Mathematical Problems in Engineering

Volume 2015, Article ID 989260, 13 pages

http://dx.doi.org/10.1155/2015/989260

## A Numerical Study of Natural Convection Heat Transfer in Fin Ribbed Radiator

^{1}The Province Key Laboratory of Fluid Transmission Technology, Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China^{2}Huadian Electric Power Research Institute, Hangzhou 310030, China

Received 6 March 2014; Accepted 11 April 2014

Academic Editor: Zhijun Zhang

Copyright © 2015 Hua-Shu Dou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper numerically investigates the thermal flow and heat transfer by natural convection in a cavity fixed with a fin array. The computational domain consists of both solid (copper) and fluid (air) areas. The finite volume method and the SIMPLE scheme are used to simulate the steady flow in the domain. Based on the numerical results, the energy gradient function of the energy gradient theory is calculated. It is observed from contours of the temperature and energy gradient function that the position where thermal instability takes place correlates well with the region of large values, which demonstrates that the energy gradient method reveals the physical mechanism of the flow instability. Furthermore, the effects of the fin height, the fin number, and the fin shape on the heat transfer rate are also investigated. It is found that the thermal performance of the fin array is determined by the combined effect of the fin space and fin height. It is also observed that the effect of fin shape on heat transfer is insignificant.

#### 1. Introduction

Natural convection heat transfer from finned surfaces has received much attention in both numerical simulations and experimental modeling, it is adopted extensively in various industrial devices, such as gas cooled nuclear reactors, automobile, aerospace vehicles, and electronic systems. Nowadays, as electronic equipment tends to be large as well as miniature, removing heat rapidly from the equipment is much more desirable than before, since the thermal efficiency of heat removal from the equipment can impact the life-span of the equipment. There are mainly two ways to enhance heat transfer from heat-generating electronic equipment. The first convenient method is to cool the electronic equipment by blowing air at a moderate velocity, and the second way is to use fin arrays. Since the forced convection approach has inherited problems and has to bear extra running cost, more and more researchers focus on designing optimum fin arrays, which can provide moderate heat transfer rates if designed properly.

In early stage, Starner and McManus [1] made some experiments by inserting four different rectangular fin arrays into a vertical, 45-degree plate and a horizontal plate, respectively. They obtained some free natural convection data which were widely used in the subsequent investigations. Harahap and McManus [2] obtained more detailed data by investigating free convection heat transfer from horizontal fin arrays using the schlieren shadowgraph technique. Jones and Smith [3] applied a Mach-Zender interferometer in their experiments to study the variation of local heat transfer coefficient for isothermal vertical fin arrays on a horizontal base. With a fixed fin height mm, the experimental data shows that the overall heat transfer coefficient depends strongly on the fin space but weakly on the fin height. Rammohan Rao and Venkateshan [4] experimentally investigated the interaction of free convection and radiation in a horizontal fin array. The most important conclusion made in their research was that there was a mutual interaction between free convection and radiation and hence a simplistic approach based on additivity of radiation and convection heat transfer, calculated independently based on isothermal surfaces, was unsatisfactory. Yüncü and Anbar [5] also made some experiments to research the effects of the fin space, the fin height, and the temperature difference on heat transfer with a fixed fin height and fin thickness. They found that there was an optimum fin spacing which was not related to the temperature difference. However, optimum fin spacing was inversely proportional with the fin height. Mobedi and Yüncü [6] numerically investigated the steady state natural convection heat transfer in longitudinally short rectangular fin arrays on a horizontal base. They observed two types of flow patterns. For the fin arrays with narrow fin spacing, air could only enter into the channel from the end regions. However, for the fin arrays with wide fin spacing, the air was also entrained into the channel from the region between the fins, turned 180 degrees at the base, and then moved up along the fin surface, while it flowed into the central part of the channel.

More recently, lots of researchers tried to enhance heat transfer rate of fin arrays using fin arrays. Arquis and Rady [7] investigated natural convection heat transfer and fluid flow characteristics from a horizontal fluid layer with finned bottom surface, and observed that the number of convection cells between two adjacent fins is a function of the values of the fin height and Rayleigh number. Liu [8] considered an optimum design problem for the longitudinal fin arrays with a constant heat transfer coefficient in a fuzzy environment, where the grid requirements to strictly satisfy the total fin volume and array width and maximize the heat dissipation rate are softened. Harahap et al. [9] conducted some experiments to investigate the effects of miniaturizing the base plate dimensions of vertically based straight rectangular fin arrays on the steady state heat dissipation performance under dominant natural convection conditions. They found that the relevant correlations proposed for large fin arrays were not applicable to the experimental data obtained from the miniaturized vertical rectangular fin arrays. Subsequently, Harahap et al. [10] conducted concurrent calorimetric and interferometric measurements to investigate the effect that the reduction of the base plate dimensions has on the steady state performance of the rate of natural convection heat transfer from miniaturized horizontal single plate-fin systems and plate-fin arrays. Their conclusions suggested that the fin height and the fin number are the prime geometric variables for generalization. Dogan and Sivrioglu [11] experimentally investigated mixed convection heat transfer, and the results obtained showed that the optimum fin spacing which yielded the maximum heat transfer is mm and the optimum fin spacing depends on the value of Ra. Azarkish et al. [12] used a modified genetic algorithm to maximize the objective function which is defined as the net heat transfer rate from the fin surface for a given height. Their results show that the number of the fins is not affected by the fin profile, but the heat transfer enhancement for the arrays with optimum fin profile is about 1–3 percent more than that for the arrays with conventional fin profiles. Giri and Das [13] numerically performed laminar mixed convection over shrouded vertical rectangular fin arrays attached to a vertical base. They found that the drop in pressure defect for forced convection, induced velocity for mixed convection and overall Nusselt number for mixed convection are correlated well with governing parameters of the considered problem. Wong and Huang [14] made some three-dimensional (3D) numerical simulations to investigate the effect of fin parameters on dynamic natural convection from long horizontal fin arrays. They observed that the optimum fin spacing decreases significantly with the fin height and increases slightly with the fin length. Furthermore, their observations agreed well with the results reported in the literature.

All the above experimental and numerical research focused on optimum design of the fin arrays in order to improve heat transfer rate. Almost all the related factors result in laminar convection involving low values of heat transfer coefficients. However, it is well known that the heat transfer rate will be prompted when the base flow loses its stability and flows in a turbulent manner [15]. Although some optimum fin arrays design can obtain a satisfying heat transfer rate which resulted from flow instability, the physical mechanism of flow instability of natural convection is still not fully understood. Recently, Dou et al. [16–24] suggested an energy gradient method which can reasonably reveal the physical mechanism of flow instability. Dou and Phan-Thien [16] described the rules of fluid material stability from the viewpoint of energy field. They claimed that the instability of natural convection could not be resolved by Newton’s three laws, for the reason that a material system moving in some cases is not simply due to the role of forces. This approach explains the mechanism of flow instability from physics and derives the criteria of turbulence transition. Accordingly, this method does not attribute Rayleigh-Benard problem to forces, but to energy gradient. It postulates that when the fluid is placed on a horizontal plate and is heated from below, the fluid density in the bottom becomes low which leads to an energy gradient along -coordinate. Only when is larger than a critical value will the flow become unstable and then fluid cells of vorticities will be formed. More recently, Dou et al. [25] applied the energy gradient method to natural convection and the results from numerical simulations accord well with those predicted based on the criteria originated from energy gradient method.

This study is focused on the research of effects of fin arrays parameters on convection heat transfer coefficient. Then, the energy gradient method is employed to reveal the physical mechanism of flow instability and explain the reason why the optimum fin arrays can result in better heat transfer rate.

#### 2. Computational Geometry and Numerical Procedures

##### 2.1. Computational Geometry

The computational geometry is shown in Figure 1. Here, the geometry is simplified from the 3D solid of GH-4 ribbed radiator model [26]. The simplified cavity in this study is a two-dimensional (2D) square, in which the length of the square cavity is 250 mm, and the origin of the coordinates is at the lower left corner of the cavity. The fin arrays are fixed at the hot bottom of the cavity with an equal distance. Here, is the fin space, is the fin height, and and are variable. In addition, means the thickness of the fin arrays which is fixed at 2 mm in this study and can be neglected by comparing to the fin height and the length of the cavity, while means the thickness of the bottom plate which is equal to 3 mm.