Conjugate heat transfer of a nanofluid in a vertical channel adjacent to a heat generating solid domain

Flavius Patrulescu
(original), Numerical Analysis, paper

Abstract

The effect of thermal dispersion in the conjugate steady free convection flow of a nanofluid in a vertical channel is investigated numerically using a single phase model. Considering the laminar and fully developed flow regime, a simplified mathematical model is obtained. In the particular cases when solid phase and thermal dispersion effects are neglected the problem was solved analytically. The numerical solution is shown to be in excellent agreement with the close form analytical solution. Nusselt number enhancement with the Grashof number, volume fraction, aspect ratio parameter and thermal diffusivity constant increasing has been found.

Author

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Teodor Groşan
(Babeş-Bolyai University Faculty of Mathematics and Computer Sciences)

Keywords

nanofluid; vertical channel; free convection; conjugate heat transfer; heat generation

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Cite this paper as:

F. Pătrulescu, T. Groşan, Conjugate heat transfer of a nanofluid in a vertical channel adjacent to a heat generating solid domain, Rev. Anal. Numer. Theor. Approx., vol. 39, no. 2 (2010), pp. 141-149

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Journal

Revue d’Analyse Numérique et de Théorie de l’Approximation

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Editions de l’Academie Roumaine

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https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art5

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1222-9024

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2457-8126

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https://scholar.google.ro/scholar?oi=bibs&hl=ro&cites=899108882819761590

References

[1] Aung,  Fully developed laminar free convection between vertical plates heated\linebreak asymmetrically, Int. J. Heat Mass Transfer 15 (1972), 1577-1580.
[2] Aung, L.S. Fletcher, V. Sernas, Developing laminar free convection between vertical flat plates with asymmetric heating, Int. J. Heat Mass Transfer 15 (1972), 2293-2308.
[3] Barletta, Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions, J. Heat Transfer 121 (1999),349-356.
[4] C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952), 571-581.
[5] Daungthongsuk, S. Wongwises, A critical review of convective heat transfer of nanofluids, Renewable and Sustainable Energy Reviews 11 (2007), 797-817.
[6] R.A. Khaled, K. Vafai, Heat transfer enhancement through control of thermal dispersion effects, Int. J. Heat Mass Transfer 48 (2005), 2172-2185.
[7] Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003), 3639-3663.
[8] P. Kumar, J.C. Umavathi, A.J. Chamkha, I. Pop, Fully-developed free-convective flow of micropolar and viscous fluids in a vertical channel, Appl. Math. Modell.  34 (2010), 1175-1186.
[9] Kumar, S.K. Prasad, J. Banerjee, Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model, Appl. Math. Modell. 34 (2010),573-592.
[10] Mokmeli, M. Saffar-Avval, Prediction of nanofluid convective heat transfer using the dispersion model, Int. J. Thermal Sci. 49 (2010), 471-478.
[11] F. Oztop, E. Abu-Nada,  Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow,  29 (2008),1326-1336.
[12] Vajravelu, K. Sastri, Fully developed laminar free convection flow between two parallel vertical walls-I, Int. J. Heat Mass Transfer, 20 (1997), 655-660.
[13] Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer  43 (2000), 3701–3707.
[14] Q. Wang, A. S. Mujumdar,  A review on nanofluids-part I: Theoretical and numerical investigations}, Brazilian Journal of Chemical Engineering,25 (2008), 613-630.

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2010_Patrulescu_ANTA_Conjugate_heat_transfer_vertical_channel

Conjugate heat transfer in a vertical channel filled with a nanofluid adjacent to a heat generating solid domain

F. Pătrulescu 1 1 ^(1){ }^{1}1, Teodor Groşan 2 2 ^(2){ }^{2}2Tiberiu Popoviciu Institute of Numerical AnalysisP.O. Box 68-1, 400110 Cluj-Napoca, Romania,fpatrulescu@ictp.acad.roFaculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania

Abstract

The effect of thermal dispersion in the conjugate steady free convection flow of a nanofluid in a vertical channel is investigated numerically using a single phase model. Considering the laminar and fully developed flow regime a simplified mathematical model is obtained. In the particular cases when solid phase and thermal dispersion effects are neglected the problem was solved analytically. The numerical solution is shown to be in excellent agreement with the close form analytical solution. Nusselt number enhancement with the Grashof number, volume fraction, aspect ratio parameter and thermal diffusivity constant increasing has been found.

2010 Mathematics Subject Classification : 82D80, 76R10
Keywords: nanofluid, vertical channel, free convection, conjugate heat transfer, heat generation

1 Introduction

Heat transfer in channels occurs in many industrial processes and natural phenomena. It has been the subject of many studies for different flow configurations. We mention some practical applications of convective heat transfer in channels: design of cooling systems for electronic devices, insulation, ventilation, grain storage, geothermal energy recover, solar energy collection, etc. Some classical papers, such as by Aung [1], Aung et al. [2], Barletta [3], Kumar et al. [8], Vajravelu and Sastri [12],
are concerned with the evaluation of the temperature and velocity profiles for the vertical parallel-flow fully developed regime. Enhancement of heat transfer is essential in improving performances and compactness of electronic devices. Usual cooling agents (water, oil, etc.) have relatively small thermal conductivities and therefore heat transfer is not very efficient. Suspensions of nanoparticles in fluids improve physical properties and increase the heat transfer. Small fraction of nanoparticles added in a base fluid leads to a large increase of the fluid thermal conductivity. Good description and classification of the nanofluids characteristics can be found in papers such as: Daungthongsuk and Wongwises [5], Wang and Mujumdar [14] and Kumar et al. 9. The chaotic movement of the nanoparticles and sleeping between the fine particles and fluid generate the thermal dispersion effect and this leads to an increase in the energy exchange rates in fluid. Xuan and Roetzel [13] proposed a thermal dispersion model for a single phase nanofluid. Thermal dispersion effects in nanofluids flow in enclosure using a single phase model were analyzed by Khanafer et al. [7] and Kumar et al. [9] for a differentially heated rectangular cavity, Khaled and Vafai [6] studied the heat transfer enhancement through control of thermal dispersion effects in a horizontal channel, while Mokmeli and Saffar-Avval [10] numerically studied nanofluid heat transfer in a straight tube. In all these paper the enhancement of heat transfer due to nanofluids special properties was reported. In the present paper, the effect of the thermal dispersion on the steady free convection flow in a long vertical channel, using the fully developed flow assumptions, is investigated using a single phase thermal dispersion model similar with the model considered by Khanafer et al. 77.

2 Basic equations

Consider the fully developed steady flow of an incompressible nanofluid in vertical channel. The left wall of the channel have a thickness b b bbb and thus we have to consider a conjugate heat transfer problem. The geometry of the problem, the boundary conditions, and the coordinate system is shown in Fig. 1.
The fluid flows up in the channel driven by buoyancy forces, so that the flow is due only to the difference in temperature gradient. The flow being fully developed the following relations apply here v = 0 v = 0 v=0v=0v=0 and ∂ v / ∂ y = 0 ∂ v / ∂ y = 0 del v//del y=0\partial v / \partial y=0∂v/∂y=0, where v v vvv is the velocity in the transversal direction. Thus, from the continuity equation, we get ∂ u / ∂ x = 0 ∂ u / ∂ x = 0 del u//del x=0\partial u / \partial x=0∂u/∂x=0 so that the velocity along the channel is u = u ( y ) u = u ( y ) u=u(y)u=u(y)u=u(y). Based on the fact that the flow is fully developed we can assume that the temperature depends only by y y yyy, i.e., T = T ( y ) T = T ( y ) T=T(y)T=T(y)T=T(y). The physical properties of the nanofluid are considered constant except for density, which is given by the Boussinesq approximation.
We use in this study the heat capacity and the thermal expansion coefficient of
Figure 1: Geometry of the problem and the coordinate system
the nanofluid given in Kanafer et al. [7] as:
(2.1) ( ρ C p ) n f = ( 1 − ϕ ) ( ρ C p ) f + ϕ ( ρ C p ) s (2.2) ( ρ β ) n f = ( 1 − ϕ ) ρ f β f + ϕ ρ s β s (2.1) ρ C p n f = ( 1 − ϕ ) ρ C p f + ϕ ρ C p s (2.2) ( ρ β ) n f = ( 1 − ϕ ) ρ f β f + ϕ ρ s β s {:[(2.1)(rhoC_(p))_(nf)=(1-phi)(rhoC_(p))_(f)+phi(rhoC_(p))_(s)],[(2.2)(rho beta)_(nf)=(1-phi)rho_(f)beta_(f)+phirho_(s)beta_(s)]:}\begin{gather*} \left(\rho C_{p}\right)_{n f}=(1-\phi)\left(\rho C_{p}\right)_{f}+\phi\left(\rho C_{p}\right)_{s} \tag{2.1}\\ (\rho \beta)_{n f}=(1-\phi) \rho_{f} \beta_{f}+\phi \rho_{s} \beta_{s} \tag{2.2} \end{gather*}(2.1)(ρCp)nf=(1−ϕ)(ρCp)f+ϕ(ρCp)s(2.2)(ρβ)nf=(1−ϕ)ρfβf+ϕρsβs
where ρ ρ rho\rhoρ is the density, c p c p c_(p)c_{p}cp is the specific heat at constant pressure, ϕ ϕ phi\phiϕ is the volume fraction of suspension particles, β β beta\betaβ is the expansion coefficient, while subscripts n f , f n f , f nf,fn f, fnf,f and s s sss stand for nanofluid, fluid and solid, respectively.
For the effective viscosity we consider the model proposed by Brinkman [4], which is valid for high volume fraction ( ϕ > 0.05 ϕ > 0.05 phi > 0.05\phi>0.05ϕ>0.05 ):
(2.3) μ n f = μ f ( 1 − ϕ ) 2.5 , (2.3) μ n f = μ f ( 1 − ϕ ) 2.5 , {:(2.3)mu_(nf)=(mu_(f))/((1-phi)^(2.5))",":}\begin{equation*} \mu_{n f}=\frac{\mu_{f}}{(1-\phi)^{2.5}}, \tag{2.3} \end{equation*}(2.3)μnf=μf(1−ϕ)2.5,
where μ μ mu\muμ is the dinamyc viscosity.
The effective stagnant thermal conductivity is approximated by the MaxwellGarnetts model, see Wang and Mujumdar [14], which applies for spherical type particles:
(2.4) k n f k f = k s + 2 k f − 2 ϕ ( k f − k s ) k s + 2 k f + ϕ ( k f − k s ) , (2.4) k n f k f = k s + 2 k f − 2 ϕ k f − k s k s + 2 k f + ϕ k f − k s , {:(2.4)(k_(nf))/(k_(f))=(k_(s)+2k_(f)-2phi(k_(f)-k_(s)))/(k_(s)+2k_(f)+phi(k_(f)-k_(s)))",":}\begin{equation*} \frac{k_{n f}}{k_{f}}=\frac{k_{s}+2 k_{f}-2 \phi\left(k_{f}-k_{s}\right)}{k_{s}+2 k_{f}+\phi\left(k_{f}-k_{s}\right)}, \tag{2.4} \end{equation*}(2.4)knfkf=ks+2kf−2ϕ(kf−ks)ks+2kf+ϕ(kf−ks),
where k k kkk is the thermal conductivity.
The effective thermal conductivity includes also the thermal dispersion enhancement
(2.5) k e f f = k n f + k d , (2.5) k e f f = k n f + k d , {:(2.5)k_(eff)=k_(nf)+k_(d)",":}\begin{equation*} k_{e f f}=k_{n f}+k_{d}, \tag{2.5} \end{equation*}(2.5)keff=knf+kd,
where the term due to thermal dispersion, k d k d k_(d)k_{d}kd, is given by, see Khaled and Vafai [6]:
(2.6) k d = C ( ρ C p ) n f | u | ϕ L , (2.6) k d = C ρ C p n f | u | ϕ L , {:(2.6)k_(d)=C(rhoC_(p))_(nf)|u|phi L",":}\begin{equation*} k_{d}=C\left(\rho C_{p}\right)_{n f}|u| \phi L, \tag{2.6} \end{equation*}(2.6)kd=C(ρCp)nf|u|ϕL,
where L L LLL is the thickness of the channel and C C CCC is a constant depending on the diameter of the nanoparticle and its surface geometry.
We limit the study in this paper to water based nanofluids containing C u C u CuC uCu nanoparticles. Nanofluids thermo-physical properties are shown in the Table 1, see Oztop and Abu-Nada [11].
Property H 2 O H 2 O H_(2)O\mathrm{H}_{2} \mathrm{O}H2O Cu
C p ( J / kgK ) C p ( J / kgK ) C_(p)(J//kgK)C_{p}(\mathrm{~J} / \mathrm{kgK})Cp( J/kgK) 4179 385
ρ ( kg / m 3 ) ρ kg / m 3 rho((kg)//m^(3))\rho\left(\mathrm{~kg} / \mathrm{m}^{3}\right)ρ( kg/m3) 997.1 8933
k ( W / mK 2 ) k W / mK 2 k((W)//mK^(2))k\left(\mathrm{~W} / \mathrm{mK}^{2}\right)k( W/mK2) 0.613 400
α × 10 7 ( m 2 / s ) α × 10 7 m 2 / s alpha xx10^(7)(m^(2)//s)\alpha \times 10^{7}\left(\mathrm{~m}^{2} / \mathrm{s}\right)α×107( m2/s) 1.47 1163.1
β × 10 − 5 ( 1 / K ) β × 10 − 5 ( 1 / K ) beta xx10^(-5)(1//K)\beta \times 10^{-5}(1 / \mathrm{K})β×10−5(1/K) 21 1.67
Property H_(2)O Cu C_(p)(J//kgK) 4179 385 rho((kg)//m^(3)) 997.1 8933 k((W)//mK^(2)) 0.613 400 alpha xx10^(7)(m^(2)//s) 1.47 1163.1 beta xx10^(-5)(1//K) 21 1.67| Property | $\mathrm{H}_{2} \mathrm{O}$ | Cu | | :---: | :---: | :---: | | $C_{p}(\mathrm{~J} / \mathrm{kgK})$ | 4179 | 385 | | $\rho\left(\mathrm{~kg} / \mathrm{m}^{3}\right)$ | 997.1 | 8933 | | $k\left(\mathrm{~W} / \mathrm{mK}^{2}\right)$ | 0.613 | 400 | | $\alpha \times 10^{7}\left(\mathrm{~m}^{2} / \mathrm{s}\right)$ | 1.47 | 1163.1 | | $\beta \times 10^{-5}(1 / \mathrm{K})$ | 21 | 1.67 |
Table 1: Physical properties of fluid and Cu nanoparticles
In the assumption of the fully developed flow the governing equations for the flow and heat transfer have the following form:
(2.7) α s d 2 T s d y 2 + q 0 ′ ′ ′ ( ρ C p ) s = 0 ; (2.8) μ n f d 2 u d y 2 + ( ρ β ) n f g ( T f − T 0 ) = 0 ; (2.9) d d y ( k e f f d T f d y ) = 0 ; (2.7) α s d 2 T s d y 2 + q 0 ′ ′ ′ ρ C p s = 0 ; (2.8) μ n f d 2 u d y 2 + ( ρ β ) n f g T f − T 0 = 0 ; (2.9) d d y k e f f d T f d y = 0 ; {:[(2.7)alpha_(s)(d^(2)T_(s))/(dy^(2))+(q_(0)^('''))/((rhoC_(p))_(s))=0;],[(2.8)mu_(nf)(d^(2)u)/(dy^(2))+(rho beta)_(nf)g(T_(f)-T_(0))=0;],[(2.9)(d)/(dy)(k_(eff)(dT_(f))/(dy))=0;]:}\begin{gather*} \alpha_{s} \frac{d^{2} T_{s}}{d y^{2}}+\frac{q_{0}^{\prime \prime \prime}}{\left(\rho C_{p}\right)_{s}}=0 ; \tag{2.7}\\ \mu_{n f} \frac{d^{2} u}{d y^{2}}+(\rho \beta)_{n f} g\left(T_{f}-T_{0}\right)=0 ; \tag{2.8}\\ \frac{d}{d y}\left(k_{e f f} \frac{d T_{f}}{d y}\right)=0 ; \tag{2.9} \end{gather*}(2.7)αsd2Tsdy2+q0′′′(ρCp)s=0;(2.8)μnfd2udy2+(ρβ)nfg(Tf−T0)=0;(2.9)ddy(keffdTfdy)=0;
subject to the boundary conditions:
(2.10) T s | y = 0 = T H ; T f | y = L = T C ; (2.11) T f | y = b = T s | y = b ; (2.12) k s ∂ T s ∂ y | y = b = k n f ∂ T f ∂ y | y = b ; (2.13) u ( b ) = u ( L ) = 0 ; (2.10) T s y = 0 = T H ; T f y = L = T C ; (2.11) T f y = b = T s y = b ; (2.12) k s ∂ T s ∂ y y = b = k n f ∂ T f ∂ y y = b ; (2.13) u ( b ) = u ( L ) = 0 ; {:[(2.10)T_(s)|_(y=0)=T_(H); quadT_(f)|_(y=L)=T_(C);],[(2.11)T_(f)|_(y=b)=T_(s)|_(y=b);],[(2.12)k_(s)(delT_(s))/(del y)|_(y=b)=k_(nf)(delT_(f))/(del y)|_(y=b);],[(2.13)u(b)=u(L)=0;]:}\begin{gather*} \left.T_{s}\right|_{y=0}=T_{H} ;\left.\quad T_{f}\right|_{y=L}=T_{C} ; \tag{2.10}\\ \left.T_{f}\right|_{y=b}=\left.T_{s}\right|_{y=b} ; \tag{2.11}\\ \left.k_{s} \frac{\partial T_{s}}{\partial y}\right|_{y=b}=\left.k_{n f} \frac{\partial T_{f}}{\partial y}\right|_{y=b} ; \tag{2.12}\\ u(b)=u(L)=0 ; \tag{2.13} \end{gather*}(2.10)Ts|y=0=TH;Tf|y=L=TC;(2.11)Tf|y=b=Ts|y=b;(2.12)ks∂Ts∂y|y=b=knf∂Tf∂y|y=b;(2.13)u(b)=u(L)=0;
where g g ggg is the gravitational acceleration, T T TTT is the temperature, u u uuu is the velocity, q 0 ′ ′ ′ q 0 ′ ′ ′ q_(0)^(''')q_{0}^{\prime \prime \prime}q0′′′ is the heat generation and α α alpha\alphaα is the thermal diffusivity.
In order to solve equations (2.7)-(2.9), subject to the boundary conditions (2.10)(2.13), we introduce the following non-dimensional variables used also by Kumar et al. (8):
(2.14) Θ s = k s ( T s − T 0 ) q 0 ′ ′ ′ L 2 , Θ f = k s ( T f − T 0 ) q 0 ′ ′ ′ L 2 , Y = y L , U = u U c , (2.14) Θ s = k s T s − T 0 q 0 ′ ′ ′ L 2 , Θ f = k s T f − T 0 q 0 ′ ′ ′ L 2 , Y = y L , U = u U c , {:(2.14)Theta_(s)=(k_(s)(T_(s)-T_(0)))/(q_(0)^(''')L^(2))","quadTheta_(f)=(k_(s)(T_(f)-T_(0)))/(q_(0)^(''')L^(2))","quad Y=(y)/(L)","quad U=(u)/(U_(c))",":}\begin{equation*} \Theta_{s}=\frac{k_{s}\left(T_{s}-T_{0}\right)}{q_{0}^{\prime \prime \prime} L^{2}}, \quad \Theta_{f}=\frac{k_{s}\left(T_{f}-T_{0}\right)}{q_{0}^{\prime \prime \prime} L^{2}}, \quad Y=\frac{y}{L}, \quad U=\frac{u}{U_{c}}, \tag{2.14} \end{equation*}(2.14)Θs=ks(Ts−T0)q0′′′L2,Θf=ks(Tf−T0)q0′′′L2,Y=yL,U=uUc,
where U c U c U_(c)U_{c}Uc and T 0 T 0 T_(0)T_{0}T0 are the characteristic velocity and temperature given by:
(2.15) T 0 = T H + T C 2 , U c = g β f ( q 0 ′ ′ ′ L 2 k s ) L 2 ν f (2.15) T 0 = T H + T C 2 , U c = g β f q 0 ′ ′ ′ L 2 k s L 2 ν f {:(2.15)T_(0)=(T_(H)+T_(C))/(2)","quadU_(c)=(gbeta_(f)((q_(0)^(''')L^(2))/(k_(s)))L^(2))/(nu_(f)):}\begin{equation*} T_{0}=\frac{T_{H}+T_{C}}{2}, \quad U_{c}=\frac{g \beta_{f}\left(\frac{q_{0}^{\prime \prime \prime} L^{2}}{k_{s}}\right) L^{2}}{\nu_{f}} \tag{2.15} \end{equation*}(2.15)T0=TH+TC2,Uc=gβf(q0′′′L2ks)L2νf
Using (2.14) in equations (2.7)-(2.9) we obtain the following dimensionless ordinary differential equations:
(2.16) d 2 Θ s d Y 2 + 1 = 0 ; (2.17) d 2 U d Y 2 + λ ϕ Θ f = 0 ; (2.18) d d Y [ ( k ϕ + C ϕ | U | ) d Θ f d Y ] = 0 ; (2.16) d 2 Θ s d Y 2 + 1 = 0 ; (2.17) d 2 U d Y 2 + λ ϕ Θ f = 0 ; (2.18) d d Y k ϕ + C ϕ | U | d Θ f d Y = 0 ; {:[(2.16)(d^(2)Theta_(s))/(dY^(2))+1=0;],[(2.17)(d^(2)U)/(dY^(2))+lambda_(phi)Theta_(f)=0;],[(2.18)(d)/(dY)[(k_(phi)+C_(phi)|U|)(dTheta_(f))/(dY)]=0;]:}\begin{gather*} \frac{d^{2} \Theta_{s}}{d Y^{2}}+1=0 ; \tag{2.16}\\ \frac{d^{2} U}{d Y^{2}}+\lambda_{\phi} \Theta_{f}=0 ; \tag{2.17}\\ \frac{d}{d Y}\left[\left(k_{\phi}+C_{\phi}|U|\right) \frac{d \Theta_{f}}{d Y}\right]=0 ; \tag{2.18} \end{gather*}(2.16)d2ΘsdY2+1=0;(2.17)d2UdY2+λϕΘf=0;(2.18)ddY[(kϕ+Cϕ|U|)dΘfdY]=0;
subject to
(2.19) Θ s | Y = 0 = q (2.20) Θ s | Y = r = Θ f | Y = r (2.21) d Θ s d Y | Y = r = K d Θ f d Y | Y = r (2.22) Θ f | Y = 1 = − q (2.23) U ( r ) = U ( 1 ) = 0 (2.19) Θ s Y = 0 = q (2.20) Θ s Y = r = Θ f Y = r (2.21) d Θ s d Y Y = r = K d Θ f d Y Y = r (2.22) Θ f Y = 1 = − q (2.23) U ( r ) = U ( 1 ) = 0 {:[(2.19)Theta_(s)|_(Y=0)=q],[(2.20)Theta_(s)|_(Y=r)=Theta_(f)|_(Y=r)],[(2.21)(dTheta_(s))/(dY)|_(Y=r)=K(dTheta_(f))/(dY)|_(Y=r)],[(2.22)Theta_(f)|_(Y=1)=-q],[(2.23)U(r)=U(1)=0]:}\begin{gather*} \left.\Theta_{s}\right|_{Y=0}=q \tag{2.19}\\ \left.\Theta_{s}\right|_{Y=r}=\left.\Theta_{f}\right|_{Y=r} \tag{2.20}\\ \left.\frac{d \Theta_{s}}{d Y}\right|_{Y=r}=\left.K \frac{d \Theta_{f}}{d Y}\right|_{Y=r} \tag{2.21}\\ \left.\Theta_{f}\right|_{Y=1}=-q \tag{2.22}\\ U(r)=U(1)=0 \tag{2.23} \end{gather*}(2.19)Θs|Y=0=q(2.20)Θs|Y=r=Θf|Y=r(2.21)dΘsdY|Y=r=KdΘfdY|Y=r(2.22)Θf|Y=1=−q(2.23)U(r)=U(1)=0
where:
(2.24) r = b L , λ ϕ = ( 1 − ϕ ) 2.5 [ ( 1 − ϕ ) + ϕ ρ s β s ρ f β f ] , q = k s ( T H − T C ) 2 q 0 ′ ′ ′ L 2 , (2.25) K = k n f k s , k ϕ = k n f / k f 1 − ϕ + ϕ ( ρ C p ) s ( ρ C p ) f , C ϕ = C ϕ Pr G r (2.24) r = b L , λ ϕ = ( 1 − ϕ ) 2.5 ( 1 − ϕ ) + ϕ ρ s β s ρ f β f , q = k s T H − T C 2 q 0 ′ ′ ′ L 2 , (2.25) K = k n f k s , k ϕ = k n f / k f 1 − ϕ + ϕ ρ C p s ρ C p f , C ϕ = C ϕ Pr G r {:[(2.24)r=(b)/(L)","quadlambda_(phi)=(1-phi)^(2.5)[(1-phi)+phi(rho_(s)beta_(s))/(rho_(f)beta_(f))]","quad q=(k_(s)(T_(H)-T_(C)))/(2q_(0)^(''')L^(2))","],[(2.25)K=(k_(nf))/(k_(s))","quadk_(phi)=(k_(nf)//k_(f))/(1-phi+phi((rhoC_(p))_(s))/((rhoC_(p))_(f)))","quadC_(phi)=C phi Pr Gr]:}\begin{align*} r=\frac{b}{L}, \quad \lambda_{\phi}=(1-\phi)^{2.5}\left[(1-\phi)+\phi \frac{\rho_{s} \beta_{s}}{\rho_{f} \beta_{f}}\right], \quad q=\frac{k_{s}\left(T_{H}-T_{C}\right)}{2 q_{0}^{\prime \prime \prime} L^{2}}, \tag{2.24}\\ K=\frac{k_{n f}}{k_{s}}, \quad k_{\phi}=\frac{k_{n f} / k_{f}}{1-\phi+\phi \frac{\left(\rho C_{p}\right)_{s}}{\left(\rho C_{p}\right)_{f}}}, \quad C_{\phi}=C \phi \operatorname{Pr} G r \tag{2.25} \end{align*}(2.24)r=bL,λϕ=(1−ϕ)2.5[(1−ϕ)+ϕρsβsρfβf],q=ks(TH−TC)2q0′′′L2,(2.25)K=knfks,kϕ=knf/kf1−ϕ+ϕ(ρCp)s(ρCp)f,Cϕ=CϕPrGr
are constants depending on the properties of the nanofluid and Pr = ν f / α f , G r = g β f q 0 ′ ′ ′ L 3 / ν f 2 Pr = ν f / α f , G r = g β f q 0 ′ ′ ′ L 3 / ν f 2 Pr=nu_(f)//alpha_(f),Gr=gbeta_(f)q_(0)^(''')L^(3)//nu_(f)^(2)\operatorname{Pr}=\nu_{f} / \alpha_{f}, G r= g \beta_{f} q_{0}^{\prime \prime \prime} L^{3} / \nu_{f}^{2}Pr=νf/αf,Gr=gβfq0′′′L3/νf2 are Prandtl number and Grashof number, respectively.
The physical quantity of interest in this problem is the Nusselt number, which for the conjugate wall is defined as:
(2.26) N u = h L k f | y = b (2.26) N u = h L k f y = b {:(2.26)Nu=(hL)/(k_(f))|_(y=b):}\begin{equation*} N u=\left.\frac{h L}{k_{f}}\right|_{y=b} \tag{2.26} \end{equation*}(2.26)Nu=hLkf|y=b
where the convective heat transfer coefficient, h h hhh, is obtained from the relation:
(2.27) − k e f f d T d y | y = b = h ( T | y = b − T 0 ) . (2.27) − k e f f d T d y y = b = h T y = b − T 0 . {:(2.27)-k_(eff)(dT)/(dy)|_(y=b)=h(T|_(y=b)-T_(0)).:}\begin{equation*} -\left.k_{e f f} \frac{d T}{d y}\right|_{y=b}=h\left(\left.T\right|_{y=b}-T_{0}\right) . \tag{2.27} \end{equation*}(2.27)−keffdTdy|y=b=h(T|y=b−T0).
Substituting (2.27) in (2.26) the dimensionless form of the Nusselt number becomes:
(2.28) N u = − k n f k f 1 Θ f | Y = r d Θ f d Y | Y = r (2.28) N u = − k n f k f 1 Θ f Y = r d Θ f d Y Y = r {:(2.28)Nu=-(k_(nf))/(k_(f))(1)/(Theta_(f)|_(Y=r))(dTheta_(f))/(dY)|_(Y=r):}\begin{equation*} N u=-\left.\frac{k_{n f}}{k_{f}} \frac{1}{\left.\Theta_{f}\right|_{Y=r}} \frac{d \Theta_{f}}{d Y}\right|_{Y=r} \tag{2.28} \end{equation*}(2.28)Nu=−knfkf1Θf|Y=rdΘfdY|Y=r

3 Results and discussions

In the case when thermal dispersion effect is neglected, i.e. C = 0 C = 0 C=0C=0C=0, the problem has an analytical solution, which is given by
(3.1) Θ s ( Y ) = − Y 2 2 + ( r + K a 1 ) Y + q ; (3.2) Θ f ( Y ) = a 1 Y − a 1 − q ; (3.3) U ( Y ) = − λ ϕ [ a 1 Y 3 6 − ( a 1 + q ) Y 2 2 ] + a 2 Y + a 3 ; (3.1) Θ s ( Y ) = − Y 2 2 + r + K a 1 Y + q ; (3.2) Θ f ( Y ) = a 1 Y − a 1 − q ; (3.3) U ( Y ) = − λ ϕ a 1 Y 3 6 − a 1 + q Y 2 2 + a 2 Y + a 3 ; {:[(3.1)Theta_(s)(Y)=-(Y^(2))/(2)+(r+Ka_(1))Y+q;],[(3.2)Theta_(f)(Y)=a_(1)Y-a_(1)-q;],[(3.3)U(Y)=-lambda_(phi)[a_(1)(Y^(3))/(6)-(a_(1)+q)(Y^(2))/(2)]+a_(2)Y+a_(3);]:}\begin{gather*} \Theta_{s}(Y)=-\frac{Y^{2}}{2}+\left(r+K a_{1}\right) Y+q ; \tag{3.1}\\ \Theta_{f}(Y)=a_{1} Y-a_{1}-q ; \tag{3.2}\\ U(Y)=-\lambda_{\phi}\left[a_{1} \frac{Y^{3}}{6}-\left(a_{1}+q\right) \frac{Y^{2}}{2}\right]+a_{2} Y+a_{3} ; \tag{3.3} \end{gather*}(3.1)Θs(Y)=−Y22+(r+Ka1)Y+q;(3.2)Θf(Y)=a1Y−a1−q;(3.3)U(Y)=−λϕ[a1Y36−(a1+q)Y22]+a2Y+a3;
where:
a 1 = 1 2 r 2 + 2 q r ( 1 − K ) − 1 ; a 2 = λ ϕ 6 [ a 1 ( r 2 − 2 r − 2 ) − 3 q ( r + 1 ) ] ; a 3 = λ ϕ 6 r [ − a 1 ( r − 2 ) + 3 q ] a 1 = 1 2 r 2 + 2 q r ( 1 − K ) − 1 ; a 2 = λ ϕ 6 a 1 r 2 − 2 r − 2 − 3 q ( r + 1 ) ; a 3 = λ ϕ 6 r − a 1 ( r − 2 ) + 3 q {:[a_(1)=((1)/(2)r^(2)+2q)/(r(1-K)-1);a_(2)=(lambda_(phi))/(6)[a_(1)(r^(2)-2r-2)-3q(r+1)];],[a_(3)=(lambda_(phi))/(6)r[-a_(1)(r-2)+3q]]:}\begin{gathered} a_{1}=\frac{\frac{1}{2} r^{2}+2 q}{r(1-K)-1} ; a_{2}=\frac{\lambda_{\phi}}{6}\left[a_{1}\left(r^{2}-2 r-2\right)-3 q(r+1)\right] ; \\ a_{3}=\frac{\lambda_{\phi}}{6} r\left[-a_{1}(r-2)+3 q\right] \end{gathered}a1=12r2+2qr(1−K)−1;a2=λϕ6[a1(r2−2r−2)−3q(r+1)];a3=λϕ6r[−a1(r−2)+3q]
In this particular case, the Nusselt number, has the form N u = − a k n f / k f N u = − a k n f / k f Nu=-ak_(nf)//k_(f)N u=-a k_{n f} / k_{f}Nu=−aknf/kf, where a = a 1 / ( a 1 r − a 1 − q ) a = a 1 / a 1 r − a 1 − q a=a_(1)//(a_(1)r-a_(1)-q)a=a_{1} /\left(a_{1} r-a_{1}-q\right)a=a1/(a1r−a1−q), and depends only by thermal characteristics of the nanofluid.
Equations (2.16)-(2.23) were solved numerically using finite difference discretization for different volume fractions of C u C u CuC uCu nanoparticles, ϕ = 0 , 0.05 , 0.1 ϕ = 0 , 0.05 , 0.1 phi=0,0.05,0.1\phi=0,0.05,0.1ϕ=0,0.05,0.1 and 0.2, and thermal conductivity ratio parameter, K = 0.001 , 0.01 K = 0.001 , 0.01 K=0.001,0.01K=0.001,0.01K=0.001,0.01 and 0.1 . In this study we consider fixed values for q q qqq and r r rrr (i.e. q = 1 , r = 0.1 q = 1 , r = 0.1 q=1,r=0.1q=1, r=0.1q=1,r=0.1 ) and, following Khaled and Vafai [6], the values for constant C ϕ C ϕ C_(phi)C_{\phi}Cϕ were taken 0 , 100 , 250 , 500 , 1000 , 5000 0 , 100 , 250 , 500 , 1000 , 5000 0,100,250,500,1000,50000,100,250,500,1000,50000,100,250,500,1000,5000 and 10000.
We compared the numerical method with the analytical solutions (3.1)-(3.3) and a very good agreement was found. In Figs. 2 to 4 the analytical solutions are also presented using a dot marker. Thus, we are confident that the numerical method works fine.
Tables 2 to 4 show the Nusselt number for different values of the above parameters. We mention that the value of Nusselt number increases with the increase of constant C ϕ C ϕ C_(phi)C_{\phi}Cϕ and thermal conductivity parameter K K KKK. Due to the conjugate heat transfer and thermal dispersion Nusselt number does not present a monotone behavior in respect with volume fraction ϕ ϕ phi\phiϕ.
Table 5 presents the variation of the maximum temperature in solid with C ϕ C ϕ C_(phi)C_{\phi}Cϕ and K K KKK. The maximum of the temperature in solid increases with the decrease of C ϕ C ϕ C_(phi)C_{\phi}Cϕ and K K KKK.
Figs. 2 to 4 present the velocity and temperature profiles for ϕ = 0.2 ϕ = 0.2 phi=0.2\phi=0.2ϕ=0.2 and different values of C ϕ C ϕ C_(phi)C_{\phi}Cϕ and K K KKK. The reversed character of the flow becomes less important with the increasing of parameter C ϕ C ϕ C_(phi)C_{\phi}Cϕ for K = 0.001 K = 0.001 K=0.001K=0.001K=0.001 (see Fig. 2a) while for K = 0.1 K = 0.1 K=0.1K=0.1K=0.1 the flow is down for large values of C ϕ C ϕ C_(phi)C_{\phi}Cϕ (see Fig. 2b). Figs. 3 and 4 present the temperature profiles for K = 0.1 K = 0.1 K=0.1K=0.1K=0.1 and K = 0.001 K = 0.001 K=0.001K=0.001K=0.001 in solid and nanofluid. We observe a decrease of the temperature in solid and an increase of the temperature near the cold wall with the increase of C ϕ C ϕ C_(phi)C_{\phi}Cϕ for both values of K K KKK.
C ϕ C ϕ C_(phi)C_{\phi}Cϕ ϕ ϕ phi\phiϕ
0.05 0.1 0.2
0 2.593686 2.984831 3.912962
100 4.383934 4.209832 4.597601
250 8.156642 7.365368 6.165330
500 12.796259 11.836095 9.605016
1000 19.102887 18.263184 15.442523
5000 31.875747 34.480343 37.022310
10000 34.522407 38.455023 45.131964
C_(phi) phi 0.05 0.1 0.2 0 2.593686 2.984831 3.912962 100 4.383934 4.209832 4.597601 250 8.156642 7.365368 6.165330 500 12.796259 11.836095 9.605016 1000 19.102887 18.263184 15.442523 5000 31.875747 34.480343 37.022310 10000 34.522407 38.455023 45.131964| $C_{\phi}$ | | $\phi$ | | | :---: | :---: | :---: | :---: | | | 0.05 | 0.1 | 0.2 | | 0 | 2.593686 | 2.984831 | 3.912962 | | 100 | 4.383934 | 4.209832 | 4.597601 | | 250 | 8.156642 | 7.365368 | 6.165330 | | 500 | 12.796259 | 11.836095 | 9.605016 | | 1000 | 19.102887 | 18.263184 | 15.442523 | | 5000 | 31.875747 | 34.480343 | 37.022310 | | 10000 | 34.522407 | 38.455023 | 45.131964 |
Table 2: Values of Nusselt number for K = 0.1 K = 0.1 K=0.1K=0.1K=0.1
C ϕ C ϕ C_(phi)C_{\phi}Cϕ ϕ ϕ phi\phiϕ
0.05 0.1 0.2
0 2.567849 2.955098 3.873984
100 3.878139 3.912190 4.532541
250 5.801081 5.747647 5.559927
500 7.698974 7.757198 7.510472
1000 9.637866 10.024039 10.138831
5000 12.575802 14.017470 16.581315
10000 13.106638 14.830093 18.361529
C_(phi) phi 0.05 0.1 0.2 0 2.567849 2.955098 3.873984 100 3.878139 3.912190 4.532541 250 5.801081 5.747647 5.559927 500 7.698974 7.757198 7.510472 1000 9.637866 10.024039 10.138831 5000 12.575802 14.017470 16.581315 10000 13.106638 14.830093 18.361529| $C_{\phi}$ | | $\phi$ | | | :---: | :---: | :---: | :---: | | | 0.05 | 0.1 | 0.2 | | 0 | 2.567849 | 2.955098 | 3.873984 | | 100 | 3.878139 | 3.912190 | 4.532541 | | 250 | 5.801081 | 5.747647 | 5.559927 | | 500 | 7.698974 | 7.757198 | 7.510472 | | 1000 | 9.637866 | 10.024039 | 10.138831 | | 5000 | 12.575802 | 14.017470 | 16.581315 | | 10000 | 13.106638 | 14.830093 | 18.361529 |
Table 3: Values of Nusselt number for K = 0.01 K = 0.01 K=0.01K=0.01K=0.01

References

[1] Aung W., Fully developed laminar free convection between vertical plates heated asymmetrically, Int. J. Heat Mass Transfer, 15, pp. 1577-1580, 1972.
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[5] Daungthongsuk W., Wongwises S., A critical review of convective heat transfer of nanofluids, Renewable and Sustainable Energy Reviews, 11, pp. 797-817, 2007.
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C ϕ C ϕ C_(phi)C_{\phi}Cϕ ϕ ϕ phi\phiϕ
0.05 0.1 0.2
0 2.565294 2.952158 3.870129
100 3.885839 3.920925 4.526447
250 5.813239 5.758804 5.571575
500 7.716117 7.774421 7.525006
1000 9.655428 10.044972 10.161326
5000 12.575822 14.022163 16.600783
10000 13.100287 14.825970 18.367821
C_(phi) phi 0.05 0.1 0.2 0 2.565294 2.952158 3.870129 100 3.885839 3.920925 4.526447 250 5.813239 5.758804 5.571575 500 7.716117 7.774421 7.525006 1000 9.655428 10.044972 10.161326 5000 12.575822 14.022163 16.600783 10000 13.100287 14.825970 18.367821| $C_{\phi}$ | | $\phi$ | | | :---: | :---: | :---: | :---: | | | 0.05 | 0.1 | 0.2 | | 0 | 2.565294 | 2.952158 | 3.870129 | | 100 | 3.885839 | 3.920925 | 4.526447 | | 250 | 5.813239 | 5.758804 | 5.571575 | | 500 | 7.716117 | 7.774421 | 7.525006 | | 1000 | 9.655428 | 10.044972 | 10.161326 | | 5000 | 12.575822 | 14.022163 | 16.600783 | | 10000 | 13.100287 | 14.825970 | 18.367821 |
Table 4: Values of Nusselt number for K = 0.001 K = 0.001 K=0.001K=0.001K=0.001
C ϕ C ϕ C_(phi)C_{\phi}Cϕ K K KKK
0.001 0.01 0.1
0 1.004777 1.003020 1.000000
100 1.004739 1.002726 1.000000
250 1.004678 1.002283 1.000000
500 1.004564 1.001548 1.000000
1000 1.004403 1.000760 1.000000
5000 1.003992 1.000000 1.000000
10000 1.003870 1.000000 1.000000
C_(phi) K 0.001 0.01 0.1 0 1.004777 1.003020 1.000000 100 1.004739 1.002726 1.000000 250 1.004678 1.002283 1.000000 500 1.004564 1.001548 1.000000 1000 1.004403 1.000760 1.000000 5000 1.003992 1.000000 1.000000 10000 1.003870 1.000000 1.000000| $C_{\phi}$ | | $K$ | | | :---: | :---: | :---: | :---: | | | 0.001 | 0.01 | 0.1 | | 0 | 1.004777 | 1.003020 | 1.000000 | | 100 | 1.004739 | 1.002726 | 1.000000 | | 250 | 1.004678 | 1.002283 | 1.000000 | | 500 | 1.004564 | 1.001548 | 1.000000 | | 1000 | 1.004403 | 1.000760 | 1.000000 | | 5000 | 1.003992 | 1.000000 | 1.000000 | | 10000 | 1.003870 | 1.000000 | 1.000000 |
Table 5: Maximum value for the temperature in solid
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[13] Xuan Y., Roetzel W., Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer, 43, pp. 3701-3707, 2000.
[14] Wang X.Q., Mujumdar A.S., A review on nanofluids-part I: Theoretical and nume-rical investigations, Brazilian Journal of Chemical Engineering, 25, pp. 613-630, 2008.
Figure 2: Velocity profile for different values of parameter C ϕ C ϕ C_(phi)C_{\phi}Cϕ
Figure 3: Temperature profile in solid (left) and fluid (right) for different values of parameter C ϕ C ϕ C_(phi)C_{\phi}Cϕ and K = 0.1 K = 0.1 K=0.1K=0.1K=0.1
Figure 4: Temperature profile in solid (left) and fluid (right) for different values of parameter C ϕ C ϕ C_(phi)C_{\phi}Cϕ and K = 0.001 K = 0.001 K=0.001K=0.001K=0.001
2010

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