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(Parte **1** de 2)

Proceedings of COBEM 2007 19th International Congress of Mechanical Engineering Copyright © 2007 by ABCM November 5-9, 2007, Brasília, DF

Rogério Fernandes Brito, rogbrito@unifei.edu.br Paulo Mohallem Guimarães, paulogui@uol.com.br Harley Souza Alencar, haarley@terra.com.br Lucilene de Oliveira Rodrigues, lucilener@unifei.edu.br Genésio José Menon, genesio@unifei.edu.br Marco Antônio Rosa do Nascimento, marcoantonio@unifei.edu.br Federal University of Itajubá - UNIFEI, Benedito Pereira dos Santos Av., Number 1,303, Pinheirinho District, Zip Code 37,500-903, Itajubá, MG, Brazil

Abstract. Heat transfer by laminar natural convection in confined spaces is of great interest in the engineering field. The flow that occurs in a cavity is an important physical phenomenon that must be investigated, as it can be applied to projects of electronic components of electrical circuits with heat dissipators. The objective of the numerical model consists of evaluating the amount of heat transferred by the fins and also visualizing the velocity field and the isothermal lines in the fluid (air) and solid (aluminum) domains. The surface of the electronic component is kept at a high uniform temperature. The vertical surfaces are uniformly kept at low temperatures. The inferior horizontal surface around the electronic component and the superior horizontal surface are considered adiabatic. Four fins with rectangular cross-sections are placed on the inferior surface of the electronic component. Solutions for low values of Rayleigh are obtained by keeping the Prandtl number equal to 0.70. The Computational Fluid Dynamics is used. Hence, the Finite Volume Method (MVF) with Eulerian scheme is applied to solve the conservation equations for the unsteady state. It is assembled a 3D model with width wide enough to eliminate the wall effect in the flow and then enabling one to compare the results with 2D cases from literature. The present work shows that not only the increase of the Rayleigh number, but also the presence of the fins augments the heat transfer.

Keywords: Natural convection, finned cubic cavities, heat dissipators, CFD.

1. INTRODUCTION

During the last four decades, significant attention was given to the study of natural convection in enclosures subjected to heating and cooling using four-finned dissipators placed on a heated base. This was due to the occurrence of natural convection in a wide range of application areas that include nuclear reactor design, post-accident heat removal in nuclear reactors, geophysics and underground storage of nuclear waste, energy storage systems and others. Natural convection heat transfer in enclosures containing heat generating fluids with different geometrical parameters and boundary conditions has been extensively considered in the open literature.

Several electronic equipments have been designed to be closed rectangular boxes with small openings on the bottom surface to allow natural ventilation. Components of electronic equipment are usually placed on the bottom surface of the cavity. They always dissipate heat at a constant temperature even on the standby mode. Although many works deal with laminar natural convective flow, some still consider a partially heated base.

In the case of natural convection in a two-dimensional domain, many works have been experimentally and numerically developed.

Dong and Li (2004) carried out a study of natural convection inside a cavity which is crossed by a horizontal cylinder using the Stream function Method and the Boussinesq Vorticity in the differential equations for conservation od mass, momentum and energy. The effect of the material, geometry and Rayleigh number on the heat transfer was investigated considering a regime which is approximately permanent.

Bilgen and Oztop (2005) conducted a natural convective heat transfer study in an inclined square cavity with isolated walls, being that one is partially opened. The flow is laminar and permanent with the Rayleigh number and the inclination angle varying from 103 to 106 and from 0° to 120°, respectively.

Nasr et al. (2005) proposed a case similar to the present work where one of the vertical walls of the two-dimensional model is heated in a small portion at a constant temperature. They considered a generic system of cooling and heating with a permanent laminar convective air flow inside a cavity. It is observed that convective flow is strongly affected by the geometry.

Bakkas et al. (2006) investigated the permanent laminar natural convective flow in a two-dimensional horizontal channel with rectangular blocks mounted along the bottom surface. These blocks were heated at a constant temperature and connected to the bottom surface by layers which were adiabatically isolated. Having Rayleigh number from 102 to 106 and Prandtl number equal to 0.7 (air), it was verified that the block dimensions affected significantly the temperature and velocity convective fields.

Ben-Nakhi and Chamkha (2007) focused their work on the numerical study of steady, laminar, conjugate natural convection around a finned pipe placed in the center of a square enclosure with uniform internal heat generation. Four perpendicular thin fins of arbitrary and equal dimensions are attached to the pipe whose internal surface is isothermally cooled. The sides of the enclosure are considered to have finite and equal thicknesses and their external sides are isothermally heated. The problem is put into dimensionless formulation and solved numerically by means of the finitevolume method. Representative results illustrating the effects of the finned pipe inclination angle and fins length on the streamlines and temperature contours within the enclosure are reported. In addition, results for the local and average Nusselt numbers are presented and discussed for various parametric conditions.

The study of natural convection in a three-dimensional domain is still not quite explored in literature where works such Janssen et al. (1993) and Tric et al. (2000) can be found.

Janssen et al. (1993) carried out a study of natural convection in a cubic cavity using the Finite Volume Method with permanent and transient flows. In the permanent flow case, the boundary layer along the wall was studied while in the transient regime, the convective flow periodicity generated by a 3D model was investigated. A comparison with the classic 2D model was conducted.

Tric et al. (2001) studied exact solutions to the governing equations of natural convection of air inside cubic cavities which were thermally loaded by two opposite vertical walls with different temperatures and Rayleigh numbers going up to 107. The solutions were considered exact with relative global errors below 0.03 % and 0.05 % for Rayleigh numbers 103 and 107, respectively.

In the present work, permanent and laminar natural convection study in a cubic cavity with four-finned aluminum dissipator placed on a bottom horizontal surface is carried out. Heat transfer is investigated based on temperature and velocity behaviour and on the local Nusselt number along the bottom heated surface of the base of the dissipater in contact with the bottom surface of the cavity. The remaining part of the bottom surface of the cavity is thermally isolated. The lateral vertical walls are kept with a uniform low temperature and the cavity upper horizontal wall is thermally isolated. Figure 1 shows details of the geometry. The solution to this problem is found by the application of the Fluid Dynamics Calculus using the Finite Volume Method and the Eulerian scheme to discretize the domain in space and time. This is done by integrating the transport differential equations of mass, momentum and energy. The mesh has 36,194 nodes and 174,909 elements where 160,281 are tetrahedral and 14,312 are prismatic which are placed near the lateral vertical walls in order to capture the boundary layer effects. This mesh is used in all cases. Two cases are analyzed with Rayleigh numbers 106 and 107. Temperature and velocity distributions are analyzed over a vertical section that crosses the half part of the base and fins. The Nusselt number is also calculated along the base by ranging the Rayleigh number. The machine used to run all cases has a processor Intel Pentium 4® that runs at 3.0 GHz and 2 GB of RAM. The time computing cost was approximately 49 minutes for Rayleigh number of 106 e 107, both for a physical time of 4 s. CFD software package from Ansys CFX-5.6® was used.

2. PROBLEM DESCRIPTION

Figure 1 shows the geometry with domain Ω that is filled with air (Pr = 0.7) and whose base with fins made of aluminum. It is considered a cubic cavity where S1 and S2 are the isolated surfaces. S3 and S6 are surfaces that are kept at a uniform low temperature while S7 is at a uniform high temperature. The local heating is simulated by a heat source that is located in the middle portion of the base wall, S7. The initial condition on the entire domain Ω is taken for temperature θ = 0 and air velocities u = v = w = 0. All properties are considered constant, except the density on the buoyancy forces that follows the Boussinesq approximation. Figure 2 depicts the mesh used in the computational simulation, projected over YZ plan, and the solid body mesh.

2.1. Problem hypotheses

The following hypotheses are considered: a) Three-dimensional domain; b) Unsteady regime; c) Laminar and incompressible flow; d) Viscous dissipation is neglected; e) Physical properties (K,c,,pμρ) are constant, except density in the buoyancy forces; f) No internal heat generation.

Proceedings of COBEM 2007 19th International Congress of Mechanical Engineering Copyright © 2007 by ABCM November 5-9, 2007, Brasília, DF

x y z Ω

(a) (b)

Figure 1. Computational domain: (a) Geometry in 3D; (b) Dissipator mounted on S7 surface, with four 4 rectangular fins (dimensions in m).

(a) (b)

Figure 2. Computational mesh: (a) In a vertical surface, YZ plan for x = 0.04 m, with prismatic (boundary layer) and tetrahedral elements; (b) In the aluminum dissipator.

2.2. Governing equations

With the above considerations, the conservation equations for air can be written as follows: i) Continuity:

i) Momentum:

,zuyuxuxpz uwyux uutu

zyxypzw yxu t

() ,TTgzwywxwzpz wwywx wutw

i) Energy:

,zTyTxTcKz TwyTx where u, v, and w are the flow velocity components in m/s, x, y and z are the directions in space, t is the time in s, ρ is the fluid density in kg/m3; p is the flow relative pressure in Pa, g is the local acceleration component in m/s2, and β is the volumetric coefficient of thermal expansion in K-1. The subscript f represents fluid (air). The conservation equation for the aluminum domain can be written as:

pss s zTyTxTcKt where the subscript s stands for solid (aluminum).

2.3. Boundary and initial conditions

They are as follows: i) Initial Conditions: for t = 0:

,0wu==υ= ( in Ω ), | (7) |

T+== ( in Ω ), | (8) |

i) Boundary Conditions: for t > 0:

,T= | ( on S7 ), (9) |

,T= | ( on S3 a S6 ), (10) |

Tq=∂∂= | ( on S1 and S2 ), (1) |

,0wu==υ= ( on S1,..., S7 ), | (12) |

where S1 , S2 , S3 , S4 , S5 , S6 and S7 represent the surfaces on the boundaries of the domain Ω, as shown in Fig. 1. 2.4. Dimensionless equations

In order to generalize the theorical analysis governed by equations 1 to 6, dimensionless variables are introduced:

Proceedings of COBEM 2007 19th International Congress of Mechanical Engineering Copyright © 2007 by ABCM November 5-9, 2007, Brasília, DF

T,HpP,HwW,HV−−=θνρ=ν=νυ=, | (14) |

where τ is the dimensionless time; X, Y e Z are the dimensionless coordinates; U, V, W are the dimensionless velocity components; P is the relative dimensionless pressure, and θ is the dimensionless temperature. Substituting (13) and (14) in (1) to (5):

U=∂∂+∂∂+∂∂, | (15) |

0ZWYVX

⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂+∂∂−=∂∂+∂∂+∂∂+∂τ∂ | (16) |

VUV, | (17) |

X2GrZWYWXWZPZ WWY WVX WUW 2 ∂

θ∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂+∂∂−=∂∂+∂∂+∂∂+∂τ∂, | (18) |

f2 f

ZYXPr1Zw YX

u | (19) |

For the solid domain:

1, | (20) |

ZYXPr where Gr and Pr are the Grashof and Prandtl numbers defined respectively by the following:

−β=, | (21) |

ch HTTgGr ν

cPrppρν=μ=αν= | (2) |

KcK

In order to calculate the local Nusselt number, Fig. 3 presents the energy balance along the dissipator surface. One can say that the heat received (Qw) is equal to the heat delivered (Qfin) taking the aluminum part as the control volume. So:

finwQQ= | (23) |

And then:

() chfinwsolid

TkAq−=∂∂=, | (24) |

sww TTAhAz where qw and qfin are the heat fluxes on S7 and the dissipater surface, respectively. Since:

Thsolid and f kf Tc Tc

Insulated

Insul. Insul. qwAw = Qw qfinAfin = Qfin h

Figure 3. Heat transfer balance along the dissipater surface. the average Nusselt number Nu is calculated by the expression as follows:

kA HkANu chh ffin

δ−δ= | (26) |

where kf and ks are the fluid and solid thermal conductivity, h is the local convective heat transfer coefficient, δ is a small distance from the dissipater bottom (δ/H = 0.025), Aw and Afin are the S7 and the aluminum surfaces in contact with air, respectively, and H* is any relevant measure to obtain appropriate Nu value orders.

The initial and boundary conditions are now expressed in the dimensionless form as: i) Initial conditions: for τ = 0:

,0WVU=== | ( in Ω ), (27) |

,210=θ=θ | ( in Ω ). (28) |

i) Boundary conditions: for τ > 0:

,1=θ | ( on S7 ), (29) |

,1−=θ | ( on S3,..., S6 ), (30) |

θ∂−= | ( on S1 and S2 ), (31) |

,0WVUwallwallwall=== ( on S1,..., S7 ) | (32) |

3. NUMERICAL METHOD

The solution of the partial differential equations in time and space can be solved using the Computational Fluid

Dynamics (CFD) by Finite Volume Method (FVM), which is a method of discretization in space and in time of the entire domain, which can use a mesh with finite number of volumes (Barth and Ohlberger, 2004).

In this method of discretization, the mesh can have two typical volume schemes: centralized face and centralized volume. For both schemes of control volume, the variables can be solved in terms of average values. Figure 4 shows the centered face and control volume used in the Finite Volume Method, as in Barth and Ohlberger (2004).

Figure 4. Details of the control volume used in the Finite Volume Method (Barth and Ohlberger, 2004).

The governing equations can be solved applying a suitable algorithm (Euler’s equations for inviscid flows and

Navier-Stokes’s equations for viscous flow). In particular case of phenomena with fluid flows and heat transfer, it is necessary to link the pressure and velocity. Among the algorithms that can solve all variables in the same time step with velocity and pressure linked equation, CFD has used the SIMPLE Method (Semi Implicit Method for Pressure Linked Equation) (Shaw, 1992).

This methodology is a interactive process, where the error or residual is compared to a reference error, also named “target error”. In this way, flow and heat transfer simulations require the introduction of suitable flow and heat transfer models to guarantee a satisfactory convergence.

(Parte **1** de 2)