J. R. Culham Associate Professor and Director Mem. ASME M. M. Yovanovich Distinguished Professor Emeritus Fellow ASME P. Teertstra Research Associate Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, Canada C.-S. Wang Advanced Thermal Solutions, Newton, MA G. Refai-Ahmed Solinet Systems, Ottawa, ON, Canada Mem. ASME Ra-Min Tain Nortel Metworks, Kanata, ON, Canada Simplified Analytical Models for Forced Convection Heat Transfer From Cuboids of Arbitrary Shape Three analytical models are presented for determining laminar, forced convection heat transfer from isothermal cuboids. The models can be used over a range of Reynolds number, including at the diffusive limit where the Reynolds number goes to zero, and for a range of cuboid aspect ratios from a cube to a flat plate. The models provide a simple, convenient method for calculating an average Nusselt number based on cuboid dimen- sions, thermophysical properties and the approach velocity. Both the cuboid and the equivalent flat plate models are strongly dependent upon the flow path length which is bounded between two easily calculated limits. In comparisons with numerical simulations, the models are shown to be within 66 percent over the range of 0 <Re A A <5000 and aspect ratios between 0 and 1. @DOI: 10.1115/1.1347993# Introduction Numerous practical applications in the design of electronics and telecommunications equipment depend upon low-velocity, lami- nar flow as a mechanism for dissipating heat. A diverse range of applications are typically encountered, including heat transfer from printed circuit boards with low profile chip-on-board pack- ages, cooling of electronic packages, transformers, heat sinks, thermal spreaders plus many other complex components that re- quire some means of forced convection cooling to maintain safe operating temperature limits. Several researchers have investigated forced convection from isothermal axisymmetric bodies, such as spheroids ~Beg @1#!, cyl- inders ~Refai-Ahmed and Yovanovich @2#! and disks ~Wedekind and Kobus @3# using a variety of predictive methods as reviewed in Yovanovich @4#. However, the number of studies for steady laminar forced convection heat transfer from isothermal cuboids is very limited, with most research restricted to experimental studies ~Igarashi @5,6#! leading to empirical correlations or detailed nu- merical procedures ~Wong and Chen @7#!. Few simple, analytical procedures are available because of the perceived difficulty in modeling the vertical faces of the cuboid, perpendicular to the flow direction. The principal objective of this paper is to present several, ana- lytical models for calculating the overall rate of heat transfer from isothermal cuboids of varying aspect ratios. Models will be simple functions of easily attainable information such as cuboid dimen- sions, flow conditions and thermophysical properties. The models presented are intended for isothermal boundary conditions but it is interesting to note that in spherical bodies the overall rate of heat transfer is identical for both isothermal and isoflux boundary conditions. While the rate of heat transfer will be different for isothermal and isoflux conditions as the aspect ratio is varied, the change is relatively small and in most instances the models can be used for nonisothermal conditions. Modeling Procedure The overall rate of heat transfer from an isothermal, convex body is a function of several fundamental modes of heat transfer including diffusion, convection and radiation. If we assume the radiative component to be relatively small, an overall Nusselt number based on a general characteristic length, L can be defined as a function of two limiting asymptotes, as shown in Eq. ~1!. Nu L 5@~ S * ! n 1~ Nu bl ! n # 1/n (1) The first asymptote is based on the diffusion or conduction of heat as the flow velocity approaches zero, while the second asymptotic limit is based on laminar boundary layer flow for flow velocities greater than zero. Yovanovich @8#~forced convec- tion, spheroids!, Yovanovich and Vanoverbeke @4#~mixed con- vection, spheroids!, Refai-Ahmed and Yovanovich @2#~forced convection, circular and square cylinders and toroids!, Wang et al. @9#~natural convection, heat sinks!, and Lee et al. @10#~natural convection, general body shapes! have all used this approach to obtain analytical models that are applicable over a wide range of flow conditions. The procedure of combining limiting asymptotic solutions was first employed by Churchill and Usagi @11#. The method presented by Churchill and Usagi takes two analytical solutions that are known to be ‘‘exact’’ at predefined limits and provides a means of transitioning smoothly between these two limits, thereby obtain- ing a comprehensive analytical procedure that is a function of a single unknown constant, n, that is typically referred to as the blending parameter. Equation ~1! satisfies exactly both limiting asymptotic solutions, however in the transition between the two limits the blending parameter, n can be used to minimize differ- ences between the model and known experimental or numerical data. Typical values for the blending parameter can vary between unity ~superposition! and real numbers in the range of 1–5. The higher the value of the blending parameter the greater the ten- dency for the model to map the asymptotic solutions near the intersection point of the two limiting models. Based on comparisons between numerical validation data and Contributed by the Electrical and Electronic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received at ASME Head- quarters October 13, 2000. Associate Editor: R. Schmidt. 182 Õ Vol. 123, SEPTEMBER 2001 Copyright © 2001 by ASME Transactions of the ASME