Effect of spatial side-wall temperature variation on

International Journal of Numerical Methods for Heat & Fluid Flow Effect of spatial side-wall temperature variation on transient natural convection of a nanofluid in a trapezoidal cavity Ammar I. Alsabery, Ali J Chamkha, Habibis Saleh, Ishak Hashim, B. Chanane,

Article information: To cite this document: Ammar I. Alsabery, Ali J Chamkha, Habibis Saleh, Ishak Hashim, B. Chanane, (2017) "Effect of spatial side-wall temperature variation on transient natural convection of a nanofluid in a trapezoidal cavity", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 27 Issue: 6, doi: 10.1108/HFF-11-2015-0488 Permanent link to this document: http://dx.doi.org/10.1108/HFF-11-2015-0488

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Effects of spatial side-wall temperature variations on transient natural convection of a nanofluid in a trapezoidal cavity A. I. Alsaberya∗, A. J. Chamkhab,c , H. Salehd , I. Hashima and B. Chananee a

School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia b

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Department of Mechanical Engineering, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia

c

Prince Sultan Endowment for Energy and the Environment, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al-Khobar 31952, Saudi Arabia d

Numerical Computing Group, Department of Mathematics, University of Riau, Pekanbaru 28293, Indonesia e

Mathematics and Statistics Department, King Fahd University of Petroleum & Minerals, Dhahran-31261, Saudi Arabia April 30, 2017

Abstract The problem of transient natural convection heat transfer in a trapezoidal cavity with spatial side-wall temperature variations is studied numerically using finite difference method. The left sloping wall is implicit to assume side-wall temperature T h , which is greater than the right sloping wall temperature Tc . The horizontal walls are adiabatic. Water-based nanofluids with Ag or Cu or Al2 O3 nanoparticles are chosen for investigation. The governing parameters of this study are the Rayleigh number (103 ≤ Ra ≤ 107 ), nanoparticle volume fraction (0 ≤ ϕ ≤ 0.2), the side wall inclination angle (0◦ ≤ φ ≤ 21.8◦ ) and the time parameter (0 ≤ τ ≤ 0.25). Explanations for the influence of the various parameters on streamlines, isotherms and average Nusselt number are provided with regards to thermal conductivities of nanoparticle suspended pure fluid. It is shown that the heat transfer rate increases in non-uniform heating increments, whereby low wave number values are more affected by the convection. The best heat transfer enhancement results from larger side wall ∗

Corresponding author: ammar e [email protected]

1

inclination angle, however trapezoidal cavities require longer time compared to that of square to reach a steady state.

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Keywords: Transient natural convection, Trapezoidal cavity, Nanofluid, Finite difference method, Spatial Temperature Nomenclature Cp specific heat capacity g gravitational accleration h heat transfer coefficient h average heat transfer coefficient k wave number L width and height of enclosure Nu average Nusselt number Pr Prandtl number Ra Rayleigh number Raeff effective Rayleigh number T temperature u, v velocity components in the x-direction and y-direction U, V dimensionless velocity components in the X-direction and Y -direction x, y & X, Y space coordinates & dimensionless space coordinates Greek symbols α β ε θ µ ν ρ ϕ φ λ ψ&Ψ ω&Ω

thermal diffusivity thermal expansion coefficient nondimensional amplitude dimensionless temperature dynamic viscosity kinematic viscosity density solid volume fraction side wall inclination angle thermal conductivity stream function & dimensionless stream function vorticity & dimensionless vorticity

subscript c bf h nf sp

cold base fluid hot nanofluid solid nanoparticle

1

Introduction

Laminar natural convection heat transfer is a significant phenomenon in engineering systems due to its wide applications in operations of solar collectors, heat exchangers, storage tanks, double pane windows, etc. Ostrach [1] presented several of these applications. Laminar flow has become a favorite aspect in heat storage applications, 2

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as there has been a rise in research activities regarding natural convection heat transfer [2–4]. Most of the published papers focused on natural convection heat transfer in square/rectangular cavities. In reality, natural convection in a differentially heated cavity is a prototype for numerous industrial applications and specifically, trapezoidal cavity has received considerable attention because of its applicability in various fields. The moderately concentrating solar energy collector is an important example involving a trapezoidal geometry. Studying natural convection heat transfer in a trapezoidal geometry is difficult than that of square/rectangular cavities due to the presence of sloping walls. In general, the mesh nodes do not lie along the sloping walls, and consequently, from a programming and computational point of view, the efforts required for setting flow characteristics rise significantly. Lee [5] was the first to perform numerical and experimental flow visualization and heat transfer studies in a differentially heated trapezoidal cavity filled with pure fluid. Karyakin [6] simulated the flow and temperature fields for various sloping wall inclination angles and showed that the heat transfer rate increases with an increasing sloping wall angle. Many researchers have considered nanofluids as working mediums that enhance thermal conductivity with the presence of nanoparticles, thus making nanofluids decent candidates for heat removal devices in workable, fluid-based, thermal applications. In view of the possibility of several different thermal problems studied in the above two media, we limit our survey of literature to natural convection in cavities. A comprehensive work on natural convection in partially occupying nanofluids in cavities was done by Khanafer et al. [7]. A study by Putra et al. [8] investigated the natural convection inside a horizontal cylinder filled with nanofluids. The work of Jou and Tzeng [9] considered the natural convective heat transfer in nanofluids occupying a rectangular cavity. Tiwari and Das [10] picked the natural convection heat transfer in a lid-driven flow for a differentially heated square cavity filled with nanofluid. Santra et al. [11] investigated the enhancement of heat transfer in Ostwald-de Waele nanofluid contained in a differentially heated cavity. Abouali and Falahatpisheh [12] studied numerically the natural convection in a vertical annuli filled with Al2 O3 water nanofluid. Jahanshahi et al. [13] investigated experimentally the natural convection heat transfer in a cavity filled with a water-silicon dioxide nanofluid. Heris et al. [14] conducted numerically the study of forced convection heat transfer in triangular ducts filled with Al2 O3 water nanofluid. Alloui et al. [15] picked the water-Al2 O3 nanofluid for the investigation of a shallow rectangular cavity using finite volume method with water-copper oxide nanofluid medium. Saleh et al. [16] studied numerically the natural convection heat transfer in a trapezoidal cavity filled with water-Cu and water-Al2 O3 nanofluids. Noghrehabadia et al. [17] made a compaction of the thermal enhancement of Ag-water and SiO2 -water nanofluids over an isothermal stretching sheet with suction or injection. Nasrin and Parvin [18] used the finite element method to study the buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water-Cu nanofluid. Using the Runge-Kutta-Fehlberg method, Noghrehabadi et al. [19] conducted a numerical study of the flow and heat transfer of nanofluids over stretching sheet with partial slip boundary conditions. Heris et al. [20] made a comparative experimental study on natural convection heat transfer in turbine oil-based nanofluid in an inclined cavity using different metal oxide nanoparticles. Recently, Moradi et al. [21] investigated experimentally the natural convection heat transfer in cylindrical cavities filled with Newtonian nanofluids. Behseresht et al. [22] investigated numerically the natural convection flow and heat transfer from a vertical cone in a saturated porous medium filled with nanofluids by using the fourth-order Runge–Kutta method.

3

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The problem of natural convection in closed cavities with boundary walls, including non-uniform temperatures, has been considered in a few studies. Saeid and Mohamad [23] studied numerically the natural convection in a porous cavity with spatial sidewall temperature variations. Saeid and Yaacob [24] considered numerically the natural convection in a square cavity filled with pure fluid in addition to a non-uniform hotwall temperature and a uniform cold-wall temperature. Sathiyamoorthy and Chamkha [22–24] considered natural convection in a square cavity for linearly heated side walls. Selamat et al. [28] investigated the effect of inclination angle on natural convection in a porous cavity with spatial sidewall temperature variations. However, the study of natural convection heat transfer in a trapezoidal cavity filled with nanofluid and heated by spatial sidewall temperature has not yet been undertaken. The aim of this study is to investigate the effects of spatial side-wall temperature variations on transient natural convection heat transfer in a trapezoidal cavity. The study on the effects of non-uniform temperature variations on trapezoidal domain has many important engineering applications, for example, the effect of the sinusoidal temperature variation of the solar energy collector when a periodic array of heater is placed on an inclined wall of the trapezoidal domain.

Mathematical Formulation Consider two-dimensional unsteady natural convection in a trapezoidal cavity with length L, as illustrated in Figure 1. The left sloping wall of the cavity is heated non-uniformly and the right sloping wall is maintained at a constant cold temperature Tc , while the horizontal walls are kept adiabatic. All the walls are assumed to be impermeable, the fluid within the cavity is a water–based nanofluids having Ag, Cu or Al2 O3 nanoparticles. The Boussinesq approximation is adopted and the nanofluid physical properties are constant except for the density. By considering these assumptions, the continuity, momentum and energy equations for the unsteady state natural convection can be written as follows: ∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u ∂u 1 ∂p +u +v =− + νnf ∂t ∂x ∂y ρnf ∂x ∂v ∂v 1 ∂p ∂v +u +v =− + νnf ∂t ∂x ∂y ρnf ∂y ∂T ∂T ∂T +u +v = αnf ∂t ∂x ∂y

(

( (

)

∂2u ∂2u + 2 , ∂x2 ∂y

∂2v ∂2v + ∂x2 ∂y 2

∂2T ∂2T + 2 ∂x ∂y 2

)

(2) (

)

+ βnf g T h − Tc ,

(3)

)

,

(4)

where x and y are the Cartesian coordinates measured in the horizontal and vertical directions respectively, u and v are the velocity components in the x-direction and ydirection. t denotes the time, p denotes the fluid pressure and g is the acceleration due to gravity. αnf is the effective thermal diffusivity of the nanofluids, ρnf is the effective density of the nanofluids and µnf is the effective dynamic viscosity of the nanofluids, which are defined as

αnf =

λnf , ρnf = (1 − ϕ)ρbf + ϕρsp , (ρCp)nf 4

θ = 0.5+ε[1-cos(2πkY)]

Ψ = ¶θ / ¶Y = 0

Nanoparticle

Ψ=

0

Ψ = 0; θ = -0.5

Fluid

g

L

j Ψ = ¶θ / ¶Y = 0

L

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Figure 1: Physical model of convection in a trapezoidal cavity together with the coordinate system. µnf 1 = , µbf (1 − ϕ)2.5

(5)

where, ϕ is the solid volume fraction of nanoparticles, the heat capacitance of the nanofluids given is (ρCp )nf = (1 − ϕ)(ρCp )bf + ϕ(ρCp )sp . (6) The thermal expansion coefficient of the nanofluids can be determined by: βnf = (1 − ϕ)(β)bf + ϕβsp ,

(7)

(ρβ)nf = (1 − ϕ)(ρβ)bf + ϕ(ρβ)sp .

(8)

The thermal conductivity based on Maxwell-Garnett’s (MG) model is : λsp + 2λbf − 2ϕ(λbf − λsp ) λnf = . λbf λsp + 2λbf + ϕ(λbf − λsp )

(9)

We assume that the temperature of the hot slopping wall has a sinusoidal variation about a mean value of T h in the form [24] (

Th (y) = T h + ε T h − Tc

)[

(

1 − cos

2πky L

)]

.

(10)

The stream function ψ and the vorticity ω are defined in the usual way as: ∂ψ ∂ψ , v=− , ∂y ∂x ∂v ∂u ω= − . ∂x ∂y

u=

(11) (12)

Now we introduce the following non-dimensional variables: x y ωL2 ψ , Y = , Ω= , Ψ= , L L αbf αbf αbf t T − Tc θ= , τ= 2 . L T h − Tc X=

5

(13)

This then yields the dimensionless governing equations 



µnf  Prbf ∂Ψ ∂Ω ∂Ω ∂Ψ ∂Ω  + − = sp ∂τ ∂Y ∂X ∂X ∂Y µbf (1 − ϕ) + ϕ ρρbf (

∂2Ω ∂2Ω + × ∂X 2 ∂Y 2

)

βnf + Rabf Prbf βbf

(

∂θ ∂X

)

(ρCp )bf (λ)nf ∂θ ∂Ψ ∂θ ∂Ψ ∂θ + − = ∂τ ∂Y ∂X ∂X ∂Y (ρCp )nf (λ)bf

, (

(14) ∂2θ ∂2θ + ∂X 2 ∂Y 2

)

,

(15)

where Rabf = gρbf βbf (T h − Tc )L3 /(µbf αbf ) is the Rayleigh number for the base fluid and Prbf = νbf /αbf is the Prandtl number for the base fluid. The dimensionless initial and boundary conditions of Eqs. (14) and (15) are: Ψ = 0,

θ = 0,

on the whole cavity at τ = 0,

(16)

Ψ(0, Y ) = AD = 0, θ(0, Y ) = 0.5 + ε [1 − cos(2πkY )] , Downloaded by Cornell University Library At 23:05 09 May 2017 (PT)

Ψ(1, Y ) = BC = 0, θ(1, Y ) = −0.5, Ψ(X, 0) = AB = Ψ(X, 1) = DC = 0,

∂θ(X, 0) ∂θ(X, 1) = = 0. ∂Y ∂Y

(17)

The local Nusselt number along the hot and cold slopping walls are defined respectively by (

)

hL ∂θ N uh = =− , λ ∂X X=0 ( ) ∂θ hL =− . N uc = λ ∂X X=1

(18) (19)

Finally, the average Nusselt number can be defined based on the average heat transfer coefficient as: ∫

N unf =

D

A

[

(

λnf − λbf

)

]

∂θ dY. ∂X

(20)

Numerical Method and Validation The governing equations (14)–(15) subject to the initial (16) and boundary conditions (17) are solved numerically by the Finite Difference (FD) scheme consisting of the Alternating Direction Implicit (ADI) method and the Tri-Diagonal Matrix Algorithm (TDMA). Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to ensure that the results are grid independent. A grid test was performed using sets of grids: 30 × 30, 50 × 50, 70 × 70, 90 × 90, 100 × 100, 120 × 120. Table 1 shows the calculated Nusselt number at different mesh numbers for water–Cu, Rabf = 105 , ϕ = 0.1, k = 1.2, ε = 0.5 and φ = 16.7◦ . The green results showed insignificant differences for the 100 × 100 grids and above. Therefore, for all computations in this paper for similar problems to this subsection, the 100 × 100 uniform grid was employed. The time step is selected to be uniform ∆τ = 10−4 and the steady state is achieved by the criterion: n+1 n θ i,j − θi,j max (21) ≤ϵ n θi,j where n represents the current time and ϵ is the convergence criterion to steady state. 6

Table 1: Grid testing for Ψmin and Nusselt number at different grid sizes for Rabf = 105 , ϕ = 0.1, k = 1.2, ε = 0.5 and φ = 0◦ .

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Grid size 30 × 30 50 × 50 70 × 70 90 × 90 100 × 100 120 × 120

Ψmin -15.94704 -15.63208 -15.44445 -15.31779 -15.23748 -15.22696

N unf 11.0004 11.30769 11.38103 11.38699 11.39647 11.39778

For validation, the results are compared against previous numerical results by calculating the average Nusselt number with various values of the Rayleigh number at constant left and right wall temperatures for a pure fluid (Pr = 0.71), as shown in Table 2. In addition, a comparison is made between the resulting figure and the one provided by Saeid and Yaacob [24] for a pure fluid (Pr = 0.71) in a square cavity as shown in Figure 2 for Rabf = 106 , k = 0.7, ε = 0.5 and ϕ = 0. These results provide confidence in the accuracy of the present numerical method. Table 2: Comparison of N u for different values of Rabf when ϕ = 0 and Pr = 0.71 with previous numerical results for steady case.

de Vahl Davis [29] Markatos and Pericleous [30] Barakos et al. [31] Chang and Tsay [32] D’Orazio et al. [33] Saeid and Yaacob [24] Present result

Rabf = 103 1.118 1.108 1.114 1.118 1.116 1.118 1.118

Rabf = 104 2.243 2.201 2.245 2.243 2.254 2.252 2.251

Rabf = 105 4.519 4.429 4.510 4.514 4.506 4.544 4.521

Rabf = 106 8.799 8.754 8.806 8.805 8.879 8.837 8.843

Results and Discussion In this section, we present numerical results for the streamlines and isotherms for various values of the Rayleigh number (103 ≤ Rabf ≤ 107 ), the nanoparticle volume fraction (0 ≤ ϕ ≤ 0.2), the side wall inclination angle (0◦ ≤ φ ≤ 21.8◦ ), the time dependent parameter (0 ≤ τ ≤ 0.25), and the Prandtl number Pr = 6.2. The values of the average Nusselt number are calculated for various values of Rabf , ϕ, k and τ . The thermophysical properties of the base fluid (water) and solid Ag, Cu and Al2 O3 phases are tabulated in Table 3 where the value of the Prandtl number is considered for water (Prbf = 6.2). Before we proceed to the discussion of the results, it is worthwhile knowing about the value of the effective Rayleigh number which can be defined as [34] Raeff

λbf Cpnf µbf = Rabf λnf Cpbf µnf

7

(

ρnf ρbf

)2

βnf . βbf

(22)

−4.13 −1−.38 2. −8. 75 27

−11.02

−12.4

−12.4 −13.78

−4.13 .89 −8.27 −6−9 .64 .4 −12 .78 3 −1

−16.54 −15.16

−11.02

.89.27 −6−8

−5.51 −2.75 −1 .38

−15.16

−13.78

−4.13

−12.4

−11.02 −9.64

−8.27 −6.89 −5.51

−4.13

−1.38

−9.64 .89 3 −6 4.1 −

1.01

1

0.81

0.61

0.61

0.41

0.41

0.41

0.21

0.21

0.41 0.21 0.01 −0.39−0.19

1.21 0.61

0.81

0.61

0.81

0.61

0.21

0.21

0.81

1.01 0.41

−13.78.4 2 −1

0. 81

1.01

1.21

1.01

.02

.21 41 00..01 0−0 .19

−1.38

0.8

−11

−5.51

−2.75

−2.75

7

−1.38

−15.16

−8 .2

−5.51

8 .3 −2 3 .75 .1 .51 −4 −5

16 5. −1

−11. 02

−15.16

−13.78 .4 −11.02 −12 −9.64

−2.75

−1

−1.38 −2.75 −4.13 −5.51 −6.89 −8.27 −9.64

−6.89 .64

−9

0.01

0.0 1 −0.19 −0.39

0.61 0.41 0.21 0.01

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0.01

0.01

−0.19 −0.19

−0

.19

−0.19

Figure 2: Streamlines (a), Saeid and Yaacob [24] (left), present study (right), isotherms (b), Saeid and Yaacob [24] (left), present study (right) for Rabf = 106 , k = 0.7, ε = 0.5, ϕ = 0 and Pr = 0.71. Table 3: Thermo-physical properties of water with Ag, Cu and Al2 O3 . Physical Properties Cp (J/kgK) ρ (kg/m3 ) λ (Wm−1 K−1 ) β × 10−5 (1/K)

Water 4179 997.1 0.6 21

Ag 235 10500 429 5.4

Cu 383 8954 400 1.67

Al2 O3 765 3600 46 0.63

It should be noted here that since the modified thermo-physical properties for the nanofluid are taken to be spatially and temporally invariant and only dependent on the volume fraction of the nanoparticles, Eq. 22 indicates that the effective Rayleigh number is modified by the presence of the nanoparticles volume fraction and so are the other thermo-physical properties like the Prandtl number. This effectively means that this analysis solves for single phase problems with different thermo-physical properties and different values of the effective Rayleigh and Prandtl numbers in comparison to the base fluid. This analysis is similar to many papers in the literature on nanofluids which effectively solved the problem for a Newtonian fluid with effective thermo-physical properties which are different to the base fluid. Table 4 presents the effect of the effective Rayleigh number of various nanoparticles on the convection pattern and average Nusselt number for the case ϕ = 0.1, k = 1.2, ε = 0.5 and φ = 16.7◦ . We clearly observe a strong enhancement of the effective Rayleigh number by the addition of the nanoparticles, i.e. the Rayleigh number increases from 10000 to

8

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Table 4: The strength of the flow circulation (|Ψmin |) and N u for different values of the effective Rayleigh number of various nanoparticles for the case ϕ = 1, k = 1.2, ε = 0.5 and φ = 16.7◦ Raeff -Ag |Ψmin | N u Raeff -Cu |Ψmin | N u Raeff -Al2 O3 |Ψmin | N u 18451 6.27 7.01 15399 5.56 6.72 7681.5 5.35 6.54 92257 13.13 10.42 76996 12.12 10.02 38408 11.91 9.77 184510 16.02 12.35 153990 15.04 11.87 76815 14.94 11.57 922570 23.93 18.58 769960 22.47 17.84 384080 22.37 17.38 1845100 28.57 22.18 1539900 26.86 21.29 768150 26.80 20.75 9225700 42.64 33.57 7699600 40.31 32.21 3840800 40.68 31.37 18451000 50.40 40.17 15399000 47.70 38.53 7681500 48.40 37.51 18451 for Ag and to 15399 for Cu and to 7681.5 for Al2 O3 . This is due to the high thermal conductivity of nanoparticles which enhances the values of the effective Rayleigh number for all nanoparticles. This increment in the effective Rayleigh number strongly increases the strength of the flow circulation (see |Ψmin | values), and as a result, the convection heat transfer is enhanced due to the increase in the buoyancy forces. We observe from the values of the average Nusselt number that the heat transfer rate is greatly increased by the addition of the nanoparticles to the base fluid. Adding nanoparticles to the base fluid tends to increase the effective Rayleigh number due to the high thermal conductivity of nanoparticles which obviously tends to increase the heat transfer rate. Figure 3 shows the effects of various Rayleigh numbers on the streamlines (left) and isotherms (right) for water–Cu nanoparticles for the case k = 1.2, ε = 0.5 and φ = 16.7◦ . Figure 3(a) demonstrates the effect of changing the flow motion with the boundary conditions set at a lower value of the Rayleigh number (105 ). Heating the left wall of the cavity by non-uniform temperature causes the flow within the cavity to rise, the flow begins to move from the hot sloping wall aligned with the horizontal wall (top), descending along the right sloping wall, aligned with the horizontal wall (bottom), ascending again at the hot wall, creating a clockwise rotating cell in the cavity centre. At low Rabf value, the thermal conductivity of pure fluid is higher than that of the nanofluid; as a result, the flow circulation of the streamlines cells for pure fluid is bigger than that of the nanofluid. When the streamlines circulated as vortices in the clockwise direction (negative signs of Ψ), the strength of the flow circulation is denoted as Ψmin . Due to the non-uniform heating, the isotherm patterns next to the left wall appear in a curved line; near the right wall, the isotherm patterns occur vertically, while in the middle of the cavity, the isotherm patterns take almost a horizontal line. Applying the nanoparticle volume fraction (ϕ = 0.05) tends to increase the strength of the flow circulation (see Ψmin values), due to the increase in the effective Rayleigh number. As Rabf increases (106 ), next to the hot sloping wall, the intensity of the streamlines increases; the streamline singular cell breaks and becomes two different cells (the larger one is near the hot wall and the smaller one is close to the right wall). Clearly, by increasing the value of Rabf , the strength of the flow circulation tends to increase (see Ψmin values). This happens due to the increment in the buoyancy forces which tends to increase the velocity of the nanofluid. The isotherm pattern intensity increases near the sloping walls, while in the central cavity, the isotherm patterns appear as horizontal lines, as shown in Figure 3(b). For a higher Rabf value, the nanofluid flow circulation of the streamline cells is bigger than that of a pure fluid; the two streamline circulation cells are linked and become one horizontal cell along the cavity. Due to the large intensity circulation near the central regime, the thermal energy at the central regime is strong and that reflects maximal

9

mixing of the convection at higher Rayleigh numbers, as presented in Figure 3(c). (Ψmin)bf =-13.94, (Ψmin)nf =-14.93 3 −0.1 00.102.37

2

0.6

2

0.6

0.3

37

1.12 0. 6 2

8 −0.3 −0.13 0 0.12

7

0.

2

0.1 0 0.3 .87 7

0.12

0

0 .6 2

0

−0.13

−0

.3

8

0.12 0

−0.13

(a) Rabf = 105 (Ψmin)bf =-25.97, (Ψmin)nf =-27.68

0.6

2 3

−0.1 0.12 0

0 −0 .37 .13 0.1 2 0

−0.13

0.12

00.12

0.3

7

0.6 1.11 20 0.8 .37 6

0.37

0 −0.13

−0.13

.38

−0

(b) Rabf = 106 (Ψmin)bf =-48.47, (Ψmin)nf =-51.09 0.36

0.61

0.61

0.86 0.3 6

0.12

0.36

0.6 1

3

0.36

−0.1

0.1

2

0.12

−0.13

−0.13

−0.13

0.3

6

0.12

−0 0. .1312

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00.12

0.62

−0.38

(c) Rabf = 107

Figure 3: Streamlines (left) and isotherms (right) evolution by Rayleigh number of (a) Raeff = 153990, (b) Raeff = 1539900 and (c) Raeff = 15399000 for k = 1.2, ε = 0.5, φ = 16.7◦ , ϕ = 0 (solid lines) and ϕ = 0.05 (dashed lines). Figure 4 depicts the effect of wave number on the streamlines (left) and isotherms (right) for water–Cu nanoparticles for the case Rabf = 106 (Raeff = 1539900), ε = 0.5 and φ = 16.7◦ . The effect of uniform heating (k = 0) on the flow motion with the boundary condition sets is demonstrated in Figure 4(a). The streamlines cell occupies the whole cavity. An increase in the thermal conductivity of the nanofluid by the addition 10

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of nanoparticles tends to increase the effective Rayleigh number as discussed earlier, and in turn, increases the strength of the flow circulation (see Ψmin values). The uniform heating enhances the convection heat transfer; as such, the isotherm patterns close to the sloping walls occur with vertical lines, while in the middle of the cavity, the isotherm patterns appear horizontally. As non-uniform heating is applied (k = 1), the streamlines are affected; the streamline cell tends to break into two cells for a pure fluid, while the nanofluid streamline cell shrinks from the middle. As a result, the non-uniform condition shows more effect on the pure fluid compared to the case of the nanofluid. This is due to the fact that the thermal conductivity of a pure fluid is lower than that of a nanofluid which is enhanced more by non-uniform heating. The increase in strength of the flow circulation could be observed with increasing values of k (see Ψmin values). The isotherm patterns are influenced by the increase of k value; close to the left sloping wall, the isotherm patterns arise with irregular lines, as illustrated in Figure 4(b). Increasing k to a high value of 4 significantly changes the flow motion. Next to the boundary (sloping and horizontal walls), the streamlines appear with almost a tortuous line. The intensity of the streamlines decreases compared to the previous heating case, where the two streamline cells in the cavity center reconnect and become one cell for both pure fluid and nanofluid cases. From this result, we observe that, stronger non-uniform heating shows an inhibition in the strength of the flow circulation. The isotherm patterns near the hot wall are transferred from an irregular-shape to a spot-shape due to the large variation of the boundary temperature, as displayed in Figure 4(c). Figure 5 illustrates the effects of the side wall inclination angle on the streamlines (left) and isotherms (right) for water–Cu nanoparticles, Rabf = 106 (Raeff = 1539900), k = 1.2 and ε = 0.5. The effect of changing the flow motion with the boundary condition sets, when the side wall inclination angle is φ = 0◦ , i.e. square cavity, is depicted in Figure 5(a). By applying the nanoparticle volume fraction (ϕ = 0.05), the strength of the flow circulation increases (see Ψmin values). Due to the higher properties of the nanofluid such as the effective Rayleigh number which clearly enhances the strength of the flow circulation, a singular eddy is observed within the center of the square cavity. The nonuniform heating forces the isotherm patterns to have high intensity with curved lines next to the hot wall, whereas near to the cold wall, the isotherm patterns appear with vertical lines. While, the isotherm patterns take almost a horizontal lines in the middle of the cavity. Increasing the side wall inclination angle (φ = 11.3◦ ) tends to increase the flow intensity; the singular core cell breaks into two different cells resulting in an increase in the strength of the flow circulation (see Ψmin values). Due to the presence of the slopping angle which tends to decrease the space for fluid circulation, thus the intensity of the flow increases. Due to the cavity type, the density of the isotherms at the hot and cold boundaries increases by the application of φ, as presented in Figure 5(b). Increasing φ value to 21.8◦ leads to maximum strength of the flow circulation for both pure fluid and nanofluid cases (see Ψmin values). The streamline circulation cells tend to rejoin and become one semi-triangular cell, as shown in Figure 5(c). The isotherms are observed to be more distorted by the increment of the sloping angle. Figure 6 shows the evolution of the time dependency on the streamlines (left) and isotherms (right) for water–Cu nanoparticles, Rabf = 106 (Raeff = 1539900), k = 1.2, ε = 0.5 and φ = 16.7◦ . At the starting of the heating time (τ = 0.001), the streamline circulation cell appears near the hot wall and the flow circulation of the streamline cells for nanofluids is bigger than that for pure fluids. As the nanoparticle volume fraction is applied (ϕ = 0.05), the strength of the flow circulation decreases (see Ψmin values) due to the low effectiveness of nanofluid in a short time period. The distribution of the temperature in the trapezoidal cavity is shown in the isotherms (Figure 6(a)). By 11

(Ψmin)bf =-19.82, (Ψmin)nf =-21.39 .291 003.0 −−0. .0 15 00.26

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Figure 4: Streamlines (left) and isotherms (right) evolution by wave number for Rabf = 106 (Raeff = 1539900) ε = 0.5, φ = 16.7◦ , ϕ = 0 (solid lines) and ϕ = 0.05 (dashed lines). increasing the time, τ = 0.01, we observe different behaviour for the pure fluid and the nanofluid. For the pure fluid, two core cells appear as having one next to the hot wall and one at the cavity center, whereas only one core cell occurs for the nanofluid. The strength of the flow circulation decreases for the pure fluid in comparison with the nanofluid, where the strength of the flow circulation is increased (see Ψmin values). Due to the presence of the nanoparticles, an increment in the dimensionless time tends to enhance the circulation of the nanofluid, while the circulation of the pure fluid is predicted to be inhibited with the increasing of the time. The distortion of the isotherm 12

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Figure 5: Streamlines (left) and isotherms (right) evolution by the side wall inclination angle for Rabf = 106 (Raeff = 1539900), k = 1.2, ε = 0.5, ϕ = 0 (solid lines) and ϕ = 0.05 (dashed lines). patterns is aggravated by the increase of time; distortion takes a vertical line close to the cold wall, while it appears with non-uniform semi-curved lines next to the hot wall, as depicted in Figure 6(b). As time progresses (τ = 0.22), the weakness of the circulation can be observed until the system reaches steady state. The two core cells can be realized for both pure fluid and nanofluid; achieving steady state tends to decrease the strength of the flow circulation (see Ψmin values), as illustrated in Figure 6(c).

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Figure 6: Streamlines (left) and isotherms (right) evolution by time dependent for Rabf = 106 (Raeff = 1539900), k = 1.2, ε = 0.5, φ = 16.7◦ , ϕ = 0 (solid lines) and ϕ = 0.05 (dashed lines). Figure 7(a) summarizes the variations in the average Nusselt number with the Rayleigh number for different wave numbers of water–Cu at ϕ = 0.05, ε = 0.5 and φ = 16.7◦ . The heat transfer rate is increased by increasing the Rayleigh number in which at high Rayleigh number values (Ra ≥ 105 ), a significant enhancement in the convection heat transfer can be observed. This is due to the gravitational forces increment compared to the viscous forces. The maximum wave number (k = 5) leads to stronger heat transfer with the maximum value of the average Nusselt number, due to the high non-uniform 14

heating which generates greater energy. However, we clearly observe a different behaviour in the heat transfer rate at a sufficiently large Rayleigh number. The wave number 0.7 produces higher heat transfer rate compared to a wave number of 1.2. This happens due to a competition between the buoyancy force and thermal generation from the nonuniform heating. Whereas, the highest wave number distinctly generates a major amount of thermal energy. Figure 7(b) shows the effect of various side wall inclination angles on the average Nusselt number with the Rayleigh number for water–Cu at k = 1.2 at ϕ = 0.05 and ε = 0.5. Obviously, the enhancement of heat transfer rate occurs by increasing the Rayleigh number, which tends to increase the average Nusselt number. A significant heat transfer enhancement appears at high Rayleigh number value (Ra ≥ 105 ). Stronger enhancement in convection heat transfer appears with a higher φ value (φ = 21.8◦ ) which leads to the maximum value of the average Nusselt number. Due to the presence of the sloping walls, the area that is exposed to the heat transfer is decreased which tends to decrease the heat dissipation, and as a result, the temperature gradient is increased. 45

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Figure 7: Variation of N u with Ra for different (a) k, (b) φ at ϕ = 0.05 and ε = 0.5. The variations of the average Nusselt number with the nanoparticle volume fraction for different nanoparticles at Rabf = 106 (Raeff = 1539900), ε = 0.5, k = 1.2 and φ = 16.7◦ are presented in Figure 8(a). Clearly, Ag helps the fluid to transport more heat and has the maximum value of the average Nusselt number due to higher thermal conductivity of Ag. The weaker enhancement in the heat transfer rate is obtained for Al2 O3 nanoparticles as the nanoparticle volume fraction increases. Further, we observe that Cu nanoparticles can transport more heat compared to that of Al2 O3 nanoparticles. Figure 8(b) depicts the effect of various non-dimensional amplitudes on the average Nusselt number with nanoparticle volume fraction of water–Cu at Ra = 106 , k = 1.2 and φ = 16.7◦ . Due to the increase in convection by the addition of nanoparticles which leads to higher effective Rayleigh numbers, the average Nusselt number increases. A significant heat transfer rate enhancement is obtained with a higher nondimensional amplitude ε = 1 which has the maximum value of the average Nusselt number. This is due to the higher energy provided by higher amplitude that clearly enhances the temperature gradient. Figure 9(a) displays the variations in the average Nusselt number together with the wave number for different nondimensional amplitudes for water–Cu at Rabf = 106 , ϕ = 0.05 and φ = 16.7◦ . Increasing the wave number enhances convection heat transfer; significant enhancement is obtained using low k values (k < 2). This indicates that in this region the heat transfer rate is less affected by heat generation from the high 15

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Figure 8: Variation of N u with ϕ for different (a) sp, (b) ε at Rabf = 106 (Raeff = 1539900), k = 1.2 and φ = 16.7◦ . non-uniform heating. Larger values of the average Nusselt number appear together with higher values of the nondimensional amplitude owing to the rise of the wave number. A higher nondimensional amplitude leads to a higher energy which strongly increases the convection heat transfer. Figure 9(b) presents the effect of the nanoparticle volume fractions on the average Nusselt number with wave number for water–Cu at Ra = 106 (Raeff = 1539900), ε = 0.5 and φ = 16.7◦ . The convection heat transfer increases by increasing the values of the wave number, as such the low non-uniform heating (k ≤ 2.5) leads to the best heat transfer enhancement. Due to higher thermal conductivity of the nanoparticles, the maximum value of the average Nusselt number can be obtained using higher concentrations of nanoparticle volume fraction (ϕ = 0.2). 35 50 45

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the average Nusselt number can be obtained by the higher wave number (k = 5). Figure 10(b) shows the effect of side wall inclination angles on the unsteady average Nusselt number with time for water–Cu at Rabf = 106 (Raeff = 1539900), k = 1.2, ϕ = 0.05, ε = 0.5 and ε = 0.5. The values of the average Nusselt number clearly show that the heat transfer rate enhances as time progresses. Generally, the convection heat transfer alteration decreases by increasing the time; the best enhancement appears for higher side wall inclination angle (φ = 21.8◦ ). From the average Nusselt number values, we observe that the higher side wall inclination angle requires a longer time compared to that of a square (φ = 0) to reach steady state. 80 60

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Conclusions In this study, the finite difference method was applied to solve the dimensionless form of the governing equations for transient natural convection of a nanofluid in a trapezoidal cavity with spatial side-wall temperature variations. Detailed computational results within the cavity for flow, temperature field in laminar flow, and the average Nusselt number have been shown graphically. The important conclusions of the study are: 1. Since the modified thermo-physical properties for the nanofluid are taken to be spatially and temporally invariant and only dependent on the volume fraction of the nanoparticles, increasing the nanoparticles volume faction clearly enhances the value of the effective Rayleigh number. This enhancement increases the convection heat transfer and thereby the strength of the flow circulation and the values of average Nusselt number increases. 2. Due to the effect of nonuniform heating, the increase in the strength of the flow circulation is observed with increasing values of the wave number, but at a high wave number, the strength of the flow circulation decreases. 3. Qualitatively, the enhanced-heat transfer situation is seen in all the three nanofluids employed in this study compared to that of the base fluid but the following general result holds: N uwater−Ag > N uwater−Cu > N uwater−Al2 O3 .

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4. At the unsteady state, we observe that a low wave number leads to larger average Nusselt number, whereas after reaching steady state, the maximum value of the average Nusselt number can be obtained by a higher wave number.

Acknowledgments The authors are grateful to the anonymous referees for the comments which improved the quality of the paper. The work was supported by the Ministry of Higher Education, Malaysia grant GSP/1/2015/SG04/UKM/01/1.

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