- Email: [email protected]
Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions
Alexandria Engineering Journal (2015) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions Padam Singh *, Manoj Kumar Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India Received 28 January 2014; revised 18 March 2015; accepted 11 April 2015
KEYWORDS Magnetohydrodynamic; Alumina water nanofluid; Boundary layer slip; Volume fraction; Concentration
Abstract Present paper deals with analysis of mass transfer in two dimensional magnetohydrodynamic (MHD) slip flow of an incompressible, electrically conducting, viscous and steady flow of alumina water nanofluid in the presence of magnetic field over a flat plate. The governing equations representing flow are transformed into a set of simultaneous ordinary differential equations by using appropriate similarity transformation. The set of equations thus obtained has been solved using adaptive Runge–Kutta method with shooting technique. The effects of solutal Grashof number and Schmidt number on velocity distribution profile, concentration profile, shear stress and concentration gradient have depicted graphically and analyzed. It is found that the concentration increases and concentration gradient decreases with increasing value of solutal Grashof number. Also, the concentration decreases with an increase in Schmidt number. ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Xuan [1] studied mass transfer in binary nanofluids and found that the nanoparticles improved the mass transfer inside binary nanofluids. Bhattacharyya [2] studied the boundary layer flow over a porous flat plate with diffusion of chemically reactive species undergoing first order reaction and subject to suction/injection. The author found that for increasing of suction at the plate both velocity and concentration boundary layer thicknesses decrease and for increasing injection both thicknesses increase. The rate of solute transfer from the plate increases with reaction rate parameter and power-law * Corresponding author. E-mail address: [email protected] (P. Singh). Peer review under responsibility of Faculty of Engineering, Alexandria University.
exponent. Sridhara and Satapathy [3] made review to summarize recent developments in research on the stability of nanofluids, enhancement of thermal conductivity, viscosity and heat transfer characteristics of alumina based nanofluids. Uddin et al. [4] investigated the steady laminar combined convective flow with heat and mass transfer of a Newtonian viscous incompressible fluid over a permeable flat plate with linear hydrodynamic and thermal slips. The effects of various parameters were investigated and presented graphically. Pang et al. [5] studied the mass transfer enhancement by binary nanofluids for bubble absorption process. The results were shown that the mass transfer enhancement with coolant is more than compared to without coolant. Kumar et al. [6] studied MHD flow and heat transfer along a porous flat plate in the presence of mass transfer. The results were obtained for velocity, temperature and concentration profiles of the flow field and shown graphically. Rout et al. [7] investigated the
http://dx.doi.org/10.1016/j.aej.2015.04.005 1110-0168 ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
2
P. Singh, M. Kumar
Nomenclature B Gc g L M Sc u1 u, v x, y vsf ðbs Þnf
magnetic field solutal Grashof number gravitational acceleration reference length magnetic parameter Schmidt number uniform velocity at infinity velocity components in x and y direction coordinate axes velocity slip factor solutal expansion coefficient
influence of chemical reaction and the combined effects of internal heat generation and a convective boundary condition on the laminar boundary layer MHD heat and mass transfer flow over a moving vertical flat plate. The discussion was focused on the physical interpretation of the results as well as their comparison with past studies. Steady laminar natural convection flow over a semi-infinite moving vertical plate with internal heat generation and convective surface boundary condition in the presence of thermal radiation, viscous dissipation and chemical reaction analyzed by Ibrahim and Reddy [8]. Authors utilized similarity variable to transform the governing nonlinear partial differential equations into a system of ordinary differential equations. Gangadhar et al. [9] studied the hydromagnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field, chemical reaction and a convective heat exchange at the surface using similarity transformation. The present investigation deals with analysis of mass transfer in MHD slip flow of an incompressible, steady, electrically conducting and viscous alumina-water nanofluid in the presence of magnetic field over a flat plate. 2. Mathematical description The physical model is given here along with flow configuration and coordinate system (Fig. 1). The system deals with two dimensional MHD boundary layer slip flow of water based nanofluid. The continuity, momentum and concentration equations representing flow are following: @u @v þ ¼0 @x @y
velocity slip parameter density of water nanofluid density density of alumina particles electrical conductivity of nanofluid dynamic viscosity of nanofluid volume fraction of nanoparticles stream function kinematic viscosity of nanofluid sphericity
c qf qnf qs rnf lnf ; w tnf x
ð1Þ
u
@u @u @ 2 u rnf B þv ¼ tnf 2 þ ðu1 uÞ þ gðbs Þnf ðC C1 Þ @x @y @y qnf
ð2Þ
u
@C @C @2C þv ¼ Dnf 2 @x @y @y
ð3Þ
With the boundary conditions: u ¼ vsf
@u ; v ¼ 0 and C ¼ CW ¼ C1 þ C0 xk at y ¼ 0 @y ðon the wall of the plateÞ
u ! u1 and C ! C1 when y ! 1 ðfar from the wallÞ qffiffiffiffiffiffi Here, vsf ¼ v0 ut1nfx is velocity slip factor with initial value v0 (see Tables 1 and 2). 3. Method for solution To solve the governing Eqs. (2) and (3) with the boundary conditions (4), following similarity transformation has been introduced: rffiffiffiffiffiffiffiffiffi y xu1 C C1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; vðgÞ ¼ w ¼ u1 tnf x fðgÞ; g ¼ ; x tnf Cw C1 M¼
rxB2 u1 qnf
Table 1
and
Nanofluid properties. lnf ¼ lf =ð1 ;Þ2:5 qnf ¼ ð1 ;Þqf þ ;qs Dnf ¼ ð1 ;ÞDf tnf ¼ lnf =qnf
Dynamic viscosity [10] Density [11] Mass diffusivity [12] Kinematic viscosity [13]
y C
v
ð4Þ
u
Table 2 The physical properties of alumina and base fluid water at 20 C.
Cw B0
Figure 1
u
x Water Alumina
Density (kg/m3)
Dynamic viscosity (N s/m2)
Mass diffusivity (m2/sec)
1000.52 3970.0
0.001002 –
0.0000242 –
Physical model of the flow.
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
Mass transfer in MHD flow of alumina water nanofluid Gc ¼
3
gðbs Þnf ðCw C1 Þx3 xu1 B0 tnf ; Re ¼ where B ¼ pffiffiffi ; Sc ¼ t2nf tnf Dnf x
where B0 is constant and the stream function w satisfies Eq. (1) and v ¼ @w . with u ¼ @w @y @x Eqs. (2) and (3) reduce to the non-linear differential equations using above transformation as follows: 1 f 000 ðgÞ þ fðgÞf 00 ðgÞ þ Mð1 f 0 ðgÞÞ þ ðGc =Re2 ÞvðgÞ ¼ 0 2
ð5Þ
1 v 00 ðgÞ þ ScfðgÞv 0 ðgÞ ¼ 0 2
ð6Þ
and the boundary conditions (4) reduce as follows: f 0 ð0Þ ¼ cf 00 ð0Þ and vð0Þ ¼ 1
fð0Þ ¼ 0;
f 0 ðgÞ ! 1 and vðgÞ ! 0 when g ! 1 Here c ¼
V0 u1 tnf
ð7Þ
is the velocity slip parameter with initial value
V0 . To solve the non-linear differential Eqs. (5) and (6) subject to the boundary conditions (7) adaptive Runge–Kutta method with shooting technique has been applied. This method is based on the discretization of the problem domain and the calculation of unknown boundary conditions. The domain of the problem is discretized and the boundary conditions for g ¼ 1 are replaced by f 0 ðgmax Þ ¼ 1 and hðgmax Þ ¼ 0 where gmax is
sufficiently large value of g at which the boundary conditions (7) for fðgÞ are satisfied. The nonlinear Eqs. (5) and (6) first converted into first order ordinary linear differential equations as follows: f 0 ðgÞ ¼ yð2Þ; f 00 ðgÞ ¼ yð3Þ; 1 f 000 ðgÞ ¼ yð1Þyð3Þ Mf1 yð2Þg ðGc =Re2 Þyð4Þ ; 2 1 0 v ¼ yð5Þ; v 00 ðgÞ ¼ Scyð1Þyð5Þ 2 There are three conditions on the boundary g ¼ 0 and two conditions at g ! 1. To get the solution of the problem two more initial conditions have been found by shooting technique. The value of unknowns f 00 ð0Þ and h 0 ð0Þ at which solution agrees with the prescribed boundary conditions (7) has been tabulated in Table 3. Adaptive Runge–Kutta method estimates the truncation error at each integration step and automatically adjusts the step size to keep the error within prescribed limits. The formula used to measure the actual conservative error is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P defined by eðhÞ ¼ 15 5i¼1 E2i ðhÞ here Ei ðhÞ is the measure of error in the dependent variable yi . Per-step error control is achieved by adjusting the increment h by hiþ1 ¼ 0:9hi ðe=eðhi ÞÞ1=5 ; i ¼ 1; 2; 3; 4; 5 so that it is approximately equal to the prescribed tolerance e ¼ 106 . The assumed initial step size is h ¼ 0:01. 4. Results and discussion
Table 3
Values of f 00 ð0Þ, and v 0 ð0Þ obtained. f 00 ð0Þ
v 0 ð0Þ
0.1
0.8305
0.1591
0.3 0.5 0.7 1.0 0.0
0.8146 0.8068 0.8015 0.7962 0.7228
0.2541 0.3131 0.3588 0.4127 0.3050
0.2 0.5 0.7 1.0
0.8879 1.1216 1.2701 1.4831
0.3211 0.3408 0.3525 0.3685
Parameters Schmidt number (Sc) ½c ¼ 0:1; m ¼ 0:5; Gc ¼ 0:1, Re = 1]
Solutal Grashof Number ðGc Þ ½c ¼ 0:1; m ¼ 0:5; Sc ¼ 0:1, Re = 1]
Table 4
The numerical computations have been made for velocity, shear stress, concentration and concentration gradient profiles and other parameters involved in the flow. Table 4 contains the values of physical parameters corresponding to volume fraction of the alumina water nanofluid. The mass diffusivity of the nanofluid decreases and the Schmidt number increases as volume fraction increases. The results have depicted graphically and their physical explanations have also been given corresponding to different values of solutal Grashof number and Schmidt number. Figs. 2–5 exhibit the velocity, shear stress, concentration and concentration gradient profiles respectively for various values of solutal Grashof number. The solutal Grashof number defines the ratio of the species buoyancy force to the viscous force. It is noticed that the velocity increases with an
Physical parameters for Al2O3 Nanofluid at 20C.
Vol. Fr. /
Density ð103 Þqnf
Dynamic viscosity lnf
Kinematic viscosity tnf
Mass diffusivityð104 ÞDnf
Schmidt number ð104 ÞSc
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
1.0005200 1.0302148 1.0599096 1.0896044 1.1192992 1.1489940 1.1786888 1.2083836 1.2380784 1.2677732 1.2974680
0.0010020000 0.0010274950 0.0010539075 0.0010812805 0.0011096592 0.0011390917 0.0011696288 0.0012013245 0.0012342358 0.0012684233 0.0013039515
1.0014792307 0.9973600090 0.9943372436 0.9923606560 0.9913875155 0.9913818130 0.9923135705 0.9941582656 0.9968963511 1.0005128571 1.0049970627
0.2420000 0.2395800 0.2371600 0.2347400 0.2323200 0.2299000 0.2274800 0.2250600 0.2226400 0.2202200 0.2178000
4.1383439289 4.1629518700 4.1926852912 4.2274885236 4.2673360690 4.3122305917 4.3622013829 4.4173032331 4.4776156625 4.5432424718 4.6143115828
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
4
P. Singh, M. Kumar 0.00
1.2
-0.05 1.0
-0.10
0.8
f '( )
-0.15
Gc
0.6
0, 0.2, 0.5, 0.7,1.0
0.1, M
'( ) -0.20
0.5, Sc 0.5
0, 0.2, 0.5, 0.7,1.0
-0.30
0.2 0.0
Gc
-0.25
0.4
0.1, M
-0.35 0
2
4
6
8
0.5, Sc 0.5
-0.40 1
2
3
4
5
6
7
8
Effect of solutal Grashof number on velocity profile.
Figure 2
Figure 5 Effect of solutal Grashof number on concentration gradient profile. 1.5 1.0 1.0
0.1, M f ''( )
Gc
0.5
0.5, Sc 0.5
0.8
Sc 0.1, 0.3, 0.5, 0.7,1.0
0, 0.2, 0.5, 0.7, 1.0
0.6
0.1, M
f '( )
0.5, Gc
0.1
0.4
0.0
0.2 0
1
2
3
4
5
6
7
8 0.0
Figure 3 profile.
Effect of solutal Grashof number on Shear stress
0
Figure 6
1
2
3
4
5
6
7
8
9
10
Effect of Schmidt number on velocity profile.
1.0
0.1, M
0.5, Sc 0.5
0.8
0.6
Gc
( )
0, 0.2, 0.5, 0.7,1.0
0.4
0.2
0.0
Figure 4 profile.
0
1
2
3
4
5
6
7
8
Effect of solutal Grashof number on concentration
increase in the Grashof number and the shear stress increases up to g ¼ 1:80266667 and then decreases asymptotically as solutal Grashof number increases. The concentration increases and concentration gradient profile decreases with an increase in the solutal Grashof number as shown in Figs. 4 and 5. Figs. 6–9 show the effect of Schmidt number on velocity, shear stress, concentration and concentration gradient profiles respectively. Schmidt number is used to characterize fluid flow in which there are simultaneous momentum and mass diffusion process. It physically relates the relative thickness of the hydrodynamic layer and mass transfer boundary layer. It has been observed that the velocity decreases with an increase in the Schmidt number as shown in Fig. 6, and shear stress profiles decreases up to g ¼ 2:28666667 with an increase in the Schmidt number and then increases asymptotically. The effect of Schmidt number on concentration and concentration gradient has also been studied through Figs. 8 and 9. It is noticed that the concentration decreases with an increase in the
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
Mass transfer in MHD flow of alumina water nanofluid
5
0.8
5. Conclusion
0.1, M
0.5, Gc
0.1 Mass transfer in an electrically conducting nanofluid over a flat plate with magnetic field under slip condition is considered. The non-linear partial differential equations are transformed into a system of ordinary differential equations by using similarity transformation and then solved numerically using the fifth order Runge–Kutta method along with the shooting technique. From the numerical computation of wall shear stress and wall concentration gradient it is concluded that.
Sc 0.1, 0.3, 0.5, 0.7,1.0
0.6
f ''( ) 0.4
0.2
0.0
0
1
2
3
4
5
Effect of Schmidt number on shear stress profile.
Figure 7
An increase in solutal Grashof number decreases the concentration gradient but enhances the velocity and concentration. The shear stress increases up to a fixed value of solutal Grashof number and after that it decreases. For the Schmidt number, velocity shows the opposite behavior to the concentration. It is concluded that the concentration gradient decreases up to some fixed range of Schmidt number and then increases asymptotically.
1.0
References 0.8
Sc ( )
0.1, M 0.5, G c 0.1 0.1, 0.3, 0.5, 0.7,1.0
0.6 0.4 0.2 0.0
0
2
4
6
8
10
12
14
Effect of Schmidt number on concentration profile.
Figure 8
0.0
-0.1
'( )
-0.2
-0.3
Sc 0.1, 0.3, 0.5, 0.7,1.0 0.1, M 0.5, Gc 0.1
-0.4 0
Figure 9 profile.
2
4
6
8
10
12
14
Effect of Schmidt number on concentration gradient
Schmidt number and concentration gradient decreases up to g ¼ 2:64 as the value of Schmidt number increases and then shows the opposite behavior.
[1] Y. Xuan, Conception for enhanced mass transport in binary nanofluids, Heat Mass Transfer 46 (2009) 277–279. [2] K. Bhattacharyya, Boundary layer flow with diffusion and first order chemical reaction over a porous flat plate subject to suction/injection and with variable wall concentration, Chem. Eng. Res. Bull. 15 (2011) 6–11. [3] V. Sridhara, L.N. Satapathy, Al2O3-based nanofluids: a review, Nanoscale Res. Lett. 456 (6) (2011) 1–16. [4] Md.J. Uddin, M.A.A. Hamad, A.I.Md. Ismail, Investigation of heat mass transfer for combined convective slips flow: a lie group analysis, Sains Malaysiana 41 (9) (2012) 1139–1148. [5] C. Pang, W. Wu, W. Sheng, H. Zhang, Y.T. Kang, Mass transfer enhancement by binary nanofluids for bubble absorption process, Int. J. Refrig. 35 (2012) 2240–2247. [6] D. Kumar, M.B. Sharma, B.S. Yadav, A study on MHD flow and heat transfer along a porous flat plate with mass transfer, Ultra Sci. 24b (2) (2012) 327–334. [7] B.R. Rout, S.K. Parida, S. Panda, MHD heat and mass transfer of chemical reaction fluid flow over a moving vertical plate in presence of heat source with convective surface boundary condition, Int. J. Chem. Eng. (2013) 1–11, Hindawi publishing corporation, Article ID 296834. [8] S.M. Ibrahim, N.B. Reddy, Similarity solution of heat and mass transfer for natural convection over a moving vertical plate with internal heat generation and a convective boundary condition in the presence of thermal radiation, viscous dissipation, and chemical reaction, ISRN Thermodyn. (2013) 1–10. Article ID 790604. [9] K. Gangadhar, N.B. Reddy, P.K. Kameswaran, Similarity solution of hydro magnetic heat and mass transfer over a vertical plate with a convective surface boundary condition and chemical reaction, Int. J. Nonlinear Sci. 13 (3) (2012) 298–307. [10] A. Einstein, Investigation on the Theory of the Brownian Movement, Dover, New York, 1956. [11] D.A. Drew, S.L. Passman, Theory of Multicomponent Fluids, Springer, 1999. [12] E. Mamut, Characterization of heat and mass transfer properties of nanofluids, Rom. J. Phys. 51 (2006) 5–12. [13] Y.T. Kang, J. King, C.K. Choi, Analysis of convective instabilities of binary nanofluids, in: Int. Refrigeration and Air Conditioning Conference at Purdue, July 2004, pp. 12–15.
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions Padam Singh *, Manoj Kumar Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India Received 28 January 2014; revised 18 March 2015; accepted 11 April 2015
KEYWORDS Magnetohydrodynamic; Alumina water nanofluid; Boundary layer slip; Volume fraction; Concentration
Abstract Present paper deals with analysis of mass transfer in two dimensional magnetohydrodynamic (MHD) slip flow of an incompressible, electrically conducting, viscous and steady flow of alumina water nanofluid in the presence of magnetic field over a flat plate. The governing equations representing flow are transformed into a set of simultaneous ordinary differential equations by using appropriate similarity transformation. The set of equations thus obtained has been solved using adaptive Runge–Kutta method with shooting technique. The effects of solutal Grashof number and Schmidt number on velocity distribution profile, concentration profile, shear stress and concentration gradient have depicted graphically and analyzed. It is found that the concentration increases and concentration gradient decreases with increasing value of solutal Grashof number. Also, the concentration decreases with an increase in Schmidt number. ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Xuan [1] studied mass transfer in binary nanofluids and found that the nanoparticles improved the mass transfer inside binary nanofluids. Bhattacharyya [2] studied the boundary layer flow over a porous flat plate with diffusion of chemically reactive species undergoing first order reaction and subject to suction/injection. The author found that for increasing of suction at the plate both velocity and concentration boundary layer thicknesses decrease and for increasing injection both thicknesses increase. The rate of solute transfer from the plate increases with reaction rate parameter and power-law * Corresponding author. E-mail address: [email protected] (P. Singh). Peer review under responsibility of Faculty of Engineering, Alexandria University.
exponent. Sridhara and Satapathy [3] made review to summarize recent developments in research on the stability of nanofluids, enhancement of thermal conductivity, viscosity and heat transfer characteristics of alumina based nanofluids. Uddin et al. [4] investigated the steady laminar combined convective flow with heat and mass transfer of a Newtonian viscous incompressible fluid over a permeable flat plate with linear hydrodynamic and thermal slips. The effects of various parameters were investigated and presented graphically. Pang et al. [5] studied the mass transfer enhancement by binary nanofluids for bubble absorption process. The results were shown that the mass transfer enhancement with coolant is more than compared to without coolant. Kumar et al. [6] studied MHD flow and heat transfer along a porous flat plate in the presence of mass transfer. The results were obtained for velocity, temperature and concentration profiles of the flow field and shown graphically. Rout et al. [7] investigated the
http://dx.doi.org/10.1016/j.aej.2015.04.005 1110-0168 ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
2
P. Singh, M. Kumar
Nomenclature B Gc g L M Sc u1 u, v x, y vsf ðbs Þnf
magnetic field solutal Grashof number gravitational acceleration reference length magnetic parameter Schmidt number uniform velocity at infinity velocity components in x and y direction coordinate axes velocity slip factor solutal expansion coefficient
influence of chemical reaction and the combined effects of internal heat generation and a convective boundary condition on the laminar boundary layer MHD heat and mass transfer flow over a moving vertical flat plate. The discussion was focused on the physical interpretation of the results as well as their comparison with past studies. Steady laminar natural convection flow over a semi-infinite moving vertical plate with internal heat generation and convective surface boundary condition in the presence of thermal radiation, viscous dissipation and chemical reaction analyzed by Ibrahim and Reddy [8]. Authors utilized similarity variable to transform the governing nonlinear partial differential equations into a system of ordinary differential equations. Gangadhar et al. [9] studied the hydromagnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field, chemical reaction and a convective heat exchange at the surface using similarity transformation. The present investigation deals with analysis of mass transfer in MHD slip flow of an incompressible, steady, electrically conducting and viscous alumina-water nanofluid in the presence of magnetic field over a flat plate. 2. Mathematical description The physical model is given here along with flow configuration and coordinate system (Fig. 1). The system deals with two dimensional MHD boundary layer slip flow of water based nanofluid. The continuity, momentum and concentration equations representing flow are following: @u @v þ ¼0 @x @y
velocity slip parameter density of water nanofluid density density of alumina particles electrical conductivity of nanofluid dynamic viscosity of nanofluid volume fraction of nanoparticles stream function kinematic viscosity of nanofluid sphericity
c qf qnf qs rnf lnf ; w tnf x
ð1Þ
u
@u @u @ 2 u rnf B þv ¼ tnf 2 þ ðu1 uÞ þ gðbs Þnf ðC C1 Þ @x @y @y qnf
ð2Þ
u
@C @C @2C þv ¼ Dnf 2 @x @y @y
ð3Þ
With the boundary conditions: u ¼ vsf
@u ; v ¼ 0 and C ¼ CW ¼ C1 þ C0 xk at y ¼ 0 @y ðon the wall of the plateÞ
u ! u1 and C ! C1 when y ! 1 ðfar from the wallÞ qffiffiffiffiffiffi Here, vsf ¼ v0 ut1nfx is velocity slip factor with initial value v0 (see Tables 1 and 2). 3. Method for solution To solve the governing Eqs. (2) and (3) with the boundary conditions (4), following similarity transformation has been introduced: rffiffiffiffiffiffiffiffiffi y xu1 C C1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; vðgÞ ¼ w ¼ u1 tnf x fðgÞ; g ¼ ; x tnf Cw C1 M¼
rxB2 u1 qnf
Table 1
and
Nanofluid properties. lnf ¼ lf =ð1 ;Þ2:5 qnf ¼ ð1 ;Þqf þ ;qs Dnf ¼ ð1 ;ÞDf tnf ¼ lnf =qnf
Dynamic viscosity [10] Density [11] Mass diffusivity [12] Kinematic viscosity [13]
y C
v
ð4Þ
u
Table 2 The physical properties of alumina and base fluid water at 20 C.
Cw B0
Figure 1
u
x Water Alumina
Density (kg/m3)
Dynamic viscosity (N s/m2)
Mass diffusivity (m2/sec)
1000.52 3970.0
0.001002 –
0.0000242 –
Physical model of the flow.
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
Mass transfer in MHD flow of alumina water nanofluid Gc ¼
3
gðbs Þnf ðCw C1 Þx3 xu1 B0 tnf ; Re ¼ where B ¼ pffiffiffi ; Sc ¼ t2nf tnf Dnf x
where B0 is constant and the stream function w satisfies Eq. (1) and v ¼ @w . with u ¼ @w @y @x Eqs. (2) and (3) reduce to the non-linear differential equations using above transformation as follows: 1 f 000 ðgÞ þ fðgÞf 00 ðgÞ þ Mð1 f 0 ðgÞÞ þ ðGc =Re2 ÞvðgÞ ¼ 0 2
ð5Þ
1 v 00 ðgÞ þ ScfðgÞv 0 ðgÞ ¼ 0 2
ð6Þ
and the boundary conditions (4) reduce as follows: f 0 ð0Þ ¼ cf 00 ð0Þ and vð0Þ ¼ 1
fð0Þ ¼ 0;
f 0 ðgÞ ! 1 and vðgÞ ! 0 when g ! 1 Here c ¼
V0 u1 tnf
ð7Þ
is the velocity slip parameter with initial value
V0 . To solve the non-linear differential Eqs. (5) and (6) subject to the boundary conditions (7) adaptive Runge–Kutta method with shooting technique has been applied. This method is based on the discretization of the problem domain and the calculation of unknown boundary conditions. The domain of the problem is discretized and the boundary conditions for g ¼ 1 are replaced by f 0 ðgmax Þ ¼ 1 and hðgmax Þ ¼ 0 where gmax is
sufficiently large value of g at which the boundary conditions (7) for fðgÞ are satisfied. The nonlinear Eqs. (5) and (6) first converted into first order ordinary linear differential equations as follows: f 0 ðgÞ ¼ yð2Þ; f 00 ðgÞ ¼ yð3Þ; 1 f 000 ðgÞ ¼ yð1Þyð3Þ Mf1 yð2Þg ðGc =Re2 Þyð4Þ ; 2 1 0 v ¼ yð5Þ; v 00 ðgÞ ¼ Scyð1Þyð5Þ 2 There are three conditions on the boundary g ¼ 0 and two conditions at g ! 1. To get the solution of the problem two more initial conditions have been found by shooting technique. The value of unknowns f 00 ð0Þ and h 0 ð0Þ at which solution agrees with the prescribed boundary conditions (7) has been tabulated in Table 3. Adaptive Runge–Kutta method estimates the truncation error at each integration step and automatically adjusts the step size to keep the error within prescribed limits. The formula used to measure the actual conservative error is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P defined by eðhÞ ¼ 15 5i¼1 E2i ðhÞ here Ei ðhÞ is the measure of error in the dependent variable yi . Per-step error control is achieved by adjusting the increment h by hiþ1 ¼ 0:9hi ðe=eðhi ÞÞ1=5 ; i ¼ 1; 2; 3; 4; 5 so that it is approximately equal to the prescribed tolerance e ¼ 106 . The assumed initial step size is h ¼ 0:01. 4. Results and discussion
Table 3
Values of f 00 ð0Þ, and v 0 ð0Þ obtained. f 00 ð0Þ
v 0 ð0Þ
0.1
0.8305
0.1591
0.3 0.5 0.7 1.0 0.0
0.8146 0.8068 0.8015 0.7962 0.7228
0.2541 0.3131 0.3588 0.4127 0.3050
0.2 0.5 0.7 1.0
0.8879 1.1216 1.2701 1.4831
0.3211 0.3408 0.3525 0.3685
Parameters Schmidt number (Sc) ½c ¼ 0:1; m ¼ 0:5; Gc ¼ 0:1, Re = 1]
Solutal Grashof Number ðGc Þ ½c ¼ 0:1; m ¼ 0:5; Sc ¼ 0:1, Re = 1]
Table 4
The numerical computations have been made for velocity, shear stress, concentration and concentration gradient profiles and other parameters involved in the flow. Table 4 contains the values of physical parameters corresponding to volume fraction of the alumina water nanofluid. The mass diffusivity of the nanofluid decreases and the Schmidt number increases as volume fraction increases. The results have depicted graphically and their physical explanations have also been given corresponding to different values of solutal Grashof number and Schmidt number. Figs. 2–5 exhibit the velocity, shear stress, concentration and concentration gradient profiles respectively for various values of solutal Grashof number. The solutal Grashof number defines the ratio of the species buoyancy force to the viscous force. It is noticed that the velocity increases with an
Physical parameters for Al2O3 Nanofluid at 20C.
Vol. Fr. /
Density ð103 Þqnf
Dynamic viscosity lnf
Kinematic viscosity tnf
Mass diffusivityð104 ÞDnf
Schmidt number ð104 ÞSc
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
1.0005200 1.0302148 1.0599096 1.0896044 1.1192992 1.1489940 1.1786888 1.2083836 1.2380784 1.2677732 1.2974680
0.0010020000 0.0010274950 0.0010539075 0.0010812805 0.0011096592 0.0011390917 0.0011696288 0.0012013245 0.0012342358 0.0012684233 0.0013039515
1.0014792307 0.9973600090 0.9943372436 0.9923606560 0.9913875155 0.9913818130 0.9923135705 0.9941582656 0.9968963511 1.0005128571 1.0049970627
0.2420000 0.2395800 0.2371600 0.2347400 0.2323200 0.2299000 0.2274800 0.2250600 0.2226400 0.2202200 0.2178000
4.1383439289 4.1629518700 4.1926852912 4.2274885236 4.2673360690 4.3122305917 4.3622013829 4.4173032331 4.4776156625 4.5432424718 4.6143115828
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
4
P. Singh, M. Kumar 0.00
1.2
-0.05 1.0
-0.10
0.8
f '( )
-0.15
Gc
0.6
0, 0.2, 0.5, 0.7,1.0
0.1, M
'( ) -0.20
0.5, Sc 0.5
0, 0.2, 0.5, 0.7,1.0
-0.30
0.2 0.0
Gc
-0.25
0.4
0.1, M
-0.35 0
2
4
6
8
0.5, Sc 0.5
-0.40 1
2
3
4
5
6
7
8
Effect of solutal Grashof number on velocity profile.
Figure 2
Figure 5 Effect of solutal Grashof number on concentration gradient profile. 1.5 1.0 1.0
0.1, M f ''( )
Gc
0.5
0.5, Sc 0.5
0.8
Sc 0.1, 0.3, 0.5, 0.7,1.0
0, 0.2, 0.5, 0.7, 1.0
0.6
0.1, M
f '( )
0.5, Gc
0.1
0.4
0.0
0.2 0
1
2
3
4
5
6
7
8 0.0
Figure 3 profile.
Effect of solutal Grashof number on Shear stress
0
Figure 6
1
2
3
4
5
6
7
8
9
10
Effect of Schmidt number on velocity profile.
1.0
0.1, M
0.5, Sc 0.5
0.8
0.6
Gc
( )
0, 0.2, 0.5, 0.7,1.0
0.4
0.2
0.0
Figure 4 profile.
0
1
2
3
4
5
6
7
8
Effect of solutal Grashof number on concentration
increase in the Grashof number and the shear stress increases up to g ¼ 1:80266667 and then decreases asymptotically as solutal Grashof number increases. The concentration increases and concentration gradient profile decreases with an increase in the solutal Grashof number as shown in Figs. 4 and 5. Figs. 6–9 show the effect of Schmidt number on velocity, shear stress, concentration and concentration gradient profiles respectively. Schmidt number is used to characterize fluid flow in which there are simultaneous momentum and mass diffusion process. It physically relates the relative thickness of the hydrodynamic layer and mass transfer boundary layer. It has been observed that the velocity decreases with an increase in the Schmidt number as shown in Fig. 6, and shear stress profiles decreases up to g ¼ 2:28666667 with an increase in the Schmidt number and then increases asymptotically. The effect of Schmidt number on concentration and concentration gradient has also been studied through Figs. 8 and 9. It is noticed that the concentration decreases with an increase in the
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005
Mass transfer in MHD flow of alumina water nanofluid
5
0.8
5. Conclusion
0.1, M
0.5, Gc
0.1 Mass transfer in an electrically conducting nanofluid over a flat plate with magnetic field under slip condition is considered. The non-linear partial differential equations are transformed into a system of ordinary differential equations by using similarity transformation and then solved numerically using the fifth order Runge–Kutta method along with the shooting technique. From the numerical computation of wall shear stress and wall concentration gradient it is concluded that.
Sc 0.1, 0.3, 0.5, 0.7,1.0
0.6
f ''( ) 0.4
0.2
0.0
0
1
2
3
4
5
Effect of Schmidt number on shear stress profile.
Figure 7
An increase in solutal Grashof number decreases the concentration gradient but enhances the velocity and concentration. The shear stress increases up to a fixed value of solutal Grashof number and after that it decreases. For the Schmidt number, velocity shows the opposite behavior to the concentration. It is concluded that the concentration gradient decreases up to some fixed range of Schmidt number and then increases asymptotically.
1.0
References 0.8
Sc ( )
0.1, M 0.5, G c 0.1 0.1, 0.3, 0.5, 0.7,1.0
0.6 0.4 0.2 0.0
0
2
4
6
8
10
12
14
Effect of Schmidt number on concentration profile.
Figure 8
0.0
-0.1
'( )
-0.2
-0.3
Sc 0.1, 0.3, 0.5, 0.7,1.0 0.1, M 0.5, Gc 0.1
-0.4 0
Figure 9 profile.
2
4
6
8
10
12
14
Effect of Schmidt number on concentration gradient
Schmidt number and concentration gradient decreases up to g ¼ 2:64 as the value of Schmidt number increases and then shows the opposite behavior.
[1] Y. Xuan, Conception for enhanced mass transport in binary nanofluids, Heat Mass Transfer 46 (2009) 277–279. [2] K. Bhattacharyya, Boundary layer flow with diffusion and first order chemical reaction over a porous flat plate subject to suction/injection and with variable wall concentration, Chem. Eng. Res. Bull. 15 (2011) 6–11. [3] V. Sridhara, L.N. Satapathy, Al2O3-based nanofluids: a review, Nanoscale Res. Lett. 456 (6) (2011) 1–16. [4] Md.J. Uddin, M.A.A. Hamad, A.I.Md. Ismail, Investigation of heat mass transfer for combined convective slips flow: a lie group analysis, Sains Malaysiana 41 (9) (2012) 1139–1148. [5] C. Pang, W. Wu, W. Sheng, H. Zhang, Y.T. Kang, Mass transfer enhancement by binary nanofluids for bubble absorption process, Int. J. Refrig. 35 (2012) 2240–2247. [6] D. Kumar, M.B. Sharma, B.S. Yadav, A study on MHD flow and heat transfer along a porous flat plate with mass transfer, Ultra Sci. 24b (2) (2012) 327–334. [7] B.R. Rout, S.K. Parida, S. Panda, MHD heat and mass transfer of chemical reaction fluid flow over a moving vertical plate in presence of heat source with convective surface boundary condition, Int. J. Chem. Eng. (2013) 1–11, Hindawi publishing corporation, Article ID 296834. [8] S.M. Ibrahim, N.B. Reddy, Similarity solution of heat and mass transfer for natural convection over a moving vertical plate with internal heat generation and a convective boundary condition in the presence of thermal radiation, viscous dissipation, and chemical reaction, ISRN Thermodyn. (2013) 1–10. Article ID 790604. [9] K. Gangadhar, N.B. Reddy, P.K. Kameswaran, Similarity solution of hydro magnetic heat and mass transfer over a vertical plate with a convective surface boundary condition and chemical reaction, Int. J. Nonlinear Sci. 13 (3) (2012) 298–307. [10] A. Einstein, Investigation on the Theory of the Brownian Movement, Dover, New York, 1956. [11] D.A. Drew, S.L. Passman, Theory of Multicomponent Fluids, Springer, 1999. [12] E. Mamut, Characterization of heat and mass transfer properties of nanofluids, Rom. J. Phys. 51 (2006) 5–12. [13] Y.T. Kang, J. King, C.K. Choi, Analysis of convective instabilities of binary nanofluids, in: Int. Refrigeration and Air Conditioning Conference at Purdue, July 2004, pp. 12–15.
Please cite this article in press as: P. Singh, M. Kumar, Mass transfer in MHD flow of alumina water nanofluid over a flat plate under slip conditions, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.04.005