Numerical investigation of heat transfer enhancement of nanofluids in an inclined lid-driven triangular enclosure

International Communications in Heat and Mass Transfer 38 (2011) 1360–1367

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Numerical investigation of heat transfer enhancement of nanofluids in an inclined lid-driven triangular enclosure☆ M.M. Rahman a, M.M. Billah b, A.T.M.M. Rahman c,⁎, M.A. Kalam d, A. Ahsan e a

Department of Mathematics, Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladesh Department of Arts and Sciences, Ahsanullah University of Science and Technology (AUST), Dhaka-1208, Bangladesh Department of Computer Science and Engineering, Dhaka International University, Dhaka-1205, Bangladesh d Department of Mechanical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia e Department of Civil Engineering, Faculty of Engineering, (and Green Engineering and Sustainable Technology Lab, Institute of Advanced Technology), University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia b c

a r t i c l e

i n f o

Available online 29 August 2011 Keywords: Penalty finite element method Nanofluid Lid-driven triangular enclosure Solid volume fraction

a b s t r a c t The behavior of nanofluids is investigated numerically in an inclined lid-driven triangular enclosure to gain insight into convective recirculation and flow processes induced by a nanofluid. The present model is developed to examine the behavior of nanofluids taking into account the solid volume fraction δ. Fluid mechanics and conjugate heat transfer, described in terms of continuity, linear momentum and energy equations, were predicted by using the Galerkin finite element method. Comparisons with previously published work on the basis of special cases are performed and found to be in excellent agreement. Numerical results are obtained for a wide range of parameters such as the Richardson number, and solid volume fraction. Copper–water nanofluids are used with Prandtl number, Pr = 6.2 and solid volume fraction δ is varied as 0%, 4%, 8% and 10%. The streamlines, isotherm plots and the variation of the average Nusselt number at the hot surface as well as average fluid temperature in the enclosure are presented and discussed in detailed. It is observed that solid volume fraction strongly influenced the fluid flow and heat transfer in the enclosure at the three convective regimes. Moreover, the variation of the average Nusselt number and average fluid temperature in the cavity is linear with the solid volume fraction. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The research topic of nanofluids has been receiving enlarged attention worldwide. Many studies on nanofluids are being conducted by talented and studious thermal scientists and engineers all over the world, and they have made the scientific breakthrough not only in discovering unexpected thermal properties of nanofluids, but also in proposing new mechanisms behind the enhanced thermal properties of nanofluids and thus identifying unusual opportunities to develop them as next generation coolants for computers and safe coolants for nuclear reactors. Regarding the various applications of nanofluids, the cooling applications of nanofluids include silicon mirror cooling, electronics cooling, vehicle cooling, and heat engine cooling and so on. Nanofluid technology can help to develop better oils and lubricants. Nanofluids are now being developed for medical applications, including cancer therapy and safer surgery by cooling. Enormous amount of research interests has been sparked in their potential applications of realistic problems [1–6]. Phenomena of natural convection in a triangular enclosure are conducted in the literature ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (A.T.M.M. Rahman). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.08.011

[7,8]. Chen and Cheng [9] numerically studied the effects of lid oscillation on the periodic flow pattern and convection heat transfer in a triangular cavity. Kent et al. [10] investigated natural convection in different triangular enclosures with boundary conditions representing the winter-time heating of an attic space. Natural convection heat transfer in a triangular enclosure with flush mounted heater on the wall [11] and the protruding heaters [12] are investigated. Koca et al. [13] analyzed the effect of Prandtl number on natural convection heat transfer and fluid flow in triangular enclosures with localized heating. Basak et al. [14] investigated the effects of uniform and non-uniform heating of inclined walls on natural convection flows within an isosceles triangular enclosure using a penalty finite element method with bi-quadratic elements. Rousse and Asllanaj [15] present a first-order skewed upwinding procedure for application to discretization numerical methods in the context of radiative transfer involving gray participating media. Yu et al.[16] investigated the effects of Prandtl number on laminar natural convection heat transfer in a horizontal equilateral triangular cylinder with a coaxial circular cylinder using the finite volume approach. They found that the flow patterns and temperature distributions are unique for low-Prandtl-number fluids (Pr ≤ 0.1), and are nearly independent of Prandtl-number when (Pr ≥ 0.7). Natural convection in a porous triangular cavity has been analyzed using penalty finite-element method with biquadratic elements by Basak et al.[17]. A conservative and

M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 1360–1367

Nomenclature cp g H k L Nu p P Pr Re Ri T u U v V V0 V x X y Y

specific heat at constant pressure gravitational acceleration (ms − 2) enclosure height (m) thermal conductivity (Wm − 1 K − 1) length of the cavity (m) Nusselt number dimensional pressure (Nm − 2) dimensionless pressure Prandtl number Reynolds number Richardson number temperature (K) horizontal velocity component (ms − 1) dimensionless horizontal velocity component vertical velocity component (ms − 1) dimensionless vertical velocity component lid velocity (ms − 1) cavity volume (m 3) horizontal coordinate (m) dimensionless horizontal coordinate vertical coordínate (m) dimensionless vertical coordinate

Greek symbols α thermal diffusivity (m 2s − 1) β thermal expansion coefficient (K − 1) δ solid volume fraction μ dynamic viscosity (kg m − 1 s − 1) ν kinematic viscosity (m 2 s − 1) θ non-dimensional temperature ρ density (kg m − 3) ψ stream function ϕ tilt angle, degree (°) γ penalty parameter Γ general dependent variable

Subscripts av average h hot c cold f fluid nf nanofluid s solid nanoparticle

consistent implementation of higher-order convection schemes and systematic comparison of four convection schemes with the available benchmark solution for various two-dimensional, steady-state, liddriven cavity flow problems are conducted by Paramane and Sharma [18]. Talukder and Shah [19] made numerical simulations for fluid flow and heat transfer in triangular ducts. Heat transfer by both forced and natural convection is considered in their investigation. They showed isotherm and secondary velocity profile formed because of natural convection at different locations with varying Rayleigh number. The effect of the apex angle of the triangular duct on Nusselt number and bulk mean temperature is also investigated. The fluids that have been traditionally used for heat transfer applications, namely water, mineral oils and ethylene glycol have a rather low thermal conductivity and do not meet the rising demand as an efficient heat transfer agent. In order to make up for the growing demands of modern technology such as chemical production, power stations and

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microelectronics, it is very important to develop new types of fluids that will be more effective in terms of heat exchange performance. Such new types of fluids are nanofluids, which are new sort of heat transfer fluids containing a small quantity of nanosized particles that are uniformly and stably suspended in a liquid. It is worth to note that these nanofluids come out to have a very high thermal conductivity which can meet the intensifying demand as an efficient heat transfer agent. Researchers have started showing interest in the study of heat transfer characteristics of these nanofluids in recent years. But a clear illustration about the heat transfer through these nanofluids is yet to emerge. The convective heat transfer feature of nanofluids is influenced by the thermo-physical properties of the base fluid and nano particles. The function of a meticulous nanofluid for a heat transfer intention can be traditional by properly modeling the convective transportation in the nanofluid [20]. Khanafer et al. [21] presented a numerical model to find out natural convection heat transfer in nanofluids. The authors investigated the effect of suspended nanoparticles on the buoyancy-driven heat transfer process and found that in any given Grashof number, heat transfer in the enclosure increased with the volumetric fraction of the copper nanoparticles in water. Jou and Tzeng [22] examined the heat transfer enhancement utilizing nanofluids in a two-dimensional enclosure for different pertinent parameters. Tiwari and Das [23] did a numerical study of heat transfer augmentation in a lid-driven square cavity filled with nanofluids. The authors found that both the Richardson number and the direction of the moving walls affect the fluid flow and heat transfer in the cavity. A numerical study is performed to analyze the transport mechanism of mixed convection in a lid-driven enclosure packed with nanofluids by Muthtamilselvan et al. [24]. Ghasemi and Aminossadati [25] considered mixed convection heat transfer in a lid-driven triangular enclosure filled with a water–Al2O3 nanofluid. They found that the addition of Al2O3 nanoparticles enhances the heat transfer rate for different values of Richardson number and for each direction of the sliding wall motion. A parametric study on mixed convection flow in a lid-driven inclined square enclosure filled with water–Al2O3 nanofluid was performed by Nada and Chamkha [26]. Mansour et al. [27] conducted a numerical simulation on mixed convection flow in a square lid-driven cavity partially heated from below using nanofluid. Eastman et al. [28] considered pure copper nanoparticles of less than 10 nm size and achieved 40% increase in thermal conductivity for only 0.3% volume fraction of the solid dispersed in ethylene glycol. The particle size effect and potential of nanofluids with smaller particles are presented in their results. Corcione [29] investigated theoretically the heat transfer features of buoyancy-driven nanofluids inside rectangular enclosures differentially heated at the vertical walls. An experimental investigation of flow and heat transfer characteristics for copper–water based nanofluids through a straight tube with a constant heat flux at the wall is conducted by Xuan and Li [30]. Their results demonstrate that the nanofluids give substantial enhancement of heat transfer rate compared to pure water. Recently, Saleh et al. [31] investigated heat transfer enhancement utilizing nanofluids in a trapezoidal enclosure for various pertinent parameters. Wang et al. [32] performed a numerical investigation on the effective thermal conductivity enhancement of carbon fiber composites. Talebi et al. [33] numerically studied mixed convection flows in a square lid-driven cavity utilizing nanofluid. Tzeng et al. [34] investigated the effect of nanofluids when used as engine coolants. CuO and Al2O3 and antifoam were individually mixed with automatic transmission oil. Owing to the rapid development of computer and computational techniques, finite element methods have provided an alternative in dealing with the mixed convection heat transfer in enclosures using nanofluids. To the best knowledge of the authors, no attention has been paid to investigate the heat transfer characteristics of the nanofluids contained in an inclined lid-driven triangular enclosure with different solid volume fractions. Hence, in this paper the effect of solid volume fractions of fluid in an inclined lid-

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driven triangular enclosure filled with Cu-water nanofluid is investigated numerically.

Table 1 Thermophysical properties of water and copper [33].

2. Mathematical analysis The physical model under present study with the system of coordinates is sketched in Fig. 1. The problem deals with a steady twodimensional flow of nanofluid contained in an inclined lid-driven triangular enclosure. The length of the base wall and height of the sliding wall of the enclosure are denoted by L and H, respectively. In addition, the sliding wall of the cavity is kept adiabatic and allowed to move from bottom to top at a constant speed V0. Moreover, it is assumed that the temperature (Th) of the bottom wall is higher than the temperature (Tc) of the right inclined wall. The free space in the enclosure is filled with copper water nanofluids. The present geometry is special in mixed convection; using nanofluids could produce considerable enhancement of the heat transfer coefficients. The nanofluid in the enclosure is Newtonian, incompressible and laminar. The nanoparticles are assumed to have uniform shape and size. It is considered that thermal equilibrium exists between the base fluid and nanoparticles, and no slip occurs between the two media. The thermo-physical properties [33] of the nanofluid are listed in Table 1. The physical properties of the nanofluid are considered to be constant except the density variation in the body force term of the momentum equation which is satisfied by the Boussinesq's approximation. Under the above assumptions, the system of equations governing the two-dimensional motion of a nanofluid is as follows: ∂u ∂v þ ¼0 ∂x ∂y

u

! 2 2 ðρβÞnf ∂ u ∂ u þ þ ðT−Tc Þg sinϕ ρnf ∂x2 ∂y2

∂u ∂u 1 ∂p μnf þv ¼− þ ρnf ∂x ρnf ∂x ∂y

u

∂v ∂v 1 ∂p μnf þv ¼− þ ρnf ∂y ρnf ∂x ∂y

u

∂T ∂T ∂2 T ∂2 T þ þv ¼ αnf ∂x ∂y ∂x2 ∂y2

2

2

∂ v ∂ v þ ∂x2 ∂y2

! þ

ðρβÞnf ðT−Tc Þg cosϕ ρnf

Property

water

copper

cp ρ k β

4179 997.1 0.613 2.1 × 10− 4

385 8933 401 1.67 × 10− 5

where δ is the solid volume fraction of nanoparticles. In addition, the thermal diffusivity αnf of the nanofluid can be expressed as: k αnf ¼  nf ρcp

ð6Þ

nf

The heat capacitance of nanofluids can be defined as:   ρcp

nf

    ¼ ð1−δÞ ρcp þ δ ρcp f

ð7Þ

s

Additionally, (ρβ)nf is the thermal expansion coefficient of the nanofluid and it can be determined by ðρβÞnf ¼ ð1−δÞðρβÞf þ δðρβÞs :

ð8Þ

Furthermore, μnf is the dynamic viscosity of the nanofluid introduced by Brikman [35] as: μf ð1−δÞ2:5

ð1Þ

μnf ¼

ð2Þ

The effective thermal conductivity of nanofluid was introduced by Kanafer et al. [21] as:

ð3Þ

  knf ks þ 2kf −2δ kf −ks   ¼ kf ks þ 2kf þ δ kf −ks

! ð4Þ

The effective density ρnf of the nanofluid is defined by

ð9Þ

ð10Þ

where, ks is the thermal conductivity of the nanoparticles and kf is the thermal conductivity of base fluid. Introducing the following dimensionless variables: x y u v ðp þ ρgyÞL2 ðT−Tc Þ X ¼ ;Y ¼ ;U ¼ ;V ¼ ;P ¼ ;θ¼ L L V0 V0 ðTh −Tc Þ ρnf V0 2

ρnf ¼ ð1−δÞρf þ δρs

ð11Þ

ð5Þ The governing equations may be written in the dimensionless form as:

Y

∂U ∂V þ ¼0 ∂X ∂Y

V0

adiabatic Tc

U = V = 0, θ = 0

ð12Þ

∂U ∂U ∂P 1 ρf 1 þV ¼− þ U ∂X ∂Y ∂X Re ρnf ð1−δÞ2:5 ðρβÞnf Ri θ sinϕ þ ρnf βf

∂2 U ∂2 U þ ∂X 2 ∂Y 2

∂V ∂V ∂P 1 ρf 1 þV ¼− þ ∂X ∂Y ∂Y Re ρnf ð1−δÞ2:5 ðρβÞnf Ri θ cosϕ þ ρnf βf

∂2 V ∂2 V þ ∂X 2 ∂Y 2

!

ð13Þ

g

H

U U = V = 0, θ = 1.0 Th

L φ X

Fig. 1. Schematic of the problem with the domain and boundary conditions.

2

2

∂θ ∂θ αnf 1 ∂ θ ∂ θ U þ þV ¼ αf RePr ∂X 2 ∂Y 2 ∂X ∂Y

!

ð14Þ

! ð15Þ

M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 1360–1367

The nondimensional numbers that appear in Eqs. (13)–(15) are as follows: Reynoldsnumber Re ¼ V0 L=υf ; Prandtl number Pr ¼ νf =αf and Richardson number Ri ¼ gβf ðTh −Tc ÞL=V0

2

The appropriate boundary conditions for the governing equations are on the bottom wall: U ¼ V ¼ 0; θ ¼ 1

U

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  ∂V ∂V ∂ ∂U ∂V 1 ρf 1 þV ¼γ þ þ Re ρnf ð1−δÞ2:5 ∂X ∂Y ∂Y ∂X ∂Y ðρβÞnf Ri θ cosϕ: þ ρnf βf

∂2 V ∂2 V þ ∂X 2 ∂Y 2

!

ð21Þ

Expanding the velocity components (U, V), and temperature (θ) using basis set {Φk}kN= 1 as N

N

N

k¼1

k¼1

k¼1

U ≈ ∑ Uk Φk ðX; Y Þ; V≈ ∑ Vk Φk ðX; Y Þ; and θ≈ ∑ θk Φk ðX; Y Þ:

ð22Þ

on the left wall: U ¼ 0; V ¼ 1;

Then the Galerkin finite element technique yields the subsequent nonlinear residual equations for the Eqs. (15), (20) and (21) respectively at nodes of the internal domain Ω:

∂θ ¼0 ∂N

on the right inclined wall: ð1Þ

Ri

U ¼ V ¼ 0; θ ¼ 0 where N is the non-dimensional distances either X or Y direction acting normal to the surface. The average Nusselt number at the heated surface of the cavity may be expressed as

ð2Þ

Ri 1

Nuav ¼ −

knf ∂θ dX ∫ kf 0 ∂Y

ð16Þ

and average fluid temperature in the enclosure may be defined as θav ¼ ∫θ d V =V :

∂ψ ∂ψ U¼ ;V ¼− ∂Y ∂X

ð18Þ

3. Numerical solution 3.1. Solution method The Galerkin finite element method is discussed to solve the non-dimensional governing equations along with boundary conditions for the present problem. The equation of continuity has been used as a constraint due to mass conservation and this restriction may be used to find the pressure distribution. The penalty finite element method [14] is used to solve the Eqs. (13)–(15), where the pressure P is eliminated by a penalty constraint γ and the incompressibility criteria given by Eq. (12) consequences in   ∂U ∂V P ¼ −γ þ : ∂X ∂Y

ð19Þ

The continuity equation is automatically fulfilled for large values of γ. The momentum equations reduce to   ∂U ∂U ∂ ∂U ∂V 1 ρf 1 þV ¼γ þ þ Re ρnf ð1−δÞ2:5 ∂X ∂Y ∂X ∂X ∂Y ðρβÞnf Ri θ sinϕ þ ρnf βf

# ! ! N N ∂Φk ∂Φk þ ∑Vk Φk Φi dXdY ∑Uk Φk ∂X ∂Y k¼1 k¼1 k¼1 Ω   N αnf ∂Φi ∂Φk ∂Φi ∂Φk þ dXdY − ∑θ ∫ αf RePr k¼1 k Ω ∂X ∂X ∂Y ∂Y "

N

¼ ∑ Uk ∫ k¼1

N

∑ Uk Φ k

∂Φk þ ∂X

!

N

∑Vk Φk

2

2

∂ U ∂ U þ ∂X 2 ∂Y 2

!

ð20Þ

ð23Þ

# ∂Φk Φi dXdY ∂Y

k¼1 # N ∂Φi ∂Φk ∂Φi ∂Φk 1 ρf 1 dXdY þ ∑Vk ∫ dXdY − −γ ∑Uk ∫ ρ Re ∂X ∂X ∂X ∂Y ð 1−δ Þ2:5 nf k¼1 k¼1 Ω Ω !   N N ∂Φi ∂Φk ∂Φi ∂Φk ðρβÞnf þ dXdY− Ri sinϕ ∫ ∑θk Φk Φi dXdY ∑ Uk ∫ ρ β ∂X ∂X ∂Y ∂Y nf f k¼1 Ω Ω k¼1

"

k¼1

!

Ω

N

ð24Þ

ð17Þ

The fluid motion is displayed using the stream function ψ obtained from velocity components U and V. The relationships between stream function and velocity components [36] for two dimensional flows can be expressed as:

U

"

N

¼ ∑θk ∫

ð3Þ Ri

# ∂Φk ¼ ∑Vk ∫ Φi dXdY ∑Vk Φk ∑Uk Φk ∂Y k¼1 k¼1 k¼1 " Ω # N N ∂Φi ∂Φk ∂Φi ∂Φk −γ ∑Uk ∫ dXdY þ ∑Vk ∫ dXdY ∂Y ∂X ∂Y ∂Y k¼1 k¼1 Ω Ω   N 1 ρf 1 ∂Φi ∂Φk ∂Φi ∂Φk − V ∫ þ dXdY ∑ k Re ρnf ð1−δÞ2:5 k¼1 Ω ∂X ∂X ∂Y ∂Y ! N ðρβÞnf − Ri cosϕ ∫ ∑θk Φk Φi dXdY ρnf βf Ω k¼1 N

"

N

!

∂Φk þ ∂X

N

!

ð25Þ

Three points Gaussian quadrature is used to evaluate the integrals in the residual equations. The non-linear residual equations (Eqs. (23)–(25)) are solved using Newton–Raphson method to determine the coefficients of the expansions in Eq. (22). The convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the convergence criterion ε such that |Γ m + 1 − Γ m| ≤ 10 − 4, where m is number of iteration and Γ is the general dependent variable. 3.2. Grid independence study A grid refinement study has been performed for Re = 100, Ri = 1, ϕ = 60 and δ = 0.04 in an inclined lid-driven triangular enclosure. Five different non-uniform grid systems with the following number of elements within the resolution field: 1486, 2808, 3490, 4894 and 5588 are examined. The numerical design is carried out for highly precise key in the average Nusselt number (Nuav) for the aforesaid elements to develop an understanding of the grid fineness as shown in Fig. 2. The scale of Nuav for 4894 elements shows a little difference with the results obtained for the other elements. Hence a grid size of 4894 elements is found to meet the requirements of both the grid independency study and the computational time limits.

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M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 1360–1367

b)

3.3. Code validation The computational model is validated against the problem of mixed convection in a lid-driven enclosure filled with nanofluids [24]. The cavity was heated at the top wall and cooled at the bottom side while the rest of the boundaries was insulated. The comparison of the average Nusselt number (at the hot surface) between the result of the present code and the results found in the literature [24] for different solid volume fractions are documented in Table 2. The comparisons reveal an excellent agreement with the reported studies. This validation boosts the confidence in the numerical outcome of the present study.

δ= 0.04

Fig. 2. Grid independency study for Re = 100, Ri = 1.0, ϕ = 60 and δ = 0.04.

δ= 0.08

δ= 0.1

a)

The current numerical study is carried out for copper–water nanofluids as working fluid with Prandtl number of 6.2. In this investigation, our attention is taken into account to investigate the effects of controlling parameters namely the solid volume fraction (δ), and Richardson number (Ri). Here, the effect of the solid volume fractions is investigated in the range of 0%–10% while the Re and tilt angle ϕ are kept fixed at 100 and 60, respectively. It is worth to note that the value of Ri is varied from 0.1 to 5 by changing Grashof number Gr to cover forced convection dominated region, pure mixed convection and free convection dominated region. Moreover, the results of this study are presented in terms of streamlines and isotherms. Furthermore, the heat transfer effectiveness of the enclosure is displayed in terms of average Nusselt number Nuav and the dimensionless average bulk temperature θav. 4.1. Flow and thermal fields Fig. 3 shows the streamlines and isotherms in an inclined liddriven triangular enclosure for various values of the solid volume fractions (δ = 0, 0.04, 0.08, and 0.1) at Ri = 0.1. The fluid flow in a two dimensional lid-driven triangular cavity is characterized by a primary circulating cell (major cell) near the vicinity of the sliding wall in the enclosure generated by the lid and a weaker anticlockwise Table 2 Comparison of Nu with those of Muthtamilselvan et al. [24]. δ

Nu

0.0 0.02 0.04 0.06 0.08

Ref [24]

Present study

% Increase

2.26 2.40 2.56 2.73 2.91

2.43 2.60 2.77 2.96 3.17

7.52 8.33 8.20 8.42 8.93

δ= 0

4. Results and discussion

Fig. 3. (a) Streamlines and (b) isotherms for different values of solid volume fraction δ and Richardson number Ri = 0.1.

rotating cell close to the right bottom corner for all values of δ. It can easily be seen from the left column of Fig. 3 that the major cell is generated by the lid dragging the neighboring fluid. Though the flow strength of the main cell is same, the size of the main cell is affected for all values of δ. It is also found from the streamlines that the flow strength of the anticlockwise rotating cell is decreasing very slowly when δ is increased. This is due to an increase in the volume fraction as a result of high-energy transport through the flow associated with the irregular motion of the ultrafine particles. The isotherm plots indicate the lines with equal intervals between unity (hot wall) and zero (cold wall). It is noticed that isotherm lines become denser towards the hot surface for each value of δ at the considered value of Ri = 0.1. The effect of various solid volume fraction on the flow and thermal fields at Ri = 1 is exposed in Fig. 4. A clockwise rotating major cell is observed near the sliding wall of the enclosure, and a counter clockwise-minor cell is seen at the lower right foot corner of the enclosure for all values of δ. The size of the small cell decreases with a decrease in δ. While the flow pattern at Ri = 1 is compared

M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 1360–1367

δ = 0.1

b)

δ = 0.08

δ= 0.08 δ= 0

δ=0

δ= 0.04

a)

δ = 0.04

b)

δ= 0.1

a)

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Fig. 5. (a) Streamlines and (b) isotherms for different values of solid volume fraction δ and Richardson number Ri = 5. Fig. 4. (a) Streamlines and (b) isotherms for different values of solid volume fraction δ and Richardson number Ri = 1.

with that of at Ri = 0.1 as shown in Fig. 3, it is observed that the minor cell is bigger in size at Ri = 1 indicating the flow pattern strongly affected in the mixed convection regime. Fig. 4 indicates that mixed convection is the dominating mode in the triangular enclosure. In this case, the isotherms become denser gradually towards the heated base surface of the enclosure for the increasing value of δ, which indicates the steeper temperature gradient in the horizontal direction in this region. At the upper part of the cavity, the temperature gradients are very small due to the mechanically-driven circulations. Fig. 5 illustrates the effect for considered values of δ on streamlines and isotherms at Ri = 5. In this case, the clockwise rotating major cell decreases mildly with the decreasing solid volume fraction. As a result the anti clockwise minor eddy increases very slowly for the lower values of δ. It is observed that the anti clockwise vortex dominates the main cell. It is evident to the shape change of the core major vortices. As the value of δ increases the core vortices expand vertically. It indicates the reduction of the flow strength of those vortices which occurs due to the movement of the lid.

Comparing Figs. 3–5, the thicken boundary layer is observed near the vicinity of the base surface when Ri = 5. This thicken boundary layer is reduced when the Richardson number Ri decreases. It is interesting to note that, with the increased concentration inside the

Fig. 6. Effect of solid volume fraction δ on average Nusselt number at the heated surface in the cavity, while ϕ = 60.

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Fig. 7. Effect of solid volume fraction δ on average fluid temperature in the cavity, while ϕ = 60.

fluid. Nanofluid helps in minimizing the natural convection effect and increase the forced convection effect. 4.2. Heat transfer The average Nusselt number (Nuav) at the hot surface, which is a measure of the overall heat transfer rate as a function of Richardson number for the abovementioned values of the solid volume fractions (δ) is shown in Fig. 6. This figure shows a linear variation of the average Nusselt number with the solid volume fraction. Clearly it can be seen that the heat transfer increases with increasing δ. When the volume fraction is increases from 0% to 10% the heat transfer increases very swiftly. However, the values of Nuav are always maximum for the highest value of δ (=10%). The effect of the solid fractions (δ) on average fluid temperature (θav) in the enclosure is depicted in Fig. 7. It is observed that θav decreases significantly with the increasing Ri for all values of δ. It is clearly seen that θav is maximum for the lowest value of solid volume fraction (δ = 0%) up to the mixed convection region but beyond Ri = 1, θav is maximum for the highest value of solid volume fraction (δ = 10%). 5. Conclusions Mixed convection in a lid-driven inclined triangular enclosure filled with nanofluids is studied numerically. Results for various parametric conditions are presented and discussed. From the above study, the following conclusions are made: • The inclusion of nanoparticles into the base fluid has produced an augmentation of the heat transfer coefficient, which increases appreciably with an increase of nanoparticles volume concentration. • The solid volume fraction has more significant effect on the flow field than thermal field. • The flow and thermal fields as well as the heat transfer rate inside the enclosure are strongly dependent on the Richardson number. • Nanofluids are capable to modify the flow pattern. • The solid volume fraction is a good control parameter for both pure and nanofluid filled enclosures. References [1] G.A. Holtzman, R.W. Hill, K.S. Bal, Laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above, Journal of Heat Transfer—Transactions of the ASME 122 (2000) 485–491. [2] P.M. Haese, M.D. Teubner, Heat exchange in an attic space, International Journal of Heat and Mass Transfer 45 (25) (2002) 4925–4936. [3] K.A. Joudi, I.A. Hussein, A.A. Farhan, Computational model for a prism shaped storage solar collector with a right triangular cross section, Energy Conversion and Management 45 (2004) 337–342.

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