thermophoretic MHD flow over an inclined radiate isothermal permeable surface in the presence of heat source/sink N.F.M. Noor a, S. Abbasbandy b, I. Hashim c,d,⇑ a
Department of Mechatronic, Faculty of Engineering, Universiti Selangor, 45600 Berjuntai Bestari, Selangor, Malaysia Department of Mathematics, Imam Khomeini International University, Ghazvin 34149-16818, Iran c School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia d Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia b
a r t i c l e
i n f o
Article history: Received 9 June 2011 Accepted 20 November 2011 Available online 5 January 2012 Keywords: Boundary layer MHD Thermophoresis Heat source
a b s t r a c t In this paper, a free convection thermophoretic hydromagnetic flow over a radiate isothermal inclined plate with heat source/sink effect is considered. The shooting method is employed to yield the numerical solutions for the model. The effects of thermophoretic parameter and internal heat generation/absorption for both suction and injection cases are discussed and presented graphically. The values of skin friction, wall heat flux and wall deposition flux are also tabulated with the variation of thermophoresis, heat source/sink and suction/injection parameters. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Thermophoresis has been the subject of abundant studies for many years [1–3]. It describes the migration of suspended small micron sized particles in a non-isothermal gas to the direction with decreasing thermal gradient. The velocity acquired by the particle is known as thermophoretic velocity while the force experienced by the suspended particles due to the temperature differences is known as thermophoretic force [4,5]. Being a potential mechanism for particles collection on cool surfaces, thermophoresis has numerous practical applications such as in aerosol particles sampling, air cleaning, scale formation on surfaces of heat exchangers, modified chemical vapor deposition, particulate material deposition on turbine blades, microelectronics manufacturing, nuclear reactor safety and removal of soot particles for combustion exhaust gas systems. One of the earliest studies on the role of thermophoresis in laminar flow over a horizontal plate with analysis on cold and hot plate conditions is given by Goren [6]. Some recent studies on the effect of thermophoresis in laminar flow are reported in [7–10]. The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal-working processes. The study of MHD flow and heat transfer are deemed as of great interest due to the effect of ⇑ Corresponding author at: School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia. Tel.: +60 3 8921 5758; fax: +60 3 8925 4519. E-mail address: [email protected] (I. Hashim). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.12.015
magnetic field on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. Some of the engineering applications are in MHD generators, plasma studies, nuclear reactor, geothermal energy extractions, purifications of metal from non-metal enclosures, polymer technology and metallurgy. Thermophoresis is also a key mechanism of study in semi-conductor technology especially controlled high-quality wafer production as well as MHD energy generation system operations. Since various industrial heat transfer process involved both the hydromagnetic flows and thermophoresis such as in MHD energy systems, many numerical studies on magnetohydrodynamic heat and mass transfer have been reported with buoyancy, Joule heating effects and heat source/sink parameters. The hydromagnetic free convection boundary layers in non-Darcian regimes via Keller-Box method was considered by Bég and Takhar [11]. Other hydromagnetic flow studies by Bég et al. include [12] on non-Newtonian wedge flow hydromagnetics and [13] on dissipative hydromagnetic boundary layers with wall suction/injection in porous media. The effects of suction and injection along a flat plate for free convection flow have attract interest of many researchers [9,14] due to the double impacts projected with respect to heat transfer. The temperature profile is influenced by the changed velocity field in the boundary layer leading to a change in the heat conduction at the wall. On the other hand, convective heat transfer occurs at the wall along with heat conduction for various values of suction/injection parameters. A computational analysis of heat generation/absorption and thermophoresis on hydromagnetic flow over a flat surface was presented by Chamkha and Camille [15].
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Using a perturbation approach, the transient oscillatory MHD convection past a flat plate adjacent to a porous medium with heat generation effects was solved in [16] and transient MHD natural convection with viscous heating was considered by Zueco [17]. Recently the effects of variable suction and thermophoresis on steady MHD flow over a permeable inclined plate was analyzed by Alam et al. [18] while the heat and mass transfer of thermophoretic hydromagnetic flow with lateral mass flux, heat source, Ohmic heating and thermal conductivity effects using the network simulation method was studied by Zueco et al. [19]. To our best knowledge, study on MHD flow over an inclined plate with thermophoresis and heat source has never been considered before. Therefore, in this paper, we will extend the previous work on a steady MHD flow with thermophoresis over a permeable radiate inclined plate by Alam et al. [20] to include the heat source/sink parameter. The present objective is to investigate the effects of thermophoretic parameter with the impact of heat source/sink in the model for both suction and injection cases. The values of skin friction, wall heat transfer and wall deposition flux are also tabulated. 2. Problem formulation 2.1. Governing equations and boundary conditions Consider a two-dimensional steady laminar flow of an incompressible electrically conducting fluid over a continuously moving semi-infinite inclined permeable flat plate with an acute angle a to the vertical. With the x-axis measured along the plate, a magnetic field B(x) is applied in the y-direction that is normal to the flow direction. Suction or injection is imposed on the permeable plate. The temperature of the surface is held uniform at Tw which is higher than the ambient temperature T1. The species concentration at the surface is maintained uniform at Cw = 0 while the ambient fluid concentration is assumed to be C1. The presence of uniform internal heat source/sink and thermophoresis are considered to study the variation of velocity, heat transfer and concentration deposition on the inclined surface. The following assumptions are made: (a) the concentration flux of particles is sufficiently small so that the main stream velocity and temperature fields are not affected by the thermophysical processes experienced by the relatively small number of particles, (b) the temperature gradient in the y-direction is larger than that in the x-direction due to the boundary layer behaviour. Hence only the thermophoretic velocity component normal to the surface is of importance, (c) the fluid has constant kinematic viscosity and thermal diffusivity and that the Boussinesq approximation may be adopted for steady laminar flow, (d) the particle diffusivity is assumed to be constant and the particles concentration is sufficiently dilute to assume the particle coagulation in the boundary layer is negligible, (e) the magnetic Reynolds is assumed to be sufficiently small so that the induced magnetic field is negligible in comparison to the applied magnetic field, and (f) the fluid is considered to be gray; absorbing-emitting radiation but non-scattering medium and the Rosseland approximation is used to describe the radioactive heat flux in the x-direction which is considered negligible in comparison to the y-direction. Under the above assumptions, the governing equations for this problem are written as @u @ v þ ¼ 0; @x @y
ð1Þ
u
@u @u @2u rB2 ðxÞ þv ¼ m 2 þ gbðT T 1 Þ cos a u; @x @y @y q
u
@T @T kg @ 2 T 1 @qr QðxÞ l @u 2 rB2 ðxÞ 2 þv ¼ ðTT 1 Þþ þ u ; ð3Þ þ 2 @x @y qcp @y qcp @y qcp qcp @y qc p
ð2Þ
u
@C @C @ 2 C @ðV T CÞ þv ¼D 2 ; @x @y @y @y
ð4Þ
where u and v are the velocity components in the x, y directions, m is the kinematic viscosity, g is the acceleration due to gravity, b is the volumetric coefficient of thermal expansion, T, Tw and T1 are the temperatures of thermal boundary layer fluid, the inclined plate and the free respectively, r is the electrical conductivity, pffiffiffiffiffistream ffi BðxÞ ¼ B0 = 2x is the magnetic induction, kg is the fluid thermal conductivity, q is the fluid density, cp is the specific heat at constant pressure, qr is the radiative heat flux in the y-direction, Q(x) = Q0/x is the internal heating, l is the dynamic viscosity, D is the molecular diffusivity of the species concentration and VT is the thermophoretic velocity. The boundary conditions for the model are as follows [20]:
v ¼ v w ðxÞ;
u ¼ U0 ; u ¼ 0;
T ¼ T1;
T ¼ Tw;
C ¼ C1;
C ¼ C w ¼ 0 at y ¼ 0;
as y ! 1;
ð5Þ ð6Þ
where U0 is the uniform plate velocity and vw(x) represents fluid suction/injection on the porous surface. The transpiration function variable vw(x) of order x1/2 is considered [20]. The radiative heat flux qr under Rosseland approximation has the form
qr ¼
4r1 @T 4 ; 3v @y
ð7Þ
where r1 is the Stefan–Boltzmann constant and v is the mean absorption coefficient. The temperature differences within the flow are assumed to be sufficiently small such that T4 may be expressed as a linear function of temperature. Expanding T4 using Taylor series and neglecting higher order terms yields
T 4 ffi 4T 31 T 3T 41 :
ð8Þ
Using Eqs. (7) and (8), we have
u
@T @T kg @ 2 T 16r1 T 31 @ 2 T Q ðxÞ þ þ ðT T 1 Þ þv ¼ @x @y qcp @y2 3qcp v @y2 qc p l @u 2 rB2 ðxÞ 2 þ u : þ qcp @y qcp
ð9Þ
The second, third, fourth and fifth terms on the RHS of the Eq. (9) denote the thermal radiation, internal heating, viscous and magnetic heating terms respectively. Now the thermophoretic velocity VT which appears in the Eq. (4) can be written as [22]:
V T ¼ km
rT Tr
¼
km @T ; T r @y
ð10Þ
where Tr is a reference temperature and k is the termophoretic coefficient with range of value from 0.2 to 1.2 as indicated by Batchelor and Chen [21] and is defined from the theory of [22] by
k¼
2C s ðkg =kp þ C t KnÞ½C 1 þ C 2 eC3 =Kn ; ð1 þ 3C m KnÞð1 þ 2kg =kp þ 2C t KnÞ
ð11Þ
where C1, C2, C3, Cm, Cs, Ct are constants, kg and kp are the thermal conductivities of the fluid and diffused particles respectively and Kn is the Knudsen number. A thermophoretic parameter s can be defined [7,23] as follows:
s¼
kðT w T 1 Þ : Tr
ð12Þ
Typical values of s are 0.01, 0.05 and 0.1 corresponding to approximate values of k(Tw T1) equal to 3 K, 15 K and 30 K for a reference temperature of Tr = 300 K.
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2.2. Similarity transformation The governing Eqs. (2)–(4) can be transformed to a set of nonlinear ordinary differential equations by introducing the following non-dimensional variables [20]:
g¼y
rffiffiffiffiffiffiffiffi U0 ; 2mx
w¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2U 0 mxf ðgÞ;
hðgÞ ¼
T T1 ; Tw T1
/ðgÞ ¼
C ; C1 ð13Þ
where w is the stream function that satisfies the continuity Eq. (1) with
u¼
@w ¼ U 0 f 0 ðgÞ and @y
v¼
rffiffiffiffiffiffiffiffiffi @w U0 m ½f ðgÞ gf 0 ðgÞ: ¼ @x 2x
ð14Þ
Using (13) and (14), the following similarity equations with the corresponding boundary conditions are obtained: 00
0
f 000 þ ff þ ch cos a Mf ¼ 0; h i ð3R þ 4Þh00 þ 3RPr f h0 þ Ecðf 00 Þ2 þ EcMðf 0 Þ2 þ 2dh ¼ 0; 00
0
0
00
/ þ Scðf sh Þ/ Scsh / ¼ 0;
ð15Þ ð16Þ ð17Þ
subject to
f ¼ fw ;
f 0 ¼ 1;
h ¼ 1;
f 0 ¼ 0;
; h ¼ 0;
/ ¼ 1;
/ ¼ 0 at g ¼ 0; as g ! 1:
ð18Þ ð19Þ
Here c ¼ Grx =Re2x is the local buoyancy parameter, Grx = gb (Tw T1)(2x)3/m2 is the local Grashof number, Rex = 2xU0/m is the local Reynolds number, M ¼ rB20 =ðqU 0 Þ is the Hartmann number, R ¼ kg v=ð4r1 T 31 Þ is the conduction-radiation parameter, Pr = m qcp/ kg is the Prandtl number, Ec ¼ U 20 =ðcp ðT w T 1 ÞÞ is the Eckert number, d = Q0/ (qcpU0) is the internal heat source/sink, Sc = m/D is the Schmidt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number and fw ¼ v w ðxÞ 2x=mU 0 is the permeability of the porous surface with positive value indicates suction while negative value indicates injection. The physical quantities of interest are the local skin friction coefficient, the wall heat transfer coefficient (or the local Nusselt number) and the wall deposition flux (or the local Stanton number) which are defined as
Cfx ¼
sw qw x J ; Nux ¼ ; Stx ¼ s ; kg ðT w T 1 Þ U0 C1 qU 20 =2
ð20Þ
respectively where the skin friction sw, the heat transfer from the wall qw and the deposition flux from wall Js are given by
! @u @T 4r1 @T 4 sw ¼ l ; qw ¼ kg ; @y y¼0 @y y¼0 3v @y y¼0 @C J s ¼ D : @y y¼0
ð21Þ
Hence the expressions for the skin friction, the rate of heat transfer and the deposition flux for general flow over a radiative surface are written as
Cfx Rex1=2 ¼ 2f 00 ð0Þ;
Nux Rex1=2
Stx ScRe1=2 ¼ /0 ð0Þ: x
3R 1 ¼ h0 ð0Þ; 3R þ 4 2
ð22Þ
3. Method of solution The system of ordinary differential Eqs. (15)–(17) subject to the boundary conditions (18), (19) are solved numerically using shooting method [18,24]. A step size of Dg = 0.01 was selected to be satisfactory for a convergence criterion of 106 in all cases. The value of g1 was found to each iteration loop by the statement g1 = g1 + Dg. The maximum value of g1 to each group of param-
eters c, Pr, M, R, a, Ec, Sc, s, fw and d determined when the value of the unknown boundary conditions at g = 0 not change to successful loop with error less than 106. Our computations show that step size Dg = 0.01 is sufficient to getting a convergent solution. The value of c = 10 which corresponds to pure free convection and the value of Pr = 0.7 which represents air at 293 K and 1 atmosphere of pressure are applied in all calculations done in this paper. The default values of other parameters are M = 0.5, R = 1, Ec = 0.1 and Sc = 0.6. On the other hand, the values of the parameters a, fw, s and d will be specified later in the discussion.
4. Results and discussion The skin friction jf00 (0)j value and the local Stanton numbers obtained in this study are compared with the results of other researchers in Tables 1 and 2. In order to show the accuracy of the computations done, we plotted some figures of the residual error for the Eqs. (15)–(17). Fig. 1 shows the residual errors for the suction case while Fig. 2 is for the injection case. For simplicity, only a few set of data considered for showing the residual errors although the same figures for other parameters’ values can be obtained. The values of the skin friction f00 (0), the wall heat transfer h0 (0) and the wall deposition flux /0 (0) with the variations of the suction/injection parameter fw, the heat source/sink d and the thermophoretic parameter s are listed in Tables 3–5 respectively. Based on Table 3, the skin friction is higher when there is an injection fw = 0.5 on the inclined permeable surface. However, the heat transfer from the surface and the deposition flux are greater for the suction case fw = 0.5. According to mass conservation theory in three dimensions, change of force in the flow direction causes change of forces in other directions including in the direction of normal to the surface. When fw = 0.5, suction caused change of force from the bottom of the inclined plate. This downward force combined with force in the direction of the flow is greater than the one in the opposite direction of the flow. Hence the skin friction decreases. With greater forces in the direction of the flow and downward the surface, greater amount of mass can be deposed and greater amount of heat can be transferred to the surroundings. Thus h0 (0) and /0 (0) increase when fw = 0.5. The influence of heat source/sink parameter d when suction parameter fw = 0.5 is observed from Table 4. Here d = 1 represents heat sink, d = 0 is without heat source/sink parameter whereas d = 1 represents heat source. With heat source in the model, greater amount of heat is generated by the flow which reflects greater amount of heat can be transferred from the wall. However, when there is a heat sink in the flow, more heat is absorbed and more fluid concentration is deposed on the wall which increases the skin friction. Table 5 shows the evidence that the thermophoretic parameter s does not has any impact in the velocity and temperature profiles. This is due to the unique appearance of the parameter in the Eq. (17). It can be assumed that higher value of the thermophoretic parameter contributes higher deposition flux on the surface. This phenomenon is caused by the characteristics of thermophoresis where small or nano particles tend to be driven away from hot surface to a cooler one. Since the wall temperature Tw is higher than the ambient temperature T1, fluid mass is deposed away from the wall to the surroundings.
Table 1 Comparison of jf00 (0)j value for c = M = fw = 0. Cortell [25]
Javed et al. [26]
Yazdi et al. [27]
Present work
0.6275
0.6275
0.6275
0.6276
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a
Table 2 Comparison of local Stanton number for Sc = 1000, Pr = 0.7, a = 90°, s = 1 and c = M = 1/R = Ec = d = 0. fw
Alam et al. [20]
Present work
1.000 0.500 0.000 0.004 0.005 0.250
0.8691 0.5359 0.2076 0.2070 0.2065 0.0349
0.9532 0.5997 0.2461 0.2433 0.2426 0.0693
a
b
c
b
c
Fig. 2. Residual error: c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1, fw = 0.5, d = 1 for (15)–(17).
Table 3 Values of f00 (0), h0 (0) and /0 (0) for c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1 and d = 1. fw
f00 (0)
h0 (0)
/0 (0)
0.5 0 0.5
3.39924 3.40786 3.31519
0.739454 0.823000 0.912593
0.643325 0.799261 0.972771
Table 4 Values of f00 (0), h0 (0) and /0 (0) for c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1 and fw = 0.5.
Fig. 1. Residual error: c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1, fw = 0.5, d = 1 for (15)–(17).
d
f00 (0)
h0 (0)
/0 (0)
1 0 1
6.51396 4.28480 3.31519
0.47729 0.468335 0.912593
1.163990 1.050370 0.972771
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Table 5 Values of f00 (0), h0 (0) and /0 (0) for c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, d = 1 and fw = 0.5.
s
f00 (0)
h0 (0)
/0 (0)
0.1 0.5 1.0
3.31519 3.31519 3.31519
0.912593 0.912593 0.912593
0.924569 0.946524 0.972771
a
a
b
b
c
c Fig. 4. (a) Velocity, (b) temperature and (c) concentration profiles for both suction and injection cases: c = 10, Pr = 0.7, M = 0.5, R = 1, d = 1, Ec = 0.1, Sc = 0.6, a = p/6 and s = 0.1, 0.5, 1.
Fig. 3. (a) Velocity, (b) temperature and (c) concentration profiles for both suction and injection cases: c = 10, Pr = 0.7, M = 0.5, R = 1, a = p/6, Ec = 0.1, Sc = 0.6, s = 1 and d = 1, 0, 1.
The effects of suction/injection parameter fw when heat source/ sink d, the inclined plate angle a and the thermophoretic parameter s are varied can be observed from Figs. 3–5 respectively. Lower skin friction indicates lower flow velocity. Since the skin friction value is
smaller for suction compared to injection case as enlisted in Table 3, then the flow velocity with suction (fw = 0.5) is less than the velocity with injection (fw = 0.5) as depicted in Figs. 3(a), 4(a) and 5(a). Furthermore, more heat is transferred to surroundings when suction is imposed on the surface. As a result, the flow becomes cooler than the one with injection. These consistent occurrences are revealed in Figs. 3(b), 4(b) and 5(b), respectively. Oppositely, as the wall deposition flux /0 (0) is greater for suction than injection as indicated in the Table 3, the concentration profile for suction are always greater than injection as shown in Fig. 3(c), 4(c) and 5(c). The profiles of velocity, temperature and concentration in Fig. 3 are consistently lower for heat source impact when d = 1 compared to heat sink when d = 1. These correspond to smaller values of the skin friction f00 (0), the heat transfer h0 (0) and the deposition flux / 0 (0) as given in the Table 4. As discussed previously in the Table 5, the thermophoretic parameter s can only affect the wall deposition
N.F.M. Noor et al. / International Journal of Heat and Mass Transfer 55 (2012) 2122–2128
a
2127
values of skin friction, wall heat flux and wall deposition flux with variation of the suction, thermophoretic and heat source parameters are tabulated. Furthermore the effects of these parameters to the velocity, temperature and concentration profiles are presented graphically. The distributions of velocity and temperature for the flow with suction are smaller than the flow with injection. Opposite occurrence is observed for the concentration distribution. On the other hand, with the effect of heat source, the velocity, temperature and concentration profiles for the flow are lesser compared to the flow with heat sink. Finally, higher value of thermophoretic parameter contributes to lower fluid flow concentration on the inclined surface. Acknowledgments
b
We gratefully acknowledge the comments of the referee which lead to an improvement in the paper. Financial support received under the grant FRGS/2/2010/SG/UNISEL/03/1 is greatly appreciated. References
c
Fig. 5. (a) Velocity, (b) temperature and (c) concentration profiles for both suction and injection cases: c = 10, Pr = 0.7, M = 0.5, R = 1, d = 1, Ec = 0.1, Sc = 0.6, s = 1 and a = 0, p/9, p/6.
flux. From Fig. 4(c), the distribution of fluid concentration obviously declines as the value of s raises. This is influenced by the nature of thermophoresis where more fluid particles are driven away from the hot inclined surface to cooler surroundings. Due to the first appearance of a in the Eq. (15), the inclined plate angle contributes greater impact to the velocity profile than the temperature and concentration profiles. Based on Fig. 5, it can be concluded that higher angle of the inclined surface denotes greater gravity force acting on the surface. Thus the velocity, the temperature and the concentration of the flow decrease as the angle a increases from 0 to p/6. 5. Concluding remarks The effects of thermophoretic and heat source/sink parameters for both suction and injection cases on MHD flow over an inclined radiate isothermal permeable surface have been discussed. The
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