Results in Physics 2 (2012) 5–13
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Numerical study of mixed convection flows in a lid-driven enclosure filled with nanofluid using variable properties G.A. Sheikhzadeh, M. Ebrahim Qomi, N. Hajialigol ⇑, A. Fattahi Department of Mechanical Engineering, University of Kashan, Ghotb Ravandi Blvd, Kashan 87317-51167, Iran
a r t i c l e
i n f o
Article history: Received 27 September 2011 Accepted 30 January 2012 Available online 4 February 2012 Keywords: Variable properties Lid-driven enclosure Laminar mixed convection Nanofluids
a b s t r a c t This paper focuses on the study of mixed convection heat transfer characteristics in a lid-driven enclosure filled with nanofluids using variable thermal conductivity and variable viscosity. The fluid in the enclosure is a water-based nanofluid containing Al2O3 nanoparticles. The top and bottom horizontal walls are insulated, while the vertical walls are kept at different constant temperatures with the top surface moving at a constant speed. The study has been carried out for the Richardson numbers of 0.01–100, the solid volume fraction of 0–0.06 and the Grashof number of 104. Various results for the streamlines and isotherms as well as the local and average Nusselt numbers are presented. The variable viscosity and thermal conductivity of both the Brinkman and the Maxwell–Garnett model were compared. Significant differences are found between the magnitudes of heat transfer enhancement in the enclosure for two employed models. Ó 2012 Elsevier B.V. Open access under CC BY-NC-ND license.
1. Introduction Mixed convection of heat transfer has been a subject of interest in recent years due to its applications, especially those related to lubrication technologies, electronic cooling, food processing and nuclear reactors [1–2]. But, low thermal properties of working fluids are a main limitation. Suspending different types of small solid particles is an innovative way to improve the heat transfer. A dilute suspension of solid nanoparticles called a nanofluid, a term firstly used by Choi [3]. Mixed convection heat transfer is affected by nanofluid properties, such as viscosity and thermal conductivity. Up to now, most studies have used the Brinkman model for viscosity and Maxwell–Garnett (MG) model for thermal conductivity. These models have some defects. The Brinkman model does not consider the effect of nanofluid temperature or nanoparticles size and the Maxwell–Garnett model does not emphasize important mechanisms for heat transfer in nanofluids such as Brownian motion. The effect of nanoparticle concentration and nanoparticle size on nanofluids viscosity under a wide range of temperatures was experimentally studied by Nguyen et al. [4] and Angue Minsta et al. [5]. They found that viscosity drops sharply with increasing temperature, especially for high concentrations of nanoparticles. In addition, Chon et al. [6] experimentally studied the combined ⇑ Corresponding author. E-mail addresses:
[email protected] (G.A. Sheikhzadeh), qomeima@yahoo. com (M. Ebrahim Qomi),
[email protected] (N. Hajialigol), fattahi@ grad.kashanu.ac.ir (A. Fattahi). 2211-3797 Ó 2012 Elsevier B.V. Open access under CC BY-NC-ND license. doi:10.1016/j.rinp.2012.01.001
effect of temperature, nanoparticle size and nanoparticle volume fraction on the thermal conductivity of nanofluids. Abu-Nada [7,8] studied the effect of variable properties of Al2O3–water and CuO–water nanofluids on natural convection in an annular region. He found that for Ra P 104 the heat transfer was elevated by increasing the concentration of nanoparticles. Additionally, Abu-Nada et al. [9] investigated the role of nanofluid variable properties in differentially heated enclosures and found that the effect of nanofluid variable properties play a major role in the prediction of heat transfer enhancement. Sensibility of mixed convection heat transfer to variable viscosity and thermal conductivity of nanofluids in a lid-driven enclosure is the aim of this work. When nanofluid viscosity is a function of temperature and nanoparticles concentration, experimental results of Nguyen et al. [4] are adopted. For the thermal conductivity, the model derived by Chon et al. [6] is used. Under a wide range of volume fractions of nanoparticles and different Richardson numbers, the enhancement of heat transfer will be evaluated. 2. Physical model and governing equations Fig. 1 shows a lid-driven square enclosure filled with a nanofluid. The top and bottom horizontal walls are insulated, while the vertical walls are kept at different constant temperatures with the top surface moving at a constant speed. The fluid in the enclosure is a water-based nanofluid containing Al2O3 nanoparticles. The nanofluid in the enclosure is Newtonian, incompressible and laminar. In addition, it is assumed that both the fluid phase and nanoparticles are in the thermal equilibrium state and they
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G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
Nomenclature g Gr H Nu k p P Pr Ra Re Ri T u, v U, V Up x, y X, Y
Greek symbols thermal diffusivity b thermal expansion coefficient u solid volume fraction l dynamic viscosity m kinematics viscosity q density h dimensionless temperature
gravitational acceleration Grashof number enclosure height Nusselt number thermal conductivity pressure dimensionless pressure Prandtl number Rayleigh number Reynolds number Richardson number temperature components of velocity dimensionless of velocity component velocity of the moving lid Cartesian coordinates dimensionless of Cartesian coordinates
a
Subscript avg c eff f h nf p
flow with the same velocity. The nanoparticles are assumed to have uniform shape and size. The density variation in the body force term of the momentum equation is satisfied by Boussinesq’s approximation. The thermal conductivity and the viscosity of the nanofluid are taken into consideration as variable properties; both of them change with volume fraction and temperature of nanoparticles. Under the above assumptions, the system of governing equations is [10]: Continuity equation
@u @ v þ ¼0 @x @y
ð1Þ
average cold wall effective fluid hot wall nanofluid particle
y-momentum equation:
qnf;0 u
@v @v þv @x @y
@p @ @v @ @v þ lnf lnf þ @y @x @y @x @y @ lnf @ v @ lnf @u þ þ @y @y @x @y þ ½/qp;0 bs þ ð1 /Þqf ;0 bf gðT T c Þ
¼
ð3Þ
Energy equation:
@T @T @ @T @ @T þv ¼ knf þ knf ðqcp Þnf u @x @y @x @x @y @y
ð4Þ
The effective density of the nanofluid at reference temperature is
x-momentum equation:
@u @u @p @ @u @ @u ¼ þ þ qnf;0 u þ v lnf lnf @x @y @x @x @x @y @y @ l @ v @ lnf @u þ þ nf @y @x @x @x
qnf;0 ¼ ð1 /Þqf ;0 þ /qp;0
ð5Þ
and the specific heat capacity of nanofluid is
ð2Þ
ðqbÞnf ¼ ð1 uÞðqbÞf þ uðqbÞp
ð6Þ
ðqcp Þnf ¼ ð1 uÞðqcp Þf þ uðqcp Þp
ð7Þ
as given by Xuan and Li [11]. The effective thermal conductivity of the nanofluid calculated by the Chon et al. model [6] is
0:3690 0:7476 df knf kp ¼ 1 þ 64:7u0:4076 Pr0:9955 Re1:2321 T kf dp kf
Adiabatic
ð8Þ
Here PrT and Re are defined by
y
PrT ¼
lf qf af
ð9Þ
Re ¼
qf kb T 3plf lf
ð10Þ
kb = 1.3807 1023 J/K is the Boltzmann constant and lf = 0.17 nm is the mean path of fluid particles [6]. Accuracy of this model was confirmed by the experiments of Angue Minsta et al. [5]. The results of Eq. (8), will be compared to the Maxwell–Garnett (MG) model given by [12]
Adiabatic x Fig. 1. Geometry and coordinate system.
Table 1 Grid independence study. Grid size
21 21
61 61
81 81
101 101
121 121
141 141
161 161
Nuavg
6.743
7.534
8.788
9.011
9.264
9.228
9.228
G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
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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. [13] Barakos and Mitsoulis [14] Markatos and Pericleous [15] De Vahl Davis [16] Fusegi et al. [17]
Ra = 103
Ra = 104
Ra = 105
Ra = 106
1.120 1.118 1.114 1.108 1.118 1.052
2.242 2.245 2.245 2.201 2.243 2.302
4.514 4.522 4.51 4.43 4.519 4.646
8.79 8.826 8.806 8.754 8.799 9.012
knf ðkp þ 2kf Þ þ 2uðkf kpÞ ¼ kf ðkp þ 2kf Þ þ uðkf kp Þ
ð11Þ
The viscosity of the nanoparticle (Al2O3) as given by Nguyen et al. [4] is
lnf ¼ expð3:003 0:04203T 0:5445/ þ 0:0002553T 2 þ 0:0524u2 1:622u1 Þ 103
ð12Þ
The temperature in Eq. (12) is expressed in °C. The results of Eq. (12) will be compared with the Brinkman model is given by [12]
lnf ¼
lf
ð13Þ
ð1 /Þ2:5
The viscosity of the base fluid (water) is considered as a function of temperature. The equation is used to obtain the viscosity of water [4]:
lf ¼ ð1:2723ln5 T 8:736ln4 T þ 33:708ln3 T 246:6ln2 T þ 518:78 ln T þ 1153:9Þ 106
ð14Þ
Using the dimensionless variables
x X¼ ; L
y Y¼ ; L
U¼
u ; Up
V¼
v Up
;
P¼
p 2 f UP
q
;
h¼
T Tc Th Tc
The governing equations are written in the dimensionless form: Continuity equation:
@U @V þ ¼0 @X @Y
ð15Þ
x-momentum equation:
U
qf ;0 @P @U @U 1 @ @U þV ¼ þ lnf @X @Y @X qnf;0 @X Re tf qnf;0 @X @ @U @ lnf @V @ lnf @U þ þ lnf þ @Y @Y @Y @X @X @X
ð16Þ
y-momentum equation:
U
qf ;0 @P @V @V 1 @ @V þV ¼ þ lnf @X @Y @X qnf;0 @Y Re tf qnf;0 @X @ @V @ lnf @V @ lnf @U ðqbÞnf þ þ þ lnf Rih þ @Y @Y @Y @Y @X @Y bf qnf;0
ð17Þ
Energy equation:
U
@h @h 1 @ @h @ @h þV ¼ knf þ knf @X @Y Re Pr af ðqcp Þnf @X @X @Y @Y ð18Þ
The Grashohf, Reynolds, Richardson and Prandtl numbers in the above equations are defined as
Gr ¼
gbf H3 ðT h T c Þ
m2f
;
Re ¼
Up H
tf
;
Ri ¼
Gr Re2
;
Pr ¼
mf af
ð19Þ
Fig. 2. Average Nusselt number on the hot wall: comparison between the present results and results of Abu-Nada et al. [9].
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G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13 Table 3 Thermophysical properties of water and Al2O3 [7]. Property
Water
Al2O3
cp
4179 997.1 0.6 2.1 104 0.384
765 3970 25 0.85 105 47
q k b dp (nm)
ϕ=0
ϕ=0 ϕ = 0.03
ϕ = 0.03 φ = 0.06 Fig. 3. Streamlines (right) and isotherms (left) for Al2O3–water nanofluid at Ri = 0.01.
Dimensionless boundary conditions are as follows:
X ¼ 0; 0 6 Y 6 1 : U ¼ V ¼ 0; h ¼ 1 X ¼ 1; 0 6 Y 6 1 : U ¼ V ¼ 0; h ¼ 0 @h Y ¼ 0; 0 6 X 6 1 : U ¼ V ¼ 0; ¼0 @Y @h ¼0 Y ¼ 1; 0 6 X 6 1 : U ¼ 1; V ¼ 0; @Y
ð20Þ
φ = 0.06
Nusselt number on the vertical hot wall is calculated as follows:
Fig. 4. Streamlines (right) and isotherms (left) for Al2O3–water nanofluid at Ri = 0.1.
knf @h Nu ¼ kf @X X¼0
3. Numerical method
ð21Þ
where knf/kf is calculated using Eq. (8) or (11). To see how knf/kf behaves on the hot wall (using Eq. (8)), the following equation is presented:
knf kf
¼
Z 0
mean
1
knf kf
ðYÞdY
ð22Þ
X¼0
Finally, the average Nusselt number is determined from
Nuavg ¼
Z
1
NuðYÞdY 0
ð23Þ
The governing equations associated with the boundary conditions were numerically solved using the control-volume based finite volume method. The SIMPLE algorithm is used to solve the coupled system of governing equations. The set of algebraic equations are iteratively solved. A second-order upwind scheme and central difference are utilized to discretize the convection terms. A uniform grid mesh was employed in the present paper. To obtain a suitable grid, various grids (21 21, 61 61, 81 81, 121 121, 141 141 and 161 161) at Ri = 0.1 and / ¼ 0:02 were examined. The average Nusselt number along the hot wall
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G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
ϕ=0
ϕ=0
ϕ = 0.03
ϕ = 0.03
φ = 0.06
φ = 0.06
Fig. 5. Streamlines (right) and isotherms (left) for Al2O3–water nanofluid at Ri = 10.
Fig. 6. Streamlines (right) and isotherms (left) for Al2O3–water nanofluid at Ri = 100.
obtained by each grid is listed in Table 1. As shown from Table 1, the grid size of 141 141 ensures a grid independent solution. To validate the computer code, the average Nusselt number on the hot wall is compared to the results of other researchers as shown in Table 2. Very good agreement is observed between the Nusselt numbers obtained by the present simulation and the other works. Another test for validation of this numerical work is shown in Fig. 2. In this test case, the average Nusselt number on the left wall has been compared with those of Abu-Nada et al. [9]. It is observed that the current results fairly coincide with the results of [9]. Based on this successful validation, the code can be used to simulate the present problem. 4. Results and discussion The fluid inside the enclosure is Al2O3–water nanofluid and Grashof number of flow is assumed to be constant at Gr = 104. To study the effects of the flow regime, the Richardson number is considered between 0.01 and 100. To study the effects of volume fraction of nanoparticles / is chosen at 0, 0.01, 0.02, 0.03, 0.04, 0.05, and 0.06. The Reynolds number varies due to variation of the Richardson number. The thermo physical properties of fluid and nanoparticles are presented in Table 3 [7].
Streamlines and the isotherms for different volume fractions of the nanoparticles at different Richardson numbers are shown in Figs. 3–6, respectively. At Ri = 0.01 and 0.1, streamlines show that the forced convection plays a dominant role and the thermal boundary layer forms at the vicinity of vertical walls. As illustrated from isotherm plots, the large core region of the cavity has the same temperature. Due to moving the fluid with the lid, a relative vacuum zone occurred at the left top corner, and then, streamlines deviated to this region. According to Figs. 5 and 6, at Ri = 10 and 100, natural convection is much more effective than forced convection. It is obviously recognized from these figures, when forced convection is governed, the effect of increasing volume fraction of nanoparticles is prominent. The variation of vertical velocity at the middle of enclosure for base fluid and nanofluid at different Richardson numbers is illustrated in Fig. 7. According to the results at Ri = 0.01 and 0.1, forced convection predominates. Consequently, in this situation V-velocity does not have a symmetric profile. It is obviously recognized from the case of Ri = 10 and 100 in which natural convection is dominant that the vertical component of velocity has a symmetric manner. High values of / cause the fluid becomes more viscous which causes the velocity to attenuate consequently. It can be
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G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
0.3
0.4 0% 6%
0.3
0% 6%
0.2
0.2
0.1
0.1
0
0
V
V
-0.1
-0.1 -0.2
-0.2
-0.3
-0.3
-0.4
-0.4 -0.5
0
0.25
0.5
0.75
1
-0.5
0
0.25
0.5
X
X
Ri=0.01
Ri=0.1
0.4
0.75
1
1 0% 6%
0.3
0% 6%
0.75
0.2
0.5
0.1
0.25
V
V
0 0
-0.1 -0.25
-0.2
-0.5
-0.3
-0.75
-0.4 -0.5
0
0.25
0.5
0.75
1
X
Ri=10
-1
0
0.25
0.5
0.75
1
X
Ri=100
Fig. 7. Comparison V-velocity at the mid-plane of the cavity for base fluid and nanofluid at various Ri.
observed that maximum of vertical velocity at mid-plane is decreased when nanofluid is used. Variations of local Nusselt number along the heated surface for various volume fractions and different Ri are shown in Fig. 8. The figure shows that the behavior of local Nusselt number for Ri = 0.01 and 0.1 are similar. At Ri = 0.01 and 0.1, by increasing Y from 0 to 0.4, local Nusselt number increases to a relative maximum value and then by increasing Y to 0.8, it decreases to a relative minimum value. By increasing Y to vicinity the top wall, it increases to a maximum value and finally, it decreases to up wall. Enhancement of Nu takes place clearly by decreasing of the thickness of the thermal boundary layer (see Figs. 3 and 4). By increasing Ri, temperature gradient and then Nu decreases, so maximum value of Nu at Y = 0.4 is about 20 for Ri = 0.01 and is 10 for Ri = 0.1. At the left top corner, at Ri = 0.01 and 0.1, deviation of streamlines causes compressing isotherms (see Figs. 3 and 4). As a result of this, in near the top wall and in vicinity of Y = 1, Nu is maximum. The value of Nu is about 36 for Ri = 0.01 and 19 for Ri = 0.1. This figure illustrates an enhancement in Nu by increasing the volume fraction of nanoparticles at Ri = 0.01 and 0.1.
Fig. 8 reveals that the behavior of Nu at Ri = 10 is similar to that of Ri = 100. In these cases Nu increases for Y < 0.1 and in 0.1 < Y Nu reduces. In this region natural convection causes isotherms to become distant from each other and as a result of that, Nu decreases. Maximum value of Nu is about 8 for Ri = 10 and 100. Nu in the range of 0 < Y < 0.1 decreases by increasing the volume fraction of the nanoparticles. To explain what is happening an attention to Fig. 9 is needed. Temperature gradient along the hot wall and thermal conductivity ratio (using Eq. (8)) versus volume fraction of nanoparticles at Ri = 10 are shown in Fig. 6. This figure reveals that by increasing / the temperature gradient term decreases. More reduction of temperature gradient occurs in 0 < Y < 0.1 as / is raised. From Fig. 9 it is clear that increasing of (8), (11) causes enhancement of thermal conductivity ratio. According to Eq. (21), Nu is the multiplication of the temperature gradient term and the thermal conductivity ratio term. Thus, both of these terms affect Nu. In 0 < Y < 0.1, decreasing temperature gradient with enhancement / is more effective than increasing thermal conductivity ratio with /. Consequently, Nu in this part of the wall decreases with increasing /.
G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
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Fig. 8. Local Nusselt number distribution along the heated surface.
Fig. 10 shows the average Nusselt number (Nuavg) on the hot wall with respect to the volume fraction of the nanoparticles at different Richardson numbers using the various models used for thermal conductivity and viscosity. The first approach is MG model for the thermal conductivity and Brinkman model for the viscosity of the nanofluid. Up to now most studies have utilized these models. The second approach, used in this paper as the base case, is the Chon et al. model for thermal conductivity and Nguyen et al. model for viscosity. The third approach is the MG model for the thermal conductivity and the experimental data of Nguyen et al. for viscosity. The fourth approach is the Chon et al. model for thermal conductivity and the Brinkman model for the viscosity of the nanofluid. As outlined in Fig. 10, it is clear that when Chon et al. and Brinkman models are used, Nuavg has maximum values in all Ri numbers and /. As it is observed for Ri = 0.01 and 0.1, the average Nusselt number of the hot wall obtained by the Chon et al. and Nguyen et al. correlations is quite higher than that obtained by the MG and Brinkman formula. The differences between Nuavg are obtained using the different formulas. With increase in nanoparticles volume fraction, the difference becomes more significant. Moreover, by decreasing Richardson number that results in increasing shear
force and forced convection, for a constant nanoparticles volume fraction the difference between average Nusselt numbers calculated by four combinations of formula increases. It is obviously recognized from Fig. 10, when the MG and Brinkman formula are used, the heat transfer increases with increasing solid volume fraction and Nuavg has an irregular manner when Chon et al. and Nguyen et al. correlations are utilized. It is found that at low Richardson number the average Nusselt number is more sensitive to viscosity models and thermal conductivity models. 5. Conclusions In this study, Laminar mixed convection flows of Al2O3–water nanofluid in a square enclosure were numerically investigated using the finite volume method. The forced convective flow within the enclosure is attained by moving the enclosure top wall, while the natural convective effect is obtained by subjecting the left wall to a higher temperature than the right wall which is kept at the same temperature. The results of this paper show that using various models for thermal conductivity and viscosity have different values of average Nusselt number for a constant solid volume fraction. For Ri = 0.01 and 0.1 that is a dominant case in forced
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G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
Fig. 9. Temperature gradient at the hot wall (left) and thermal conductivity ratio versus volume fraction of nanoparticles (right) at Ri = 10.
Fig. 10. Average Nusselt number distribution along the heated wall.
G.A. Sheikhzadeh et al. / Results in Physics 2 (2012) 5–13
convection, the difference between the two models is higher than that of Ri = 10 and 100. It was recognized that at low Richardson numbers, Nuavg was more sensitive to the viscosity and the thermal conductivity models. References [1] Koseff JR, Prasad AK. The lid-driven cavity flow: a synthesis of quantitative and qualitative observations. J Fluids Eng Trans ASME 1984;106:390–8. [2] Morzinski M, Popiel CO. Laminar heat transfer in a two-dimensional cavity covered by a moving wall. Numer Heat Transfer 1988;12:265–73. [3] Choi US. Enhancing thermal conductivity of fluids with nanoparticles. ASME Fluids Eng Div 1995;231:99–105. [4] Nguyen CT, Desgranges F, Roy G, Galanis N, Mare T, Boucher S, et al. Temperature and particle-size dependent viscosity data for water-based nanofluids – hysteresis phenomenon. Int J Heat Fluid Flow 2007;28:1492–506. [5] Angue Minsta H, Roy G, Nguyen CT, Doucet D. New temperature and conductivity data for water-based nanofluids. Int J Therm Sci 2009;48(2):363–71. [6] Chon CH, Kihm KD, Lee SP, Choi SUS. Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement. Appl Phys Lett 2005;87:153107. [7] Abu-Nada E. Effects of variable viscosity and thermal conductivity of Al2O3– water nanofluid on heat transfer enhancement in natural convection. Int J Heat Fluid Flow 2009;30:679–90.
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