ar ticle

Copyright © 2013 by American Scientific Publishers

Journal of Nanofluids Vol. 2, pp. 297–302, 2013 (www.aspbs.com/jon)

All rights reserved. Printed in the United States of America

Free Convection Past a Vertical Plate Embedded in a Porous Medium Saturated with a Non-Newtonian Nanofluid Rama Subba Reddy Gorla1, ∗ and Ali Chamkha2 1 2

Cleveland State University, Cleveland, Ohio 44115, USA Public Authority for Applied Education and Training, Shuweikh, 70654, Kuwait

KEYWORDS: Free Convection, Porous Medium, Nanofluid, Non-Newtonian Fluid.

Delivered by Publishing Technology to: Guest flow of nanofluids in User a porous medium. Gorla and Kumari8 IP: 49.50.98.136 On: Sun,investigated 13 Oct 2013 10:36:55 the mixed convection boundary layer flow of Copyright: American Publishers Heat transfer rates can be augmented by using solid Scientific non-Newtonian nanofluids over a stretching sheet. particles dispersed in base fluids. Nanofluids are engiThe present work has been undertaken in order to ananeered stable colloidal suspensions of nanometric metallic/ lyze the natural convection past a non-isothermal verticeramic solids (particles, rods and fibers) in conventional cal plate in a porous medium saturated by a power law heat transfer fluids in a small quantity (< 1% volume) type non-Newtonian nanofluid. The effects of Brownian maintaining a quasi-single phase state. Thermal properties motion and thermophoresis are included for the nanofluid. of nanofluids, as summarized by Eastman et al. and Choi Numerical solutions of the boundary layer equations are et al.1 2 are characterized by many fold increase in therobtained and discussion is provided for several values of mal conductivity compared to that of the base fluids as the nanofluid parameters governing the problem. a function of volume fraction of nanoparticles and temperature of the medium. This property makes nanofluids 1.1. Analysis a potential candidate for application in the fields of key engineering sectors such as microelectronics, automobiles, We consider the steady free convection boundary layer power generation, transportation, aerospace and nuclear flow past a vertical plate placed in a nano-fluid saturated power plants. porous medium. The co-ordinate system is selected such Prabhat et al.3 discussed the enhancement in heat transthat x-axis is aligned vertically upwards. We consider the fer by using nanofluids. Murshed et al.4 formulated a two-dimensional problem. We consider a vertical plate at Brownian motion-based model for the prediction of the y = 0. At this boundary the heat flux and mass flux take thermal conductivity of nanofluids. Murshed et al.5 invesconstant values qw and mw , respectively. The ambient valtigated the effect of surfactant and nanoparticle clustering ues of T and are denoted by T and , respectively. on thermal conductivity of nanofluids. Calvin et al.6 anaThe Oberbeck-Boussinesq approximation is employed lyzed the dual role of nanoparticles on thermal conductivand the homogeneity and local thermal equilibrium in ity enhancement of nanofluids. Kumari and Gorla7 studied the porous medium are assumed. We consider the porous the combined forced and natural convective boundary layer medium whose porosity is denoted by and permeability by K. ∗ Author to whom correspondence should be addressed. Here f , and are the density, viscosity and voluEmail: [email protected] metric volume expansion coefficient of the fluid; p the Received: 12 May 2013 Accepted: 22 May 2013 density of the particles; g the gravitational acceleration;

1. INTRODUCTION

J. Nanofluids 2013, Vol. 2, No. 4

2169-432X/2013/2/297/006

doi:10.1166/jon.2013.1064

297

ARTICLE

In this paper, a boundary layer analysis is presented for the free convection past a vertical plate in a porous medium saturated with a power law type non-Newtonian nano fluid. The boundary conditions of prescribed surface heat and mass flux are considered. Numerical results for friction factor, surface heat transfer rate and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter Nr , Brownian motion parameter Nb , thermophoresis parameter Nt , Lewis number Le and the power law exponent n. The dependency of the friction factor, surface heat transfer rate (Nusselt number) and mass transfer rate on these parameters has been discussed.

Free Convection Past a Vertical Plate Embedded in a Porous Medium Saturated with a Non-Newtonian Nanofluid

c m the effective heat capacity and km effective thermal conductivity of the porous medium and DB the Brownian diffusion coefficient and DT the thermophoretic diffusion coefficient. We now make the standard boundary layer approximation based on a scale analysis and write the governing equations.

u v + =0

x y

un 1− f gK T P −f gK = · − ·

y

y

y

T

T u· +v ·

x

y

2T

T DT T 2 = m 2 + DB +

y

y y T y 1

2 DT 2 T +v = DB 2 + u

x

y

y T y 2

1+n L ·S ·f 2n + 1 e n N − Le · S · f + t = 0 2n + 1 Nb

f +

(1)

Nr =

P − f mw k f 1 − DB qw

Nb =

c P mw Ra−n/2n+1 x c f m

Nt =

(3)

· x2n · U 2 · n Pr = m

(4)

Rex =

ARTICLE

where

m =

= 0 S = 0 = −1 f = −1 m

q

T =− w (5) =− w by kf Delivered

y D Publishing Technology to:Guest

y → User S = 0 = 0 f = 0 IP: 49.50.98.136 On: Sun, 13 Oct 2013 10:36:55 American Publishers u = 0 T → T → Copyright: as y → (6) Scientific The heat transfer rate is given by: We can introduce a stream line function defined by

T qw = −kf

y y=0

v=− u= (7) dy dx The heat transfer coefficient is given by: qW The continuity Eq. (1) is automatically satisfied. Proh= ceeding with the analysis we introduce the following TW − T dimensionless variables: Local Nusselt number is given by: y n/2n+1 = · Rax h·x 1 x Nux = = Ran/2n+1 · x 1/n kf 0 x 1 − f gKqw Rax = The Mass Transfer rate is given by: k m

S= N = −D = hm W − W n/2n+1

y y=0 m · Rax T − T TW − T

f=

− W −

+ Nb · f · + Nt 2 = 0 298

(13)

(14)

(15)

(16)

(17)

where hm = mass transfer coefficient, Local Sherwood number is given by: Sh =

(8)

Substituting the transformations in Eq. (8) into the governing Eqs. (1)–(4) we have: nS n−1 S − + Nr · f = 0 1+n n S − + 2n + 1 2n + 1

(12)

The boundary conditions are:

v = 0

=

(11)

c P DT qw Ra−n/2n+1 x c f m T k m Le = · DB x 1 − f gKqw 1/n Rax = k m

(2)

c p km = c f c f The boundary conditions are taken to be

Gorla and Chamkha

(9)

hm · x 1 = Ran/2n+1 · x D f 0

(18)

The wall shear stress and local friction Factor Cfx may be written as: n

u

W = · (19)

y y=0 2

(10)

Cfx =

/2n+1

W 2 · Ra3n · S 0 n x = U 2 /2 Rex · Pr n

(20)

J. Nanofluids, 2, 297–302, 2013

Gorla and Chamkha


2. NUMERICAL METHOD

Nt

S 0 n

1/0

1/f 0

0.1 0.2 0.3 0.4 0.5

8.621568E-01 8.622349E-01 8.599678E-01 8.563197E-01 8.525910E-01

6.975108E-01 6.846317E-01 6.735741E-01 6.638066E-01 6.547510E-01

2.725506 2.529002 2.351873 2.188917 2.040151

Table III. Effects of Nb on S 0 n , 1/0 and 1/f 0 for n = 07, Nr = 05, Nt = 03 and Le = 10. Nb

S 0 n

1/0

1/f 0

0.1 0.2 0.3 0.4 0.5

8.322425E-01 8.517124E-01 8.599678E-01 8.658313E-01 8.705959E-01

6.631348E-01 6.749043E-01 6.735741E-01 6.693159E-01 6.638361E-01

1.475477 2.051698 2.351873 2.543306 2.681394

Table IV. Effects of Le on S 0 n , 1/0 and 1/f 0 for n = 07, Nb = 03, Nr = 05 and Nt = 01. Le

S 0 n

1/0

1/f 0

5

8.537991E-01

6.713466E-01

1789430

Delivered by Publishing Technology to:8.621568E-01 Guest User 10 6.975108E-01 2725506 3. RESULTS AND DISCUSSION IP: 49.50.98.136 On: Sun,100 13 Oct 2013 10:36:55 8.745834E-01 7.335733E-01 9741728 Copyright: American Scientific Publishers 1000 8.768299E-01 7.431004E-01 31968710 Equations (9)–(11) were solved numerically to satisfy the boundary conditions (13) for parametric values of Le, Nr (buoyancy ratio number), Nb (Brownian motion parameTable V. Effects of n on S 0 n , 1/(0) and 1/f 0 for Nb = 03, Nr = ter), Nt (thermophoresis parameter) and power law expo05, Nt = 01 and Le = 10. nent n using finite difference method. Tables I–V indicate n S 0 n 1/0 1/f 0 results for wall values for the gradients of velocity, temperature and concentration functions which are proportional 0.5 1.140356 6.945069E-01 2.841735 0.7 8.621568E-01 6.975108E-01 2.725506 to the friction factor, Nusselt number and Sherwood num1.0 5.141028E-01 7.052962E-01 2.649577 ber, respectively. From Tables I–III, we notice that as Nr 1.2 3.480087E-01 7.111436E-01 2.624259 and Nt increase, the heat transfer rate (Nusselt number) 1.5 1.838417E-01 7.191658E-01 2.604907 and mass transfer rate (Sherwood number) decrease. As Nb increases, the friction factor and surface mass transfer rates increase. 1.8 Results from Tables IV indicate that as Le increases, n = 0.7 1.6 Nb = 0.3 the heat and mass transfer rates increase. From Table V, Nr = 0, 0.1, 0.2, 0.3, 0.4, 0.5 Nt = 0.1 1.4 we observe that as n increases, the heat transfer rate Le = 10 increases and the mass transfer rates decrease. Therefore, 1.2

Nr

S 0 n

1/0

1/f 0

0 0.1 0.2 0.3 0.4 0.5

1.44867 1.337995 1.223858 1.107332 9.874866E-01 8.621568E-01

7.188717E-01 7.147066E-01 7.104437E-01 7.063260E-01 7.020382E-01 6.975108E-01

2.860762 2.834730 2.808479 2.780565 2.752911 2.725506

1.0

S′

Table I. Effects of Nr on S 0 n , 1/0 and 1/f 0 for n = 07, Nb = 03, Nt = 01 and Le = 10.

0.8 0.6 0.4

J. Nanofluids, 2, 297–302, 2013

0.2 0.0 0

1

2

η

3

4

Fig. 1. Effects of Nr on velocity profiles.

299

ARTICLE

The system of Eqs. (9)–(11) with the boundary conditions (13) is solved numerically by means of an efficient, iterative, tri-diagonal implicit finite-difference method discussed previously by Blottner.9 Equations (19)–(21) are discretized using three-point central difference formulae with S replaced by another variable V . The direction is divided into 196 nodal points and a variable step size is used to account for the sharp changes in the variables in the region close to the surface where viscous effects dominate. The initial step size used is 1 = 0001 and the growth factor K = 1037 such that j = Kj−1 (where the subscript j is the number of nodes minus one). This gives max ≈ 35 which represents the edge of the boundary layer at infinity. The ordinary differential equations are then converted into linear algebraic equations that are solved by the Thomas algorithm discussed by Blottner.9 Iteration is employed to deal with the nonlinear nature of the governing equations. The convergence criterion employed in this work was based on the relative difference between the current and the previous iterations. When this difference or error reached 10−5 , the solution was assumed converged and the iteration process was terminated.

Table II. Effects of Nt on S 0 n , 1/0 and 1/f 0 for n = 07, Nb = 03, Nr = 05 and Le = 10.


Gorla and Chamkha

1.6

1.5 n = 0.7 Nb = 0.3 Nt = 0.1 Le = 10

1.2

n = 0.7 Nb = 0.3 Nr = 0.5 Le = 10

1.2

0.9

θ

θ

Nr = 0, 0.1, 0.2, 0.3, 0.4, 0.5

Nt = 0.1, 0.2, 0.3, 0.4, 0.5

0.8

0.6 0.4 0.3 0.0

0.0 0

Fig. 2.

1

2

η

3

4

0

5

η

3

4

5

0.5 n = 0.7 Nb = 0.3 Nt = 0.1 Le = 10

0.35 0.30

n = 0.7 Nb = 0.3 Nr = 0.5 Le = 10

0.4

0.25

0.3

0.20

Nr = 0, 0.1, 0.2, 0.3, 0.4, 0.5

f

f

2

Fig. 5. Effects of Nt on temperature profiles.

Effects of Nr on temperature profiles.

0.40

0.2

0.15

Nt = 0.1, 0.2, 0.3, 0.4, 0.5

0.10 0.1

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0.05 0.00 0

1

2

3

4

0

η Fig. 3.

Effects of Nr on volume fraction profiles.

1

2

η

3

4

5

Fig. 6. Effects of Nt on volume fraction profiles.

our current results indicate that non-Newtonian nanoflids promote higher heat and mass transfer rates in nano fluid boundary layers. Figures 1–3 indicate that as Nr increases, the velocity decreases and the temperature and concentration increase. Similar effects are observed from Figures 4–9 as Nt and Nb vary. Figure 10 illustrates the variation of velocity within the boundary layer as Le increases. The velocity increases as Le increases. From

Figures 11 and 12, we observe that as Le increases, the temperature and concentration within the boundary layer decrease and the thermal and concentration boundary later thicknesses decrease. Figures 13–15 indicate that as the power-law exponent n increases, the velocity close to the surface decreases whereas the temperature decreases and concentration within the boundary layer increases. 1.5

1.5 n = 0.7 Nb = 0.3 Nr = 0.5 Le = 10

1.2 Nt = 0.1, 0.2, 0.3, 0.4, 0.5

n = 0.7 Nr = 0.5 Nt = 0.3 Le = 10

1.2 Nb = 0.1, 0.2, 0.3, 0.4, 0.5

0.9

S′

0.9

S′

ARTICLE

1

0.6

0.6

0.3

0.3

0.0

0.0 0

Fig. 4.

300

1

2

η

Effects of Nt on velocity profiles.

3

4

5

0

1

2

3

4

η Fig. 7. Effects of Nb on velocity profiles. J. Nanofluids, 2, 297–302, 2013

Gorla and Chamkha


1.6

1.5 n = 0.7 Nr = 0.5 Nt = 0.3 Le = 10

1.2

n = 0.7 Nb = 0.3 Nr = 0.5 Nt = 0.1

1.2

Le = 5, 10, 100, 1000

θ

θ

0.9 Nb = 0.1, 0.2, 0.3, 0.4, 0.5

0.8

0.6 0.4 0.3

0.0

0.0 0

1

2

3

4

5

0

1

2

Fig. 8.

3

η

η

4

Fig. 11. Effects of Le on temperature profiles.

Effects of Nb on temperature profile.

0.7

0.6

n = 0.7 Nr = 0.5 Nt = 0.3 Le = 10

0.6 0.5

0.5

n = 0.7 Nb = 0.3

0.4

Nt = 0.1

Nr = 0.5

f

f

Nb = 0.1, 0.2, 0.3, 0.4, 0.5

0.3

0.3 Le = 5, 10, 100, 1000

0.2

0.2

0.1 Delivered by Publishing Technology to: Guest User IP: 49.50.98.136 On: Sun, 13 Oct 2013 10:36:55 Copyright: American Scientific Publishers 0.0

0.1 0.0 0

1

2

3

4

5

0

6

1

2

Fig. 9.

4

Fig. 12. Effects of Le on volume fraction profiles.

Effects of Nb on volume fraction profiles.

The influence of nanoparticles on natural convection is modeled by accounting for Brownian motion and thermophoresis as well as non-isothermal boundary conditions. The thickness of the boundary layer for the mass fraction is smaller than the thermal boundary layer thickness for Large values of Lewis number, Le. The contribution of Nt to heat and mass transfer does not depend on 1.6

3

η

η

the value of Le. The Brownian motion and thermophoresis of nano particles increases the effective thermal conductivity of the nanofluid. Both Brownian diffusion and thermophoresis give rise to cross diffusion terms that are similar to the familiar Soret and Dufour cross diffusion terms that arise with a binary fluid discussed by Lakshmi Narayana et al.10 1.6

Le = 5, 10, 100, 1000

n = 0.7 Nb = 0.3 Nr = 0.5

1.2

Nb = 0.3 Nr = 0.5 Nt = 0.1 Le = 10

1.4 n = 0.5, 0.7, 1.0, 1.2, 1.5

1.2

Nt = 0.1

S′

S′

1.0 0.8

0.8 0.6

0.4

0.4 0.2

0.0

0.0 0

1

2

η Fig. 10.

Effects of Le on velocity profiles.

J. Nanofluids, 2, 297–302, 2013

3

4

0

1

2

3

η

4

5

6

Fig. 13. Effects of n on velocity profiles.

301

ARTICLE

0.4


1.5 Nb = 0.3 Nr = 0.5 Nt = 0.1 Le = 10

1.2

θ

0.9

0.6

n = 0.5, 0.7, 1.0, 1.2, 1.5

0.3

0.0 0

Fig. 14.

1

2

η

3

4

5

Effects of n on temperature profiles.

0.4 Nb = 0.3 Nr = 0.5 Nt = 0.1

0.3

km — Effective thermal conductivity of the porous medium K— Permeability of porous medium K ∗ — Inertial coefficient in Ergun model Le— Lewis number Nr — Buoyancy Ratio Nb — Brownian motion parameter Nt — Thermophoresis parameter Nu— Nusselt number n— Power law index P — Pressure q — Wall heat flux Rax — Local Rayleigh number Re— Reynolds number S— Dimensionless stream function T — Temperature TW — Wall temperature of the vertical plate T — Ambient temperature U — Reference velocity u v— Darcy velocity components x y — Cartesian coordinates

Greek Symbols m — Thermal diffusivity of porous medium — Volumetric expansion coefficient of fluid — Porosity 0.1 — Dimensionless distance Delivered by Publishing Technology to: Guest User IP: 49.50.98.136 On: Sun, 13 Oct 10:36:55temperature — 2013 Dimensionless Copyright: American Scientific Publishers 0.0 — Viscosity of fluid 0 1 2 3 4 — Fluid density f η p — Nano-particle mass density Fig. 15. Effects of n on volume fraction profiles. c f — Heat capacity of the fluid c m — Effective heat capacity of porous medium 4. CONCLUDING REMARKS c p — Effective heat capacity of nano-particle material

— Ratio between the effective heat capacitis of the In this paper, we presented a boundary layer analysis for the nano particle material and fluid natural convection past a vertical plate in a porous medium — Nano-particle volume fraction saturated with a non-Newtonian nano fluid. Numerical w — Nano-particle volume fraction at the wall of the results for friction factor, surface heat transfer rate and vertical plate mass transfer rate have been presented for parametric varia — Ambient nano-particle volume fraction tions of the buoyancy ratio parameter Nr , Brownian motion — Stream function. parameter Nb , thermophoresis parameter Nt and Lewis number Le. The results indicate that as Nr and Nt increase, the friction factor increases whereas the heat transfer rate References and Notes (Nusselt number) and mass transfer rate (Sherwood num1. J. A. Eastman, S. U. S. Choi, S. Li, W. Yu, and L. J. Thompson, ber) decrease. As Nb increases, the friction factor and Appl. Phys. Lett. 78, 718 (2001). surface mass transfer rates increase whereas the surface 2. S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood, and E. A. Grulke, Appl. Phys. Lett. 79, 2252 (2001). heat transfer rate decreases. As Le increases, the heat and 3. N. Prabhat, J. Buongiorno, and L. W. Hu, J. Nanofluids 1, 155 mass transfer rates increase. As the power law exponent n (2012). increases, the heat and mass transfer rates increase. f

ARTICLE

Le = 10

Gorla and Chamkha

0.2

n = 0.5, 0.7, 1.0, 1.2, 1.5

NOMENCLATURE DB — Brownian diffusion coefficient DT — Thermophoretic diffusion coefficient f — Rescaled nano-particle volume fraction g— Gravitational acceleration vector 302

4. S. M. S. Murshed and C. A. N. Castro, J. Nanofluids 1, 180 (2012). 5. S. M. S. Murshed, C. A. N. D. Castro, and M. J. V. Lourenco, J. Nanofluids 1, 175 (2012). 6. H. Calvin, P. J. Li, and G. P. Peterson, J. Nanofluids 2, 20 (2013). 7. M. Kumari and R. S. R. Gorla, J. Nanofluids 1, 166 (2012). 8. R. S. R. Gorla and M. Kumari, J. Nanofluids 1, 186 (2012). 9. F. G. Blottner, AIAA J. 8, 193 (1970). 10. P. A. Lakshmi Narayana, P. V. S. N. Murthy, and R. S. R. Gorla, Journal of Fluid Mechanics 612, 1 (2008). J. Nanofluids, 2, 297–302, 2013