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Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center
Superlattices and Microstructures 52 (2012) 261–275
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Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center Mostafa Mahmoodi a, Saeed Mazrouei Sebdani b,⇑ a b
Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran Department of Mechanical Engineering, University of Kashan, Kashan, Iran
a r t i c l e
i n f o
Article history: Received 23 March 2012 Accepted 7 May 2012 Available online 15 May 2012 Keywords: Finite volume method Free convection Nanofluid Square cavity Square body
a b s t r a c t The problem of free convection fluid flow and heat transfer of Cu– water nanofluid inside a square cavity having adiabatic square bodies at its center has been investigated numerically. The governing equations have been discretized using the finite volume method. The SIMPLER algorithm was employed to couple velocity and pressure fields. Using the developed code, a parametric study was conducted and the effects of pertinent parameters such as Rayleigh number, size of the adiabatic square body, and volume fraction of the Cu nanoparticles on the fluid flow and thermal fields and heat transfer inside the cavity were investigated. The obtained results show that for all Rayleigh numbers with the exception of Ra = 104 the average Nusselt number increases with increase in the volume fraction of the nanoparticles. At Ra = 104 the average Nusselt number is a decreasing function of the nanoparticles volume fraction. Moreover at low Rayleigh numbers (103 and 104) the rate of heat transfer decreases when the size of the adiabatic square body increases while at high Rayleigh numbers (105 and 106) it increases. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In recent years nanofluids, which are a suspension of nano-sized solid particles in a base fluid, with thermal conductivity higher than the based fluid, are used to enhance the rate of heat transfer in many practical engineering applications [1]. Free convection heat transfer inside nanofluid filled rectangular cavities with different boundary conditions on the side walls has been studied by many ⇑ Corresponding author. E-mail address: [email protected] (S.M. Sebdani). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.05.007
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Nomenclature cp g h H k l Nu p P Pr q Ra T u,v U,V x,y X,Y
specific heat, J kg1 K1 gravitational acceleration, m s2 heat transfer coefficient, W m2 K enclosure height, m thermal conductivity, W m1 K1 width of the adiabatic square body, m Nusselt number pressure, N m2 dimensionless pressure Prandtl number heat flux, W m2 Rayleigh number dimensional temperature, K dimensional velocities components in x and y direction, m s1 dimensionless velocities components in X and Y direction dimensional Cartesian coordinates, m dimensionless Cartesian coordinates
Greek symbols a thermal diffusivity, m2 s b thermal expansion coefficient, K1 h dimensionless temperature l dynamic viscosity, kg m1 s m kinematic viscosity, m2 s q density, kg m3 u volume fraction of the nanoparticles Subscripts c cold f fluid h hot nf nanofluid s solid particles w wall
researchers. Khanafer et al. [2] conducted a numerical study on free convection inside nanofluid filled rectangular cavities with cold right wall, hot left wall and insulated horizontal walls. Their results showed that rate of heat transfer increased with increase in nanoparticles volume fraction for entire range of Grashof number considered. Similar results were found in work of Jou and Tzeng [3] on numerical study of free convection in differentially heated rectangular cavities filled with a nanofluid. Santra et al. [4] studied free convection of Cu–water nanofluid in a differentially heated square cavity with consideration of Ostwald–de Waele non-Newtonian behavior of the nanofluid. They found that heat transfer decreased with increase in the nanoparticles volume fraction for a particular Rayleigh number. Effects due to uncertainties in effective dynamic viscosity and thermal conductivity of alumina-water nanofluid on free convection heat transfer in a differentially heated square cavity were investigated by Ho et al. [5]. Their results demonstrated that heat transfer across the cavity can be found to be enhanced or mitigated with respect to the base fluid via the used dynamic viscosity formula. Oztop and Abu-nada [6] carried out a numerical study on free convection of nanofluid in partially heated rectangular cavities. The cavities had a cold vertical wall, a localized heater on the other vertical wall and insulated horizontal walls. They considered effects of Rayleigh number,
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aspect ratio of cavities, size and location of the heater and different types of water based nanofluids. Their results showed an increase in average Nusselt number with increase in the volume fraction of the nanoparticles and increase in the height of heater. Numerical study on free convection of nanofluid in a square cavity cooled from two vertical and top horizontal walls and heated by a constant heat flux heater on its horizontal bottom wall has been done by Aminossadati and Ghasemi [7]. They investigated the effects of Rayleigh number, nanoparticles volume fraction, size and location of heater and type of nanofluid. The obtained results indicated that type of nanoparticles and the length and location of the heat source affected significantly the heat source maximum temperature. Abunada and Oztop [8] investigated effect of inclination angle of a square cavity on free convection of the Cu–water nanofluid inside it. They observed that the inclination angle can be used as a control parameter for fluid flow and heat transfer inside the cavity. Moreover their results showed that effects of inclination angle on percentage of heat transfer enhancement became insignificant at low Rayleigh number. Free convection heat transfer of water-based nanofluids in an inclined square cavity with right cold wall, a constant heat flux heater at the center of its left wall and adiabatic other sides, was studied numerically by Ogut [9]. He investigated effects of inclination angle of the cavity, solid volume fractions, length of the constant heat flux heater, and the Rayleigh number on flow and temperature field inside the cavity. The obtained results showed that the average heat transfer rate increased significantly as nanoparticle volume fraction and Rayleigh number increased. Moreover the results showed that the length of the heater was an important parameter affecting the flow and temperature fields. Numerical investigation of periodic free convection fluid flow and heat transfer inside a nanofluid filled square cavity was done by Ghasemi and Aminossadati [10]. They considered a square cavity with insulated top and bottom walls, cold right vertical wall and a heater with oscillating heat flux on its left vertical wall. A periodic behavior of the flow and temperature fields was observed as a result of the oscillating heat flux. Moreover it was found that optimum position of the heat source was a function of Rayleigh number. Kumar et al. [11] analyzed free convection of nanofluid in a differentially heated square cavity using single phase thermal dispersion model. Simulations incorporating the thermal dispersion model showed increment in local thermal conductivity at locations with maximum velocity. Moreover their results showed that the average Nusselt number increased with the solid volume fraction. In a numerical study, Lin and Violi [12] analyzed free convection of Al2O3-water nanofluid in a differentially heated cavity with consideration of slip mechanism in nanofluids. They investigated effects of pertinent parameters such as non-uniform nanoparticle size, mean nanoparticle diameter, nanoparticle volume fraction, Prandtl number, and Grashof number on flow field and temperature distribution inside the cavity. They found enhanced and mitigated heat transfer due to the presence of nanoparticles. Sheikhzadeh et al. [13] conducted a numerical simulation to study free convection of Cu–water nanofluid inside a square cavity with partially thermally active side walls. The active parts of the left and the right side walls of the cavity were maintained hot and cold respectively while the cavity’s top and bottom walls as well as the inactive parts of the side walls were kept insulated. They found that maximum average Nusselt number for high Rayleigh numbers occurred when the hot and cold parts were located in the bottom and middle region of the vertical walls respectively. Saleh et al. [14] investigated numerically free convection of nanofluid in a trapezoidal enclosure numerically and developed a correlation for the average Nusselt number as a function of the angle of the sloping wall, effective thermal conductivity and viscosity as well as Grashof number. Sheikhzadeh et al. [15] conducted a numerical simulation to investigate the problem of free convection of the TiO2-water nanofluid in rectangular cavities differentially heated on adjacent walls. The left and the top walls of the cavities were heated and cooled respectively; while, the cavities right and bottom walls were kept insulated. They found that by increase in the volume fraction of the nanoparticles, mean Nusselt number of the hot wall increased for the shallow cavities; while, a reverse trend occurred for tall cavities. Very recently numerical results of a study on free convection in a nanofluid filled square cavity by using heating and cooling by sinusoidal temperature profiles on one side were reported by Oztop et al. [16]. They considered effects of various inclination angles of the cavity, different types of water based nanofluids, volume fraction of nanoparticles, and the Rayleigh number on the heat transfer rate. They found that addition of nanoparticle into the water affected the fluid flow and temperature distribution especially for higher Rayleigh numbers.
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Apart from application of nanofluids in buoyancy driven heat transfer, free convection in cavities having an inside body has received considerable attention in the recent years due to its practical engineering applications, such as solar collectors, thermal insulation, cooling of electronic components and designing building [17]. Mezrhab et al. [18] investigated radiation-natural convection interactions in a differentially heated cavity with an inner body numerically. They found that the radiation exchange homogenized the temperature inside the cavity and produced an increase in the average Nusselt number. Free convection heat transfer between inner hot sphere and outer vertically eccentric cold cylinder was investigated by Chen [19] numerically. Yu et al. [20] conducted a numerical simulation to study of transient free convective heat transfer of a liquid gallium inside a horizontal circular cylinder with an inner coaxial triangular cylinder. Their results showed that the time averaged Nusselt number was apparently increased by horizontally placing the top side of the inner triangular cylinder for Grashof numbers greater than 105. Based on literature reviews, despite a large number of numerical studies on free convection of nanofluids inside rectangular cavities with different boundary conditions, there is no study on free convection in square cavities with an inside adiabatic body. This problem may be occurred in a number of technical applications such as solar collectors, heat exchangers, and cooling of electronic equipment and chips using nanofluids. The adiabatic body can be considered as a model of heat transfer controller or modifier device. In a heat exchanger an adiabatic body can be a model of baffle which manages the flow rate and heat transport process. In view point of cooling of electronic chips an adiabatic body can be used to increase or decrease of heat transfer from special part of the hot chip. In the present paper the problem of free convection heat transfer and fluid flow in a square cavity containing nanofluid and with an adiabatic square body at its center, is investigated using the finite volume method and SIMPLER algorithm. The results in the form of streamlines and isotherms plots, average Nusselt number and local Nusselt number are presented for a wide range of Rayleigh numbers, size of the adiabatic square body and volume fraction of the nanoparticles. 2. Mathematical modeling A schematic view of the square cavity with an adiabatic square body at its center is shown in Fig. 1. The height and the width of the cavity are denoted by H. An adiabatic square body with the length of l is located at the center of the square cavity. Aspect ratio (dimensionless size of the adiabatic square body) is defined as AR = l/W. The left wall is kept at high temperature Th, while, the right wall is kept at cold temperature Tc. The horizontal top and bottom walls of the cavity are kept insulated. The length of the cavity perpendicular to its plane is assumed to be long enough; hence, the problem is
y
l
H
Th
Tc
x H Fig. 1. A schematic view of the considered cavity in the present study.
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considered two dimensional. The cavity is filled with Cu–water nanofluid. It is assumed that the nanoparticles and the base fluid are in thermal equilibrium and there is no slip between them. The thermophysical properties of the Cu nanoparticles and the water as the base fluid are listed in Table 1. The nanofluid is considered Newtonian, laminar, and incompressible. The thermophysical properties of the nanofluid are considered constant with the exception of the density which varies according to the Boussinesq approximation [21]. The steady state continuity, momentum and energy equations for two-dimensional laminar free convection fluid flow and heat transfer with the Boussinesq approximation in y-direction are as following:
@u @ v þ ¼0 @x @y
ð1Þ
@u @u 1 @p lnf þv ¼ þ u @x @y qnf @x qnf
! @2u @2u þ @x2 @y2
@v @v 1 @p lnf u þ þv ¼ @x @y qnf @y qnf
@2v @2v þ @x2 @y2
u
@T @T @2T @2T þ þv ¼ anf @x @y @x2 @y2
ð2Þ
! þ
ðqbÞnf
qnf
gðT T c Þ
ð3Þ
! ð4Þ
where the density, heat capacity, thermal expansion coefficient, and thermal diffusivity of the nanofluid are as follow, respectively, [2]
qnf ¼ ð1 uÞqf þ uqs
ð5Þ
ðqC p Þnf ¼ ð1 uÞðqC p Þf þ uðqC p Þs
ð6Þ
ðqbÞnf ¼ ð1 uÞðqbÞf þ uðqbÞs
ð7Þ
anf ¼
knf ðqC p Þnf
ð8Þ
To estimation of the dynamic viscosity of the nanofluid the Brinkman model [22] is employed.
leff ¼
lf
ð9Þ
ð1 uÞ2:5
This formula has been used to calculation of the dynamic viscosity of nanofluid in numerical simulation of free convection in recently published articles [2,3,6–10,13,14,16]. The effective thermal conductivity of the nanofluid is determined according to Maxwell [23].
knf ðks þ 2kf Þ 2uðkf ks Þ ¼ kf ðkf þ 2ks Þ þ uðkf ks Þ
ð10Þ
Table 1 Thermo-physical properties of water and solid nanoparticles [6]. Physical properties
Fluid phase
Cu
CP (J/kg k) q (kg/m3) k (W/m k) b (K1)
4179 997.1 0.613 21 105
385 8933 401 1.67 105
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This relation has been used recently in literature related to numerical simulation of free convection of nanofluid [2,3,6–10,13,14,16]. Using the following dimensionless parameters the governing equation can be converted to dimensionless form.
X¼
x y uH vH pH2 T Tc ; Y¼ ; U¼ ; V¼ ; P¼ ; h¼ H H af af Th Tc qnf a2f
ð11Þ
The dimensionless forms of the governing equations are
@U @V þ ¼0 @X @Y
ð12Þ
lnf @U @U @P þV ¼ þ U @X @Y @X qnf af
@2U
U
lnf @V @V @P þV ¼ þ @X @Y @Y qnf af
@2V
U
@h @h anf þV ¼ @X @Y af
@X 2
@X
@2h
@2h
@X
@Y 2
þ 2
2
þ
þ
@2U
! ð13Þ
@Y 2 @2V @Y
2
! þ
ðqbÞnf Ra Pr h qnf bf
ð14Þ
! ð15Þ
where the Rayleigh number Ra, and the Prandtl number Pr are
Ra ¼
gbf ðT h T c Þ H3
af mf
; Pr ¼
mf af
ð16Þ
The Prandtl number of water is Pr = 6.8. The boundary conditions for Eqs. (12)–(15) are
8 > < on the left wall U ¼ V ¼ 0; h ¼ 1 on the right wall U ¼ V ¼ 0; h ¼ 0 > : on the adiabatic body : U ¼ V ¼ 0; @h=@n ¼ 0
ð17Þ
Where n is normal direction to the walls. The local Nusselt number of the heat source is expressed as
Nulocal ¼
hH kf
ð18Þ
where the heat transfer coefficient is
h¼
qw Th Tc
ð19Þ
The thermal conductivity is calculated as following
qw @T=@XjX¼0
knf ¼
ð20Þ
By substituting Eqs. (20) and (19) in Eq. (18), the Nusselt number can be written as
Nu ¼
knf @h kf @X X¼0
ð21Þ
The average Nusselt number of the hot wall is obtained by integrating the local Nusselt number along the hot wall as follows
Nuav g ¼
Z
1 0
Nu dY
X¼0
ð22Þ
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3. Numerical scheme The governing equations are discretized using the finite volume method. Coupling between the pressure and the velocity is done using the SIMPLER algorithm. The diffusion terms in the equations are discretized by a second order central difference scheme, while a hybrid scheme (a combination of the central difference scheme and the upwind scheme) is employed to approximate the convection terms. The set of discretized equations are solved by TDMA line by line method [24]. In order to validate the proposed numerical scheme, two different test cases are chosen. The first test case is free convection in a Cu–water filled square cavity with cold right wall, partially heated left wall and insulated horizontal walls. The obtained results using the presented code for the first case are compared with the results of Oztop and Abu-Nada [6] for the same problem. Fig. 2 shows the average Nusselt number of the heated portion of the left wall obtained by the present simulation and the results of Oztop and Abu-Nada [6]. As can be observed from the figure, very good agreements exist between the two results. The second test case is natural convection of air in an annulus between two concentric isothermal square ducts with AR = 0.6. The average Nusselt numbers of outer square which are obtained by the present code are compared with results of Asan [25] in Table 2. It can be seen from the table that a good agreement exist between the present results and those obtained by Asan [25]. In order to determine a proper grid for the numerical simulation, a square cavity with an inside adiabatic square body with AR = 0.5 at Ra = 106 is chosen. Six different uniform grids, namely, 21 21, 41 41, 61 61, 81 81, 100 100 and 121 121 are employed for the numerical simulation. The average Nusselt numbers of hot left wall for these grids are shown in Table 3. As can be observed from the table, a uniform 81 81 grid is sufficiently fine for the numerical calculation. 2.5
5.5
3 2.8 Present study Oztop and Abu-Nada [6]
2
Present study Oztop and Abu-Nada [6]
2.6
Present study Oztop and Abu-Nada [6]
5
Nu
Nu
Nu
2.4 1.5
2.2
4.5
2 1
4
1.8 1.6
0.5 0
0.05
0.1
ϕ
0.15
0.2
0
0.05
0.1
ϕ
3
0.15
0.2
3.5 0
0.05
4
(b) Ra =10
(a) Ra =10
0.1
ϕ
0.15
0.2
5
(c) Ra =10
Fig. 2. Comparison of the average Nusselt number of the present code with the results of Oztop and Abu-Nada [6].
Table 2 Average Nusselt number along the hot wall for different grids.
Present study Asan [25]
Ra = 103
Ra = 104
Ra = 105
Ra = 106
3.655 3.790
3.749 3.832
3.760 3.899
5.873 5.926
Table 3 Average Nusselt number along the hot wall for different grids. Grid size Nu
u=0 u = 0.05 u = 0.1
21 21
41 41
61 61
81 81
101 101
121 121
6.1574 7.5912 7.3015
8.5409 9.0225 9.3591
9.5214 10.1905 10.7539
9.6757 10.2629 10.8640
9.6762 10.2635 10.8658
9.6788 10.2639 10.8665
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4. Results and discussion In this section, results of numerical simulation of free convection fluid flow and heat transfer of the Cu–water nanofluid in cavities having an inside adiabatic square body are presented. The study focuses on effects of the Rayleigh number, size of the adiabatic square body, and volume fraction of the nanoparticles on the flow and temperature fields. The Rayleigh number, the aspect ratio (dimensionless size) of the adiabatic square body, and the nanoparticles volume fraction are ranging from 103 to 106, 0.4 to 0.6 and 0 to 0.1, respectively. Fig. 3 shows the streamlines and isotherms for the three different aspect ratios of the adiabatic square body for the cavity filled with pure fluid at Ra = 106. For all cases the heated fluid ascends along the hot wall, then moves horizontally, is cooled and descends at the vicinity of the cold right wall, hence a primary clockwise eddy is developed inside the cavity, regardless the aspect ratio of the adiabatic square body. Secondary clockwise eddies are developed in the left and right side of the adiabatic body via movement of the primary eddy and existence of the adiabatic body. The corresponding isotherms are nearly located adjacent to the horizontal isothermal walls. This phenomenon depicts existence of thermal boundary layers. Moreover uniformly distributed parallel horizontal isotherms are formed in the right and left hand side of the square body. Also the isotherms in Fig. 3 indicate that the size of the adiabatic square body does not affect the temperature distribution significantly. Effects of Rayleigh number and volume fraction of the nanoparticles on the flow pattern and temperature distribution inside the cavity with an inside adiabatic square body with AR = 0.4, 0.5 and 0.6 are depicted in Figs. 4–6, respectively. At Ra = 103 and for all values of AR, via domination of conduction heat transfer the isotherms are nearly parallel with the vertical walls, for the cavity filled with pure fluid and nanofluid. From the streamlines a single clockwise eddy is observed inside the cavity for all values of AR at Ra = 103. The symmetric eddy demonstrates a low velocity and low intensity flow at low Rayleigh numbers. From the streamlines a similar behavior is found for the cavities filled with nanofluid. By increase of the buoyant force via increase in the Rayleigh number, the flow intensity increases and the streamlines closes to the side walls. At Ra = 105 and for AR = 0.4 two secondary eddies are developed in the lower left and upper right sides of the adiabatic square body. These secondary eddies are not developed for the cavity filled with nanofluid because of increase in the viscosity of nanofluid via increase in the nanoparticles volume fraction.
(b) AR=0.5
(c) AR=0.6
Streamlines
(a) AR=0.4
0.91
0.91
0.82 0.73
Isotherms
0.64
0.64
0.56
0.64
0.54 0.45
0.45
0.45 0.36
0.36 0.27
0.27
0.27 0.18
0.18
Fig. 3. Variation of the streamlines (up) and isotherms (down) with the aspect ratio of the adiabatic body for the cavity filled with the pure fluid at Ra = 106.
269
(d) Ra=106
(c) Ra=105
(b) Ra=104
(a) Ra=103
M. Mahmoodi, S.M. Sebdani / Superlattices and Microstructures 52 (2012) 261–275
Isotherms
Streamlines
Fig. 4. Streamline (on the right) and isotherm (on the left) for Cu–water nanofluid with u = 0.1(——) and water (___) for AR = 0.4 at different Rayleigh numbers.
At Ra = 106 the streamlines are located close to the isothermal side walls and distinct velocity boundary layers form in this region. For all values of AR, inside the cavity filled with pure fluid the secondary eddies are elongated from down to the top of the side walls of the adiabatic square body. From the streamlines at every Rayleigh number it can be observed that the velocity boundary layers adjacent to the isothermal walls are thicken when the nanoparticles volume fraction increases. It is because of increase in diffusion of mass via increase in viscosity of nanofluid. The isotherms in Figs. 3–5 indicate that with increase in the Rayleigh number, the effect of free convection increases and the isotherms are condensed next to the isothermal side walls. Moreover, thermal stratification is observed in the left and right sides of the square body. Formation of the thermal boundary layers can be observed from the isotherms at Ra = 105 and 106. It can be seen from the isotherms that at each Rayleigh number, with increase in the nanoparticles volume fraction the thermal boundary layers thicken via increase in the thermal conductivity of the nanofluid, reduction of the temperature gradient adjacent to the side walls and increase in diffusion of heat.
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(d) Ra=106
(c) Ra=105
(b) Ra=104
(a) Ra=103
270
Isotherms
Streamlines
Fig. 5. Streamline (on the right) and isotherm (on the left) for Cu–water nanofluid with u = 0.1(——) and water (___) for AR = 0.5 at different Rayleigh numbers.
Effects of increase in the aspect ratio of the adiabatic square body on the local Nusselt number along the left wall of the cavity filled with pure fluid at different Rayleigh numbers are illustrated in Fig. 7. At Ra = 103 for all aspect ratios, maximum local Nusselt number occurs at the lower end of the hot wall. At this region the cold fluid faces the hot wall; hence maximum temperature gradient occurs at this region. When the fluid ascends adjacent to the hot wall the fluid temperature increases, then the temperature gradient decreases; hence the local Nusselt number decreases. For AR = 0.4 minimum local Nusselt number occurs at the upper part of the hot wall. Also for this size of the adiabatic body, a uniform Nusselt number distribution is observed along whole upper half of the hot wall. At Ra = 103 the local Nusselt number decreases when the size of the adiabatic inside body increases. For a differentially heated square cavity without an inside body, at low Rayleigh numbers the fluid flows to the middle of the cavity [26]. When an adiabatic body is located in the center of cavity, the fluid movement is damped; hence rate of free convection decreases. As size of the adiabatic body in-
271
(d) Ra=106
(c) Ra=105
(b)
Ra=104
(a)
Ra=103
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Isotherms
Streamlines
Fig. 6. Streamline (on the right) and isotherm (on the left) for Cu–water nanofluid with u = 0.1(——) and water (___) for AR = 0.6 at different Rayleigh numbers.
creases, the reduction in the rate heat transfer is augmented. For AR = 0.5 and 0.6 the local Nusselt number decreases from down to the middle section of the hot wall and then increases by moving towards the top of the wall. The Location in which the minimum local Nusselt number occurs is about Y = 0.55. At Ra = 104 numbers, maximum rate of heat transfer occurs at the lower end of the hot wall. Maximum local Nusselt number at the lower end of the hot wall occurs for the cavity with AR = 0.6. With decrease in AR, the maximum local Nusselt number decreases for Y 6 0.2. A reverse behavior is found for the major portion of the wall. For Y P 0.2 the local Nusselt number decreases when AR increases. It is because of increase in blockage of fluid flow via increase in size of the adiabatic body. At Ra = 105 and 106 the size of the adiabatic square body does not affect the Nusselt number distribution significantly. For a differentially heated square cavity without an inside adiabatic body, at high Rayleigh numbers, boundary layers are formed adjacent to the isothermal side walls and the fluid existing in the core of the cavity in nearly stagnant. Therefore existence of an adiabatic body at the
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1
1
AR = 0.4 AR = 0.5 AR = 0.6
AR = 0.4 AR = 0.5 AR = 0.6
0.8
Y
(b) Ra=10 4
0.6
0.6
Y
(a) Ra=10 3
0.8
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
0
1.4
0
1
2
1
4
5
1
AR = 0.4 AR = 0.5 AR = 0.6
0.8
AR = 0.4 AR = 0.5 AR = 0.6
0.8
0.6
Y
(d) Ra=106
0.6
Y
(c) Ra=10 5
3
Nu
Nu
0.4
0.4
0.2
0.2
0
0
2
4
6
Nu
8
10
12
0
0
5
10
15
20
Nu
Fig. 7. Variation of the local Nusselt number with the aspect ratio of the adiabatic body for the cavity filled with the pure fluid at different Rayleigh numbers.
center of the cavity does not affect the flow pattern, temperature distribution and the heat transfer rate significantly [26]. Fig. 8 illustrates variation of the Nusselt number distribution along the left hot wall of the cavity with respect to the volume fraction of the nanoparticles for different aspect ratios of the adiabatic square body and at different Rayleigh numbers. As it can be seen from the figure at Ra = 103 local Nusselt number increases with increase in the nanoparticles volume fraction for all range of the aspect ratios considered. A different behavior is observed at Ra = 104. At this Rayleigh number for the cavity with an adiabatic square body with AR = 0.4 the increase in the nanoparticles volume fraction does not affect local Nusselt number distribution significantly. With increase in the size of the adiabatic square body, the effect of the nanoparticles volume fraction on the local Nusselt number increases, when the Rayleigh number is kept at 104. For AR = 0.5 and 0.6 a decreasing trend in local Nusselt number with increase in the nanoparticles volume fraction is observed. At Ra = 104 the buoyancy force is not strong and the contribution of convection in heat transfer process is higher than conduction; hence the increase in the viscosity of nanofluid via increase in the nanoparticles volume fraction so weaken the fluid flow and convection heat transfer that the increase in heat transfer via increase in thermal conductivity does not retrieve it. Therefore at Ra = 104 the net rate of heat transfer decreases with increase in nanoparticles concentration. At Ra = 105 and 106 the fluid flow is stronger than that at Ra = 104. At these Rayleigh numbers the local Nusselt number at the lower end of the hot wall increases when the nanoparticles volume fraction increases while with moving upward along the wall, the effect of the
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(a) AR=0.4
(b) AR=0.5 1
0.8
0.8
0.6
Y 0.4
0.4
0.2
0.2
0.2
0.2
0.4
0.6
0.8
1
0
1.6
ϕ=0 ϕ = 0.05 ϕ = 0.1
0
0.2
0.4
0.6
0.8
1
0.6
1.2
1.4
0
1.6
Nu
0
ϕ=0 ϕ = 0.05 ϕ = 0.1
Y 0.4
0.4
0.2
0.2
0.2
1
2
Nu
3
4
0
5
0
1
2
1
ϕ=0 ϕ = 0.05 ϕ = 0.1
0.8
Nu
3
4
ϕ=0 ϕ = 0.05 ϕ = 0.1
0
0.4
0.4
0.2
0.2
0.2
0
2
4
Nu
6
8
0
10
0
2
4
6
Nu
8
10
0
12
ϕ=0 ϕ = 0.05 ϕ = 0.1
ϕ=0 ϕ = 0.05 ϕ = 0.1
0.8
0.6
0.2
0.2
Nu
15
20
3
4
5
ϕ=0 ϕ = 0.05 ϕ = 0.1
2
4
6
8
10
12
ϕ=0 ϕ = 0.05 ϕ = 0.1
Y
Y
0.2
10
Nu
0.6
0.4
5
1.6
Nu
0.4
0
2
0.8
0.6
Y
0
0.4
0
1
1
1
0.8
1.4
0.6
0.4
0
1.2
ϕ=0 ϕ = 0.05 ϕ = 0.1
0.8
Y
Y
0
5
0.6
1
1
Nu
1
0.8
0.6
0.8
Y
0
0.6
0.6
0.6
1
0.4
0.8
0.4
0
0.2
1
0.8
Y
Ra=104
1.4
1
0.8
Ra=105
1.2
Nu
Y
0
1
ϕ=0 ϕ = 0.05 ϕ = 0.1
0.6
0.4
0
Ra=106
ϕ=0 ϕ = 0.05 ϕ = 0.1
0.6
Y
ϕ=0 ϕ = 0.05 ϕ = 0.1
Y
Ra=103
0.8
(c) AR=0.6
1
1
0
0
5
10
Nu
15
20
0
0
5
10
15
20
Nu
Fig. 8. Variation of the local Nusselt number with the volume fraction of the nanoparticles at different Rayleigh numbers and for different aspect ratio of the adiabatic body.
nanoparticles concentration on increase of the rate of heat transfer decreases and finally a similar local Nusselt number for the nanofluid and pure fluid occurs in the upper end of the hot wall. Variations of the average Nusselt number with the volume fraction of the nanoparticles for different aspect ratios at different Rayleigh numbers are shown in Fig. 9. The figure shows that the rate of heat transfer increases with increase in the volume fraction of the nanoparticles at all Rayleigh numbers with the exception of Ra = 104. At Ra = 104 the average Nusselt number is a decreasing function of nanoparticles volume fraction and the size of the adiabatic square body. As it was previously said, at Ra = 104 the buoyancy force is not strong and the increase in viscosity of nanofluid deteriorates the rate of heat transfer. At this Rayleigh number as the adiabatic square body becomes bigger the flow
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Fig. 9. Variation of the average Nusselt number with the volume fraction of the nanoparticles at different Rayleigh numbers and for different aspect ratio of the adiabatic body.
weakens and the effect of nanoparticles concentration on deterioration of heat transfer increases. At Ra = 103, 105 and 106, the average Nusselt number is found an increasing function of the size of the adiabatic square body and the volume fraction of nanoparticles. At these Rayleigh numbers the rate of increase of the average Nusselt number with increase in the nanoparticles concentration is same for all cavities with different sizes of the adiabatic square body. It should be noted that effect of nanofluid with variable properties will be considered in the continue of the present study and will be presented in another paper. 5. Conclusion In the present paper the problem of free convection of Cu–water nanofluid inside differentially heated square cavity with an adiabatic square body located in its center was investigated numerically using the finite volume method and SIMPLER algorithm. A parametric study was undertaken and effects of the Rayleigh number, the volume fraction of the Cu nanoparticles, and the aspect ratio of the adiabatic square body on the fluid flow, temperature field and rate of heat transfer were investigated and the following results were obtained. For all considered cases, when the volume fraction of the nanoparticles is kept constant, the rate of heat transfer increases by increase of the Rayleigh number. At Ra = 103, 105, and 106, when the Rayleigh number is kept constant, the average Nusselt number increases by increase in the volume fraction of the nanoparticles. At Ra = 104, for a constant aspect ratio, the average Nusselt number is found a decreasing function of the nanoparticles volume fraction. As AR increases, effect of volume fraction of nanoparticles on deterioration of heat transfer increases. At Ra = 103 and 104, for a constant nanoparticles volume fraction, the rate of heat transfer decreases with increasing the size of the adiabatic square body. At Ra = 105 and 106, when the nanoparticles volume fraction is kept constant, increase in the size of the inside square body does not affect the rate of heat transfer significantly.
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References [1] L. Godson, B. Raja, D.M. Lal, S. Wongwises, Enhancement of heat transfer using nanofluids—An overview, Renewable Sustainable Energy Reviews 14 (2010) 629–641. [2] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluid, International Journal of Heat and Mass Transfer 46 (2003) 3639–3653. [3] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, International Communication in Heat and Mass Transfer 33 (2006) 727–736. [4] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer augmentation in a differentially heated square cavity using copper–water nanofluid, International Journal of Thermal Science 47 (2008) 1113–1122. [5] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, International Journal of Heat and Mass Transfer 51 (2008) 4506–4516. [6] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, International Journal of Heat and Fluid Flow 29 (2008) 1326–1336. [7] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanic B/Fluids 28 (2009) 630–640. [8] E. Abu-nada, H. Oztop, Effect of inclination angle on natural convection in enclosures filled with Cu–water nanofluid, International Journal of Heat and Fluid Flow 30 (2009) 669–678. [9] E.B. Ogut, Natural convection of water-based nanofluids in an inclined enclosure with a heat source, International Journal of Thermal Science 48 (2009) 2063–2073. [10] B. Ghasemi, S.M. Aminossadati, Periodic natural convection in a nanofluid-filled enclosure with oscillating heat flux, International Journal of Thermal Science 49 (2010) 1–9. [11] S. Kumar, S.K. Prasad, J. Banerjee, Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model, Applied Mathematical Modeling 34 (2010) 573–592. [12] K.C. Lin, A. Violi, Natural convection heat transfer of nanofluids in a vertical cavity: effects of non-uniform particle diameter and temperature on thermal conductivity, International Journal of Heat and Fluid Flow 31 (2010) 236–245. [13] G.A. Sheikhzadeh, A. Arefmanesh, M.H. Kheirkhah, R. Abdollahi, Natural convection of Cu–water nanofluid in a cavity with partially active side walls, European Journal of Mechanics B/Fluids 30 (2011) 166–176. [14] H. Saleh, R. Roslan, I. Hashim, Natural convection heat transfer in a nanofluid-filled trapezoidal enclosure, International Journal of Heat and Mass Transfer 54 (2011) 194–201. [15] G.A. Sheikhzadeh, A. Arefmanesh, M. Mahmoodi, Numerical study of natural convection in a differentially-heated rectangular cavity filled with TiO2–Water nanofluid, Journal of Nano Research 13 (2011) 75–80. [16] H.F. Oztop, E. Abu-Nada, Y. Varol, Kh. Al-Salem, Computational analysis of non-isothermal temperature distribution on natural convection in nanofluid filled enclosures, Superlattices and Microstructures 49 (2011) 453–467. [17] S.H. Tasnim, M.R. Collins, Suppressing natural convection in a differentially heated square cavity with an arc shaped baffle, International Communication in Heat and Mass Transfer 32 (2005) 94–106. [18] A. Mezrhab, H. Bouali, H. Amaoui, M. Bouzidi, Computation of combined natural-convection and radiation heat-transfer in a cavity having a square body at its center, Applied Energy 83 (2006) 1004–1023. [19] W.R. Chen, Natural convection heat transfer between inner sphere and outer vertically eccentric cylinder, International Journal of Heat and Mass Transfer 53 (2010) 5147–5155. [20] Z.T. Yu, X. Xub, Y.C. Hu, L.W. Fan, K.F. Cen, Transient natural convective heat transfer of a low-Prandtl-number fluid inside a horizontal circular cylinder with an inner coaxial triangular cylinder, International Journal of Heat and Mass Transfer 53 (2010) 5102–5110. [21] A. Bejan, Convection heat transfer, John Wiley & Sons, Inc., Hoboken, New Jersey, USA, 2004. [22] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, Journal of Chemical Physics 20 (1952) 571–581. [23] J.C. Maxwell-Garnett, Colours in metal glasses and in metallic films, Philosophical Transactions of the Royal Society A 203 (1904) 385–420. [24] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis Group, New York, 1980. [25] H. Asan, Natural convection in an annulus between two isothermal concentric square ducts, International Communication in Heat and Mass Transfer 27 (2000) 367–376. [26] G.V. Davis, Natural convection of air in a square cavity- a benchmark numerical solution, International Journal for Numerical Methods in fluids 3 (1983) 249–264.
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center Mostafa Mahmoodi a, Saeed Mazrouei Sebdani b,⇑ a b
Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran Department of Mechanical Engineering, University of Kashan, Kashan, Iran
a r t i c l e
i n f o
Article history: Received 23 March 2012 Accepted 7 May 2012 Available online 15 May 2012 Keywords: Finite volume method Free convection Nanofluid Square cavity Square body
a b s t r a c t The problem of free convection fluid flow and heat transfer of Cu– water nanofluid inside a square cavity having adiabatic square bodies at its center has been investigated numerically. The governing equations have been discretized using the finite volume method. The SIMPLER algorithm was employed to couple velocity and pressure fields. Using the developed code, a parametric study was conducted and the effects of pertinent parameters such as Rayleigh number, size of the adiabatic square body, and volume fraction of the Cu nanoparticles on the fluid flow and thermal fields and heat transfer inside the cavity were investigated. The obtained results show that for all Rayleigh numbers with the exception of Ra = 104 the average Nusselt number increases with increase in the volume fraction of the nanoparticles. At Ra = 104 the average Nusselt number is a decreasing function of the nanoparticles volume fraction. Moreover at low Rayleigh numbers (103 and 104) the rate of heat transfer decreases when the size of the adiabatic square body increases while at high Rayleigh numbers (105 and 106) it increases. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In recent years nanofluids, which are a suspension of nano-sized solid particles in a base fluid, with thermal conductivity higher than the based fluid, are used to enhance the rate of heat transfer in many practical engineering applications [1]. Free convection heat transfer inside nanofluid filled rectangular cavities with different boundary conditions on the side walls has been studied by many ⇑ Corresponding author. E-mail address: [email protected] (S.M. Sebdani). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.05.007
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Nomenclature cp g h H k l Nu p P Pr q Ra T u,v U,V x,y X,Y
specific heat, J kg1 K1 gravitational acceleration, m s2 heat transfer coefficient, W m2 K enclosure height, m thermal conductivity, W m1 K1 width of the adiabatic square body, m Nusselt number pressure, N m2 dimensionless pressure Prandtl number heat flux, W m2 Rayleigh number dimensional temperature, K dimensional velocities components in x and y direction, m s1 dimensionless velocities components in X and Y direction dimensional Cartesian coordinates, m dimensionless Cartesian coordinates
Greek symbols a thermal diffusivity, m2 s b thermal expansion coefficient, K1 h dimensionless temperature l dynamic viscosity, kg m1 s m kinematic viscosity, m2 s q density, kg m3 u volume fraction of the nanoparticles Subscripts c cold f fluid h hot nf nanofluid s solid particles w wall
researchers. Khanafer et al. [2] conducted a numerical study on free convection inside nanofluid filled rectangular cavities with cold right wall, hot left wall and insulated horizontal walls. Their results showed that rate of heat transfer increased with increase in nanoparticles volume fraction for entire range of Grashof number considered. Similar results were found in work of Jou and Tzeng [3] on numerical study of free convection in differentially heated rectangular cavities filled with a nanofluid. Santra et al. [4] studied free convection of Cu–water nanofluid in a differentially heated square cavity with consideration of Ostwald–de Waele non-Newtonian behavior of the nanofluid. They found that heat transfer decreased with increase in the nanoparticles volume fraction for a particular Rayleigh number. Effects due to uncertainties in effective dynamic viscosity and thermal conductivity of alumina-water nanofluid on free convection heat transfer in a differentially heated square cavity were investigated by Ho et al. [5]. Their results demonstrated that heat transfer across the cavity can be found to be enhanced or mitigated with respect to the base fluid via the used dynamic viscosity formula. Oztop and Abu-nada [6] carried out a numerical study on free convection of nanofluid in partially heated rectangular cavities. The cavities had a cold vertical wall, a localized heater on the other vertical wall and insulated horizontal walls. They considered effects of Rayleigh number,
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aspect ratio of cavities, size and location of the heater and different types of water based nanofluids. Their results showed an increase in average Nusselt number with increase in the volume fraction of the nanoparticles and increase in the height of heater. Numerical study on free convection of nanofluid in a square cavity cooled from two vertical and top horizontal walls and heated by a constant heat flux heater on its horizontal bottom wall has been done by Aminossadati and Ghasemi [7]. They investigated the effects of Rayleigh number, nanoparticles volume fraction, size and location of heater and type of nanofluid. The obtained results indicated that type of nanoparticles and the length and location of the heat source affected significantly the heat source maximum temperature. Abunada and Oztop [8] investigated effect of inclination angle of a square cavity on free convection of the Cu–water nanofluid inside it. They observed that the inclination angle can be used as a control parameter for fluid flow and heat transfer inside the cavity. Moreover their results showed that effects of inclination angle on percentage of heat transfer enhancement became insignificant at low Rayleigh number. Free convection heat transfer of water-based nanofluids in an inclined square cavity with right cold wall, a constant heat flux heater at the center of its left wall and adiabatic other sides, was studied numerically by Ogut [9]. He investigated effects of inclination angle of the cavity, solid volume fractions, length of the constant heat flux heater, and the Rayleigh number on flow and temperature field inside the cavity. The obtained results showed that the average heat transfer rate increased significantly as nanoparticle volume fraction and Rayleigh number increased. Moreover the results showed that the length of the heater was an important parameter affecting the flow and temperature fields. Numerical investigation of periodic free convection fluid flow and heat transfer inside a nanofluid filled square cavity was done by Ghasemi and Aminossadati [10]. They considered a square cavity with insulated top and bottom walls, cold right vertical wall and a heater with oscillating heat flux on its left vertical wall. A periodic behavior of the flow and temperature fields was observed as a result of the oscillating heat flux. Moreover it was found that optimum position of the heat source was a function of Rayleigh number. Kumar et al. [11] analyzed free convection of nanofluid in a differentially heated square cavity using single phase thermal dispersion model. Simulations incorporating the thermal dispersion model showed increment in local thermal conductivity at locations with maximum velocity. Moreover their results showed that the average Nusselt number increased with the solid volume fraction. In a numerical study, Lin and Violi [12] analyzed free convection of Al2O3-water nanofluid in a differentially heated cavity with consideration of slip mechanism in nanofluids. They investigated effects of pertinent parameters such as non-uniform nanoparticle size, mean nanoparticle diameter, nanoparticle volume fraction, Prandtl number, and Grashof number on flow field and temperature distribution inside the cavity. They found enhanced and mitigated heat transfer due to the presence of nanoparticles. Sheikhzadeh et al. [13] conducted a numerical simulation to study free convection of Cu–water nanofluid inside a square cavity with partially thermally active side walls. The active parts of the left and the right side walls of the cavity were maintained hot and cold respectively while the cavity’s top and bottom walls as well as the inactive parts of the side walls were kept insulated. They found that maximum average Nusselt number for high Rayleigh numbers occurred when the hot and cold parts were located in the bottom and middle region of the vertical walls respectively. Saleh et al. [14] investigated numerically free convection of nanofluid in a trapezoidal enclosure numerically and developed a correlation for the average Nusselt number as a function of the angle of the sloping wall, effective thermal conductivity and viscosity as well as Grashof number. Sheikhzadeh et al. [15] conducted a numerical simulation to investigate the problem of free convection of the TiO2-water nanofluid in rectangular cavities differentially heated on adjacent walls. The left and the top walls of the cavities were heated and cooled respectively; while, the cavities right and bottom walls were kept insulated. They found that by increase in the volume fraction of the nanoparticles, mean Nusselt number of the hot wall increased for the shallow cavities; while, a reverse trend occurred for tall cavities. Very recently numerical results of a study on free convection in a nanofluid filled square cavity by using heating and cooling by sinusoidal temperature profiles on one side were reported by Oztop et al. [16]. They considered effects of various inclination angles of the cavity, different types of water based nanofluids, volume fraction of nanoparticles, and the Rayleigh number on the heat transfer rate. They found that addition of nanoparticle into the water affected the fluid flow and temperature distribution especially for higher Rayleigh numbers.
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Apart from application of nanofluids in buoyancy driven heat transfer, free convection in cavities having an inside body has received considerable attention in the recent years due to its practical engineering applications, such as solar collectors, thermal insulation, cooling of electronic components and designing building [17]. Mezrhab et al. [18] investigated radiation-natural convection interactions in a differentially heated cavity with an inner body numerically. They found that the radiation exchange homogenized the temperature inside the cavity and produced an increase in the average Nusselt number. Free convection heat transfer between inner hot sphere and outer vertically eccentric cold cylinder was investigated by Chen [19] numerically. Yu et al. [20] conducted a numerical simulation to study of transient free convective heat transfer of a liquid gallium inside a horizontal circular cylinder with an inner coaxial triangular cylinder. Their results showed that the time averaged Nusselt number was apparently increased by horizontally placing the top side of the inner triangular cylinder for Grashof numbers greater than 105. Based on literature reviews, despite a large number of numerical studies on free convection of nanofluids inside rectangular cavities with different boundary conditions, there is no study on free convection in square cavities with an inside adiabatic body. This problem may be occurred in a number of technical applications such as solar collectors, heat exchangers, and cooling of electronic equipment and chips using nanofluids. The adiabatic body can be considered as a model of heat transfer controller or modifier device. In a heat exchanger an adiabatic body can be a model of baffle which manages the flow rate and heat transport process. In view point of cooling of electronic chips an adiabatic body can be used to increase or decrease of heat transfer from special part of the hot chip. In the present paper the problem of free convection heat transfer and fluid flow in a square cavity containing nanofluid and with an adiabatic square body at its center, is investigated using the finite volume method and SIMPLER algorithm. The results in the form of streamlines and isotherms plots, average Nusselt number and local Nusselt number are presented for a wide range of Rayleigh numbers, size of the adiabatic square body and volume fraction of the nanoparticles. 2. Mathematical modeling A schematic view of the square cavity with an adiabatic square body at its center is shown in Fig. 1. The height and the width of the cavity are denoted by H. An adiabatic square body with the length of l is located at the center of the square cavity. Aspect ratio (dimensionless size of the adiabatic square body) is defined as AR = l/W. The left wall is kept at high temperature Th, while, the right wall is kept at cold temperature Tc. The horizontal top and bottom walls of the cavity are kept insulated. The length of the cavity perpendicular to its plane is assumed to be long enough; hence, the problem is
y
l
H
Th
Tc
x H Fig. 1. A schematic view of the considered cavity in the present study.
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considered two dimensional. The cavity is filled with Cu–water nanofluid. It is assumed that the nanoparticles and the base fluid are in thermal equilibrium and there is no slip between them. The thermophysical properties of the Cu nanoparticles and the water as the base fluid are listed in Table 1. The nanofluid is considered Newtonian, laminar, and incompressible. The thermophysical properties of the nanofluid are considered constant with the exception of the density which varies according to the Boussinesq approximation [21]. The steady state continuity, momentum and energy equations for two-dimensional laminar free convection fluid flow and heat transfer with the Boussinesq approximation in y-direction are as following:
@u @ v þ ¼0 @x @y
ð1Þ
@u @u 1 @p lnf þv ¼ þ u @x @y qnf @x qnf
! @2u @2u þ @x2 @y2
@v @v 1 @p lnf u þ þv ¼ @x @y qnf @y qnf
@2v @2v þ @x2 @y2
u
@T @T @2T @2T þ þv ¼ anf @x @y @x2 @y2
ð2Þ
! þ
ðqbÞnf
qnf
gðT T c Þ
ð3Þ
! ð4Þ
where the density, heat capacity, thermal expansion coefficient, and thermal diffusivity of the nanofluid are as follow, respectively, [2]
qnf ¼ ð1 uÞqf þ uqs
ð5Þ
ðqC p Þnf ¼ ð1 uÞðqC p Þf þ uðqC p Þs
ð6Þ
ðqbÞnf ¼ ð1 uÞðqbÞf þ uðqbÞs
ð7Þ
anf ¼
knf ðqC p Þnf
ð8Þ
To estimation of the dynamic viscosity of the nanofluid the Brinkman model [22] is employed.
leff ¼
lf
ð9Þ
ð1 uÞ2:5
This formula has been used to calculation of the dynamic viscosity of nanofluid in numerical simulation of free convection in recently published articles [2,3,6–10,13,14,16]. The effective thermal conductivity of the nanofluid is determined according to Maxwell [23].
knf ðks þ 2kf Þ 2uðkf ks Þ ¼ kf ðkf þ 2ks Þ þ uðkf ks Þ
ð10Þ
Table 1 Thermo-physical properties of water and solid nanoparticles [6]. Physical properties
Fluid phase
Cu
CP (J/kg k) q (kg/m3) k (W/m k) b (K1)
4179 997.1 0.613 21 105
385 8933 401 1.67 105
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This relation has been used recently in literature related to numerical simulation of free convection of nanofluid [2,3,6–10,13,14,16]. Using the following dimensionless parameters the governing equation can be converted to dimensionless form.
X¼
x y uH vH pH2 T Tc ; Y¼ ; U¼ ; V¼ ; P¼ ; h¼ H H af af Th Tc qnf a2f
ð11Þ
The dimensionless forms of the governing equations are
@U @V þ ¼0 @X @Y
ð12Þ
lnf @U @U @P þV ¼ þ U @X @Y @X qnf af
@2U
U
lnf @V @V @P þV ¼ þ @X @Y @Y qnf af
@2V
U
@h @h anf þV ¼ @X @Y af
@X 2
@X
@2h
@2h
@X
@Y 2
þ 2
2
þ
þ
@2U
! ð13Þ
@Y 2 @2V @Y
2
! þ
ðqbÞnf Ra Pr h qnf bf
ð14Þ
! ð15Þ
where the Rayleigh number Ra, and the Prandtl number Pr are
Ra ¼
gbf ðT h T c Þ H3
af mf
; Pr ¼
mf af
ð16Þ
The Prandtl number of water is Pr = 6.8. The boundary conditions for Eqs. (12)–(15) are
8 > < on the left wall U ¼ V ¼ 0; h ¼ 1 on the right wall U ¼ V ¼ 0; h ¼ 0 > : on the adiabatic body : U ¼ V ¼ 0; @h=@n ¼ 0
ð17Þ
Where n is normal direction to the walls. The local Nusselt number of the heat source is expressed as
Nulocal ¼
hH kf
ð18Þ
where the heat transfer coefficient is
h¼
qw Th Tc
ð19Þ
The thermal conductivity is calculated as following
qw @T=@XjX¼0
knf ¼
ð20Þ
By substituting Eqs. (20) and (19) in Eq. (18), the Nusselt number can be written as
Nu ¼
knf @h kf @X X¼0
ð21Þ
The average Nusselt number of the hot wall is obtained by integrating the local Nusselt number along the hot wall as follows
Nuav g ¼
Z
1 0
Nu dY
X¼0
ð22Þ
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3. Numerical scheme The governing equations are discretized using the finite volume method. Coupling between the pressure and the velocity is done using the SIMPLER algorithm. The diffusion terms in the equations are discretized by a second order central difference scheme, while a hybrid scheme (a combination of the central difference scheme and the upwind scheme) is employed to approximate the convection terms. The set of discretized equations are solved by TDMA line by line method [24]. In order to validate the proposed numerical scheme, two different test cases are chosen. The first test case is free convection in a Cu–water filled square cavity with cold right wall, partially heated left wall and insulated horizontal walls. The obtained results using the presented code for the first case are compared with the results of Oztop and Abu-Nada [6] for the same problem. Fig. 2 shows the average Nusselt number of the heated portion of the left wall obtained by the present simulation and the results of Oztop and Abu-Nada [6]. As can be observed from the figure, very good agreements exist between the two results. The second test case is natural convection of air in an annulus between two concentric isothermal square ducts with AR = 0.6. The average Nusselt numbers of outer square which are obtained by the present code are compared with results of Asan [25] in Table 2. It can be seen from the table that a good agreement exist between the present results and those obtained by Asan [25]. In order to determine a proper grid for the numerical simulation, a square cavity with an inside adiabatic square body with AR = 0.5 at Ra = 106 is chosen. Six different uniform grids, namely, 21 21, 41 41, 61 61, 81 81, 100 100 and 121 121 are employed for the numerical simulation. The average Nusselt numbers of hot left wall for these grids are shown in Table 3. As can be observed from the table, a uniform 81 81 grid is sufficiently fine for the numerical calculation. 2.5
5.5
3 2.8 Present study Oztop and Abu-Nada [6]
2
Present study Oztop and Abu-Nada [6]
2.6
Present study Oztop and Abu-Nada [6]
5
Nu
Nu
Nu
2.4 1.5
2.2
4.5
2 1
4
1.8 1.6
0.5 0
0.05
0.1
ϕ
0.15
0.2
0
0.05
0.1
ϕ
3
0.15
0.2
3.5 0
0.05
4
(b) Ra =10
(a) Ra =10
0.1
ϕ
0.15
0.2
5
(c) Ra =10
Fig. 2. Comparison of the average Nusselt number of the present code with the results of Oztop and Abu-Nada [6].
Table 2 Average Nusselt number along the hot wall for different grids.
Present study Asan [25]
Ra = 103
Ra = 104
Ra = 105
Ra = 106
3.655 3.790
3.749 3.832
3.760 3.899
5.873 5.926
Table 3 Average Nusselt number along the hot wall for different grids. Grid size Nu
u=0 u = 0.05 u = 0.1
21 21
41 41
61 61
81 81
101 101
121 121
6.1574 7.5912 7.3015
8.5409 9.0225 9.3591
9.5214 10.1905 10.7539
9.6757 10.2629 10.8640
9.6762 10.2635 10.8658
9.6788 10.2639 10.8665
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4. Results and discussion In this section, results of numerical simulation of free convection fluid flow and heat transfer of the Cu–water nanofluid in cavities having an inside adiabatic square body are presented. The study focuses on effects of the Rayleigh number, size of the adiabatic square body, and volume fraction of the nanoparticles on the flow and temperature fields. The Rayleigh number, the aspect ratio (dimensionless size) of the adiabatic square body, and the nanoparticles volume fraction are ranging from 103 to 106, 0.4 to 0.6 and 0 to 0.1, respectively. Fig. 3 shows the streamlines and isotherms for the three different aspect ratios of the adiabatic square body for the cavity filled with pure fluid at Ra = 106. For all cases the heated fluid ascends along the hot wall, then moves horizontally, is cooled and descends at the vicinity of the cold right wall, hence a primary clockwise eddy is developed inside the cavity, regardless the aspect ratio of the adiabatic square body. Secondary clockwise eddies are developed in the left and right side of the adiabatic body via movement of the primary eddy and existence of the adiabatic body. The corresponding isotherms are nearly located adjacent to the horizontal isothermal walls. This phenomenon depicts existence of thermal boundary layers. Moreover uniformly distributed parallel horizontal isotherms are formed in the right and left hand side of the square body. Also the isotherms in Fig. 3 indicate that the size of the adiabatic square body does not affect the temperature distribution significantly. Effects of Rayleigh number and volume fraction of the nanoparticles on the flow pattern and temperature distribution inside the cavity with an inside adiabatic square body with AR = 0.4, 0.5 and 0.6 are depicted in Figs. 4–6, respectively. At Ra = 103 and for all values of AR, via domination of conduction heat transfer the isotherms are nearly parallel with the vertical walls, for the cavity filled with pure fluid and nanofluid. From the streamlines a single clockwise eddy is observed inside the cavity for all values of AR at Ra = 103. The symmetric eddy demonstrates a low velocity and low intensity flow at low Rayleigh numbers. From the streamlines a similar behavior is found for the cavities filled with nanofluid. By increase of the buoyant force via increase in the Rayleigh number, the flow intensity increases and the streamlines closes to the side walls. At Ra = 105 and for AR = 0.4 two secondary eddies are developed in the lower left and upper right sides of the adiabatic square body. These secondary eddies are not developed for the cavity filled with nanofluid because of increase in the viscosity of nanofluid via increase in the nanoparticles volume fraction.
(b) AR=0.5
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(a) AR=0.4
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0.36 0.27
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Fig. 3. Variation of the streamlines (up) and isotherms (down) with the aspect ratio of the adiabatic body for the cavity filled with the pure fluid at Ra = 106.
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Isotherms
Streamlines
Fig. 4. Streamline (on the right) and isotherm (on the left) for Cu–water nanofluid with u = 0.1(——) and water (___) for AR = 0.4 at different Rayleigh numbers.
At Ra = 106 the streamlines are located close to the isothermal side walls and distinct velocity boundary layers form in this region. For all values of AR, inside the cavity filled with pure fluid the secondary eddies are elongated from down to the top of the side walls of the adiabatic square body. From the streamlines at every Rayleigh number it can be observed that the velocity boundary layers adjacent to the isothermal walls are thicken when the nanoparticles volume fraction increases. It is because of increase in diffusion of mass via increase in viscosity of nanofluid. The isotherms in Figs. 3–5 indicate that with increase in the Rayleigh number, the effect of free convection increases and the isotherms are condensed next to the isothermal side walls. Moreover, thermal stratification is observed in the left and right sides of the square body. Formation of the thermal boundary layers can be observed from the isotherms at Ra = 105 and 106. It can be seen from the isotherms that at each Rayleigh number, with increase in the nanoparticles volume fraction the thermal boundary layers thicken via increase in the thermal conductivity of the nanofluid, reduction of the temperature gradient adjacent to the side walls and increase in diffusion of heat.
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(d) Ra=106
(c) Ra=105
(b) Ra=104
(a) Ra=103
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Isotherms
Streamlines
Fig. 5. Streamline (on the right) and isotherm (on the left) for Cu–water nanofluid with u = 0.1(——) and water (___) for AR = 0.5 at different Rayleigh numbers.
Effects of increase in the aspect ratio of the adiabatic square body on the local Nusselt number along the left wall of the cavity filled with pure fluid at different Rayleigh numbers are illustrated in Fig. 7. At Ra = 103 for all aspect ratios, maximum local Nusselt number occurs at the lower end of the hot wall. At this region the cold fluid faces the hot wall; hence maximum temperature gradient occurs at this region. When the fluid ascends adjacent to the hot wall the fluid temperature increases, then the temperature gradient decreases; hence the local Nusselt number decreases. For AR = 0.4 minimum local Nusselt number occurs at the upper part of the hot wall. Also for this size of the adiabatic body, a uniform Nusselt number distribution is observed along whole upper half of the hot wall. At Ra = 103 the local Nusselt number decreases when the size of the adiabatic inside body increases. For a differentially heated square cavity without an inside body, at low Rayleigh numbers the fluid flows to the middle of the cavity [26]. When an adiabatic body is located in the center of cavity, the fluid movement is damped; hence rate of free convection decreases. As size of the adiabatic body in-
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(d) Ra=106
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Isotherms
Streamlines
Fig. 6. Streamline (on the right) and isotherm (on the left) for Cu–water nanofluid with u = 0.1(——) and water (___) for AR = 0.6 at different Rayleigh numbers.
creases, the reduction in the rate heat transfer is augmented. For AR = 0.5 and 0.6 the local Nusselt number decreases from down to the middle section of the hot wall and then increases by moving towards the top of the wall. The Location in which the minimum local Nusselt number occurs is about Y = 0.55. At Ra = 104 numbers, maximum rate of heat transfer occurs at the lower end of the hot wall. Maximum local Nusselt number at the lower end of the hot wall occurs for the cavity with AR = 0.6. With decrease in AR, the maximum local Nusselt number decreases for Y 6 0.2. A reverse behavior is found for the major portion of the wall. For Y P 0.2 the local Nusselt number decreases when AR increases. It is because of increase in blockage of fluid flow via increase in size of the adiabatic body. At Ra = 105 and 106 the size of the adiabatic square body does not affect the Nusselt number distribution significantly. For a differentially heated square cavity without an inside adiabatic body, at high Rayleigh numbers, boundary layers are formed adjacent to the isothermal side walls and the fluid existing in the core of the cavity in nearly stagnant. Therefore existence of an adiabatic body at the
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1
1
AR = 0.4 AR = 0.5 AR = 0.6
AR = 0.4 AR = 0.5 AR = 0.6
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Fig. 7. Variation of the local Nusselt number with the aspect ratio of the adiabatic body for the cavity filled with the pure fluid at different Rayleigh numbers.
center of the cavity does not affect the flow pattern, temperature distribution and the heat transfer rate significantly [26]. Fig. 8 illustrates variation of the Nusselt number distribution along the left hot wall of the cavity with respect to the volume fraction of the nanoparticles for different aspect ratios of the adiabatic square body and at different Rayleigh numbers. As it can be seen from the figure at Ra = 103 local Nusselt number increases with increase in the nanoparticles volume fraction for all range of the aspect ratios considered. A different behavior is observed at Ra = 104. At this Rayleigh number for the cavity with an adiabatic square body with AR = 0.4 the increase in the nanoparticles volume fraction does not affect local Nusselt number distribution significantly. With increase in the size of the adiabatic square body, the effect of the nanoparticles volume fraction on the local Nusselt number increases, when the Rayleigh number is kept at 104. For AR = 0.5 and 0.6 a decreasing trend in local Nusselt number with increase in the nanoparticles volume fraction is observed. At Ra = 104 the buoyancy force is not strong and the contribution of convection in heat transfer process is higher than conduction; hence the increase in the viscosity of nanofluid via increase in the nanoparticles volume fraction so weaken the fluid flow and convection heat transfer that the increase in heat transfer via increase in thermal conductivity does not retrieve it. Therefore at Ra = 104 the net rate of heat transfer decreases with increase in nanoparticles concentration. At Ra = 105 and 106 the fluid flow is stronger than that at Ra = 104. At these Rayleigh numbers the local Nusselt number at the lower end of the hot wall increases when the nanoparticles volume fraction increases while with moving upward along the wall, the effect of the
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(a) AR=0.4
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1
1
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10
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15
20
0
0
5
10
15
20
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Fig. 8. Variation of the local Nusselt number with the volume fraction of the nanoparticles at different Rayleigh numbers and for different aspect ratio of the adiabatic body.
nanoparticles concentration on increase of the rate of heat transfer decreases and finally a similar local Nusselt number for the nanofluid and pure fluid occurs in the upper end of the hot wall. Variations of the average Nusselt number with the volume fraction of the nanoparticles for different aspect ratios at different Rayleigh numbers are shown in Fig. 9. The figure shows that the rate of heat transfer increases with increase in the volume fraction of the nanoparticles at all Rayleigh numbers with the exception of Ra = 104. At Ra = 104 the average Nusselt number is a decreasing function of nanoparticles volume fraction and the size of the adiabatic square body. As it was previously said, at Ra = 104 the buoyancy force is not strong and the increase in viscosity of nanofluid deteriorates the rate of heat transfer. At this Rayleigh number as the adiabatic square body becomes bigger the flow
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Fig. 9. Variation of the average Nusselt number with the volume fraction of the nanoparticles at different Rayleigh numbers and for different aspect ratio of the adiabatic body.
weakens and the effect of nanoparticles concentration on deterioration of heat transfer increases. At Ra = 103, 105 and 106, the average Nusselt number is found an increasing function of the size of the adiabatic square body and the volume fraction of nanoparticles. At these Rayleigh numbers the rate of increase of the average Nusselt number with increase in the nanoparticles concentration is same for all cavities with different sizes of the adiabatic square body. It should be noted that effect of nanofluid with variable properties will be considered in the continue of the present study and will be presented in another paper. 5. Conclusion In the present paper the problem of free convection of Cu–water nanofluid inside differentially heated square cavity with an adiabatic square body located in its center was investigated numerically using the finite volume method and SIMPLER algorithm. A parametric study was undertaken and effects of the Rayleigh number, the volume fraction of the Cu nanoparticles, and the aspect ratio of the adiabatic square body on the fluid flow, temperature field and rate of heat transfer were investigated and the following results were obtained. For all considered cases, when the volume fraction of the nanoparticles is kept constant, the rate of heat transfer increases by increase of the Rayleigh number. At Ra = 103, 105, and 106, when the Rayleigh number is kept constant, the average Nusselt number increases by increase in the volume fraction of the nanoparticles. At Ra = 104, for a constant aspect ratio, the average Nusselt number is found a decreasing function of the nanoparticles volume fraction. As AR increases, effect of volume fraction of nanoparticles on deterioration of heat transfer increases. At Ra = 103 and 104, for a constant nanoparticles volume fraction, the rate of heat transfer decreases with increasing the size of the adiabatic square body. At Ra = 105 and 106, when the nanoparticles volume fraction is kept constant, increase in the size of the inside square body does not affect the rate of heat transfer significantly.
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References [1] L. Godson, B. Raja, D.M. Lal, S. Wongwises, Enhancement of heat transfer using nanofluids—An overview, Renewable Sustainable Energy Reviews 14 (2010) 629–641. [2] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluid, International Journal of Heat and Mass Transfer 46 (2003) 3639–3653. [3] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, International Communication in Heat and Mass Transfer 33 (2006) 727–736. [4] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer augmentation in a differentially heated square cavity using copper–water nanofluid, International Journal of Thermal Science 47 (2008) 1113–1122. [5] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, International Journal of Heat and Mass Transfer 51 (2008) 4506–4516. [6] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, International Journal of Heat and Fluid Flow 29 (2008) 1326–1336. [7] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanic B/Fluids 28 (2009) 630–640. [8] E. Abu-nada, H. Oztop, Effect of inclination angle on natural convection in enclosures filled with Cu–water nanofluid, International Journal of Heat and Fluid Flow 30 (2009) 669–678. [9] E.B. Ogut, Natural convection of water-based nanofluids in an inclined enclosure with a heat source, International Journal of Thermal Science 48 (2009) 2063–2073. [10] B. Ghasemi, S.M. Aminossadati, Periodic natural convection in a nanofluid-filled enclosure with oscillating heat flux, International Journal of Thermal Science 49 (2010) 1–9. [11] S. Kumar, S.K. Prasad, J. Banerjee, Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model, Applied Mathematical Modeling 34 (2010) 573–592. [12] K.C. Lin, A. Violi, Natural convection heat transfer of nanofluids in a vertical cavity: effects of non-uniform particle diameter and temperature on thermal conductivity, International Journal of Heat and Fluid Flow 31 (2010) 236–245. [13] G.A. Sheikhzadeh, A. Arefmanesh, M.H. Kheirkhah, R. Abdollahi, Natural convection of Cu–water nanofluid in a cavity with partially active side walls, European Journal of Mechanics B/Fluids 30 (2011) 166–176. [14] H. Saleh, R. Roslan, I. Hashim, Natural convection heat transfer in a nanofluid-filled trapezoidal enclosure, International Journal of Heat and Mass Transfer 54 (2011) 194–201. [15] G.A. Sheikhzadeh, A. Arefmanesh, M. Mahmoodi, Numerical study of natural convection in a differentially-heated rectangular cavity filled with TiO2–Water nanofluid, Journal of Nano Research 13 (2011) 75–80. [16] H.F. Oztop, E. Abu-Nada, Y. Varol, Kh. Al-Salem, Computational analysis of non-isothermal temperature distribution on natural convection in nanofluid filled enclosures, Superlattices and Microstructures 49 (2011) 453–467. [17] S.H. Tasnim, M.R. Collins, Suppressing natural convection in a differentially heated square cavity with an arc shaped baffle, International Communication in Heat and Mass Transfer 32 (2005) 94–106. [18] A. Mezrhab, H. Bouali, H. Amaoui, M. Bouzidi, Computation of combined natural-convection and radiation heat-transfer in a cavity having a square body at its center, Applied Energy 83 (2006) 1004–1023. [19] W.R. Chen, Natural convection heat transfer between inner sphere and outer vertically eccentric cylinder, International Journal of Heat and Mass Transfer 53 (2010) 5147–5155. [20] Z.T. Yu, X. Xub, Y.C. Hu, L.W. Fan, K.F. Cen, Transient natural convective heat transfer of a low-Prandtl-number fluid inside a horizontal circular cylinder with an inner coaxial triangular cylinder, International Journal of Heat and Mass Transfer 53 (2010) 5102–5110. [21] A. Bejan, Convection heat transfer, John Wiley & Sons, Inc., Hoboken, New Jersey, USA, 2004. [22] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, Journal of Chemical Physics 20 (1952) 571–581. [23] J.C. Maxwell-Garnett, Colours in metal glasses and in metallic films, Philosophical Transactions of the Royal Society A 203 (1904) 385–420. [24] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis Group, New York, 1980. [25] H. Asan, Natural convection in an annulus between two isothermal concentric square ducts, International Communication in Heat and Mass Transfer 27 (2000) 367–376. [26] G.V. Davis, Natural convection of air in a square cavity- a benchmark numerical solution, International Journal for Numerical Methods in fluids 3 (1983) 249–264.