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Non-Darcy mixed convection from a horizontal plate embedded in a nanofluid saturated porous media
International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
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Non-Darcy mixed convection from a horizontal plate embedded in a nanofluid saturated porous media☆ A.V. Rosca a, N.C. Rosca b, T. Grosan b,⁎, I. Pop b a b
Faculty of Economics and Business Administration, Babes-Bolyai University, 400084 Cluj-Napoca, Romania Faculty of Mathematics and Computer Science, Babes-Bolyai University, 400084 Cluj-Napoca, Romania
a r t i c l e
i n f o
Available online 30 June 2012 Keywords: Non-Darcy Porous medium Nanofluid Horizontal flat plate Mixed convection
a b s t r a c t The problem of steady mixed convection boundary-layer flow over an impermeable horizontal flat plate embedded in a porous medium saturated by a nanofluid is numerically studied. The model used for the nanofluid incorporates only the effect of the volume fraction parameter. The surface of the plate is maintained at a constant temperature and a constant nano-particle volume fraction. The resulting governing partial differential equations are transformed into a set of two ordinary (similar) equations, which are solved using the bvp4c function from Matlab. A comparison is made with the available results in the literature, and the present results are in very good agreement with the known results. A representative set of numerical results for the reduced heat transfer from the plate, dimensionless velocity and temperature profiles is graphically and tabularly presented. Also, the salient features of the results are analyzed and discussed. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Fluid flow and heat transfer in porous media have received considerable attention during the last several decades because of the numerous practical applications such as, for example, storage of radioactive nuclear waste, transpiration cooling, separation processes in chemical industries, filtration, transport processes in aquifers, groundwater pollution, geothermal extraction, fiber insulation, etc. Theories and experiments of thermal convection in porous media and state-of-the-art reviews, with special emphasis on practical applications are presented in the books by Nield and Bejan [1], Vafai [2,3], Pop and Ingham [4], Ingham and Pop [5] and Vadasz [6]. Fluid heating and cooling are important in many industrial and engineering applications, such as aerodynamic extrusion of plastic sheet, and the cooling of metallic plate in a cooling bath, and in a thin film solar energy collector device. The effective cooling techniques are needed for cooling any sort of high energy device. Conventional heat transfer fluids such as water, ethylene glycol mixture and engine oil have limited heat transfer capabilities due to their low thermal conductivity in enhancing the performance and compactness of many engineering electronic devices. In contrast, metals have thermal conductivities that are up to three times higher than these fluids. Thus, it is naturally desirable to combine the two substances to produce a medium for heat transfer that would behave like a fluid, but has the thermal conductivity of a metal. Therefore, there is a strong need to develop advanced heat transfer fluids with substantially ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (T. Grosan). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2012.06.025
higher conductivities to enhance thermal characteristics. Small particles (nanoparticles) stay suspended much longer than larger particles. The presence of the nanoparticles in the fluids increases appreciably the effective thermal conductivity and viscosity of the base fluid and consequently enhances the heat transfer characteristics. Thus, nanofluids may be used in various applications which include electronic cooling, vehicle cooling transformer and coolant for nuclear reactors. It seems that Choi [7] was the first who introduced the term nanofluids for reference to a liquid containing a dispersion of submicronic solid particles (nanoparticles). The characteristic feature of a nanofluid is the thermal conductivity enhancement, a phenomenon observed by Masuda et al. [8]. This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems [9]. Choi's [7] results have been supported by other researchers from time to time. Eastman et al. [10] reported an increase of 40% in the effective thermal conductivity of ethylene-glycol with 0.3% volume of copper nanoparticles of 10 nm diameter. Further 10–30% increase of the effective thermal conductivity in alumina/water nanofluids with 1–4% of alumina was reported by Das et al. [11]. These reports led Buongiorno and Hu [9] to suggest the possibility of using nanofluids in advanced nuclear systems. Another recent application of the nanofluid flow is in the delivery of nano-drug as suggested by Kleinstreuer et al. [12]. There are several numerical and experimental studies on the forced and natural convection using nanofluids related with differentially heated enclosures and we mention here those by Khanafer et al. [13], Tiwari and Das [14], Oztop and Abu-Nada [15], etc. Recently, Ahmad and Pop [16] have considered the steady mixed convection boundary layer flow over a vertical flat plate embedded in a porous medium filled with a nanofluid using the nanofluid model proposed by Tiwari and Das [14]. The book by Das et al. [17] and the review
A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
Nomenclature A Cp (ρ Cp)nf g G k K K′ knf Nux Nur p Pex Rax qw T Tw T∞ u, v U∞ x, y
constant defined in Eq. (9) specific heat at a constant temperature heat capacitance of the nanofluid acceleration due to gravity parameter of the inertial effect on mixed convection thermal conductivity permeability inertial coefficient of the Ergun equation thermal conductivity of the nanofluid local Nusselt number for the porous medium reduced Nusselt number pressure local Péclet number for the porous medium local Rayleigh number for the porous medium heat flux from the surface of the plate fluid temperature surface temperature ambient fluid temperature velocity components along the x and y directions free stream velocity Cartesian coordinates along the plate and normal to it, respectively
Greek letters αnf thermal diffusivity of the nanofluid βnf thermal expansion coefficient of the nanofluid ϕ nanoparticle volume fraction λ constant mixed convection parameter η similarity variable μ dynamic viscosity ν kinematic viscosity θ dimensionless temperature ρ density ρnf density of the nanofluid ψ stream function
Subscripts f fluid fraction s solid fraction nf nanofluid
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combination with Ergun's [25] non-Darcy porous medium model. Similarity solutions are obtained for the convective flow above a heated plate, where the wall temperature varies as x 1/2 and x is the distance along the plate measured from the origin. The similarity equations are solved numerically for copper (Cu) nanoparticles. 2. Basic equations Consider the problems of steady mixed convection flow in a saturated porous medium above a heated horizontal impermeable surface. The physical model for the case of free convection flow is shown in Fig. 1 where y is the coordinate measured normal to the surface, with the x axis pointing toward the porous medium and the dashed lines refer to the momentum boundary layer and the full lines refer to the thermal boundary layer, respectively. It is assumed that the constant velocity outside the boundary layer is U∞ and the temperature of the plate is Tw(x) > T∞, where T∞ is the uniform temperature of the ambient nanofluid. The Boussinesq approximation is employed and the homogeneity and local thermal equilibrium in the porous medium are assumed. It is also assumed that the nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. Adopting the nanofluid model proposed by Tiwari and Das [14] along with the non-Darcy law (with Ergun's [25] term), the basic equations can be written in the Cartesian coordinates x and y as (see Nield and Bejan [1]) ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
u þ ρnf ; ∞
K′ 2 K ∂p u ¼− μ nf ∂x μ nf
v þ ρnf ; ∞
K′ 2 K v ¼− μ nf μ nf
u
ð2Þ
∂p þ ρnf g ∂y
∂T ∂T ∂2 T ∂2 T þ þv ¼ α nf ∂x ∂y ∂x2 ∂y2
ð3Þ
! ð4Þ
where papers by Wang and Mujumdar [18], Ding et al. [19] and, Kakaç and Pramunjaroenkij [20] present excellent collection of up to now published papers on nanofluids. One of the fundamental problems concerning heat transfer in porous media is the mixed convection boundary layer flow from a horizontal flat plate. It seems that the first study of this problem was reported by Cheng [21]. The study was followed by a series of investigations, out of which the notable similarity solutions are that by Lai and Kulacki [22], Magyari et al. [23], etc. Very recently, Arifin et al. [24] have investigated the steady free and mixed convection boundary layer flow past a horizontal flat plate embedded in a porous medium saturated by a nanofluid using the Tiwari and Das [14] nanofluid model along with Darcy law. The focus of the study was to analyze the effects of several pertinent parameters for three types of nanopaticles such as copper (Cu), alumina (Al2O3) and titania (TiO2). Motivated by these studies, an analysis of the steady mixed convection boundary layer flow past a horizontal impermeable surface embedded in a porous medium filled by a nanofluid is performed in this paper using the model proposed by Tiwari and Das [14] in
h i ρnf ¼ ρnf ;∞ 1−βnf ðT−T ∞ Þ
ð5Þ
y g
v u
δm U
δT x
Tw(x) > T Fig. 1. Physical model and coordinate system.
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Table 1 Physical properties of fluid and nanoparticles.
Table 3 Values of the heat transfer − θ′(0) and sleep velocity f′(0) when ϕ = 0 (Newtonian fluid) G = 0, 0.1, 1, 10 and λ= 0.1, 0.5, 1, 2, 4, 8, 10, 20, 30, 40, 50 compared with the results of Lai and Kulacki [22].
Property
Water
Cu
Al2O3
TiO2
cp (J/kg K) ρ (kg/m3) k (W/m K) α × 107 (m2/s) β × 10−5 (1/K)
4179 997.1 0.613 1.47 21
385 8933 400 1163.1 1.67
765 3970 40 131.7 0.85
686.2 4250 8.9538 30.7 0.9
is the Boussinesq approximation. From Eqs. (2) and (3) one obtains 3 2 2 ∂ v2 gρ K βnf ∂T ∂u ∂v ρnf ; ∞ ′ 4∂ u 5 ¼ − nf ;∞ K − þ : − μ nf μ nf ∂y ∂x ∂x ∂y ∂x
ð6Þ
Using the boundary layer approximation and introducing the stream function ψ defined as u = ∂ ψ/∂ y and v = − ∂ ψ/∂ x, Eqs. (6) and (4) become: ∂2 ψ ρnf ; ∞ ′ ∂ þ K μ nf ∂y ∂y2
"
# g ρnf ;∞ K βnf ∂T ∂ψ 2 ¼− μ nf ∂y ∂x
ð7Þ
∂ψ ∂T ∂ψ ∂T ∂2 T − ¼ α nf 2 : ∂y ∂x ∂x ∂y ∂y
ð8Þ
We assume that the boundary conditions for these equations are v ¼ 0; T w ¼ T ∞ þ Ax1=2 at y ¼ 0 u→U ∞ ; T→T ∞ as y→∞:
ð9Þ
For the sake of simplicity, in the following we will denote ρnf,∞ by ρnf. The physical characteristics of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [15]) knf
; α nf ¼ ρ Cp
ρnf ¼ ð1−ϕÞρf þ ϕ ρs ;
nf
ρ Cp
nf
¼ ð1−ϕÞ ρ C p þ ϕ ρ C p ; f
s
μ nf ¼
μf
ð1−ϕÞ2:5 ks þ 2 kf −2ϕ kf −ks knf : ¼ kf ks þ 2 kf þ ϕ kf −ks ð10Þ
λ
G=0 − θ′(0)
0.1 0.9137 (0.9137 0.5 1.0076 (1.0077 1 1.1020 (1.1020 2 1.2495 (1.2495 4 1.4645 (1.4645 8 1.7609 (1.7610 10 1.8763 (1.8763 20 2.3059 (2.3059 30 2.6139 (2.6139 40 2.8614 (2.8614 50 3.0715 (3.0713
G = 0.1
G=1
G= 10
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
1.0865 1.0865) 1.4014 1.4015) 1.7474 1.7474) 2.3480 2.3479) 3.3542 3.3543) 4.9986 4.9990) 5.7190 5.7191) 8.8011 8.8010) 11.3978 11.3978) 13.7195 13.7193) 15.8551 15.8529)
0.9092 (0.9092 0.9878 (0.9878 1.0666 (1.0666 1.1884 (1.1883 1.3614 (1.3614 1.5904 (1.5904 1.6763 (1.6763 1.9820 (1.9820 2.1884 (2.1884 2.3474 (2.3474 2.4780 (2.4780
1.0720 1.0720) 1.3306 1.3306) 1.6082 1.6082) 2.0730 2.0729) 2.8078 2.8077) 3.9123 3.9125) 4.3658 4.3657) 6.1501 6.1500) 7.5060 7.5060) 8.6340 8.6340) 9.6154 9.6144)
0.8955 (0.8956 0.9295 (0.9295 0.9661 (0.9661 1.0266 (1.0266 1.1180 (1.1180 1.2447 (1.2446 1.2932 (1.2932 1.4688 (1.4688 1.5888 (1.5888 1.6820 (1.6820 1.7588 (1.7588
1.0290 1.0291) 1.1362 1.1362) 1.2548 1.2547) 1.4569 1.4569) 1.7789 1.7788) 2.2596 2.2595) 2.4551 2.4551) 3.2146 3.2145) 3.7833 3.7829) 4.2524 4.2528) 4.6582 4.6582)
0.8876 (0.8876 0.8929 (0.8929 0.8994 (0.8994 0.9117 (0.9117 0.9343 (0.9343 0.9735 (0.9735 0.9908 (0.9907 1.0620 (1.0620 1.1174 (1.1174 1.1633 (1.1633 1.2027 (1.2027
1.0042 1.0042) 1.0208 1.0207) 1.0409 1.0409) 1.0796 1.0795) 1.1512 1.1511) 1.2773 1.2773) 1.3338 1.3337) 1.5740 1.5740) 1.7688 1.7688) 1.9357 1.9357) 2.0834 2.0834)
( ) results by Lai and Kulacki [22]
Following Lai and Kulacki [22], we introduce the similar variables 1=2 ψ ¼ αf U∞ x f ðηÞ; θðηÞ ¼ ðT−T ∞ Þ=ðT w −T ∞ Þ; 1=2 ðy=xÞ η ¼ U ∞ x=α f
ð11Þ
so that the partial differential Eqs. (7) and (8) are transformed to the following ordinary (similarity) equations 2 ′ f ″ þ ð1−ϕÞ2:5 1−ϕ þ ϕρs =ρf G f ′ h i 1 2:5 ′ ¼ ð1−ϕÞ 1−ϕ þ ϕ ðρs βs Þ= ρf βf λ η θ −θ 2
ð12Þ
k =k 1 ′ ′ nf f . θ″ ¼ θ f −f θ 2 ρC p 1−ϕ þ ϕ ρC p
ð13Þ
s
f
It should be mentioned that Maxwell [26] first published a theoretical work for the effective thermal conductivity k of two-phase mixtures containing powders with particle diameters in the order of millimeters or even micrometers. Maxwell's model predicted that the effective thermal conductivity of suspended spherical particles increases with an increase in the volume fraction of the solid particles. Table 2 Values of the heat transfer − θ′(0) and sleep velocity f′(0) for ϕ = 0 (Newtonian fluid), G = 0 (Darcy case) and several values of λ compared with the results of Arifin et al. [24]. λ
0 0.6 1 2 5 8 15
Present
Arifin et al. [24]
− θ′(0)
f′(0)
− θ′(0)
f′(0)
0.8862 1.0281 1.1020 1.2495 1.5503 1.7609 2.1137
1.0000 1.4741 1.7474 2.3480 3.7995 4.9986 7.3446
0.8862 1.0282 1.1020 1.2493 1.5502 1.7609 2.1133
1.0000 1.4741 1.7474 2.3471 3.7992 4.9984 7.3422
Fig. 2. Dimensionless temperature profiles θ(η) for λ= 0, 0.5, 1, 2, 4, 8, 16 when G= 1 and ϕ = 0.1.
A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
Fig. 3. Dimensionless velocity profiles f′(η) for λ = 0, 0.5, 1, 2, 4, 8, 16 when G= 1 and ϕ = 0.1.
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Fig. 5. Dimensionless velocity profiles f′(η) for G= 0, 0.1, 1, 5, 8, 10 when λ= 1 and ϕ = 0.1.
Using Eqs. (11), (16) and (17), we obtain along with the corresponding boundary conditions f ð0Þ ¼ 0; θð0Þ ¼ 1 at f ′ ðηÞ→1; θðηÞ→0 as
−1=2
η¼0 η→∞:
ð14Þ
Here G is the non-Darcy or inertial parameter and λ is the mixed convection parameter, which are defined as ′ K ′ U ∞ K U ∞ ρf G¼ ¼ ; νf μf
λ¼
Rax
ð15Þ
3=2
Pex
where Rax = g ρ f K β f (Tw − T∞)x/(μ f α f) is the local Rayleigh number and Pex = U∞ x/αf is the local Péclet number for the porous medium. A physical quantity of practical interest is the local Nusselt number Nux, which is defined as Nux ¼
x qw kf ðT w −T ∞ Þ
ð16Þ
where qw is the heat flux from the surface of the plate, which is given by qw ¼ −knf
∂T : ∂y y¼0
ð17Þ
1
Pex
Nux ¼ −
knf ′ θ ð0Þ: kf
ð18Þ
3. Results and discussion The problem is formulated so that we can consider different types of nanoparticles (e.g. Cu, Al2O3, TiO2, etc.) and water as a base fluid. However, in order to save space, we have considered here only the case of Cu nanoparticles. The system of the ordinary differential Eqs. (12) and (13) which is subject to the boundary conditions (Eq. 14) was solved using the function bvp4c from Matlab. The relative tolerance was set to 10−10. The obtained results are presented for several values of the governing parameters in some figures and tables. As in Oztop and Abu-Nada [15], we take the values of the solid volume fraction ϕ in the range of 0≤ϕ≤0.2. However, Muthtamilselvan et al. [27] have considered for the convective flow in a lid driven cavity, the values of the solid volume fraction in the range of 0 ≤ϕ≤0.08. If the concentration exceeds the maximum level of 0.08, sedimentation could take place. In order to verify the numerical method the results were compared with those reported by Arifin et al. [24] for the Darcy model (G=0) (see Table 2) and Lai and Kulacki [22] for regular fluid (ϕ=0) as can be seen in Table 3. We notice that the results are in very good agreement and therefore we are confident that the present results are accurate. Figs. 2 and 3 present the dimensionless temperature θ(η) and velocity f′(η) profiles for ϕ = 0.1 when G = 1 and λ = 0, 0.5, 1, 2, 4, 8, 16.
0.9 0.8 0.7
θ(η)
0.6 0.5
G = 0, 0.1, 1, 5, 8, 10 0.4 0.3 0.2 0.1 0
0
2
4
η
6
8
10
Fig. 4. Dimensionless temperature profiles θ(η) for G = 0, 0.1, 1, 5, 8, 10 when λ = 1 and ϕ = 0.1.
Fig. 6. Dimensionless temperature profiles θ(η) for ϕ = 0, 0.05, 0.1, 0.15, 0.2 when λ = 1and G = 1.
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A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
Fig. 7. Dimensionless velocity profiles f′(η) for ϕ= 0, 0.05, 0.1, 0.15, 0.2 when λ= 1and G = 1.
Fig. 9. Variation of the local Nusselt number with λ for ϕ = 0, 0.1, 0.2 when G = 1.
Following Nield and Kunznetsov [28], we refer to the reduced Nusselt number Nur given by It can be seen that the thermal boundary layer thickness decreases while the momentum boundary layer increases with the increase of the mixed convection parameter λ. This happens because the flow is assisting. Figs. 4 to 5 display the variation of the dimensionless temperature and velocity profiles for G = 0, 0.1, 1, 5, 10 when λ = 1 and ϕ = 0.1. The thickness of the temperature boundary layer is greater for the non-Darcy fluid (G ≠ 0) than for the Darcy fluid (G = 1), while the momentum boundary layer thickness decreases with the inertial parameter G. The behavior of the profiles shown in Figs. 2, 3, 4, and 5 is similar with those reported by Lai and Kulacki [22]. Further, Figs. 6 and 7 present the dimensionless temperature and velocity profiles for ϕ = 0, 0.05, 0.1, 0.15, 0.2 when λ = 1 and G = 1. The thermal boundary layer thickness increases with the increase of the nanoparticle concentration parameter ϕ, while the momentum boundary layer thickness decreases near the plate and then increases at some distance from the plate. From the physical point of view, the increases of the parameter ϕ lead to an increase of the viscosity and thermal conductivity of the nanofluid. This behavior of the thermal boundary layer is in agreement with the decrease of the heat transfer rate from the surface of the plate to the nanofluid. Finally, Figs. 8 to 9 illustrate the variation of the local Nusselt number Nux/Pe 1/2 with λ when ϕ = 0, 0.1 and 0.2 when G = 0, 0.1, 1 and 10. One can notice from these figures that Nux/Pe 1/2 increases monotonically with λ and also increases with ϕ.
Nur ¼ Nux =Pex
1=2
ð19Þ
and we perform a multiple linear regression in order to obtain a correlation between Nur and ϕ, and G and λ. A multiple linear regression of the form Nurest ¼ a þ bϕ þ cG þ dλ
ð20Þ
was considered, where Nurest is the response variable, while ϕ, G and λ are the independent variables. The following values for ϕ = 0, 0.1, 0.15, 0.2, G = 0, 0.1, 0.5, 1 and λ = 0.1, 0.5, 0.75, 1 are considered. In total for carrying out the regression, we used 64, 3-tuple of the form (ϕ, G, λ) with the corresponding values of the response variable Nur given by Eq. (19). We used the function regress from Matlab, and it gives the following form of Nurest: Nurest ¼ 0:9337 þ 1:2163ϕ−0:0827G þ 0:1276λ:
ð21Þ
The coefficient of multiple determination is R 2 = 0.9603 and the maximum relative error defined by ε = |(Nurest − Nur)/Nur| is ε = 0.0424. We observe from Eq. (21) that an increase in the parameters ϕ and λ leads to an increase in the value of Nurest, while a decrease in the parameter G leads to an increase of Nurest. This behavior is in agreement with the numerical values of Nur shown in Tables 2, 3, 4 and 5.
Table 4 Values of the heat transfer − θ′(0) and sleep velocity f′(0), when ϕ= 0.1, G= 0, 0.1, 1, 10 and λ= 0.1, 0.5, 1, 2, 4, 8, 10, 20, 30, 40, 50 in the case of Cu nanoparticles. λ
Fig. 8. Variation of the local Nusselt number with λ for G= 0, 0.1, 1, 10 when ϕ = 0.1.
0.1 0.5 1 2 4 8 10 20 30 40 50
G=0
G = 0.1
G= 1
G = 10
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
0.7804 0.8475 0.9170 1.0281 1.1933 1.4247 1.5153 1.8547 2.0990 2.2957 2.4628
1.0700 1.3286 1.6175 2.1250 2.9840 4.3970 5.0178 7.6779 9.9220 11.9293 13.7763
0.7763 0.8292 0.8841 0.9713 1.0983 1.2693 1.3341 1.5656 1.7226 1.8438 1.9435
1.0548 1.2548 1.4736 1.8450 2.4383 3.3350 3.7036 5.1538 6.2545 7.1696 7.9652
0.7664 0.7858 0.8077 0.8454 0.9055 0.9930 1.0275 1.1550 1.2439 1.3135 1.3712
1.0188 1.0897 1.1707 1.3138 1.5506 1.9164 2.0677 2.6634 3.1142 3.4877 3.8117
0.7619 0.7647 0.7680 0.7745 0.7868 0.8091 0.8194 0.8635 0.8997 0.9305 0.9576
1.0025 1.0124 1.0245 1.0481 1.0931 1.1756 1.2137 1.3818 1.5235 1.6476 1.7590
A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085 Table 5 Values of the heat transfer − θ′(0) and sleep velocity f′(0), when ϕ = 0.2, G= 0, 0.1, 1, 10 and λ = 0.1, 0.5, 1, 2, 4, 8, 10, 20, 30, 40, 50 in the case of Cu nanoparticles. λ
0.1 0.5 1 2 4 8 10 20 30 40 50
G=0
G = 0.1
G=1
G = 10
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
0.6717 0.7177 0.7668 0.8477 0.9714 1.1483 1.2183 1.4824 1.6736 1.8280 1.9594
1.0538 1.2557 1.4855 1.8957 2.5994 3.7682 4.2837 6.4990 8.3716 10.0479 11.5910
0.6688 0.7045 0.7428 0.8056 0.9003 1.0314 1.0818 1.2641 1.3889 1.4857 1.5656
1.0414 1.1956 1.3683 1.6679 2.1574 2.9111 3.2238 4.4634 5.4104 6.2001 6.8880
0.6622 0.6746 0.6890 0.7145 0.7568 0.8209 0.8468 0.9444 1.0137 1.0685 1.1141
1.0136 1.0660 1.1270 1.2375 1.4260 1.7261 1.8524 2.3562 2.7419 3.0631 3.3426
0.6594 0.6610 0.6631 0.6672 0.6751 0.6896 0.6964 0.7266 0.7521 0.7744 0.7943
1.0018 1.0088 1.0176 1.0347 1.0677 1.1294 1.1584 1.2894 1.4028 1.5037 1.5953
Acknowledgment This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS‐UEFISCDI, project number PN-II-RU-TE-2011-3-0013. References [1] D.A. Nield, A. Bejan, Convection in Porous Media, 3rd ed. Springer, New York, 2006. [2] K. Vafai, Handbook of Porous Media, 2nd ed. Taylor & Francis, New York, 2005. [3] K. Vafai, Porous Media: Applications in Biological Systems and Biotechnology, CRC Press, 2010. [4] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001. [5] D.B. Ingham, I. Pop, Transport Phenomena in Porous Media III, Elsevier, Oxford, 2005. [6] P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media, Springer, New York, 2008. [7] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, In: in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows, FED-vol. 231/MD-vol. 66, ASME, New York, 1995, pp. 99–105. [8] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu Bussei 7 (1993) 227–233. [9] J. Buongiorno, W. Hu, Nanofluid coolants for advanced nuclear power plants, In: Proceedings of ICAPP ’05, Seoul, May 15–19, 2005, Paper no. 5705, 2007.
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[10] J.A. Eastman, S.U.S. Choi, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Applied Physics Letters 78 (2001) 718–720. [11] S.K. Das, N. Putra, P. Thiesen, W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, ASME Journal of Heat Transfer 125 (2003) 567–574. [12] C. Kleinstreuer, J. Li, J. Koo, Microfluidics of nano-drug delivery, International Journal of Heat and Mass Transfer 51 (2008) 5590–5597. [13] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, International Journal of Heat and Mass Transfer 46 (2003) 3639–3653. [14] 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. [15] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, International Journal of Heat and Fluid Flow 29 (2008) 1326–1336. [16] 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. [17] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley, New Jersey, 2007. [18] X.-Q. Wang, A.S. Mujumdar, A review on nanofluids — part II: experiments and applications, Brazilian Journal of Chemical Engineering 25 (2008) 631–648. [19] Y. Ding, H. Chen, L. Wang, C.-Y. Yang, Y. Hel, W. Yang, W.P. Lee, L. Zhang, R. Huo, Heat transfer intensification using nanofluids, Kona 25 (2007) 23–38. [20] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, International Journal of Heat and Mass Transfer 52 (2009) 3187–3196. [21] P. Cheng, Similarity solutions for mixed convection from horizontal impermeable surfaces in saturated porous media, International Journal of Heat and Mass Transfer 20 (1977) 893–898. [22] F.C. Lai, F.A. Kulacki, Non-Darcy convection from horizontal impermeable surfaces in saturated porous media, International Journal of Heat and Mass Transfer 30 (1987) 2189–2192. [23] E. Magyari, I. Pop, B. Keller, New similarity solutions for boundary-layer flow on a horizontal surface in a porous medium, Transport in Porous Media 51 (2003) 123–140. [24] N. Arifin, R. Nazar, I. Pop, Free and mixed convection boundary layer flow past a horizontal surface embedded in a porous medium filled with a nanofluid, AIAA Journal of Thermophysics and Heat Transfer 26 (2012) 375–382. [25] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress 48 (1952) 89–94. [26] J. Maxwell, A Treatise on Electricity and Magnetism, 2nd ed. Oxford University Press, Cambridge, 1904. [27] M. Muthtamilselvan, P. Kandaswamy, J. Lee, Heat transfer enhancement of copper–water nanofluids in a lid-driven enclosure, Communications in Nonlinear Science and Numerical Simulation 15 (2010) 1501–1510. [28] D.A. Nield, A.V. Kuznetsov, The Cheng–Minkowycz problem for natural convection boundary-layer flow in a porous medium saturated by a nanofluid, International Journal of Heat and Mass Transfer 52 (2009) 5792–5795.
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Non-Darcy mixed convection from a horizontal plate embedded in a nanofluid saturated porous media☆ A.V. Rosca a, N.C. Rosca b, T. Grosan b,⁎, I. Pop b a b
Faculty of Economics and Business Administration, Babes-Bolyai University, 400084 Cluj-Napoca, Romania Faculty of Mathematics and Computer Science, Babes-Bolyai University, 400084 Cluj-Napoca, Romania
a r t i c l e
i n f o
Available online 30 June 2012 Keywords: Non-Darcy Porous medium Nanofluid Horizontal flat plate Mixed convection
a b s t r a c t The problem of steady mixed convection boundary-layer flow over an impermeable horizontal flat plate embedded in a porous medium saturated by a nanofluid is numerically studied. The model used for the nanofluid incorporates only the effect of the volume fraction parameter. The surface of the plate is maintained at a constant temperature and a constant nano-particle volume fraction. The resulting governing partial differential equations are transformed into a set of two ordinary (similar) equations, which are solved using the bvp4c function from Matlab. A comparison is made with the available results in the literature, and the present results are in very good agreement with the known results. A representative set of numerical results for the reduced heat transfer from the plate, dimensionless velocity and temperature profiles is graphically and tabularly presented. Also, the salient features of the results are analyzed and discussed. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Fluid flow and heat transfer in porous media have received considerable attention during the last several decades because of the numerous practical applications such as, for example, storage of radioactive nuclear waste, transpiration cooling, separation processes in chemical industries, filtration, transport processes in aquifers, groundwater pollution, geothermal extraction, fiber insulation, etc. Theories and experiments of thermal convection in porous media and state-of-the-art reviews, with special emphasis on practical applications are presented in the books by Nield and Bejan [1], Vafai [2,3], Pop and Ingham [4], Ingham and Pop [5] and Vadasz [6]. Fluid heating and cooling are important in many industrial and engineering applications, such as aerodynamic extrusion of plastic sheet, and the cooling of metallic plate in a cooling bath, and in a thin film solar energy collector device. The effective cooling techniques are needed for cooling any sort of high energy device. Conventional heat transfer fluids such as water, ethylene glycol mixture and engine oil have limited heat transfer capabilities due to their low thermal conductivity in enhancing the performance and compactness of many engineering electronic devices. In contrast, metals have thermal conductivities that are up to three times higher than these fluids. Thus, it is naturally desirable to combine the two substances to produce a medium for heat transfer that would behave like a fluid, but has the thermal conductivity of a metal. Therefore, there is a strong need to develop advanced heat transfer fluids with substantially ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (T. Grosan). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2012.06.025
higher conductivities to enhance thermal characteristics. Small particles (nanoparticles) stay suspended much longer than larger particles. The presence of the nanoparticles in the fluids increases appreciably the effective thermal conductivity and viscosity of the base fluid and consequently enhances the heat transfer characteristics. Thus, nanofluids may be used in various applications which include electronic cooling, vehicle cooling transformer and coolant for nuclear reactors. It seems that Choi [7] was the first who introduced the term nanofluids for reference to a liquid containing a dispersion of submicronic solid particles (nanoparticles). The characteristic feature of a nanofluid is the thermal conductivity enhancement, a phenomenon observed by Masuda et al. [8]. This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems [9]. Choi's [7] results have been supported by other researchers from time to time. Eastman et al. [10] reported an increase of 40% in the effective thermal conductivity of ethylene-glycol with 0.3% volume of copper nanoparticles of 10 nm diameter. Further 10–30% increase of the effective thermal conductivity in alumina/water nanofluids with 1–4% of alumina was reported by Das et al. [11]. These reports led Buongiorno and Hu [9] to suggest the possibility of using nanofluids in advanced nuclear systems. Another recent application of the nanofluid flow is in the delivery of nano-drug as suggested by Kleinstreuer et al. [12]. There are several numerical and experimental studies on the forced and natural convection using nanofluids related with differentially heated enclosures and we mention here those by Khanafer et al. [13], Tiwari and Das [14], Oztop and Abu-Nada [15], etc. Recently, Ahmad and Pop [16] have considered the steady mixed convection boundary layer flow over a vertical flat plate embedded in a porous medium filled with a nanofluid using the nanofluid model proposed by Tiwari and Das [14]. The book by Das et al. [17] and the review
A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
Nomenclature A Cp (ρ Cp)nf g G k K K′ knf Nux Nur p Pex Rax qw T Tw T∞ u, v U∞ x, y
constant defined in Eq. (9) specific heat at a constant temperature heat capacitance of the nanofluid acceleration due to gravity parameter of the inertial effect on mixed convection thermal conductivity permeability inertial coefficient of the Ergun equation thermal conductivity of the nanofluid local Nusselt number for the porous medium reduced Nusselt number pressure local Péclet number for the porous medium local Rayleigh number for the porous medium heat flux from the surface of the plate fluid temperature surface temperature ambient fluid temperature velocity components along the x and y directions free stream velocity Cartesian coordinates along the plate and normal to it, respectively
Greek letters αnf thermal diffusivity of the nanofluid βnf thermal expansion coefficient of the nanofluid ϕ nanoparticle volume fraction λ constant mixed convection parameter η similarity variable μ dynamic viscosity ν kinematic viscosity θ dimensionless temperature ρ density ρnf density of the nanofluid ψ stream function
Subscripts f fluid fraction s solid fraction nf nanofluid
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combination with Ergun's [25] non-Darcy porous medium model. Similarity solutions are obtained for the convective flow above a heated plate, where the wall temperature varies as x 1/2 and x is the distance along the plate measured from the origin. The similarity equations are solved numerically for copper (Cu) nanoparticles. 2. Basic equations Consider the problems of steady mixed convection flow in a saturated porous medium above a heated horizontal impermeable surface. The physical model for the case of free convection flow is shown in Fig. 1 where y is the coordinate measured normal to the surface, with the x axis pointing toward the porous medium and the dashed lines refer to the momentum boundary layer and the full lines refer to the thermal boundary layer, respectively. It is assumed that the constant velocity outside the boundary layer is U∞ and the temperature of the plate is Tw(x) > T∞, where T∞ is the uniform temperature of the ambient nanofluid. The Boussinesq approximation is employed and the homogeneity and local thermal equilibrium in the porous medium are assumed. It is also assumed that the nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. Adopting the nanofluid model proposed by Tiwari and Das [14] along with the non-Darcy law (with Ergun's [25] term), the basic equations can be written in the Cartesian coordinates x and y as (see Nield and Bejan [1]) ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
u þ ρnf ; ∞
K′ 2 K ∂p u ¼− μ nf ∂x μ nf
v þ ρnf ; ∞
K′ 2 K v ¼− μ nf μ nf
u
ð2Þ
∂p þ ρnf g ∂y
∂T ∂T ∂2 T ∂2 T þ þv ¼ α nf ∂x ∂y ∂x2 ∂y2
ð3Þ
! ð4Þ
where papers by Wang and Mujumdar [18], Ding et al. [19] and, Kakaç and Pramunjaroenkij [20] present excellent collection of up to now published papers on nanofluids. One of the fundamental problems concerning heat transfer in porous media is the mixed convection boundary layer flow from a horizontal flat plate. It seems that the first study of this problem was reported by Cheng [21]. The study was followed by a series of investigations, out of which the notable similarity solutions are that by Lai and Kulacki [22], Magyari et al. [23], etc. Very recently, Arifin et al. [24] have investigated the steady free and mixed convection boundary layer flow past a horizontal flat plate embedded in a porous medium saturated by a nanofluid using the Tiwari and Das [14] nanofluid model along with Darcy law. The focus of the study was to analyze the effects of several pertinent parameters for three types of nanopaticles such as copper (Cu), alumina (Al2O3) and titania (TiO2). Motivated by these studies, an analysis of the steady mixed convection boundary layer flow past a horizontal impermeable surface embedded in a porous medium filled by a nanofluid is performed in this paper using the model proposed by Tiwari and Das [14] in
h i ρnf ¼ ρnf ;∞ 1−βnf ðT−T ∞ Þ
ð5Þ
y g
v u
δm U
δT x
Tw(x) > T Fig. 1. Physical model and coordinate system.
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Table 1 Physical properties of fluid and nanoparticles.
Table 3 Values of the heat transfer − θ′(0) and sleep velocity f′(0) when ϕ = 0 (Newtonian fluid) G = 0, 0.1, 1, 10 and λ= 0.1, 0.5, 1, 2, 4, 8, 10, 20, 30, 40, 50 compared with the results of Lai and Kulacki [22].
Property
Water
Cu
Al2O3
TiO2
cp (J/kg K) ρ (kg/m3) k (W/m K) α × 107 (m2/s) β × 10−5 (1/K)
4179 997.1 0.613 1.47 21
385 8933 400 1163.1 1.67
765 3970 40 131.7 0.85
686.2 4250 8.9538 30.7 0.9
is the Boussinesq approximation. From Eqs. (2) and (3) one obtains 3 2 2 ∂ v2 gρ K βnf ∂T ∂u ∂v ρnf ; ∞ ′ 4∂ u 5 ¼ − nf ;∞ K − þ : − μ nf μ nf ∂y ∂x ∂x ∂y ∂x
ð6Þ
Using the boundary layer approximation and introducing the stream function ψ defined as u = ∂ ψ/∂ y and v = − ∂ ψ/∂ x, Eqs. (6) and (4) become: ∂2 ψ ρnf ; ∞ ′ ∂ þ K μ nf ∂y ∂y2
"
# g ρnf ;∞ K βnf ∂T ∂ψ 2 ¼− μ nf ∂y ∂x
ð7Þ
∂ψ ∂T ∂ψ ∂T ∂2 T − ¼ α nf 2 : ∂y ∂x ∂x ∂y ∂y
ð8Þ
We assume that the boundary conditions for these equations are v ¼ 0; T w ¼ T ∞ þ Ax1=2 at y ¼ 0 u→U ∞ ; T→T ∞ as y→∞:
ð9Þ
For the sake of simplicity, in the following we will denote ρnf,∞ by ρnf. The physical characteristics of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [15]) knf
; α nf ¼ ρ Cp
ρnf ¼ ð1−ϕÞρf þ ϕ ρs ;
nf
ρ Cp
nf
¼ ð1−ϕÞ ρ C p þ ϕ ρ C p ; f
s
μ nf ¼
μf
ð1−ϕÞ2:5 ks þ 2 kf −2ϕ kf −ks knf : ¼ kf ks þ 2 kf þ ϕ kf −ks ð10Þ
λ
G=0 − θ′(0)
0.1 0.9137 (0.9137 0.5 1.0076 (1.0077 1 1.1020 (1.1020 2 1.2495 (1.2495 4 1.4645 (1.4645 8 1.7609 (1.7610 10 1.8763 (1.8763 20 2.3059 (2.3059 30 2.6139 (2.6139 40 2.8614 (2.8614 50 3.0715 (3.0713
G = 0.1
G=1
G= 10
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
1.0865 1.0865) 1.4014 1.4015) 1.7474 1.7474) 2.3480 2.3479) 3.3542 3.3543) 4.9986 4.9990) 5.7190 5.7191) 8.8011 8.8010) 11.3978 11.3978) 13.7195 13.7193) 15.8551 15.8529)
0.9092 (0.9092 0.9878 (0.9878 1.0666 (1.0666 1.1884 (1.1883 1.3614 (1.3614 1.5904 (1.5904 1.6763 (1.6763 1.9820 (1.9820 2.1884 (2.1884 2.3474 (2.3474 2.4780 (2.4780
1.0720 1.0720) 1.3306 1.3306) 1.6082 1.6082) 2.0730 2.0729) 2.8078 2.8077) 3.9123 3.9125) 4.3658 4.3657) 6.1501 6.1500) 7.5060 7.5060) 8.6340 8.6340) 9.6154 9.6144)
0.8955 (0.8956 0.9295 (0.9295 0.9661 (0.9661 1.0266 (1.0266 1.1180 (1.1180 1.2447 (1.2446 1.2932 (1.2932 1.4688 (1.4688 1.5888 (1.5888 1.6820 (1.6820 1.7588 (1.7588
1.0290 1.0291) 1.1362 1.1362) 1.2548 1.2547) 1.4569 1.4569) 1.7789 1.7788) 2.2596 2.2595) 2.4551 2.4551) 3.2146 3.2145) 3.7833 3.7829) 4.2524 4.2528) 4.6582 4.6582)
0.8876 (0.8876 0.8929 (0.8929 0.8994 (0.8994 0.9117 (0.9117 0.9343 (0.9343 0.9735 (0.9735 0.9908 (0.9907 1.0620 (1.0620 1.1174 (1.1174 1.1633 (1.1633 1.2027 (1.2027
1.0042 1.0042) 1.0208 1.0207) 1.0409 1.0409) 1.0796 1.0795) 1.1512 1.1511) 1.2773 1.2773) 1.3338 1.3337) 1.5740 1.5740) 1.7688 1.7688) 1.9357 1.9357) 2.0834 2.0834)
( ) results by Lai and Kulacki [22]
Following Lai and Kulacki [22], we introduce the similar variables 1=2 ψ ¼ αf U∞ x f ðηÞ; θðηÞ ¼ ðT−T ∞ Þ=ðT w −T ∞ Þ; 1=2 ðy=xÞ η ¼ U ∞ x=α f
ð11Þ
so that the partial differential Eqs. (7) and (8) are transformed to the following ordinary (similarity) equations 2 ′ f ″ þ ð1−ϕÞ2:5 1−ϕ þ ϕρs =ρf G f ′ h i 1 2:5 ′ ¼ ð1−ϕÞ 1−ϕ þ ϕ ðρs βs Þ= ρf βf λ η θ −θ 2
ð12Þ
k =k 1 ′ ′ nf f . θ″ ¼ θ f −f θ 2 ρC p 1−ϕ þ ϕ ρC p
ð13Þ
s
f
It should be mentioned that Maxwell [26] first published a theoretical work for the effective thermal conductivity k of two-phase mixtures containing powders with particle diameters in the order of millimeters or even micrometers. Maxwell's model predicted that the effective thermal conductivity of suspended spherical particles increases with an increase in the volume fraction of the solid particles. Table 2 Values of the heat transfer − θ′(0) and sleep velocity f′(0) for ϕ = 0 (Newtonian fluid), G = 0 (Darcy case) and several values of λ compared with the results of Arifin et al. [24]. λ
0 0.6 1 2 5 8 15
Present
Arifin et al. [24]
− θ′(0)
f′(0)
− θ′(0)
f′(0)
0.8862 1.0281 1.1020 1.2495 1.5503 1.7609 2.1137
1.0000 1.4741 1.7474 2.3480 3.7995 4.9986 7.3446
0.8862 1.0282 1.1020 1.2493 1.5502 1.7609 2.1133
1.0000 1.4741 1.7474 2.3471 3.7992 4.9984 7.3422
Fig. 2. Dimensionless temperature profiles θ(η) for λ= 0, 0.5, 1, 2, 4, 8, 16 when G= 1 and ϕ = 0.1.
A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
Fig. 3. Dimensionless velocity profiles f′(η) for λ = 0, 0.5, 1, 2, 4, 8, 16 when G= 1 and ϕ = 0.1.
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Fig. 5. Dimensionless velocity profiles f′(η) for G= 0, 0.1, 1, 5, 8, 10 when λ= 1 and ϕ = 0.1.
Using Eqs. (11), (16) and (17), we obtain along with the corresponding boundary conditions f ð0Þ ¼ 0; θð0Þ ¼ 1 at f ′ ðηÞ→1; θðηÞ→0 as
−1=2
η¼0 η→∞:
ð14Þ
Here G is the non-Darcy or inertial parameter and λ is the mixed convection parameter, which are defined as ′ K ′ U ∞ K U ∞ ρf G¼ ¼ ; νf μf
λ¼
Rax
ð15Þ
3=2
Pex
where Rax = g ρ f K β f (Tw − T∞)x/(μ f α f) is the local Rayleigh number and Pex = U∞ x/αf is the local Péclet number for the porous medium. A physical quantity of practical interest is the local Nusselt number Nux, which is defined as Nux ¼
x qw kf ðT w −T ∞ Þ
ð16Þ
where qw is the heat flux from the surface of the plate, which is given by qw ¼ −knf
∂T : ∂y y¼0
ð17Þ
1
Pex
Nux ¼ −
knf ′ θ ð0Þ: kf
ð18Þ
3. Results and discussion The problem is formulated so that we can consider different types of nanoparticles (e.g. Cu, Al2O3, TiO2, etc.) and water as a base fluid. However, in order to save space, we have considered here only the case of Cu nanoparticles. The system of the ordinary differential Eqs. (12) and (13) which is subject to the boundary conditions (Eq. 14) was solved using the function bvp4c from Matlab. The relative tolerance was set to 10−10. The obtained results are presented for several values of the governing parameters in some figures and tables. As in Oztop and Abu-Nada [15], we take the values of the solid volume fraction ϕ in the range of 0≤ϕ≤0.2. However, Muthtamilselvan et al. [27] have considered for the convective flow in a lid driven cavity, the values of the solid volume fraction in the range of 0 ≤ϕ≤0.08. If the concentration exceeds the maximum level of 0.08, sedimentation could take place. In order to verify the numerical method the results were compared with those reported by Arifin et al. [24] for the Darcy model (G=0) (see Table 2) and Lai and Kulacki [22] for regular fluid (ϕ=0) as can be seen in Table 3. We notice that the results are in very good agreement and therefore we are confident that the present results are accurate. Figs. 2 and 3 present the dimensionless temperature θ(η) and velocity f′(η) profiles for ϕ = 0.1 when G = 1 and λ = 0, 0.5, 1, 2, 4, 8, 16.
0.9 0.8 0.7
θ(η)
0.6 0.5
G = 0, 0.1, 1, 5, 8, 10 0.4 0.3 0.2 0.1 0
0
2
4
η
6
8
10
Fig. 4. Dimensionless temperature profiles θ(η) for G = 0, 0.1, 1, 5, 8, 10 when λ = 1 and ϕ = 0.1.
Fig. 6. Dimensionless temperature profiles θ(η) for ϕ = 0, 0.05, 0.1, 0.15, 0.2 when λ = 1and G = 1.
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A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085
Fig. 7. Dimensionless velocity profiles f′(η) for ϕ= 0, 0.05, 0.1, 0.15, 0.2 when λ= 1and G = 1.
Fig. 9. Variation of the local Nusselt number with λ for ϕ = 0, 0.1, 0.2 when G = 1.
Following Nield and Kunznetsov [28], we refer to the reduced Nusselt number Nur given by It can be seen that the thermal boundary layer thickness decreases while the momentum boundary layer increases with the increase of the mixed convection parameter λ. This happens because the flow is assisting. Figs. 4 to 5 display the variation of the dimensionless temperature and velocity profiles for G = 0, 0.1, 1, 5, 10 when λ = 1 and ϕ = 0.1. The thickness of the temperature boundary layer is greater for the non-Darcy fluid (G ≠ 0) than for the Darcy fluid (G = 1), while the momentum boundary layer thickness decreases with the inertial parameter G. The behavior of the profiles shown in Figs. 2, 3, 4, and 5 is similar with those reported by Lai and Kulacki [22]. Further, Figs. 6 and 7 present the dimensionless temperature and velocity profiles for ϕ = 0, 0.05, 0.1, 0.15, 0.2 when λ = 1 and G = 1. The thermal boundary layer thickness increases with the increase of the nanoparticle concentration parameter ϕ, while the momentum boundary layer thickness decreases near the plate and then increases at some distance from the plate. From the physical point of view, the increases of the parameter ϕ lead to an increase of the viscosity and thermal conductivity of the nanofluid. This behavior of the thermal boundary layer is in agreement with the decrease of the heat transfer rate from the surface of the plate to the nanofluid. Finally, Figs. 8 to 9 illustrate the variation of the local Nusselt number Nux/Pe 1/2 with λ when ϕ = 0, 0.1 and 0.2 when G = 0, 0.1, 1 and 10. One can notice from these figures that Nux/Pe 1/2 increases monotonically with λ and also increases with ϕ.
Nur ¼ Nux =Pex
1=2
ð19Þ
and we perform a multiple linear regression in order to obtain a correlation between Nur and ϕ, and G and λ. A multiple linear regression of the form Nurest ¼ a þ bϕ þ cG þ dλ
ð20Þ
was considered, where Nurest is the response variable, while ϕ, G and λ are the independent variables. The following values for ϕ = 0, 0.1, 0.15, 0.2, G = 0, 0.1, 0.5, 1 and λ = 0.1, 0.5, 0.75, 1 are considered. In total for carrying out the regression, we used 64, 3-tuple of the form (ϕ, G, λ) with the corresponding values of the response variable Nur given by Eq. (19). We used the function regress from Matlab, and it gives the following form of Nurest: Nurest ¼ 0:9337 þ 1:2163ϕ−0:0827G þ 0:1276λ:
ð21Þ
The coefficient of multiple determination is R 2 = 0.9603 and the maximum relative error defined by ε = |(Nurest − Nur)/Nur| is ε = 0.0424. We observe from Eq. (21) that an increase in the parameters ϕ and λ leads to an increase in the value of Nurest, while a decrease in the parameter G leads to an increase of Nurest. This behavior is in agreement with the numerical values of Nur shown in Tables 2, 3, 4 and 5.
Table 4 Values of the heat transfer − θ′(0) and sleep velocity f′(0), when ϕ= 0.1, G= 0, 0.1, 1, 10 and λ= 0.1, 0.5, 1, 2, 4, 8, 10, 20, 30, 40, 50 in the case of Cu nanoparticles. λ
Fig. 8. Variation of the local Nusselt number with λ for G= 0, 0.1, 1, 10 when ϕ = 0.1.
0.1 0.5 1 2 4 8 10 20 30 40 50
G=0
G = 0.1
G= 1
G = 10
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
0.7804 0.8475 0.9170 1.0281 1.1933 1.4247 1.5153 1.8547 2.0990 2.2957 2.4628
1.0700 1.3286 1.6175 2.1250 2.9840 4.3970 5.0178 7.6779 9.9220 11.9293 13.7763
0.7763 0.8292 0.8841 0.9713 1.0983 1.2693 1.3341 1.5656 1.7226 1.8438 1.9435
1.0548 1.2548 1.4736 1.8450 2.4383 3.3350 3.7036 5.1538 6.2545 7.1696 7.9652
0.7664 0.7858 0.8077 0.8454 0.9055 0.9930 1.0275 1.1550 1.2439 1.3135 1.3712
1.0188 1.0897 1.1707 1.3138 1.5506 1.9164 2.0677 2.6634 3.1142 3.4877 3.8117
0.7619 0.7647 0.7680 0.7745 0.7868 0.8091 0.8194 0.8635 0.8997 0.9305 0.9576
1.0025 1.0124 1.0245 1.0481 1.0931 1.1756 1.2137 1.3818 1.5235 1.6476 1.7590
A.V. Rosca et al. / International Communications in Heat and Mass Transfer 39 (2012) 1080–1085 Table 5 Values of the heat transfer − θ′(0) and sleep velocity f′(0), when ϕ = 0.2, G= 0, 0.1, 1, 10 and λ = 0.1, 0.5, 1, 2, 4, 8, 10, 20, 30, 40, 50 in the case of Cu nanoparticles. λ
0.1 0.5 1 2 4 8 10 20 30 40 50
G=0
G = 0.1
G=1
G = 10
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
− θ′(0)
f′(0)
0.6717 0.7177 0.7668 0.8477 0.9714 1.1483 1.2183 1.4824 1.6736 1.8280 1.9594
1.0538 1.2557 1.4855 1.8957 2.5994 3.7682 4.2837 6.4990 8.3716 10.0479 11.5910
0.6688 0.7045 0.7428 0.8056 0.9003 1.0314 1.0818 1.2641 1.3889 1.4857 1.5656
1.0414 1.1956 1.3683 1.6679 2.1574 2.9111 3.2238 4.4634 5.4104 6.2001 6.8880
0.6622 0.6746 0.6890 0.7145 0.7568 0.8209 0.8468 0.9444 1.0137 1.0685 1.1141
1.0136 1.0660 1.1270 1.2375 1.4260 1.7261 1.8524 2.3562 2.7419 3.0631 3.3426
0.6594 0.6610 0.6631 0.6672 0.6751 0.6896 0.6964 0.7266 0.7521 0.7744 0.7943
1.0018 1.0088 1.0176 1.0347 1.0677 1.1294 1.1584 1.2894 1.4028 1.5037 1.5953
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