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Free convection of non-Newtonian nanofluids about a vertical truncated cone in a porous medium
International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
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Free convection of non-Newtonian nanofluids about a vertical truncated cone in a porous medium☆ Ching-Yang Cheng Department of Mechanical Engineering, Southern Taiwan University of Science and Technology, Yungkang 71005, Taiwan
a r t i c l e
i n f o
Available online 30 August 2012 Keywords: Free convection Nanofluid Non-Newtonian Vertical truncated cone Porous medium
a b s t r a c t This work studies the free convection heat transfer over a truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effects of the power-law index, Brownian motion parameter, thermophoresis parameter and buoyancy ratio on the temperature, nanoparticle volume fraction and velocity profiles are discussed. The reduced Nusselt numbers are plotted as functions of the power-law index, thermophoresis parameter, Brownian parameter, Lewis number, and buoyancy ratio. Results show that increasing the thermophoresis parameter or the Brownian parameter tends to decrease the reduced Nusselt number. Moreover, the reduced Nusselt number increases as the power-law index is increased. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The non-linear characteristics of non-Newtonian power-law fluids in porous media are quite different from those of Newtonian fluids in porous media. It is of much importance to study the heat and mass transfer characteristics about free convection of non-Newtonian power-law fluids in porous media because of practical engineering applications, such as oil recovery and food processing. Chen and Chen [1] obtained similarity solutions for free convection of a non-Newtonian fluid over a vertical plate in a porous medium. Nakayama and Koyama [2] studied the natural convection a non-Newtonian fluid over a non-isothermal body of arbitrary shape embedded in a porous medium. Rastogi and Poulikakos [3] examined the problem of double diffusion from a vertical surface in a porous medium saturated with a non-Newtonian power law fluid. Getachew et al. [4] performed a numerical and theoretical study of double-diffusive natural convection in a rectangular porous cavity saturated by a non-Newtonian power law fluid. Benhadj and Vasseur [5] studied the double diffusive convection in a shallow porous cavity filled with a non-Newtonian fluid. Kim and Hyun [6] studied the natural convection heat transfer of power law fluid in an enclosure filled with heat-generating porous media. Hadim [7] examined the non-Darcy natural convection of a non-Newtonian fluid in a porous cavity. Nanofluids refer to a liquid containing a dispersion of nanoparticles. The nanoparticles are different from conventional particles in that they keep suspended in the base fluid without sedimentation. The heat transfer and flow characteristics of nanofluids are of growing interest ☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.08.004
and of much importance because these fluids represent a possible way to enhance the heat transfer for mixed convection and forced convection [8–11]. This feature of nanofluids can be used to enhance the heat transfer in cylindrical heat pipes [8]. Buongiorno [9] made a comprehensive survey of convection in nanofluids and wrote down conservation equations for nanofluids based on the effects of Brownian motion and thermophoresis. Das et al. [10] made a comprehensive review on the heat transfer in nanofluids. Tiwari and Das [11] studied the heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. There are a lot of works on the convection heat transfer in a porous medium saturated with nanofluids. Nield and Kuznetsov [12] studied the Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid considering the effects of Brownian motion and thermophoresis. Nield and Kuznetsov [13] examined the thermal instability in a porous medium layer saturated by a nanofluid with the Brinkman model. Ahmad and Pop [14] studied the mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Gorla and Chamkha [15] studied the natural convective boundary layer flow over a nonisothermal vertical plate embedded in a porous medium saturated with a nanofluid. Khan and Pop [16] examined the free convection boundary layer flow past a horizontal flat plate embedded in a porous medium filled with a nanofluid. Chamkha et al. [17] presented the non-similar solutions for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Rashad et al. [18] studied the free convection heat transfer of a non-Newtonian fluid about a permeable vertical cone in a porous medium saturated with a nanofluid.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
Nomenclature A DB DT f g h km K Le n Nb Nr Nt Nu r Ra T u,v x,y
half angle of the truncated cone Brownian diffusion coefficient thermophoretic diffusion coefficient dimensionless stream function acceleration due to gravity heat transfer coefficient effective thermal conductivity of the porous medium permeability of the porous medium Lewis number power-law index Brownian motion parameter buoyancy ratio thermophoresis parameter local Nusselt number local radius of the cone Rayleigh number temperature dimensional velocity components along x and y axes dimensional Cartesian coordinates along and normal to the cone
Greek symbols αm thermal diffusivity of the porous medium β volumetric expansion coefficient of the fluid ε porosity of the porous medium η nondimensional coordinate θ dimensionless temperature μeff effective viscosity of the fluid ξ nondimensional coordinate ρf fluid density ρp nanoparticle mass density (ρc)f heat capacity of the fluid (ρc)p effective heat capacity of the nanoparticle material τ heat capacity ratio ϕ nanoparticle volume fraction Φ dimensionless nanoparticle volume fraction ψ stream function
collocation method. The effects of the power-law index, thermophoresis parameter, Brownian motion parameter, Lewis number, and buoyancy ratio on the heat transfer and fluid flow characteristics have been studied. 2. Analysis We consider the boundary layer flow due to free convection over a downward-pointing vertical truncated cone of half angle A embedded in a porous medium saturated with a non-Newtonian power-law nanofluid. The origin of the coordinate system is placed at the vertex of the full cone, with x being the coordinate along the surface of the cone measured from the origin and y being the coordinate perpendicular to the conical surface, as shown in Fig. 1. The surface of the cone is maintained at a constant temperature Tw, which is different from the porous medium temperature sufficiently far from the surface of the cone. The nanoparticle volume fraction on the surface of the cone is ϕw and the ambient value of the nanoparticle volume fraction is denoted by ϕ∞. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermal and nanoparticle volume fraction boundary layers are assumed to be sufficiently thin compared with the local radius. The governing equations for the conservation of total mass, momentum, energy, and nanoparticles within the boundary layer near the vertical truncated cone in a porous medium saturated by a nonNewtonian power-law nanofluid can be written in two-dimensional Cartesian coordinates (x,y) as [12,20] ∂ðruÞ ∂ðrvÞ þ ¼0 ∂x ∂y
ð1Þ
∂un K ∂T K ∂ϕ ð1−ϕ∞ Þρf∞ βg cos A − ρp −ρf∞ g cos A ¼ μ eff ∂y ∂y μ eff ∂y
ð2Þ
" # 2 ∂T ∂T ∂ T ∂ϕ ∂T DT ∂T 2 þv ¼ α m 2 þ τ DB þ u ∂x ∂y ∂y ∂y T ∞ ∂y ∂y
ð3Þ
2 1 ∂ϕ ∂ϕ ∂2 ϕ DT ∂ T u : þv ¼ DB 2 þ ε T ∞ ∂y2 ∂x ∂y ∂y
ð4Þ
Subscripts w condition at wall ∞ condition at infinity
There are a lot of works on the free convection about a cone in a porous medium saturated with Newtonian or non-Newtonian fluids. Cheng et al. [19] studied the free convection of a Darcian fluid about a cone. Yih [20] studied the coupled heat and mass transfer by free convection over a truncated cone in porous media with variable wall temperature and concentration. Cheng [21] used an integral approach for heat and mass transfer by free convection from truncated cones in porous media with variable wall temperature and concentration. Cheng [22] studied the free convection heat and mass transfer from a vertical truncated cone in a porous medium saturated with non-Newtonian power-law fluid with variable wall temperature and concentration. This work aims to study the free convection heat transfer from a vertical truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid using the model proposed by Nield and Kuznetsov [12]. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline
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Fig. 1. Physical model and coordinates for a truncated cone.
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C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
Here u and v are the volume-averaged velocity components in the x and y directions, respectively. T is the volume-averaged temperature. ϕ is the nanoparticle volume fraction. n is the power-law index of the non-Newtonian fluid. ρf, μeff and β are the density, effective viscosity, and volumetric volume expansion coefficient of the fluid. ϕp is the density of the particles. K and ε are the permeability and porosity of the porous medium. Furthermore, (ρc)f and (ρc)p are the heat capacity of the fluid and the effective heat capacity of the nanoparticle, respectively. αm is the thermal diffusivity of the porous medium. DB and DT are the Brownian diffusion coefficient and the thermophoresis diffusion coefficient, respectively. The heat capacity ratio between nanoparticle and fluid is defined as τ ¼ εðρcÞp =ðρcÞf . The gravitational acceleration is denoted by g. The boundary conditions are given by y ¼ 0 : v ¼ 0 T ¼ Tw y→∞ : u→0
T→T ∞
ϕ ¼ ϕw on y ¼ 0; x0 ≤x≤∞ ϕ ¼ ϕ∞ as y→∞ :
ð5Þ ð6Þ
Because the boundary layer thickness is small, the local radius to a point in the boundary layer r can be represented by the local radius of the truncated cone, r ¼ x sin A:
ð7Þ
We introduce the nondimensional variables: ξ¼
x x−x0 y 1=ð2nÞ ψ T−T ∞ ϕ−ϕ∞ ¼ ; η ¼ Ra ; f ð ηÞ ¼ ;θ ¼ ;Φ ¼ x x0 T w −T ∞ ϕw −ϕ∞ x0 α m rRa1=ð2nÞ
ð8Þ where ψ is the stream function defined as u¼
1 ∂ψ 1 ∂ψ ;v ¼ − r ∂y r ∂x
ð9Þ
and the Rayleigh number is given by ρf∞ βgK xn ð1−ϕ∞ ÞðT w −T ∞ Þ cos A : μα nm
Ra ¼
ð10Þ
Upon using these variables, the basic equations of the boundary layer for the problem under consideration can be written in a non-dimensional form as n ′ f −θ þ NrΦ ¼ 0 ″
θ þ
1 ξ ′ ′ ′ ′2 ′ ∂θ ′ ∂f þ f θ þ NbΦ θ þ Ntθ ¼ ξ f −θ 2 1þξ ∂ξ ∂ξ
″ Φ 1 ξ 1 Nt ″ ′ ′ ∂Φ ′ ∂f þ fΦ þ θ ¼ξ f þ −Φ 2 1þξ Le Nb Le ∂ξ ∂ξ
ð11Þ ð12Þ
ð13Þ
Nt ¼
εðρcÞp DT ðT w −T ∞ Þ ðρcÞf α m T ∞
:
ð17Þ
Here Le, Nb, Nr, and Nt denote the Lewis number, the Brownian motion parameter, the buoyancy ratio, and the thermophoresis parameter, respectively. The boundary conditions are transformed to η¼0:
f ¼0
θ¼1
Φ¼1
ð18Þ
η→∞ :
f ′ →0
θ→0
Φ→0 :
ð19Þ
The quantity of practical interest is the local Nusselt number given by Nu ′ ¼ −θ ðξ; 0Þ: Ra1=ð2nÞ
ð20Þ
In Eq. (20), Nu ¼ hx=km , where h is the local heat transfer coefficient and km is the effective thermal conductivity of the porous medium. 3. Results and discussion The transformed governing equations, Eqs. (12) and (13), and the associated boundary conditions, Eqs. (18) and (19), can be solved by the cubic spline collocation method [23,24]. The velocity f′ is calculated from the momentum equation, Eq. (11). Moreover, the Simpson's rule for variable grids is used to calculate the value of f at every position from the boundary conditions, Eqs. (18) and (19). At every position, the iteration process continues until the convergence criterion for all the variables, 10 −6, is achieved. Variable grids with 300 grid points are used in the η-direction. The optimum value of boundary layer thickness is used. To assess the accuracy of the solution, the present results are compared with the results obtained by other researchers. Table 1 shows the numerical values of − θ′(0,0) for different values of n with Nr = 0, Nb = 0, and Nt = 0, the conditions for natural convection heat of a vertical plate of non-Newtonian fluids in porous media with constant wall temperature. It is shown that the present results are in excellent agreement with the results reported by Chen and Chen [1]. Fig. 2 shows the effect of the thermophoresis parameter Nt, buoyancy ratio Nr, the Brownian motion parameter Nb and power-law index n on the dimensionless temperature profile θ. An increase in the thermophoresis parameter or buoyancy ratio tends to thicken the thermal boundary layer, thus decreasing the temperature gradient at the wall, as shown in Fig. 2a. Moreover, Fig. 2b shows that, as the Brownian motion is decreased, the thermal boundary layer thickness decreases, thus increasing the temperature gradient at the wall. Furthermore, it is noted that an increase in the power-law index tends to increase the temperature gradient at the wall. Fig. 3 shows the effect of the thermophoresis parameter Nt, buoyancy ratio Nr, the Brownian motion parameter Nb and power-law index n on the nanoparticle volume fraction profile ϕ. Increasing the thermophoresis parameter or buoyancy ratio tends to decrease the boundary-layer thickness of the nanoparticle volume fraction, thus
where primes denote differentiation with respect to η. Note that Eq. (11) has been obtained by integrating the transformed momentum equation. Moreover, the four important parameters are defined as Le ¼
αm εDB
Nb ¼
εðρcÞp DB ðϕw −ϕ∞ Þ ðρcÞf α m
Nr ¼
ð14Þ
ρp −ρf∞ ðϕw −ϕ∞ Þ
ρf∞ βðT w −T ∞ Þð1−ϕ∞ Þ
ð15Þ
ð16Þ
Table 1 Comparison of values of −θ′(0,0) for various values of n with Nr=0, Nb=0, and Nt=0. n
Chen and Chen [1]
Present results
0.4 0.5 0.8 1.0 1.2 1.5 2.0
0.3533 0.3769 0.4238 0.4437 0.4588 0.4753 0.4937
0.3531 0.3771 0.4240 0.4439 0.4590 0.4755 0.4938
C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
a
a
1.0
1351
1.0 Nr=0.4 Nt=0.1
Nr=0.4 Nt=0.1 Nr=0.4 Nt=0.3
Nr=0.4 Nt=0.3
Nr=0.8 Nt=0.1 n=0.8 Nb=0.2 Le=9 ξ=1
0.8
Nr=0.8 Nt=0.1
0.8
n=0.8 Nb=0.2 Le=9
0.6
0.6
φ
θ 0.4
0.4
0.2
0.2
0.0
0.0 0
1
2
3
4
5
6
0
1
2
η
b
ξ=1
3
4
b
1.0
1.0
Nb=0.2
n=0.8
Nb=0.7 Nr=0.4 Nt=0.3 Le=9 ξ=1
0.8
5
η
n=1.2
0.8
Nr=0.4 Nt=0.3 ξ=1 Le=9
0.6
0.6
φ
θ
0.4
0.4
0.2
0.2
n=0.8
Nb=0.2
n=1.2
Nb=0.7
0.0
0.0 0
1
2
3
4
5
6
η
0
1
2
η
3
4
Fig. 2. The effect of (a) thermophoresis parameter and buoyancy ratio; and (b) Brownian motion parameter and power-law index on the temperature profile.
Fig. 3. The effect of (a) thermophoresis parameter and buoyancy ratio; and (b) Brownian motion parameter and power-law index on the nanoparticle volume fraction profile.
increasing the nanoparticle volume fraction gradient at the wall, as shown in Fig. 3a. Moreover, as the Brownian motion parameter or the power-law index is increased, the boundary layer thickness of the nanoparticle volume fraction decreases, thus increasing the nanoparticle volume fraction gradient at the wall, as shown in Fig. 3b. Fig. 4 shows the effect of the thermophoresis parameter Nt, buoyancy ratio Nr, the Brownian motion parameter Nb and power-law index n on the streamwise velocity profile f′. Increasing the Brownian motion parameter or the power-law index tends to increase the maximum streamwise velocity, thus enhancing the fluid flow, as shown in Fig. 4a. Moreover, an increase in the buoyancy ratio tends to increase the maximum streamwise velocity, as shown in Fig. 4b. The effect of the thermophoresis parameter on the maximum streamwise velocity is not significant. Fig. 5 depicts the variation of the reduced Nusselt number NuRa − 1/(2n) with the streamwise coordinate ξ for various values of thermophoresis parameter Nt and power-law index n for Le = 9, Nb = 0.2, and Nr = 0.4. Fig. 5 shows that the reduced Nusselt number increases as the streamwise coordinate is increased. Moreover, the reduced Nusselt number tends to decrease as the thermophoresis parameter increases. As the thermophoresis parameter is increased, the thermal boundary layer thickness increases, thus decreasing the temperature gradient at the wall and the reduced Nusselt number, as
shown in Fig. 2. Moreover, an increase in the power-law index tends to increase the reduced Nusselt number. Fig. 6 shows the variation of the reduced Nusselt number NuRa − 1/(2n) with the streamwise coordinate ξ for various values of buoyancy ratio Nr and power-law index n for Le = 9, Nb = 0.2, and Nt = 0.3. It is clearly seen that the reduced Nusselt number tends to decrease as the buoyancy ratio increases. Increasing the buoyancy tends to thicken the thermal boundary layer, thus decreasing the temperature gradient at the wall and the reduced Nusselt number, as shown in Fig. 2. Fig. 7 depicts the variation of the reduced Number NuRa −1/(2n) with the streamwise coordinate ξ for various values of Brownian motion parameter Nb and power-law index n with Le= 9, Nr = 0.4, and Nt = 0.3. Results show that the reduced Nusselt number tends to decrease as the Brownian motion parameter increases. An increase in the Brownian motion parameter tends to increase the thermal boundary layer thickness, thus decreasing the temperature gradient at the wall and the reduced Nusselt number, as shown in Fig. 2. Moreover, Fig. 8 depicts the variation of the reduced Nusselt number NuRa −1/(2n) with the streamwise coordinate ξ for various values of Lewis number Le and power-law index n with Nb = 0.2, Nr = 0.4, and Nt = 0.3. Fig. 8 shows that an increase in the Lewis number tends to increase the reduced Nusselt number.
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C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
a
0.7 0.7 0.6
Le=9 Nb=0.2 Nt=0.3
Nr=0.2 Nr=0.4
Nr=0.8 Nt=0.3 Nr=0.4 Nt=0.3 Nr=0.4 Nt=0.1 n=0.8 Nb=0.2 Le=9 ξ=1
0.6
n=1.2
0.5 0.5
Nu
0.4
f ′
Ra
n=0.8
1 (2 n )
0.4
0.3 0.2
0.3 0.1 0.2 0.0 0
1
2
3
4
5
1E-3
6
1E-2
1E-1
1E+1
1E+2
1E+3
ξ
η
b
1E+0
Fig. 6. The effect of the buoyancy ratio on the reduced Nusselt number for Le=9, Nb=0.2, and Nt=0.3.
1.0 Nb=0.2 Nb=0.7 Nr=0.4 Nt=0.3 Le=9 ξ=1
0.8
4. Conclusions 0.6
f′ 0.4
n=1.2
0.2
n=0.8 0.0 0
1
2
3
4
η
5
6
Fig. 4. The effect of (a) thermophoresis parameter and buoyancy ratio; and (b) Brownian motion parameter and power-law index on the streamwise velocity profile.
This work studies the free convection heat transfer from a truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The model used for nanofluids incorporates the effects of Brownian motion and thermophoresis. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effects of the power-law index, thermophoresis parameter, Brownian motion parameter, Lewis number, and buoyancy ratio on the heat transfer and fluid flow characteristics have been studied. Results show that increasing the thermophoresis parameter or the Brownian motion parameter tends to decrease the reduced Nusselt number. Moreover, an increase in the power-law index tends to increase the reduced Nusselt number.
0.7
0.6 Nt=0.3
n=1.2 n=0.8
Le=9 Nb=0.2 Nr=0.4
0.6
Nt=0.5
0.5
0.5 0.4
Nu Ra
1 (2 n )
Le=9 Nr=0.4 Nt=0.3
Nb=0.2
Nu
0.4
Ra
1 (2 n )
0.3
n=1.2 0.2 0.3
Nb=0.6 0.1
n=0.8 0.0
0.2 1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
ξ Fig. 5. The effect of the thermophoresis parameter on the reduced Nusselt number for Le = 9, Nb = 0.2, and Nr = 0.4.
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
ξ Fig. 7. The effect of the Brownian motion parameter on the reduced Nusselt number for Le = 9, Nr= 0.4, and Nt = 0.3.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
0.6
Le=5 Le=9
Nb=0.2 Nt=0.3 Nr=0.4
0.5
Nu Ra
0.4
1 (2 n )
n=1.2 0.3
n=0.8 0.2 1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
ξ Fig. 8. The effect of the Lewis number on the reduced Nusselt number for Nb = 0.2, Nr = 0.4, and Nt = 0.3.
Acknowledgement This work was supported by National Science Council of Republic of China under Grant No. NSC 100-2221-E-218-045. References [1] H.T. Chen, C.K. Chen, Free convection of non-Newtonian fluids along a vertical plate embedded in a porous medium, ASME Journal of Heat Transfer 110 (1988) 257–260. [2] A. Nakayama, H. Koyama, Buoyancy induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium, Applied Scientific Research 48 (1991) 55–70. [3] S.K. Rastogi, D. Poulikakos, Double-diffusion from a vertical surface in a porous region saturated with a non-Newtonian fluid, International Journal of Heat and Mass Transfer 38 (1995) 935–946. [4] D. Getachew, D. Poulikakos, W.J. Minkowycz, Double diffusion in a porous cavity saturated with non-Newtonian fluid, Journal of Thermophysics and Heat Transfer 12 (1998) 437–446. [5] K. Benhadj, P. Vasseur, Double diffusive convection in a shallow porous cavity filled with a non-Newtonian fluid, International Communications in Heat and Mass Transfer 28 (2001) 723–732.
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[6] G.B. Kim, J.M. Hyun, Buoyant convection of power-law fluid in an enclosure filled with heat-generating porous media, Numerical Heat Transfer Part A: Applications 45 (2004) 569–582. [7] H. Hadim, Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity, International Communications in Heat and Mass Transfer 33 (2006) 1179–1189. [8] K.N. Shukla, A. Brusty Solomon, B.C. Pillai, Mohammed Ibrahim, Thermal performance of cylindrical heat pipe using nanofluids, Journal of Thermophysics and Heat Transfer 24 (2010) 796–802. [9] J. Buongiorno, Convective transport in nanofluids, ASME Journal of Heat Transfer 128 (2006) 240–250. [10] S.K. Das, S.U.S. Choi, H.E. Patel, Heat transfer in nanofluids — a review, Heat Transfer Engineering 27 (2006) 3–19. [11] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, International Journal of Heat and Mass Transfer 50 (2007) 2002–2018. [12] D.A. Nield, A.V. Kuznetsov, The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, International Journal of Heat and Mass Transfer 52 (2009) 5792–5795. [13] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model, Transport in Porous Media 81 (2010) 409–422. [14] S. Ahmad, I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, International Communications in Heat and Mass Transfer 37 (2010) 987–991. [15] R.S.R. Gorla, A. Chamkha, Natural convective boundary layer flow over a nonisothermal vertical plate embedded in a porous medium saturated with a nanofluid, Nanoscale and Microscale Thermophysical Engineering 15 (2011) 81–94. [16] W.A. Khan, I. Pop, Free convection boundary layer flow past a horizontal flat plate embedded in a porous medium filled with a nanofluid, ASME Journal of Heat Transfer 133 (2011) 094501.1–094501.4. [17] A. Chamkha, R.S.R. Gorla, K. Ghodeswar, Non-similar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid, Transport in Porous Media 86 (2011) 13–22. [18] A.M. Rashad, M.a. El-Hakiem, M.M.M. Abdou, Natural convection boundary layer of a non-Newtonian fluid about a permeable vertical cone embedded in a porous medium saturated with a nanofluid, Computers and Mathematics with Applications 62 (2011) 3140–3151. [19] P. Cheng, T.T. Le, I. Pop, Natural convection of a Darcian fluid about a cone, International Communications in Heat and Mass Transfer 12 (1985) 705–717. [20] K.A. Yih, Coupled heat and mass transfer by free convection over a truncated cone in porous media: VWT/VWC or VHF/VMF, Acta Mechanica 137 (1999) 83–97. [21] C.Y. Cheng, An integral approach for heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and concentration, International Communications in Heat and Mass Transfer 27 (2000) 437–548. [22] C.Y. Cheng, Natural convection heat and mass transfer from a vertical truncated cone in a porous medium saturated with non-Newtonian fluid with variable wall temperature and concentration, International Communications in Heat and Mass Transfer 36 (2009) 585–589. [23] S.G. Rubin, R.A. Graves, Viscous flow solution with a cubic spline approximation, Computers and Fluids 3 (1975) 1–36. [24] P. Wang, R. Kahawita, Numerical integration of a partial differential equations using cubic spline, International Journal of Computer Mathematics 13 (1983) 271–286.
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Free convection of non-Newtonian nanofluids about a vertical truncated cone in a porous medium☆ Ching-Yang Cheng Department of Mechanical Engineering, Southern Taiwan University of Science and Technology, Yungkang 71005, Taiwan
a r t i c l e
i n f o
Available online 30 August 2012 Keywords: Free convection Nanofluid Non-Newtonian Vertical truncated cone Porous medium
a b s t r a c t This work studies the free convection heat transfer over a truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effects of the power-law index, Brownian motion parameter, thermophoresis parameter and buoyancy ratio on the temperature, nanoparticle volume fraction and velocity profiles are discussed. The reduced Nusselt numbers are plotted as functions of the power-law index, thermophoresis parameter, Brownian parameter, Lewis number, and buoyancy ratio. Results show that increasing the thermophoresis parameter or the Brownian parameter tends to decrease the reduced Nusselt number. Moreover, the reduced Nusselt number increases as the power-law index is increased. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The non-linear characteristics of non-Newtonian power-law fluids in porous media are quite different from those of Newtonian fluids in porous media. It is of much importance to study the heat and mass transfer characteristics about free convection of non-Newtonian power-law fluids in porous media because of practical engineering applications, such as oil recovery and food processing. Chen and Chen [1] obtained similarity solutions for free convection of a non-Newtonian fluid over a vertical plate in a porous medium. Nakayama and Koyama [2] studied the natural convection a non-Newtonian fluid over a non-isothermal body of arbitrary shape embedded in a porous medium. Rastogi and Poulikakos [3] examined the problem of double diffusion from a vertical surface in a porous medium saturated with a non-Newtonian power law fluid. Getachew et al. [4] performed a numerical and theoretical study of double-diffusive natural convection in a rectangular porous cavity saturated by a non-Newtonian power law fluid. Benhadj and Vasseur [5] studied the double diffusive convection in a shallow porous cavity filled with a non-Newtonian fluid. Kim and Hyun [6] studied the natural convection heat transfer of power law fluid in an enclosure filled with heat-generating porous media. Hadim [7] examined the non-Darcy natural convection of a non-Newtonian fluid in a porous cavity. Nanofluids refer to a liquid containing a dispersion of nanoparticles. The nanoparticles are different from conventional particles in that they keep suspended in the base fluid without sedimentation. The heat transfer and flow characteristics of nanofluids are of growing interest ☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.08.004
and of much importance because these fluids represent a possible way to enhance the heat transfer for mixed convection and forced convection [8–11]. This feature of nanofluids can be used to enhance the heat transfer in cylindrical heat pipes [8]. Buongiorno [9] made a comprehensive survey of convection in nanofluids and wrote down conservation equations for nanofluids based on the effects of Brownian motion and thermophoresis. Das et al. [10] made a comprehensive review on the heat transfer in nanofluids. Tiwari and Das [11] studied the heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. There are a lot of works on the convection heat transfer in a porous medium saturated with nanofluids. Nield and Kuznetsov [12] studied the Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid considering the effects of Brownian motion and thermophoresis. Nield and Kuznetsov [13] examined the thermal instability in a porous medium layer saturated by a nanofluid with the Brinkman model. Ahmad and Pop [14] studied the mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Gorla and Chamkha [15] studied the natural convective boundary layer flow over a nonisothermal vertical plate embedded in a porous medium saturated with a nanofluid. Khan and Pop [16] examined the free convection boundary layer flow past a horizontal flat plate embedded in a porous medium filled with a nanofluid. Chamkha et al. [17] presented the non-similar solutions for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Rashad et al. [18] studied the free convection heat transfer of a non-Newtonian fluid about a permeable vertical cone in a porous medium saturated with a nanofluid.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
Nomenclature A DB DT f g h km K Le n Nb Nr Nt Nu r Ra T u,v x,y
half angle of the truncated cone Brownian diffusion coefficient thermophoretic diffusion coefficient dimensionless stream function acceleration due to gravity heat transfer coefficient effective thermal conductivity of the porous medium permeability of the porous medium Lewis number power-law index Brownian motion parameter buoyancy ratio thermophoresis parameter local Nusselt number local radius of the cone Rayleigh number temperature dimensional velocity components along x and y axes dimensional Cartesian coordinates along and normal to the cone
Greek symbols αm thermal diffusivity of the porous medium β volumetric expansion coefficient of the fluid ε porosity of the porous medium η nondimensional coordinate θ dimensionless temperature μeff effective viscosity of the fluid ξ nondimensional coordinate ρf fluid density ρp nanoparticle mass density (ρc)f heat capacity of the fluid (ρc)p effective heat capacity of the nanoparticle material τ heat capacity ratio ϕ nanoparticle volume fraction Φ dimensionless nanoparticle volume fraction ψ stream function
collocation method. The effects of the power-law index, thermophoresis parameter, Brownian motion parameter, Lewis number, and buoyancy ratio on the heat transfer and fluid flow characteristics have been studied. 2. Analysis We consider the boundary layer flow due to free convection over a downward-pointing vertical truncated cone of half angle A embedded in a porous medium saturated with a non-Newtonian power-law nanofluid. The origin of the coordinate system is placed at the vertex of the full cone, with x being the coordinate along the surface of the cone measured from the origin and y being the coordinate perpendicular to the conical surface, as shown in Fig. 1. The surface of the cone is maintained at a constant temperature Tw, which is different from the porous medium temperature sufficiently far from the surface of the cone. The nanoparticle volume fraction on the surface of the cone is ϕw and the ambient value of the nanoparticle volume fraction is denoted by ϕ∞. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermal and nanoparticle volume fraction boundary layers are assumed to be sufficiently thin compared with the local radius. The governing equations for the conservation of total mass, momentum, energy, and nanoparticles within the boundary layer near the vertical truncated cone in a porous medium saturated by a nonNewtonian power-law nanofluid can be written in two-dimensional Cartesian coordinates (x,y) as [12,20] ∂ðruÞ ∂ðrvÞ þ ¼0 ∂x ∂y
ð1Þ
∂un K ∂T K ∂ϕ ð1−ϕ∞ Þρf∞ βg cos A − ρp −ρf∞ g cos A ¼ μ eff ∂y ∂y μ eff ∂y
ð2Þ
" # 2 ∂T ∂T ∂ T ∂ϕ ∂T DT ∂T 2 þv ¼ α m 2 þ τ DB þ u ∂x ∂y ∂y ∂y T ∞ ∂y ∂y
ð3Þ
2 1 ∂ϕ ∂ϕ ∂2 ϕ DT ∂ T u : þv ¼ DB 2 þ ε T ∞ ∂y2 ∂x ∂y ∂y
ð4Þ
Subscripts w condition at wall ∞ condition at infinity
There are a lot of works on the free convection about a cone in a porous medium saturated with Newtonian or non-Newtonian fluids. Cheng et al. [19] studied the free convection of a Darcian fluid about a cone. Yih [20] studied the coupled heat and mass transfer by free convection over a truncated cone in porous media with variable wall temperature and concentration. Cheng [21] used an integral approach for heat and mass transfer by free convection from truncated cones in porous media with variable wall temperature and concentration. Cheng [22] studied the free convection heat and mass transfer from a vertical truncated cone in a porous medium saturated with non-Newtonian power-law fluid with variable wall temperature and concentration. This work aims to study the free convection heat transfer from a vertical truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid using the model proposed by Nield and Kuznetsov [12]. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline
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Fig. 1. Physical model and coordinates for a truncated cone.
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C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
Here u and v are the volume-averaged velocity components in the x and y directions, respectively. T is the volume-averaged temperature. ϕ is the nanoparticle volume fraction. n is the power-law index of the non-Newtonian fluid. ρf, μeff and β are the density, effective viscosity, and volumetric volume expansion coefficient of the fluid. ϕp is the density of the particles. K and ε are the permeability and porosity of the porous medium. Furthermore, (ρc)f and (ρc)p are the heat capacity of the fluid and the effective heat capacity of the nanoparticle, respectively. αm is the thermal diffusivity of the porous medium. DB and DT are the Brownian diffusion coefficient and the thermophoresis diffusion coefficient, respectively. The heat capacity ratio between nanoparticle and fluid is defined as τ ¼ εðρcÞp =ðρcÞf . The gravitational acceleration is denoted by g. The boundary conditions are given by y ¼ 0 : v ¼ 0 T ¼ Tw y→∞ : u→0
T→T ∞
ϕ ¼ ϕw on y ¼ 0; x0 ≤x≤∞ ϕ ¼ ϕ∞ as y→∞ :
ð5Þ ð6Þ
Because the boundary layer thickness is small, the local radius to a point in the boundary layer r can be represented by the local radius of the truncated cone, r ¼ x sin A:
ð7Þ
We introduce the nondimensional variables: ξ¼
x x−x0 y 1=ð2nÞ ψ T−T ∞ ϕ−ϕ∞ ¼ ; η ¼ Ra ; f ð ηÞ ¼ ;θ ¼ ;Φ ¼ x x0 T w −T ∞ ϕw −ϕ∞ x0 α m rRa1=ð2nÞ
ð8Þ where ψ is the stream function defined as u¼
1 ∂ψ 1 ∂ψ ;v ¼ − r ∂y r ∂x
ð9Þ
and the Rayleigh number is given by ρf∞ βgK xn ð1−ϕ∞ ÞðT w −T ∞ Þ cos A : μα nm
Ra ¼
ð10Þ
Upon using these variables, the basic equations of the boundary layer for the problem under consideration can be written in a non-dimensional form as n ′ f −θ þ NrΦ ¼ 0 ″
θ þ
1 ξ ′ ′ ′ ′2 ′ ∂θ ′ ∂f þ f θ þ NbΦ θ þ Ntθ ¼ ξ f −θ 2 1þξ ∂ξ ∂ξ
″ Φ 1 ξ 1 Nt ″ ′ ′ ∂Φ ′ ∂f þ fΦ þ θ ¼ξ f þ −Φ 2 1þξ Le Nb Le ∂ξ ∂ξ
ð11Þ ð12Þ
ð13Þ
Nt ¼
εðρcÞp DT ðT w −T ∞ Þ ðρcÞf α m T ∞
:
ð17Þ
Here Le, Nb, Nr, and Nt denote the Lewis number, the Brownian motion parameter, the buoyancy ratio, and the thermophoresis parameter, respectively. The boundary conditions are transformed to η¼0:
f ¼0
θ¼1
Φ¼1
ð18Þ
η→∞ :
f ′ →0
θ→0
Φ→0 :
ð19Þ
The quantity of practical interest is the local Nusselt number given by Nu ′ ¼ −θ ðξ; 0Þ: Ra1=ð2nÞ
ð20Þ
In Eq. (20), Nu ¼ hx=km , where h is the local heat transfer coefficient and km is the effective thermal conductivity of the porous medium. 3. Results and discussion The transformed governing equations, Eqs. (12) and (13), and the associated boundary conditions, Eqs. (18) and (19), can be solved by the cubic spline collocation method [23,24]. The velocity f′ is calculated from the momentum equation, Eq. (11). Moreover, the Simpson's rule for variable grids is used to calculate the value of f at every position from the boundary conditions, Eqs. (18) and (19). At every position, the iteration process continues until the convergence criterion for all the variables, 10 −6, is achieved. Variable grids with 300 grid points are used in the η-direction. The optimum value of boundary layer thickness is used. To assess the accuracy of the solution, the present results are compared with the results obtained by other researchers. Table 1 shows the numerical values of − θ′(0,0) for different values of n with Nr = 0, Nb = 0, and Nt = 0, the conditions for natural convection heat of a vertical plate of non-Newtonian fluids in porous media with constant wall temperature. It is shown that the present results are in excellent agreement with the results reported by Chen and Chen [1]. Fig. 2 shows the effect of the thermophoresis parameter Nt, buoyancy ratio Nr, the Brownian motion parameter Nb and power-law index n on the dimensionless temperature profile θ. An increase in the thermophoresis parameter or buoyancy ratio tends to thicken the thermal boundary layer, thus decreasing the temperature gradient at the wall, as shown in Fig. 2a. Moreover, Fig. 2b shows that, as the Brownian motion is decreased, the thermal boundary layer thickness decreases, thus increasing the temperature gradient at the wall. Furthermore, it is noted that an increase in the power-law index tends to increase the temperature gradient at the wall. Fig. 3 shows the effect of the thermophoresis parameter Nt, buoyancy ratio Nr, the Brownian motion parameter Nb and power-law index n on the nanoparticle volume fraction profile ϕ. Increasing the thermophoresis parameter or buoyancy ratio tends to decrease the boundary-layer thickness of the nanoparticle volume fraction, thus
where primes denote differentiation with respect to η. Note that Eq. (11) has been obtained by integrating the transformed momentum equation. Moreover, the four important parameters are defined as Le ¼
αm εDB
Nb ¼
εðρcÞp DB ðϕw −ϕ∞ Þ ðρcÞf α m
Nr ¼
ð14Þ
ρp −ρf∞ ðϕw −ϕ∞ Þ
ρf∞ βðT w −T ∞ Þð1−ϕ∞ Þ
ð15Þ
ð16Þ
Table 1 Comparison of values of −θ′(0,0) for various values of n with Nr=0, Nb=0, and Nt=0. n
Chen and Chen [1]
Present results
0.4 0.5 0.8 1.0 1.2 1.5 2.0
0.3533 0.3769 0.4238 0.4437 0.4588 0.4753 0.4937
0.3531 0.3771 0.4240 0.4439 0.4590 0.4755 0.4938
C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
a
a
1.0
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1.0 Nr=0.4 Nt=0.1
Nr=0.4 Nt=0.1 Nr=0.4 Nt=0.3
Nr=0.4 Nt=0.3
Nr=0.8 Nt=0.1 n=0.8 Nb=0.2 Le=9 ξ=1
0.8
Nr=0.8 Nt=0.1
0.8
n=0.8 Nb=0.2 Le=9
0.6
0.6
φ
θ 0.4
0.4
0.2
0.2
0.0
0.0 0
1
2
3
4
5
6
0
1
2
η
b
ξ=1
3
4
b
1.0
1.0
Nb=0.2
n=0.8
Nb=0.7 Nr=0.4 Nt=0.3 Le=9 ξ=1
0.8
5
η
n=1.2
0.8
Nr=0.4 Nt=0.3 ξ=1 Le=9
0.6
0.6
φ
θ
0.4
0.4
0.2
0.2
n=0.8
Nb=0.2
n=1.2
Nb=0.7
0.0
0.0 0
1
2
3
4
5
6
η
0
1
2
η
3
4
Fig. 2. The effect of (a) thermophoresis parameter and buoyancy ratio; and (b) Brownian motion parameter and power-law index on the temperature profile.
Fig. 3. The effect of (a) thermophoresis parameter and buoyancy ratio; and (b) Brownian motion parameter and power-law index on the nanoparticle volume fraction profile.
increasing the nanoparticle volume fraction gradient at the wall, as shown in Fig. 3a. Moreover, as the Brownian motion parameter or the power-law index is increased, the boundary layer thickness of the nanoparticle volume fraction decreases, thus increasing the nanoparticle volume fraction gradient at the wall, as shown in Fig. 3b. Fig. 4 shows the effect of the thermophoresis parameter Nt, buoyancy ratio Nr, the Brownian motion parameter Nb and power-law index n on the streamwise velocity profile f′. Increasing the Brownian motion parameter or the power-law index tends to increase the maximum streamwise velocity, thus enhancing the fluid flow, as shown in Fig. 4a. Moreover, an increase in the buoyancy ratio tends to increase the maximum streamwise velocity, as shown in Fig. 4b. The effect of the thermophoresis parameter on the maximum streamwise velocity is not significant. Fig. 5 depicts the variation of the reduced Nusselt number NuRa − 1/(2n) with the streamwise coordinate ξ for various values of thermophoresis parameter Nt and power-law index n for Le = 9, Nb = 0.2, and Nr = 0.4. Fig. 5 shows that the reduced Nusselt number increases as the streamwise coordinate is increased. Moreover, the reduced Nusselt number tends to decrease as the thermophoresis parameter increases. As the thermophoresis parameter is increased, the thermal boundary layer thickness increases, thus decreasing the temperature gradient at the wall and the reduced Nusselt number, as
shown in Fig. 2. Moreover, an increase in the power-law index tends to increase the reduced Nusselt number. Fig. 6 shows the variation of the reduced Nusselt number NuRa − 1/(2n) with the streamwise coordinate ξ for various values of buoyancy ratio Nr and power-law index n for Le = 9, Nb = 0.2, and Nt = 0.3. It is clearly seen that the reduced Nusselt number tends to decrease as the buoyancy ratio increases. Increasing the buoyancy tends to thicken the thermal boundary layer, thus decreasing the temperature gradient at the wall and the reduced Nusselt number, as shown in Fig. 2. Fig. 7 depicts the variation of the reduced Number NuRa −1/(2n) with the streamwise coordinate ξ for various values of Brownian motion parameter Nb and power-law index n with Le= 9, Nr = 0.4, and Nt = 0.3. Results show that the reduced Nusselt number tends to decrease as the Brownian motion parameter increases. An increase in the Brownian motion parameter tends to increase the thermal boundary layer thickness, thus decreasing the temperature gradient at the wall and the reduced Nusselt number, as shown in Fig. 2. Moreover, Fig. 8 depicts the variation of the reduced Nusselt number NuRa −1/(2n) with the streamwise coordinate ξ for various values of Lewis number Le and power-law index n with Nb = 0.2, Nr = 0.4, and Nt = 0.3. Fig. 8 shows that an increase in the Lewis number tends to increase the reduced Nusselt number.
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C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
a
0.7 0.7 0.6
Le=9 Nb=0.2 Nt=0.3
Nr=0.2 Nr=0.4
Nr=0.8 Nt=0.3 Nr=0.4 Nt=0.3 Nr=0.4 Nt=0.1 n=0.8 Nb=0.2 Le=9 ξ=1
0.6
n=1.2
0.5 0.5
Nu
0.4
f ′
Ra
n=0.8
1 (2 n )
0.4
0.3 0.2
0.3 0.1 0.2 0.0 0
1
2
3
4
5
1E-3
6
1E-2
1E-1
1E+1
1E+2
1E+3
ξ
η
b
1E+0
Fig. 6. The effect of the buoyancy ratio on the reduced Nusselt number for Le=9, Nb=0.2, and Nt=0.3.
1.0 Nb=0.2 Nb=0.7 Nr=0.4 Nt=0.3 Le=9 ξ=1
0.8
4. Conclusions 0.6
f′ 0.4
n=1.2
0.2
n=0.8 0.0 0
1
2
3
4
η
5
6
Fig. 4. The effect of (a) thermophoresis parameter and buoyancy ratio; and (b) Brownian motion parameter and power-law index on the streamwise velocity profile.
This work studies the free convection heat transfer from a truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The model used for nanofluids incorporates the effects of Brownian motion and thermophoresis. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effects of the power-law index, thermophoresis parameter, Brownian motion parameter, Lewis number, and buoyancy ratio on the heat transfer and fluid flow characteristics have been studied. Results show that increasing the thermophoresis parameter or the Brownian motion parameter tends to decrease the reduced Nusselt number. Moreover, an increase in the power-law index tends to increase the reduced Nusselt number.
0.7
0.6 Nt=0.3
n=1.2 n=0.8
Le=9 Nb=0.2 Nr=0.4
0.6
Nt=0.5
0.5
0.5 0.4
Nu Ra
1 (2 n )
Le=9 Nr=0.4 Nt=0.3
Nb=0.2
Nu
0.4
Ra
1 (2 n )
0.3
n=1.2 0.2 0.3
Nb=0.6 0.1
n=0.8 0.0
0.2 1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
ξ Fig. 5. The effect of the thermophoresis parameter on the reduced Nusselt number for Le = 9, Nb = 0.2, and Nr = 0.4.
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
ξ Fig. 7. The effect of the Brownian motion parameter on the reduced Nusselt number for Le = 9, Nr= 0.4, and Nt = 0.3.
C.-Y. Cheng / International Communications in Heat and Mass Transfer 39 (2012) 1348–1353
0.6
Le=5 Le=9
Nb=0.2 Nt=0.3 Nr=0.4
0.5
Nu Ra
0.4
1 (2 n )
n=1.2 0.3
n=0.8 0.2 1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E+3
ξ Fig. 8. The effect of the Lewis number on the reduced Nusselt number for Nb = 0.2, Nr = 0.4, and Nt = 0.3.
Acknowledgement This work was supported by National Science Council of Republic of China under Grant No. NSC 100-2221-E-218-045. References [1] H.T. Chen, C.K. Chen, Free convection of non-Newtonian fluids along a vertical plate embedded in a porous medium, ASME Journal of Heat Transfer 110 (1988) 257–260. [2] A. Nakayama, H. Koyama, Buoyancy induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium, Applied Scientific Research 48 (1991) 55–70. [3] S.K. Rastogi, D. Poulikakos, Double-diffusion from a vertical surface in a porous region saturated with a non-Newtonian fluid, International Journal of Heat and Mass Transfer 38 (1995) 935–946. [4] D. Getachew, D. Poulikakos, W.J. Minkowycz, Double diffusion in a porous cavity saturated with non-Newtonian fluid, Journal of Thermophysics and Heat Transfer 12 (1998) 437–446. [5] K. Benhadj, P. Vasseur, Double diffusive convection in a shallow porous cavity filled with a non-Newtonian fluid, International Communications in Heat and Mass Transfer 28 (2001) 723–732.
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