Transp Porous Med (2013) 99:73–83 DOI 10.1007/s11242-013-0174-6
Onset of Convection with Internal Heating in a Porous Medium Saturated by a Nanofluid D. A. Nield · A. V. Kuznetsov
Received: 3 April 2013 / Accepted: 6 May 2013 / Published online: 25 June 2013 © Springer Science+Business Media Dordrecht 2013
Abstract Linear stability analysis was applied to the onset of convection due to internal heating in a porous medium saturated by a nanofluid. A model in which the effects of thermophoresis and Brownian motion are taken into account is employed. We utilized more realistic boundary conditions than in the previous work on this subject; now the nanofluid particle fraction is allowed to adapt to the temperature profile induced by the internal heating, subject to the requirement that there is zero perturbation flux across a boundary. The results show that the presence of the nanofluid particles leads to increased instability of the system. We identified two combinations of dimensionless parameters that are the major controllers of convection instability in the layer. Keywords
Thermal instability · Nanofluid · Porous medium
List of Symbols a D DB DT F(z) g G(z) km
Horizontal overall wavenumber d/dz Brownian diffusion coefficient Thermophoresis diffusion coefficient Negative basic temperature gradient Gravitational acceleration F(z)/Ra Effective thermal conductivity
D. A. Nield Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand e-mail:
[email protected] 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]
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K H Le
Permeability Layer width Lewis number,
NA
Modified diffusivity ratio,
NB NB ∗ p p∗ Q Ra
Modified particle-density scaling parameter, RaNB Dimensionless pressure, p ∗ K /μαm Pressure Volumetric heat source strength K H3Q Internal Rayleigh number, ρ0 gβ 2μkαm
Rm Rn t t∗ T∗ T0 u v v∗D w x x∗ y y∗ z z∗
αm DB
DT ρ0 gβ K H DB T0 φ0 μαm
ε(ρc)p (ρc)f φ0
ρ0 gK H μαm (ρ −ρ)φ gK H Concentration Rayleigh number, p μαm0 Dimensionless time, t ∗ αm /σ H 2
Basic-density Rayleigh number,
Time Temperature Temperature at the upper and lower boundary Dimensionless x-velocity component, u ∗ H/αm Dimensionless y-velocity component, v ∗ H/αm Vector of Darcy velocity, (u ∗ , v ∗ , w ∗ ) Dimensionless z-velocity component, w ∗ H/αm x ∗ /H Coordinate in the horizontal plane y ∗ /H Coordinate in the horizontal plane z ∗ /H Upward vertical coordinate
Greek Symbols αm β ε θ μ ρ ρc ρ0 σ φ φ∗ φ0
Effective thermal diffusivity, (ρckmP )f Fluid volumetric expansion coefficient Porosity gβ K H Dimensionless temperature, ρ0μα (T ∗ − T0 ) m Fluid viscosity Density Heat capacity Fluid density at temperature T0 m Heat capacity ratio, (ρc) (ρc) f ∗
0 Rescaled nanofluid particle fraction, φ φ−φ 0 Nanofluid particle fraction Nanofluid particle fraction at the upper and lower boundary
Subscripts 0 c
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Reference quantity Critical value
Onset of Convection with Internal Heating
f m p
75
Fluid Porous medium Particles
Superscripts
∗
Perturbation variable Dimensional variable
1 Introduction Recently there have been a number of theoretical papers published on the onset of convection in a layer of a porous medium saturated by a nanofluid, and the most of these have involved extensions of the work by Nield and Kuznetsov (2009), who adopted the Buongiorno (2006) model incorporating the effects of thermophoresis and Brownian motion, with the convection being induced by heating from below. An example is the paper by Yadav et al. (2012), who added the effect of a uniform volumetric heat source. Thus, their paper was an extension of the well-known Horton–Rogers–Lapwood problem (see, for example, Sect. 6.1 of Nield and Bejan 2013). They expressed their analysis in terms of a standard external Rayleigh–Darcy number, one based on the applied temperature difference across layer, and they used the same boundary conditions on the nanofluid fraction as those used by Nield and Kuznetsov (2009). These boundary conditions were of limited practical validity, but their use could be justified in a pioneering analytical paper (Nield and Kuznetsov 2009) in which mathematical convenience was important. Further studies of convection in a nanofluid-saturated porous medium are surveyed in Sect. 9.7 of Nield and Bejan (2013). In the present paper we consider the case of purely internal heating and we re-examine the boundary conditions. The internal heating problem is distinctive in that the basic temperature profile in not only nonlinear but also the gradient in the upwards direction changes sign across the layer, being positive in the lower portion but negative in the upper portion. Thus the effect of buoyancy is destabilizing in the upper portion only, and thus the bulk of the resulting convection takes place in the upper portion of the layer. Also, there is no applied temperature difference across the layer and so the external Rayleigh number is no longer appropriate. The relevant parameter is now an internal Rayleigh number, one based on the volumetric source strength. The case of a homogeneous porous medium with uniform volumetric heating was first studied by Gasser and Kazimi (1976); later studies are surveyed in Sect. 6.11.2 of Nield and Bejan (2013). Relevant papers are Buretta and Berman (1976), Hardee and Nilson (1977), Kulacki and Freeman (1979), Kulacki and Ramchandani (1975), Nield (2008), Nield and Kuznetsov (2007), Rhee et al. (1978), Royer and Flores (1994), Rudraiah et al. (1982), Tveitereid (1977).
2 Analysis As in Nield and Kuznetsov (2009), we assume that in order to prevent nanoparticles from agglomeration and deposition on the porous matrix, the nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology.
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Asterisks are used to denote dimensional variables. The z ∗ -axis is taken in the upward vertical direction, and the porous medium is unbounded in the x ∗ and y ∗ directions. The convection is driven by a volumetric heat source of strength Q. 2.1 Basic Equations We select a coordinate frame in which the z-axis is aligned vertically upwards. We consider a horizontal layer of a porous medium confined between the planes z ∗ = 0 and z ∗ = H . For simplicity, Darcy’s law is assumed to hold and the Oberbeck-Boussinesq approximation is employed. Homogeneity and local thermal equilibrium in the porous medium are assumed. The reference temperature is taken to be T0 , the temperature at the boundaries. In the linear theory being applied here the temperature change in the fluid is assumed to be small in comparison with T0 . The mass, momentum, thermal energy, and nanofluid particle conservation equations take the form ∗ ∇ ∗ · vD = 0,
0 = −∇ ∗ p ∗ − (ρc)m
μ ∗ ∗ vD + φ ρp + (1 − φ ∗ ) ρ0 [1 − β(T ∗ − T0 )] g, K
∂T ∗ 2 ∗ + (ρc)f vD · ∇ ∗ T ∗ = km ∇ ∗ T ∗ + ε(ρc)p ∂t ∗ × DB ∇ ∗ φ ∗ · ∇ ∗ T ∗ + (DT /T0 )∇ ∗ T ∗ · ∇ ∗ T ∗ + Q, ∂φ ∗ 1 2 2 + vD ∗ · ∇ ∗ φ ∗ = DB ∇ ∗ φ ∗ + (DT /T0 )∇ ∗ T ∗ . ∂t ∗ ε
(1) (2)
(3) (4)
∗ is the Darcy velocity, p ∗ is the pressure, φ ∗ is the nanofluid particle fraction, T ∗ is the Here vD temperature, t ∗ is the time, μ is the fluid viscosity, β is the volumetric expansion coefficient, K is the permeability, ε is the porosity, g is the gravitational acceleration, km is the effective thermal conductivity, ρ is the density, and ρc is the heat capacity. The subscripts p, f, m, and zero refer to the particles, the fluid, the porous medium, and a reference quantity, respectively. The other quantities are the Brownian diffusion coefficient DB and the thermophoresis diffusion coefficient DT . ∗ = (u ∗ , v ∗ , w ∗ ). We write vD We assume that the temperature and the volumetric fraction of the nanoparticles are constant on the boundaries. Thus, the boundary conditions are
w ∗ = 0, T ∗ = T0 , φ ∗ = φ0 at z ∗ = 0,
(5)
w ∗ = 0, T ∗ = T0 , φ ∗ = φ0 at z ∗ = H.
(6)
We introduce dimensionless variables as follows. We define (x, y, z) = (x ∗ , y ∗ , z ∗ )/H, t = t ∗ αm /σ H 2 , (u, v, w) = (u ∗ , v ∗ , w ∗ )H/αm , φ ∗ − φ0 ρ0 gβ K H ∗ ,θ = (T − T0 ), p = p ∗ K /μαm , φ = φ0 μαm
(7)
where φ0 is a reference particle fraction, ρ0 is the density at temperature T0 and αm =
123
km (ρcP )m , σ = . (ρcP )f (ρcP )f
(8)
Onset of Convection with Internal Heating
77
Then Eqs. (1)–(6) take the form: ∇ · v = 0,
(9)
0 = −∇ p − v − Rmˆez + θ eˆ z − Rnφ eˆ z ,
(10)
∂θ NB NA NB + v · ∇θ = ∇ 2 θ + ∇φ · ∇θ + ∇θ · ∇θ + 2Ra, ∂t Le Le
(11)
NA 2 1 ∂φ 1 1 2 + v · ∇φ = ∇ φ+ ∇ θ, σ ∂t ε Le Le
(12)
w = 0, θ = 0, φ = 0 at z = 0,
(13)
w = 0, θ = 0, φ = 0 at z = 1.
(14)
We have introduced an internal Rayleigh number Ra defined by Ra =
ρ0 gβ K H 3 Q . 2μkαm
(15)
We have also introduced αm , DB
(16)
ρ0 gK H , μαm
(17)
Rn =
(ρp − ρ)φ0 gK H , μαm
(18)
NA =
DT ρ0 gβ K H , DB T0 φ0 μαm
(19)
ε(ρc)p φ0 , (ρc)f
(20)
Le =
Rm =
NB =
The parameter Le is a Lewis number. The parameters Rm and Rn may be regarded as a basicdensity Rayleigh number and a concentration Rayleigh number, respectively. The parameter NA is a modified diffusivity ratio and is somewhat similar to the Soret parameter that arises in cross-diffusion phenomena in solutions, while NB is a modified particle-density scaling parameter. In writing Eq. (10) the buoyancy term involving the temperature has been further linearized. The notation used for the nanofluid parameters is that introduced by Nield and Kuznetsov (2009).
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2.2 Basic Solution We now obtain a steady-state zero-flow solution that is independent of x and y. Equations (10)–(12) reduce to the ordinary differential equations dp = −Rm + θ − Rnφ, dz d2 θ NB dφ dθ NA NB + + dz 2 Le dz dz Le
dθ dz
(21)
2 + 2Ra = 0,
(22)
d2 θ d2 φ + NA 2 = 0. 2 dz dz
(23)
θ ≡ θB (z) = Ra(z − z 2 ),
(24)
φ ≡ φB (z) = −NA Ra(z − z 2 ).
(25)
One finds that the basic solution is
(The reader should note that, because of Eq. (23) and the boundary conditions (13) and (14), the group φ + NA θ is now zero and as a result the second and third terms in Eq. (22) cancel.) Thus the solution for the basic temperature profile is the same as for the regular fluid case, and the basic particle fraction profile conforms with this. Of particular interest for the stability problem is the distribution of the negative basic temperature gradient, which we denote by F(z) and which is given here by the linear expression F(z) = Ra(2z − 1).
(26)
2.3 Perturbation Equations We now define perturbation variables, denoted by primes, by v = vB + εv , θ = θB + εθ , φ = φB + εφ , p = pB + εp ,
(27)
where ε is a small quantity. We substitute Eq. (27) into the governing equations, and neglect terms of order ε 2 . After elimination of p we obtained the following. Equations (9), (11), (12), (10) give
123
∂u ∂v ∂w + + = 0, ∂x ∂y ∂z
(28)
∂v ∂θ ∂φ ∂w − = − Rn , ∂y ∂z ∂y ∂y
(29)
∂w ∂θ ∂φ ∂u − =− + Rn , ∂z ∂x ∂x ∂x
(30)
∂u ∂v − = 0, ∂x ∂y
(31)
Onset of Convection with Internal Heating
− F(z)w =
79
∂ 2θ ∂ 2θ ∂ 2θ NA NB ∂θ NB ∂φ + + − F(z) − F(z) , 2 2 2 ∂x ∂y ∂z Le ∂z Le ∂z 1 1 2 NA 2 w = ∇ φ + ∇ θ. ε Le Le
Using Eq. (28) to eliminate u and v from Eqs. (29)–(31), we get 2 ∂ 2 w ∂ 2 w ∂ 2 w ∂ 2θ ∂ 2θ ∂ φ ∂ 2φ + + = + − Rn + . ∂x2 ∂ y2 ∂z 2 ∂x2 ∂ y2 ∂x2 ∂ y2
(32)
(33)
(34)
In writing Eqs. (33) and (34) we have assumed that the critical perturbation solution is timeindependent, corresponding to non-oscillatory instability. In the present problem there are no two independent agencies that generate opposing buoyancy forces capable of producing oscillatory convection. In distinction from what was done in Nield and Kuznetsov (2009) we no longer have independent applied thermal and nanofluid concentration gradients. We now introduce normal modes by (w , θ ) = [W (z), (z)] exp[i(kx + ly)],
(35)
and the horizontal overall wavenumber a = (k 2 + l 2 )1/2 .
(36)
(D 2 − a 2 )W + a 2 − Rn a 2 = 0,
(37)
NB NA NB F(z)W + D 2 − F(z)D − a 2 − F(z)D = 0, Le Le
(38)
Here D denotes the operator d/dz.
−
Le W + NA (D 2 − a 2 ) + (D 2 − a 2 ) = 0. ε
(39)
Equations (37)–(39) can now be solved subject to appropriate boundary conditions. We assume that the boundaries are impermeable, and hence the vertical velocity is zero on the boundaries. Then the natural practical condition for the nanoparticle fraction is that there is no perturbation particle flux at the boundary. For the temperature perturbation we have a choice, and in this paper we assume that the boundaries are perfectly conducting, and so the perturbation temperature is zero on the boundaries. Hence the boundary conditions take the form W = 0, = 0, D = 0 at z = 0 and at z = 1.
(40)
2.4 Solution of the Stability Problem It is known that one gets an approximation to within 5 % using a single term Galerkin expansion. In that case a single mode (an odd one) is the prime contributor to instability. However, in the case of internal heating the basic temperature gradient changes sign across the layer, the prime contributor to instability is a combination of an odd mode and an even mode. As a result it is necessary to proceed to a second-order Galerkin approximation in order to get equivalent accuracy. We choose trial functions WTi, Ti and Tι (i = 1, 2) that satisfy the boundary conditions exactly, take W = A1 WT1 + A2 WT2 ,
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D. A. Nield, A. V. Kuznetsov
= B1 T1 + B2 T2 , = C1 T1 + C2 T2 , and then solve the differential equations approximately by substituting and making the residuals orthogonal to the respective trial functions. This gives six homogeneous linear equations in the six constants A1 , A2 , B1 , B2 , C1 , C2 . The vanishing of the determinant of coefficients yields an eigenvalue equation with Ra as the eigenvalue. In this case we take as the trial functions WT1 = z − z 2 ,
(41a)
WT2 = z 2 − z 3 ,
(41b)
T1 = z − z 2 ,
(41c)
T2 = z − z ,
(41d)
2
3
T1 = 1,
(41e)
T2 = 3z − 2z . 2
3
(41f)
Starting from Eqs. (37)–(39) the Galerkin method gives ⎤ ⎡ M11 M12 M13 det ⎣ M21 M22 M23 ⎦ = 0 M31 M32 M33 where
WT1 D 2 − a 2 WT1 WT1 D 2 − a 2 WT2 = WT2 D 2 − a 2 WT1 WT2 D 2 − a 2 WT2
M11
M12 = M13 = M21 = T1 D 2 − a 2 − =⎣ T2 D 2 − a 2 − ⎡
M22
(42)
a 2 WT1 T1 a 2 WT1 T2 a 2 WT2 T1 a 2 WT2 T2
−Rna 2 WT1 T1 −Rna 2 WT1 T2 −Rna 2 WT2 T1 −Rna 2 WT2 T2
(43b)
RaG(z)WT1 T1 RaG(z)WT2 T1 RaG(z)WT1 T2 RaG(z)WT2 T2
NA NB∗ Le G(z)D T1 NA NB∗ Le G(z)D T1
(43a)
(43c)
T1 D 2 − a 2 − T2 D 2 − a 2 −
(43d)
⎤
NA NB∗ Le G(z)D T2 ⎦ NA NB∗ Le G(z)D T2
(43e) M23 =
−(NB∗ /Le)G(z) T1 D T1 −(NB∗ /Le)G(z) T1 D T2 −(NB∗ /Le)G(z) T2 D T1 −(NB∗ /Le)G(z) T2 D T2
M31 = M32 =
(43f)
(43g)
NA T1 D 2 − a 2 T1 NA T1 D 2 − a 2 T2 NA T2 D 2 − a 2 T1 NA T2 D 2 − a 2 T2
(43h)
T1 D 2 − a 2 T1 T1 D 2 − a 2 T2 T2 D 2 − a 2 T1 ) T2 D 2 − a 2 T2
(43i)
M33 =
123
−(Le/ε)WT1 T1 −(Le/ε)WT2 T1 −(Le/ε)WT1 T2 −(Le/ε)WT2 T2
Onset of Convection with Internal Heating
81
where, substituting the expression from Eq. (26), we have G(z) = F(z)/Ra = 2z − 1
(44)
NB∗ = RaNB ,
(45)
and we use the shorthand 1 (•)dz.
• =
(46)
0
We can now see that it is NB ∗ rather than NB that is the appropriate independent parameter representing the scale for the nanofluid particle concentration in the case where we have internal heating rather than bottom heating. Equation (42) is thus a quadratic equation in Ra, and the critical Rayleigh number Rac is the minimum, as a varies, of the smaller root of this equation.
3 Results and Discussion The eigenvalue Eq. (42) gives the critical Rayleigh number as a function of the five parameters Rn, Le, NA , NB , and ε. It was noted by Nield and Kuznetsov (2009) that for a typical nanofluid Le is of the order 104 or 105 . Since NB ∗ enters Eq. (42) only in the combination NB ∗ /Le, this means that normally the effect of NB ∗ will be small; it is zero in the case investigated by Nield and Kuznetsov (2009). Hence we now consider only the case NB ∗ = 0 (and so NB = 0) and then we observe that Le and ε enter Eq. (42) in the combination Le/ε only. Hence we are left with only three parameters, Rn, Le/ε and NA , to investigate in detail. (Again this is in accord with the results for the non-oscillatory case obtained by Nield and Kuznetsov (2009).) In fact, one can use Eq. (37) to eliminate from Eqs. (38) and (39) and the resulting two differential equations are
2 2 Le D − a2 − (47) Rna 2 W + (NA Rn + 1) a 2 = 0, ε
RaG(z)W + D 2 − a 2 = 0,
(48)
which we observe involve just two parameters, RnLe/ε and RnNA , although the boundary conditions still implicitly involve Rn on its own. Further, a single term Galerkin procedure gives the estimate 2
WT D 2 − a 2 − (Le/ε)Rna 2 WT T D 2 − a 2 T Ra = , (49) (1 + RnNA ) a 2 G(z)WT T WT T which indicates increasing instability as RnLe/ε and RnNA each increase. This argument suggests that, at least for moderate values of Rn, the Rayleigh number Rac depends on just RnLe/ε and RnNA , and hence without much loss of generality one can usefully perform calculations with arbitrarily fixed Rn =1 and ε = 1 and vary just Le and NA . This we have done, with results shown in Table 1. (We also did some separate calculations to check that the two independent parameters are sufficient here.) As expected, Rac decreases as Le increases, and also decreases as NA increases. The change with NA is particularly rapid. It appears that the reduction in stability is associated
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82 Table 1 Values of the critical internal Rayleigh number Rac and the corresponding critical wavenumber ac for various values of the Lewis number Le and the diffusivity ratio NA , for NB ∗ = 0, Rn = 1, ε = 1
D. A. Nield, A. V. Kuznetsov Le
NA
Rac
ac
0
0
246.02
4.52
0
0.1
210.29
4.23
0
1
85.72
3.18
0
10
11.70
2.63
0.1
0
245.58
4.52
0.1
0.1
209.88
4.23
0.1
1
85.51
3.18
0.1
10
11.67
2.62
1
0
241.57
4.48
1
0.1
206.21
4.18
1
1
83.59
3.13
1
10
11.37
2.59
10
0
196.40
3.83
10
0.1
164.12
3.48
10
1
59.79
2.29
10
10
7.56
1.85
with an increasing role for the buoyancy effect due to the nanofluid particles becoming concentrated in the upper portion of the upper half of the layer (where the basic temperature gradient is negative) and so producing a top-heavy situation. (We have in mind the way in which Eq. (25) follows from Eq. (24) as a result of Eq. (23).) The establishment of this particle gradient is aided if the Brownian diffusion coefficient is small, so that the Lewis number Le is large and the modified diffusivity ratio NA is also large. This effect is amplified if the parameter Rn is large.
4 Conclusions We have taken a fresh look at the onset of convection in a layer of a porous medium saturated by a nanofluid for which thermophoresis and Brownian motion are important. In contrast to our previous study (Nield and Kuznetsov 2009), we have now examined internal heating instead of heating from below. Furthermore, we now consider more realistic boundary conditions on the nanofluid particle fraction. Instead of that quantity being held strictly fixed on the boundary as in the old approach, it is now allowed to adapt to the temperature profile induced by the internal heating, subject to the requirement that there is zero perturbation flux across a boundary, something expressed by Eq. (40). (We regard Eqs. (5) and (6) as specifying a reference particle fraction.) This has required a rescaling of various parameters. In particular, it is now an internal Rayleigh number rather than an external Rayleigh number value of which characterizes the onset of convection. It is found that it is the combinations RnLe/ε and RnNA of redefined quantities that are now important, and increase of either of these combinations leads to increased instability.
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83
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