Laminar Mixed Convection in Inclined Triangular Enclosures Filled

Article pubs.acs.org/IECR

Laminar Mixed Convection in Inclined Triangular Enclosures Filled with Water Based Cu Nanofluid M. M. Rahman,*,†,‡ Hakan F. Ö ztop,§ A. Ahsan,⊥ R. Saidur,†,# Khaled Al-Salem,∥ and N. A. Rahim† †

Centre of Research UMPEDAC, Level 4, Engineering Tower, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia ‡ Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh § Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey ∥ Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia ⊥ 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 # Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia ABSTRACT: This paper presents the results on mixed convection in an inclined lid-driven triangular enclosure filled with water based Cu nanofluid. The enclosure is cooled at the inclined surface and simultaneously heated at the base surface. The vertical wall is adiabatic and moving at a constant speed. The governing equations are solved numerically by the Galerkin finite element method. The effects of parameters such as the Richardson number, nanoparticle volume fraction, and tilt angle on the flow and thermal fields as well as the heat transfer rate of the heated surface are taken into account. The overall heat transfer rate of the heated surface is characterized by the average Nusselt number at the heated surface. It has been observed that the effects of the tilt angle and solid volume fractions are significant on the flow and thermal fields. Besides, an optimum value for the solid volume fraction is found, which results in the maximum heat transfer rate at the considered values of the Richardson numbers. The rate of the increase of the average Nusselt number is slow for higher values of the tilt angle while it is much quicker for lower values of the tilt angle. Koca et al.13 analyzed the effect of Prandtl number on natural convection heat transfer and fluid flow in triangular enclosures with localized heating. The authors found that both flow and temperature fields are affected by the change of Prandtl number, Rayleigh number, and the location and length of the heater. Basak et al.14 investigated the effects of uniform and nonuniform heating of inclined walls on natural convection flows within an isosceles triangular enclosure using a penalty finite element method with biquadratic elements. They observed that nonuniform heating produces higher heat transfer rates at the center of the walls than is the case with uniform heating. Nanofluids are a new class of fluids, which consist of a base fluid in the presence of nanoscale materials.15 The convective heat transfer feature of nanofluids is affected by the thermophysical properties of the base fluid and nanoparticles. The function of a meticulous nanofluid for a heat transfer intention can be dealt with quite accurately by modeling the convective transportation in the nanofluid.16 In recent years, convective heat transfer using nanofluids has received attention because of the promising prospect of nanofluids in enhancing heat transfer. Khanafer et al.17 numerically studied natural convection of copper−water nanofluid in a two-dimensional enclosure. The authors found that, at any given Grashof

1. INTRODUCTION The problem of convection (forced, mixed, or natural) in a two-dimensional cross sectional heated triangular enclosure is of great interest to the computational fluid dynamics community for their practical applications such as attic space heating or cooling of electronic equipment, and solar air heaters.1−6 Previous studies were mostly performed on the different flow systems in triangular enclosures. Basak et al.7 numerically studied the phenomena of natural convection in a right-angled triangular enclosure. The objective of their study was to examine the flow and thermal fields with comprehensive study of heat transfer estimate for natural convection in a triangular enclosure. Omri et al.8 analyzed natural convection flows using a control volume finite element method in the isosceles triangular cavities. Chen and Cheng9 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 wintertime heating of an attic space. Varol et al.11 numerically analyzed natural convection heat transfer in a triangular enclosure with a flush mounted heater on the wall. They used the finite difference method in their computations. The authors found that both the position and location of the heater affect the flow circulation and heat transfer in the enclosure. Later, Varol et al.12 made a numerical study of natural convection heat transfer from a protruding heater located in a triangular enclosure. © 2012 American Chemical Society

Received: Revised: Accepted: Published: 4090

June 9, 2011 January 21, 2012 February 7, 2012 February 7, 2012 dx.doi.org/10.1021/ie201235p | Ind. Eng. Chem. Res. 2012, 51, 4090−4100

Industrial & Engineering Chemistry Research

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Figure 2. Grid independency study for Re = 100, Ri = 1, ϕ = 60, and δ = 0.04.

Table 2. Comparison of Nu Values with Those of Muthtamilselvan et al.21 Nu

Figure 1. Schematic of configuration and boundary conditions.

Table 1. Thermophysical Properties of Water and Copper property

water

copper

cp ρ k β

4179 997.1 0.613 2.1 × 10−4

385 8933 401 1.67 × 10−5

27

δ

ref 21

present study

% increase

0.0 0.02 0.04 0.06 0.08

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

Abu-Nada and Oztop,29 Abu-Nada et al.,30 and Oztop et al.31 All of these studies were done to simulate cavities with smooth or regular boundaries. The influence of the inclination angle on buoyancy-driven convection in a triangular enclosure filled with a fluid-saturated porous medium was studied by Varol et al.32 It is clear that the mixed convection heat transfer in enclosures using nanofluids has received great interest in recent years. However, to the best knowledge of the authors, mixed convection in a triangular enclosure using nanofluids has received little attention in the literature. Thus, the purpose of this work is to investigate the effects of the solid volume fraction and tilt angle of the enclosure on the flow and temperature fields as well as the heat transfer rate for mixed convection in an inclined lid-driven triangular enclosure filled with Cu−water nanofluid. In order to develop an understanding of the fluid flow and mixed convection heat transfer characteristics in such processes, this study presents a numerical simulation to the complete Navier−Stokes and energy equations for steady-state laminar mixed convection flow in lid-driven inclined triangular nanofluid filled enclosures. Because of the specifity of the triangular geometry, the inclined boundary and moving lid may play important roles in the heat transfer and fluid flow by adding nanoparticles into the base fluid. These coupled equations are solved using the finite element scheme based on the Galerkin method of weighted residuals.

number, heat transfer within the enclosure increased with the increase of volumetric fraction of the copper nanoparticles in water. Jou and Tzeng18 investigated heat transfer enhancement utilizing nanofluids in a two-dimensional enclosure for various pertinent parameters. Tiwari and Das19 performed numerical modeling of mixed convection in a two-sided lid-driven differentially heated square cavity filled with nanofluid. 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. The authors also concluded that the variation of the average Nusselt number is nonlinearly proportional to the solid volume fractions. Aminossadatia and Ghasemi20 investigated natural convection cooling of a heat source embedded on the bottom wall of an enclosure filled with nanofluids. Muthtamilselvan et al.21 conducted a numerical study to investigate the transport mechanism of mixed convection in a lid-driven enclosure filled with nanofluids. Ghasemi and Aminossadati22 considered mixed convection in a lid-driven triangular enclosure filled with water−Al2O3 nanofluid. They found that the addition of Al2O3 nanoparticles enhances the heat transfer rate for different values of the Richardson number and for both directions of motion of the sliding wall. A parametric study on mixed convection flow in a lid-driven inclined square enclosure filled with water− Al2O3 nanofluid was conducted by Abu-Nada and Chamkha.23 Mansour et al.24 performed a numerical simulation on mixed convection flow in a square nanofluid-filled lid-driven cavity partially heated from below. Corcione25 investigated theoretically the heat transfer features of buoyancy-driven nanofluids inside rectangular enclosures differentially heated at the vertical walls. Saleh et al.26 investigated heat transfer enhancement utilizing nanofluids in a trapezoidal enclosure for various pertinent parameters. Talebi et al.27 numerically studied mixed convection flows in a square lid-driven cavity utilizing nanofluid. Other studies can be found in the literature by Oztop and Abu-Nada,28

2. PROBLEM FORMULATION AND GOVERNING EQUATIONS A schematic of the considered model is shown in Figure 1 with the coordinate system. It is a two-dimensional inclined liddriven triangular enclosure. The length of the base wall and height of the sliding wall of the enclosure are depicted by L and H, respectively. The tilt angle is given by ϕ and the aspect ratio of the enclosure is considered unity (L/H = 1). The temperature (θh) of the isothermal wall is higher than the temperature (θc) of the inclined wall. The sliding wall of the cavity is kept insulated and is assumed to slide from bottom to 4091

dx.doi.org/10.1021/ie201235p | Ind. Eng. Chem. Res. 2012, 51, 4090−4100


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top at a constant speed V0. The enclosure is filled with a copper−water nanofluid that is modeled as a Newtonian fluid having constant physical properties except for the density variation, which is considered to be temperature dependent according to the Boussinesq approximation. The flow is assumed to be laminar and incompressible. It is considered that thermal equilibrium exists between the base fluid and nanoparticles, and no slip occurs between the two media. The thermophysical properties27 of the nanofluid are listed in Table 1. Under these assumptions, the mathematical model for steady laminar mixed convection flow using conservation of mass, momentum, and energy can be written as ∂u ∂v + =0 ∂x ∂y

u

P=

(ρβ)nf (T − Tc)g sin ϕ ρnf

(ρβ)nf (T − Tc)g cos ϕ ρnf

∂U ∂U ∂P 1 ρf 1 +V =− + ∂X ∂Y ∂X Re ρnf (1 − δ)2.5 ⎛ ∂ 2U ∂ 2U ⎞ (ρβ)nf ⎟⎟ + × ⎜⎜ 2 + Ri θ sin ϕ ρnf βf ⎝ ∂X ∂Y 2 ⎠

U

(2)

∂V ∂V ∂P 1 ρf 1 + +V =− U ∂X ∂Y ∂Y Re ρnf (1 − δ)2.5 ⎛ ∂ 2V ∂ 2V ⎞ (ρβ)nf ⎟⎟ + × ⎜⎜ 2 + Ri θ cos ϕ ρnf βf ⎝ ∂X ∂Y 2 ⎠ (14)

U (4)

(5)

k nf (ρcp)nf

α ∂θ ∂θ 1 ⎛ ∂ 2θ ∂ 2θ ⎞ ⎟⎟ ⎜⎜ +V = nf + ∂X ∂Y α f Re Pr ⎝ ∂X 2 ∂Y 2 ⎠

(15)

Here the Reynolds number, Prandtl number, and Richardson number are defined as Re = V0L /υf , Pr = νf /α f ,

and δ is the solid volume fraction of the nanoparticles. In addition, the thermal diffusivity αnf of the nanofluid is given by

Ri = g βf (Th − Tc)L /V02

The appropriate boundary conditions for the governing equations are

on the base wall: U = V = 0,

(6)

θ=1

on the sliding wall:

The heat capacitance of the nanofluids can be expressed as

U = 0,

(7)

Additionally, (ρβ)nf is the thermal expansion coefficient of the nanofluid and it can be determined by

∂θ =0 ∂N

V = 1,

on the inclined wall: U = V = 0,

(8)

Furthermore, μnf is the dynamic viscosity of the nanofluid introduced by Brinkman33 as μf μnf = (1 − δ)2.5 (9)

θ=0

where N is the nondimensional distance in either the X or Y direction acting normal to the surface. The average Nusselt number at the heated surface of the cavity may be expressed by 1 ∂θ k Nu = − nf dX k f 0 ∂Y

∫

The effective thermal conductivity of nanofluid was given by Khanafer et al.17 and Wasp34 as follows: k + 2k f − 2δ(k f − k s) k nf = s kf k s + 2k f + δ(k f − k s)

(11)

(12)

(3)

ρnf = (1 − δ)ρf + δρs

(ρβ)nf = (1 − δ)(ρβ)f + δ(ρβ)s

T − Tc Th − Tc

(1)

where the effective density ρnf of the nanofluid is defined by

(ρcp)nf = (1 − δ)(ρcp)f + δ(ρcp)s

θ=

,

(13)

⎛ ∂ 2T ∂T ∂T ∂ 2T ⎞ +v = α nf ⎜⎜ 2 + 2 ⎟⎟ u ∂x ∂y ∂y ⎠ ⎝ ∂x

α nf =

ρnf V0

2

∂U ∂V + =0 ∂X ∂Y

μ ⎛ ∂ 2v ∂v ∂v 1 ∂p ∂ 2v ⎞ +v =− + nf ⎜⎜ 2 + 2 ⎟⎟ ∂x ∂y ρnf ∂y ρnf ⎝ ∂x ∂y ⎠ +

(p + ρgy)L2

Using the definitions in 11, eqs 1−4 are transformed into the following nondimensional forms:

μ ⎛ ∂ 2u ∂u ∂u 1 ∂p ∂ 2u ⎞ +v =− + nf ⎜⎜ 2 + 2 ⎟⎟ ∂x ∂y ρnf ∂x ρnf ⎝ ∂x ∂y ⎠ +

u

nondimensional governing equations, the following dimensionless variables are used: y x u v Y= , U= V= X= , , , L L V0 V0

(16)

and the average fluid temperature in the enclosure is defined by (10)

Θ=

where ks is the thermal conductivity of the nanoparticles and kf is the thermal conductivity of the base fluid. To obtain

∫ θ dV̅ /V̅

(17)

where V̅ is the cavity volume. 4092



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Figure 3. Streamlines and isotherms for δ = 0.08: (a) present code; (b) Muthtamilselvan et al.21

Figure 4. Streamlines for different values of tilt angle ϕ and Richardson number Ri, with δ = 0.04. 4093



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Figure 5. Isotherms for different values of tilt angle ϕ and Richardson number Ri, with δ = 0.04.

The fluid motion is displayed using the stream function ψ obtained from velocity components U and V. The relationships between the stream function and the velocity components35 for two-dimensional flows are U=

∂ψ , ∂Y

V=−

∂ψ ∂X

The continuity equation is automatically fulfilled for large values of γ. Using eq 19, the momentum equations (eqs 13 and 14) reduce to

U (18)

∂U ∂U ∂ ⎛⎜ ∂U ∂V ⎞⎟ + +V =γ ⎝ ∂X ∂Y ∂X ∂X ∂Y ⎠ +

⎛ ∂ 2U 1 ρf 1 ∂ 2U ⎞ ⎟⎟ ⎜⎜ + Re ρnf (1 − δ)2.5 ⎝ ∂X 2 ∂Y 2 ⎠

+

(ρβ)nf Ri θ sin ϕ ρnf βf

3. NUMERICAL SOLUTION 3.1. Solution Method. In this section, the Galerkin finite element method is discussed to solve the nondimensional governing equations along with boundary conditions for the considered 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. To solve eqs 13−15, the penalty finite element method7 is used, where the pressure P is eliminated by a penalty constraint γ and the incompressibility criterion given by eq 12 results in ⎛ ∂U ∂V ⎞⎟ + P = −γ⎜ ⎝ ∂X ∂Y ⎠

U

(19) 4094

(20)

∂V ∂V ∂ ⎛⎜ ∂U ∂V ⎞⎟ +V =γ + ⎝ ∂X ∂Y ∂Y ∂X ∂Y ⎠ +

⎛ ∂ 2V 1 ρf 1 ∂ 2V ⎞ ⎜⎜ ⎟⎟ + Re ρnf (1 − δ)2.5 ⎝ ∂X 2 ∂Y 2 ⎠

+

(ρβ)nf Ri θ cos ϕ ρnf βf

(21)



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Figure 7. Effect of tilt angle ϕ on (a) average Nusselt number at the heated surface and (b) average fluid temperature in the cavity, with δ = 0.04. ⎡ N

N

R i(2) =

∑

Uk

k=1

∫Ω ⎢⎢( ∑ Uk Φk) ⎣ k=1

N ⎤ ∂Φk ∂Φk ⎥ + ( ∑ Vk Φk ) Φi dX dY ∂X ∂Y ⎥⎦ k=1

⎡ N ∂Φi ∂Φk − γ⎢ ∑ Uk dX dY ⎢ ∂X ∂X Ω ⎣k = 1

∫

N

∑

+

k=1 N

Figure 6. Local heat transfer variation on heated surface for various tilt angles ϕ at (a) Ri = 0.1, (b) Ri = 1, and (c) Ri = 10, with δ = 0.04.

V≈

θ≈

∑ θkΦk(X , Y )

N

−

∫

⎡ N

∑

α nf α f Re Pr

⎣ k=1

N ⎤ ∂Φk ∂Φk ⎥ + ( ∑ Vk Φk ) Φi dX dY ∂X ∂Y ⎥⎦ k=1

∫

N

+

∑

×

⎡ ∂Φi ∂Φk ∂Φi ∂Φk ⎤ + ⎥ dX dY ⎢⎣ ∂Y ∂Y ⎦ ∂ ∂ X X Ω

∑ k=1

⎤ ∂Φi ∂Φk 1 ρf 1 dX dY ⎥ − ⎥ ∂Y ∂Y Re ρnf (1 − δ)2.5 ⎦

Vk

∫Ω

Vk

∫Ω ⎢⎣ ∂Xi

k=1 N

∑ θk ∫ k=1

∫Ω ⎢⎢( ∑ Uk Φk)

(24)

⎡ N ∂Φi ∂Φk − γ⎢ ∑ Uk dX dY ⎢ ∂Y ∂X Ω ⎣k = 1

N ⎡ N ⎤ ⎢( ∑ U Φ ) ∂Φk + ( ∑ V Φ ) ∂Φk ⎥Φ dX dY k k k k i ∂Y ⎥⎦ ∂X Ω⎢ ⎣ k=1 k=1 N

Vk

k=1

(22)

∑ θk∫ k=1

(ρβ)nf Ri sin ϕ ( ∑ θkΦk )Φi dX dY ρnf βf Ω

N

R i(3) =

Then the Galerkin finite element technique yields the subsequent nonlinear residual equations for eqs 15, 20, and 21, respectively, at nodes of the internal domain Ω: R i(1) =

⎡ ∂Φ ∂Φk ∂Φi ∂Φk ⎤ + ⎥ dX dY ∂X ∂Y ∂Y ⎦

∫Ω ⎢⎣ ∂Xi

k=1

∑ Vk Φk(X , Y ),

k=1

⎤ ∂Φi ∂Φk 1 ρf 1 dX dY ⎥ − ⎥ ∂X ∂Y Re ρnf (1 − δ)2.5 ⎦

N

−

k=1

k=1 N

∫Ω

Uk

k=1

N

N

∑ Uk Φk(X , Y ),

∑

×

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

Vk

⎡ ∂Φ ∂Φk ∂Φi ∂Φk ⎤ + ⎥ dX dY ∂X ∂Y ∂Y ⎦ N

(ρβ)nf Ri cos ϕ ( ∑ θkΦk )Φi dX dY − ρnf βf Ω

∫

(23)

k=1

4095

(25)



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Figure 8. Streamlines for different values of solid volume fractions δ and Richardson number Ri, with ϕ = 60°.

First, the comparison of the average Nusselt number (at the hot wall) between the outcome of the present code and the results found in the literature21 for various solid volume fractions is documented in Table 2. The obtained results are a little higher than those from the literature, but they are acceptable. This stems from the method used. Finally, the streamlines and isotherms for the solid volume fractions (δ = 0.08) are presented in Figure 3. It can be seen from Figure 3 that the present results and those reported by Muthtamilselvan et al.21 are in excellent agreement. This validation boosts the confidence in the numerical outcome of the present work.

Bi-quadratic basis functions with three point Gaussian quadrature are used to evaluate the integrals in the residual equations, except for the second term in eqs 24 and 25. The penalty parameter (γ) containing these equations are evaluated using reduced integration penalty formulation.36 The nonlinear residual equations (eqs 23−25) are solved using the Newton− Raphson method to determine the coefficients of the expansions in eq 22. The solution process is iterated until the subsequent convergence condition is satisfied: |Γ m + 1 − Γ m| ≤ 10−6

4. RESULTS AND DISCUSSION

where m is the number of iterations and Γ is a general dependent variable. 3.2. Grid Independence Study. A grid independence study has been carried out for Re = 100, Ri = 1, ϕ = 60°, and δ = 0.04 with average Nusselt number (Nu) for five different grid sizes as shown in Figure 2. A grid size of 3490 elements is found to meet the requirements of both the grid independence study and the computational time limits. 3.3. Code Validation. The present numerical code was validated against the problem of mixed convection in a lid-driven square enclosure filled with nanofluids studied by Muthtamilselvan et al.21 The cavity was heated at the top wall and cooled at the bottom, while the rest of the boundaries were insulated.

In the studied configuration, numerical analysis has been performed on laminar mixed convection in a two-dimensional inclined lid-driven triangular enclosure filled with water based copper nanofluid. In this investigation, an attention is given to the effects of the parameters, namely, the tilt angle ϕ, solid volume fraction δ, and Richardson number Ri. Simulations are made for various values of the solid volume fractions (0 ≤ δ ≤ 0.1) and tilt angle (0° ≤ ϕ ≤ 270°). Here, the Re is fixed at 100 and the value of Ri is varied from 0.1 to 10 by changing the Grashof number Gr to cover the forced convection dominated region, pure mixed convection, and the free convection dominated region. 4096



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Figure 9. Isotherms for different values of solid volume fractions δ and Richardson number Ri, with ϕ = 60°.

clockwise circulation cell is decreased, and that of the anticlockwise cell is increased while the tilt angle increases to ϕ = 90°. For ϕ = 270°, a single cell is formed again and the results are very similar to those in the case of ϕ = 0°. Similarly, the influence of the tilt angle on the temperature distribution, which is plotted as isotherms, is shown in Figure 5 for the same values of Ri and ϕ in Figure 4. Here, isotherms represent the lines with equal intervals between unity (hot wall) and zero (cold wall). As illustrated in Figure 5, as ϕ increases, the isotherm lines are denser toward the warm wall for all considered values of Ri. These results are supported by Varol et al.32 for pure fluid filled enclosures. In this case, Rayleigh− Bénard type flow occurs due to bottom wall heating at ϕ = 0°. However, this effect disappears with increasing inclination angle. Inclination angle is not an effective parameter on temperature distribution for Ri = 0.1 and 1, because the flow inside the cavity mostly moves by the lid-driven wall. When Ri = 10, buoyancy induced flow motion becomes effective. This motion causes the distortion of isotherms. However, conductive heat transfer becomes dominant near the cold wall for ϕ = 0° and ϕ = 270°. This behavior is also related to increasing flow strength as seen from Figure 4. Local heat transfer variation on a heated surface for various tilt angles ϕ at different Richardson numbers is presented in Figure 6. In Figure 6, the nanofluid volume fraction is taken as δ = 0.04.

Figure 4 shows the streamlines for three different values of the Richardson number Ri (=0.1, 1, and 10 from left to right) and four tilt angles ϕ (=0, 45, 90, and 270° from bottom to top) at δ = 4% solid volume fraction. As seen clearly from the definition of Ri, the Richardson number provides a measure of the importance of buoyancy-driven natural convection relative to the lid-driven forced convection. For the dominating forced convection case with Ri = 0.1 and tilt angle ϕ = 0°, the streamlines in the cavity are characterized by a primary recirculating clockwise cell occupying the whole of the cavity generated by the moving lid. In this case, the streamlines still show the formation of the clockwise recirculating cell of the size of the cavity at higher values of the Richardson number Ri (=1 and 10). It is also observed that the core of the recirculating cell goes through the middle of the triangle with increasing Richardson number. The cell turns from ellipsoidal to circular. On the other hand, the flow strength of the recirculating cell in the core becomes stronger and changes from ψ = −0.06 to ψ = −0.23 with the increasing value of the Richardson number. Then, for ϕ = 45° and Ri = 0.1, two opposite direction vortices are developed inside the cavity as seen from the third line from the top. Moreover, the size of the clockwise cell decreases in comparison with the previous case of ϕ = 0° as a result of the formation of the counterclockwise cell. Furthermore, for the same Richardson number, the size of the 4097



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Figure 11. Effect of solid volume fractions δ on (a) average Nusselt number at the heated surface and (b) average fluid temperature in the cavity, with ϕ = 60°.

angle. In any case, Figure 7 summarizes that the inclination angle of the triangular cavity with the moving lid can be a control parameter for heat transfer. The effect of the cavity inclination angle on the average fluid temperature in the enclosure is shown in Figure 7b for δ = 0.04. The average fluid temperature Θ increases moderately for the lowest and highest values of ϕ (=0 and 270°). It decreases fairly for other values as ϕ (=45 and 90°) with the increase of Ri. Based on the inclination angle, aiding and opposing flow mechanisms are obtained. Thus, heat and fluid flow inside the cavity can be controlled with the inclination angle. In addition, the values of Θ are always the same with the lowest value of the Richardson number. It changes a little at Ri = 1, which is in the mixed convection regime. Effect of the solid volume fractions δ on streamlines is presented in Figure 8 for ϕ = 60° and the three different values of the Richardson number. The solid volume fraction affects the viscosity of the flow. However, lower values of the volume fraction are chosen in this study. It is assumed that the viscosity value becomes constant with temperature. Visualizations are given from δ = 0 to δ = 0.1. Based on the literature on nanofluids cited in the reference list (refs 21−30), the maximum value of the nanoparticle fraction was taken as 0.1. It can be seen from Figure 8 that mainly two vortices are formed for all values of the Richardson number and solid volume fractions. A cell rotating in the clockwise direction is observed near the left wall because the wall moves from bottom to top. On the other hand, an anticlockwise rotating cell near the right bottom corner is developed because the heated air moves upward from the heated bottom wall and impinges on the cold inclined wall. The size of the clockwise vortex decreases and the anticlockwise vortex increases with the increasing values of the Richardson number. As Ri increases, the influence of sliding

Figure 10. Local heat transfer variation on heated surface for various solid volume fractions δ at (a) Ri = 0.1, (b) Ri = 1, and (c) Ri = 10, with ϕ = 60°.

In all cases, local Nusselt number increases along the X-direction due to the heating surface. When forced convection is dominant, effects of the inclination angle on heat transfer become insignificant. For higher values of Richardson number, namely Ri = 1 and 10, the highest values of the local Nusselt number are formed at ϕ = 270° (for Ri = 1) and the lowest values are formed at ϕ = 45° (for Ri = 10). The average Nusselt number Nu is calculated by using eq 16 along the hot surface, which is a measure of the overall heat transfer rate, plotted in Figure 7 as a function of the Richardson number for the aforementioned values of tilt angle at δ = 0.04. The effect of the inclination angle on the overall heat transfer process is clearly depicted in Figure 7. The average Nusselt number increases sharply for the highest value of ϕ (=270°) and mildly for the lower values of ϕ (=0, 45, and 90°) with increasing Ri. Moreover, the values of Nu are always highest for the highest value of ϕ (=270°). As seen from Figure 7, the inclination angle becomes insignificant for low values of the Richardson number due to the domination of forced convection. Mean Nusselt values are almost the same for Ri = 0.1, 1, and 5 at ϕ = 45°. This means that it is a specific case that the flow due to the moving lid is not affected at this inclination 4098



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wall motion becomes insignificant and natural convection heat transfer becomes dominant. Addition of nanoparticles is not effective on the stream function for lower values of Ri number, but stream function at the right circulation cell decreases with increasing volume fraction for Ri = 10. The effect of the solid volume fractions on isotherms is shown in Figure 9, for ϕ = 60° and Ri = 0.1, 1 and 10. At Ri = 0.1 and the considered values of δ (=0, 0.04, 0.08, and 0.1), conduction is the dominant mode of heat transfer and the contribution due to natural convection is relatively lower, with the result that the isotherms for each value of the solid volume fraction are closer to the pure conduction pattern with a small twist near the center of the cavity. For Ri = 10, wavy variation is observed in the isotherms and then changing the nanoparticle volume fraction becomes effective. The twist becomes more noticeable as the Richardson number is increased for the aforesaid values of δ. Isotherms are cumulated in the corner between the hot and cold walls. Figure 10 illustrates the variation of the local Nusselt number on the heated surface for various solid volume fractions δ and Richardson numbers at ϕ = 60°. Local heat transfer increases with increasing solid volume fraction for all values of the Richardson number. Higher local Nusselt number values are obtained for higher values of the Richardson number. A linear increase is obtained for lower values of the Richardson number due to domination of the forced convection mode of heat transfer. Higher heat transfer enhancement is observed around X = 0.7 and 0.8 due to a higher temperature difference between walls. Parts a and b of Figure 11 show the variation of the average Nusselt number Nu and average fluid temperature Θ with different Ri numbers for the four different values of the solid volume fractions. The average Nusselt number increases very slowly with increasing Richardson number. In addition, the values of Nu are always higher for the higher values of δ, which is expected, and the maximum value of the Nusselt number is obtained at δ = 0.1. The average fluid temperature, Θ, inside the enclosure decreases monotonically with the Richardson number for the considered values of δ. Moreover, the values of Θ are the highest in the forced convection dominated region and lowest in the free convection dominated region for the lowest values of δ. However, at the pure mixed convection, the values of Θ are the same at the aforementioned values of δ. Both the nanoparticle volume fraction and the Richardson number are not effective in the temperature distribution for Ri = 0.1.

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• • • •

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that the inclination angle is a good control parameter for both pure fluid and nanofluid filled enclosures. Flow and thermal fields are influenced by the solid volume fraction, because raising the solid volume fraction leads to reduction of both the action of the fluid movement and the temperature. Nevertheless, it leads to increasing the corresponding average Nusselt number at the heated surface. Moreover, at the highest value of the solid volume fractions the average fluid temperature is least at the forced convection dominated region and most at the free convection dominated region. The finite element method gives good results for curvilinear problems. The flow and thermal fields as well as the heat transfer rate inside the enclosure are strongly dependent on the Richardson number. Both the nanoparticle volume fraction and the inclination angle are more effective for the natural convection dominated regime. Variation of the Richardson number becomes insignificant on heat transfer at lower values of Ri.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge the financial support from the High Impact Research Grant (HIRG) scheme (UM-MoHE) project to carry out this research.

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5. CONCLUSIONS Steady laminar mixed convective flow and heat transfer in an inclined lid-driven triangular enclosure filled with Cu−water nanofluid is formulated and solved using the Galerkin finite element method. The developed code is validated by comparisons with earlier published work, and the results are found to be in good agreement. The flow and heat transport structures are described by the streamlines and isotherms, respectively. Graphical results for the different parametric situations, namely, the tilt angle (ϕ), solid volume fraction (δ), and Richardson number (Ri), are presented and discussed. From this investigation, the following conclusions can be drawn: • The inclination angle affects the fluid flow significantly, but not much so for the heat transfer. Moreover, effects of the inclination angle on the percentage of heat transfer enhancement become trivial. Finally, it has been found

NOMENCLATURE cp = specific heat at constant pressure (J kg−1 K−1) g = gravitational acceleration (m s−2) H = enclosure height (m) k = thermal conductivity (W m−1 K−1) L = length of the cavity (m) Nu = average Nusselt number p = dimensional pressure (kg m−1 s−2) P = dimensionless pressure Pr = Prandtl number Re = Reynolds number Ri = Richardson number T = temperature (K) u = horizontal velocity component (m s−1) U = dimensionless horizontal velocity component v = vertical velocity component (m s−1) V = dimensionless vertical velocity component V0 = lid velocity (m s−1) V̅ = cavity volume (m3) x = horizontal coordinate (m) X = dimensionless horizontal coordinate y = vertical coordinate (m) Y = dimensionless vertical coordinate

Greek Symbols

α = thermal diffusivity (m2 s−1) β = thermal expansion coefficient (K−1) δ = solid volume fraction μ = dynamic viscosity (kg m−1 s−1) ν = kinematic viscosity (m2 s−1)

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Θ = average fluid temperature θ = nondimensional temperature ρ = density (kg m−3) ψ = stream function ϕ = tilt angle, deg γ = penalty parameter

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Subscripts

h = hot c = cold f = fluid nf = nanofluid s = solid nanoparticle

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Laminar Mixed Convection in Inclined Triangular Enclosures Filled

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