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Natural convection of Cu–water nanofluid in a cavity with partially active side walls
European Journal of Mechanics B/Fluids 30 (2011) 166–176
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Natural convection of Cu–water nanofluid in a cavity with partially active side walls G.A. Sheikhzadeh ∗ , A. Arefmanesh, M.H. Kheirkhah, R. Abdollahi Department of Mechanical Engineering, University of Kashan, Kashan, Iran
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Article history: Received 12 April 2010 Received in revised form 14 August 2010 Accepted 12 October 2010 Available online 20 October 2010 Keywords: Nanofluid Natural convection Numerical study Partially active walls Cavity
abstract The buoyancy-driven fluid flow and heat transfer in a square cavity with partially active side walls filled with Cu–water nanofluid is investigated numerically. The active parts of the left and the right side walls of the cavity are maintained at temperatures Th and Tc , respectively, with Th > Tc . The enclosure’s top and bottom walls as well as the inactive parts of its side walls are kept insulated. The governing equations in the two-dimensional space are discretized using the control volume method. A proper upwinding scheme is employed to obtain stabilized solutions. Using the developed code, a parametric study is undertaken, and the effects of the Rayleigh number, the locations of the active parts of the side walls, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the cavity are investigated. It is observed from the results that the average Nusselt number increases with increasing both the Rayleigh number and the volume fraction of the nanoparticles. Moreover, the maximum average Nusselt number for the high and the low Rayleigh numbers occur for the bottom–middle and the middle–middle locations of the thermally active parts, respectively. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction Natural convection heat transfer is an important phenomenon in engineering and industry with widespread applications in diverse fields, such as, geophysics, solar energy, electronic cooling, and nuclear energy to mention a few. A major limitation against enhancing the heat transfer in such engineering systems is the inherently low thermal conductivity of the commonly used fluids, such as, air, water, and oil. Nanofluids were introduced in order to circumvent the above limitation [1]. A nanofluid is a dilute suspension of solid nanoparticles with the average size below 100 nm in a base fluid, such as, water, oil, and ethylene glycol. Nanofluids exhibit thermal properties superior to those of the base fluids of the conventional particle–fluid suspensions [2]. There are a number of studies on the thermal properties of the nanofluids. Xuan and Li [3] used the hot-wire apparatus to measure the thermal conductivity of nanofluids. They investigated the effects of the size, the shape, the volume fraction, and the thermo-physical properties of the nanoparticles on the thermal conductivity of the nanofluids. In another experimental work, Xie et al. [4] investigated the dependence of the thermal conductivity of the nanoparticle–fluid mixture on the thermo-physical properties of the base fluid. They showed that nanoparticle–fluid mixtures, containing only a small amount of the nanosized Al2 O3
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particles, had substantially higher thermal conductivities compared to that of the pure fluid. The forced convection, as well as the buoyancy-driven fluid flow and heat transfer in enclosures filled with nanofluids have been studied by several investigations. Khanafer et al. [5] considered the buoyancy-driven heat transfer in a two-dimensional cavity filled with nanofluids for the Grashof numbers of 103 , 104 and 105 using the finite volume method. They concluded that the existence of the nanoparticles resulted in an increase in the rate of heat transfer for all the Grashof numbers considered. Using the finite volume method, Maiga et al. [6] studied the problem of laminar forced convection fluid flow and heat transfer of nanofluids in a uniformly heated tube, as well as, a system of parallel, coaxial, heated disks. Their results showed that the inclusion of Al2 O3 nanoparticles in water and ethylene-glycol resulted in a considerable augmentation of the heat transfer coefficient in the considered geometrical configurations. Jou and Tzeng [7] conducted a numerical study of the natural convection heat transfer in rectangular enclosures filled with a nanofluid using the finite difference method with the stream functionvorticity formulation. They investigated the effects of the Rayleigh number, the aspect ratio of the enclosure, and the volume fraction of the nanoparticles on the heat transfer inside the enclosures. Their results showed that the average heat transfer coefficient increased with increasing the volume fraction of the nanoparticles. More recently, Polidori et al. [8] used the integral formulation approach to investigate the natural convection heat transfer of Alumina–water nanofluid in a laminar, external, boundary layer
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Nomenclature cp g h H L k N Nu Nuavg p P Pr Ra T u, v U, V
Specific heat (kJ kg−1 K−1 ) Gravitational acceleration (m s−2 ) Heat transfer coefficient (W m−2 K−1 ) Enclosure height (m) Enclosure width (m) Thermal conductivity (W m−1 K−1 ) Number of grid points in each direction Nusselt number Average Nusselt number Pressure (Nm−2 ) Dimensionless pressure Prandtl number Rayleigh number Temperature (K) Velocity components Dimensionless velocity components
Greek symbols
α β ϕ ν θ ψ ρ µ
Thermal diffusivity (m2 s−1 ) Thermal expansion coefficient (K−1 ) Nanoparticle volume fraction Kinematic viscosity (m2 s−1 ) Dimensionless temperature Stream function (m2 s−1 ) Density (kg m−3 ) Dynamic viscosity (N s m−2 )
Fig. 1. Domain and boundary conditions for a square cavity with partially active side walls.
and the volume fraction of the nanoparticles on the heat transfer inside the enclosures. They found that the mean Nusselt number increased with increasing the volume fraction of the nanoparticles. Moreover, the location of the heater affected the flow and temperature fields inside the cavities. In the present study, the natural convection in a square cavity with partially active side walls filled with the Cu–water nanofluid is studied numerically using the finite volume method. A parametric study is performed and the effects of pertinent parameters, such as the locations of the heat source and the heat sink on the side walls, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the enclosure are investigated. 2. The problem formulation
Subscripts nf f h c s
Nanofluid Fluid Hot Cold Solid
flow. They examined different viscosity models, and concluded that, in addition to the nanofluid thermal conductivity, its viscosity also played a significant role in the heat transfer enhancement. The buoyancy-driven heat transfer in square cavities filled with air with partially active vertical walls was studied numerically by Valencia and Frederick [9]. They considered different relative positions of the active parts of the walls for the Rayleigh numbers of 103 –107 . Their results showed that the maximum heat transfer occurred for the heat source positioned at the middle of the hot wall. Deng et al. [10] used the finite volume method to study the laminar natural convection in a rectangular enclosure with discrete heat sources on its side walls. They concluded that an isothermal heat source had a stronger effect on the fluid flow and heat transfer inside the cavity compared to a constant heat flux one. The natural convection heat transfer in a square cavity filled with air with partially active side walls was studied by Nithyadevi et al. [11] using the finite volume method. They investigated the effects of the aspect ratio of the cavity and different relative positions of the active zones on the heat transfer inside the cavity. Their results showed that the heat transfer rate was high for the bottom–top thermally active location, while it was low for the top–bottom thermally active location. Very recently, Oztop and Abu-Nada [12] employed the finite volume method to study the natural convection in rectangular enclosures with heaters on their left vertical walls filled with nanofluids. They investigated the effects of the Rayleigh number, heater height, location of the heater, aspect ratio of the cavity,
A schematic diagram of the cavity with the width L and the height H (L = H) is shown in Fig. 1. An isothermal heat source with the temperature Th , and a heat sink with a constant temperature of Tc (Th > Tc ) are placed along the left and the right side walls of the cavity, respectively. The length of the heat source which is the same as that of the heat sink is equal to L/2 (Fig. 1). The positions of the heat source and the heat sink along the side walls are symmetric with respect to the horizontal centerline of the cavity for the case shown in this figure. This configuration of the source and the sink is called middle–middle for brevity. Other configurations are generated by changing the relative positions of the source and the sink on the side walls. The top and the bottom walls of the cavity, as well as, the inactive portions of the side walls are kept insulated. The cavity is filled with the Cu–water nanofluid which is assumed to be Newtonian. The nanoparticles are assumed to be uniform in shape and size. Moreover, the base fluid and the nanoparticles are assumed to be in thermal equilibrium. Other assumptions are that the flow is incompressible and laminar. Moreover, there is no slip between the water and the nanoparticles. The thermo-physical properties of the nanofluid are considered to be constant with the exception of its density which varies according to the Boussinesq approximation. The steady-state continuity, momentum, and energy equations for the buoyancy-driven fluid flow and heat transfer of the nanofluid inside the cavity employing the Boussinesq approximation for the buoyancy term are given by
∂ u ∂v + = 0, ∂x ∂y 2 ∂u ∂u ∂p ∂ u ∂ 2u ρnf u +v = − + µnf + , ∂x ∂y ∂x ∂ x2 ∂ y2 2 ∂v ∂v ∂p ∂ v ∂ 2v ρnf u +v = − + µnf + ∂x ∂y ∂y ∂ x2 ∂ y2 + g (ρβ)nf (T − Tc ) ,
(1)
(2)
(3)
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Table 1 Thermo-physical properties of the base fluid and the nanoparticles. Properties
Base fluid (water)
Nanoparticles (Cu)
Nanoparticles (Al2 O3 )
µ (Ns/m2 ) cp (J/kg K) ρ (kg/m3 ) k (W/mK) β (K−1 ) α (m2 /s)
8.55 × 10−4 4179 997.1 0.613 2.761 × 10−4 1.47 × 107
– 385 8933 401 1.67 × 10−5 1.17 × 10−4
765 3970 40 0.85 × 10−5 1.32 × 10−5
2 ∂T ∂T ∂ 2T ∂ T +v = αnf + , (4) ∂x ∂y ∂ x2 ∂ y2 where u and v are the velocity components in the x and y-
u
directions, respectively. p is the pressure, T is the temperature, ϕ is the volume fraction of the nanoparticles, and β is the thermal expansion coefficient. The thermo-physical properties of the nanofluid are obtained from the following relations [5,13]:
ρnf = (1 − ϕ) ρf + ϕρs , ρ cp nf = (1 − ϕ) ρ cp f + ϕ ρ cp s ,
(5)
(ρβ)nf = (1 − ϕ) (ρβ)f + ϕ (ρβ)s ,
(7)
knf kf
ks + 2kf − 2ϕ(kf − ks )
=
ks + 2kf + ϕ(kf − ks )
(6)
,
(8)
, (ρ cp )nf µf µnf = . (1 − ϕ)2.5
(9) (10)
The thermo-physical properties of the base fluid and the nanoparticles (at 300 K) are presented in Table 1. Using the dimensionless variables X = P =
x L
,
pL2
Y =
, 2
ρnf αf
y L
,
U =
θ=
T − Tc Th − Tc
uL
αf
,
V =
vL , αf
(11)
,
the governing equations are written in the following dimensionless form:
∂U ∂V + = 0, ∂X ∂Y ∂U ∂U ρf ∂ P µnf ∂ 2 U ∂ 2U U +V =− + + , ∂X ∂Y ρnf ∂ X αρnf ∂ X 2 ∂Y 2 2 ∂V ρf ∂ P µnf ∂ 2V ∂V ∂ V U +V = − + + ∂X ∂Y ρnf ∂ Y αf ρnf ∂ X 2 ∂Y 2 (ρβ)nf + RaPrθ , βf ρnf ∂θ ∂θ αnf ∂ 2 θ ∂ 2θ U +V = + . ∂X ∂Y αf ∂ X 2 ∂Y 2
(12) (13)
(14)
(15)
In the above equations, the Prandtl number, Pr, and the Rayleigh number, Ra, are defined as: Pr =
νf , αf
Ra =
g ρf βf (Th − Tc )L3
µf αf
.
(16)
The boundary conditions for Eqs. (12)–(15) in the dimensionless form are given by U = V = 0, θ = 0, θ = 1, ∂θ/∂ n = 0,
where ∂/∂ n denotes the derivative with respect to the normal direction. The average Nusselt number is calculated from the following relation Nuavg =
hL kf
where Y1 = in Fig. 1.
=2 y1 L
knf kf
∫
Y2 Y1
and Y2 =
∂θ dY , ∂ X X =0 y2 . L
(18)
The coordinates y1 and y2 are defined
3. Numerical procedure
knf
αnf =
Fig. 2. Domain and boundary conditions for the first test case.
on the cavity walls on the heat sink on the heat source on the insulated walls
(17)
The governing equations and the associated boundary conditions are solved numerically using the finite volume method. The diffusion terms in the governing equations are discretized using a second-order central difference scheme; while, an upwind scheme is employed to discretize the convective terms. A staggered grid system together with the SIMPLER-algorithm is adopted to solve for the pressure and the velocity components. The coupled set of discretized equations is solved iteratively using the TDMA method [14]. To obtain converged solutions, an under-relaxation scheme is employed. In order to validate the numerical procedure, different test cases are solved using the developed code, and the results are compared with the existing results in the literature. A grid independence study is also undertaken in order to determine the proper mesh for the numerical calculations. 4. Benchmarking of the code In order to validate the numerical procedure, two test cases are examined using the proposed code, and the results are compared with the existing numerical results in the literature. The first test case considered is the buoyancy-driven fluid flow and heat transfer in a differentially-heated square cavity filled with air (Fig. 2). The left and the right side walls of the cavity are maintained at constant temperatures Th and Tc , respectively, with Th > Tc ; while, its top and bottom walls are insulated (Fig. 2). An 80 × 80 uniform grid is used for the numerical calculations. Table 2 shows the average Nusselt number of the hot side wall for different Rayleigh numbers obtained by the present simulation. The results of other investigators for the same problem are also presented in the table. As it can be observed from this table, very good agreements exist between the Nusselt numbers obtained by the present simulation and the results of other investigators for the Rayleigh numbers between 103 and 106 . The second test case considered is the buoyancy-driven fluid flow and heat transfer in a square cavity with a heat source placed in the middle of its left wall filled with the nanofluid. The length of the heat source is equal to half of the length of the cavity (Fig. 3). The heat source and the right wall of the cavity are maintained at constant temperatures Th and Tc , respectively, with Th > Tc ; while
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169
Table 2 The average Nusselt number of the hot wall for the first test code, comparisons of the present results with the results of other investigators.
Present work Khanafer et al. [5] Barakos and Mitsoulis [16] Markatos and Pericleous [17] De Vahl Davis [18] Fusegi et al. [19]
Ra = 103
Ra = 104
Ra = 105
Ra = 106
1.148 1.118 1.114
2.311 2.245 2.245
4.651 4.522 4.510
9.010 8.826 8.806
1.108
2.201
4.430
8.754
1.118 1.105
2.243 2.302
4.519 4.646
8.799 9.012
the remaining portion of the cavity’s left wall, as well as, its top and bottom walls are insulated (Fig. 3). An 80 × 80 uniform grid is employed for the numerical simulation. Fig. 4(a)–(c) show the variation of the average Nusselt number of the heat source with respect to the volume fraction of the nanoparticles for different Rayleigh numbers. The results of Oztop and Abu-Nada [12] and Aminossadati and Ghasemi [15] for the same problem are also shown in these figures. For the three Rayleigh numbers considered, Ra = 103 , 104 , and 105 , excellent agreements are observed between the average Nusselt number of the heat source obtained by the present simulation with the results of Aminossadati and Ghasemi [15] for different volume fractions of the nanoparticles (Fig. 4). Moreover, as it is observed from Fig. 4, the variation of the average Nusselt number of the heat source with respect to the volume fraction of the nanoparticles obtained by the present
(a) Ra = 103 .
Fig. 3. Domain and boundary conditions for the second test case.
simulation has a trend similar to the results of Oztop and AbuNada [12]. The maximum difference between the average Nusselt number obtained by the present study and the results of Oztop and Abu-Nada [12] is 2.2%. 5. Grid independence study In order to determine a proper grid for the numerical simulation, a grid independence study is undertaken for the buoyancy-driven fluid flow and heat transfer inside a square cavity with partially active side walls filled with the Cu–water nanofluid.
(b) Ra = 104 .
(c) Ra = 105 . Fig. 4. Comparison of the average Nusselt number of the heat source at different volume fraction of nanoparticles for the second test case with the results of other investigators.
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Fig. 5. Effect of the grid density on the vertical centerline velocity, V (X , 0.5), for the middle–middle case, Ra = 106 and ϕ = 0.05.
Eight different uniform grids, namely, 20 × 20, 40 × 40, 60 × 60, 80 × 80, 100 × 100, 120 × 120, 140 × 140, and 160 × 160 are employed to simulate the natural convection inside a square cavity with the heat source and the heat sink positioned at the middle of the vertical walls (the middle–middle case) for ϕ = 0.05. Fig. 5 shows the cavity’s vertical centerline velocity, V (X , 0.5), for different grids obtained by the current simulation for Ra = 106 . The variations of the average Nusselt number of the heat source with the respect to the number of grid points in each direction for different Rayleigh numbers are depicted in Fig. 6. As it can be observed from these figures, a 80 × 80 uniform grid is sufficiently fine to ensure a grid independent solution. Hence, this grid is used to perform all the subsequent calculations. 6. Results and discussions Having performed the grid independence study, and verified the present code performance via solving different test case, the code is then employed to study the natural convection in a rectangular cavity with partially active side walls filled with the nanofluids. Using the code, a parametric is performed and the effects of the pertinent parameters, such as, the Rayleigh number, the position of the heat source and the heat sink, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the cavity are investigated. Nine different configurations of the source
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
Fig. 6. Variation of the average Nusselt number of the heat source for the middle–middle case with respect to the grid size for different Ra, ϕ = 0.05.
and the sink as far as their positions on the side walls of the cavity, namely, top–top, top–middle, top–bottom, middle–top, middle–middle, middle–bottom, bottom–top, bottom–middle, and bottom–bottom, are considered in this study. For all of the nine configurations, the length of the heat source, which is the same as that of the heat sink, is equal to half of the cavity’s height (width). All the results presented below are for the Prandtl number of the base fluid equal to 5.83. Figs. 7 and 8 show the streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the source and the sink on the vertical walls of the cavity at Ra = 103 , respectively. For the sake of comparisons, the streamlines and the isotherms for pure water (ϕ = 0.0) are also shown in these figures. For Ra = 103 , the heat transfer inside the cavity is mainly through conduction. A clockwise vortex develops inside the cavity for these cases. At this Rayleigh number, changing the locations of active portions of the side walls does not have a significant effect on the streamline patterns. However, as it can be seen from Fig. 8, changing the positions of the heat source and the heat sink on the side walls of the cavity have a dramatic effect on the temperature distribution inside the cavity at Ra = 103 . Figs. 9 and 10 show the streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the source and the sink on the vertical walls of the cavity at Ra = 104 , respectively. For the sake of comparisons, the streamlines and the isotherms for pure water (ϕ = 0.0) are also shown in these figures.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 7. Streamlines for different position of the source and the sink at Ra = 103 , — ϕ = 0.0, - - - ϕ = 0.15.
G.A. Sheikhzadeh et al. / European Journal of Mechanics B/Fluids 30 (2011) 166–176
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
171
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 8. Isotherms for different position of the source and the sink at Ra = 103 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 9. Streamlines for different position of the source and the sink at Ra = 104 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(g) Bottom–top.
(c) Top–bottom.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 10. Isotherms for different position of the source and the sink at Ra = 104 , — ϕ = 0.0, - - - ϕ = 0.15.
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(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(e) Middle–middle.
(h) Bottom–middle.
(i) Bottom–bottom.
Fig. 11. Streamlines for different position of the source and the sink at Ra = 105 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 12. Isotherms for different position of the source and the sink at Ra = 105 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 13. Streamlines for different position of the source and the sink at Ra = 106 , — ϕ = 0.0, - - - ϕ = 0.15.
G.A. Sheikhzadeh et al. / European Journal of Mechanics B/Fluids 30 (2011) 166–176
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
173
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 14. Isotherms for different position of the source and the sink at Ra = 106 , — ϕ = 0.0, - - - ϕ = 0.15.
Fig. 15. Variations of the average Nusselt number of the heat source with respect to the volume fraction of the nanoparticles for different configurations of the source and the sink at different Rayleigh numbers.
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Fig. 16. Variations of the stream function with respect to the volume fraction of the nanoparticles for different configurations of the source and the sink at different Rayleigh numbers.
With increasing the Rayleigh number, the effect of the convection heat transfer becomes more significant, and distinct boundary layers form along the active walls of the cavity. The compact streamlines and isotherms in the boundary layers along the active side walls indicate higher velocities and intense convection heat transfer in these areas. The core region of the cavity is relatively stagnant. The isotherms in the stratified core region are nearly parallel to the horizontal walls of the cavity. The streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the active portions of the side walls of the cavity at Ra = 105 , are shown in Figs. 11 and 12, respectively. The results for pure water (ϕ = 0.0) are also shown in these figures for comparison purposes. As it can be seen from Fig. 11(a), in the top–top configuration, a primary recirculation zone forms inside the cavity with its core being near the heat sink and expanding towards the heat source. Moving the heat sink from the top to the middle of the left wall of the cavity (Fig. 11(b)) enlarges the central vortex. Moreover, two secondary inner vortices form in the latter case with the inner vortex near the hot active wall being smaller than the one near the cold active wall. Two distinct inner vortices near the heat source and the heat sink for the top–bottom case are observed in Fig. 11(c). For the middle–top configuration, the core of the vortex for the nanofluid is larger than the one for the base fluid (Fig. 11(d)). By moving the heat sink from the middle to the bottom of the cavity’s left wall (Fig. 11(f)), the core region of the streamlines divides into two vortices of different sizes. As Fig. 11(g) shows, for
the bottom–top configuration, a single vortex with a protracted center exists in the cavity. The maximum and the minimum absolute values of the stream functions (|ψ|max = 12.8, |ψ|max = 3.4) occur for the bottom–top and the top–bottom configurations, respectively. As Fig. 11(h) and (i) show the central vortex becomes smaller for the bottom–middle and the –bottom configurations. As it is observed from Fig. 12, the temperature gradients are quite large in the vicinity of the active portions of the side walls of the cavity. The heat transfer in these regions is mainly through convection. Moreover, in the stratified core region of the cavity, the isotherms are nearly horizontal, and the heat transfer is conduction dominated. The isotherms are more condensed in the core region for the middle–middle configuration (Fig. 12(e)). The streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the active portions of the side walls of the cavity at Ra = 106 are shown in Figs. 13 and 14, respectively. The results for pure water (ϕ = 0.0) are also shown in these figures for comparison purposes. As it is observed from these figures, for this convection-dominated heat transfer regime, the streamlines and the isotherms are more packed next to the active portions of side walls of the cavity. Moreover, the core region is protracted along the horizontal direction for all the cases considered in the figures. In the bottom–top configuration, a single protracted central vortex develops inside the cavity for the nanofluid, but three secondary inner vortices form for the base fluid. The inner vortices near the active walls of the cavity are of the same size (Fig. 13(g)). The isotherms are condensed in
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(a) Ra = 103 .
(b) Ra = 104 .
175
(c) Ra = 105 .
(d) Ra = 106 . Fig. 17. Variations of the average Nusselt number for the middle–middle location of the active parts with respect to the volume fraction of nanoparticles at different Rayleigh numbers.
the cavity center and near the active walls. The maximum and the minimum absolute values of the stream functions (|ψ|max = 25.0, |ψ|max = 5.6) in this case occur for the bottom–top and the top–bottom configurations, respectively. The variations of the average Nusselt number of the heat source and the maximum absolute value of the stream function with respect to the volume fraction of the nanoparticles for different positions of the heat source and the heat sink on the side walls of the cavity at the different Rayleigh numbers are displayed in Figs. 15 and 16, respectively. As it can be observed from Fig. 15, the heat transfer increases with increasing the volume fraction of the nanoparticles for all the configurations, and for the entire range of the Rayleigh numbers considered. Moreover, for Ra = 103 and 104 , the average Nusselt number of the heat source is maximum when the heat sink is located at the middle of the right vertical wall of the cavity irrespective of the position of the heat source. For Ra = 104 , 105 , and 106 , the average Nusselt number of the heat source is minimum when the heat sink is located at the bottom of the right vertical wall of the cavity (Fig. 15). Moreover, Fig. 15 shows that for Ra = 103 and 104 the maximum heat transfer occurs for the middle–middle configuration; while, for Ra = 105 and 106 , the bottom–middle configuration has the maximum heat transfer coefficient. The heat transfer inside the cavity occurs mainly through conduction at Ra = 103 . By increasing the volume fraction of nanoparticles, the viscosity and the thermal conductivity of the nanofluid, and, consequently, the diffusion of heat inside the cavity are increased. Therefore, the temperature gradient and the buoyancy force are decreased. Similar conclusions can be drawn from Fig. 16 which shows the variations of the stream functions with respect to the volume fraction of the nanoparticles. The natural convection inside the cavity diminishes with increasing the volume fractions of the nanoparticles. However, the average
Nusselt number increases under these conditions. The increase of the average Nusselt number is attributed to the increases of the thermal conductivity of the nanofluid with increasing the volume fraction of the nanoparticles. For a moderate Rayleigh number (Ra = 104 ), the conduction and convection heat transfers are comparable. It is observed from Fig. 16 that, for the top–bottom and the bottom–top configurations, the maximum absolute value of the stream function (|ψ|max ) increases and decreases, respectively, with increasing the volume fraction of the nanoparticles for Ra = 104 . However, for the rest of the configurations shown in Fig. 16, |ψ|max initially increases and then decreases with increasing the volume fraction of the nanoparticles for Ra = 104 . The maximum absolute values of the stream function in Fig. 16 show that, for Ra = 104 , the free convection heat transfer has the minimum and the maximum strengths in the top–bottom and bottom–top configurations, respectively. For low and moderate Rayleigh numbers (Ra = 103 , 104 ), the heat transfer inside the cavity occurs mainly through conduction. At these Rayleigh numbers, the convection strength is weak. The values of |ψ|max show that the middle–middle configuration has the most convection strength compared to the other configurations (Fig. 16). Therefore, the middle–middle configuration has the maximum average Nusselt number at these Rayleigh numbers (Fig. 15). The convection heat transfer mechanism becomes dominant at high Rayleigh numbers (Ra = 105 and 106 ). As it is observed from Fig. 16, the bottom–top configuration has the highest rate of convection heat transfer for high Rayleigh numbers. However, the rate of heat transfer through conduction is the lowest for this configuration. Keeping the heat source at the bottom location, the convection heat transfer increases and the conduction decreases by moving the heat sink from the bottom to
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inside the cavity were investigated. The results show that, for all the Rayleigh numbers considered, the minimum average Nusselt number occurred for the top–bottom case of the active portions of the side walls, but the maximum average Nusselt number occurred for the middle–middle case for Ra = 103 and 104 and the middle–bottom case for Ra = 105 and 106 . Moreover, the heat transfer rate increases with increasing the volume fraction of the nanoparticles. References
Fig. 18. The correlation results for the average Nusselt number of the heat source for the middle–middle configuration.
the top of the right wall of the cavity. Therefore, as it is observed from Fig. 15, the maximum average Nusselt number occurs for the bottom–middle configuration at Ra = 105 and 106 . Fig. 17 shows the variation of the average Nusselt number with respect to the volume fraction of the nanoparticles for the middle–middle configuration of the active parts at different Rayleigh numbers. The presented results in this figure are for the Cu–water and Al2 O3 –water nanofluids. As the figure shows, the average Nusselt number for Cu–water nanofluid is higher than that of the Al2 O3 –water nanofluid. Using the numerical results, the following correlation is obtained for the average Nusselt number for middle–middle configuration in terms of the Rayleigh number (103 ≤ Ra ≤ 106 ), and the volume fraction of the nanoparticles (0.0 ≤ ϕ ≤ 0.15). Nuavg = (0.4778ϕ 1.0 + 0.3554)Ra0.25 .
(19)
The average Nusselt number obtained from Eq. (19) together with the numerical results are presented in Fig. 18. 7. Conclusions The two-dimensional buoyancy-driven fluid flow and heat transfer in a square cavity with partially active side walls filled with Cu–water nanofluid have been simulated numerically using the finite volume method. A parametric study was undertaken and the effects of the position of active portions of the side walls, and the volume fraction of the nanoparticles on the heat transfer
[1] U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Div. 231 (1995) 99–105. [2] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, John Wiley & Sons Inc., New York, 2007. [3] Y. Xuan, Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow 21 (2000) 58–64. [4] H.Q. Xie, J.C. Wang, T.G. Xi, Y. Li, F. Ai, Dependence of the thermal conductivity of nanoparticle–fluid mixture on the base fluid, J. Mater. Sci. Lett. 21 (2002) 1469–1471. [5] 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. [6] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [7] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Commun. Heat Mass Transfer 33 (2006) 727–736. [8] G. Polidori, S. Fohanno, C.T. Nguyen, A note on heat transfer modeling of Newtonian nanofluids in laminar free convection, Int. J. Therm. Sci. 46 (2007) 739–744. [9] A. Valencia, R.L. Frederick, Heat transfer in square cavities with partially active vertical walls, Int. J. Heat Mass Transfer 32 (1998) 1567–1574. [10] Q.H. Deng, G.F. Tang, Y. Li, A combined temperature scale for analyzing natural convection in rectangular enclosures with discrete wall heat sources, Int. J. Heat Mass Transfer 45 (2002) 3437–3446. [11] N. Nithyadevi, P. Kandaswamy, J. Lee, Natural convection in a rectangular cavity with partially active side walls, Int. J. Heat Mass Transfer 50 (2007) 4688–4697. [12] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [13] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [14] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, Washington, DC, 1980. [15] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B Fluids 28 (2009) 630–640. [16] G. Barakos, E. Mitsoulis, Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions, Internat. J. Numer. Methods Fluids 18 (1994) 695–719. [17] N.C. Markatos, K.A. Pericleous, Laminar and turbulent natural convection in an enclosed cavity, Int. J. Heat Mass Transfer 27 (1984) 772–775. [18] G. De Vahl Davis, Natural convection of air in a square cavity, a benchmark numerical solution, Internat. J. Numer. Methods Fluids 3 (1983) 249–264. [19] T. Fusegi, J.M. Hyun, K. Kuwahara, B. Farouk, A numerical study of three dimensional natural convection in a differentially heated cubical enclosure, Int. J. Heat Mass Transfer 34 (1991) 1543–1557.
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Natural convection of Cu–water nanofluid in a cavity with partially active side walls G.A. Sheikhzadeh ∗ , A. Arefmanesh, M.H. Kheirkhah, R. Abdollahi Department of Mechanical Engineering, University of Kashan, Kashan, Iran
article
info
Article history: Received 12 April 2010 Received in revised form 14 August 2010 Accepted 12 October 2010 Available online 20 October 2010 Keywords: Nanofluid Natural convection Numerical study Partially active walls Cavity
abstract The buoyancy-driven fluid flow and heat transfer in a square cavity with partially active side walls filled with Cu–water nanofluid is investigated numerically. The active parts of the left and the right side walls of the cavity are maintained at temperatures Th and Tc , respectively, with Th > Tc . The enclosure’s top and bottom walls as well as the inactive parts of its side walls are kept insulated. The governing equations in the two-dimensional space are discretized using the control volume method. A proper upwinding scheme is employed to obtain stabilized solutions. Using the developed code, a parametric study is undertaken, and the effects of the Rayleigh number, the locations of the active parts of the side walls, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the cavity are investigated. It is observed from the results that the average Nusselt number increases with increasing both the Rayleigh number and the volume fraction of the nanoparticles. Moreover, the maximum average Nusselt number for the high and the low Rayleigh numbers occur for the bottom–middle and the middle–middle locations of the thermally active parts, respectively. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction Natural convection heat transfer is an important phenomenon in engineering and industry with widespread applications in diverse fields, such as, geophysics, solar energy, electronic cooling, and nuclear energy to mention a few. A major limitation against enhancing the heat transfer in such engineering systems is the inherently low thermal conductivity of the commonly used fluids, such as, air, water, and oil. Nanofluids were introduced in order to circumvent the above limitation [1]. A nanofluid is a dilute suspension of solid nanoparticles with the average size below 100 nm in a base fluid, such as, water, oil, and ethylene glycol. Nanofluids exhibit thermal properties superior to those of the base fluids of the conventional particle–fluid suspensions [2]. There are a number of studies on the thermal properties of the nanofluids. Xuan and Li [3] used the hot-wire apparatus to measure the thermal conductivity of nanofluids. They investigated the effects of the size, the shape, the volume fraction, and the thermo-physical properties of the nanoparticles on the thermal conductivity of the nanofluids. In another experimental work, Xie et al. [4] investigated the dependence of the thermal conductivity of the nanoparticle–fluid mixture on the thermo-physical properties of the base fluid. They showed that nanoparticle–fluid mixtures, containing only a small amount of the nanosized Al2 O3
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particles, had substantially higher thermal conductivities compared to that of the pure fluid. The forced convection, as well as the buoyancy-driven fluid flow and heat transfer in enclosures filled with nanofluids have been studied by several investigations. Khanafer et al. [5] considered the buoyancy-driven heat transfer in a two-dimensional cavity filled with nanofluids for the Grashof numbers of 103 , 104 and 105 using the finite volume method. They concluded that the existence of the nanoparticles resulted in an increase in the rate of heat transfer for all the Grashof numbers considered. Using the finite volume method, Maiga et al. [6] studied the problem of laminar forced convection fluid flow and heat transfer of nanofluids in a uniformly heated tube, as well as, a system of parallel, coaxial, heated disks. Their results showed that the inclusion of Al2 O3 nanoparticles in water and ethylene-glycol resulted in a considerable augmentation of the heat transfer coefficient in the considered geometrical configurations. Jou and Tzeng [7] conducted a numerical study of the natural convection heat transfer in rectangular enclosures filled with a nanofluid using the finite difference method with the stream functionvorticity formulation. They investigated the effects of the Rayleigh number, the aspect ratio of the enclosure, and the volume fraction of the nanoparticles on the heat transfer inside the enclosures. Their results showed that the average heat transfer coefficient increased with increasing the volume fraction of the nanoparticles. More recently, Polidori et al. [8] used the integral formulation approach to investigate the natural convection heat transfer of Alumina–water nanofluid in a laminar, external, boundary layer
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Nomenclature cp g h H L k N Nu Nuavg p P Pr Ra T u, v U, V
Specific heat (kJ kg−1 K−1 ) Gravitational acceleration (m s−2 ) Heat transfer coefficient (W m−2 K−1 ) Enclosure height (m) Enclosure width (m) Thermal conductivity (W m−1 K−1 ) Number of grid points in each direction Nusselt number Average Nusselt number Pressure (Nm−2 ) Dimensionless pressure Prandtl number Rayleigh number Temperature (K) Velocity components Dimensionless velocity components
Greek symbols
α β ϕ ν θ ψ ρ µ
Thermal diffusivity (m2 s−1 ) Thermal expansion coefficient (K−1 ) Nanoparticle volume fraction Kinematic viscosity (m2 s−1 ) Dimensionless temperature Stream function (m2 s−1 ) Density (kg m−3 ) Dynamic viscosity (N s m−2 )
Fig. 1. Domain and boundary conditions for a square cavity with partially active side walls.
and the volume fraction of the nanoparticles on the heat transfer inside the enclosures. They found that the mean Nusselt number increased with increasing the volume fraction of the nanoparticles. Moreover, the location of the heater affected the flow and temperature fields inside the cavities. In the present study, the natural convection in a square cavity with partially active side walls filled with the Cu–water nanofluid is studied numerically using the finite volume method. A parametric study is performed and the effects of pertinent parameters, such as the locations of the heat source and the heat sink on the side walls, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the enclosure are investigated. 2. The problem formulation
Subscripts nf f h c s
Nanofluid Fluid Hot Cold Solid
flow. They examined different viscosity models, and concluded that, in addition to the nanofluid thermal conductivity, its viscosity also played a significant role in the heat transfer enhancement. The buoyancy-driven heat transfer in square cavities filled with air with partially active vertical walls was studied numerically by Valencia and Frederick [9]. They considered different relative positions of the active parts of the walls for the Rayleigh numbers of 103 –107 . Their results showed that the maximum heat transfer occurred for the heat source positioned at the middle of the hot wall. Deng et al. [10] used the finite volume method to study the laminar natural convection in a rectangular enclosure with discrete heat sources on its side walls. They concluded that an isothermal heat source had a stronger effect on the fluid flow and heat transfer inside the cavity compared to a constant heat flux one. The natural convection heat transfer in a square cavity filled with air with partially active side walls was studied by Nithyadevi et al. [11] using the finite volume method. They investigated the effects of the aspect ratio of the cavity and different relative positions of the active zones on the heat transfer inside the cavity. Their results showed that the heat transfer rate was high for the bottom–top thermally active location, while it was low for the top–bottom thermally active location. Very recently, Oztop and Abu-Nada [12] employed the finite volume method to study the natural convection in rectangular enclosures with heaters on their left vertical walls filled with nanofluids. They investigated the effects of the Rayleigh number, heater height, location of the heater, aspect ratio of the cavity,
A schematic diagram of the cavity with the width L and the height H (L = H) is shown in Fig. 1. An isothermal heat source with the temperature Th , and a heat sink with a constant temperature of Tc (Th > Tc ) are placed along the left and the right side walls of the cavity, respectively. The length of the heat source which is the same as that of the heat sink is equal to L/2 (Fig. 1). The positions of the heat source and the heat sink along the side walls are symmetric with respect to the horizontal centerline of the cavity for the case shown in this figure. This configuration of the source and the sink is called middle–middle for brevity. Other configurations are generated by changing the relative positions of the source and the sink on the side walls. The top and the bottom walls of the cavity, as well as, the inactive portions of the side walls are kept insulated. The cavity is filled with the Cu–water nanofluid which is assumed to be Newtonian. The nanoparticles are assumed to be uniform in shape and size. Moreover, the base fluid and the nanoparticles are assumed to be in thermal equilibrium. Other assumptions are that the flow is incompressible and laminar. Moreover, there is no slip between the water and the nanoparticles. The thermo-physical properties of the nanofluid are considered to be constant with the exception of its density which varies according to the Boussinesq approximation. The steady-state continuity, momentum, and energy equations for the buoyancy-driven fluid flow and heat transfer of the nanofluid inside the cavity employing the Boussinesq approximation for the buoyancy term are given by
∂ u ∂v + = 0, ∂x ∂y 2 ∂u ∂u ∂p ∂ u ∂ 2u ρnf u +v = − + µnf + , ∂x ∂y ∂x ∂ x2 ∂ y2 2 ∂v ∂v ∂p ∂ v ∂ 2v ρnf u +v = − + µnf + ∂x ∂y ∂y ∂ x2 ∂ y2 + g (ρβ)nf (T − Tc ) ,
(1)
(2)
(3)
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Table 1 Thermo-physical properties of the base fluid and the nanoparticles. Properties
Base fluid (water)
Nanoparticles (Cu)
Nanoparticles (Al2 O3 )
µ (Ns/m2 ) cp (J/kg K) ρ (kg/m3 ) k (W/mK) β (K−1 ) α (m2 /s)
8.55 × 10−4 4179 997.1 0.613 2.761 × 10−4 1.47 × 107
– 385 8933 401 1.67 × 10−5 1.17 × 10−4
765 3970 40 0.85 × 10−5 1.32 × 10−5
2 ∂T ∂T ∂ 2T ∂ T +v = αnf + , (4) ∂x ∂y ∂ x2 ∂ y2 where u and v are the velocity components in the x and y-
u
directions, respectively. p is the pressure, T is the temperature, ϕ is the volume fraction of the nanoparticles, and β is the thermal expansion coefficient. The thermo-physical properties of the nanofluid are obtained from the following relations [5,13]:
ρnf = (1 − ϕ) ρf + ϕρs , ρ cp nf = (1 − ϕ) ρ cp f + ϕ ρ cp s ,
(5)
(ρβ)nf = (1 − ϕ) (ρβ)f + ϕ (ρβ)s ,
(7)
knf kf
ks + 2kf − 2ϕ(kf − ks )
=
ks + 2kf + ϕ(kf − ks )
(6)
,
(8)
, (ρ cp )nf µf µnf = . (1 − ϕ)2.5
(9) (10)
The thermo-physical properties of the base fluid and the nanoparticles (at 300 K) are presented in Table 1. Using the dimensionless variables X = P =
x L
,
pL2
Y =
, 2
ρnf αf
y L
,
U =
θ=
T − Tc Th − Tc
uL
αf
,
V =
vL , αf
(11)
,
the governing equations are written in the following dimensionless form:
∂U ∂V + = 0, ∂X ∂Y ∂U ∂U ρf ∂ P µnf ∂ 2 U ∂ 2U U +V =− + + , ∂X ∂Y ρnf ∂ X αρnf ∂ X 2 ∂Y 2 2 ∂V ρf ∂ P µnf ∂ 2V ∂V ∂ V U +V = − + + ∂X ∂Y ρnf ∂ Y αf ρnf ∂ X 2 ∂Y 2 (ρβ)nf + RaPrθ , βf ρnf ∂θ ∂θ αnf ∂ 2 θ ∂ 2θ U +V = + . ∂X ∂Y αf ∂ X 2 ∂Y 2
(12) (13)
(14)
(15)
In the above equations, the Prandtl number, Pr, and the Rayleigh number, Ra, are defined as: Pr =
νf , αf
Ra =
g ρf βf (Th − Tc )L3
µf αf
.
(16)
The boundary conditions for Eqs. (12)–(15) in the dimensionless form are given by U = V = 0, θ = 0, θ = 1, ∂θ/∂ n = 0,
where ∂/∂ n denotes the derivative with respect to the normal direction. The average Nusselt number is calculated from the following relation Nuavg =
hL kf
where Y1 = in Fig. 1.
=2 y1 L
knf kf
∫
Y2 Y1
and Y2 =
∂θ dY , ∂ X X =0 y2 . L
(18)
The coordinates y1 and y2 are defined
3. Numerical procedure
knf
αnf =
Fig. 2. Domain and boundary conditions for the first test case.
on the cavity walls on the heat sink on the heat source on the insulated walls
(17)
The governing equations and the associated boundary conditions are solved numerically using the finite volume method. The diffusion terms in the governing equations are discretized using a second-order central difference scheme; while, an upwind scheme is employed to discretize the convective terms. A staggered grid system together with the SIMPLER-algorithm is adopted to solve for the pressure and the velocity components. The coupled set of discretized equations is solved iteratively using the TDMA method [14]. To obtain converged solutions, an under-relaxation scheme is employed. In order to validate the numerical procedure, different test cases are solved using the developed code, and the results are compared with the existing results in the literature. A grid independence study is also undertaken in order to determine the proper mesh for the numerical calculations. 4. Benchmarking of the code In order to validate the numerical procedure, two test cases are examined using the proposed code, and the results are compared with the existing numerical results in the literature. The first test case considered is the buoyancy-driven fluid flow and heat transfer in a differentially-heated square cavity filled with air (Fig. 2). The left and the right side walls of the cavity are maintained at constant temperatures Th and Tc , respectively, with Th > Tc ; while, its top and bottom walls are insulated (Fig. 2). An 80 × 80 uniform grid is used for the numerical calculations. Table 2 shows the average Nusselt number of the hot side wall for different Rayleigh numbers obtained by the present simulation. The results of other investigators for the same problem are also presented in the table. As it can be observed from this table, very good agreements exist between the Nusselt numbers obtained by the present simulation and the results of other investigators for the Rayleigh numbers between 103 and 106 . The second test case considered is the buoyancy-driven fluid flow and heat transfer in a square cavity with a heat source placed in the middle of its left wall filled with the nanofluid. The length of the heat source is equal to half of the length of the cavity (Fig. 3). The heat source and the right wall of the cavity are maintained at constant temperatures Th and Tc , respectively, with Th > Tc ; while
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Table 2 The average Nusselt number of the hot wall for the first test code, comparisons of the present results with the results of other investigators.
Present work Khanafer et al. [5] Barakos and Mitsoulis [16] Markatos and Pericleous [17] De Vahl Davis [18] Fusegi et al. [19]
Ra = 103
Ra = 104
Ra = 105
Ra = 106
1.148 1.118 1.114
2.311 2.245 2.245
4.651 4.522 4.510
9.010 8.826 8.806
1.108
2.201
4.430
8.754
1.118 1.105
2.243 2.302
4.519 4.646
8.799 9.012
the remaining portion of the cavity’s left wall, as well as, its top and bottom walls are insulated (Fig. 3). An 80 × 80 uniform grid is employed for the numerical simulation. Fig. 4(a)–(c) show the variation of the average Nusselt number of the heat source with respect to the volume fraction of the nanoparticles for different Rayleigh numbers. The results of Oztop and Abu-Nada [12] and Aminossadati and Ghasemi [15] for the same problem are also shown in these figures. For the three Rayleigh numbers considered, Ra = 103 , 104 , and 105 , excellent agreements are observed between the average Nusselt number of the heat source obtained by the present simulation with the results of Aminossadati and Ghasemi [15] for different volume fractions of the nanoparticles (Fig. 4). Moreover, as it is observed from Fig. 4, the variation of the average Nusselt number of the heat source with respect to the volume fraction of the nanoparticles obtained by the present
(a) Ra = 103 .
Fig. 3. Domain and boundary conditions for the second test case.
simulation has a trend similar to the results of Oztop and AbuNada [12]. The maximum difference between the average Nusselt number obtained by the present study and the results of Oztop and Abu-Nada [12] is 2.2%. 5. Grid independence study In order to determine a proper grid for the numerical simulation, a grid independence study is undertaken for the buoyancy-driven fluid flow and heat transfer inside a square cavity with partially active side walls filled with the Cu–water nanofluid.
(b) Ra = 104 .
(c) Ra = 105 . Fig. 4. Comparison of the average Nusselt number of the heat source at different volume fraction of nanoparticles for the second test case with the results of other investigators.
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Fig. 5. Effect of the grid density on the vertical centerline velocity, V (X , 0.5), for the middle–middle case, Ra = 106 and ϕ = 0.05.
Eight different uniform grids, namely, 20 × 20, 40 × 40, 60 × 60, 80 × 80, 100 × 100, 120 × 120, 140 × 140, and 160 × 160 are employed to simulate the natural convection inside a square cavity with the heat source and the heat sink positioned at the middle of the vertical walls (the middle–middle case) for ϕ = 0.05. Fig. 5 shows the cavity’s vertical centerline velocity, V (X , 0.5), for different grids obtained by the current simulation for Ra = 106 . The variations of the average Nusselt number of the heat source with the respect to the number of grid points in each direction for different Rayleigh numbers are depicted in Fig. 6. As it can be observed from these figures, a 80 × 80 uniform grid is sufficiently fine to ensure a grid independent solution. Hence, this grid is used to perform all the subsequent calculations. 6. Results and discussions Having performed the grid independence study, and verified the present code performance via solving different test case, the code is then employed to study the natural convection in a rectangular cavity with partially active side walls filled with the nanofluids. Using the code, a parametric is performed and the effects of the pertinent parameters, such as, the Rayleigh number, the position of the heat source and the heat sink, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the cavity are investigated. Nine different configurations of the source
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
Fig. 6. Variation of the average Nusselt number of the heat source for the middle–middle case with respect to the grid size for different Ra, ϕ = 0.05.
and the sink as far as their positions on the side walls of the cavity, namely, top–top, top–middle, top–bottom, middle–top, middle–middle, middle–bottom, bottom–top, bottom–middle, and bottom–bottom, are considered in this study. For all of the nine configurations, the length of the heat source, which is the same as that of the heat sink, is equal to half of the cavity’s height (width). All the results presented below are for the Prandtl number of the base fluid equal to 5.83. Figs. 7 and 8 show the streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the source and the sink on the vertical walls of the cavity at Ra = 103 , respectively. For the sake of comparisons, the streamlines and the isotherms for pure water (ϕ = 0.0) are also shown in these figures. For Ra = 103 , the heat transfer inside the cavity is mainly through conduction. A clockwise vortex develops inside the cavity for these cases. At this Rayleigh number, changing the locations of active portions of the side walls does not have a significant effect on the streamline patterns. However, as it can be seen from Fig. 8, changing the positions of the heat source and the heat sink on the side walls of the cavity have a dramatic effect on the temperature distribution inside the cavity at Ra = 103 . Figs. 9 and 10 show the streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the source and the sink on the vertical walls of the cavity at Ra = 104 , respectively. For the sake of comparisons, the streamlines and the isotherms for pure water (ϕ = 0.0) are also shown in these figures.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 7. Streamlines for different position of the source and the sink at Ra = 103 , — ϕ = 0.0, - - - ϕ = 0.15.
G.A. Sheikhzadeh et al. / European Journal of Mechanics B/Fluids 30 (2011) 166–176
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
171
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 8. Isotherms for different position of the source and the sink at Ra = 103 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 9. Streamlines for different position of the source and the sink at Ra = 104 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(g) Bottom–top.
(c) Top–bottom.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 10. Isotherms for different position of the source and the sink at Ra = 104 , — ϕ = 0.0, - - - ϕ = 0.15.
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(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(e) Middle–middle.
(h) Bottom–middle.
(i) Bottom–bottom.
Fig. 11. Streamlines for different position of the source and the sink at Ra = 105 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 12. Isotherms for different position of the source and the sink at Ra = 105 , — ϕ = 0.0, - - - ϕ = 0.15.
(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
(e) Middle–middle.
(i) Bottom–bottom.
Fig. 13. Streamlines for different position of the source and the sink at Ra = 106 , — ϕ = 0.0, - - - ϕ = 0.15.
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(a) Top–top.
(b) Top–middle.
(f) Middle–bottom.
(c) Top–bottom.
(g) Bottom–top.
(d) Middle–top.
(h) Bottom–middle.
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(e) Middle–middle.
(i) Bottom–bottom.
Fig. 14. Isotherms for different position of the source and the sink at Ra = 106 , — ϕ = 0.0, - - - ϕ = 0.15.
Fig. 15. Variations of the average Nusselt number of the heat source with respect to the volume fraction of the nanoparticles for different configurations of the source and the sink at different Rayleigh numbers.
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Fig. 16. Variations of the stream function with respect to the volume fraction of the nanoparticles for different configurations of the source and the sink at different Rayleigh numbers.
With increasing the Rayleigh number, the effect of the convection heat transfer becomes more significant, and distinct boundary layers form along the active walls of the cavity. The compact streamlines and isotherms in the boundary layers along the active side walls indicate higher velocities and intense convection heat transfer in these areas. The core region of the cavity is relatively stagnant. The isotherms in the stratified core region are nearly parallel to the horizontal walls of the cavity. The streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the active portions of the side walls of the cavity at Ra = 105 , are shown in Figs. 11 and 12, respectively. The results for pure water (ϕ = 0.0) are also shown in these figures for comparison purposes. As it can be seen from Fig. 11(a), in the top–top configuration, a primary recirculation zone forms inside the cavity with its core being near the heat sink and expanding towards the heat source. Moving the heat sink from the top to the middle of the left wall of the cavity (Fig. 11(b)) enlarges the central vortex. Moreover, two secondary inner vortices form in the latter case with the inner vortex near the hot active wall being smaller than the one near the cold active wall. Two distinct inner vortices near the heat source and the heat sink for the top–bottom case are observed in Fig. 11(c). For the middle–top configuration, the core of the vortex for the nanofluid is larger than the one for the base fluid (Fig. 11(d)). By moving the heat sink from the middle to the bottom of the cavity’s left wall (Fig. 11(f)), the core region of the streamlines divides into two vortices of different sizes. As Fig. 11(g) shows, for
the bottom–top configuration, a single vortex with a protracted center exists in the cavity. The maximum and the minimum absolute values of the stream functions (|ψ|max = 12.8, |ψ|max = 3.4) occur for the bottom–top and the top–bottom configurations, respectively. As Fig. 11(h) and (i) show the central vortex becomes smaller for the bottom–middle and the –bottom configurations. As it is observed from Fig. 12, the temperature gradients are quite large in the vicinity of the active portions of the side walls of the cavity. The heat transfer in these regions is mainly through convection. Moreover, in the stratified core region of the cavity, the isotherms are nearly horizontal, and the heat transfer is conduction dominated. The isotherms are more condensed in the core region for the middle–middle configuration (Fig. 12(e)). The streamlines and the isotherms for the Cu–water nanofluid (ϕ = 0.15) for different positions of the active portions of the side walls of the cavity at Ra = 106 are shown in Figs. 13 and 14, respectively. The results for pure water (ϕ = 0.0) are also shown in these figures for comparison purposes. As it is observed from these figures, for this convection-dominated heat transfer regime, the streamlines and the isotherms are more packed next to the active portions of side walls of the cavity. Moreover, the core region is protracted along the horizontal direction for all the cases considered in the figures. In the bottom–top configuration, a single protracted central vortex develops inside the cavity for the nanofluid, but three secondary inner vortices form for the base fluid. The inner vortices near the active walls of the cavity are of the same size (Fig. 13(g)). The isotherms are condensed in
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(a) Ra = 103 .
(b) Ra = 104 .
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(c) Ra = 105 .
(d) Ra = 106 . Fig. 17. Variations of the average Nusselt number for the middle–middle location of the active parts with respect to the volume fraction of nanoparticles at different Rayleigh numbers.
the cavity center and near the active walls. The maximum and the minimum absolute values of the stream functions (|ψ|max = 25.0, |ψ|max = 5.6) in this case occur for the bottom–top and the top–bottom configurations, respectively. The variations of the average Nusselt number of the heat source and the maximum absolute value of the stream function with respect to the volume fraction of the nanoparticles for different positions of the heat source and the heat sink on the side walls of the cavity at the different Rayleigh numbers are displayed in Figs. 15 and 16, respectively. As it can be observed from Fig. 15, the heat transfer increases with increasing the volume fraction of the nanoparticles for all the configurations, and for the entire range of the Rayleigh numbers considered. Moreover, for Ra = 103 and 104 , the average Nusselt number of the heat source is maximum when the heat sink is located at the middle of the right vertical wall of the cavity irrespective of the position of the heat source. For Ra = 104 , 105 , and 106 , the average Nusselt number of the heat source is minimum when the heat sink is located at the bottom of the right vertical wall of the cavity (Fig. 15). Moreover, Fig. 15 shows that for Ra = 103 and 104 the maximum heat transfer occurs for the middle–middle configuration; while, for Ra = 105 and 106 , the bottom–middle configuration has the maximum heat transfer coefficient. The heat transfer inside the cavity occurs mainly through conduction at Ra = 103 . By increasing the volume fraction of nanoparticles, the viscosity and the thermal conductivity of the nanofluid, and, consequently, the diffusion of heat inside the cavity are increased. Therefore, the temperature gradient and the buoyancy force are decreased. Similar conclusions can be drawn from Fig. 16 which shows the variations of the stream functions with respect to the volume fraction of the nanoparticles. The natural convection inside the cavity diminishes with increasing the volume fractions of the nanoparticles. However, the average
Nusselt number increases under these conditions. The increase of the average Nusselt number is attributed to the increases of the thermal conductivity of the nanofluid with increasing the volume fraction of the nanoparticles. For a moderate Rayleigh number (Ra = 104 ), the conduction and convection heat transfers are comparable. It is observed from Fig. 16 that, for the top–bottom and the bottom–top configurations, the maximum absolute value of the stream function (|ψ|max ) increases and decreases, respectively, with increasing the volume fraction of the nanoparticles for Ra = 104 . However, for the rest of the configurations shown in Fig. 16, |ψ|max initially increases and then decreases with increasing the volume fraction of the nanoparticles for Ra = 104 . The maximum absolute values of the stream function in Fig. 16 show that, for Ra = 104 , the free convection heat transfer has the minimum and the maximum strengths in the top–bottom and bottom–top configurations, respectively. For low and moderate Rayleigh numbers (Ra = 103 , 104 ), the heat transfer inside the cavity occurs mainly through conduction. At these Rayleigh numbers, the convection strength is weak. The values of |ψ|max show that the middle–middle configuration has the most convection strength compared to the other configurations (Fig. 16). Therefore, the middle–middle configuration has the maximum average Nusselt number at these Rayleigh numbers (Fig. 15). The convection heat transfer mechanism becomes dominant at high Rayleigh numbers (Ra = 105 and 106 ). As it is observed from Fig. 16, the bottom–top configuration has the highest rate of convection heat transfer for high Rayleigh numbers. However, the rate of heat transfer through conduction is the lowest for this configuration. Keeping the heat source at the bottom location, the convection heat transfer increases and the conduction decreases by moving the heat sink from the bottom to
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inside the cavity were investigated. The results show that, for all the Rayleigh numbers considered, the minimum average Nusselt number occurred for the top–bottom case of the active portions of the side walls, but the maximum average Nusselt number occurred for the middle–middle case for Ra = 103 and 104 and the middle–bottom case for Ra = 105 and 106 . Moreover, the heat transfer rate increases with increasing the volume fraction of the nanoparticles. References
Fig. 18. The correlation results for the average Nusselt number of the heat source for the middle–middle configuration.
the top of the right wall of the cavity. Therefore, as it is observed from Fig. 15, the maximum average Nusselt number occurs for the bottom–middle configuration at Ra = 105 and 106 . Fig. 17 shows the variation of the average Nusselt number with respect to the volume fraction of the nanoparticles for the middle–middle configuration of the active parts at different Rayleigh numbers. The presented results in this figure are for the Cu–water and Al2 O3 –water nanofluids. As the figure shows, the average Nusselt number for Cu–water nanofluid is higher than that of the Al2 O3 –water nanofluid. Using the numerical results, the following correlation is obtained for the average Nusselt number for middle–middle configuration in terms of the Rayleigh number (103 ≤ Ra ≤ 106 ), and the volume fraction of the nanoparticles (0.0 ≤ ϕ ≤ 0.15). Nuavg = (0.4778ϕ 1.0 + 0.3554)Ra0.25 .
(19)
The average Nusselt number obtained from Eq. (19) together with the numerical results are presented in Fig. 18. 7. Conclusions The two-dimensional buoyancy-driven fluid flow and heat transfer in a square cavity with partially active side walls filled with Cu–water nanofluid have been simulated numerically using the finite volume method. A parametric study was undertaken and the effects of the position of active portions of the side walls, and the volume fraction of the nanoparticles on the heat transfer
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