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Heat Transfer Research 45(4), 309–338 (2014)

MIXED CONVECTION FLOW AND HEAT TRANSFER IN A VENTILATED INCLINED CAVITY CONTAINING HOT OBSTACLES SUBJECTED TO A NANOFLUID Mohammad Hemmat Esfe,1 Sina Niazi,1 Seyed Sadegh Mirtalebi Esforjani,2 & Mohammad Akbari2,* 1

Department of Mechanical Engineering, University of Kashan, Kashan, Iran

2

Department of Mechanical Engineering, Najafabad Branch,  Islamic Azad University, Isfahan, Iran

*Address

all correspondence to Mohammad Akbari E-mail: [email protected]

The present study focuses on the problem of mixed convection fluid flow and heat transfer of nanofluid in a ventilated square cavity. The cavity contains two heated blocks subjected to external Al2O3–water (with particle diameter of 47 nm)nanofluid, while temperature and nanoparticle concentration are dependent on thermal conductivity and effective viscosity insidea square cavity. The governing equations have been solved utilizing the finite volume method,while the SIMPLER algorithm is used to couple velocity and pressure fields. The natural convection effect is obtained by heating from the blocks on the bottom wall and cooling from the inlet flow. Using the developed code, the effect of Richardson number, the aspect ratio of hot blocks, solid volume fraction, and cavity inclination angles on the thermal behavior and fluid flow inside the cavity are studied. The study has been executed for the Richardson number in the range of 0.1 ≤ Ri ≤ 10, solid volume fraction 0 ≤ φ ≤ 0.06, aspect ratio 0.5 ≤ AR ≤ 1, and cavity inclination angles between 0° and 90°.The obtained results are presented in the form of streamline and isotherm counter and Nusselt diagrams. It was observed from the results that for both obstacles and in all ranges of the parameters in this study, adding nanoparticles to the base fluid or increasing the volume fraction of the nanoparticles causes the Nusselt number increases. Also, the rate of heat transfer increases when the Reynolds number increases.

KEY WORDS: nanoscale particle, nanofluid, mixed convection, heat transfer

1. INTRODUCTION Cooling systems can be taken into account as one of the most important concerns in the industries or generally everywhere that heat transfer is important. In most cases,

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NOMENCLATURE cp

specific heat, J·kg−1·K−1

Gr

Grashof number −2

x, y

dimensional Cartesian coordinates, m

X, Y

dimensionless Cartesian coordinates

g

gravitational acceleration, m·s

Greek Symbols

h

heat transfer coefficient, W·m-2·K−1

α

thermal diffusivity, m2·s

L

enclosure length, m

β

thermal expansion coefficient, K−1

k

thermal conductivity, W m−1·K−1

θ

dimensionless temperature

Nu

Nusselt number

μ

dynamic viscosity, kg·m−1·s−1

p

pressure, N·m−2

ν

kinematic viscosity, m2·s−1

P

dimensionless pressure

ρ

density, kg·m−3

Pr

Prandtl number

φ

volume fraction of the nanoparticles

−2

q

heat flux, W·m

Subscripts

Re

Reynolds number

c

cold

Ri

Richardson number

eff

effective

T

dimensional temperature, K

f

fluid

u, v

dimensional velocities components

h

hot

in x and y directions, m·s

nf

nanofluid

dimensionless velocities components

s

solid particles

in X and Y directions

w

wall

−1

U, V

cooling optimization of existing heat transfer systems is done by increasing their heat transfer surfaces, which increases the size of these devices, which is not desirable. Besides, conventional fluids such as water, ethylene glycol, and mineral oils have a rather low thermal conductivity. Heat transfer in such devices can be enhanced by utilizing the nanofluid media. In recent years, nanofluids, which are a suspension of nano-sized solid particles in a base fluid, with thermal conductivity higher than the base fluid, have been used to enhance the rate of heat transfer in many practical engineering applications (Choi et al., 2004; Godson et al., 2010; Sarkar, 2011). Many researchers have investigated various features of nanofluids(Godson et al., 2010). Some of them have been conducted about nanofluids in an enclosure. Wen and Ding (2005) studied a numerical model in a cavity filled with water–TiO2 nanofluid heated from below to survey its heat transfer behavior. They used different models to evaluate the viscosity and effective thermal conductivity. Hwang et al. (2007) discussed the improvement of heat transfer in a nanofluid-filled rectangular enclosure

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heated from below. They illustrated that Al2O3–water nanofluids are more stable than base fluid as the size of nanoparticles decreases, the volume fraction of nanoparticles increases, or the average temperature of nanofluids increases. Numerical simulation of natural convection in a two-dimensional cavity utilizing water–copper nanofluid, with a cold right wall, hot left wall, and insulated horizontal walls, has been executed by Khanafer et al. (2003). They showed that rate of heat transfer increased with an increase in nanoparticle volume fraction for the entire range of Grashof numbers considered. Similar results were found in the work of Jou and Tzeng (2006) on numerical study of free convection for various pertinent parameters in differentially heated rectangular cavities filled with a nanofluid. They also represented that increasing the buoyancy parameter and volume fraction of nanofluids causes an increase in the average heat transfer coefficient. Abu-nada and Oztop (2009) analyzed numerically the effect of inclination angle of a square cavity on natural convection of the Cu–water nanofluid. They observed that the inclination angle can be used as a control parameter for fluid flow and heat transfer. In addition, their results showed that effects of inclination angle on percentage of heat transfer enhancement became insignificant at lower Rayleigh number. Ogut (2009) studied numerically natural convection heat transfer of water-based nanofluids in an inclined square cavity where the left vertical side is heated with a constant heat flux, the right side is cooled, and the other sides are kept adiabatic. He investigated effects of inclination angle of the cavity, solid volume fractions, length of the constant heat flux heater, and the Rayleigh number on flow and temperature field inside the cavity. Their results showed that as particle volume fraction and Rayleigh number increase, the average heat transfer rate increases. Despite application of nanofluids in heat transfer, convection in cavities having an inside hot obstacle has attracted considerable attention due to its practicality in engineering applications such as solar collectors, heat exchangers, thermal insulation, and cooling of electronic equipment and chips using nanofluids. The obstacle can be assumed as a model of heat transfer controller or modifier device. In a heat exchanger, a hot obstacle can be a model of baffle which controls the fluid flow rate and heat transfer process. For cooling of electronic chips, a hot obstacle can be used to increase or decrease heat transport from a particular part of the hot chip. Mezrhab et al. (2006) numerically described radiation–natural convection interactions in a differentially heated cavity with an inner obstacle. They concluded that the radiation exchange homogenized the temperature inside the square cavity and generated an increase in the average Nusselt number. Chen (2010) investigated natural convection heat transfer between the inner hot sphere and outer vertically eccentric cold cylinder. His obtained results indicated that heat and fluid flow patterns in the annulus are primarily dependent on the Rayleigh number, eccentricity, and geometric configurations. A numerical simulation to study of transient natural convective heat transfer of a liquid gallium inside a horizontal circular cylinder with an inner coaxial triangular cylinder is performed by Yu et al. (2010). They concluded that the time averaged Nusselt number

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was apparently increased by horizontally placing the top side of the inner triangular cylinder for Grashof numbers greater than 105. Mixed convection, which is a combination of both natural and forced convection, is a significant heat transfer mechanism that occurs in many applications. Most applications of the mixed convection flow with lid-driven effects include cooling of the electronic devices, lubrication technologies, drying technologies, chemical processing equipment, and so on. Owing to interaction of buoyancy and shear forces, mixed convection heat transfer is a complex phenomenon. With regard to the importance of this phenomenon, there are a large number of recent investigations on the mixed convection heat transfer in cavities utilizing nanofluids. For the first time, Tiwari and Das (2007) conducted a numerical study on mixed convection heat transfer in a two-sided lid-driven square cavity filled with the Cu–water nanofluid with differentially heated moving sidewalls and insulated top and bottom walls. They found that both the Richardson number and the direction of the moving walls affect the fluid flow and heat transfer in the cavity. It is also proved that with increasing volume fraction of the nanoparticles, for Richardson number equal to unity, the average Nusselt number augments. In another study, a numerical simulation on mixed convection flow and heat transfer of a Cu–water nanofluid in a lid-driven rectangular enclosure has been done by Muthtamilselvan et al. (2010). Their results indicated that both the aspect ratio and solid volume fraction influence the fluid flow and heat transfer in the enclosure. Also, the variation of the average Nusselt number is linear with solid volume fraction. Guo and Sharif (2004) used the finite volume method (FVM) and the well-known SIMPLER method to survey the mixed convection in rectangular cavities at different aspect ratios with moving isothermal sidewalls and a constant heat flux source on the bottom wall. Their results represented the average Nusselt number enhanced by moving the heat source toward the sidewalls. Mixed convection in a vented, partially heated from below square cavity was investigated numerically by Shahi et al. (2010). The cavity had an inlet and outlet in the lower corner of the left wall and the upper corner of the right wall, respectively, and a constant heat flux heater on the middle of the bottom wall. They considered effects of Richardson number and nanoparticle concentration and found that an increase in solid concentration led to an increase in the average Nusselt number of the heat source. The problem of the available classical models is their inability to evaluate the effective viscosity and thermal conductivity of the nanofluids (Morshed et al., 2008). Ho et al. (2008) distinguished four recent models for the effective dynamic viscosity and thermal conductivity of an alumina–water nanofluid on the natural convection in a square cavity. They concluded that the model used for the viscosity and the thermal conductivity of the nanofluid plays an important role in predicting that the heat transfer inside the enclosure could be either increased or decreased with respect to that of the base fluid. Effects of inlet and outlet location on mixed convection of a nanofluid in a square cavity were investigated by Mahmoudi et al. (2010). They considered four

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different combinations of inlet and outlet location and found that the flow pattern and temperature distribution inside the cavity were dependent on outlet and inlet location. Abu-Nada et al. (2010) investigated the effects of variable properties (variable thermal conductivity and variable viscosity) on natural convection in cavities subjected to Al2O3–water and CuO–water nanofluids. Their results indicated that the average Nusselt number is more affected by viscosity models than by thermal conductivity models at high Rayleigh numbers. In another study, Mahmoodi (2011) performed a numerical simulation on mixed convection of Al2O3–water nanofluid in rectangular cavities with hot moving bottom lid and cold right, left, and top walls. Also, the effect of nanofluid variable properties on mixed convection in a square cavity has been investigated by Mazrouei Sebdani et al. (2012). Recently, a numerical study has been performed by Arefmanesh and Mahmoodi (2011) to investigate uncertainty effects of dynamic viscosity models for a Al2O3–water nanofluid on mixed convection in a square enclosure consisting of two cold walls (left and right), a cold horizontal top wall, and a moving hot bottom wall. They used two different viscosity models and studied the effects of the solid volume fraction and Richardson number on the fluid flow and heat transfer inside the enclosure. They found that for both of the viscosity models, with increasing volume fraction of the nanoparticles, the average Nusselt number of the hot wall also increases. Our studies are focused on the mixed convection in a square double lid-driven inclined cavity filled with a Al2O3–water nanofluid based on a recent model. The latest proposed models by Jang et al. (2007) and Abu-Nada et al. (2010) have been used to calculate the dynamic viscosity. Also, the model of Xu et al. (2006) is used to calculate the thermal conductivity of nanofluid in the current study. These models are dependent on parameters such as diameter of nanoparticles, diameter ratio, temperature, and volume fraction of nanoparticles. In this study, heat transfer and fluid flow characteristics on the mixed convection inside an enclosure with entrance and exit of nanofluid are investigated. On the basis of the author’s knowledge in this field, no investigation has been provided using new models of variable properties so far. Existence of two hot blocks in an inclined square cavity utilizing nanofluid and investigation of the effects due to variation in size of these blocks on heat transfer and temperature and flow fields are other aspects that distinguished this survey from other numerical studies in this domain. 2. MATHEMATICAL MODELING Figure 1 shows a two-dimensional square cavity considered for the present study with physical dimensions. The cavity is subjected to an external cold nanofluid entering into the cavity from the bottom of the left insulated wall and leaving it from the top of the right wall. Two hot square obstacles are located at the bottom wall of the square cavity. The height and width of these obstacles are denoted by h and d.

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FIG. 1: Schematic of the current study

The length of the cavity perpendicular to its plane is assumed to be long enough; hence the study is considered two-dimensional. The cavity is subjected to a suspension of Al2O3 nanoparticles (with particle diameter of 47 nm) in water that there is no slip between them. The nanofluid is assumed to be incompressible, and nanoparticles and the base fluid are in thermal equilibrium. The thermophysical properties of nanoparticles and the water as the base fluid at T = 25°C are presented in Table 1. The governing equations for a steady, two-dimensional laminar and incompressible flow are expressed as ∂u ∂v + = 0, (1) ∂x ∂y ∂u ∂u 1 ∂p +v = − + υnf ∇ 2u , ∂x ∂y ρ nf ∂x

(2)

(ρβ) nf ∂v ∂v 1 ∂p +v = − + υnf ∇ 2 v + g ΔT , ∂x ∂y ρ nf ∂y ρ nf

(3)

u

u

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TABLE 1: Thermophysical properties of water and Al2O3 Physical property

Fluid phase (water)

Solid phase (Al2O3)

Cp (J/kg·k)

4179

765

ρ (kg/m3)

997.1

3970

−1

K (W·m ·K )

0.6

25

β × 10−5 (1/K)

21.

0.85

8.9





47

−1

μ × 10

−4

(kg/ms)

Diameter (nm)

u

∂T ∂T +v = α nf ∇ 2T . ∂x ∂y

(4)

The dimensionless parameters may be presented as

X =

x , L

Y =

ΔT = Th − Tc ,

y , L

V =

v , u0

T − Tc θ = , ΔT

U =

u , u0 (5)

P =

p ρ nf u02

.

Hence

Re =

ρ f u0 L μf

,

Ri =

Ra Pr . Re

2

,

Ra =

g β f ΔTL3 υfαf

,

Pr =

υf αf

.

(6)

The dimensionless forms of the preceding governing equations (1)–(4) become

∂U ∂V + = 0, ∂X ∂Y U

U

∂U ∂U ∂P υnf 1 +V = − + .∇ 2U , ∂X ∂Y ∂X υ f Re

∂V ∂V ∂P υnf 1 Ri β nf +V = − + .∇ 2V + . Δθ , ∂X ∂Y ∂Y υ f Re Pr β f U

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∂θ ∂θ nf +V = ∇ θ ∂X ∂Y α

(7) (8)

(9) (10)

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2.1 Thermal Diffusivity and Effective Density Thermal diffusivity and effective density of the nanofluid are, respectively, α nf =

k nf (ρc p ) nf

,

(11)

ρ nf = ϕρ s + (1 − ϕ)ρ f .

(12)

2.2 Heat Capacity and Thermal Expansion Coefficient Heat capacity and thermal expansion coefficient of the nanofluid are, respectively, (ρc p ) nf = ϕ(ρc p ) s + (1 − ϕ)(ρc p ) f ,

(13)

(ρβ) nf = ϕ(ρβ) s + (1 − ϕ)(ρβ) f .

2.3 Viscosity The effective viscosity of nanofluid was computed by −2ε 2 ⎡ ⎤ ⎛dp ⎞ ⎢ 3 (ε + 1)⎥ . (14) μ eff = μ f (1 + 2.5ϕ) 1 + η ⎜ ϕ ⎟ ⎢ ⎥ ⎝ L ⎠ ⎣ ⎦ This well-validated model is proposed by Jang et al. (2007) for a nanofluid with spherical particles. The empirical constant ε and η are −0.25 and 280 for Al2O3, respectively. Also, the viscosity of water is considered to change with temperature, and the flowing equation is used to evaluate the viscosity of base fluid:

μ H 2O = (1.2723 × Trc 5 − 8.736 × Trc 4 + 33.708 × Trc 3 2

(15)

6

−246.6 × Trc + 518.78 × Trc + 1153.9) × 10 , where Trc = log (T – 273). 2.4 Dimensionless Stagnant Thermal Conductivity The effective thermal conductivity of the nanoparticles in the liquid as stationary is evaluated by the model proposed by Hamilton and Crosser (1962) (the H–C model): k stationary kf

=

k s + 2k f − 2ϕ(k f − k s ) . k s + 2k f + ϕ(k f − k s )

(16)

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2.5 Total Dimensionless Thermal Conductivity of Nanofluids The following model was proposed by Xu et al. (2006), and it has been chosen in this study to describe the thermal conductivity of nanofluids:

k nf kf

+c

=

k stationary kf

k s + 2k f − 2ϕ(k f − k s ) k + c = kf k s + 2k f + ϕ(k f − k s )

Nu p d f ( 2 − D f Pr (1 − D f

)

)

1− Df ⎡⎛ d ⎤ ⎞ ⎢⎜ max ⎟ − 1⎥ ⎥ D f ⎢⎣⎝ d min ⎠ ⎦

2

⎛ d max ⎞ ⎜ ⎟ ⎝ d min ⎠

2 − Df

(17)

2

−1

1 . dp

The first term is the H–C model, and the second term is the thermal conductivity based on heat convection due to Brownian motion, where c is an empirical constant that is relevant to the thermal boundary layer and dependent on different fluids (e.g., c = 85 for the deionized water and c = 280 for ethylene glycol) but independent of the type of nanoparticles. Nup is the Nusselt number for liquid flowing around a spherical particle and is equal to 2 for a single particle in this work. The fluid molecular diameter df = 4.5⋅10−10(m) for water in the present study. The Pr is the Prandtl number, and and dp are the nanoparticle volume fraction and mean nanoparticle diameter, respectively. The fractal dimension Df is determined by Df = 2 −

ln ϕ ⎛ d p, min ⎞ ln ⎜ ⎜ d p, max ⎟⎟ ⎝ ⎠

(18)

where dp,max and dp,min are the maximum and minimum diameters of nanoparticles, respectively. With the measured ratio of dp,min/dp,max, the minimum and maximum diameters of nanoparticles can be obtained with mean nanoparticle diameter dp from the statistical property of fractal media. The ratio of minimum to maximum nanoparticles dp,min/dp,max is R: −1 D f − 1 ⎛ d p, min ⎞ d p, max = d p . ⎜ ⎟ , D f ⎜⎝ d p, max ⎟⎠ (19) Df − 1 d p, min = d p . . Df 2.6 Nusselt Number The Nusselt number is evaluated from the following relation:

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Nu =

hd , kf

(20)

where the heat transfer coefficient h is defined as

h =

qw Th − Tc

(21)

and the thermal conductivity may be expressed as k nf =

−qw . ∂T ∂X

(22)

By substituting Eqs. (18) and (19) into Eq. (17), the Nusselt number for the left hot wall can be written as ⎛ k nf ⎞ ⎛ ∂θ ⎞ Nu = − ⎜ . (23) ⎜ k f ⎟⎟ ⎜⎝ ∂x ⎟⎠ ⎝ ⎠ The average Nusselt number calculated over the hot surface by Eq. (18) becomes

Nu m =

1 L ∫ NudY . L 0

(24)

3. NUMERICAL IMPLEMENTATION Governing equations (including continuity, momentum, and energy equations) are discretized using the finite volume method and with a staggered grid system. The SIMPLER algorithm introduced by Patankar (1980) has been adopted for the pressure velocity coupling. The convection terms are approximated by a blend of central difference scheme and upwind scheme (hybrid scheme), which is conducive to a stable solution. Besides, a second-order central differencing scheme is served to the diffusion terms. The algebraic system arising from numerical discretization is computed using the tridiagonal matrix algorithm (Versteeg and Malalasekera, 1995). The solution process is repeated until an acceptable convergence criterion is reached. A FORTRAN computer code has been developed to solve the equations, as described earlier. The process is repeated until the following convergence criterion is satisfied: error =

j = M i = N n +1 − λn ∑ j =1 ∑ i =1 λ j=M i=N ∑ j =1 ∑ i =1

λ

n +1

< 10 −7.

(25)

Here M and N correspond to the number of grid points in x and y directions, respectively, n is the number of the iteration, and λ denotes any scalar transport quantity. To verify

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FIG. 2: Grid study

FIG. 3: Comparison of average Nusselt numbers between Lin and Violi (2010) and the present result for Gr = 105, dp = 5 nm, and ϕ = 0.05 for different values of Pr number

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grid independence, the numerical procedure was examined for nine different mesh sizes, namely, 21 × 21, 31 × 31, 41 × 41, 51 × 51, 61 × 61, 71 × 71, 81 × 81, 91 × 91, and 101 × 101. Average Nu at Re = 10, AR = 0.75, γ = 60D , and ϕ = 0.06 is attained for each grid size, as shown in Fig. 2. As can be observed, an 91 × 91 uniform grid size yields the required accuracy and was hence applied for all simulation exercises in this work, as presented in the following section. TABLE 2: Code validation showsthecomparison between the results in present study and the results of other research. Present study

Lin and Violi (2010)

Tiwari and Das (2007)

Hadjisophocleous et al. (1998)

(a) Ra = 103 umax

3.619

3.597

3.642

3.544

Y

0.811

0.819

0.804

0.814

vmax

3.697

3.690

3.7026

3.586

X

0.180

0.181

0.178

0.186

Nuave

1.114

1.118

1.0871

1.141

4

(b) Ra = 10 umax

16.052

16.158

16.1439

15.995

Y

0.817

0.819

0.822

0.814

vmax

19.528

19.648

19.665

18.894

X

0.110

0.112

0.110

0.103

Nuave

2.215

2.243

2.195

2.29

5

(c) Ra = 10 umax

36.812

36.732

34.30

37.144

Y

0.856

0.858

0.856

0.855

vmax

68.791

68.288

68.7646

68.91

X

0.062

0.063

0.05935

0.061

Nuave

4.517

4.511

4.450

4.964

6

(d) Ra = 10 umax

66.445

66.46987

65.5866

66.42

Y

0.873

0.86851

0.839

0.897

vmax

221.748

222.33950

219.7361

226.4

X

0.0398

0.03804

0.04237

0.0206

Nuave

8.795

8.757933

8.803

10.39

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(a)

(b)

FIG. 4: Dimensionless effective thermal conductivity of Al2O3–water nanofluid versus concentration of nanoparticles with different mean nanoparticle diameters and fractal distributions: (a) T = 300 and (b) R = 0.004

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The proposed numerical scheme is validated by comparing the present code results for Gr = 105, dp = 5 nm, ϕ = 0.05, and zero inclination angle for different values of Pr numbers against the numerical simulation published by Linand Violi (2010) in Fig. 3. It is clear that the present code is in good agreement with another work reported in the literature, as shown in Fig. 3. Also, to ensure the accuracy and validity of the new model, we analyze a square cavity filled with base fluid with Pr = 0.7 and different Ra numbers. This system has been previously investigated by other researchers such as Lin and Violi (2010), Tiwari and Das (2007), and Hadjisophocleous et al. (1998). Table 2 shows the comparison between the results obtained with the new model and the values presented in the literature. The quantitative comparisons for the average Nusselt numbers indicate an excellent agreement between them. To check Xu’s model, Fig. 4 indicates the characteristics of the effective thermal conductivity, which is a function of the practical parameters T, R = dp,min/dp,max, and dp. As indicated in Fig. 4a, R has a relatively high impact for small average nanoparticle diameters. The temperature effect of nanofluids is illustrated in Fig. 4b. It is found that the existence of nanoparticles affects extensively the heat conductivity of the nanofluid at high temperatures. Thus, compared with the H–C model with the assumption of uniform nanoparticle size, Xu’s model proposed a better flexibility in evaluating the heat transfer characteristics. 4. RESULTS AND DISCUSSION Figure 5 illustrates the flow pattern changes with respect to the variation in both Richards on number (Ri) and aspect ratio (AR) at Re = 100, particle diameter of 47 nm, and horizontal position. Solid lines show water-based fluid and dashed lines display a nanofluid with the volume fraction of 0.06. With Ri = 0.1, the effect of the entrance fluid force predominates the buoyancy effect, and streamlines have been expanded after entering the cavity so that, consequently, velocity has been decreased and pressure has been dropped. In this case, the flow pattern shows three weak vortices in different positions among the distance between two blocks, the distance between the second block and wall, and the upper right wall. In creasing aspect ratio and length of blocks lead to an increase in intensity of these vortexes. Also, a substantial difference in flow pattern within base fluid and nanofluid have not been seen. With increasing Ri and reinforcement of buoyancy force than fluid force, the counterclockwise vortex near the upper surface is reinforced and occupies a considerable portion of the cavity. The streamlines are accumulated near the blocks with a widened vortex in the upper half of the cavity, and as a result, flow over hot surfaces of blocks is increased. Temperature fields for aspect ratios and Richardson numbers corresponding to Fig. 5 and Re = 100, γ = 0 are shown for nanofluid and base fluid in Fig. 6. With increasing Ri and overcoming buoyancy force on shear force, isotherm lines ac-

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FIG. 5: Streamline variation with respect to the variation in both Richardson number (Ri) and aspect ratio (AR) at Re = 100 and horizontal position

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FIG. 6: Temperature field variation with respect to the variation in both Ri and AR at Re = 100 and horizontal position

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cumulate near the block walls and the temperature gradient surges, which leads to reinforced convection heat transfer. According to the figures, a considerable portion of the upper left cavity is isothermal. The existence of nanofluid due to rising thermal conductivity reduces the temperature gradient and diffuses isothermal lines near the block walls. Flow field variation with Re and the inclination angle of the cavity-filled nanofluid are shown in Fig. 7 at Ri = 7, AR = 0.75, with Al2O3 particle diameter of 47 nm. With changing inclination angle from horizontal position to 30° at Ri = 1, no significant variation inflow pattern is created. However, the increase of Reynolds number (Re) leads to the emergence of more variation at different angles. An increase in cavity angle causes an aid to the buoyancy force due to temperature difference and shear force in the cavity. Also, it strengthens flow near blocks. At Re = 10, with increasing inclination angle of the cavity, the upper vortex is strengthened, and the compression of streamlines on the blocks (especially the left) is increased to γ = 90 . At Re = 100, with increase of angle, compression of streamlines and mass flow are grown on the blocks. Almost in all examined cases in this figure, adding nanofluid reinforces and expands the sizes of the upper vortex and also increases the compression of streamlines at the lower areas of the cavity. This accumulation leads to increased mass flow, and it is expected that it will increase heat transfer on the blocks. Figure 8 shows temperature fields with variation of Re and inclination angle of cavity at Ri = 7, AR = 0.75. As can be seen, in the range of these parameters, heat transfer mainly is done through conduction heat transfer. Isotherms demonstrate that there is not much of a temperature gradient near the isotherm walls, and distribution of these lines is uniform in the cavity, which is indicative of conduction heat transfer. The rise of cavity inclination angle in this case does not show significant changes in temperature gradient. Also, the existence of nanoparticles in fluid does not cause a palpable difference in thermal behavior of fluid and cavity. With increasing Re, isotherm lines near the isotherm walls are accumulated, and also the temperature gradient is increased. At Re = 100, the compression of isotherm lines is clearly visible near the wall blocks. In this case, with increasing cavity inclination angle, the temperature gradient near the walls increases very slightly. The existence of nanoparticles at high Reynolds number, owing to increasing thermal conductivity, leads to a slight reduction in temperature gradient, which is visible for all examined inclination angles, especially high inclination angles (60° and 90°). As can be seen in Fig. 9, flow patterns in the base of changing both Re and Ri at AR = 1 and the horizontal position of the cavity are displayed. In this figure, the temperature field is investigated at Re = 1, 10, 100 and Ri = 0.1, 3, 7, 10. The general pattern of flow in all figures indicates three vortexes that are located in the upper block, the distance between the two blocks, and the interspace between the block and right wall. The size, strength, and weakness of these vortices are different for various

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FIG. 7.

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FIG. 7: Flow field variation with Re and inclination angle of cavity-filled nanofluid

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FIG. 8.

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FIG. 8: Temperature fields with variation of Re and inclination angle of cavity at Ri = 7, AR = 0.75

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FIG. 9

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Mixed Convection Flow and Heat Transfer in a Ventilated Inclined Cavity

FIG. 9: Streamlines at different Re and Ri at AR = 1 for horizontal position of cavity

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values of Reynolds and Richardson numbers. Rising values of Ri at low values of the Reynolds number (Re = 1) does not change much in the flow behavior. In this case, each of the three vortices is weak and streamlines are nearly distributed uniformly throughout the cavity. At Re = 10, with growth of Ri, buoyancy force is increased. Also, its interaction with input fluid motion strengthens the upper vortex. Counter clockwise vortex growth in the upper are as of the block leads to agglomerate stream lines and increases mass flow close to the hot block walls. In addition, in this case, heat transfer is increased. At Re = 100, also, streamlines have similar behavior with rising Richardson number, like Re = 10, but with the difference that the upper vortex is grown more and accumulation entrance streamlines in the lower of cavity are more too. Therefore, with growth of mass flow on the hot blocks rises heat transfer. The existence of suspension nanoparticles in the base fluid does not noticeably affect flow behavior at Re = 1. Figure 10 illustrates isotherm changes with variation of Re and Ri at AR = 1 and horizontal position of the nanofluid-filled cavity. At Ri = 1, the uniform distribution of isotherm contours indicates that there is a significant conduction heat transfer in this case. Furthermore, in this case, Richardson number variation and using nanoparticles do not cause change in the temperature field. With increasing values of Re, accumulation of isotherm lines leads to a temperature gradient near the isotherm walls that indicates that convection heat transfer is enhanced. A nanofluid with a volume fraction of 0.06 reduced the temperature gradient rather than the base fluid. But naturally increasing the thermal conductivity of fluid leads to an increase in heat transfer in this case. Nusselt number diagrams in terms of Ri in the various cavity inclination angles for nanofluid with ϕ = 0.06 , AR = 0.5 at different Reynolds numbers separately for left and right blocks are displayed in Fig. 11. As expected, an increase in Richardson number in all cases, except for the right block, leads to an increase in heat transfer at Re = 1 in the enclosure. As mentioned earlier, this increase is due to an intense upper vortex in the upper cavity and enhancement of the flow passing over the heated surfaces and also an increase in the temperature gradient near the blocks. In addition, cavity inclination angle enhancement leads to the alignment between the buoyancy force and entrance fluid force and rising Nusselt number, especially on the left block. Nusselt number variations with cavity inclination angles are irregular for the right block because of thermal behavior and the severe influence of fluids near this wall by the upper and lower vortices. Owing to a small distance between the left wall and the fluid in let, and high velocity and mass flow rate over the surfaces of the block, major differences within the Nusselt numbers and right and left walls are observed that indicate that the left wall is important for all Reynolds numbers, Richardson numbers, and inclination angles in the considered configuration. Figure 12 also shows the changes of Nusselt number versus Richardson number for different volume fractions on both the left and right blocks. In these figures, the cav-

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Mixed Convection Flow and Heat Transfer in a Ventilated Inclined Cavity

FIG. 10.

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FIG. 10: Isotherm lines at different Re and Ri at AR = 1 for horizontal position of cavity

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FIG. 11: Nusselt number versus Ri number in the various cavity inclination angles for nanofluid with φ = 0.06, AR = 0.5 at different Reynolds numbers separately for left and right blocks

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FIG. 12: Nusselt number versus Ri number for different volume fractions on both the left and right blocks.

ity is in a horizontal position, and Re = 10 and AR = 0.5. It is clear that, with adding nanoparticles to the base fluid, heat transfer has increased in all cases on the both of blocks. 5. CONCLUSIONS Heat transfer and fluid flow characteristics in mixed heat transfer inside a cavity with two hot square obstacles along with nanofluid inlet and outlet were investigated. The following items are extracted from analysis of streamlines and temperature lines and also Nusselt diagrams, which are obtained from the numerical simulation. 1. For both obstacles and in all ranges of the parameters in this study, adding nanoparticles to the base fluid or increasing the volume fraction of the nanoparticles causes the Nusselt number and, consequently, the heat transfer to increase. 2. In all the cases, except for the right-hand side obstacle at Re = 1, an increase in Richardson number causes the heat transfer inside the cavity to increase. 3. Increasing inclination of the cavity causes the Nusselt number at the left-hand side obstacle to increase due to agreement (alignment) of buoyancy force and the force created by the entering fluid. 4. In all ranges of the parameters in this study, increasing Reynolds number makes the temperature lines near the nonadiabatic lids become more intense, which causes heat transfer to increase. 5. Adding nanoparticles to the base fluid reduces the temperature gradient while considerably increasing the heat transfer coefficient and thus improving the heat transfer.

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