Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid

International Communications in Heat and Mass Transfer 37 (2010) 79–90

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid☆ Farhad Talebi, Amir Houshang Mahmoudi, Mina Shahi ⁎ Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran

a r t i c l e

i n f o

Available online 1 October 2009 Keywords: Nanofluid Mixed convection Square lid-driven cavity Numerical study

a b s t r a c t A numerical investigation of laminar mixed convection flows through a copper–water nanofluid in a square lid-driven cavity has been executed. In the present study, the top and bottom horizontal walls are insulated while the vertical walls are maintained at constant but different temperatures. The study has been carried out for the Rayleigh number 104 to 106, Reynolds number 1 to 100 and the solid volume fraction 0 to 0.05. The thermal conductivity and effective viscosity of nanofluid have been calculated by Patel and Brinkman models, respectively. The effects of solid volume fraction of nanofluids on hydrodynamic and thermal characteristics have been investigated and discussed. It is found that at the fixed Reynolds number, the solid concentration affects on the flow pattern and thermal behavior particularly for a higher Rayleigh number. In addition it is observed that the effect of solid concentration decreases by the increase of Reynolds number. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The primary limitation of conventional fluids such as water, ethylene glycol or propylene glycol is their low thermal conductivity. Therefore in recent years, nanofluids have attracted more attention for cooling in various industrial applications. Such fluids consist of suspended nanoparticles which have a better suspension stability compared to millimetre or micrometer sized ones. Use of metallic nanoparticles with high thermal conductivity will increase the effective thermal conductivity of these types of fluid remarkably. For instance just 0.3% volume fraction of copper nanoparticles with 10 nm diameter led to an increase of up to 40% in the thermal conductivity of ethylene glycol [1]. Indeed, when nanosized particles are added to liquid flow, scalar transport properties can be considerably enhanced. Lee et al. [2] measured the thermal conductivity of Al2o3–water and Cu–water nanofluids and indicated that the thermal conductivity of nanofluids increases with solid volume fraction. He concluded that any new models of nanofluid thermal conductivity should contain the effect of surface area and structure dependent behavior as well as the size effect. The dependence of thermal conductivity of nanoparticles– fluid mixture was estimated by Xie et al. [3]. Due to the lack of a sophisticated theory for estimating the thermal conductivity of nanofluids many models developed that mostly focused on several parameters, such as: temperature or Brownian motion, geometry of

☆ Communicated by: W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (M. Shahi). 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.08.013

nanoparticles and interaction between nanoparticles and base fluid etc. The first model is given by Maxwell [4]; he proposed a model to predict the thermal conductivity of mixtures that contain solid particles. Maxwell's model shows that the effective thermal conductivity of suspensions that contain spherical particles increases with the volume fraction of the solid particles. The proposed model was valid only for low dense mixtures and for micro sized particles. Thus several researchers (for example, Yu and Choi [5], Kumar et al. [6], Prasher et al. [7], Hemanth et al. [8], etc.) tried to improve the Maxwell's model. Due to the Brownian motion of nanoparticles in base fluid, a model was proposed by Xu [9] that predicted the thermal conductivity of nanofluids by considering heat convection between nanoparticles and fluid. The proposed model is dependent upon the average size of nanoparticles, fractal dimension, temperature, physical properties of fluids and concentration of nanoparticles. Patel et al. [10] have improved the model given by Hemanth et al. [8]. They have considered three contributions for heat flow: conduction through liquid and through solid and advection due to Brownian motion of the particles. This model is able to predict the thermal conductivity over a wide range of particles size (10–100) nm, particle concentrations (1– 8) %, different base fluids and temperatures. Considering that each nanofluid exhibits different rheological properties, various viscosity correlations have been developed for several nanofluids [11–13]. They have observed that the viscosity of nanofluid increases with an increase in solid concentration. Since nanofluid consists of very small sized solid particles, therefore in low solid concentration it is reasonable to consider nanofluid as single phase flow [4]. Contrary to numerous experimental performances there have been few numerical works to study of the enhancement heat transfer by nanofluid.

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Nomenclature cp Gr g H k kb Nu p P pr Ra Re T u,v U,V x,y X,Y W

Specific heat capacity/(J/K) Grashof number, βgH3ΔT/υ2 Gravitational acceleration/(m/s2) Height of cavity/(m) Thermal conductivity/(W/m K) Boltzmann′s constant, 1.38065 × 10− 23 Nusselt number Pressure/(N/m2) Dimensionless pressure, p/ρnfU20 Prandtl number, υf/αf Rayleigh number, βgH3ΔT/(υα) Reynolds number, ρU0H/μ Temperature/(K) Components of velocity/(m/s) Dimensionless of velocity component, (U = u/U0, V = v/U0) Cartesian coordinates/(m) Dimensionless of Cartesian coordinates/(m) width of cavity/(m)

Greek letters α Thermal diffusivity, k/(ρcp) (m2/s) β Coefficient of volume expansion/(K− 1) ϕ Solid volume fraction μ Dynamic viscosity/(Pa s) υ Kinematics viscosity/(m2/s) ρ Density/(kg/m3) θ Dimensionless temperature

Subscript f m nf o s w

The presented work has been concerned with the mixed convection flows of copper–water nanofluid in a square cavity with a moving lid that moves uniformly in the horizontal plane while all other walls of the cavity are fixed. The natural convection has been induced by subjecting the left vertical wall to a higher temperature than right one. In addition, both the top and bottom walls are insulated. The effective thermal conductivity of nanofluid has been calculated with a model that was proposed by Patel [10]. To determine the viscosity of nanofluid, a model that was given by Brinkman [11] has been used. The consequence of varying the Reynolds number, Rayleigh number and the nanoparticle concentration on the hydrodynamic and thermal characteristics have been investigated and discussed.

2. Mathematical formulation Fig. 1 shows a two-dimensional square cavity of length W and height H which its aspect ratio is taken to be equal to one unit. The cavity is filled with a suspension of copper nanoparticles in water. The shape and size of solid particles are assumed to be uniform and the diameter of them to be equal to 100 nm. Both the vertical walls are maintained at constant temperature. In order to induce the buoyancy effect, the left vertical wall is kept at a higher temperature. The two horizontal walls are insulated and the top wall slides from left to right with uniform velocity. It is assumed that both the fluid phase and nanoparticles are in thermal equilibrium. Except for the density the properties of nanoparticles and fluid are taken to be constant. Table 1 presents thermo physical properties of water and copper at the reference temperature. It is further assumed that the Boussinesq approximation is valid for buoyancy force. The governing equations (continuity, momentum and energy equations) for a steady,

Fluid Average Nanofluid Reference state solid wall

A numerical study of natural convection of copper–water nanofluid in a two-dimensional enclosure was conducted by Khanafer et al. [14]. It was found in any given Grashof number, heat transfer in the enclosure increased with the volumetric fraction of the copper nanoparticles in water. Ho et al. [15] presented a two-dimensional numerical simulation of buoyancy-driven convection in the enclosure filled with alumina–water nanofluid. The effects of adopting different formulas for the effective viscosity and thermal conductivity have been identified. A significant difference was found in the effective dynamic viscosity enhancement calculated from considered formulas other than increment of thermal conductivity. Santra et al. [16] numerically investigated the laminar natural convection heat transfer in a differentially heated square cavity filled with copper–water nanofluid. They considered a two parameter power law model for an incompressible non-Newtonian fluid. Other researches have been conducted that simulate the natural convection heat transfer using nanofluid in the other geometrical configurations [17–21]. Fluid flow and heat transfer in a cavity filled by pure fluid which is driven by buoyancy and shear have been studied extensively in literature [22–24]. The most usage of the mixed convection flow with lid-driven effect is to include the cooling of the electronic devices, lubrication technologies, drying technologies, etc.

Fig. 1. Problem geometry.

Table 1 Thermophysical properties of water and copper. Property

Water

Copper

cp ρ k β

4179 997.1 0.6 2.1 × 10− 4

383 8954 400 1.67 × 10− 5

F. Talebi et al. / International Communications in Heat and Mass Transfer 37 (2010) 79–90 2

two-dimensional laminar and incompressible flow are expressed as below:

u

∂u ∂v + =0 ∂x ∂y

u

ð1Þ

81 2

∂u ∂u 1 ∂p ∂ u ∂ u + +v =− + υnf ρnf ∂x ∂x ∂y ∂x2 ∂y2

! ð2Þ

! ∂v ∂v 1 ∂p ∂2 v ∂2 v + +v =− + υnf ρnf ∂y ∂x ∂y ∂x2 ∂y2 h i g ðT−T∞ Þ ϕρs;0 βs + ð1−ϕÞρf;0 βf + ρnf

∂T ∂T ∂2 T ∂2 T + u +v = αnf ∂x ∂y ∂x2 ∂y2

ð3Þ

! ð4Þ

where αnf = knf/(ρcp)nf. The effective density of nanofluid at the reference temperature can be defined as: ρnf;0 = ð1−ϕÞρf;0 + ϕρs;0

ð5Þ

Table 2 Comparison of results obtained in this study by de Vahl Davis [27]. Nu

4

Ra = 10 Ra = 105 Ra = 106

Fig. 2. Grid study.

Present

de Vahl Davis [27]

Error (%)

2.248 4.5028 9.147

2.242 4.523 9.035

0.267 0.447 1.24

Fig. 3. Comparison between a) horizontal component of velocity b) vertical component of velocity with those of de Ghia et al. at Re = 100 [28].

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which ρnf,0, ρf,0, ρs,0 and ϕ are the density of nanofluid, density of base fluid, density of nanoparticle and volume fraction of the nanoparticles, respectively. The heat capacitance of nanofluid can be given as: ðρcp Þnf = ð1−ϕÞðρcp Þf + ϕðρcp Þs :

ð6Þ

The effective thermal conductivity of nanofluid was given by Patel et al. [10] as follows:

k p Ap Ap keff =1+ + ckp Pe kf kf Af k f Af

ð7Þ

Fig. 4. The effects of solid volume fraction and Rayleigh number on the stream line for Re = 1 a) Ra = 1.47 × 104, ϕ = 0 b) Ra = 1.47 × 104, ϕ = 5% c) Ra = 1.47 × 105, ϕ = 0 d) Ra = 1.47 × 105, ϕ = 5% e) Ra = 1.47 × 106, ϕ = 0 f) Ra = 1.47 × 106, ϕ = 5%.

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where c is constant and must be determined experimentally, Ap/Af and Pe here is defined as:

taken as 2 Å for water. Also up is the Brownian motion velocity of nanoparticle which is defined as:

Ap dp ϕ up dp ; Pe = = : Af df ð1−ϕÞ αf

up =

ð8Þ

Where dp is diameter of solid particles that in this study is assumed to be equal to 100 nm, df is the molecular size of liquid that is

2kb T πμf d2p

ð9Þ

where kb is the Boltzmann constant.

Fig. 5. The effects of solid volume fraction and Rayleigh number on the stream line for Re = 10 a) Ra = 1.47 × 104, ϕ = 0 b) Ra = 1.47 × 104, ϕ = 5% c) Ra = 1.47 × 105, ϕ = 0 d) Ra = 1.47 × 105, ϕ = 5% e) Ra = 1.47 × 106, ϕ = 0 f) Ra = 1.47 × 106, ϕ = 5%.

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The effective viscosity of nanofluid was introduced by Brinkman [11] as below,

The boundary conditions are in the following forms: u=v=0

μnf =

μf : ð1−ϕÞ2:5

ð10Þ

at x = 0; W

0≤y≤H;

u=v=0

at y = 0

0≤x≤W

u = U0 v = 0

at y = H

0≤x≤W

ð11Þ

Fig. 6. The effects of solid volume fraction and Rayleigh number on the stream line for Re = 100 a) Ra = 1.47 × 104, ϕ = 0 b) Ra = 1.47 × 104, ϕ = 5% c) Ra = 1.47 × 105, ϕ = 0 d) Ra = 1.47 × 105, ϕ = 5% e) Ra = 1.47 × 106, ϕ = 0 f) Ra = 1.47 × 106, ϕ = 5%.

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T = TH at x = 0 0≤y≤H; T = TC at x = W 0≤y≤H; ∂T = 0 at y = 0; H 0≤x≤W: ∂y

ð12Þ

85

In order to estimate the heat transfer enhancement, we have calculated Nu (Nusselt number) and Num (average Nusselt number) for the vertical hot wall as: NuðXÞ =

knf ∂θ kf ∂X

j

X =0

ð13Þ

Fig. 7. The effect of solid volume fraction on the isotherms at various Rayleigh and Reynolds numbers: a) Re = 1, Ra = 1.47 × 104 b) Re = 1, Ra = 1.47 × 106 c) Re = 10, Ra = 1.47 × 104 d) Re = 10, Ra = 1.47 × 106 e) Re = 100, Ra = 1.47 × 104 f) Re = 100, Ra = 1.47 × 106.

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F. Talebi et al. / International Communications in Heat and Mass Transfer 37 (2010) 79–90 W

Num =

∫0 NuðXÞdX W ∫0 dX

ð14Þ

Eqs. (1)–(4) can be converted to the dimensionless forms by definition of the following parameters as: X=

x y u v T−TC p ; V= ; θ= ; P= : ; Y= ; U= TH −TC H H U0 U0 ρnf U02

ð15Þ

Therefore using the above parameters leads to dimensionless forms of the governing equations as below: ∂U ∂V + =0 ∂X ∂Y U

∂U ∂U ∂P 1 ρf 1 +V =− + Re ρnf ð1−ϕÞ2:5 ∂X ∂Y ∂X

ð16Þ ∂2 U ∂2 U + ∂X 2 ∂Y 2

! ð17Þ

Fig. 8. The effect of solid volume fraction on the vertical component of velocity at the middle section of cavity for various Re and Ra a) Re = 1, Ra = 1.47 × 104 b) Re = 1, Ra = 1.47 × 105 c) Re = 1, Ra = 1.47 × 106 d) Re = 10, Ra = 1.47 × 104 e) Re = 10, Ra = 1.47 × 105 f) Re = 10, Ra = 1.47 × 106 g) Re = 100, Ra = 1.47 × 104 h) Re = 100, Ra = 1.47 × 105 i) Re = 100, Ra = 1.47 × 106.

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3. Numerical method The above equations have been solved numerically based on the finite volume method using a collocated grid system. The resulting discretized equations have been solved iteratively through strongly implicit procedure (SIP) [25]. The SIMPLE algorithm [26] has been adopted for the pressure velocity coupling. In this study we have used a non-uniform grid mesh which is finer in vicinity of horizontal walls in order to increase the accuracy of the results. To allow gridindependent examination, the numerical procedure has been conducted for different grid resolutions. Fig. 2 demonstrates the influence of number of grid points for a test case of fluid confined within the present configuration. From this figure it is clear that the grid system of 41 × 41 is fine enough to obtain accurate results. We therefore adopted a grid system of 41 × 41. First the governing equations have been solved for the natural convection flow in an enclosed cavity filled by pure fluid, in order to compare the results with those obtained by de Vahl Davis [27]. This comparison revealed good agreements between results which are shown in Table 2. Another test for validation of this numerical method has been performed for the lid-driven square cavity filled by pure fluid (see Fig. 3). In this test case, the results have been compared with those of Ghia et al. [28]. 4. Discussion and results

Fig. 8 (continued).

∂V ∂V ∂P 1 ρf 1 ∂2 V ∂2 V U + +V =− + 2:5 2 Re ρnf ð1−ϕÞ ∂X ∂Y ∂Y ∂X ∂Y 2   Ra ρf;0 ρβ 1−ϕ + ϕ s s θ + ρf βf Re2 Pr ρnf;0

U

! 2 2 ∂θ ∂θ k ðρcp Þf 1 ∂ θ ∂ θ : + +V = nf kf ðρcp Þnf Re⋅Pr ∂X 2 ∂X ∂Y ∂Y 2

! ð18Þ

ð19Þ

Two-dimensional mixed convection is studied for a copper–water nanofluid in a square lid-driven cavity for Ra = 104–106, Re = 1–102 and solid volume fraction 0 to 0.05. Fig. 4 illustrates the stream lines at Ra = 1.47 × 104–1.47 × 106 and Re = 1 for a pure fluid and nanofluid with ϕ = 5%. Fig. 4(a) demonstrates that the effect of the natural convection flow dominates lid-driven flow for Ra = 1.47 × 104. The stream lines are mostly symmetric with respect to the center of the cavity. As can be seen from Fig. 4(b), the solid volume fraction does not have effect on the flow pattern but it augments the flow intensity. So that the value of the stream function at the center of cavity gets from − 0.8 to − 1.1 for a 5% concentration. As the value of Rayleigh number increases to 1.47 × 105, the intensity of buoyancy and hence the intensity of natural convection within the cavity increases as shown in Fig. 4(c). In this case, as the solid concentration increases the value of the stream function increases by about 41% at the center of the cavity (see Fig. 4(d)). Fig. 4(e) displays that the effect of lid-driven flow is negligible for Ra = 1.47 × 106. In this case, the value of the stream function increases by about 66% as a result of an increase about 5% in the solid concentration as shown in Fig. 4(f). Fig. 5(a)–(f) demonstrates the stream lines of both pure fluid and nanofluid for Re = 10 and for the various Rayleigh numbers. As can be seen the increase in Re augments the effect of lid-driven and hence forced convection flow. But it gradually vanishes with increase in the value of Rayleigh number, so that at Ra = 1.47 × 106 the natural convection becomes dominant mode as can be found from comparison between Fig. 5(a), (c) and (e). For smaller Rayleigh number, the motion of the upper lid stretches the stream lines towards the right vertical wall and it causes the asymmetric flow pattern. Increase in solid concentration enhances the value of the stream function by about 25%, 47% and 66% respectively at Ra = 1.47 × 104 to Ra = 1.47 × 106 as seen in Fig. 5(b), (d) and (f). The stream lines for Re = 100 at various Rayleigh numbers and for two case of pure fluid and nanofluid are presented in Fig. 6(a)–(f). As can be seen from Fig. 6(a) for Ra = 1.47 × 104 the effect of lid-driven flow is more dominant. In this case the increment of solid concentration does not have considerable effect on the flow pattern because the buoyancy effect is insignificant. Increase in Rayleigh number enhances the buoyancy effect and so the intensity of flow; but the dominant effect of the force convection is still observed. The effect of the presence of nanoparticle on the thermal behavior, temperature distribution contours for pure are overlaid with that for

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nanofluid with ϕ of 0.05. The results for Re = 1 to 100 at the two Rayleigh numbers are illustrated in Fig. 7. As can be seen at the lower Rayleigh number, the solid concentration has more effect to increase the heat penetration; because the conduction heat transfer has more effective role at the lower Rayleigh number. On the other hand the increase of the effective thermal conductivity of the nanofluid with solid concentration leads to enhance the conduction mode. But the

effect of conduction heat transfer decreases with the increase in Ra, so the solid concentration has a smaller effect on the thermal distribution. Fig. 8 demonstrates the effect of solid concentration on the vertical velocity distribution for various Reynolds and Rayleigh numbers. It is obvious from this figure that for Re = 1 and Ra = 1.47 × 104 the effect of the buoyancy predominates the forced convection effect. Upward flow and downward flow are symmetric

Fig. 9. The effect of solid volume fraction on the horizontal component of velocity at the middle section of cavity for various Re and Ra a) Re = 1, Ra = 1.47 × 104 b) Re = 1, Ra = 1.47 × 105 c) Re = 1, Ra = 1.47 × 106 d) Re = 10, Ra = 1.47 × 104 e) Re = 10, Ra = 1.47 × 105 f) Re = 10, Ra = 1.47 × 106 g) Re = 100, Ra = 1.47 × 104 h) Re = 100, Ra = 1.47 × 105 i) Re = 100, Ra = 1.47 × 106.

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is no symmetric behavior respect to the center point of the cavity. The increase in the solid concentration therefore does not have considerable effect on the intensity of flow; so that as the solid volume fraction increases from 0 to 5%, the enhancement of the vertical velocity near the wall is approximately 28%. But increase in a Ra suppresses the effect of lid-driven flow. At Re = 100 the flow pattern varies considerably, so that for Ra = 1.47 × 104 the flow is nearly transformed to the lid-driven cavity flow and hence the solid

Fig. 9 (continued).

with respect to the center of the cavity. As can be seen the increase in solid concentration leads to enhance the flow intensity. So that the maximum value of the velocity is obtained at the maximum solid concentration. The increase of both Rayleigh number and solid concentration augments the strength of buoyancy. So at higher Rayleigh number, the solid concentration has more effect to enhance the velocity with respect to the pure fluid. The increase in Re, augments the effect of forced convection. So for Ra = 1.47 × 104, there

Fig. 10. The average Nusselt number at left wall for the various Rayleigh numbers a) Re = 1 b) Re = 10 c) Re = 100.

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concentration does not affect on the velocity significantly. But with the increase in Ra, it is observed a mixed effect of buoyancy and liddriven within the cavity. The effect of the solid concentration on the horizontal velocity distribution for various Reynolds and Rayleigh numbers is shown in Fig. 9. From this figure, it can be found that as solid volume fraction increases from 0 to 5% for Re = 1, the horizontal component of velocity increases by about 37% for Ra = 1.47 × 104 and about 52% for Ra = 1.47 × 106. It must be noted that for Re = 10 and Ra = 1.47 × 105 the lid-driven flow has more effect on the horizontal velocity as compared to the vertical one. Fig. 10 illustrates the variation of the average Nusselt number on the left wall (hot wall). The increase of the solid concentration augments the effective thermal conductivity and hence the energy transfer. The increase of the solid concentration also augments the flow intensity. Both of two factors enhance the average Nusselt number with solid concentration. The increase of Rayleigh number also increases the heat transfer rate and hence average Nusselt number. It is obvious that at higher Ra the solid concentration has more effect on the average Nusselt number which is similar to those of Figs. 8 and 9. So that at Re = 1, the increase of the solid concentration from 0 to 5% enhances the average Nusselt number by about 44% for Ra = 1.47 × 104 and about 52% for Ra = 1.47 × 106. 5. Conclusion The model was applied to simulate the mixed convection flows of copper–water nanofluid in a square cavity with a moving lid that moves uniformly in the horizontal plane while all other walls of the cavity are fixed. The natural convection has been induced by subjecting the left vertical wall to the higher temperature than the right one. In addition, both the top and bottom walls are insulated. The results are presented for the Reynolds number 1 to 100, Rayleigh number 104 to 106 and for the different values of solid concentration (0 to 5%). In this study the effect of solid volume fraction, Rayleigh number and Reynolds number on the flow pattern and heat characteristics were investigated. The results showed at a given Reynolds number and Rayleigh number, solid concentration has a positive effect on heat transfer enhancement. The results also indicated that for a given Reynolds number the increase in the solid concentration augments the stream function particularly at the higher Rayleigh number. This point is also observed in the computation of the average Nusselt number. But the increase in Re decreases the effect of solid concentration on the stream function particularly for lower Ra. Acknowledgments This work is supported by the talented office of Semnan University. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows, FED, vol. 231/MD–vol. 66, ASME, New York, 1995, pp. 99–105. [2] S. Lee, SUS. Choi, S.Li, J>A>Eastman, Measuring Thermal Conductivity of Fluids Containing Oxide Nanoparticles, Journal of Heat and Mass Transfer 121:280-289. [3] 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, Journal of Materials Science Letters 21 (2002) 1469–1471.

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