Über die Autorin bzw. den Autor

Allan D. Kraus, PhD, is Professor of Mechanical Engineering at the University of Akron, Ohio, and is principal associate at Allan D. Kraus Associates. He is the author of many works on thermal systems.

Abdul Aziz, PhD, is Professor of Mechanical Engineering at Gonzaga University in Spokane, Washington.

James Welty, PhD, Professor of Mechanical Engineering at Oregon State University in Corvallis, is co-author of Fundamentals of Momentum, Heat, and Mass Transfer (Wiley).

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Extended Surface Heat Transfer

John Wiley & Sons

Chapter One

1.1 INTRODUCTION

Three quarters of a century ago, a paper by Harper and Brown (1922) appeared as an NACA report. It was an elegant piece of work and appears to be the first really significant attempt to provide a mathematical analysis of the interesting interplay between convection and conduction in and upon a single extended surface. Harper and Brown called this a cooling fin, which later became known merely as a fin. It is most probable that Harper and Brown were the pioneers even though Jakob (1949) pointed out that published mathematical analyses of extended surfaces can be traced all the way back to 1789. At that time, Ingenhouss demonstrated the differences in thermal conductivity of several metals by fabricating rods, coating them with wax, and then observing the melting pattern when the bases of the rods were heated. Jakob also pointed out that Fourier (1822) and Despretz (1822, 1828a,b) published mathematical analyses of the temperature variation of the metal bars or rods.

Although these ancient endeavors may have been quite significant at the time they were written, it appears that the Harper-Brown work should be considered as the forerunner of what has become a burgeoning literature that pertains to a very significant subject area in the general field of heat transfer.

The NACA report of Harper and Brown was inspired by a request from the Engineering Division of the U.S. Army and the U.S. Bureau of Standards in connection with the heat-dissipating features of air-cooled aircraft engines. It is interesting to note that this request came less than halfway through the time period between the first flight of the Wright Brothers at Kitty Hawk and the actual establishment of the U.S. Air Force. The work considered longitudinal fins of rectangular profile and trapezoidal profile (which Harper and Brown called wedge-shaped fins) and radial fins of rectangular profile (which Harper and Brown called circumferential fins). It also introduced the concept of fin efficiency, although the expression employed by Harper and Brown was called the fin effectiveness. From this modest, yet masterful beginning, the analysis and evaluation of the performance of individual components of extended surface and arrays of extended surface where individual components are assembled into complicated configurations has become an art.

Harper and Brown (1922) provided thorough analytical solutions for the two-dimensional model for both rectangular and wedge-shaped longitudinal fins and the circumferential fin of uniform thickness. The solutions culminated in expressions for the fin efficiency (called the effectiveness) or in correction factors that adjusted the efficiency of the rectangular profile longitudinal fin. They concluded that the use of a one-dimensional model was sufficent and they proposed that the tip heat loss could be accounted for through the use of a corrected fin height, which increases the fin height by a value equal to half of the fin thickness. Lost in the shuffle, however, was the interesting observation that with dx as the differential element of fin height, the differential face surface area of the element is dx/ cos [kappa], where [kappa] is the taper angle, which is 90 for rectangular profile straight and circumferential fins as well as for spines of constant cross section.

Schmidt (1926) covered the three profiles considered by Harper and Brown from the standpoint of material economy. He stated that the least material is required for given conditions if the fin temperature gradient (from base to tip) is linear, and he showed how the fin thickness of each type of fin must vary to produce this result. Finding, in general, that the calculated shapes were impractical to manufacture, he proceeded to show the optimum dimensions for longitudinal and radial fins of constant thickness (rectangular profile) and the longitudinal fin of trapezoidal profile. He also considered the longitudinal fin of triangular profile as the case of the longitudinal fin of trapezoidal profile with zero tip thickness.

The case of integral pin fins of different profiles was considered by Bueche and Schau (1936). They determined for conical fins that the heat dissipation was a function of the Biot modulus based on the base radius and aspect ratio of fin height to base radius. They also showed that a weight optimization could be effected.

Murray (1938) considered the problem of the radial fin of uniform thickness (the radial fin of rectangular profile) presenting equations for the temperature gradient and effectiveness under conditions of a symmetrical temperature distribution around the base of the fin. He also proposed that the analysis of extended surface should be based on a set of assumptions that have been known since 1945 as the Murray-Gardner assumptions. These assumptions are deemed to be of considerable importance because their elimination, either one at a time or in combination, provided a series of paths for subsequent investigators to follow.

A stepwise procedure for calculating the temperature gradient and efficiency of fins whose thickness varies in any manner was presented by Hausen (1940). The temperature gradient in conical and cylindrical spines was determined by Focke (1942) who, like Schmidt, showed how the spine thickness must vary to keep the material required to a minimum. He too, found the resulting shape impractical and went on to determine the optimum cylindrical and conical spine dimensions.

Avrami and Little (1942) derived equations for the temperature gradient in thick bar fins and showed under what conditions fins might act as insulators on the base or prime surface. Carrier and Anderson (1944) discussed straight fins of constant thickness, radial fins of constant thickness, and radial fins of constant cross-sectional area, presenting equations for the fin efficiency of each. However, in the latter two cases, the efficiencies are given in the form of an infinite series.

Gardner (1945), in a giant leap forward, derived general equations for the temperature excess profile and fin efficiency for any form of extended surface for which the Murray-Gardner assumptions are applicable and whose thickness varies as some power of the distance measured along an axis normal to the base or prime surface (the fin height). He proposed the profile function (his nomenclature)

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for the straight or longitudinal fins,

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for spines and

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for radial or circumferential (radial) fins.

These equations depend on the assignment of some number to n; for example, the straight fin of rectangular profiles results when n = 0 in the first of these. This also serves to show that in Gardner's profile functions, the positive sense of the height coordinate x is in a direction from fin tip to fin base.

With the foregoing profile functions in hand and working with a general differential equation that he derived, Gardner was able to provide solutions for the temperature excess profile in terms of modified Bessel functions. For n equal to zero or an integer,

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and for n equal to a...