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Three-dimensional natural convection in a porous enclosure filled with a nanofluid using Buongiorno’s mathematical model
International Journal of Heat and Mass Transfer 82 (2015) 396–405
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Three-dimensional natural convection in a porous enclosure filled with a nanofluid using Buongiorno’s mathematical model M.A. Sheremet a,b,⇑, I. Pop c, M.M. Rahman d a
Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Applied Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania d Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, P.C. 123 Al-Khod, Muscat, Oman b
a r t i c l e
i n f o
Article history: Received 12 October 2014 Received in revised form 19 November 2014 Accepted 19 November 2014
Keywords: Natural convection Three-dimensional cavity Porous medium Nanofluids Brownian motion Thermophoresis Numerical results
a b s t r a c t Steady-state natural convection heat transfer in a three-dimensional porous enclosure filled with a nanofluid using the mathematical nanofluid model proposed by Buongiorno is presented. The nanofluid model takes into account two important slip mechanisms in nanofluids like Brownian diffusion and thermophoresis. The study is formulated in terms of the dimensionless vector potential functions, temperature and concentration of nanoparticles. The governing equations were solved by finite difference method on nonuniform mesh and solution of algebraic equations was made on the basis of successive under relaxation method. Effort has been focused on the effects of six types of influential factors such as the Rayleigh and Lewis numbers, the buoyancy-ratio parameter, the Brownian motion parameter, the thermophoresis parameter and the aspect ratio on the fluid flow, heat and mass transfer. Three-dimensional velocity, temperature and nanoparticle volume fraction fields, average Nusselt numbers are presented. It is found that low Rayleigh and Lewis numbers and high thermophoresis parameter reflect essential non-homogeneous distribution of nanoparticles inside the cavity, hence a non-homogeneous model is more appropriate for the description of the system. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Nanofluid, a term introduced by Choi [1], represents a fluid in which nano-scale particles are suspended in a base fluid with a low thermal conductivity such as water, ethylene glycol, oils, etc. This is a two-phase mixture in which the solid phase consists of nano-sized particles. Since the size of the particles is less than 100 nm, nanofluids behave much more like a fluid than a mixture (Xuan and Roetzel [2], Maliga et al. [3]). Xuan and Roetzel [2] proposed a homogeneous flow model where the convective transport equations of pure fluids are directly extended to nanofluids. This means that all the traditional heat transfer correlations can be used for nanofluids provided the properties of pure fluids are replaced by those of nanofluids involving the volume fraction of the nanoparticles (Kumar et al. [4]). These homogeneous flow models are, however, in conflict with the experimental observations of Maliga et al. [3] who considered forced convection flow in a channel, as they under predict the heat transfer coefficients of nanofluids. ⇑ Corresponding author at: Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.066 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
Masuda et al. [5] experimentally found that the fluids containing thermally conducting submicron solid particles have excellent thermal conductivity. Nanofluids perform a crucial role in the development of newer technologies ideal for industrial purposes. Such fluids have been studied extensively in recent years, in view of their great potential as a higher-energy carrier due to their promising feature of high effective thermal conductivity (Sakai et al. [6]). Materials with sizes of nanometers possess unique physical and chemical properties (Das et al. [7]). They can flow smoothly through micro-channels without clogging because they are sufficiently small to behave similar to liquid molecules (Khanafer et al. [8]). It has been found that the presence of nanoparticles within the fluid can appreciably increase the effective thermal conductivity of the fluid and, as a consequence, enhance the heat transfer characteristics. An excellent collection of articles on this topic can be found in the book by Das et al. [7], and Nield and Bejan [9], and in the review papers by Buongiorno [10], Kakaç and Pramuanjaroenkij [11], Lee et al. [12], Eagen et al. [13], Wong and Leon [14], Fan and Wang [15], Mahian et al. [16,17], etc. In a very recently interesting paper by Mehrali et al. [18], new experimental results on the
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397
Nomenclature A C C0 Cp DB DT g jp K L Ly Le Nb Nr Nt Nu Nu p Ra Sh Sh t T T0 Tc
aspect ratio parameter nanoparticle volume fraction reference value of nanoparticle volume fraction heat capacity Brownian diffusion coefficient thermophoretic diffusion coefficient gravitational acceleration nanoparticles mass flux defined by jp ¼ qp ½DB rCþ ðDT =T c ÞrT permeability of the porous medium size of the cavity along x and z axes axis size of the cavity along y Lewis number Brownian motion parameter buoyancy-ratio parameter thermophoresis parameter local Nusselt number mean Nusselt number fluid pressure Rayleigh number for the porous medium local Sherwood number mean Sherwood number time fluid temperature mean temperature of heated and cooled walls defined by T 0 ¼ ðT h þ T c Þ=2 temperature of the cooled wall
thermal conductivity, viscosity, stability, and electrical conductivity at different concentrations of the Nitrogen-doped graphene (NDG) nanofluids. The feasibility of water-based NDG nanofluids for use as innovative heat transfer fluid in medium-temperature systems was demonstrated. Convective flow within porous media both with or without nanoparticles have a wide range of practical and engineering applications see, for example the books by Nield and Bejan [9], Ingham and Pop [19], Pop and Ingham [20], Nakayama [21], Vadasz [22], Vafai [23,24] and de Lemos [25]. In a series of pioneering papers, Nield and Kuznetsov [26,27] have used the mathematical nanofluid model proposed by Buongiorno [10] to study the problem of free convective boundary layer flow past a vertical flat plate embedded in a porous medium and the problem of thermal instability in a porous medium layer saturated by a nanofluid. The authors have 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. On the other hand, it is very important to explain how nanofluid flows is possible in a porous medium. This situation has been described in the paper by Sakai et al. [6]. It should be noticed at this end that Bianco et al. [28,29] found that the two-components mixture model is quite adequate for describing the nanofluid heat transfer, as supported by Buongiorno [10] using a magnitudes analysis. Most researchers neglected the spatial variations of thermophysical properties including Brownian and thermophoretic diffusion coefficients. Such analytical treatments could lead to substantial errors. Sakai et al. [6] have derived a macroscopic set of the governing equations for describing heat transfer in nanofluid saturated porous media using a volume average theory by considering all these variations. Here we study the steady natural convection heat transfer in a three-dimensional porous enclosure filled with a nanofluid using the mathematical nanofluid model proposed by Buongiorno [10]
Th ; v ; w u V x; y ; z x, y, z
temperature of the hot wall ; z, dimensional velocity components along the axes x; y respectively Darcy velocity vector dimensional Cartesian coordinates dimensionless Cartesian coordinates
Greek letters am effective thermal diffusivity of the porous medium b volumetric expansion coefficient of the fluid d parameter defined by d ¼ eðqC p Þp =ðqC p Þf e porosity of the porous medium / rescaled nanoparticle volume fraction l dynamic viscosity h dimensionless temperature qf fluid density qf0 reference fluid density qp nanoparticle mass density (qCp)f volumetric heat capacity of the base fluid (qCp)p effective volumetric heat capacity of the nanoparticle material (qCp)m effective volumetric heat capacity of the porous medium r parameter defined by r ¼ ðqC p Þm =ðqC p Þf dimensional vector potential x; w y; w z dimensional vector potential functions w wx ; wy ; wz dimensionless vector potential functions
in combination with Darcy’s law for the flow in the porous medium and the Boussinesq approximation for the buoyancy forces. 2. Basic equations Consider the steady free convection in a three-dimensional porous cavity filled with a water based nanofluid. It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. A schematic geometry of the and z problem under investigation is shown in Fig. 1, where x; y are the Cartesian coordinates and L and Ly are the sizes of the cavity. It is assumed that the vertical surface x ¼ 0 is heated and maintained at the constant temperature Th, while the opposite vertical wall x ¼ L is cooled and has the constant temperature Tc. The other walls are adiabatic ð@T=@~ nÞ ¼ 0, where T is the fluid temperature. The basic equations for the flow, heat transfer and nanoparticles can be written in the following form (Nield and Bejan [9])
Fig. 1. Physical model and coordinate system.
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rV ¼0 0 ¼ rp
l K
h
i V þ C qp þ ð1 CÞqf 0 ð1 bðT T c ÞÞ g
@T þ ðV rÞT ¼ am r2 T þ d½DB rC rT þ ðDT =T c ÞrT rT @t @C 1 qp þ ðV rÞC ¼ r jp @t e
r
ð1Þ
l @ 2 w y
ð2Þ
K
0 ¼ rp
K
h
i V þ Cðqp qf 0 Þ þ qf 0 ð1 bðT T c Þð1 C 0 ÞÞ g
ð4Þ
ð5Þ
Eqs. (1) and (3)–(5) for the problem under consideration can be ; z as written in dimensional Cartesian coordinates x; y
@ v @ w @u þ þ ¼0 @ z @ x @ y @p l ¼ u @ x K @p l ¼ v @y K i @p l h ¼ w Cðqp qf 0 Þ þ qf 0 ð1 bðT T c Þð1 C 0 ÞÞ g @z K ! @T @T @T @T @2T @2T @2T r þ u þ v þ w ¼ am þ þ 2 @z2 @t @x @y @z @ x2 @ y @C @T @C @T @C @T þ d DB þ þ @y @z @ z @ x @ x @ y " 2 2 2 #) DT @T @T @T þ þ þ @ x @y @z Tc ! @C 1 @C @C @C @2C @2C @2C þ v þ w ¼ DB þ u þ þ 2 @z2 @t e @ x @y @z @ x2 @ y ! 2 DT @ T @2T @2T þ 2þ 2 þ @ x2 @ y @z TC
ð6Þ
K
x @2w x @2w þ þ 2 @ x2 @y @z2
@T @C gðqp qf 0 Þ @ x @ x
y @T @ w x @T @ w y @w x @T @w þ þ @z @z @ x @z @ y @ x @y ! 2 2 2 @ T @ T @ T ¼ am þ 2þ 2 @ x2 @ y @ z @C @T @C @T @C @T þ þ þ d DB @y @z @z @ x @ x @ y " 2 2 2 #) DT @T @T @T þ þ þ @ x @y @ z Tc
y @C @ w x @C @ w y @w x @C @w þ þ @z e @ z @ x @z @ y @ x @y ! ! 2 2 2 @ C @ C @ C DT @2T @2T @2T þ 2þ 2 þ þ 2þ 2 ¼ DB @ x2 @ y @ z @ x2 @ y @ z TC
ð8Þ
x =am ; w ¼ w y =am ; =L; z ¼ z=L; wx ¼ w x ¼ x=L; y ¼ y y z =am ; h ¼ ðT T 0 Þ=ðT h T c Þ; / ¼ C=C 0 wz ¼ w
ð9Þ
ð10Þ
ð11Þ
ð15Þ
1
Introducing the following dimensionless variables
ð16Þ
ð17Þ
where T0 = (Th + Tc)/2 is the mean temperature of heated and cooled walls, and substituting (17) into Eqs. (13)–(16), we obtain
@ 2 wx @ 2 wx @ 2 wx @h @/ þ 2 þ 2 ¼ Ra þ Ra Nr ð18Þ @y @y @x2 @y @z 2 2 2 @ wy @ wy @ wy @h @/ þ 2 þ 2 ¼ Ra Ra Nr ð19Þ @x @x @x2 @y @z 2 2 2 @wy @h @wx @h @wy @wx @h @ h @ h @ h þ þ ¼ þ þ @z @x @z @y @x @y @z " @x2 @y2 @z2 2 2 2 #) @/ @h @/ @h @/ @h @h @h @h þ Nt þ Nb þ þ þ þ @x @x @y @y @z @z @x @y @z ! @wy @/ @wx @/ @wy @wx @/ 1 @ 2 / @ 2 / @ 2 / þ þ ¼ þ þ @z @x @z @y @x @y @z Le @x2 @y2 @z2 ! 1 Nt @ 2 h @ 2 h @ 2 h þ þ þ Le Nb @x2 @y2 @z2
ð20Þ
ð21Þ
The corresponding boundary conditions for these equations are given by
@wx @/ @h ¼ wy ¼ 0; h ¼ 0:5; Nb þ Nt ¼ 0 on x ¼ 0 @x @x @x @wx @/ @h ¼ wy ¼ 0; h ¼ 0:5; Nb þ Nt ¼ 0 on x ¼ 1 @x @x @x @wy @h @/ wx ¼ ¼ 0; ¼ 0; ¼ 0 on y ¼ 0 and y ¼ A @y @y @y @h @/ wx ¼ wy ¼ 0; ¼ 0; ¼ 0 on z ¼ 0 and z ¼ 1 @z @z
ð22Þ
where key parameters are defined as
@w @w v ¼ x z ; @z @x
y @w x @w ¼ w @ x @y
ð12Þ
so that Eq. (6) is satisfied identically. We are then left with the following equations taking into account steady-state regime
l @ 2 w x
¼ qf 0 bð1 C 0 Þg
ð7Þ
A transformation of the formulated system of differential Eqs. (6)– (11) to a form eliminating direct search of the pressure field (Horne [30] and Sheremet [31,32]) is represented as the most reasonable because the aim of the present investigation is the analysis of a thermal state of the system in conditions of natural convection inside the cavity filled with fluid-saturated porous medium having nanoparticles. For this purpose we shall enter into consideration the vector potential. Moreover, Horne [30] has shown that by introducing a vector potential of the form V ¼ r into the formulation, the resulting equations may be solved numerically faster and more accurately than with the formulation using the primitive variables in Eqs. (1)–(5). It has been shown early (Aziz and Hellums [33], and Hirasaki and Hellums [34]) that the potential is also solenoidal since the velocity is solenoidal (incompressible flow), therefore r ¼ 0. The velocity components will be defined as follows:
V ¼r z @w y @w ¼ ) u ; @y @ z
!
ð14Þ ð3Þ
The flow is assumed to be slow so that an advective term and a Forchheimer quadratic term do not appear in the momentum equation. In keeping with the 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 [26,27]
l
y @2w y @2w þ þ 2 @ x2 @y @z2
!
¼ gðqp qf 0 Þ
@C @T qf 0 bð1 C 0 Þg @y @y ð13Þ
ðqp qf 0 ÞC 0 ; qf 0 bDTð1 C 0 Þ dDB C 0 dDT DT am Ly Nb ¼ ; Nt ¼ ; Le ¼ ; A¼ am am T c eDB L
Ra ¼
ð1 C 0 ÞgK qf 0 bDTL
am l
;
Nr ¼
ð23Þ
As regards the boundary conditions (22) it is worth pointing out that Nield and Kuznetsov [26,27] have 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
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no indication was given of how this could be done in practice. Thus, in the recently published papers by Kuznetsov and Nield [35], and Nield and Kuznetsov [36] they have replaced the boundary conditions by a set that are more realistic physically, assuming that there is no nanoparticle flux at the plate and that the particle fraction value there adjusts accordingly. The physical quantities of interest are the local Nusselt ðNul ; Nur Þ and Sherwood ðShl ; Shr Þ numbers, which are defined as
@h @h ; Nur ¼ ; @x x¼0 @x x¼1 @/ @/ ¼ ; Shr ¼ @x x¼0 @x x¼1
Nul ¼
Shl ð24Þ
and the average Nusselt ðNul ; Nur Þ and Sherwood ðShl ; Shr Þ numbers, which are given by
Nul ¼ Shl ¼
Z Z
1 0 1
0
Z Z
Z
1
Nul dydz;
Nur ¼
0 1
Shl dydz;
Shr ¼
0
Z 0
0 1
Z
1
Z
1
Nur dydz;
0 1
Shr dydz
ð25Þ
0
Table 1 Comparison of the average Nusselt number of the hot wall. Authors
Ra
Walker and Homsy [42] Bejan [43] Beckerman et al. [44] Gross et al. [45] Manole and Lage [46] Baytas and Pop [47] Present results
10
100
1000
10000
– – – – – 1.079 1.079
3.097 4.2 3.113 3.141 3.118 3.16 3.115
12.96 15.8 – 13.448 13.637 14.06 13.667
51.0 50.8 48.9 42.583 48.117 48.33 48.823
399
3. Numerical method The partial differential Eqs. (18)–(21) with corresponding boundary conditions (22) are solved by the finite difference method with the second order central differencing schemes using non-uniform mesh (Aleshkova and Sheremet [37], Sheremet and Trifonova [38], Sheremet and Pop [39,40], and Sheremet et al. [41]. The solution for the corresponding linear algebraic equations is obtained through the successive under relaxation method. Optimum value of the relaxation parameter is chosen on the basis of computing experiments. The computation is terminated when the residuals for the vector potential functions get bellow 106. The present model, in the form of an in-house computational fluid dynamics (CFD) code, has been validated successfully against the works of Walker and Homsy [42], Bejan [43], Beckerman et al. [44], Gross et al. [45], Manole and Lage [46], and Baytas and Pop [47] for steady-state natural convection in a square porous cavity with isothermal vertical and adiabatic horizontal walls. Table 1 shows the values of the average Nusselt number computed for various Rayleigh numbers in the range 10–104 in comparison with other authors. In case of 3D analysis, well-known benchmark is the 3D natural convection in a differentially heated cubical enclosure filled with fluid saturated porous medium (Sharma and Sharma [48], and Kramer et al. [49]). Figs. 2 and 3 show a good agreement between the obtained streamlines and temperature contour plots at middle cross-section and the results by Kramer et al. [49] for Ra = 1000, and different values of the Darcy number. Table 2 presents the average Nusselt number values for the cubical enclosure for Ra = 1000, e = 0.8, and different values of the Darcy number. The results are compared to the study of Sharma and Sharma [48], and Kramer et al. [49]. Very good agreement between these results can be observed in Table 2.
Fig. 2. Comparison of streamlines and isotherms for Da = 104: numerical results of Kramer et al. [49] – a, present study – b.
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For the purpose of obtaining grid independent solution, a grid sensitivity analysis is performed. The grid independent solution was performed by preparing the solution for steady-state free convection in a cubical porous cavity filled with a water based nanofluid at Ra = 500, Le = 1, Nr = 0.1, Nb = 0.1, Nt = 0.1, A = 1. Four cases of non-uniform grid are tested: a grid of 50 50 50 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:00684 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0437Þ, a grid of 80 80 80 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:00193 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0388Þ, a grid of 100 100 100 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:00087 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0376, and a much finer grid of 150 150 150 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:0002 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0328Þ. Table 3 shows an effect of the mesh on the average Nusselt number of the hot wall. On the basis of the conducted verifications the non-uniform grid of 100 100 100 points has been selected for the following analysis.
Table 2 Variations of the average Nusselt number with the Darcy number.
4. Results and discussion
vortex structure, reflecting an appearance of both upflows near the hot surface and downflows near the cold surface with transverse flows of small sizes from the adiabatic walls, is formed inside the three-dimensional enclosure. It should be noted that these transverse flows from vertical adiabatic walls collide with each other at central part of the cavity and are involved in the main circulation flow under an effect of horizontal temperature and concentration differences. At the same time the temperature field is a result of interaction of two thermal boundary layers formed at isothermal walls. An increase in Ra leads to both an essential intensification of convective flow inside the cavity and an extending of the flow cell along the x-axis with decreasing of the thermal boundary layers thickness. At the same time an essential modification of nanoparticle volume fraction field occurs
Numerical investigation of the boundary value problem (18)– (22) has been carried out at the following values of key parameters: Rayleigh number (Ra = 30–500), Lewis number (Le = 1–10), the buoyancy-ratio parameter (Nr = 0.1–0.4), the Brownian motion parameter (Nb = 0.1–0.4), the thermophoresis parameter (Nt = 0.1– 0.4) and the aspect ratio parameter (A = 0.2–5.0). Particular efforts have been focused on the effects of these key parameters on the fluid flow, heat and mass transfer characteristics. Fig. 4 shows three-dimensional velocity, temperature and nanoparticle volume fraction fields for nanofluid under different values of the Rayleigh number at A = 1.0, Le = 1.0, Nr = Nb = Nt = 0.1. Regardless of the Rayleigh number values the
1
Da = 10 Da = 102 Da = 103 Da = 104
Present results
Sharma and Sharma [48]
Kramer et al. [49]
1.854 3.755 6.820 10.838
– 3.99 6.95 10.14
1.855 3.770 6.922 10.558
Table 3 Variations of the average Nusselt number of the heat wall with the non-uniform grid. Non-uniform grids 50 50 50 80 80 80 100 100 100 150 150 150
Nul
D¼
9.453 8.958 8.898 8.861
5.87 0.67 – 0.42
jNul ij Nul 100100100 j
Fig. 3. Comparison of streamlines and isotherms for Da = 102: numerical results of Kramer et al. [49] – a, present study – b.
Nul ij
100%
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401
Fig. 4. Velocity, temperature and concentration fields at Le = 1.0, Nr = Nb = Nt = 0.1: Ra = 30 – a, Ra = 100 – b, Ra = 300 – c.
with the Rayleigh number. At Ra = 30 (Fig. 4a) the main heat transfer mechanism is a heat conduction therefore values of the vertical velocity are rather small and isotherms have small distortion. For large Rayleigh numbers (Ra = 300) there is a stratification of the flow and temperature. The third row of Fig. 4 shows the distributions of the nanoparticles volume fraction inside the cubical cavity. The green color inside the nanoparticles distributions frames characterizes those regions where the nanoparticles volume fraction values belong to the neighborhood of the value / = 1. When this green color region occupies a small or a negligible part of the volume, the distribution of nanoparticles has to be considered as nonhomogeneous (Celli [50]). On the other hand, when the green color region occupies most part of the cavity, the distribution of the nanoparticles can be considered as homogeneous. Since the conduction regime enhances the effect of the thermophoresis phenomenon, therefore the green color region appears only in a small central part of the volume of the enclosure and the nanoparticles distribution is highly non-homogeneous (Fig. 4a). An increase in the Rayleigh number leads to an expansion of the homogeneous area. Simultaneously one can find a decrease in the non-homogeneous zones and their locations are close to the bottom part of the hot wall and the upper part of the cold wall. Therefore it is worth noting here, that an increase in the Rayleigh number leads to an intensification of convective flow and as a result to more homogeneous distribution of nanoparticles. It should be noted that non-homogeneous areas reflect direction of convective heat transfer as heatlines defined by Kimura and Bejan
[51]. The main reason for such behaviour is the thermophoresis effect inside the thermal plume. Fig. 5 demonstrates a comparison of the streamlines, isotherms and isoconcentrations for three-dimensional (solid lines in xz-midplane) and two-dimensional (dashed lines) models at A = 1.0, Le = 1.0, Nr = Nb = Nt = 0.1 and different values of the Rayleigh number. It is worth noting here that for the considered range of the Rayleigh number there is no significant differences between local fields for these models. Main reason for such situation is the slow flows through porous medium having low permeability where linear Darcy law applies. Some differences between 3D and 2D data are observed in isoconcentrations (Fig. 5b and c). The main features of the effect of Ra on the fluid flow, heat and mass transfer are described above, Fig. 5 confirms all of these mentioned features. The effect of the Rayleigh number and the aspect ratio at Ra = 100 on the average Nusselt number at hot wall for Le = 1.0, Nr = Nb = Nt = 0.1 is presented in Fig. 6. An increase in Ra from 30 to 500 leads to an essential increase in Nul . Taking into account boundary conditions for the nanoparticles volume fraction (22) dependences of Shl on Ra are similar to Fig. 6a and b respectively. As for an effect of the aspect ratio (Fig. 6b) one can find here that due to weak convective heat transfer for Ra = 100 the average Nusselt number at the hot surface can be defined on the basis of the two-dimensional model for A P 0.2 while distributions of streamlines and isoconcentrations are different. Figs. 5b and 7 illustrate streamlines, isotherms and isoconcentrations at different values of the Lewis number Le = 1.0 and Le = 10.0 respectively. An increase in Le leads to more intensive
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Fig. 5. Streamlines w, isotherms h and isoconcentrations / at Le = 1.0, Nr = Nb = Nt = 0.1: Ra = 30 – a, Ra = 100 – b, Ra = 300 – c (solid lines – 3D case for cross-section y = 0.5; dashed lines – 2D case).
convective heat transfer inside the volume owing to high differences between heat and mass transfer. It should be noted that, with respect to Fig. 5b, low values of the Lewis number characterize non-homogeneous distributions of the nanoparticles inside the cavity. Also an increase in Lewis number leads to an essential decrease in the thickness of concentration boundary layers at vertical walls. It physically means that flow with large Lewis number prevent spreading the nanoparticles in the nanofluid and as a result one can find essential homogeneous distributions of nanoparticles volume fraction inside the cavity in comparison with the case for low values of the Lewis number (Fig. 5b). An increase in Nr leads to an attenuation of the convective flow inside the cavity while temperature and nanoparticles volume fraction do not change for the considered values of the key parameters. It is worth noting, that an increase in the buoyancy-ratio parameter leads to the insignificant decrease in Nul and Shl (e.g. an increase in Nr from 0.1 to 0.4 at A = 1, Ra = 100, Le = 1.0,
Nb = Nt = 0.1 leads to a reduction of the average Nusselt and Sherwood numbers up to 4%). In case of constant values for nanoparticles volume fraction at vertical isothermal walls [41] an increase in the buoyancy-ratio parameter from 0.1 to 0.4 leads to an essential decrease in the average Nusselt and Sherwood numbers up to 27.6%. Figs. 5b and 8 illustrate streamlines, isotherms and isoconcentrations at xz-mid-plane for different values of the Brownian motion parameter (Nb = 0.1 and Nb = 0.4). It can be seen from these figures that an increase in the Brownian motion parameter Nb leads to both an intensification of convective flow inside the cavity and more homogeneous distribution of nanoparticles. The abovementioned changes lead to an essential decrease in the average Sherwood number and insignificant increase in the average Nusselt number at hot surface (e.g. an increase in Nb from 0.1 to 0.4 at A = 1, Ra = 100, Le = 1.0, Nr = Nt = 0.1 leads to an increment in Nul up to 1% and a reduction of Shl up to 76%).
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403
Fig. 6. Dependences of average Nusselt number at the hot vertical surface on Rayleigh number (a) and on the aspect ratio parameter for Ra = 100 (b) at Le = 1.0, Nr = Nb = Nt = 0.1.
Fig. 7. Streamlines w, isotherms h and isoconcentrations / at Ra = 100, Le = 10.0, Nr = Nb = Nt = 0.1.
Fig. 8. Streamlines w, isotherms h and isoconcentrations / at Ra = 100, Le = 1.0, Nr = 0.1, Nb = 0.4, Nt = 0.1.
Figs. 5b and 9 show streamlines, isotherms and isoconcentrations at different values of the thermophoresis parameter. An increase in Nt leads to an attenuation of convective flow inside
the volume and essential non-homogeneous distribution of nanoparticles. It is worth noting here, that an increase in Nt leads to a significant increase in the average Sherwood number and
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Fig. 9. Streamlines w, isotherms h and isoconcentrations / at Ra = 100, Le = 1.0, Nr = 0.1, Nb = 0.1, Nt = 0.4.
insignificant decrease in the average Nusselt number (e.g. an increase in Nt from 0.1 to 0.4 at A = 1, Ra = 100, Le = 1.0, Nr = Nb = 0.1 leads to a reduction of Nul up to 11% and 3.6 times increment in Shl ). 5. Conclusions Free convection in a three-dimensional porous cavity with two isothermal vertical surfaces and adiabatic rest walls filled with nanofluid has been studied numerically. Particular efforts have been focused on the effects of the Rayleigh and Lewis numbers, buoyancy-ratio parameter, Brownian motion parameter, thermophoresis parameter and aspect ratio parameter on flow field, temperature and nanoparticles volume fraction distributions, average Nusselt number in comparison with data for two-dimensional model. It is found that the average Nusselt number at the hot surface is an increasing function of the Rayleigh number and Brownian motion parameter, and a decreasing function of the Lewis number, buoyancy-ratio and thermophoresis parameters. The average Sherwood number at the hot surface is an increasing function of the Rayleigh number and thermophoresis parameter, and a decreasing function of the Lewis number, buoyancy-ratio and Brownian motion parameters. It is also found that low Rayleigh and Lewis numbers, and high thermophoresis parameter reflect essential non-homogeneous distribution of nanoparticles inside the cavity. Therefore, for such range of Ra, Le and Nt a non-homogeneous model is more appropriate for the description of the system. Conflict of interest None declared. Acknowledgement This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA. ASME, FED 231/MD 66, 1995, pp. 99–105. [2] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707. [3] S.E.B. Maliga, S.M. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement using nanofluid in forced convection flow, Int. J. Heat Fluid Flow 26 (2005) 530–546.
[4] S. Kumar, S.K. Prasad, J. Banerjee, Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model, Appl. Math. Model. 34 (2010) 573–592. [5] H. Masuda, A. Ebata, K. Teramae, N. Hishimura, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu Bussei (Japan) 4 (1993) 227–233. [6] F. Sakai, W. Li, A. Nakayama, A rigorous derivation and its applications of volume averaged transport equations for heat transfer in nanofluids saturated metal foam, in: Proc. 15th International Heat Transfer Conference, IHTC-15, August 10–15, Kyoto, Japan, 2014, 5p. [7] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley, New Jersey, 2007. [8] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653. [9] D.A. Nield, A. Bejan, Convection in Porous Media, 4th ed., Springer, New York, 2013. [10] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [11] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [12] J.H. Lee, S.H. Lee, C.J. Choi, S.P. Jang, S.U.S. Choi, A review of thermal conductivity data, mechanics and models for nanofluids, Int. J. Micro-Nano Scale Transp. 1 (2010) 269–322. [13] J. Eagen, R. Rusconi, R. Piazza, S. Yip, The classical nature of thermal conduction in nanofluids, ASME J. Heat Transfer 132 (2010) 102402. 14 pages. [14] K.F.V. Wong, O.D. Leon, Applications of nanofluids: current and future, Adv. Mech. Eng. (2010) 519659. 11 pages. [15] J. Fan, L. Wang, Review of heat conduction in nanofluids, ASME J. Heat Transfer 133 (2011) 040801. 14 pages. [16] O. Mahian, A. Kianifar, S.A. Kalogirou, I. Pop, S. Wongwises, A review of the applications of nanofluids in solar energy, Int. J. Heat Mass Transfer 57 (2013) 582–594. [17] O. Mahian, C. Kleinstreuer, M.A. Al-Nimr, I. Pop, S. Wongwises, A review of entropy generation in nanofluid flow, Int. J. Heat Mass Transfer 65 (2013) 514– 532. [18] M. Mehrali, E. Sadeghinezhad, S.T. Latibari, M. Mehrali, H. Togun, M.N.M. Zubir, S.N. Kazi, H.S.C. Metselaar, Preparation, characterization, viscosity, and thermal conductivity of nitrogen-doped grapheme aqueous nanofluids, J. Mater. Sci. 49 (2014) 7156–7171. [19] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, vol. III, Elsevier, Oxford, 2005. [20] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001. [21] A. Nakayama, PC-Aided Numerical Heat Transfer and Convective Flow, CRC Press, Tokyo, 1995. [22] P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media, Springer, New York, 2008. [23] K. Vafai (Ed.), Handbook of Porous Media, second ed., Taylor and Francis, New York, 2005. [24] K. Vafai, Porous Media: Applications in Biological Systems and Biotechnology, CRC Press, Tokyo, 2010. [25] M.J.S. de Lemos, Turbulence in Porous Media: Modeling and Applications, second ed., Elsevier, Oxford, 2012. [26] 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. [27] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5796–5801. [28] V. Bianco, O. Manca, S. Nardini, Numerical simulation of water/Al2O3 nanofluid turbulent convection, Adv. Mech. Eng. (2010) 10.
M.A. Sheremet et al. / International Journal of Heat and Mass Transfer 82 (2015) 396–405 [29] V. Bianco, O. Manca, S. Nardini, Numerical investigation on nanofluids turbulent convection heat transfer inside a circular tube, Int. J. Therm. Sci. 50 (2011) 341–349. [30] R.N. Horne, Three-dimensional natural convection in a confined porous medium heated from below, J. Fluid Mech. 92 (1979) 751–766. [31] M.A. Sheremet, Laminar natural convection in an inclined cylindrical enclosure having finite thickness walls, Int. J. Heat Mass Transfer 55 (2012) 3582–3600. [32] M.A. Sheremet, Numerical simulation of 3D unsteady natural convection in a porous enclosure having finite thickness walls, in: Int. Conf. on Applications of Porous Media, vol. 5, 2013, pp. 385–393. [33] K. Aziz, J.D. Hellums, Numerical solution of three-dimensional equations of motion for laminar natural convection, Phys. Fluids 10 (1967) 314–324. [34] G.J. Hirasaki, J.D. Hellums, A general formulation of the boundary conditions on the vector potential in three dimensional hydrodynamics, Q. Appl. Math. 16 (1968) 331–342. [35] A.V. Kuznetsov, D.A. Nield, The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: a revised model, Int. J. Heat Mass Transfer 65 (2013) 682–685. [36] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid: a revised model, Int. J. Heat Mass Transfer 68 (2014) 211–214. [37] I.A. Aleshkova, M.A. Sheremet, Unsteady conjugate natural convection in a square enclosure filled with a porous medium, Int. J. Heat Mass Transfer 53 (2010) 5308–5320. [38] M.A. Sheremet, T.A. Trifonova, Unsteady conjugate natural convection in a vertical cylinder partially filled with a porous medium, Numer. Heat Transfer, Part A 64 (2013) 994–1015. [39] M.A. Sheremet, I. Pop, Conjugate natural convection in a square porous cavity filled by a nanofluid using Buongiorno’s mathematical model, Int. J. Heat Mass Transfer 79 (2014) 137–145.
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[40] M.A. Sheremet, I. Pop, Thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms, Transp. Porous Media 103 (2014) 191–205. [41] M.A. Sheremet, T. Grosan, I. Pop, Free convection in shallow and slender porous cavities filled by a nanofluid using Buongiorno’s model, ASME J. Heat Transfer 136 (2014) 082501. [42] K.L. Walker, G.M. Homsy, Convection in a porous cavity, J. Fluid Mech. 87 (1978) 338–363. [43] A. Bejan, On the boundary layer regime in a vertical enclosure filled with a porous medium, Lett. Heat Mass Transfer 6 (1979) 82–91. [44] C. Beckermann, R. Viskanta, S. Ramadhyani, A numerical study of non-Darcian natural convection in a vertical enclosure filled with a porous medium, Numer. Heat Transfer 10 (1986) 446–469. [45] R. Gross, M.R. Bear, C.E. Hickox, The application of flux-corrected transport (FCT) to high Rayleigh number natural convection in a porous medium, in: Proc. 7th International Heat Transfer Conference, San Francisco, CA, 1986. [46] D.M. Manole, J.L. Lage, Numerical benchmark results for natural convection in a porous medium cavity, ASME Conference on Heat Mass Transfer Porous Media, vol. 105, 1992, pp. 44–59. [47] A.C. Baytas, I. Pop, Free convection in oblique enclosures filled with a porous medium, Int. J. Heat Mass Transfer 42 (1999) 1047–1057. [48] R.V. Sharma, R.P. Sharma, Non-Darcy effects on three-dimensional natural convection in a porous box, in: Proceedings of International Heat Transfer Conference, NCV-10, 2006. [49] J. Kramer, J. Ravnik, R. Jecl, L. Skerget, Simulation of 3D flow in porous media by boundary element method, Eng. Anal. Boundary Elem. 35 (2011) 1256–1264. [50] M. Celli, Non-homogeneous model for a side heated square cavity filled with a nanofluid, Int. J. Heat Fluid Flow 44 (2013) 327–335. [51] S. Kimura, A. Bejan, The ‘‘heatline’’ visualization of convective heat transfer, ASME J. Heat Transfer 105 (1983) 916–919.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Three-dimensional natural convection in a porous enclosure filled with a nanofluid using Buongiorno’s mathematical model M.A. Sheremet a,b,⇑, I. Pop c, M.M. Rahman d a
Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Applied Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania d Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, P.C. 123 Al-Khod, Muscat, Oman b
a r t i c l e
i n f o
Article history: Received 12 October 2014 Received in revised form 19 November 2014 Accepted 19 November 2014
Keywords: Natural convection Three-dimensional cavity Porous medium Nanofluids Brownian motion Thermophoresis Numerical results
a b s t r a c t Steady-state natural convection heat transfer in a three-dimensional porous enclosure filled with a nanofluid using the mathematical nanofluid model proposed by Buongiorno is presented. The nanofluid model takes into account two important slip mechanisms in nanofluids like Brownian diffusion and thermophoresis. The study is formulated in terms of the dimensionless vector potential functions, temperature and concentration of nanoparticles. The governing equations were solved by finite difference method on nonuniform mesh and solution of algebraic equations was made on the basis of successive under relaxation method. Effort has been focused on the effects of six types of influential factors such as the Rayleigh and Lewis numbers, the buoyancy-ratio parameter, the Brownian motion parameter, the thermophoresis parameter and the aspect ratio on the fluid flow, heat and mass transfer. Three-dimensional velocity, temperature and nanoparticle volume fraction fields, average Nusselt numbers are presented. It is found that low Rayleigh and Lewis numbers and high thermophoresis parameter reflect essential non-homogeneous distribution of nanoparticles inside the cavity, hence a non-homogeneous model is more appropriate for the description of the system. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Nanofluid, a term introduced by Choi [1], represents a fluid in which nano-scale particles are suspended in a base fluid with a low thermal conductivity such as water, ethylene glycol, oils, etc. This is a two-phase mixture in which the solid phase consists of nano-sized particles. Since the size of the particles is less than 100 nm, nanofluids behave much more like a fluid than a mixture (Xuan and Roetzel [2], Maliga et al. [3]). Xuan and Roetzel [2] proposed a homogeneous flow model where the convective transport equations of pure fluids are directly extended to nanofluids. This means that all the traditional heat transfer correlations can be used for nanofluids provided the properties of pure fluids are replaced by those of nanofluids involving the volume fraction of the nanoparticles (Kumar et al. [4]). These homogeneous flow models are, however, in conflict with the experimental observations of Maliga et al. [3] who considered forced convection flow in a channel, as they under predict the heat transfer coefficients of nanofluids. ⇑ Corresponding author at: Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.066 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
Masuda et al. [5] experimentally found that the fluids containing thermally conducting submicron solid particles have excellent thermal conductivity. Nanofluids perform a crucial role in the development of newer technologies ideal for industrial purposes. Such fluids have been studied extensively in recent years, in view of their great potential as a higher-energy carrier due to their promising feature of high effective thermal conductivity (Sakai et al. [6]). Materials with sizes of nanometers possess unique physical and chemical properties (Das et al. [7]). They can flow smoothly through micro-channels without clogging because they are sufficiently small to behave similar to liquid molecules (Khanafer et al. [8]). It has been found that the presence of nanoparticles within the fluid can appreciably increase the effective thermal conductivity of the fluid and, as a consequence, enhance the heat transfer characteristics. An excellent collection of articles on this topic can be found in the book by Das et al. [7], and Nield and Bejan [9], and in the review papers by Buongiorno [10], Kakaç and Pramuanjaroenkij [11], Lee et al. [12], Eagen et al. [13], Wong and Leon [14], Fan and Wang [15], Mahian et al. [16,17], etc. In a very recently interesting paper by Mehrali et al. [18], new experimental results on the
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397
Nomenclature A C C0 Cp DB DT g jp K L Ly Le Nb Nr Nt Nu Nu p Ra Sh Sh t T T0 Tc
aspect ratio parameter nanoparticle volume fraction reference value of nanoparticle volume fraction heat capacity Brownian diffusion coefficient thermophoretic diffusion coefficient gravitational acceleration nanoparticles mass flux defined by jp ¼ qp ½DB rCþ ðDT =T c ÞrT permeability of the porous medium size of the cavity along x and z axes axis size of the cavity along y Lewis number Brownian motion parameter buoyancy-ratio parameter thermophoresis parameter local Nusselt number mean Nusselt number fluid pressure Rayleigh number for the porous medium local Sherwood number mean Sherwood number time fluid temperature mean temperature of heated and cooled walls defined by T 0 ¼ ðT h þ T c Þ=2 temperature of the cooled wall
thermal conductivity, viscosity, stability, and electrical conductivity at different concentrations of the Nitrogen-doped graphene (NDG) nanofluids. The feasibility of water-based NDG nanofluids for use as innovative heat transfer fluid in medium-temperature systems was demonstrated. Convective flow within porous media both with or without nanoparticles have a wide range of practical and engineering applications see, for example the books by Nield and Bejan [9], Ingham and Pop [19], Pop and Ingham [20], Nakayama [21], Vadasz [22], Vafai [23,24] and de Lemos [25]. In a series of pioneering papers, Nield and Kuznetsov [26,27] have used the mathematical nanofluid model proposed by Buongiorno [10] to study the problem of free convective boundary layer flow past a vertical flat plate embedded in a porous medium and the problem of thermal instability in a porous medium layer saturated by a nanofluid. The authors have 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. On the other hand, it is very important to explain how nanofluid flows is possible in a porous medium. This situation has been described in the paper by Sakai et al. [6]. It should be noticed at this end that Bianco et al. [28,29] found that the two-components mixture model is quite adequate for describing the nanofluid heat transfer, as supported by Buongiorno [10] using a magnitudes analysis. Most researchers neglected the spatial variations of thermophysical properties including Brownian and thermophoretic diffusion coefficients. Such analytical treatments could lead to substantial errors. Sakai et al. [6] have derived a macroscopic set of the governing equations for describing heat transfer in nanofluid saturated porous media using a volume average theory by considering all these variations. Here we study the steady natural convection heat transfer in a three-dimensional porous enclosure filled with a nanofluid using the mathematical nanofluid model proposed by Buongiorno [10]
Th ; v ; w u V x; y ; z x, y, z
temperature of the hot wall ; z, dimensional velocity components along the axes x; y respectively Darcy velocity vector dimensional Cartesian coordinates dimensionless Cartesian coordinates
Greek letters am effective thermal diffusivity of the porous medium b volumetric expansion coefficient of the fluid d parameter defined by d ¼ eðqC p Þp =ðqC p Þf e porosity of the porous medium / rescaled nanoparticle volume fraction l dynamic viscosity h dimensionless temperature qf fluid density qf0 reference fluid density qp nanoparticle mass density (qCp)f volumetric heat capacity of the base fluid (qCp)p effective volumetric heat capacity of the nanoparticle material (qCp)m effective volumetric heat capacity of the porous medium r parameter defined by r ¼ ðqC p Þm =ðqC p Þf dimensional vector potential x; w y; w z dimensional vector potential functions w wx ; wy ; wz dimensionless vector potential functions
in combination with Darcy’s law for the flow in the porous medium and the Boussinesq approximation for the buoyancy forces. 2. Basic equations Consider the steady free convection in a three-dimensional porous cavity filled with a water based nanofluid. It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. A schematic geometry of the and z problem under investigation is shown in Fig. 1, where x; y are the Cartesian coordinates and L and Ly are the sizes of the cavity. It is assumed that the vertical surface x ¼ 0 is heated and maintained at the constant temperature Th, while the opposite vertical wall x ¼ L is cooled and has the constant temperature Tc. The other walls are adiabatic ð@T=@~ nÞ ¼ 0, where T is the fluid temperature. The basic equations for the flow, heat transfer and nanoparticles can be written in the following form (Nield and Bejan [9])
Fig. 1. Physical model and coordinate system.
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rV ¼0 0 ¼ rp
l K
h
i V þ C qp þ ð1 CÞqf 0 ð1 bðT T c ÞÞ g
@T þ ðV rÞT ¼ am r2 T þ d½DB rC rT þ ðDT =T c ÞrT rT @t @C 1 qp þ ðV rÞC ¼ r jp @t e
r
ð1Þ
l @ 2 w y
ð2Þ
K
0 ¼ rp
K
h
i V þ Cðqp qf 0 Þ þ qf 0 ð1 bðT T c Þð1 C 0 ÞÞ g
ð4Þ
ð5Þ
Eqs. (1) and (3)–(5) for the problem under consideration can be ; z as written in dimensional Cartesian coordinates x; y
@ v @ w @u þ þ ¼0 @ z @ x @ y @p l ¼ u @ x K @p l ¼ v @y K i @p l h ¼ w Cðqp qf 0 Þ þ qf 0 ð1 bðT T c Þð1 C 0 ÞÞ g @z K ! @T @T @T @T @2T @2T @2T r þ u þ v þ w ¼ am þ þ 2 @z2 @t @x @y @z @ x2 @ y @C @T @C @T @C @T þ d DB þ þ @y @z @ z @ x @ x @ y " 2 2 2 #) DT @T @T @T þ þ þ @ x @y @z Tc ! @C 1 @C @C @C @2C @2C @2C þ v þ w ¼ DB þ u þ þ 2 @z2 @t e @ x @y @z @ x2 @ y ! 2 DT @ T @2T @2T þ 2þ 2 þ @ x2 @ y @z TC
ð6Þ
K
x @2w x @2w þ þ 2 @ x2 @y @z2
@T @C gðqp qf 0 Þ @ x @ x
y @T @ w x @T @ w y @w x @T @w þ þ @z @z @ x @z @ y @ x @y ! 2 2 2 @ T @ T @ T ¼ am þ 2þ 2 @ x2 @ y @ z @C @T @C @T @C @T þ þ þ d DB @y @z @z @ x @ x @ y " 2 2 2 #) DT @T @T @T þ þ þ @ x @y @ z Tc
y @C @ w x @C @ w y @w x @C @w þ þ @z e @ z @ x @z @ y @ x @y ! ! 2 2 2 @ C @ C @ C DT @2T @2T @2T þ 2þ 2 þ þ 2þ 2 ¼ DB @ x2 @ y @ z @ x2 @ y @ z TC
ð8Þ
x =am ; w ¼ w y =am ; =L; z ¼ z=L; wx ¼ w x ¼ x=L; y ¼ y y z =am ; h ¼ ðT T 0 Þ=ðT h T c Þ; / ¼ C=C 0 wz ¼ w
ð9Þ
ð10Þ
ð11Þ
ð15Þ
1
Introducing the following dimensionless variables
ð16Þ
ð17Þ
where T0 = (Th + Tc)/2 is the mean temperature of heated and cooled walls, and substituting (17) into Eqs. (13)–(16), we obtain
@ 2 wx @ 2 wx @ 2 wx @h @/ þ 2 þ 2 ¼ Ra þ Ra Nr ð18Þ @y @y @x2 @y @z 2 2 2 @ wy @ wy @ wy @h @/ þ 2 þ 2 ¼ Ra Ra Nr ð19Þ @x @x @x2 @y @z 2 2 2 @wy @h @wx @h @wy @wx @h @ h @ h @ h þ þ ¼ þ þ @z @x @z @y @x @y @z " @x2 @y2 @z2 2 2 2 #) @/ @h @/ @h @/ @h @h @h @h þ Nt þ Nb þ þ þ þ @x @x @y @y @z @z @x @y @z ! @wy @/ @wx @/ @wy @wx @/ 1 @ 2 / @ 2 / @ 2 / þ þ ¼ þ þ @z @x @z @y @x @y @z Le @x2 @y2 @z2 ! 1 Nt @ 2 h @ 2 h @ 2 h þ þ þ Le Nb @x2 @y2 @z2
ð20Þ
ð21Þ
The corresponding boundary conditions for these equations are given by
@wx @/ @h ¼ wy ¼ 0; h ¼ 0:5; Nb þ Nt ¼ 0 on x ¼ 0 @x @x @x @wx @/ @h ¼ wy ¼ 0; h ¼ 0:5; Nb þ Nt ¼ 0 on x ¼ 1 @x @x @x @wy @h @/ wx ¼ ¼ 0; ¼ 0; ¼ 0 on y ¼ 0 and y ¼ A @y @y @y @h @/ wx ¼ wy ¼ 0; ¼ 0; ¼ 0 on z ¼ 0 and z ¼ 1 @z @z
ð22Þ
where key parameters are defined as
@w @w v ¼ x z ; @z @x
y @w x @w ¼ w @ x @y
ð12Þ
so that Eq. (6) is satisfied identically. We are then left with the following equations taking into account steady-state regime
l @ 2 w x
¼ qf 0 bð1 C 0 Þg
ð7Þ
A transformation of the formulated system of differential Eqs. (6)– (11) to a form eliminating direct search of the pressure field (Horne [30] and Sheremet [31,32]) is represented as the most reasonable because the aim of the present investigation is the analysis of a thermal state of the system in conditions of natural convection inside the cavity filled with fluid-saturated porous medium having nanoparticles. For this purpose we shall enter into consideration the vector potential. Moreover, Horne [30] has shown that by introducing a vector potential of the form V ¼ r into the formulation, the resulting equations may be solved numerically faster and more accurately than with the formulation using the primitive variables in Eqs. (1)–(5). It has been shown early (Aziz and Hellums [33], and Hirasaki and Hellums [34]) that the potential is also solenoidal since the velocity is solenoidal (incompressible flow), therefore r ¼ 0. The velocity components will be defined as follows:
V ¼r z @w y @w ¼ ) u ; @y @ z
!
ð14Þ ð3Þ
The flow is assumed to be slow so that an advective term and a Forchheimer quadratic term do not appear in the momentum equation. In keeping with the 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 [26,27]
l
y @2w y @2w þ þ 2 @ x2 @y @z2
!
¼ gðqp qf 0 Þ
@C @T qf 0 bð1 C 0 Þg @y @y ð13Þ
ðqp qf 0 ÞC 0 ; qf 0 bDTð1 C 0 Þ dDB C 0 dDT DT am Ly Nb ¼ ; Nt ¼ ; Le ¼ ; A¼ am am T c eDB L
Ra ¼
ð1 C 0 ÞgK qf 0 bDTL
am l
;
Nr ¼
ð23Þ
As regards the boundary conditions (22) it is worth pointing out that Nield and Kuznetsov [26,27] have 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
M.A. Sheremet et al. / International Journal of Heat and Mass Transfer 82 (2015) 396–405
no indication was given of how this could be done in practice. Thus, in the recently published papers by Kuznetsov and Nield [35], and Nield and Kuznetsov [36] they have replaced the boundary conditions by a set that are more realistic physically, assuming that there is no nanoparticle flux at the plate and that the particle fraction value there adjusts accordingly. The physical quantities of interest are the local Nusselt ðNul ; Nur Þ and Sherwood ðShl ; Shr Þ numbers, which are defined as
@h @h ; Nur ¼ ; @x x¼0 @x x¼1 @/ @/ ¼ ; Shr ¼ @x x¼0 @x x¼1
Nul ¼
Shl ð24Þ
and the average Nusselt ðNul ; Nur Þ and Sherwood ðShl ; Shr Þ numbers, which are given by
Nul ¼ Shl ¼
Z Z
1 0 1
0
Z Z
Z
1
Nul dydz;
Nur ¼
0 1
Shl dydz;
Shr ¼
0
Z 0
0 1
Z
1
Z
1
Nur dydz;
0 1
Shr dydz
ð25Þ
0
Table 1 Comparison of the average Nusselt number of the hot wall. Authors
Ra
Walker and Homsy [42] Bejan [43] Beckerman et al. [44] Gross et al. [45] Manole and Lage [46] Baytas and Pop [47] Present results
10
100
1000
10000
– – – – – 1.079 1.079
3.097 4.2 3.113 3.141 3.118 3.16 3.115
12.96 15.8 – 13.448 13.637 14.06 13.667
51.0 50.8 48.9 42.583 48.117 48.33 48.823
399
3. Numerical method The partial differential Eqs. (18)–(21) with corresponding boundary conditions (22) are solved by the finite difference method with the second order central differencing schemes using non-uniform mesh (Aleshkova and Sheremet [37], Sheremet and Trifonova [38], Sheremet and Pop [39,40], and Sheremet et al. [41]. The solution for the corresponding linear algebraic equations is obtained through the successive under relaxation method. Optimum value of the relaxation parameter is chosen on the basis of computing experiments. The computation is terminated when the residuals for the vector potential functions get bellow 106. The present model, in the form of an in-house computational fluid dynamics (CFD) code, has been validated successfully against the works of Walker and Homsy [42], Bejan [43], Beckerman et al. [44], Gross et al. [45], Manole and Lage [46], and Baytas and Pop [47] for steady-state natural convection in a square porous cavity with isothermal vertical and adiabatic horizontal walls. Table 1 shows the values of the average Nusselt number computed for various Rayleigh numbers in the range 10–104 in comparison with other authors. In case of 3D analysis, well-known benchmark is the 3D natural convection in a differentially heated cubical enclosure filled with fluid saturated porous medium (Sharma and Sharma [48], and Kramer et al. [49]). Figs. 2 and 3 show a good agreement between the obtained streamlines and temperature contour plots at middle cross-section and the results by Kramer et al. [49] for Ra = 1000, and different values of the Darcy number. Table 2 presents the average Nusselt number values for the cubical enclosure for Ra = 1000, e = 0.8, and different values of the Darcy number. The results are compared to the study of Sharma and Sharma [48], and Kramer et al. [49]. Very good agreement between these results can be observed in Table 2.
Fig. 2. Comparison of streamlines and isotherms for Da = 104: numerical results of Kramer et al. [49] – a, present study – b.
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For the purpose of obtaining grid independent solution, a grid sensitivity analysis is performed. The grid independent solution was performed by preparing the solution for steady-state free convection in a cubical porous cavity filled with a water based nanofluid at Ra = 500, Le = 1, Nr = 0.1, Nb = 0.1, Nt = 0.1, A = 1. Four cases of non-uniform grid are tested: a grid of 50 50 50 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:00684 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0437Þ, a grid of 80 80 80 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:00193 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0388Þ, a grid of 100 100 100 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:00087 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0376, and a much finer grid of 150 150 150 points ðDxmin ¼ Dymin ¼ Dzmin ¼ 0:0002 and Dxmax ¼ Dymax ¼ Dzmax ¼ 0:0328Þ. Table 3 shows an effect of the mesh on the average Nusselt number of the hot wall. On the basis of the conducted verifications the non-uniform grid of 100 100 100 points has been selected for the following analysis.
Table 2 Variations of the average Nusselt number with the Darcy number.
4. Results and discussion
vortex structure, reflecting an appearance of both upflows near the hot surface and downflows near the cold surface with transverse flows of small sizes from the adiabatic walls, is formed inside the three-dimensional enclosure. It should be noted that these transverse flows from vertical adiabatic walls collide with each other at central part of the cavity and are involved in the main circulation flow under an effect of horizontal temperature and concentration differences. At the same time the temperature field is a result of interaction of two thermal boundary layers formed at isothermal walls. An increase in Ra leads to both an essential intensification of convective flow inside the cavity and an extending of the flow cell along the x-axis with decreasing of the thermal boundary layers thickness. At the same time an essential modification of nanoparticle volume fraction field occurs
Numerical investigation of the boundary value problem (18)– (22) has been carried out at the following values of key parameters: Rayleigh number (Ra = 30–500), Lewis number (Le = 1–10), the buoyancy-ratio parameter (Nr = 0.1–0.4), the Brownian motion parameter (Nb = 0.1–0.4), the thermophoresis parameter (Nt = 0.1– 0.4) and the aspect ratio parameter (A = 0.2–5.0). Particular efforts have been focused on the effects of these key parameters on the fluid flow, heat and mass transfer characteristics. Fig. 4 shows three-dimensional velocity, temperature and nanoparticle volume fraction fields for nanofluid under different values of the Rayleigh number at A = 1.0, Le = 1.0, Nr = Nb = Nt = 0.1. Regardless of the Rayleigh number values the
1
Da = 10 Da = 102 Da = 103 Da = 104
Present results
Sharma and Sharma [48]
Kramer et al. [49]
1.854 3.755 6.820 10.838
– 3.99 6.95 10.14
1.855 3.770 6.922 10.558
Table 3 Variations of the average Nusselt number of the heat wall with the non-uniform grid. Non-uniform grids 50 50 50 80 80 80 100 100 100 150 150 150
Nul
D¼
9.453 8.958 8.898 8.861
5.87 0.67 – 0.42
jNul ij Nul 100100100 j
Fig. 3. Comparison of streamlines and isotherms for Da = 102: numerical results of Kramer et al. [49] – a, present study – b.
Nul ij
100%
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Fig. 4. Velocity, temperature and concentration fields at Le = 1.0, Nr = Nb = Nt = 0.1: Ra = 30 – a, Ra = 100 – b, Ra = 300 – c.
with the Rayleigh number. At Ra = 30 (Fig. 4a) the main heat transfer mechanism is a heat conduction therefore values of the vertical velocity are rather small and isotherms have small distortion. For large Rayleigh numbers (Ra = 300) there is a stratification of the flow and temperature. The third row of Fig. 4 shows the distributions of the nanoparticles volume fraction inside the cubical cavity. The green color inside the nanoparticles distributions frames characterizes those regions where the nanoparticles volume fraction values belong to the neighborhood of the value / = 1. When this green color region occupies a small or a negligible part of the volume, the distribution of nanoparticles has to be considered as nonhomogeneous (Celli [50]). On the other hand, when the green color region occupies most part of the cavity, the distribution of the nanoparticles can be considered as homogeneous. Since the conduction regime enhances the effect of the thermophoresis phenomenon, therefore the green color region appears only in a small central part of the volume of the enclosure and the nanoparticles distribution is highly non-homogeneous (Fig. 4a). An increase in the Rayleigh number leads to an expansion of the homogeneous area. Simultaneously one can find a decrease in the non-homogeneous zones and their locations are close to the bottom part of the hot wall and the upper part of the cold wall. Therefore it is worth noting here, that an increase in the Rayleigh number leads to an intensification of convective flow and as a result to more homogeneous distribution of nanoparticles. It should be noted that non-homogeneous areas reflect direction of convective heat transfer as heatlines defined by Kimura and Bejan
[51]. The main reason for such behaviour is the thermophoresis effect inside the thermal plume. Fig. 5 demonstrates a comparison of the streamlines, isotherms and isoconcentrations for three-dimensional (solid lines in xz-midplane) and two-dimensional (dashed lines) models at A = 1.0, Le = 1.0, Nr = Nb = Nt = 0.1 and different values of the Rayleigh number. It is worth noting here that for the considered range of the Rayleigh number there is no significant differences between local fields for these models. Main reason for such situation is the slow flows through porous medium having low permeability where linear Darcy law applies. Some differences between 3D and 2D data are observed in isoconcentrations (Fig. 5b and c). The main features of the effect of Ra on the fluid flow, heat and mass transfer are described above, Fig. 5 confirms all of these mentioned features. The effect of the Rayleigh number and the aspect ratio at Ra = 100 on the average Nusselt number at hot wall for Le = 1.0, Nr = Nb = Nt = 0.1 is presented in Fig. 6. An increase in Ra from 30 to 500 leads to an essential increase in Nul . Taking into account boundary conditions for the nanoparticles volume fraction (22) dependences of Shl on Ra are similar to Fig. 6a and b respectively. As for an effect of the aspect ratio (Fig. 6b) one can find here that due to weak convective heat transfer for Ra = 100 the average Nusselt number at the hot surface can be defined on the basis of the two-dimensional model for A P 0.2 while distributions of streamlines and isoconcentrations are different. Figs. 5b and 7 illustrate streamlines, isotherms and isoconcentrations at different values of the Lewis number Le = 1.0 and Le = 10.0 respectively. An increase in Le leads to more intensive
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Fig. 5. Streamlines w, isotherms h and isoconcentrations / at Le = 1.0, Nr = Nb = Nt = 0.1: Ra = 30 – a, Ra = 100 – b, Ra = 300 – c (solid lines – 3D case for cross-section y = 0.5; dashed lines – 2D case).
convective heat transfer inside the volume owing to high differences between heat and mass transfer. It should be noted that, with respect to Fig. 5b, low values of the Lewis number characterize non-homogeneous distributions of the nanoparticles inside the cavity. Also an increase in Lewis number leads to an essential decrease in the thickness of concentration boundary layers at vertical walls. It physically means that flow with large Lewis number prevent spreading the nanoparticles in the nanofluid and as a result one can find essential homogeneous distributions of nanoparticles volume fraction inside the cavity in comparison with the case for low values of the Lewis number (Fig. 5b). An increase in Nr leads to an attenuation of the convective flow inside the cavity while temperature and nanoparticles volume fraction do not change for the considered values of the key parameters. It is worth noting, that an increase in the buoyancy-ratio parameter leads to the insignificant decrease in Nul and Shl (e.g. an increase in Nr from 0.1 to 0.4 at A = 1, Ra = 100, Le = 1.0,
Nb = Nt = 0.1 leads to a reduction of the average Nusselt and Sherwood numbers up to 4%). In case of constant values for nanoparticles volume fraction at vertical isothermal walls [41] an increase in the buoyancy-ratio parameter from 0.1 to 0.4 leads to an essential decrease in the average Nusselt and Sherwood numbers up to 27.6%. Figs. 5b and 8 illustrate streamlines, isotherms and isoconcentrations at xz-mid-plane for different values of the Brownian motion parameter (Nb = 0.1 and Nb = 0.4). It can be seen from these figures that an increase in the Brownian motion parameter Nb leads to both an intensification of convective flow inside the cavity and more homogeneous distribution of nanoparticles. The abovementioned changes lead to an essential decrease in the average Sherwood number and insignificant increase in the average Nusselt number at hot surface (e.g. an increase in Nb from 0.1 to 0.4 at A = 1, Ra = 100, Le = 1.0, Nr = Nt = 0.1 leads to an increment in Nul up to 1% and a reduction of Shl up to 76%).
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Fig. 6. Dependences of average Nusselt number at the hot vertical surface on Rayleigh number (a) and on the aspect ratio parameter for Ra = 100 (b) at Le = 1.0, Nr = Nb = Nt = 0.1.
Fig. 7. Streamlines w, isotherms h and isoconcentrations / at Ra = 100, Le = 10.0, Nr = Nb = Nt = 0.1.
Fig. 8. Streamlines w, isotherms h and isoconcentrations / at Ra = 100, Le = 1.0, Nr = 0.1, Nb = 0.4, Nt = 0.1.
Figs. 5b and 9 show streamlines, isotherms and isoconcentrations at different values of the thermophoresis parameter. An increase in Nt leads to an attenuation of convective flow inside
the volume and essential non-homogeneous distribution of nanoparticles. It is worth noting here, that an increase in Nt leads to a significant increase in the average Sherwood number and
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Fig. 9. Streamlines w, isotherms h and isoconcentrations / at Ra = 100, Le = 1.0, Nr = 0.1, Nb = 0.1, Nt = 0.4.
insignificant decrease in the average Nusselt number (e.g. an increase in Nt from 0.1 to 0.4 at A = 1, Ra = 100, Le = 1.0, Nr = Nb = 0.1 leads to a reduction of Nul up to 11% and 3.6 times increment in Shl ). 5. Conclusions Free convection in a three-dimensional porous cavity with two isothermal vertical surfaces and adiabatic rest walls filled with nanofluid has been studied numerically. Particular efforts have been focused on the effects of the Rayleigh and Lewis numbers, buoyancy-ratio parameter, Brownian motion parameter, thermophoresis parameter and aspect ratio parameter on flow field, temperature and nanoparticles volume fraction distributions, average Nusselt number in comparison with data for two-dimensional model. It is found that the average Nusselt number at the hot surface is an increasing function of the Rayleigh number and Brownian motion parameter, and a decreasing function of the Lewis number, buoyancy-ratio and thermophoresis parameters. The average Sherwood number at the hot surface is an increasing function of the Rayleigh number and thermophoresis parameter, and a decreasing function of the Lewis number, buoyancy-ratio and Brownian motion parameters. It is also found that low Rayleigh and Lewis numbers, and high thermophoresis parameter reflect essential non-homogeneous distribution of nanoparticles inside the cavity. Therefore, for such range of Ra, Le and Nt a non-homogeneous model is more appropriate for the description of the system. Conflict of interest None declared. Acknowledgement This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA. ASME, FED 231/MD 66, 1995, pp. 99–105. [2] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707. [3] S.E.B. Maliga, S.M. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement using nanofluid in forced convection flow, Int. J. Heat Fluid Flow 26 (2005) 530–546.
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