Baixe 2007-Numerical Simulation of Fluid Flow in a Cubic Cavity with-Revista-RETERM e outras Notas de estudo em PDF para Engenharia Mecânica, somente na Docsity! Ciência/Science Engenharia Térmica (Thermal Engineering), Vol. 6 • No 02 • December 2007 • p. 54-61 54 NUMERICAL SIMULATION OF FLUID FLOW IN A CUBIC CAVITY WITH A FOUR-FINNED DISSIPATOR PLACED ON THE BOTTOM SURFACE R. F. Brito a , H. S. Alencar b , L. O. Rodrigues c , G. J. Menon c , and M. A. R. ascimento c a,cUniversidade Federal de Itajubá Departamento de Engenharia Mecânica Bairro Pinheirinho CP. 50, Itajubá, MG, CEP: 37500-903, Brasil rogbrito@unifei.edu.br lucilener@unifei.edu.br genesio@unifei.edu.br marcoantonio@unifei.edu.br bAlstom Equipamentos do Brasil Centro de Tecnologia Av. Charles Sheineider, s/ nº Bairro Barranco Taubaté, SP, CEP: 12400-040, Brasil haarley@terra.com.br 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 (CFD) is used. Hence, the Finite Volume Method (FVM) 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, fins, cavities, heat dissipators, CFD OME CLATURE A area, m 2 a, b dimensions of the surface at an uniform high temperature, m CFD computational fluid dynamics cp fluid specific heat at constant pressure, J/(kg.K) FVM finite volume method g acceleration of gravity, m/s 2 Gr Grashof number H geometry height, m h heat transfer coefficient, W/(m 2 K) k thermal conductivity, W/(m.K) L geometry length, m n normal direction N normal direction Nu average Nusselt number NuL local Nusselt number P dimensionless pressure p pressure, N/m 2 Pr Prandtl number Q heat flux rate, W q heat flux, W/m 2 Qfi heat flux rate delivered, W Qw heat flux rate received, W Ra Rayleigh number RAM random access memory RMS root mean square S1, …, S7 surfaces of the geometry SIMPLE semi implicit method for pressure linked equation T temperature, K t time, s U, V, W dimensionless velocity components u, υ, w velocity components, m/s x,y,z cartesian coordinates, m X,Y,Z dimensionless cartesian coordinates Greek symbols ρ density, kg/m3 θ dimensionless temperature τ dimensionless time ν kinematic viscosity, m2/s δ lenght, m α thermal diffusivity, m2/s µ dynamic viscosity, N s/m 2 β volumetric coefficient of thermal expansion, K -1 Ciência/Science Brito et al. umerical Simulation of… Engenharia Térmica (Thermal Engineering), Vol. 6 • No 02 • December 2007 • p. 54-61 55 Ω computational domain Subscripts 0 average value c cold or contact f fluid fi fin h hot L local s solid w wall I TRODUCTIO 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 10 3 to 10 6 and from 0° to 120°, respectively. Nasr et al. (2006) 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 10 2 to 10 6 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 finite-volume 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. (2000) 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 10 7 . The solutions were considered exact with relative global errors below 0.03 % and 0.05 % for Rayleigh numbers 10 3 and 10 7 , 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 Ciência/Science Brito et al. umerical Simulation of… Engenharia Térmica (Thermal Engineering), Vol. 6 • No 02 • December 2007 • p. 54-61 58 oh o 2 2 TT TT , Hp P − − =θ νρ = (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): 0 Z W Y V X U = ∂ ∂ + ∂ ∂ + ∂ ∂ (15) X P Z U W Y U V X U U U ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂τ ∂       ∂ ∂ + ∂ ∂ + ∂ ∂ + 2 2 2 2 2 2 Z U Y U X U (16) Y P Z V W Y V V X V U V ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂τ ∂       ∂ ∂ + ∂ ∂ + ∂ ∂ + 2 2 2 2 2 2 Z V Y V X V (17) Z P Z W W Y W V X W U W ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂τ ∂ X2 Gr Z W Y W X W 2 2 2 2 2 2 ∂ θ∂ +      ∂ ∂ + ∂ ∂ + ∂ ∂ + (18) Z w YX u ffff ∂ ∂θ + ∂ ∂θ υ+ ∂ ∂θ + ∂τ ∂θ       ∂ θ∂ + ∂ θ∂ + ∂ θ∂ = 2 f 2 2 f 2 2 f 2 ZYXPr 1 (19) For the solid domain:       ∂ θ∂ + ∂ θ∂ + ∂ θ∂ = ∂τ ∂θ 2 s 2 2 s 2 2 s 2 s ZYXPr 1 (20) where Gr and Pr are the Grashof and Prandtl numbers defined respectively by the following: ( ) 2 3 ch HTTgGr ν −β = (21) K c K c Pr pp ρν = µ = α ν = (22) 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 (Qfi) taking the aluminum part as the control volume. So: fiw QQ = (23) And then: ( )chfiw s sww TTAhA z T kAq −=⋅ ∂ ∂ = (24) where qw and qfi are the heat fluxes on S7 and the dissipater surface, respectively. Since: ( )[ ] δ δ− −= ∂ ∂ TT z T h s (25a) and, f * L k Hh Nu = (25b) The average Nusselt number Nu is calculated by the expression as follows: ( )[ ] ( )       − δ− δ = ch h ffi * sw TT TT kA HkA Nu (26) Figure 3. Heat transfer balance along the dissipater surface. 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 Afi are the S7 and the aluminum surfaces in contact with air, respectively, and H* is any relevant measure to obtain appropriate Nu value orders. kf h h Tc Tc ks T(δ) δ Th Insulated Insul. Insul. z qfi · Afi = Qfi qw · Aw = Qw Ciência/Science Brito et al. umerical Simulation of… Engenharia Térmica (Thermal Engineering), Vol. 6 • No 02 • December 2007 • p. 54-61 59 The initial and boundary conditions are now expressed in the dimensionless form as: i) Initial conditions: for τ = 0: 0WVU === ( in Ω ) (27) 2 1 0 =θ=θ ( in Ω ) (28) ii) Boundary conditions: for τ > 0: 1=θ ( on S7 ) (29) 1−=θ ( on S3,..., S6 ) (30) 0 N Q = ∂ θ∂ −= ( on S1 and S2 ) (31) 0WVU www === ( on S1,..., S7 ) (32) UMERICAL 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). Figure 4. Details of the control volume used in the Finite Volume Method (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). 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 (Shaw, 1992). This methodology is an 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. RESULT A ALYSIS Through the iterative calculation using CFD, it was observed the convergence in relation to the deviation calculated by the goal error equal to 10 6 for the two cases where Ra = 10 6 and 10 7 . Figure 5 shows the Convergence Curves for the local Nusselt number on S2 and S7 for Ra = 10 6 and 10 7 . Accumulated Time Step 0 2 4 6 8 10 12 14 16 18 20 R es id u a l R M S 0.000 0.050 0.100 0.150 0.200 0.250 0.300 Ra = 10 6 Ra = 10 7 Figure 5. Convergence curves for the RMS deviation on heat transfer rate for Ra = 10 6 and 10 7 . It can be noticed that convergence is rapidly reached for lower Rayleigh numbers where the internal flow is significantly laminar and the viscosity effects are stronger. Table 1 shows the results for the average Nusselt number given by Eq. (26) at δ = 0.003 with H = 0.05 m, where H is the height of dissipator placed on the bottom surface. It can be observed that Nu increase as Rayleigh number goes up to 10 7 , due to the higher temperature gradient between the surfaces S7 and S8. To observe the effect of Rayleigh number on heat transfer in the cubic cavity, Figs. 6 and 7 depict the temperature and velocity vectors distributions Centralized face Centralized volume Control volume Point of calculation Ciência/Science Brito et al. umerical Simulation of… Engenharia Térmica (Thermal Engineering), Vol. 6 • No 02 • December 2007 • p. 54-61 60 plotted on the ZX plan at (x / L = 0.40; 0 ≤ y / L ≤ 1, 0 ≤ z / H ≤ 1). From Figs. 6(a) and 7(a), it is noted that the ascending convective flow increases with Rayleigh number increase. For Ra = 10 6 , two opposite vortices of the same intensity are formed on the reference plan at x / L = 0.40; 0 ≤ y / L ≤ 1, 0 ≤ z / H ≤ 1, progressively nearer the finned surface. Table 1. Average Nusselt number for Ra = 10 6 and 10 7 at the last time instant. Ra = 10 6 Ra = 10 7 Aw [m 2 ] in S7 2.30E-03 2.30E-03 Ac [m 2 ] in S8 8.06E-03 8.06E-03 ks (Al) [W m^-1 K^-1] 237.0 237.0 kf (air) [W m^-1 K^-1] 0.026100 0.026100 H [m] 0.12 0.12 δi [m] 0.003 0.003 Th [K] 673.2 673.2 Tc [K] 667.1 618.1 T(δi) [K] 673.1 654.1 Nu 518.8 35874.9 Figure 6a. Case 1 – Ra =10 6 at the final instant for contour number 100: temperature distribution, plan ZX. Figure 6b. Case 1 – Ra =10 6 at the final instant for contour number 100: velocity vector field at = 0.012 m, plan ZX. Figure 7a. Case 2 – Ra =10 7 at the final instant for contour number 100: temperature distribution, plan ZX. Figure 7b. Case 2 – Ra =10 7 at the final instant for contour number 100: velocity vector field at = 0.012 m, plan ZX. For Ra = 10 6 and 10 7 , a typical behavior takes place. This behavior represents the strength order of the ascending convective flow between the fin heated surface and the upper isolated horizontal surface. This can also be seen in Figs 6(b) and 7(b), where the progressive increase of the maximum velocity vector is noted when Rayleigh number is higher. One can also note that in Figs. 6(a) and 6(b), the isotherms find themselves in a more uniform distribution and deformed for Ra = 10 6 , along the fin and in the fluid domain, thus enhancing the heat transfer. For Ra = 10 7 , the heat transfer is predominantly featured by conductive effect, as depicted in Figs. 7(a) and 7(b). Significantly, for Figs 6 and 7, the calculus indicates that the temperature gradient becomes stronger near the fin region as Ra increases. An analysis in other flow plans is strongly recommended to analyze the intensity of the vortices generated in the third dimension. According to Bilgen and Oztop (2005) and Nasr et al. (2006), the temperature distributions as well as the vortices intensities are affected by the secondary effects of the ascending convective flow given by the third