Natural convection in cavities with a thin fin on the hot wall

Abstract

A numerical study has been carried out in differentially heated square cavities, which are formed by horizontal adiabatic walls and vertical isothermal walls. A thin fin is attached on the active wall. Heat transfer by natural convection is studied by numerically solving equations of mass, momentum and energy. Streamlines and isotherms are produced, heat and mass transfer is calculated. A parametric study is carried out using following parameters: Rayleigh number from 10⁴ to 10⁹, dimensionless thin fin length from 0.10 to 0.90, dimensionless thin fin position from 0 to 0.90, dimensionless conductivity ratio of thin fin from 0 (perfectly insulating) to 60. It is found that Nusselt number is an increasing function of Rayleigh number, and a decreasing function of fin length and relative conductivity ratio. There is always an optimum fin position, which is often at the center or near center of the cavity, which makes heat transfer by natural convection minimized. The heat transfer may be suppressed up to 38% by choosing appropriate thermal and geometrical fin parameters.

Introduction

Natural convection heat transfer in differentially heated, partitioned cavities are encountered in various industrial applications, such as heating and ventilating of living spaces, fire in buildings, solar thermal collector systems, electronic cooling devices, in storage of radioactive wastes. Studies of various aspects of this problem have been carried out by many researchers both theoretically and experimentally.

Excluding studies with multiple fins in tall systems (see, for example, [1], [2], [3]), thick fin(s) in square and tall cavities (see, for example, [4], [5], [6]), the specific problem studied in this work involving a single thin fin attached to the hot or cold wall makes a distinct category, which has been studied by various researchers that we will briefly review [7], [8], [9], [10], [11]. Oosthuizen and Paul [7] studied differentially heated cavities with aspect ratio between 3 and 7 with a horizontal plate attached to the center of the cold vertical wall. They found the local heat transfer rate on the upper portion of the hot wall increased, but the heat transfer rate near the center of the hot wall decreased. Frederick [8] studied a similar situation in a inclined cavity with diathermal partition at Rayleigh number between 10³ and 10⁵. The partition was attached to the cold wall at its center, its relative length was 0.25 and 0.50. The inclination angle was from 90° (corresponding to horizontal partition) to 45°. His results showed that the partition caused suppression of convection and the heat transfer relative to that in an identical cavity without partition was reduced considerably. Frederick and Valencia [9] studied natural convection in a square cavity with a conducting partition at the center of its hot wall and with perfectly conducting horizontal walls. Partition length and its conductivity were variable. For low values of relative conductivity, they reported reduced heat transfer with respect to that in an identical cavity without partition at Rayleigh numbers between 10⁴ and 10⁵. Nag et al. [10] studied numerically in a differentially heated square cavity where a horizontal plate was attached on the hot wall. The length and the position of the partition were varied and Rayleigh was between 10³ and 10⁶. They considered two cases, one with adiabatic partition and the other with perfectly conducting partition. They found that with the perfectly conducting partition the heat transfer at the cold wall increased irrespective of its position or length and it is attenuated with the adiabatic partition, which was more pronounced when the position of the partition was higher. Shi and Khodadadi [11] studied numerically the same problem reported by Nag et al. with almost perfectly conducting partition on the hot wall, but with more extensive parametric details. Their dimensionless fin length was between 0.20 and 0.50, which had seven positions along the hot wall and Rayleigh number was from 10⁴ to 10⁷. Since the fin was almost perfectly conducting and attached to the hot wall, the fin’s heating enhanced the convection while its blockage of the flow field suppressed it. The contribution of these two counter-acting mechanisms were not clearly quantified. Based on the numerical data, they proposed correlations to calculate Nusselt number as a function of relevant parameters for this particular case.

We see from this brief review that excluding that by Frederick and Valencia [9], all other studies considered only the perfectly conducting or insulated partitions. Yet, in practical engineering problems, often partitions are made from materials with finite conductivities. In this paper, we will study the case with dimensionless fin length from 0.10 to 0.90 at dimensionless fin position from 0.10 to 0.90 and fin to air conductivity ratio from 0 (perfectly insulating materials) to 60 (construction and fabrication materials) and examine heat transfer by conduction and convection due to fins with finite conductivity ratios.