Transp Porous Med (2010) 83:425–436 DOI 10.1007/s11242-009-9452-8

Effect of Local Thermal Non-equilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid A. V. Kuznetsov · D. A. Nield

Received: 11 May 2009 / Accepted: 8 July 2009 / Published online: 31 July 2009 © Springer Science+Business Media B.V. 2009

Abstract The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is analytically studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model is employed. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases is investigated using a three-temperature model. The analysis reveals that in some circumstances the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with small particle-to-fluid heat capacity ratio) the effect is small. Keywords Local thermal non-equilibrium · Nanofluid · Porous medium · Instability · Natural convection

List of Symbols DB Brownian diffusion coefficient DT Thermophoretic diffusion coefficient g Gravitational acceleration g Gravitational acceleration vector H Dimensional layer depth k Thermal conductivity K Permeability of the porous medium Le Lewis number, defined by Eq. 20 NA Modified thermophoresis to Brownian-motion diffusivity ratio, defined by Eq. 24

A. V. Kuznetsov (B) Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA e-mail: [email protected] D. A. Nield Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand e-mail: [email protected]

123

426

NB NHP NHS p∗ p Ra Rm Rn t∗ t T∗ T Tc∗ Th∗ (u, v, w) v ∗ vD (x, y, z) (x ∗ , y ∗ , z ∗ )

A. V. Kuznetsov, D. A. Nield

Modified particle-density increment, defined by Eq. 25 Nield number for the fluid/particle interface, defined by Eq. 26a Nield number for the fluid/solid-matrix interface, defined by Eq. 26b Pressure Dimensionless pressure, p ∗ K /μαf Thermal Rayleigh–Darcy number, defined by Eq. 21 Basic-density Rayleigh number, defined by Eq. 22 Concentration Rayleigh number, defined by Eq. 23 Time Dimensionless time, t ∗ αf /H 2 Temperature T ∗ −T ∗ Dimensionless temperature, T ∗ −Tc∗ c h Temperature at the upper wall Temperature at the lower wall Dimensionless Darcy velocity components, (u ∗ , v ∗ , w ∗ )H/αf Dimensionless Darcy velocity, H v∗D /αf Dimensional Darcy velocity, (u ∗ , v ∗ , w ∗ ) Dimensionless Cartesian coordinates, (x ∗ , y ∗ , z ∗ )/H ; z is the vertically upward coordinate Cartesian coordinates

Greek Symbols kf Thermal diffusivity of the fluid, (ρc) αf f β Volumetric expansion coefficient of the fluid γP Modified thermal capacity ratio defined by Eq. 27a γS Modified thermal capacity ratio defined by Eq. 27b ε Porosity εP Modified thermal diffusivity ratio defined by Eq. 28a εS Modified thermal diffusivity ratio defined by Eq. 28b μ Viscosity of the fluid ρf Fluid density ρp Nanoparticle mass density (ρc)f Heat capacity of the fluid (ρc)p Heat capacity of the particle material (ρc)s Heat capacity of the solid-matrix material φ∗ Nanoparticle volume fraction φ ∗ −φ ∗ φ Relative nanoparticle volume fraction, φ ∗ −φ0∗ 1

Superscripts ∗ Dimensional variable  Perturbation variable Subscripts b Basic solution f Fluid phase p Particle phase s Solid-matrix phase

123

0

Effect of Local Thermal Non-equilibrium in a Porous Medium Layer

427

1 Introduction The term “nanofluid” refers to a liquid containing a dispersion of submicronic solid particles (nanoparticles). This term was coined by Choi (1995). The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. (1993). This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu 2005). A comprehensive survey of convective transport in nanofluids was made by Buongiorno (2006), who says that a satisfactory explanation for the abnormal increase of the thermal conductivity and viscosity is yet to be found. He focused on the further heat transfer enhancement observed in convective situations. Buongiorno notes that several authors have suggested that convective heat transfer enhancement could be due to the dispersion of the suspended nanoparticles, but he argues that this effect is too small to explain the observed enhancement. Buongiorno also concludes that turbulence is not affected by the presence of the nanoparticles so this cannot explain the observed enhancement. Particle rotation has also been proposed as a cause of heat transfer enhancement, but Buongiorno calculates that this effect is too small to explain the effect. With dispersion, turbulence and particle rotation ruled out as significant agencies for heat transfer enhancement; Buongiorno proposed a new model based on the mechanics of the nanoparticle/base-fluid relative velocity. Buongiorno (2006) noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling. After examining each of these in turn, he concluded that in the absence of turbulent effects it is the Brownian diffusion and the thermophoresis that will be important. Buongiorno proceeded to write down conservation equations on the basis of these two effects. The Bénard problem (the onset of convection in a horizontal layer uniformly heated from below) for a nanofluid was studied by Tzou (2008a,b) and Nield and Kuznetsov (2009a) on the basis of the transport equations of Buongiorno (2006). The corresponding problem for flow in a porous medium (the Horton–Rogers–Lapwood problem) was studied by Nield and Kuznetsov (2009b) using the Darcy model. The extension to the Brinkman model was made by Kuznetsov and Nield (2009). Local thermal equilibrium was assumed in the above theoretical studies. However, thermal lagging between the particles and the fluid has been proposed as an explanation of the increased thermal conductivity that has been observed in nanofluids by Vadasz (2006). This has motivated the present study. The analysis of Nield and Kuznetsov (2009b) for a porous medium has been extended to include the effects of local thermal non-equilibrium. The corresponding problem in a fluid without a solid matrix has been studied by Nield and Kuznetsov (2009c) using a two-temperature model. The present study involves a three-temperature model. We are not aware of any published work involving three temperatures.

2 Analysis It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents particles from agglomeration and deposition on the porous matrix. We select a coordinate frame in which the z-axis is aligned vertically upward. We consider a horizontal layer of a porous medium confined between the planes z ∗ = 0 and z ∗ = H . Asterisks are used to denote dimensional variables. Each boundary

123

428

A. V. Kuznetsov, D. A. Nield

wall is assumed to be impermeable and perfectly thermally conducting. The temperatures at the lower and upper walls are taken to be Th∗ and Tc∗ , the former being the greater. The Oberbeck–Boussinesq approximation is employed. The reference temperature is taken to be Tc∗ . In the linear theory being applied here, the temperature change in the fluid is assumed to be small in comparison with Tc∗ . The boundary values of the nanoparticle volume fraction are taken to be φ0∗ at the at the bottom and φ1∗ at the at the top of the layer, and it is assumed that the increment φ1∗ − φ0∗ is small in comparison with φ0∗ . Homogeneity in the porous medium is assumed. We consider a porous medium whose porosity is denoted by ε and permeability by K . The Darcy velocity is denoted by vD . The following six field equations embody the conservation of total mass, momentum, thermal energy in the fluid phase, thermal energy in the particle phase, thermal energy in the solidmatrix phase, and nanoparticles, respectively. The field variables are the Darcy velocity vD , the temperature in the fluid phase Tf∗ , the temperature in the particle phase Tp∗ , the temperature in the solid-matrix phase Ts∗ , and the nanoparticle volume fraction φ ∗ . ∗ ∇ ∗ · vD = 0,

μ ∗  ∗ vD + φ ρp K   +(1 − φ ∗ ) ρ(1 − β(Tf∗ − Tc∗ )) g,

(1)

0 = −∇ ∗ p ∗ −

ε(1 − φ0∗ )(ρc)f



 ∂ Tf∗ 1 ∗ ∗ v = ε(1 − φ0∗ )kf ∇ ∗2 Tf∗ + · ∇ T D f ∂t ∗ ε  ∗ +ε(1 − φ0 )(ρc)p DB ∇ ∗ φ ∗ · ∇ ∗ Tf∗  ∇ ∗ Tf∗ · ∇ ∗ Tf∗ +DT Tf∗ +h fp (Tp∗ − Tf∗ ) + h fs (Ts∗ − Tf∗ ),

εφ0∗ (ρc)p



(2)

∂ Tp∗ ∂t ∗

 1 ∗ + vD · ∇ ∗ Tp∗ = εφ0∗ kp ∇ ∗2 Tp∗ + h fp (Tf∗ − Tp∗ ), ε

(3)

(4)

∂ Ts∗ = (1 − ε)ks ∇ 2 Ts∗ + h fs (Tf∗ − Ts∗ ), ∂t ∗

(5)

∂φ ∗ 1 ∗ + vD · ∇ ∗ φ ∗ = DB ∇ ∗2 φ ∗ + (DT /Tc∗ )∇ ∗2 Tf∗ ∗ ∂t ε

(6)

(1 − ε)(ρc)s

∗ = (u ∗ , v ∗ , w ∗ ). Here, ρ , μ, and β are the density, viscosity, and volumetric We write vD f volume expansion coefficient of the fluid, whereas ρp is the density of the particles. The gravitational acceleration is denoted by g. We have introduced the effective heat capacities (ρc)f , (ρc)p , (ρc)s and the thermal conductivities kf , kp , ks of the fluid, particle and solid-matrix phases, respectively. The coefficients that appear in Eqs. 2 and 6 are the Brownian diffusion coefficient DB and the thermophoretic diffusion coefficient DT . Details of the derivation of Eqs. 2 and 6 are given in the articles by Buongiorno (2006), Tzou (2008a,b) and Nield and Kuznetsov (2009a,b). The interface heat transfer coefficients (incorporating the specific surface area) between the fluid/particle phases and the fluid/solid-matrix phases are denoted

123

Effect of Local Thermal Non-equilibrium in a Porous Medium Layer

429

by h fp and h fs , respectively. The flow is assumed to be slow so that an advective term and a Forchheimer quadratic drag term do not appear in the momentum equation. In the limit of local thermal equilibrium, one has Tf = Tp = Ts = T . The addition of Eqs. 22–24 then gives   ∂T ∗   ∗ ε(1−φ0∗ )(ρc)f +εφ0∗ (ρc)p +(1−ε)(ρc)s + (1−φ0∗ )(ρc)f +φ0∗ (ρc)p vD · ∇∗T ∗ ∂t ∗ = [ε(1 − φ0∗ )kf + (1 − ε)ks ]∇ ∗2 T ∗   ∇∗T ∗ · ∇∗T ∗ ∗ ∗ ∗ ∗ + ε(1 − φ0 )(ρc)p DB ∇ φ · ∇ T + DT . (7) T∗ Then approximating φ0∗ by zero (something that is valid for a dilute fluid) one gets ∂T ∗ ∗ + (ρc)f vD · ∇T ∗ ∂t ∗   = km ∇ ∗2 T ∗ + ε(ρc)p DB ∇ ∗ φ ∗ · ∇ ∗ T ∗ + (DT /Tc∗ )∇ ∗ T ∗ · ∇ ∗ T ∗ ,

(ρc)m

(8)

where (ρc)m = ε(ρc)f + (1 − ε)(ρc)s and km = εkf + (1 − ε)ks ,

(9)

in accord with Eq. 22 of Nield and Kuznetsov (2009b). It has been assumed that the particles move with the fluid and that the fluid is sufficiently dilute (φ0∗ close to zero) so that the intrinsic fluid velocity can be approximated by vD /ε rather than the precise value vD /(1 − φ0∗ )ε. We assume that the temperature and the volumetric fraction of the nanoparticles are constant on the boundaries and that there is local thermal equilibrium there. Thus, the boundary conditions are w ∗ = 0, Tf∗ = Th∗ , Tp∗ = Th∗ , Ts∗ = Th∗ , φ ∗ = φ0∗ at z ∗ = 0,

(10)

w ∗ = 0, Tf∗ = Tc∗ , Tp∗ = Tc∗ , Ts∗ = Tc∗ , φ ∗ = φ1∗ at z ∗ = H.

(11)

We introduce dimensionless variables as follows. We define (x, y, z) = (x ∗ , y ∗ , z ∗ )/H, t = t ∗ αf /H 2 , (u, v, w) = (u ∗ , v ∗ , w ∗ )H/αf , Tp∗ − Tc∗ φ ∗ − φ0∗ T ∗ − Tc∗ , Tf = f∗ , T = , p = p ∗ K /μαf , φ = ∗ p ∗ φ1 − φ0 Th − Tc∗ Th∗ − Tc∗ Ts =

Ts∗ − Tc∗ , Th∗ − Tc∗

αf =

kf . (ρc) f

(12)

where (13)

Equations 1–6 take the form: ∇ ·v =0

(14)

123

430

A. V. Kuznetsov, D. A. Nield

0 = −∇ p − v − Rm eˆ z + Ra Tf eˆ z − Rn φ eˆ z

(15)

∂ Tf NB NA NB v + · ∇Tf = ∇ 2 Tf + ∇φ · ∇Tf + ∇Tf · ∇Tf ∂t ε Le Le +NHP (Tp − Tf ) + NHS (Ts − Tf )

(16)

∂ Tp v + · ∇Tp = εP ∇ 2 Tp + γP NHP (Tf − Tp ) ∂t ε

(17)

∂ Ts = εS ∇ 2 Ts + γS NHS (Tf − Ts ) ∂t

(18)

v 1 NA 2 ∂φ + · ∇φ = ∇2φ + ∇ Tf ∂t ε Le Le

(19)

where αf DB

(20)

ρgβ K H (Th∗ − Tc∗ ) μαf

(21)

Rm =

[ρp φ1∗ + ρ(1 − φ1∗ )]gK H μαf

(22)

Rn =

(ρp − ρ)(φ1∗ − φ0∗ )gK H μαf

(23)

NA =

DT (Th∗ − Tc∗ ) DB Tc∗ (φ1∗ − φ0∗ )

(24)

NB =

(ρc)p ∗ (φ − φ0∗ ) (ρc)f 1

(25)

Le =

Ra =

NHP =

γP =

h fp H 2 , ε(1 − φ0∗ )kf

NHS =

h fs H 2 ε(1 − φ0∗ )kf

(26a,b)

(1 − φ0∗ ) (ρc)f ε(1 − φ0∗ ) (ρc)f , γ = S φ0∗ (ρc)p (1 − ε) (ρc)s

(27a,b)

kp /(ρc)p ks /(ρc)s , εS = kf /(ρc)f kf /(ρc)f

(28a,b)

εP = The boundary conditions become

123

w = 0, Tf = 1, Tp = 1, Ts = 1, φ = 0 at z = 0,

(29a,b,c,d)

w = 0, Tf = 0, Tp = 0, Ts = 0, φ = 1 at z = 1.

(30a,b,c,d)

Effect of Local Thermal Non-equilibrium in a Porous Medium Layer

431

The parameter Le is a Lewis number and Ra is the familiar thermal Rayleigh number. The new parameters Rm and Rn may be regarded as a basic-density Rayleigh number and a concentration Rayleigh number, respectively. The parameter NA is a modified thermophoresis to Brownian-motion diffusivity ratio and is somewhat similar to the Soret parameter that arises in cross-diffusion phenomena in solutions, NB is a modified particle-density increment, while NHP and NHS are interface heat transfer parameters. Vadasz (2006) called this type of parameter as the Nield number, citing Nield (1998), while elsewhere in the recent literature it has been called as the Sparrow number. Because two other parameters have been called Sparrow numbers (an internal heating parameter and a radiation resistance parameter) in the heat transfer literature, we will follow Vadasz and adopt the terminology the Nield number. Finally, γP and γS are modified thermal capacity ratios and εP and εS are modified thermal diffusivity ratios. In the spirit of the Oberbeck–Boussinesq approximation, Eq. 14 has been linearized by the neglect of a term proportional to the product of φ and Tf . This assumption is likely to be valid in the case of small temperature gradients in a dilute suspension of nanoparticles. 2.1 Basic Solution We seek a time-independent quiescent solution of Eqs. 29–32 with temperature and nanoparticle volume fraction varying in the z-direction only, which is a solution of the form v = 0, p = pb (z), Tf = Tfb (z), Tp = Tpb (z), Ts = Tsb (z), φ = φb (z). Equations 14–18 reduce to 0=−

d pb − Rm + Ra Tfb − Rn φb , dz

 d2 Tfb NB dφb dTfb NA NB dTfb 2 + + dz 2 Le dz dz Le dz +NHP (Tp − Tf ) + NHS (Ts − Tf ) = 0,

(31)

(32)

εP

d2 Tpb + γP NHP (Tf − Tp ) = 0, dz 2

(33)

εS

d2 Tsb + γS NHS (Tf − Ts ) = 0, dz 2

(34)

d2 Tfb d2 φb + N = 0. A dz 2 dz 2

(35)

Using the boundary conditions (29) and (30), Eq. 34 may be integrated to give φb = −NA Tfb + (1 − NA )z + NA ,

(36)

and substitution of this into Eq. 32 gives d2 Tfb (1 − NA )NB dTfb + + NHP (Tpb − Tfb ) + NHS (Tsb − Tfb ) = 0. 2 dz Le dz

(37)

The solution of Eqs. 32, 33 and 37 for Tfb and Tpb satisfying Eqs. 29 and 30 is readily found but is quite complicated in form in the general case. The remainder of the basic solution can

123

432

A. V. Kuznetsov, D. A. Nield

then be found by first substituting in Eq. 36 to obtain φb and then using integration of Eq. 31 to obtain pb . According to Buongiorno (2006), for most nanofluids investigated so far Le is large, of order 102 –103 , while NA is no greater than about 10. Then the coefficient in Eq. 37, (1 − NA )NB /Le is negligible and so to a good approximation one obtains the results Tfb = Tpb = Tsb = 1 − z,

(38)

φb = z.

(39)

and so

Thus, for the case of large Lewis numbers, the basic solution is one in which the temperature and volume fraction gradients are linear and there is local thermal equilibrium. 2.2 Perturbation Solution We now superimpose perturbations on the basic solution. We write v = v , p = pb + p  , Tf = Tfb + Tf , Tp = Tpb + Tp , Ts = Tsb + Ts , φ = φb + φ  , (40) substitute in Eqs. 14–18, and linearize by neglecting products of primed quantities. The following equations are obtained when Eqs. 38 and 39 are used: ∇ · v = 0,

(41)

0 = −∇ p  − v + Ra Tf eˆ z − Rn φ  eˆ z ,

(42)

 ∂ Tf 1 ∂φ  2NA NB ∂ Tf NB ∂ Tf − w  = ∇ 2 Tf + − − ∂t ε Le ∂z ∂z Le ∂z +NHP (Tp − Tf ) + NHS (Ts − Tf ),

(43)

∂ Tp ∂t

1 − w  = εP ∇ 2 Tp + γP NHP (Tf − TP ), ε

(44)

∂ Ts = εS ∇ 2 Ts + γS NHS (Tf − Ts ), ∂t

(45)

∂φ  1 2  NA 2  1 + w = ∇ φ + ∇ Tf , ∂t ε Le Le

(46)

w  = 0, Tf = 0, Tp = 0, Ts = 0, φ  = 0 at z = 0 and at z = 1.

(47)

It will be noted that the parameter Rm is not involved in these and subsequent equations. It is just a measure of the basic static pressure gradient. The eight unknowns u  , v  , w  , p  , Tf , Tp , Ts , φ  can be reduced to five by operating on Eq. 42 with eˆ z · curlcurl and using Eq. 41. The result is ∇ 2 w  = Ra ∇H2 Tf − Rn ∇H2 φ  . Here,

∇H2

is the two-dimensional Laplacian operator on the horizontal plane.

123

(48)

Effect of Local Thermal Non-equilibrium in a Porous Medium Layer

433

The differential equations (48), (43)–(46) and the boundary conditions (47) constitute a linear boundary-value problem that can be solved using the method of normal modes. We write (w  , Tf , Tp , Ts , φ  ) = [W (z), f (z), p (z), s (z), (z)] exp(st + ilx + imy),

(49)

and substitute into the differential equations to obtain (D 2 − α 2 )W + Ra α 2 f − Rn α 2 = 0,  NB 1 2NA NB 2 2 W+ D + D− D − α − s − NHP − NHS f ε Le Le NB D = 0 + NHP p + NHS s − Le

(50)

(51)

  1 W + γP NHP f + εP (D 2 − α 2 ) − s − γP NHP p = 0, ε

(52)

  γS NHS f + εS (D 2 − α 2 ) − s − γS NHS s = 0,

(53)

NA 2 1 W− (D − α 2 ) f − ε Le



1 2 2 (D − α ) − s = 0, Le

W = 0, f = 0, p = 0, s = 0, = 0 at z = 0 and at z = 1.

(54) (55)

where D≡

d and α = (l 2 + m 2 )1/2 . dz

(56)

Thus, α is a dimensionless horizontal wavenumber. For neutral stability, the real part of s is zero. Hence, we now write s = iω, where ω is real and is a dimensionless frequency. We now employ a Galerkin-type weighted residuals method to obtain an approximate solution to the system of Eqs. 50–54. We choose as trial functions (satisfying the boundary conditions) Wn , fn , pn , sn , n ; n = 1, 2, 3, . . . , N and write

N

N

N An Wn , f = n=1 Bn fn , p = n=1 Cn pn , W = n=1

N

N (57) p = n=1 Dn pn , = n=1 E n n , substitute into Eqs. 50–54, and make the expressions on the left-hand sides of those equations (the residuals) orthogonal to the trial functions, thereby obtaining a system of 5N linear algebraic equations in the 5N unknowns An , Bn , Cn , Dn , E n ; n = 1, 2, . . . , N . The vanishing of the determinant of coefficients produces the eigenvalue equation for the system. One can regard Ra as the eigenvalue. Thus Ra is found in terms of the other parameters.

3 Results and Discussion In the case of local thermal equilibrium investigated by Nield and Kuznetsov (2009b) there are five parameters of interest, namely, Ra, Rn, Le, NA and NB . The introduction of LTNE has led to the appearance of seven more parameters, namely, ε, NHP , NHS , γP , γS , εP , and

123

434

A. V. Kuznetsov, D. A. Nield

εS . Because of the large parameter space, we simplify the analysis. In Nield and Kuznetsov (2009b) both non-oscillatory and oscillatory convection cases were investigated, but in this article for simplicity we confine ourselves to the case of non-oscillatory instability, so that we can set s = 0 in the above equations. Further, in order to have a tractable analysis, we employ a one-term (N = 1) Galerkin approximation. In our case, the trial functions satisfying the boundary conditions can be taken as Wn = fn = pn = sn = n = sin nπ z; n = 1, 2, 3, . . .

(58)

For shorthand we write F = π 2 + α 2 . The following stability boundary is obtained: F(εP F + [γP + 1]NHP )(εS F + γS NHS )Ra ⎧ ⎫ ⎨ Le (F + NHP + NHS )(εP F + γP NHP )(εS F + γS NHS ) ⎬ + +NA F(εP F + [γP + 1]NHP )(εS F + γS NHS ) Rn ⎩ 2 (ε F + γ N ) − Le γ N 2 (ε F + γ N ) ⎭ −Le γS NHS P P HP P HP S S HS

(59)

  ε 2 (F + NHP + NHS )(εP F + γP NHP )(εS F + γS NHS ) = 2F 2 (ε F + γ N ) − γ N 2 (ε F + γ N ) −γS NHS α P P HP P HP S S HS This simplifies to P Ra + (P NA + Q Le)Rn = Q

εF2 α2

(60)

where P = (εP F + [γP + 1]NHP )(εS F + γS NHS )

(61)

Q = εP εS F 2 + [εP εS NHP + εP εS NHS + γP εS NHP + γS εP NHS ]F   + γP γS + γS εP + γP εS NHP NHS

(62)

In each of the special cases (1) NHP = NHS = 0 or (2) γP , γS , NHP , NHS → ∞, the stability boundary takes the form Ra + (NA + Le)Rn =

ε(π 2 + α 2 )2 . α2

(63)

The right-hand side takes a minimum as α varies when α = π, and the minimum value is 4π 2 ε. One recognizes that 4π 2 is the value of the critical Rayleigh number for the standard Horton–Rogers–Lapwood problem. The extra factor ε arises because we have defined the Rayleigh number Ra in terms of the thermal diffusivity αf = kf /(ρc)f of the fluid rather than in terms of [εkf + (1 − ε)ks ]/(ρc)f . One has reconciliation if one formally sets ks = 0. From now on we will approximate the critical wavenumber ac by π. We also write P˜ = P/4π 2 εP εS , Q˜ = Q/4π 2 εP εS , N˜ HP = NHP /2πεP , N˜ HS = NHS /2πεE , Ro = 4π 2 ε Then we have

123

 Q˜ Q˜ Le Rn = Ro, Ra + NA + P˜ P˜

(64)



(65)

Effect of Local Thermal Non-equilibrium in a Porous Medium Layer

435

where (66) P˜ = (1 + [γP + 1] N˜ HP )(1 + γS N˜ HS ),   ˜ ˜ ˜ ˜ ˜ ˜ ˜ Q = 1 + εp NHP + εS NHS + γP NHP + γS NHS + γP γS + γS εP + γP εS NHP NHS . (67) ˜ P˜ takes the value 1 as the Nield numbers tend to zero (no thermal coupling The ratio Q/ between the phases) and the value (γP γS + γS εP + γP εS )/(γP + 1)γS as they tend to infinity (enforcing LTE). This last value tends to 1 as γP and γS tend to infinity. From the definitions in Eqs. 27 and 28, we see that γP will be very large for a typical dilute nanofluid, whereas γS /εS will normally be quite large also. We conclude that the effect of LTNE at the particle/fluid interface will normally be negligible whereas the effect of LTNE at the solid-matrix/fluid interface will normally be small.

4 Conclusions We have investigated analytically the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid, using a model for the nanofluid that incorporates the effects of Brownian motion and thermophoresis, and that allows for local thermal non-equilibrium (LTNE) between the particle and fluid phases and between the solid matrix and fluid phases. The three-temperature analysis revealed that in some circumstances (when the Nield numbers are neither very small nor very large) the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with large modified fluid-to-particle heat capacity ratio) the effect is small. This provides support to our hypothesis that any enhanced effective thermal conductivity of nanofluids due to LTNE, as proposed by Vadasz (2006) and explained by him in terms of thermal lagging, is a phenomenon associated with a highly transient situation.

References Buongiorno, J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006). doi:10.1115/ 1.2150834 Buongiorno, J., Hu, W.: Nanofluid coolants for advanced nuclear power plants, Paper no. 5705. In: Proceedings of ICAPP ‘05, Seoul (2005) Choi, S.: Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, D.A., Wang, H.P. (eds.) Developments and Applications of Non-Newtonian Flows, FED—vol. 231/MD—vol. 66, pp. 99–105. ASME, New York (1995) Kuznetsov, A.V., Nield, D.A.: Thermal instability in a porous layer saturated by a nanofluid: Brinkman model. Transp. Porous Media (2009) (to appear) Masuda, H., Ebata, A., Teramae, K., Hishinuma, N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, 227–233 (1993) Nield, D.A.: Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media 1, 181–186 (1998) Nield, D.A., Kuznetsov, A.V.: The onset of convection in a nanofluid layer. ASME J. Heat Transf. (2009a) (submitted) Nield, D.A., Kuznetsov, A.V.: Thermal instability in a porous layer saturated by a nanofluid. Int. J. Heat Mass Transf. (2009b) (to appear) Nield, D.A., Kuznetsov, A.V.: The effect of local thermal non-equilibrium on the onset of convection in a nanofluid. Int. J. Therm. Sci. (2009c) (to appear) Tzou, D.Y.: Instability of nanofluids in natural convection. ASME J. Heat Transf. 130 (072401) (2008a)

123

436

A. V. Kuznetsov, D. A. Nield

Tzou, D.Y.: Thermal instability of nanofluids in natural convection. Int. J. Heat Mass Transf. 51, 2967– 2979 (2008). doi:10.1016/j.ijheatmasstransfer.2007.09.014 Vadasz, P.: Heat conduction in nanofluid suspensions. ASME J. Heat Transf. 128, 465–477 (2006). doi:10. 1115/1.2175149

123