A much-needed reference focusing on the theory, design, and applications of a broad range of surface types. * Written by three of the best-known experts in the field. * Covers compact heat exchangers, periodic heat flow, boiling off finned surfaces, and other essential topics. 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). Extended Surface Heat Transfer By Allan D. Kraus Abdul Aziz James Welty John Wiley & Sons Copyright © 2001 Allan D. Kraus All right reserved. ISBN: 978-0-471-39550-8 Chapter One CONVECTION WITH SIMPLIFIED CONSTRAINTS 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 triangu