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Effects of the form factor on natural convection heat transfer and fluid flow in a two-dimensional cavity filled with Al
_{2}O
_{3}-nanofluid has been analyzed numerically. A model was developed to explain the behavior of nanofluids taking account of the volume fraction
*φ*. The Navier-Stokes equations are solved numerically by alternating an implicit method (Method ADI) for various Rayleigh numbers varies as 10
^{3}, 10
^{4} and 10
^{5}. The nanofluid used is aluminum oxide with water
*Pr* = 6.2; solid volume fraction
*φ* is varied as 0%, 5% and 10%. Inclination angle Φ varies from 0
° to 90
° with a step the 15
° and the form report varies as
*R *= 0.25, 0.5, 1 and 4. The problem considered is a two-dimensional heat transfer enclosure. The vertical walls are differentially heated; the right is cold when the left is hot. The horizontal walls are assumed to be insulated. The nanofluid in the cavity is considered as incompressible, Newtonian and laminar flow. The nanoparticles are assumed to have a shape and a uniform size. However, it is supposed that the two fluid phases and nanoparticles are in a state of thermal equilibrium and they sink at the same speed. The thermophysical properties of nanofluids are assumed to be constant at the exception of the variation of density in the force of buoyancy, which is based on the approximation of Boussinesq values.

The improvement of the heat transfer by natural convection is the main object of several studies; many researchers have conducted a series of tests numerical and experimental of the description of the phenomena of convection: the manager of fact, the nature of the systems (in particular the geometry) and the properties of fluids involved (physico-chemical properties).

Bairi et al. [_{2}O_{3} and TiO_{2}). The left wall is maintained at a constant temperature higher than the temperature of the right wall. They have shown that the value of the Rayleigh number, the size of the heating device and the volume fraction of the nanofluids effect on the strength of heat transfer. Kamal Raj Kumar Tiwari and ManabDas [

G. de Vahl Davis et al. [^{4}. G. de Vahl Davis [^{3} and Ra = 10^{5}. Markatos and Pericleous [^{3} ≤ Ra ≤ 10^{6}. Jou and Tzeng [^{10}. The solution captures very well all flow and heat transfer phenomena, especially near the walls where dense, non-uniform grids are used in the thin boundary layers constituted there. Elsherbiny [

The configuration has studied is show in _{2}O_{3}, including two walls are maintained at the respective temperatures T_{H} hot and cold T_{C}, the other walls are supposed, adiabatic. The form factor of the cavity is defined as R = H/W.

The nanofluid in the cavity is considered incompressible and Newtonian, the flow is assumed to be laminar and two-dimensional; the liquid phase and nanoparticles are in a state of thermal equilibrium. According to the approximations Boussinesq, the variation of density is negligible everywhere except in the term of buoyancy.

The thermophysical properties with which we will work are described in

property | Fluid phase (water) | Solid phase (copper) |
---|---|---|

C_{p} (J/kg∙K) | 4179 | 765 |

r (kg/m^{3}) | 997.1 | 3970 |

K (W/m∙K) | 0.613 | 40 |

α (m^{2}/s) | 1.47 × 10^{−7} | 131.7 × 10^{−7} |

β (1/K) | 21 × 10^{−3} | 0.85 × 10^{−3} |

Under the assumption of thermal properties constants, the Navier-Stokes equations for an incompressible fluid, unsteady, two-dimensional and are:

・ Continuity equation:

∂ u ∂ x + ∂ v ∂ y = 0 (1)

・ x-momentum equation:

ρ n f ( u ∂ u ∂ x + v ∂ u ∂ y ) = − ∂ P ∂ x + μ n f Δ 2 u ( ρ s β s φ + ( 1 − φ ) β f ρ f ) ⋅ T ⋅ g ⋅ sin ( ϕ ) (2)

・ y-momentum equation:

ρ n f ( u ∂ v ∂ x + v ∂ v ∂ y ) = − ∂ P ∂ y + μ n f Δ 2 v ( ρ s β s φ + ( 1 − φ ) β f ρ f ) ⋅ T ⋅ g ⋅ cos ( ϕ ) (3)

・ Energy equation:

u ∂ T ∂ x + v ∂ T ∂ y = α n f [ ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 ] (4)

・ Kinematics equation:

∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 = − ω (5)

where α n f = k n f ( ρ ⋅ c p ) n f

・ The effective viscosity of the fluid containing a dilute suspension of small particles rigid spherical is given by Brinkman:

μ n f = μ f ( 1 − φ ) 2.5 (6)

・ The effective density of the nanofluids:

ρ n f = ( 1 − φ ) ρ f + φ ρ s (7)

・ The heat capacity of the nanofluids is given as follows:

( ρ c p ) n f = φ ( ρ c p ) s + ( 1 − φ ) ( ρ c p ) f (8)

・ The thermal conductivity of the actual nanofluids is determined by the model Maxwel-Garnetts:

k n f k f = k s + ( n − 1 ) k f − ( n − 1 ) ( k f − k s ) φ k s + ( n − 1 ) k f − ( k f − k s ) φ (9)

・ The dimensionless parameters are:

X = x H ; Y = y H ; Ω = ω H 2 α f ; Ψ = ψ α f ; V = v H α f ; U = u H α f ; θ = T − T h T c − T h (10)

・ The equations can be written in the form dimensionless below

∂ ( U Ω ) ∂ X + ∂ ( V Ω ) ∂ Y = G r ⋅ P r 2 ⋅ A ⋅ [ cos ( ϕ ) ∂ θ ∂ X − sin ( ϕ ) ∂ θ ∂ Y ] + B ⋅ P r ⋅ [ ∂ 2 Ω ∂ X 2 + ∂ 2 Ω ∂ Y 2 ] (11)

where: A = 1 ( 1 − φ ) φ . ρ f ρ s + 1 β s β f + 1 φ ( 1 − φ ) . ρ s ρ f + 1 ; B = 1 ( 1 − φ ) 2.5 [ ( 1 − φ ) + φ ρ s ρ f ]

U = ∂ ψ ∂ Y ; V = ∂ ψ ∂ X (12)

∂ ∂ X ( θ ⋅ ∂ ψ ∂ Y ) − ∂ ∂ Y ( θ ⋅ ∂ ψ ∂ X ) = α n f α f ( ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 ) (13)

∂ 2 ψ ∂ X 2 + ∂ 2 ψ ∂ Y 2 = − Ω (14)

The Rayleigh number, Prandtl and Grashof are given respectively:

P r = ν f α f ; R a = g β H 3 ( T h − T c ) ν f α ; G r = g β H 3 ( T h − T c ) ν f 2

The boundary conditions no dimensional are written as:

On the left wall : X = 0 ; Ψ = 0 ; Ω = − ∂ 2 Ψ ∂ X 2 ; θ = 1 On the Right wall : X = 1 ; Ψ = 0 ; Ω = − ∂ 2 Ψ ∂ X 2 ; θ = 0 On the wall stop and bottom : Ψ = 0 ; Ω = − ∂ 2 Ψ ∂ Y 2 ; ∂ θ ∂ Y = 0 } (15)

The Nusselt number can be expressed as:

N u = h H k f (16)

The coefficient of heat transfer is given by the Equation (17):

h = q w T h − T c (17)

The thermal conductivity is expressed as:

k n f = q w ∂ T ∂ x (18)

Substituting Equations ((17), (18)) and Equation (9) in Equation (16), and using the settings dimensionless, the Nusselt number is given by the equation below:

N u = − ( k n f k f ) ⋅ ∂ θ ∂ X (19)

The average Nusselt number is:

N u ¯ = ∫ 0 1 N u ( Y ) ⋅ d Y (20)

The procedure of discretization of the equations (Guidelines Equations (11), (13), (14)) and the boundary conditions described by the Equation (15) have solved numerically using a finite difference technique. The equations of vorticity and energy are resolved using the method ADI (alternating direction implicit) and the equation of the function of current is resolved by the method SOR (successively over relaxation) while the difference in the near is used for the convective terms. The application line by line of the TDMA method (Tri-Diagonal Matrix Algorithm) (21) is applied to the equations of vorticity and energy until the sum of the residues becomes lower than 10^{−6}. The developed algorithm has been implemented in the Fortran program.

The convergence criterion is defined by the following expression:

ε = ∑ j = 1 j = M ∑ i = 1 i = N | Φ n + 1 − Φ n | ∑ j = 1 j = M ∑ i = 1 i = N | Φ n + 1 | < 10 − 6 (21)

where ε is the tolerance; M and N are the number of grid points in the x and y directions respectively.

The test of the mesh size is carried out in the case of Ra = 10^{5}, Pr = 6.2, φ = 10%, Φ = 0˚ and R = 1. The temperature and vertical speed calculated at half height of the cavity are presented in

In order to validate our code of calculation, we made a comparison with the previous results (

Once the code of calculation has been validated, we studied the influence of different

parameters on the thermal transfer. The results will be given in the form of the streamlines, and isotherms, average Nusselt as a function of correlation of the Rayleigh number between 10^{3} ≤ Ra ≤ 10^{5}, for different volume fractions of 0% ≤ φ ≤ 10% and different angles of tilt angle 0˚ ≤ f ≤ 90˚. For the form factor (R = 0.25, 0.5, 1 and 4), the results are presented as the streamlines and isotherms, the Nusselt number, temperature and velocity profiles.

For a horizontal cavity without nanofluid and R = 1, being observed in ^{4} and Ra = 10^{5} but a conductive schema for Ra = 10^{3}.

For a Rayleigh number (Ra = 10^{4}), the isotherms become more deflected by report to the waves of Ra = 10^{3}, and the volume fractions higher always cause a slight deviation of the current lines and the isotherms between the nanofluid and the pure fluid. For a Rayleigh number Ra = 10^{5}, the isotherms become deflected by report to the curves of the two previous cases. The increase of the volume fractions entails a significant deviation of the isotherms and of the streamlines between the nanofluid and the pure fluid. The flow passes to a type of boundary layer (

for different angles of inclinations. It is observed that the shape of the main cell is sensitive to the inclination angle. For Φ = 90˚, the flow of Rayleigh-Bénard type is formed for the inner cavity because of the existence of the gravity in the enclosure.

^{5}, Pr = 6.2, φ = 10% and the form factor equal to 0.5. It is observed that the shape of the main cell of the streamlines decreases with the increase of angle of tilt and deflected to the left. Small cells of recirculation in the lower right corner and in the upper left corner for Φ = 90˚.

^{5}, Pr = 6.2, φ = 10% and R = 0.25.It is noticed that the shape of the cell is reduced with the increase of the inclination angle. It is observed a change in the cell structure of cells for Φ = 90˚. That is why the transfer of heat is low, as shown in

^{5}, Pr = 6.2, φ = 10% and the form factor equal to 4. We see that the isotherms are insensitive to the inclination angle with the exception of the angle Φ = 90˚ to the cause of the decline in heat transfer, as shown in

In this work, it has been demonstrated the influence of the inclination angle

as well as the volume fraction of nanoparticles and the form factor in cavity. The results have indicated that the addition of the nanoparticles of Al_{2}O_{3} has produced a remarkable enhancement on heat transfer with respect to that of the pure fluid. The heat transfer increases with the increase of Rayleigh, but the effect of the concentration of nanoparticles on the the Nusselt number is more pronounced at low Rayleigh number than at high Rayleigh number. Inclination angle of the cavity is considered as a control parameter of the fluid flow and heat transfer. It has been seen that the heat transfer is more low to Φ = 90˚. But the increase of the values of the volume fraction becomes insignificant flow of fluid from this angle. Finally, the angle of tilt is a good parameter of control in the two cases of a cavity filled with pure fluid and nanofluid.

It was concluded that inclination angle where the heat transfer minimal has developed to a close relationship with the structure of the flow of transition. For the form factor of 1 and 0.5, the secondary flow which occurs in the corners of the cavities was found to cause a decrease in the transfer of heat. For the form factor of 0.25, a decrease of heat has occurred at the transformation of a unique cellular structure in a cell to seven cells. The transfer of heat is incremented for form factor equal to 4.

The authors declare no conflicts of interest regarding the publication of this paper.

Eljamali, L., Wakif, A., Boulahia, Z., Zaydan, M. and Sehaqui, R. (2019) Effects of the Form Factor and the Force of the Gravity on the Thermal Exchanges by Natural Convection in a Rectangular Cavity Filled with Nanofluid. Engineering, 11, 59-73. https://doi.org/10.4236/eng.2019.111006

C_{p} Specific heat capacity [J∙kg^{−1}∙K^{−1}]

g Gravitational acceleration [m∙s^{−2}]

Gr Grashof number G r = ( β f H 3 g ( T h − T c ) ) / ( ν f 2 )

H Height of the cavity [m]

k Thermal conductivity [W/m∙K]

N u ¯ Average Nusselt number

Nu Local Nusselt number

p Pressure [Pa]

P The dimensionless pressure

Pr

R Prandtl number Pr = α_{f}/ν_{f}

Form factor

Ra Rayleigh number R a = ( β f g H 3 ( T h − T c ) ) / ( ν f α )

T Temperature [K]

W Width of the cavity [m]

(x, y) Cartesian coordinates-dimensional [m]

(X, Y) Cartesian coordinates dimensionless

u, v Velocity components in the directions x, y [m∙s^{−1}]

U, V Dimensionless velocity components

Greek symbols

α Thermal diffusivity [m^{2}∙s^{−1})

β Thermal expansion coefficient [K^{−1}]

φ Solid volume fraction [%]

µ Dynamic viscosity [N∙s∙m^{−2}]

n Kinematic viscosity [m²∙s^{−1}]

r Density [kg∙m^{−3}]

Ω Dimensionless vorticity

Φ Inclination angle [˚]

Ψ Dimensionless stream function

q Dimensionless temperature

Indices

c Cold

eff Effective

h Hot

s Solid

f Fluid

Operators

D Laplacian