Natural Convective Boundary Layer Flow over a Horizontal Plate Embedded in a Porous Medium Saturated with a Nanofluid

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1 Journal of Modern Physics, 011,, 6-71 doi:10.46/jp Published Online February 011 ( Natural Convective Boundary Layer Flow over a Horizontal Plate Ebedded in a Porous Mediu Saturated with a Nanofluid Abstract Raa Subba Reddy Gorla 1, Ali Chakha 1 Cleveland State University, Cleveland, USA Public Authority for Applied Education and Training, Shuweikh, Kuwait E-ail: r.gorla@csuohio.edu, achakha@yahoo.co Received Septeber 5, 010; revised October 8, 010; accepted October 11, 010 A boundary layer analysis is presented for the natural convection past a horizontal plate in a porous ediu saturated with a nano fluid. Nuerical results for friction factor, surface heat transfer rate and ass transfer rate have been presented for paraetric variations of the buoyancy ratio paraeter Nr, Brownian otion paraeter Nb, therophoresis paraeter Nt and Lewis nuber Le. The dependency of the friction factor, surface heat transfer rate (Nusselt nuber) and ass transfer rate on these paraeters has been discussed. Keywords: Natural Convection, Porous Mediu, Nanofluid 1. Introduction The study of convective heat transfer in nanofluids is gaining a lot of attention. The nanofluids have any applications in the industry since aterials of nanoeter size have unique physical and cheical properties. Nanofluids are solid-liquid coposite aterials consisting of solid nanoparticles or nanofibers with sizes typically of n suspended in liquid. Nanofluids have attracted great interest recently because of reports of greatly enhanced theral properties. For exaple, a sall aount (<1% volue fraction) of Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is reported to increase the inherently poor theral conductivity of the liquid by 40% and 150%, respectively [1,]. Conventional particle-liquid suspensions require high concentrations (>10%) of particles to achieve such enhanceent. However, probles of rheology and stability are aplified at high concentrations, precluding the widespread use of conventional slurries as heat transfer fluids. In soe cases, the observed enhanceent in theral conductivity of nanofluids is orders of agnitude larger than predicted by well-established theories. Other perplexing results in this rapidly evolving field include a surprisingly strong teperature dependence of the theral conductivity [] and a three-fold higher critical heat flux copared with the base fluids [4,5]. These enhanced theral properties are not erely of acadeic interest. If confired and found consistent, they would ake nanofluids proising for applications in theral anageent. Furtherore, suspensions of etal nanoparticles are also being developed for other purposes, such as edical applications including cancer therapy. The interdisciplinary nature of nanofluid research presents a great opportunity for exploration and discovery at the frontiers of nanotechnology. Porous edia heat transfer probles have several engineering applications such as geotheral energy recovery, crude oil extraction, ground water pollution, theral energy storage and flow through filtering edia. Cheng and Minkowycz [6] presented siilarity solutions for free convective heat transfer fro a vertical plate in a fluid-saturated porous ediu. Gorla and co-workers [7,8] solved the nonsiilar proble of free convective heat transfer fro a vertical plate ebedded in a saturated porous ediu with an arbitrarily varying surface teperature or heat flux. Chen and Chen [9] and Mehta and Rao [10] presented siilarity solutions for free convection of non-newtonian fluids over horizontal surfaces in porous edia. Nakayaa and Koyaa [11] studied the natural convection over a non-isotheral body of arbitrary geoetry placed in a porous ediu. All these studies were concerned with Newtonian fluid flows. The boundary layer flows in nano fluids have been analyzed re- Copyright 011 SciRes.

2 6 cently by Nield and Kuznetsov and Kuznetsov [1] and Nield and Kuznetsov [1]. A clear picture about the nanofluid boundary layer flows is still to eerge. The present work has been undertaken in order to analyze the natural convection past an isotheral horizontal plate in a porous ediu saturated by a nanofluid. The effects of Brownian otion and therophoresis are included for the nanofluid. Nuerical solutions of the boundary layer equations are obtained and discussion is provided for several values of the nanofluid paraeters governing the proble.. Analysis We consider the steady free convection boundary layer flow past a horizontal plate placed in a nano-fluid saturated porous ediu. The co-ordinate syste is selected such that x-axis is in the horizontal direction. We consider the two-diensional proble. Figure 1 shows the coordinate syste and flow odel. At the surface, the teperature T and the nano-particle fraction take constant values T W and ϕ W, respectively. The abient values, attained as y tends to infinity, of T and are denoted by T and ϕ, respectively. The Oberbeck-Boussinesq approxiation is eployed and the hoogeneity and local theral equilibriu in the porous ediu are assued. We consider the porous ediu whose porosity is denoted by ε and pereability by K. The Darcy velocity is denoted by v. The following four field equations ebody the conservation of total ass, oentu, theral energy, and nano-particles, respectively. The field variables are the Darcy velocity v, the teperature T and the nano-particle volue fraction..v 0 (1) f v P v p 1 f 1 T T g t K () T c c v f T t () D T k T c D p BT T T T Figure 1. Coordinate syste and flow odel. 1 D T v DB T (4) t T We write v u,v. Here, μ and β are the density, viscosity and voluetric volue expansion coefficient of the fluid; p f the density of the particles; g the gravitational acceleration; c the effective heat capacity and k effective theral conductivity of the porous ediu and D B the Brownian diffusion coefficient and D T the therophoretic diffusion coefficient. The flow is assued to be slow so that an advective ter and a Forchheier quadratic drag ter do not appear in the oentu equation. The boundary conditions are taken to be v 0, T T W, W, at y 0, (5) u 0, T T,, as y (6) We consider the steady state flow. In keeping with the Oberbeck-Boussinesq approxiation and an assuption that the nano-particle concentration is dilute, the oentu equation ay be written as: 0 P v K (7) P f1ft T g We now ake the standard boundary layer approxiation based on a scale analysis and write the governing equations. u v 0 (8) x y P u (9) x K P 1 f g T T P fg y (10) T T T DT T u v T DB x y y y T y (11) 1 D T T u v DB (1) x y y T y where k c p, c c (1) f One can eliinate P fro Equations (9) and (10) by cross-differentiation. At the sae tie one can introduce a strea line function ψ such that the continuity is autoatically satisfied: f Copyright 011 SciRes.

3 64 R. S. R. GORLA ET AL. u, v (14) dy dx We are then left with the following three equations. 1 gk T P f gk f y x (15) x L e D The transfored boundary conditions are: 0: S 0, 1, f 1 : S 0, 0, f 0 B () () T T T DT T T D B y x x y y y T y (16) D T 1 T DB y x x y y T y (17) Proceeding with the analysis we introduce the following diensionless variables: Ra x y Rax x 1 1 f gkax S Ra T T T T W 1/ x f (18) W We assue that T w and are constants. Substituting the expressions in Equation (18) into the governing Equations (15)-(17) we obtain the following transfored equations: S Nr f 0 (19) 1 S N b f Nt 0 (0) 1 N t f Le S f 0 N b (1) where the four paraeters are defined as: N r N b N t P fw T T 1 f W c DB W P c, f c DT TW T P c T, f, The local friction factor ay be written as u y y0 Ra x.s" 0 Cfx U Re x.pr The heat transfer rate at the surface is given by: T qw kf y y0 The heat transfer coefficient is given by: qw h TW T Local Nusselt nuber is given by: 1 x 0 hx Nux Ra, (4) kf The ass transfer rate at the surface is given by: NW D h W y y0 where h = ass transfer coefficient, The local Sherwood nuber is given by: 1 h x Sh Rax f, 0. Results and Discussion.1. Nuerical Method (5) D The syste of Equations (19)-(1) with the boundary conditions () is solved nuerically by eans of an efficient, iterative, tri-diagonal iplicit finite-difference ethod discussed previously by Blottner [1]. Equations (19)-(1) are discretized using three-point central difference forulae 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 doinate. The initial step size used is and the growth factor K such that n Kn 1 (where the subscript n is the nuber of nodes inus one). This gives 5 which repre- ax Copyright 011 SciRes.

4 65 sents the edge of the boundary layer at infinity. The ordinary differential equations are then converted into linear algebraic equations that are solved by the Thoas algorith discussed by Blottner [14]. Iteration is eployed to deal with the nonlinear nature of the governing equations. The convergence criterion eployed 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 assued converged and the iteration process was terinated. Equations (19)-(1) were solved nuerically to satisfy the boundary conditions () for paraetric values of Le, Nr (buoyancy ratio nuber), Nb (Brownian otion paraeter) and Nt (therophoresis paraeter) using finite difference ethod. Tables 1-5 indicate results for wall values for the gradients of velocity, teperature and concentration functions which are proportional to the friction factor, Nusselt nuber and Sherwood nuber, respectively. Fro Tables 1-, we notice that as Nr and Nt increase, the friction factor increases whereas the heat transfer rate (Nusselt nuber) and ass transfer rate (Sherwood nuber) decrease. As Nb increases, the fric- Table 1. Effects of N r on S"(0), θ'(0) and f'(0) for N b = 0., N t = 0.1 and Le = 10. N r S"(0) θ'(0) f'(0) E E E-05.67E E-04.46E E E E E E E Table. Effects of N t on S"(0), θ'(0) and f'(0) for N b = 0., N r = 0.5 and Le = 10. N t S"(0) θ'(0) f'(0) E E E E E-07.95E E E E E Table. Effects of N b on S"(0), θ'(0) and f'(0) for N r =0.5, N t =0.1 and Le=10. N b S"(0) θ'(0) f'(0) E E E E E E E E E E Table 4. Effects of Le on S"(0), θ'(0) and f'(0) for N b =0., N r = 0.5, and N t = 0.1. Le S"(0) θ'(0) f'(0) E E E E E E E E E Table 5. Effects of Le on S"(0), θ'(0) and f'(0) for N b = 0, N r = 0, and N t = 0. Le S"(0) θ'(0) f'(0) E E E E E E E E E tion factor and surface ass transfer rates increase whereas the surface heat transfer rate decreases. Results fro Table 4 indicate that as Le increases, the heat and ass transfer rates increase. Fro Table 5, we observe that the nano fluids display drag reducing and heat and ass transfer rate reducing characteristics. Figures -4 indicate that as Nr increases, the velocity decreases and the teperature and concentration increase. Siilar effects are observed fro Figures 5-10 as Nt and Nb vary. Figure 11 illustrates the variation of velocity within the boundary layer as Le increases. The velocity increases as Le increases. Fro Figures 1 and 1, we ob- serve that as Le increases, the teperature and concentration within the boundary layer decrease and the theral and concentration boundary later thicknesses decrease. The influence of nanoparticles on natural convection is odeled by accounting for Brownian otion and therophoresis as well as non-isotheral boundary conditions. The thickness of the boundary layer for the ass fraction is saller than the theral boundary layer thickness for Large values of Lewis nuber Le. The contribution of N t to heat and ass transfer does not depend on the value of Le. The Brownian otion and therophoresis of nano particles increases the effective theral conductivity of the nanofluid. Both Brownian diffusion and therophoresis give rise to cross diffusion ters that are siilar to the failiar Soret and Dufour cross diffusion ters that arise with a binary fluid discussed by Lakshi Narayana et al. [15]. 4. Concluding Rearks In this paper, we presented a boundary layer analysis for the natural convection past a non-isotheral vertical Copyright 011 SciRes.

5 Copyright 011 SciRes Nr=0,0.1,0.,0.,0.4,0.5Nb=0.Nt=0.1Le=10S' 1Figure. Effects of Nr on velocity profiles Nb=0.Nt=0.1Le=10Nr=0,0.1,0.,0.,0.4,0.5 Figure. Effects of N r on teperature profiles Nb=0.Nt=0.1Le=10Nr=0,0.1,0.,0.,0.4,0.5f Figure 4. Effects of N r on volue fraction profiles.

6 Copyright 011 SciRes Nb=0.Nr=0.5Le=10Nt=0.1,0.,0.,0.4,0.5S' Figure 5. Effects of N t on velocity profiles Nb=0.Nr=0.5Le=10Nt=0.1,0.,0.,0.4,05. Figure 6. Effects of N t on teperature profiles Nb=0.Nr=0.5Le=10Nt=0.1,0.,0.,0.4,0.5f Figure 7. Effects of N t on volue fraction profiles.

7 Copyright 011 SciRes Nr=0.5Nt=0.1Le=10Nb=0.1,0.,0.,0.4,0.5S' Figure 8. Effects of N b on velocity profiles Nr=0.5Nt=0.1Le=10Nb=0.1,0.,0.,0.4,0.5 Figure 9. Effects of N b on teperature profiles Nr=0.5Nt=0.1Le=10Nb=0.1,0.,0.,0.4,0.5f Figure 10. Effects of N b on volue fraction profiles.

8 Copyright 011 SciRes Le=100,1000Le=10Nb=0.Nr=0.5Nt=0.1Le=1S' Figure 11. Effects of Le on velocity profiles Nb=0.Nr=0.5Nt=0.1Le=1,10,100,1000 Figure 1. Effects of Le on teperature profiles Nb=0.Nr=0.5Nt=0.1Le=1,10,100,1000f Figure 1. Effects of Le on volue fraction profiles.

9 plate in a porous ediu saturated with a nano fluid. Nuerical results for friction factor, surface heat transfer rate and ass transfer rate have been presented for paraetric variations of the buoyancy ratio paraeter Nr, Brownian otion paraeter Nb, therophoresis paraeter Nt and Lewis nuber Le. The results indicate that as Nr and Nt increase, the friction factor increases whereas the heat transfer rate (Nusselt nuber) and ass transfer rate (Sherwood nuber) decrease. As Nb increases, the friction factor and surface ass transfer rates increase whereas the surface heat transfer rate decreases. As Le increases, the heat and ass transfer rates increase. Nano fluids display drag reducing and heat and ass transfer rate reducing characteristics. 5. References [1] J. A. Eastan, S. U. S. Choi, S. Li, W. Yu and L. J. Thopson, Anoalously Increased Effective Theral Conductivities Containing Copper Nanoparticles, Applied Physics Letters, Vol. 78, No. 6, 001, pp doi:10.106/ [] S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood and E. A. Grulke, Anoalous Theral Conductivity Enhanceent on Nanotube Suspensions, Applied Physics Letters, Vol. 79, No. 14, 001, pp doi:10.106/ [] H. E. Patel, S. K. Das, T. Sundararajan, A. Sreekuaran, B. George and T. Pradeep, Theral Conductivities of Naked and Monolayer Protected Metal Nanoparticle Based Nanofluids: Manifestation of Anoalous Enhanceent and Cheical Effects, Applied Physics Letters, Vol. 14, No. 8, 00, pp doi:10.106/ [4] S. M. You, J. H. Ki and K. H. Ki, Effect of Nanoparticles on Critical Heat Flux of Water in Pool Boiling Heat Transfer, Applied Physics Letters, Vol. 8, No. 16, 00, pp doi:10.106/ [5] P. Vassallo, R. Kuar and S. D Aico, Pool Boiling Heat Transfer Experients in Silica-Water Nanofluids, International Journal of Heat and Mass Transfer, Vol. 47, No., 004, pp doi: /s (0)0061- [6] P. Cheng and W. J. Minkowycz, Free Convection about a Vertical Flat Plate Ebedded in a Saturated Porous Mediu with Applications to Heat Transfer fro a Dike, Journal of Geophysics Res., Vol. 8, 1977, pp doi:10.109/jb08i014p0040 [7] R. S. R. Gorla and R. Tornabene, Free Convection fro a Vertical Plate with Nonunifor Surface Heat Flux and Ebedded in a Porous Mediu, Transport in Porous Media Journal, Vol., 1988, pp doi: /bf00688 [8] R. S. R. Gorla and A. Zinolabedini, Free Convection Fro a Vertical Plate With Nonunifor Surface Teperature and Ebedded in a Porous Mediu, Journal of Energy Resources Technology, Vol. 109, 1987, pp doi: /1.119 [9] H. T. Chen and C. K. Chen, Natural Convection of Non-Newtonian Fluids about a Horizontal Surface in a Porous Mediu, Journal of Energy Resources Technology, Vol. 109, 1987, pp doi: /1.16 [10] K. N. Mehta and K. N. Rao, Buoyancy-Induced Flow of Non-Newtonian Fluids in a Porous Mediu Past a Horizontal Plate with Nonunifor Surface Heat Flux, International Journal of Engineering Science, Vol., 1994, pp [11] A. Nakayaa and H. Koyaa, Buoyancy-Induced Flow of Non-Newtonian Fluids Over a Non-Isotheral Body of Arbitrary Shape in a Fluid-Saturated Porous Mediu, Applied Scientific Research, Vol. 48, 1991, pp doi: /bf [1] D. A. Nield and A. V. Kuznetsov, The Cheng Minkowycz Proble for Natural Convective Boundary Layer Flow in a Porous Mediu Saturated by a Nanofluid, International Journal of Heat and Mass Transfer, Vol. 5, 009, pp doi: /j.ijheatasstransfer [1] D. A. Nield and A. V. Kuznetsov, Theral Instability in a Porous Mediu Layer Saturated by a Nanofluid, International Journal of Heat and Mass Transfer, Vol. 5, No. 5-6, 009, pp doi: /j.ijheatasstransfer [14] F. G. Blottner, Finite-Difference Methods of Solution of the Boundary-Layer Equations, Journal of AIAA, Vol. 8, 1970, pp doi:10.514/.564 [15] P. A. L. Narayana, P. V. S. N. Murthy and R. S. R. Gorla, Soret-Driven Therosolutal Convection Induced by Inclined Theral and Solutal Gradients in a Shallow Horizontal Layer of a Porous Mediu, Journal of Fluid Mechanics, Vol. 61, 008, pp doi: /s Copyright 011 SciRes.

10 71 Noenclature D B Brownian diffusion coefficient D T therophoretic diffusion coefficient f rescaled nano-particle volue fraction g gravitational acceleration vector k effective theral conductivity of the porous ediu K pereability of porous ediu Le Lewis nuber Nr Buoyancy Ratio Nb Brownian otion paraeter Nt therophoresis paraeter Nu Nusselt nuber P pressure q wall heat flux Ra x local Rayleigh nuber Re Reynolds nuber S diensionless strea function T teperature T W wall teperature of the vertical plate T abient teperature U reference velocity u, v Darcy velocity coponents (x,y) Cartesian coordinates α Greek Sybols: theral diffusivity of porous ediu β voluetric expansion coefficient of fluid ε porosity η diensionless distance θ diensionless teperature μ viscosity of fluid ρ f fluid density ρ p nano-particle ass density (ρc) f heat capacity of the fluid (ρc) effective heat capacity of porous ediu (ρc) p effective heat capacity of nano-particle aterial τ paraeter defined by equation (1) ϕ nano-particle volue fraction ϕ W nano-particle volue fraction at the wall of the vertical plate ϕ abient nano-particle volue fraction ψ strea function Copyright 011 SciRes.