EFFECTS OF A HEATED STRIP ARRANGEMENT ON HEAT TRANSFER RATE IN CUBICAL ENCLOSURES © 2014 S. Mellah*, N. Ben-Cheikh, B. Ben-Beya, T. Lili
Laboratory of Fluid Dynamics, Physics Department, Faculty of Science of Tunis, Campus Universitaire, 2092 El-Manar II, Tunisia *E-mail: sihem_mel@yahoo.fr
This work addresses a numerical approach based on the finite volume method and a full multi-grid technique to study three-dimensional flow structures and heat transfer rates in cubical cavity partially heated from one wall. The working fluid is air so that the Prandtl number equates to 0.71. Numerical solutions are generated for representative values of the controlling Rayleigh number inside the range 103 < Ra < 107. The heating occurs with a heated strip placed inside the enclosure. Three scenarios are investigated: (i) the heated source is mounted on the bottom horizontal wall, (ii) the heated source is mounted horizontally on the lateral wall, and (iii) the heated source is mounted vertically on the lateral wall. Effects of heated strip position in the enclosure on heat transfer are studied. It is shown that suitable configuration which generates highest heat transfer rate through the heated strip is depending on Rayleigh number. Results are presented in the form of projection of flow lines and isotherms plots as well as the variation of the Nusselt number and the average temperature at the heat source surface for all configurations computed in this study.
DOI: 10.7868/S004036441403020X
INTRODUCTION
The problem on laminar natural convection in enclosures constitutes a useful description of the heat and flow behaviour of confined fluids in many practical engineering applications. Some examples are: solar collectors, houses and buildings, electronic devices, etc. which often leads to the design of systems with heated and/or cooled surfaces of different sizes. The majority of studies on partially heated cavity have been confined to two-dimensional fluid flow, for example those of [1—6]. However, to be near the real three-dimensional situation a limited number of articles fall into this general category and have been reported in the literature. For example, Frederick and Quiroz [7] numerically studied natural convection in a cubical enclosure with a cold vertical wall and a hot square sector on the opposite wall. Sezai and Mohammad [8] examined three-dimensional natural convection from a discrete flush-mounted rectangular heat source placed at the bottom of a horizontal enclosure. The upper wall was maintained at a cold temperature and two kinds of boundary conditions were applied at the sidewalls. Tou and Zhang [9] studied the effects of inclination on the heat transport processes in a liquid-filled rectangular enclosure housing an array of discrete heaters. Oosthuizen and Paul [10] focused on the unsteady features of free convective flow in a parallelepiped enclosure having two square isothermal heated sections located on the lower surface. While the other walls were insulated, the vertical side-walls were kept at a uniform low temperature.
A marginally related publication was done by Shar-ma et al. [11] who reported the numerical results of turbulent natural convection in a square enclosure with localized heating from below and symmetrical cooling at the two vertical side walls. Ben-Cheikh et al. [12] investigated three-dimensional flow structures and heat transfer characteristics in partially heated 3D containers having a heated strip on the lower horizontal wall. Using FLUENT, Bocu and Altac [13] recently studied numerically the laminar natural convection heat transfer in 3D rectangular air filled enclosures, with pins attached to the active wall. The pin diameters and the lengths were varied, and the mean Nusselt numbers over the cold surface were computed.
The central objective of the present paper is to analyze numerically the laminar natural convection features of an incompressible fluid confined to a cubic enclosure partially heated in the central part of the wall of the enclosure. Three different arrangements of the heated source are considered: (i) the heated source is mounted on the bottom horizontal wall, (ii) the heated source is mounted horizontally on the lateral wall, and (iii) the heated source is mounted vertically on the lateral wall. For all configurations, two of the four adjacent walls to the wall with the heated strip are maintained at a constant low temperature. The remaining walls are insulated.
The working fluid is air (Pr = 0.71) and the Ray-leigh numbers spans from 103 to 107.
The layout of the paper is delineated as follows. In Section 2, we define the physical system and introduce the mathematical formulation.
The numerical procedure used to solve the full elliptic conservation equations is explained in Section 3. A discussion of the significant results is presented in Section 4. Finally, some conclusions are reported in Section 5.
PROBLEM DESCRIPTION
The physical model considered here is depicted in Fig. 1, along with the important geometric parameters. It consists of a cubic enclosure of dimension L, partially heated from one wall with an iso-flux, q", and length 40% of the dimension of the cavity. Three scenarios are investigated:
(i) The heated strip is placed on the bottom horizontal wall, two of the vertical walls are fully cooled at a uniform low temperature Tc, and the remaining walls of the enclosure are kept insulated;
(ii) the heated strip is mounted horizontally on the lateral wall, both horizontal walls are fully cooled at temperature Tc, and the remaining walls are kept insulated;
(iii) the heated strip is positioned vertically on the lateral wall, both vertical walls are cooled at temperature Tc, and remaining walls are kept insulated.
The following standard assumptions are adopted: 1) the fluid is an incompressible Newtonian fluid, 2) no phase change takes place, 3) the flow regime is laminar; 4) the Boussinesq approximation is applicable.
Framed in a Cartesian coordinate system, the di-mensionless governing equations take the following form (i = 1, 2, 3):
du, dxj
= 0,
дщ + д (UUj ) =
dt дХ :
I Ra dxj dxj
+ 68,
de , д (u,e) __
д 2e
dt dxj -v/RaPr dxjdxj '
(1) (2)
(3)
where q = 2 for configuration 1, q = 1 for configuration 2 and q = 3 for configuration 3.
In these equations, there are two parameters that control the buoyant flow, namely the Rayleigh number Ra = (gPATL3)/av and the Prandtl number Pr = v/a. Here, g is the gravitational acceleration, P is the iso-baric coefficient of thermal expansion, a is the thermal diffusivity, v the kinematic viscosity and AT is the temperature scaling defined as (q"L)/k, where k is the thermal conductivity of air. For the non-dimensional-ization in eq. 1, eq. 2 and eq. 3, the length scale is L,
the velocity scale is u0 = sjgfiATL, the time scale is t0 =
= L/u0, and the pressure scale is p0 = pu0 where p is the mass density.
The dimensionless flow boundary conditions consist of no-slip velocities, i.e., ui = (u, v, w) = 0 and impermeable walls. Upon defining the dimensionless temperature 9 as: 9 = (T — Tc)/AT. The dimensionless temperature boundary conditions for the problem are specified as follows:
9 = 1 at x = 0 and x = 1, 0 < y < 1 and 0 < z < 1 (59/5y) = -1 at y = 0, 0.5 - l/2 < x < 0.5 + l/2, 0 < z < 1, (4)
(59/5«) = 0 applies elsewhere on the remaining walls (n denotes the normal direction to the surface).
Finally, at each instant of time the average heat flux taken at a wall leads to the average Nusselt number:
0.5+// 2 1
Nu =1 i [-^-dxdz. (5)
/J J0 s(x)
0.5-//2 0 SV ^
NUMERICAL PROCEDURE
The continuity, momentum and the energy equations are discretized using staggered, non-uniform control volumes. A projection method Achdou and Guermond [14] is used to solve the Navier-Stokes equations. An intermediate velocity is first computed and later updated for satisfaction of mass continuity. In the intermediate velocity field the old pressure is used. A Poisson equation, with the divergence of the intermediate velocity field as the source term, is then solved to obtain the pressure correction and the real velocity field. A finite-volume method Patankar [15] is used to discretize the Navier-Stokes and energy equations. The advective terms are discretized using a QUICK third-order scheme proposed by Leonard [16] in the momentum equation and a second order central differencing one in the energy equation. The discretized momentum and energy equations are resolved using the red and black successive over relaxation method RBSOR Leonard [17], while the Poisson pressure correction equation is solved using a full multi-grid method as suggested by Ben-Cheikh et al. [18]. The numerical method was implemented in a FORTRAN program. The convergence of the numerical results is established at each time step according to the following criterion:
< 10-
(6)
( N
Gi,j,k - ^ j,k IVi,j,k i, j,k y where G stands for ui or 9 and m is the iteration level.
The code was validated with the numerical results of Sharif and Mohammad [19] for natural convection in rectangular cavities. They considered a constant ux heating at the bottom and isothermal cooling from the sidewalls while the top wall was adiabatic. A comparison of the computations is given in Table 1, which shows that our results do not exceed 1.03% and 0.7% of relative error, respectively, in the average Nusselt number, Nu, and in the maximum temperature at the
Table 1. Comparison of the present results with previous works
[19] Present work
Gr Mesh 70 x 70 64 x 64 96 x 96
103 Nu 4.085 4.073 4.077
/л max Hi 0.274 0.275 0.274
104 Nu 4.132 4.125 4.129
.-.max 0.275 0.275 0.275
105 Nu 6.058 6.039 6.027
max 0.21 0.209 0.21
106 Nu 10.572 10.463 10.458
max "i 0.141 0.142 0.142
Table 2. Comparison of maximum velocities umax and vmax for Ra = 105
[7] Present work
^max v max ^max vmax
35.9146 63.2177 35.9436 65.6693
heated surface, 0 s . Besides, th
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