Forced convective heat transfer across a pin fin micro heat sink

Abstract

This paper investigates heat transfer and pressure drop phenomena over a bank of micro pin fins. A simplified expression for the total thermal resistance has been derived, discussed and experimentally validated. Geometrical and thermo-hydraulic parameters affecting the total thermal resistance have been discussed. It has been found that very low thermal resistances are achievable using a pin fin heat sink. The thermal resistance values are comparable with the data obtained in microchannel convective flows. In many cases, the increase in the flow temperature results in a convection thermal resistance, which is considerably smaller than the total thermal resistance.

Introduction

Over the last decade, driven by the rapidly increasing heat dissipation predicament involving microprocessors, numerous investigations have been conducted on forced convection in microchannels concerning both single-phase [1], [2], [3], [4] and boiling (two-phase flow) [5], [6], [7], [8], [9]. Nonetheless, other microscale cooling methods have received inadequate attention. This is perhaps because channels are fundamental geometries in heat transfer and fluid flow engineering applications, and extensive literature and knowledge exist for conventional scale channels. Furthermore, the relative ease of fabricating microchannels has ensured their popularity among researchers. However, recent advances in microfabrication technology have enabled the realization of alternate microscale cooling methods, whose performance can outshine the standards set by microchannels. At the conventional scale, pin fin heat sinks, either shrouded or open ended, are widely used in the industry. Nevertheless, very limited studies of such cooling methods have been conducted at the microscale.

Over the past century, several investigators have explored various heat transfer and pressure drop characteristics for flow across a bank of tubes at the macro scale and a considerable amount of data and correlations for heat transfer coefficients (Nusselt number) and friction factors are readily available in the literature. Explicit correlations for various flow regimes (laminar, transitional, or turbulent), pin fin arrangements (staggered or in-line) and pin geometry including longitudinal/transverse pitch-to-diameter ratio and pin height-to-diameter ratio (L/D) have been developed. Since cross flow over a long array of tubes is commonly encountered in shell-and-tube heat exchangers, early studies have mainly focused on arrays of long cylinders with L/D > 8 [10], [11]. An assortment of correlations for different array configurations and flow regimes are readily available. Dimensional analysis suggests that the convective heat transfer across cylinders in cross flows varies with the Reynolds and Prandtl numbers. A commonly used relation for the average Nusselt number is in the following form $\overline{\mathit{Nu}}={\mathit{CRe}}_D^m{\mathit{Pr}}^n$, where C, m, n are constants. Various constants for different tube configurations and thermo-hydraulic conditions have been proposed, and it has been experimentally determined that the exponent n lies between 0.3 and 0.4 [11]. Pressure drops have been expressed in terms of the velocity, number of rows, fluid density, and the friction factor. Besides, various correlations have been proposed for the friction factor for flow across a bank of tubes (e.g., [12]).

The knowledge gained through studies involving long tubes has contributed immensely to the development of various pin fin heat sinks. As noted by Moores and Joshi [13], intermediate size shrouded pin fin (1/2 < L/D < 8) heat sinks are primarily encountered in applications concerning turbine blade or vane cooling, whereas short pins are commonly found in compact heat exchangers [14]. A concise review of staggered array arrangements for intermediate pin sizes is provided by Armstrong and Winstanley [15]. In general, the average heat transfer coefficient for relatively short pin fins is slightly lower than for long cylinders. Friction factors, on the other hand, display no such deviation. However, Short et al. [16] have found that at low Reynolds numbers friction factors are altered for intermediate size tubes. This has an important implication for microscale pin fin heat sinks, since laminar flow is expected to dominate in these systems. As a result, Koşar et al. [17] experimentally obtained friction factors over intermediate size 50 μm and 100 μm pin fin heat sink and demonstrated the importance of fin height-to-diameter ratio. Recently, Marques and Kelly [18] experimentally studied a heat exchanger possessing 500 μm diameter staggered micro pin fins with a pitch-to-diameter ratio of 2.5, which they fabricated using a modified LIGA (Lithography, Electroforming (German: Galvanoformung), and molding (German: Abformung)) micromachining process. An increase from 4.1 to 5.5 in the heat transfer rate has been reported with the introduction of pin fins for Reynolds number ranging from 4000 to 20 000.

The extensive knowledge of convective flow across a bank of pin fins presents an indispensable engineering tool that can be used to correlate the thermo-hydraulic field in such flow configurations. Although, it was developed mainly in the context of conventional scale systems it provides an excellent platform for extension to microscale devices, and can be employed to obtain preliminary heat transfer and pressure drop data. Due to the small pin dimensions at the microscale, the flow regime is expected to be predominantly laminar. Moreover, as stated by Incropera and DeWitt [19], a staggered fin arrangement is favored for enhanced heat transfer rates at low Reynolds numbers. The present study introduces the microscale pin fin heat sink design and concept, and analyzes its heat transfer capabilities. A test case that demonstrates the thermo-hydraulic performance of the micro heat sink is discussed in Section 3. In Section 4, preliminary experimental results, which support the analytical work, have been presented.