The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: A revised model

International Journal of Heat and Mass Transfer 65 (2013) 682–685

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: A revised model A.V. Kuznetsov a,⇑, D.A. Nield b a b

Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

a r t i c l e

i n f o

Article history: Received 8 May 2013 Received in revised form 21 June 2013 Accepted 21 June 2013

Keywords: Nanofluid Porous medium Brownian motion Thermophoresis Boundary layer Vertical plate

a b s t r a c t The classical Cheng–Minkowycz problem considers natural convection past a vertical plate in a fluid-saturated porous medium. In our previous work we extended the Cheng–Minkowycz problem to the case when a porous medium is saturated by a nanofluid. We utilized Buongiorno’s nanofluid model that includes the effects of Brownian motion and thermophoresis. The major limitation of our previous model was active control of nanoparticle volume fraction at the boundary. Here we revisited our previous model and extended it to the case when the nanofluid particle fraction on the boundary is passively rather than actively controlled. This makes the model physically more realistic than our previous model as well as models employed by other authors simulating nanofluid flow in porous media. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction

2. Analysis

The problem of natural convection in a porous medium past a vertical plate is a classical problem first studied by Cheng and Minkowycz [1]. This problem is further investigated in Kim and Vafai [2] and is discussed as textbook material in Bejan [3]. Further work on this topic is surveyed in Sections 5.1 and 9.2.1 in Nield and Bejan [4]. An extension to the case of a nanofluid, based on a model presented by Buongiorno [5], in which the effects of Brownian motion and thermophoresis are taken into account, was made by Nield and Kuznetsov [6]. In their paper it was assumed that one could control the value of the nanoparticle fraction at the boundary in the same way as the temperature there could be controlled, but no indication was given of how this could be done in practice. Thus it is advisable to replace their boundary conditions by a set that are more realistic physically. In the present paper we revisit this problem, and we now assume that there is no nanoparticle flux at the plate and that the particle fraction value there adjusts accordingly. We have prepared a paper [7] on the corresponding problem in a fluid clear of solid material.

The analysis closely follows that employed in [6] and so only a brief outline is given here. It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents nanoparticles from agglomeration and deposition on the porous matrix. We consider a two-dimensional problem. We select a coordinate frame in which the x-axis is aligned vertically upwards. We consider a vertical plate at y = 0. At this boundary the temperature T takes the constant value Tw and it is assumed that the nanopartcicle flux is zero there. The ambient values, attained as y tends to infinity, of T and / are denoted by T1 and /1, respectively. The Oberbeck–Boussinesq approximation is employed. Homogeneity and local thermal equilibrium in the porous medium is assumed. We consider a medium whose porosity is denoted by e and permeability by K. The Darcy velocity is denoted by v. The following four field equations embody the conservation of total mass, momentum, thermal energy, and nanoparticles, respectively. The field variables are the Darcy velocity v, the temperature T and the nanoparticle volume fraction /.

⇑ Corresponding author. Tel.: +1 919 515 5292; fax: +1 919 515 7968. E-mail addresses: [email protected] (A.V. Kuznetsov), [email protected] (D.A. Nield). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.06.054

r  v ¼ 0;

ð1Þ

qf @ v l ¼ rp  v þ ½/qp þ ð1  /Þfqf ð1  bðT  T 1 ÞÞgg; e @t K

ð2Þ

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A.V. Kuznetsov, D.A. Nield / International Journal of Heat and Mass Transfer 65 (2013) 682–685

Nomenclature DB DT f

Brownian diffusion coefficient thermophoretic diffusion coefficient rescaled nanoparticle volume fraction, defined by Eq. (20) gravitational acceleration vector effective thermal conductivity of the porous medium permeability of the porous medium Lewis number, defined by Eq. (27) buoyancy ratio, defined by Eq. (24) Brownian motion parameter, defined by Eq. (25) thermophoresis parameter, defined by Eq. (26) Nusselt number, defined by Eq. (32) reduced Nusselt number, Nu/Rax1=2 pressure wall heat flux local Rayleigh number, defined by Eq. (18) dimensionless stream function, defined by Eq. (20) temperature temperature at the vertical plate ambient temperature attained as y tends to infinity Darcy velocity, (u, v)

g km K Le Nr Nb Nt Nu Nur p q00 Rax s T Tw T1 v

ðqcÞm

@T þ ðqcÞf v  rT ¼ km r2 T þ eðqcÞp ½DB r/  rT @t þ ðDT =T 1 ÞrT  rT;

@/ 1 þ v  r/ ¼ DB r2 / þ ðDT =T 1 Þr2 T: @t e

T ¼ Tw;

u ¼ v ¼ 0;

DB

T ! T1;

@/ DB @T þ ¼ 0 at y ¼ 0; @y T 1 @y / ! /1 as y ! 1:

l K

f

volumetric expansion coefficient of the fluid porosity similarity variable, defined by Eq. (19) dimensionless temperature, defined by Eq. (20) viscosity of the fluid fluid density nanoparticle mass density heat capacity of the fluid effective heat capacity of the porous medium effective heat capacity of the nanoparticle material

b

e g h

l qf qp ðqcÞf ðqcÞm ðqcÞp s

eðqcÞ

parameter defined by Eq. (13), ðqcÞ p f nanoparticle volume fraction ambient nanoparticle volume fraction attained as y tends to infinity stream function, defined by Eq. (14)

/ /1

w

ð3Þ u ð4Þ

ð10Þ

"   2 # @T @T @/ @T DT @T ; þv ¼ am r2 T þ s DB þ @x @y @y @y @y T1

    2 @/ @/ @2/ DT @ T ¼ DB 2 þ u þv : @y @y e @x T 1 @y2

1

am ¼

km ; ðqcÞf

s¼

eðqcÞp : ðqcÞf

u¼

@w ; @y

v ¼

@w ; @x

@ 2 w ð1  /1 Þqf 1 bgK @T ðqp  qf 1 ÞgK @/ ¼  @y2 @y @y l l "   2 # @w @T @w @T @/ @T DT @T  ¼ am r2 T þ s DB þ ; @y @x @x @y @y @y @y T1

We now make the standard boundary-layer approximation, based on a scale analysis, and write the governing equations in the form

@u @ v þ ¼ 0; ð8Þ @x @y @p l ¼  u þ ½ð1  /1 Þqf 1 bgðT  T 1 Þ  ðqp  qf 1 Þgð/  /1 Þ ð9Þ @x K

ð13Þ

ð14Þ

ð6a; b; cÞ

ð7Þ

ð12Þ

One can eliminate p from Eqs. (9) and (10) by cross-differentiation. At the same time one can introduce a stream function w defined by

so that Eq. (8) is satisfied identically. We are then left with the following three equations.

v þ ½ðqp  qf 1 Þð/  /1 Þ þ ð1  /1 Þqf 1 bðT  T 1 Þg:

ð11Þ

where

ð5a; b; cÞ

Eq. (5c) is a statement that, with thermophoresis taken into account, the normal flus of nanoparticles is zero at the boundary. We consider a steady state flow. In keeping with the Oberbeck–Boussinesq approximation and an assumption that the nanoparticle concentration is dilute, and with a suitable choice for the reference pressure, we can linearize the momentum equation and write Eq. (2) as

0 ¼ rp 

Cartesian coordinates (x-axis is aligned vertically upwards, plate is at y = 0) Greek symbols m am thermal diffusivity of the porous medium, ðqkcÞ

@p ¼ 0; @y

We write v = (u,v). Here qf, l and b are the density, viscosity, and volumetric volume expansion coefficient of the fluid a while qp is the density of the particles. The gravitational acceleration is denoted by g. We have introduced the effective heat capacity (qc)m, and the effective thermal conductivity km of the porous medium. The coefficients that appear in Eqs. (3) and (4) are the Brownian diffusion coefficient DB and the thermophoretic diffusion coefficient DT. 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. The boundary conditions are taken to be

v ¼ 0;

(x, y)

    2 1 @w @/ @w @/ @2/ DT @ T ¼ DB 2 þ  : @y e @y @x @x @y T 1 @y2

ð15Þ

ð16Þ

ð17Þ

We now introduce the local Rayleigh number Rax defined by

Rax ¼

ð1  /1 Þqf 1 bgKx

lam

;

ð18Þ

and the similarity variable

y x

g ¼ Ra1=2 x :

ð19Þ

Since most nanofluids examined to date have large values for the Lewis number, we are interested mainly in the case Le > 1.

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We also introduce the dimensionless variables s, h, and f defined by

sðgÞ ¼

w

am Ra1=2 x

hðgÞ ¼

;

T  T1 ; Tw  T1

f ðgÞ ¼

/  /1 : /1

ð20Þ

Then, on substitution in Eqs. (15)–(17), we obtain the ordinary differential equations

s00  h0 þ Nrf 0 ¼ 0;

ð21Þ

1 h00 þ sh0 þ Nbf 0 h0 þ Nth02 ¼ 0; 2

ð22Þ

1 Nt 00 0 h ¼ 0; f 00 þ Lesf þ 2 Nb

ð23Þ

ðqp  qf 1 Þ/1 ; qf 1 bðT w  T 1 Þð1  /1 Þ

Le ¼

am : eDB

ð27Þ

Here Nr, Nb, Nt, Le denote a buoyancy ratio, a Brownian motion parameter, a thermophoresis parameter, and a Lewis number, respectively. Eqs. (21)–(23) are solved subject to the following boundary conditions:

At g ¼ 0 : As g ! 1 :

s0 ¼ 0;

h ¼ 0;

ð29a; b; cÞ

ð30Þ

(b) s ds/dη θ Le f

Le = 10

5

s, ds/dη, θ, Le f

4 s ds/dη θ Le f

3

2

1

0

2

4

6

8

0

10

0

2

4

η 2

6

8

10

η

(d)

Le = 100

1.5

2

Le = 1000

1.5 s ds/dη θ Le f

s, ds/dη, θ, Le f

s, ds/dη, θ, Le f

f ¼ 0:

ð25Þ

10

1

0.5

0

ð28a; b; cÞ

g = 0 given by Eqs. (28a,b,c) and the following boundary conditions at g ! 1 :

5

(c)

Nbf 0 þ Nth0 ¼ 0;

h ¼ 1;

s0  h þ Nrf ¼ 0;

15

0

s ¼ 0;

ð24Þ

Le = 1

25

20

s, ds/dη, θ, Le f

ð26Þ

Eqs. (30), (22) and (23) are solved subject to boundary conditions at

eðqcÞp DB /1 ; Nb ¼ ðqcÞf am

(a)

eðqcÞp DT ðT w  T 1 Þ ; ðqcÞf am T 1

Integrating Eq. (21) once and using boundary conditions (29) results in

where the four parameters are defined by

Nr ¼

Nt ¼

s ds/dη θ Le f 1

0.5

0

2

4

6

η

8

10

0

0

2

4

6

8

10

η

Fig. 1. Plots of dimensionless similarity functions s(g), s0 ðgÞ h(g), Lef(g) for the case Nr = 0.5, Nb = 0.5, Nt = 0.5 and for various values of Le. (a) Le = 1, (b) Le = 10, (c) Le = 100 and (d) Le = 1000.

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A.V. Kuznetsov, D.A. Nield / International Journal of Heat and Mass Transfer 65 (2013) 682–685

As g ! 1 :

h ¼ 0;

f ¼ 0:

ð31a; bÞ

In the case when Nr, Nb and Nt are all zero, Eqs. (21) and (22) involve just two dependent variables, namely s and h. The boundary-value problem for s and h reduces to the classical problem solved by Cheng and Minkowycz [1]. The boundary value problem for f, given by Eqs. (23), (28c) and (31b), then becomes ill-posed and is of no physical significance. The Nusselt number Nu is defined by the following equation:

Nu ¼

q00 x ; km ðT w  T 1 Þ

ð32Þ

where q00 is the wall heat flux and km is the effective thermal conductivity of the porous medium. The parameter Nu/Ra1=2 is reprex sented by h0 (0). With the new boundary condition the Sherwood number which represents the dimensionless mass flux is identically zero. One of the effects of the change in the boundary condition is immediately obvious. The parameters Nb and Nt now appear in the boundary conditions as well as in the differential equations. 3. Results and discussion In Fig. 1 we displayed plots of the similarity variables for four different values of Le. We have chosen a typical case of Nr = 0.5, Nb = 0.5, Nt = 0.5. The shape of the boundary layer profiles for the temperature function h(g) and the stream function s(g) are essentially the same as for a regular fluid. In the case of a regular fluid the profile for ds/dg, which represents the longitudinal component of the velocity, u, is identical with that for the temperature h. We now see that in the case of a nanofluid the two profiles diverge when Le is relatively small but tend to coincidence when Le increases. The form of Eq. (23) indicates that f is of order 1/Le when Le is large, and thus Lef is the appropriate function to plot in Fig. 1. It is interesting that as g increases the value of this function, which is proportional to the elevation of the nanoparticle fraction above its ambient value, rises to a maximum before decaying to zero. On the scale used the plots of h(g) and ds/dg in Fig. 1 coincide. We refer to the parameter Nu=Ra1=2 as the reduced Nusselt x number; we denote this parameter by Nur. In order to produce a regression correlation, we calculated 125 sets of values of Nr, Nb, Nt in the range [0.1, 0.2, 0.3, 0.4, 0.5]. For Le = 10 a linear regression on the results yielded the following correlation:

Nurest ¼ 0:444  0:0011Nr  0:0001Nb  0:1052Nt:

ð33Þ

Equation (33) is valid for Nr, Nb, Nt each taking values in the range [0, 0.5]. The maximum error in this case was about 1%. The maximum error occurs when (Nr, Nb, Nt) = (0.5, 0.1, 0.5), and the correlation formula overestimates the reduction from the standard value 0.444 that applies to a regular fluid in a porous medium. For comparison, the corresponding formula in [6] was

Nurest ¼ 0:444  0:111Nr  0:245Nb  0:150Nt;

ð33 Þ

which had a maximum error of about 12%. Both the new formula and the old formula indicate that an increase in either the buoyancy-ratio number or the thermophoresis parameter Nt leads to a decrease in the value of the reduced Nusselt number (corresponding to an increase in the thermal boundary-layer thickness). On the other hand there is a dramatic difference in the coefficient of Nb in the two formulas. The new

Table 1 Summary of linear regression coefficients and error bound for the reduced Nusselt number. Here Cr, Cb, Ct are the coefficients in the linear regression estimate Nuest =Ra1=2 ¼ 0:444 þ C r Nr þ C b Nb þ C t Nt, and e is the maximum relative error x defined by e ¼ jðNuest  NuÞ=Nuj, applicable for Nr, Nb, Nt each in [0, 0.5]. Le

Cr

Cb

Ct

e

5 10 20 50 100 200 500 1000

0.003 0.001 0.001 0.002 0.002 0.003 0.003 0.003

0.004 0.000 0.001 0.002 0.003 0.003 0.003 0.003

0.090 0.105 0.120 0.135 0.143 0.150 0.155 0.158

0.030 0.009 0.003 0.004 0.005 0.006 0.006 0.007

formula indicates that the Brownian motion has very little effect on the Nusselt number. The coefficient of Nr is now also small. We repeated this analysis for other values of Le, and we now present the results in Table 1. The coefficient of Nt decreases as Le increases. The relative errors presented in Table 1 suggest that for most practical purposes the simple linear regression formula in Eq. (33) should be adequate. 4. Conclusions We revisited the extension of the Cheng–Minkowycz problem to the case of a nanofluid-saturated porous medium. The major improvement achieved here is that we were able to reformulate the boundary conditions such that the nanoparticle volume fraction on the surface of the vertical plate passively adjusts itself to whatever temperature is imposed on the surface of the plate. The model in which Brownian motion and thermophoresis are accounted for permits a simple similarity solution which depends on four dimensionless groups that were first identified in Nield and Kuznetsov [6]. The dependence of the Nusselt number Nur on these four parameters is investigated. We summarized our results in the form of a linear regression correlation. The main change that we found was that the reduced Nusselt number is almost independent of the Brownian motion parameter Nb when zero nanoparticle flux is imposed on the boundary. Acknowledgment AVK gratefully acknowledges support of the Alexander von Humboldt Foundation though the Humboldt Research Award. References [1] P. Cheng, W.J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82 (1977) 2040–2044. [2] S. Kim, K. Vafai, Analysis of natural-convection about a vertical plate embedded in a porous-medium, Int. J. Heat Mass Transfer 32 (1989) 665–677. [3] A. Bejan, Convection Heat Transfer, 4th ed., Wiley, Hoboken, 2013. pp. 555–559. [4] D.A. Nield, A. Bejan, Convection in Porous Media, 4th ed., Springer, New York, 2013. [5] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [6] D.A. Nield, A.V. Kuznetsov, The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5792–5795. [7] D.A. Nield, A.V. Kuznetsov, Natural convective boundary-layer flow of a nanofluid past a vertical plate: a revised model. Int. J. Therm. Sci., submitted for publication.