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Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet
International Journal of Heat and Mass Transfer 57 (2013) 465–472
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet P.K. Kameswaran a, S. Shaw a, P. Sibanda a,⇑, P.V.S.N. Murthy b a b
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, 3209 Pietermaritzburg, South Africa Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India
a r t i c l e
i n f o
Article history: Received 24 August 2012 Received in revised form 13 October 2012 Accepted 14 October 2012 Available online 23 November 2012 Keywords: Homogeneous–heterogeneous reactions Nanofluids Volume fraction model Stretching sheet
a b s t r a c t We investigate the effects of homogeneous–heterogeneous reactions in nanofluid flow over a stretching or shrinking sheet placed in a porous medium saturated with a nanofluid. Copper–water and silver–water nanofluids are investigated in this study. The steady states of this system are analyzed in the case when the diffusion coefficients of the reactant and auto catalyst are equal. The governing partial differential equations are transformed into a system of non-linear ordinary differential equations and solved numerically. An analytical solution for the momentum equation is obtained. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The study of convective flow through porous media has received a great deal of research interest over the last three decades due to its wide and important applications in environmental, geophysical and industrial problems. Prominent applications include the utilization of geothermal energy, the migration of moisture in fibrous insulation, drying of a porous solid, food processing, casting and welding in manufacturing processes, the dispersion of chemical contaminants in different industrial processes, the design of nuclear reactors, chemical catalytic reactors, compact heat exchangers, solar power and many others. The problem of a stretching surface has many engineering applications such as in extrusion processes, melt-spinning, hot rolling, wire drawing, glass-fiber production, manufacture of plastic and rubber sheets. Other applications can be found in the manufacture of polymer sheets, food processing and in the movement of biological fluids, [1–4]. Crane [5] was the first to consider steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moves in its own plane with a velocity varying linearly with the distance from a fixed point. This was subsequently extended by many authors to explore various aspects of heat transfer in a fluid surrounding a stretching sheet [6–15]. Abel et al. [16] investigated the effect of porous medium on the flow and heat transfer of a nonuniform viscoelastic liquid over a non-isothermal stretching sheet. Abel et al. [17] presented a math⇑ Corresponding author. Tel.: +27 33 260 5626; fax: +27 33 260 5648. E-mail address: [email protected] (P. Sibanda). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.047
ematical analysis for the momentum and heat transfer characteristics of the boundary layer flow of an electrically conducting viscoelastic fluid over a linearly stretching sheet. Pal and Mondal [18] studied the influence of a chemical reaction and thermal radiation on a stretching sheet in a Darcian porous medium with Soret and Dufour effects. They found that the temperature profiles increase as the thermal radiation parameter and the Dufour number increase. Recently, Rosali et al. [19] investigated micropolar fluid flow over a stretching/shrinking sheet in a porous medium with suction. They observed that an increase in the permeability parameter led to an increase in the skin friction coefficient and the local Nusselt number. Many chemically reacting systems involve both homogeneous and heterogeneous reactions, with examples occurring in combustion, catalysis and biochemical systems. The interaction between the homogeneous reactions in the bulk of the fluid and heterogeneous reactions occurring on some catalytic surfaces is generally very complex, and is involved in the production and consumption of reactant species at different rates both within the fluid and on the catalytic surfaces. A model for isothermal homogeneous– heterogeneous reactions in boundary layer flow of a viscous fluid past a flat plate was studied by Merkin [20]. He presented the homogeneous reaction by cubic autocatalysis and the heterogeneous reaction by a first order process and showed that the surface reaction is the dominant mechanism near the leading edge of the plate. Chaudhary and Merkin [21] studied homogeneous– heterogeneous reactions in boundary layer flow. They studied the numerical solution near the leading edge of a flat plate. Ziabakhsh et al. [22] studied the problem of flow and diffusion of chemically
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reactive species over a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos [23] studied an isothermal chemical reaction on a catalytic reactor in laminar boundary layer flow. They found the actual surface concentration without introducing unnecessary assumptions related to the reaction mechanism. The effects of flow near the two dimensional stagnation point flow on an infinite permeable wall with a homogeneous–heterogeneous reaction was studied by Khan and Pop [24]. They solved the governing nonlinear equations using the implicit finite difference method and observed that the mass transfer parameter considerably affects the flow characteristics. Khan and Pop [25] studied the effects of homogeneous–heterogeneous reactions on a viscoelastic fluid toward a stretching sheet. They observed that the concentration at the surface decreased with an increase in the viscoelastic parameter. Convectional heat transfer fluids, including oil, water, and ethylene glycol mixtures are poor heat transfer fluids due to the low thermal conductivity of these fluids. The thermal conductivity of these fluids may be improved by suspending nano sized particle materials in the liquid to form a nanofluid. A characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. [26]. Thermophysical properties of nanofluids such as thermal conductivity, diffusivity and viscosity have been studied by, among others, Kang et al. [27], Velagapudi et al. [28] and Rudyak et al. [29]. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition was studied by Makinde and Aziz [30]. They found that with an increase in the Biot number, the concentration layer thickened, but the concentration layer became thinner as the Lewis number increased. Alsaedi et al. [31] studied the effects of heat generation/ absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions. Recently, Narayana and Sibanda [32] studied the effects of laminar flow of a nanoliquid film over an unsteady stretching sheet. They found that the effect of the nanoparticle volume fraction is to reduce the axial velocity and free stream velocity in the case of a Cu-water nanoliquid, but the opposite is true in the case of a Al2O3 – water nanoliquid. Kameswaran et al. [33] studied hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. They found that the velocity profile decreased with an increase in the nano particle volume fraction, while the opposite was true in the case of temperature and concentration profiles. The study further showed that liquids with nanoparticle suspensions are better suited for effective cooling of the stretching sheet problem due to their enhanced conductivity and thermal properties. In this paper we study the combined effects of a porous medium permeability parameter subject to homogeneous–heterogeneous reactions in a nanofluid flow over a stretching or shrinking sheet. The model developed by Merkin [20] for homogeneous–heterogeneous reactions in boundary layer flow with cubic autocatalysis is used in the present study. The transformed ordinary boundary layer equations of motion and concentration are solved numerically. To the best of the authors’ knowledge, this problem has not been considered before in the literature.
2. Mathematical formulation
and the nanoparticles are in thermal equilibrium and no slip occurs between them. It is assumed that a simple homogeneous– heterogeneous reaction model exists as proposed by Chaudhary and Merkin [21] in the following form: 2
A þ 2B ! 3B;
rate ¼ kc ab ;
ð1Þ
while on the catalyst surface we have the single, isothermal, first order reaction
A ! B;
rate ¼ ks a;
ð2Þ
where a and b are the concentrations of the chemical species A and B, kc and ks are the rate constants. We assume that both reaction processes are isothermal. Under these assumptions, the boundary layer equations governing the flow can be written in dimensional form [21,34];
@u @ v þ ¼ 0; @x @y @u @u lnf @ 2 u lnf 1 þv ¼ u; u @x @y qnf @y2 qnf k
ð3Þ ð4Þ
2
@a @a @ a 2 þv ¼ DA 2 kc ab ; @x @y @y @b @b @2b 2 ¼ DB 2 þ kc ab : u þv @x @y @y u
ð5Þ ð6Þ
The boundary conditions for Eqs. (3)–(6) are given in the form:
u ! 0;
@a @b ¼ ks a; DB ¼ ks a at y ¼ 0; @y @y b ! 0 as y ! 1;
v ¼ 0;
u ¼ uw ¼ cx;
a ! a0 ;
DA
ð7Þ where u, v are the velocity components in the x-and y-directions respectively, c is the stretching (c > 0) or shrinking (c < 0) rate, k is the permeability of the porous medium, DA and DB are the respective diffusion species coefficients of A and B, a0 is a positive constant. The effective dynamic viscosity of the nanofluid was given by Brinkman [35] as
lnf ¼
lf ð1 /Þ2:5
ð8Þ
;
where / is the solid volume fraction of nanoparticles. The effective density of the nanofluids is given as
qnf ¼ ð1 /Þqf þ /qs :
ð9Þ
The thermal diffusivity of the nanofluid is
anf ¼
knf ; ðqC p Þnf
ð10Þ
where the heat capacitance of the nanofluid is given by
ðqC p Þnf ¼ ð1 /ÞðqC p Þf þ /ðqC p Þs :
ð11Þ
Here, the subscripts nf, f and s represents the thermophysical properties of the nano fluid, base fluid and nano solid particles, respectively. The continuity Eq. (3) is satisfied by introducing a stream function w(x, y) such that
u¼
@w @y
v¼
and
@w ; @x
1
Consider two dimensional steady boundary layer flow of an incompressible nanofluid over a stretching sheet. A cartesian coordinate system is used with the x-axis along the sheet and the y-axis normal to the sheet. Two equal but opposite forces are applied along the sheet so that the wall is stretched, keeping the position of the origin unaltered. The fluid is a water based nanofluid containing copper (Cu) or silver (Ag) nanoparticles. The base fluid
where w ¼ ðcmf Þ2 xf ðgÞ; f ðgÞ is the dimensionless stream function 1 2
and g ¼ ðc=mf Þ y. The velocity components are then given by 0
u ¼ cxf ðgÞ and
1 2
v ¼ ðcmf Þ f ðgÞ:
ð12Þ
The concentrations of the chemical species A and B are represented as
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a ¼ a0 gðgÞ and b ¼ a0 hðgÞ;
ð13Þ
where a0 is a constant, g(g) and h(g) is the dimensionless concentration. On using Eqs. (8)–(13), Eqs. (4)–(7) transform into the following two-point boundary value problems
Cf ¼
2sw qf u2w
ð25Þ
and using Eq. (24) in Eq. (25) we obtain
pffiffiffiffiffiffiffi Rex ¼ 2f 00 ð0Þ:
f 000 þ /1 ff /1 f 02 k1 f 0 ¼ 0;
ð14Þ
C f ð1 /Þ2:5
1 00 0 2 g þ fg Kgh ¼ 0; Sc
ð15Þ
In Eq. (26), Rex represents the local Reynolds number defined by Rex = xuw/mf.
d 00 0 2 h þ fh þ Kgh ¼ 0; Sc
ð16Þ
3.1. Exact solution for momentum equation
00
f 0 ð0Þ ¼ 1;
f ð0Þ ¼ 0;
g 0 ð0Þ ¼ K s gð0Þ;
f 0 ðgÞ ! 0 as g ! 1
ð17Þ
gðgÞ ! 1 as g ! 1
0
ð18Þ
hðgÞ ! 0 as g ! 1
dh ð0Þ ¼ K s gð0Þ;
ð19Þ
The non-dimensional constants in Eqs. (14)–(16) are the porous medium parameter k1, the Schmidt number Sc, the measure of the strength of the homogeneous reaction K, and the ratio of the diffusion coefficient d. They are respectively defined as
k1 ¼
mf kc
;
Sc ¼
mf DA
;
K¼
kc a20 ; c
d¼
DB ; DA
ð26Þ
The momentum boundary layer equation is partially decoupled from the species equation. The solution is obtained by looking for an exponential function of the form f0 (g) = esg that satisfies both the differential equation and governing boundary conditions over the interval [0, 1). An exact solution to (14) and (17) is obtained as
f ðgÞ ¼
1 esg ; s
ð27Þ
where s is the parameter associated with the nanoparticle volume fraction and the porous parameter, which is defined as
s¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /1 þ k1 :
ð28Þ
where
" 2:5
/1 ¼ ð1 /Þ
q 1/þ/ s qf
!# ð20Þ
:
In most applications, we expect the diffusion coefficients of chemical species A and B to be of a comparable size. This leads us to make a further assumption that the diffusion coefficients DA and DB are equal, i.e., to take d = 1 [21]. In this case we have from Eqs. (18) and (19)
gðgÞ þ hðgÞ ¼ 1:
ð21Þ
Thus Eqs. (15) and (16) reduce to
1 00 0 g þ fg Kgð1 gÞ2 ¼ 0 Sc
ð22Þ
and are subject to the boundary conditions
g 0 ð0Þ ¼ K s gð0Þ;
gðgÞ ! 1 as g ! 1:
ð23Þ
The thermophysical properties of the nanofluid are given in Table 1 [36]. 3. Skin friction coefficient The physical quantity of interest is the skin friction coefficient Cf. It characterizes the surface drag. The shearing stress at the surface of the wall sw is given by
sw ¼ lnf
qffiffiffiffiffiffiffiffiffi @u 1 ¼ qf mf c3 xf 00 ð0Þ; 2:5 @y y¼0 ð1 /Þ
ð24Þ
where lnf is the coefficient of viscosity. The skin friction coefficient is defined as
Table 1 Thermo-physical properties of water and nanoparticles, Oztop and Abu-Nada [36]. 3
5
Properties ?
q (kg/m )
Cp (J/kg K)
k (W/m K)
b 10 (K
Pure water Cu Ag
997.1 8933 10500
4179 385 235
0.613 401 429
21 1.67 1.89
1
)
4. Results and discussion We have studied mass transfer in a nanofluid flow due to a stretching or shrinking sheet in porous medium. We considered two different nanoparticles, copper (Cu) and silver (Ag), with water as the base fluid. An analytical solution was obtained for the momentum equation. The system of ordinary differential equations (14) and (22) along with the boundary conditions (17) and (23) for some values of the physical parameters k1, Ks, K, Sc, / were solved numerically using the Matlab routine bvp4c. A comparison was also made with the analytical results and bvp4c results. The comparison showed good agreement for each value of k1 so that we may be confident that the present results are accurate. Tables 2 and 3 show the results of f00 (0) for different parameter values. Table 2 gives the comparison of analytical and numerical results in the absence of the physical parameters i.e., / = Sc = K = Ks = 0. The numerical solution results are found to be in good agreement with the exact solution. Table 3 shows the skin friction coefficient for various physical parameters in the case of a Cu–water and Ag–water nanofluid. With increasing values of the nanoparticle volume parameter /, the skin friction coefficient increases. The comparison shows good agreement between bvp4c results and the exact solution. The effects of permeability, Schmidt number, strength of the homogeneous reaction and the strength of heterogeneous reaction parameters are shown in Figs. 1–5 for various fluid flow parameters.
Table 2 Comparison of f00 (0) obtained analytically and numerically using bvp4c for different values of k1, for fixed values of Sc = 0, / = 0, K = 0 and Ks = 0. k1
0.5 1 1.5 2 5
f00 (0) Analytical
Numerical
1.22474487 1.41421356 1.58113883 1.73205081 2.44948974
1.22474487 1.41421356 1.58113883 1.73205081 2.44948974
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Table 3 Comparison of f00 (0) for various values of k1 and / for fixed values of Sc = 0, K = 0 and Ks = 0. k1
/
Analytical
Numerical
Cu–Water
Ag–Water
Cu–Water
Ag–Water
1
0.05 0.1 0.15 0.2
1.49321912 1.54273401 1.56886842 1.57595391
1.51618740 1.58138925 1.61813878 1.63203956
1.49321912 1.54273401 1.56886842 1.57595391
1.51618740 1.58138925 1.61813878 1.63203956
2
0.05 0.1 0.15 0.2
1.79713754 1.83848530 1.86046986 1.86644869
1.81626657 1.87104034 1.90220218 1.91404105
1.79713754 1.83848530 1.86046986 1.86644869
1.81626657 1.87104034 1.90220218 1.91404105
3
0.05 0.1 0.15 0.2
2.05662426 2.09285169 2.11219036 2.11745855
2.07336061 2.12150700 2.14904005 2.15952613
2.05662426 2.09285169 2.11219036 2.11745855
2.07336061 2.12150700 2.14904005 2.15952613
4
0.05 0.1 0.15 0.2
2.28685447 2.31948878 2.33695274 2.34171534
2.30191751 2.34537672 2.37031077 2.37982208
2.28685447 2.31948878 2.33695274 2.34171534
2.30191751 2.34537672 2.37031077 2.37982208
5
0.05 0.1 0.15 0.2
2.49593737 2.52587177 2.54191819 2.54629745
2.50974585 2.54966507 2.57261990 2.58138589
2.49593737 2.52587177 2.54191819 2.54629745
2.50974585 2.54966507 2.57261990 2.58138589
4.1. Stretching sheet results Fig. 1(a) and (b) show the effect of nanoparticle volume fraction parameter / on the nanofluid velocity and concentration profiles, respectively, in the case of a Cu–water and an Ag–water nanofluid. We observe that, as the nanoparticle volume fraction increases, the nanofluid velocity decreases while the concentration increases. This is consistent with the expected physical behavior. When the volume fraction of the nanoparticle increases, the thermal conductivity increases, and the thermal boundary layer increases. It is observed that the velocity in the case of a Ag–water nanofluid is less than that of Cu–water nanofluid. Since the thermal conductivity of Ag is more than that of Cu, we observe that the concentration distribution in a Ag–water nanofluid is higher than that of a Cu–water nanofluid. With increasing nanoparticle volume fraction, the concentration boundary layer thickness increases for both types of nanofluids. The effects of velocity and concentration may be analyzed from Fig. 2(a) and (b) for different values of the porous permeability parameter k1. It is observed that the velocity distribution decreases with increasing permeability parameter, whereas the reverse trend is seen in the case of the concentration distributions. This is because the presence of a porous medium increases the resistance to the flow causing a decrease in the fluid velocity. Hence a rise
Fig. 1. Effect of / on (a) the velocity, and (b) the concentration when K = 1, Sc = 1, Ks = 0.5 and k1 = 1.
Fig. 2. Effect of k1 on (a) the velocity, and (b) the concentration when K = 1, Sc = 1, Ks = 1 and / = 0.2.
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469
Fig. 3. Effect of k1 on (a) the skin friction, and (b) the concentration for K = 1, Sc = 1, Ks = 1 and / = 0.1.
Fig. 4. Effect of Ks and K on the concentration when k1 = 1, Sc = 5 and / = 0.2.
Fig. 5. The effect of variation of / on the concentration when k1 = 1, K = 1 and Ks = 1.
in the concentration is obtained when increasing the porous medium parameter. The velocity in the case of a Ag–water nanofluid is
slightly less than that of a Cu–water nanofluid, while the reverse trend is found in case of the concentration profiles. Fig. 3(a) shows the skin friction coefficient f00 (0) as a function of the permeability parameter k1. The skin friction coefficient increases with increasing values of k1. The results of the skin friction coefficient are reported for both types of nanofluids. We observe that the Ag–water nanofluid gives a higher drag force in opposition to the flow as compared to the Cu–water nanofluid. Fig. 3(b) illustrates the effect of concentration as a function of the porous permeability parameter in the case of the two nanofluids considered. We observe that the concentration decreases with an increase in permeability parameter. Further, it is observed that the Ag–water nanofluid gives a lower drag force in opposition to the flow as compared to the Cu–water nanofluid. The variation of dimensionless concentration for different values of K and Ks are shown in Fig. 4(a) and (b) respectively. From 4(a) it is observed that concentration at the surface decreases as the strength of the heterogeneous reaction increases in the case of the two different types of nanofluids. One can see from Fig. 4(b) that g(0) decreases with the increase of K and Ks in the case of both Cu–water and Ag–water nanofluids. It is also found that the decrease in the Ag–water nanofluid is less than that of the Cu–water nanofluid.
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Fig. 6. Effect of / on the velocity profiles when k1 = 2, K = 1, Sc = 1 and Ks = 0.5.
Fig. 7. Effect of / on the concentration when k1 = 2, K = 1, Sc = 1 and Ks = 0.5.
Fig. 8. Effect of k1 on the velocity when Sc = 1, K = 1, / = 0.2 and Ks = 1.
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471
Fig. 9. Effect of k1 on the concentration when Sc = 1, K = 1, / = 0.2 and Ks = 1.
The influence of Sc on g(0) for two different nanoparticle volume fractions is shown in Fig. 5. It is clear that the concentration increases with an increase in Schmidt number for the two types of nanofluids considered. The increment is more in the case of the Cu–water nanofluid compared to that of the Ag–water nanofluid. 4.2. Shrinking sheet results The shrinking sheet results for various physical parameter values and the nanoparticle volume fraction are shown in Figs. 6–9. Fig. 6 shows the effect of the nanoparticle volume fraction on the fluid velocity for Cu–water and Ag–water nanofluids. The velocity profiles decrease with increasing values of the nanoparticle volume fraction for both nanofluids. Fig. 7 shows the effect of the nanoparticle volume fraction on the fluid concentration. The fluid concentration increases with an increase in the nanoparticle volume fraction. This is explained by the fact that as the nanoparticle volume fraction increases the reaction becomes increasingly confined to a relatively narrow region far from the wall. Figs. 8 and 9 show the effect of the porous permeability parameter on the velocity and concentration profiles. We found that the Cu–water nanofluid has a higher velocity compared to the Ag– water nanofluid for increasing permeability parameter values. The reverse trend was observed in the case of a stretching sheet. 5. Conclusions The problem of steady boundary layer flow in a nanofluid due to a porous stretching sheet with homogeneous and heterogeneous reactions was studied. The governing equations were transformed into a set of coupled nonlinear differential equations and solved numerically by Matlab bvp4c. We found that the velocity profiles decrease with an increase in the nanoparticle volume fraction, while the opposite is true in the case of the concentration profiles. The skin friction coefficient increases with an increase in porous permeability parameter k1. It was observed that the concentration at the surface decreases with the strength of the heterogeneous reaction for both Cu–water and Ag–water nanofluids. In case of shrinking sheet it is noted that velocity profile decreases with an increasing values of nanoparticle volume fraction for the case of Cu–water nanofluid. The same trend is observed in the case of Ag–water nanofluid also.
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[28] V. Velagapudi, R.K. Konijeti, C.S.K. Aduru, Empirical correlation to predict thermophysical and heat transfer characteristics of nanofluids, Therm. Sci. 12 (2) (2008) 27–37. [29] V.Y. Rudyak, A.A. Belkin, E.A. Tomilina, On the thermal conductivity of nanofluids, Tech. Phys. Lett. 36 (7) (2010) 660–662. [30] O.D. Makinde, A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, Int. J. Therm. Sci. 50 (2011) 1326– 1332. [31] A. Alsaedi, M. Awais, T. Hayat, Effects of heat generation/absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4210–4223. [32] M. Narayana, P. Sibanda, Laminar flow of a nanoliquid film over an unsteady stretching sheet, Int. J. Heat Mass Transfer 55 (2012) 7552–7560. [33] P.K. Kameswaran, M. Narayana, P. Sibanda, P.V.S.N. Murthy, Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects, Int. J. Heat Mass Transfer 55 (2012) 7587–7595. [34] R.K. Tiwari, M.N. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [35] H.C. Brinkman, The viscosity of concentrated suspensions and solution, J. Chem. Phys. 20 (1952) 571–581. [36] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet P.K. Kameswaran a, S. Shaw a, P. Sibanda a,⇑, P.V.S.N. Murthy b a b
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, 3209 Pietermaritzburg, South Africa Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India
a r t i c l e
i n f o
Article history: Received 24 August 2012 Received in revised form 13 October 2012 Accepted 14 October 2012 Available online 23 November 2012 Keywords: Homogeneous–heterogeneous reactions Nanofluids Volume fraction model Stretching sheet
a b s t r a c t We investigate the effects of homogeneous–heterogeneous reactions in nanofluid flow over a stretching or shrinking sheet placed in a porous medium saturated with a nanofluid. Copper–water and silver–water nanofluids are investigated in this study. The steady states of this system are analyzed in the case when the diffusion coefficients of the reactant and auto catalyst are equal. The governing partial differential equations are transformed into a system of non-linear ordinary differential equations and solved numerically. An analytical solution for the momentum equation is obtained. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The study of convective flow through porous media has received a great deal of research interest over the last three decades due to its wide and important applications in environmental, geophysical and industrial problems. Prominent applications include the utilization of geothermal energy, the migration of moisture in fibrous insulation, drying of a porous solid, food processing, casting and welding in manufacturing processes, the dispersion of chemical contaminants in different industrial processes, the design of nuclear reactors, chemical catalytic reactors, compact heat exchangers, solar power and many others. The problem of a stretching surface has many engineering applications such as in extrusion processes, melt-spinning, hot rolling, wire drawing, glass-fiber production, manufacture of plastic and rubber sheets. Other applications can be found in the manufacture of polymer sheets, food processing and in the movement of biological fluids, [1–4]. Crane [5] was the first to consider steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moves in its own plane with a velocity varying linearly with the distance from a fixed point. This was subsequently extended by many authors to explore various aspects of heat transfer in a fluid surrounding a stretching sheet [6–15]. Abel et al. [16] investigated the effect of porous medium on the flow and heat transfer of a nonuniform viscoelastic liquid over a non-isothermal stretching sheet. Abel et al. [17] presented a math⇑ Corresponding author. Tel.: +27 33 260 5626; fax: +27 33 260 5648. E-mail address: [email protected] (P. Sibanda). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.047
ematical analysis for the momentum and heat transfer characteristics of the boundary layer flow of an electrically conducting viscoelastic fluid over a linearly stretching sheet. Pal and Mondal [18] studied the influence of a chemical reaction and thermal radiation on a stretching sheet in a Darcian porous medium with Soret and Dufour effects. They found that the temperature profiles increase as the thermal radiation parameter and the Dufour number increase. Recently, Rosali et al. [19] investigated micropolar fluid flow over a stretching/shrinking sheet in a porous medium with suction. They observed that an increase in the permeability parameter led to an increase in the skin friction coefficient and the local Nusselt number. Many chemically reacting systems involve both homogeneous and heterogeneous reactions, with examples occurring in combustion, catalysis and biochemical systems. The interaction between the homogeneous reactions in the bulk of the fluid and heterogeneous reactions occurring on some catalytic surfaces is generally very complex, and is involved in the production and consumption of reactant species at different rates both within the fluid and on the catalytic surfaces. A model for isothermal homogeneous– heterogeneous reactions in boundary layer flow of a viscous fluid past a flat plate was studied by Merkin [20]. He presented the homogeneous reaction by cubic autocatalysis and the heterogeneous reaction by a first order process and showed that the surface reaction is the dominant mechanism near the leading edge of the plate. Chaudhary and Merkin [21] studied homogeneous– heterogeneous reactions in boundary layer flow. They studied the numerical solution near the leading edge of a flat plate. Ziabakhsh et al. [22] studied the problem of flow and diffusion of chemically
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reactive species over a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos [23] studied an isothermal chemical reaction on a catalytic reactor in laminar boundary layer flow. They found the actual surface concentration without introducing unnecessary assumptions related to the reaction mechanism. The effects of flow near the two dimensional stagnation point flow on an infinite permeable wall with a homogeneous–heterogeneous reaction was studied by Khan and Pop [24]. They solved the governing nonlinear equations using the implicit finite difference method and observed that the mass transfer parameter considerably affects the flow characteristics. Khan and Pop [25] studied the effects of homogeneous–heterogeneous reactions on a viscoelastic fluid toward a stretching sheet. They observed that the concentration at the surface decreased with an increase in the viscoelastic parameter. Convectional heat transfer fluids, including oil, water, and ethylene glycol mixtures are poor heat transfer fluids due to the low thermal conductivity of these fluids. The thermal conductivity of these fluids may be improved by suspending nano sized particle materials in the liquid to form a nanofluid. A characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. [26]. Thermophysical properties of nanofluids such as thermal conductivity, diffusivity and viscosity have been studied by, among others, Kang et al. [27], Velagapudi et al. [28] and Rudyak et al. [29]. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition was studied by Makinde and Aziz [30]. They found that with an increase in the Biot number, the concentration layer thickened, but the concentration layer became thinner as the Lewis number increased. Alsaedi et al. [31] studied the effects of heat generation/ absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions. Recently, Narayana and Sibanda [32] studied the effects of laminar flow of a nanoliquid film over an unsteady stretching sheet. They found that the effect of the nanoparticle volume fraction is to reduce the axial velocity and free stream velocity in the case of a Cu-water nanoliquid, but the opposite is true in the case of a Al2O3 – water nanoliquid. Kameswaran et al. [33] studied hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. They found that the velocity profile decreased with an increase in the nano particle volume fraction, while the opposite was true in the case of temperature and concentration profiles. The study further showed that liquids with nanoparticle suspensions are better suited for effective cooling of the stretching sheet problem due to their enhanced conductivity and thermal properties. In this paper we study the combined effects of a porous medium permeability parameter subject to homogeneous–heterogeneous reactions in a nanofluid flow over a stretching or shrinking sheet. The model developed by Merkin [20] for homogeneous–heterogeneous reactions in boundary layer flow with cubic autocatalysis is used in the present study. The transformed ordinary boundary layer equations of motion and concentration are solved numerically. To the best of the authors’ knowledge, this problem has not been considered before in the literature.
2. Mathematical formulation
and the nanoparticles are in thermal equilibrium and no slip occurs between them. It is assumed that a simple homogeneous– heterogeneous reaction model exists as proposed by Chaudhary and Merkin [21] in the following form: 2
A þ 2B ! 3B;
rate ¼ kc ab ;
ð1Þ
while on the catalyst surface we have the single, isothermal, first order reaction
A ! B;
rate ¼ ks a;
ð2Þ
where a and b are the concentrations of the chemical species A and B, kc and ks are the rate constants. We assume that both reaction processes are isothermal. Under these assumptions, the boundary layer equations governing the flow can be written in dimensional form [21,34];
@u @ v þ ¼ 0; @x @y @u @u lnf @ 2 u lnf 1 þv ¼ u; u @x @y qnf @y2 qnf k
ð3Þ ð4Þ
2
@a @a @ a 2 þv ¼ DA 2 kc ab ; @x @y @y @b @b @2b 2 ¼ DB 2 þ kc ab : u þv @x @y @y u
ð5Þ ð6Þ
The boundary conditions for Eqs. (3)–(6) are given in the form:
u ! 0;
@a @b ¼ ks a; DB ¼ ks a at y ¼ 0; @y @y b ! 0 as y ! 1;
v ¼ 0;
u ¼ uw ¼ cx;
a ! a0 ;
DA
ð7Þ where u, v are the velocity components in the x-and y-directions respectively, c is the stretching (c > 0) or shrinking (c < 0) rate, k is the permeability of the porous medium, DA and DB are the respective diffusion species coefficients of A and B, a0 is a positive constant. The effective dynamic viscosity of the nanofluid was given by Brinkman [35] as
lnf ¼
lf ð1 /Þ2:5
ð8Þ
;
where / is the solid volume fraction of nanoparticles. The effective density of the nanofluids is given as
qnf ¼ ð1 /Þqf þ /qs :
ð9Þ
The thermal diffusivity of the nanofluid is
anf ¼
knf ; ðqC p Þnf
ð10Þ
where the heat capacitance of the nanofluid is given by
ðqC p Þnf ¼ ð1 /ÞðqC p Þf þ /ðqC p Þs :
ð11Þ
Here, the subscripts nf, f and s represents the thermophysical properties of the nano fluid, base fluid and nano solid particles, respectively. The continuity Eq. (3) is satisfied by introducing a stream function w(x, y) such that
u¼
@w @y
v¼
and
@w ; @x
1
Consider two dimensional steady boundary layer flow of an incompressible nanofluid over a stretching sheet. A cartesian coordinate system is used with the x-axis along the sheet and the y-axis normal to the sheet. Two equal but opposite forces are applied along the sheet so that the wall is stretched, keeping the position of the origin unaltered. The fluid is a water based nanofluid containing copper (Cu) or silver (Ag) nanoparticles. The base fluid
where w ¼ ðcmf Þ2 xf ðgÞ; f ðgÞ is the dimensionless stream function 1 2
and g ¼ ðc=mf Þ y. The velocity components are then given by 0
u ¼ cxf ðgÞ and
1 2
v ¼ ðcmf Þ f ðgÞ:
ð12Þ
The concentrations of the chemical species A and B are represented as
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a ¼ a0 gðgÞ and b ¼ a0 hðgÞ;
ð13Þ
where a0 is a constant, g(g) and h(g) is the dimensionless concentration. On using Eqs. (8)–(13), Eqs. (4)–(7) transform into the following two-point boundary value problems
Cf ¼
2sw qf u2w
ð25Þ
and using Eq. (24) in Eq. (25) we obtain
pffiffiffiffiffiffiffi Rex ¼ 2f 00 ð0Þ:
f 000 þ /1 ff /1 f 02 k1 f 0 ¼ 0;
ð14Þ
C f ð1 /Þ2:5
1 00 0 2 g þ fg Kgh ¼ 0; Sc
ð15Þ
In Eq. (26), Rex represents the local Reynolds number defined by Rex = xuw/mf.
d 00 0 2 h þ fh þ Kgh ¼ 0; Sc
ð16Þ
3.1. Exact solution for momentum equation
00
f 0 ð0Þ ¼ 1;
f ð0Þ ¼ 0;
g 0 ð0Þ ¼ K s gð0Þ;
f 0 ðgÞ ! 0 as g ! 1
ð17Þ
gðgÞ ! 1 as g ! 1
0
ð18Þ
hðgÞ ! 0 as g ! 1
dh ð0Þ ¼ K s gð0Þ;
ð19Þ
The non-dimensional constants in Eqs. (14)–(16) are the porous medium parameter k1, the Schmidt number Sc, the measure of the strength of the homogeneous reaction K, and the ratio of the diffusion coefficient d. They are respectively defined as
k1 ¼
mf kc
;
Sc ¼
mf DA
;
K¼
kc a20 ; c
d¼
DB ; DA
ð26Þ
The momentum boundary layer equation is partially decoupled from the species equation. The solution is obtained by looking for an exponential function of the form f0 (g) = esg that satisfies both the differential equation and governing boundary conditions over the interval [0, 1). An exact solution to (14) and (17) is obtained as
f ðgÞ ¼
1 esg ; s
ð27Þ
where s is the parameter associated with the nanoparticle volume fraction and the porous parameter, which is defined as
s¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /1 þ k1 :
ð28Þ
where
" 2:5
/1 ¼ ð1 /Þ
q 1/þ/ s qf
!# ð20Þ
:
In most applications, we expect the diffusion coefficients of chemical species A and B to be of a comparable size. This leads us to make a further assumption that the diffusion coefficients DA and DB are equal, i.e., to take d = 1 [21]. In this case we have from Eqs. (18) and (19)
gðgÞ þ hðgÞ ¼ 1:
ð21Þ
Thus Eqs. (15) and (16) reduce to
1 00 0 g þ fg Kgð1 gÞ2 ¼ 0 Sc
ð22Þ
and are subject to the boundary conditions
g 0 ð0Þ ¼ K s gð0Þ;
gðgÞ ! 1 as g ! 1:
ð23Þ
The thermophysical properties of the nanofluid are given in Table 1 [36]. 3. Skin friction coefficient The physical quantity of interest is the skin friction coefficient Cf. It characterizes the surface drag. The shearing stress at the surface of the wall sw is given by
sw ¼ lnf
qffiffiffiffiffiffiffiffiffi @u 1 ¼ qf mf c3 xf 00 ð0Þ; 2:5 @y y¼0 ð1 /Þ
ð24Þ
where lnf is the coefficient of viscosity. The skin friction coefficient is defined as
Table 1 Thermo-physical properties of water and nanoparticles, Oztop and Abu-Nada [36]. 3
5
Properties ?
q (kg/m )
Cp (J/kg K)
k (W/m K)
b 10 (K
Pure water Cu Ag
997.1 8933 10500
4179 385 235
0.613 401 429
21 1.67 1.89
1
)
4. Results and discussion We have studied mass transfer in a nanofluid flow due to a stretching or shrinking sheet in porous medium. We considered two different nanoparticles, copper (Cu) and silver (Ag), with water as the base fluid. An analytical solution was obtained for the momentum equation. The system of ordinary differential equations (14) and (22) along with the boundary conditions (17) and (23) for some values of the physical parameters k1, Ks, K, Sc, / were solved numerically using the Matlab routine bvp4c. A comparison was also made with the analytical results and bvp4c results. The comparison showed good agreement for each value of k1 so that we may be confident that the present results are accurate. Tables 2 and 3 show the results of f00 (0) for different parameter values. Table 2 gives the comparison of analytical and numerical results in the absence of the physical parameters i.e., / = Sc = K = Ks = 0. The numerical solution results are found to be in good agreement with the exact solution. Table 3 shows the skin friction coefficient for various physical parameters in the case of a Cu–water and Ag–water nanofluid. With increasing values of the nanoparticle volume parameter /, the skin friction coefficient increases. The comparison shows good agreement between bvp4c results and the exact solution. The effects of permeability, Schmidt number, strength of the homogeneous reaction and the strength of heterogeneous reaction parameters are shown in Figs. 1–5 for various fluid flow parameters.
Table 2 Comparison of f00 (0) obtained analytically and numerically using bvp4c for different values of k1, for fixed values of Sc = 0, / = 0, K = 0 and Ks = 0. k1
0.5 1 1.5 2 5
f00 (0) Analytical
Numerical
1.22474487 1.41421356 1.58113883 1.73205081 2.44948974
1.22474487 1.41421356 1.58113883 1.73205081 2.44948974
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Table 3 Comparison of f00 (0) for various values of k1 and / for fixed values of Sc = 0, K = 0 and Ks = 0. k1
/
Analytical
Numerical
Cu–Water
Ag–Water
Cu–Water
Ag–Water
1
0.05 0.1 0.15 0.2
1.49321912 1.54273401 1.56886842 1.57595391
1.51618740 1.58138925 1.61813878 1.63203956
1.49321912 1.54273401 1.56886842 1.57595391
1.51618740 1.58138925 1.61813878 1.63203956
2
0.05 0.1 0.15 0.2
1.79713754 1.83848530 1.86046986 1.86644869
1.81626657 1.87104034 1.90220218 1.91404105
1.79713754 1.83848530 1.86046986 1.86644869
1.81626657 1.87104034 1.90220218 1.91404105
3
0.05 0.1 0.15 0.2
2.05662426 2.09285169 2.11219036 2.11745855
2.07336061 2.12150700 2.14904005 2.15952613
2.05662426 2.09285169 2.11219036 2.11745855
2.07336061 2.12150700 2.14904005 2.15952613
4
0.05 0.1 0.15 0.2
2.28685447 2.31948878 2.33695274 2.34171534
2.30191751 2.34537672 2.37031077 2.37982208
2.28685447 2.31948878 2.33695274 2.34171534
2.30191751 2.34537672 2.37031077 2.37982208
5
0.05 0.1 0.15 0.2
2.49593737 2.52587177 2.54191819 2.54629745
2.50974585 2.54966507 2.57261990 2.58138589
2.49593737 2.52587177 2.54191819 2.54629745
2.50974585 2.54966507 2.57261990 2.58138589
4.1. Stretching sheet results Fig. 1(a) and (b) show the effect of nanoparticle volume fraction parameter / on the nanofluid velocity and concentration profiles, respectively, in the case of a Cu–water and an Ag–water nanofluid. We observe that, as the nanoparticle volume fraction increases, the nanofluid velocity decreases while the concentration increases. This is consistent with the expected physical behavior. When the volume fraction of the nanoparticle increases, the thermal conductivity increases, and the thermal boundary layer increases. It is observed that the velocity in the case of a Ag–water nanofluid is less than that of Cu–water nanofluid. Since the thermal conductivity of Ag is more than that of Cu, we observe that the concentration distribution in a Ag–water nanofluid is higher than that of a Cu–water nanofluid. With increasing nanoparticle volume fraction, the concentration boundary layer thickness increases for both types of nanofluids. The effects of velocity and concentration may be analyzed from Fig. 2(a) and (b) for different values of the porous permeability parameter k1. It is observed that the velocity distribution decreases with increasing permeability parameter, whereas the reverse trend is seen in the case of the concentration distributions. This is because the presence of a porous medium increases the resistance to the flow causing a decrease in the fluid velocity. Hence a rise
Fig. 1. Effect of / on (a) the velocity, and (b) the concentration when K = 1, Sc = 1, Ks = 0.5 and k1 = 1.
Fig. 2. Effect of k1 on (a) the velocity, and (b) the concentration when K = 1, Sc = 1, Ks = 1 and / = 0.2.
P.K. Kameswaran et al. / International Journal of Heat and Mass Transfer 57 (2013) 465–472
469
Fig. 3. Effect of k1 on (a) the skin friction, and (b) the concentration for K = 1, Sc = 1, Ks = 1 and / = 0.1.
Fig. 4. Effect of Ks and K on the concentration when k1 = 1, Sc = 5 and / = 0.2.
Fig. 5. The effect of variation of / on the concentration when k1 = 1, K = 1 and Ks = 1.
in the concentration is obtained when increasing the porous medium parameter. The velocity in the case of a Ag–water nanofluid is
slightly less than that of a Cu–water nanofluid, while the reverse trend is found in case of the concentration profiles. Fig. 3(a) shows the skin friction coefficient f00 (0) as a function of the permeability parameter k1. The skin friction coefficient increases with increasing values of k1. The results of the skin friction coefficient are reported for both types of nanofluids. We observe that the Ag–water nanofluid gives a higher drag force in opposition to the flow as compared to the Cu–water nanofluid. Fig. 3(b) illustrates the effect of concentration as a function of the porous permeability parameter in the case of the two nanofluids considered. We observe that the concentration decreases with an increase in permeability parameter. Further, it is observed that the Ag–water nanofluid gives a lower drag force in opposition to the flow as compared to the Cu–water nanofluid. The variation of dimensionless concentration for different values of K and Ks are shown in Fig. 4(a) and (b) respectively. From 4(a) it is observed that concentration at the surface decreases as the strength of the heterogeneous reaction increases in the case of the two different types of nanofluids. One can see from Fig. 4(b) that g(0) decreases with the increase of K and Ks in the case of both Cu–water and Ag–water nanofluids. It is also found that the decrease in the Ag–water nanofluid is less than that of the Cu–water nanofluid.
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Fig. 6. Effect of / on the velocity profiles when k1 = 2, K = 1, Sc = 1 and Ks = 0.5.
Fig. 7. Effect of / on the concentration when k1 = 2, K = 1, Sc = 1 and Ks = 0.5.
Fig. 8. Effect of k1 on the velocity when Sc = 1, K = 1, / = 0.2 and Ks = 1.
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471
Fig. 9. Effect of k1 on the concentration when Sc = 1, K = 1, / = 0.2 and Ks = 1.
The influence of Sc on g(0) for two different nanoparticle volume fractions is shown in Fig. 5. It is clear that the concentration increases with an increase in Schmidt number for the two types of nanofluids considered. The increment is more in the case of the Cu–water nanofluid compared to that of the Ag–water nanofluid. 4.2. Shrinking sheet results The shrinking sheet results for various physical parameter values and the nanoparticle volume fraction are shown in Figs. 6–9. Fig. 6 shows the effect of the nanoparticle volume fraction on the fluid velocity for Cu–water and Ag–water nanofluids. The velocity profiles decrease with increasing values of the nanoparticle volume fraction for both nanofluids. Fig. 7 shows the effect of the nanoparticle volume fraction on the fluid concentration. The fluid concentration increases with an increase in the nanoparticle volume fraction. This is explained by the fact that as the nanoparticle volume fraction increases the reaction becomes increasingly confined to a relatively narrow region far from the wall. Figs. 8 and 9 show the effect of the porous permeability parameter on the velocity and concentration profiles. We found that the Cu–water nanofluid has a higher velocity compared to the Ag– water nanofluid for increasing permeability parameter values. The reverse trend was observed in the case of a stretching sheet. 5. Conclusions The problem of steady boundary layer flow in a nanofluid due to a porous stretching sheet with homogeneous and heterogeneous reactions was studied. The governing equations were transformed into a set of coupled nonlinear differential equations and solved numerically by Matlab bvp4c. We found that the velocity profiles decrease with an increase in the nanoparticle volume fraction, while the opposite is true in the case of the concentration profiles. The skin friction coefficient increases with an increase in porous permeability parameter k1. It was observed that the concentration at the surface decreases with the strength of the heterogeneous reaction for both Cu–water and Ag–water nanofluids. In case of shrinking sheet it is noted that velocity profile decreases with an increasing values of nanoparticle volume fraction for the case of Cu–water nanofluid. The same trend is observed in the case of Ag–water nanofluid also.
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