Temperature distribution in a fin partially cooled by nucleate boiling

Abstract

When heat is transferred from a fin into a nearly saturated liquid, nucleate 'boiling may set in if the temperature on the fin is sufficiently higher than the liquid's saturation temperature (1). Such a mode of heat transfer would be very efficient because of the high heat transfer coefficient of nucleate boiling over an extended surface area. Prior to this report, only complicated numerical solutions for nucleate boiling from a fin had been proposed (1, 2 ) . The purposes of this report are to present a simplified model for establishing a criterion for the incipience of nucleate boiling on a straight fin, to propose an expression for the length of the boiling section, and to determine the heat flux for a iven base temperature. The the need of resorting to a tedious step-by-step numerical calculation with difference equations.The temperature profile on a straight fin was determined, and this temperature was compared with the existing criterion for the incipience of nucleate boiling. The temperature for the incipience of nucleate boiling can be determined either experimentally or by use of the equation proposed by Hsu (3) :simplified method will provi f e a quick estimate without Consider the convective section first, and assume that the heat transfer coefficient is represented by a constant value h,.The boundary conditions are Use of Equation (2a) and the previous boundary conditions gives the classical fin equation for a fin of heightIn the present study, the temperature profile distribution along the straight fin was derived to determine the location where the criterion for the inci ience of nucleate boilknown, the heat flux at the base can be determined.A simple experiment was conducted to check the theory. Also, a comparison was made between the present theory and the experimental data on nucleate boiling from a fin reported by Haley and Westwater (1 ) .ing is satisfied. Once the length o P the boiling section is ANALYSIS If nucleate boiling does occur on a fin, it will first take place at the root of the fin. As the temperature of the fin is increased, the nucleate boiling section will extend out further and further on the fin. The one fin can thus be considered as made up of two separate fins in contact with each other. The section near the root is hot enough to have nucleate boiling taking place while the end section is being cooled by convection (which could be either forced or free convection). The model is shown in Figure 1 for a straight fin of rectangular shape. Similar analyses can be made on other geometries, such as a cylindrical fin, without much additional difficulty.The fin could be either tion of the temperature profile on a straight fin were applied herein, namely, constant roperties, flat profile in t~ tion for a straight fin is direction, and negligible end e P ect. The differential equa-The heat flux at x = Li is For the boiling section, the heat transfer coefficient varies as the temperature difference 8 to the nth power. Thus, Equation (2) should take the form where and ei are the heat ...

Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems

“…In many applications, various heat transfer modes, such as convection, nucleate boiling, transition boiling, and film boiling, the heat transfer coefficient are no longer uniform. A fin with an insulated end has been studied by many investigators [5][6][7][8][9]. Most of them are immersed in the investigation of single boiling mode on an extended surface.…”

A simple design of fins for boiling heat transfer

“…Since the temperature at the junction of two segments is known, the segment length and heat transfer rate in the k-section can be calculated from Eqs. (4), (5), (8) and (11). In case the fin tip is subject to transition boiling, Eq.…”

“…In the analytical study of multi-boiling from a fin, Lai and Hsu [8] presented a one-dimensional model to study the heat transfer characteristics of a fin partly cooled by nucleate boiling and partly by convection. Good agreements were observed between their results and the experimental data of Haley and Westwater [5].…”

Analysis of the 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient: Part I – Extended inverse and direct solutions

“…(8), (11) and below # local temperature difference between a fin and environment, t À t a (K) u relative percent discrepancy between the closedform and recurrent solutions for T e and T (%) w fin aspect ratio (fin height to half thickness or half radius ratio), Bi l /Bi x = h e /h x=0 ratio of heat transfer coefficients on the fin tip and lateral surfaces, a e /a Subscripts and superscripts a ambient medium (environment) b fin base (at X = 1) e fin tip (at X = 0) * fin with an insulated tip been presented in the plots of fin efficiency vs thermo-geometrical parameter of a fin. But the latter parameter involves the heat transfer coefficient calculated locally and averaged along the fin.…”

“…The latter dependence is usually expressed as a power function of #, h = a# n with constant a and n. The 1-D steady-state problem of heat conduction, governed by the power-law type temperature dependent local heat transfer coefficient for a single straight fin of uniform cross section is reduced to the solution of ordinary nonlinear second-order differential equation with two boundary conditions. The analytical and numerical solutions of such problems for the definite values of n made their appearance in the middle 1960s-early 1970s (see, for example, papers [6][7][8][9]). Ü nal [10][11][12] showed that the explicit analytical closed-form solutions can be obtained only for several definite values of n = À4, À1, and for n = 0.…”