Unlike the investigations reported in the literature which are restricted to the simpler one-dimensional heat flow situation, the current study presents an analytical solution to the inverse problem that is applicable to two-dimensional conduction systems for geometries of arbitrary shape; heretofore intractable even for the simplest geometries. From theoretical considerations, temperatures can be predicted at discrete locations throughout the conducting medium, when input data such as thermocouple responses are known at several interior locations. Particularly, the transient temperature behavior may be readily established on any of the bounding surfaces by suitable interior thermocouple positioning. To facilitate computation of the desired temperatures, the theory allows for a temporal power series approximation of the input or thermocouple data, as is the customary practice in an experimental program. From transform techniques, the resultant theoretical expression for the prediction temperature is generated, and it appears as a summation of repeated error integrals. The form of the solution is convenient for numerical evaluation. Several numerical examples are presented as an indication of the accuracy of the theoretical results. Nomenclature ' ;>« ' .j ' = general coefficients a lf a{ = free parameters, a 2 + (tf/) 2 = 1 y(f) > g(f) = general functions of time /(A), g(d) = defined by Eq. (5) i" erfc x = repeated error integral of variable x /, m,n,q = summation indices N -number of matched points per face P = (s/a) 1 ' 2 s = Laplace transform variable t = time variable T = temperature T = Laplace transform of temperature x,y = space variables a = thermal diffusivity A = distance between thermocouples, x 2 -x 3 = distance between thermocouples, y 2 -y Subscripts 1 = property along sampling path x i or y±
For planar geometries, the present investigation develops a method for generating approximate solutions for heat conduction in solids with variable thermal conductivity, for both the direct and inverse problems. In the first portion, the direct case, from theoretical considerations, an analytical solution is generated for the original nonlinear differential system when it is replaced by a sequence of linear differential equations in some optimum fashion. The approach is unlike traditional methods appearing in the literature, since it does not require perturbation parameters and penetration distances. It entails an iterative procedure for accuracy improvement. For the second part of the study, an analytical procedure is developed for the nonlinear inverse problem. The concept of approximating the original material with a pseudolinear one is employed, and thermal diffusivity iteration is introduced. Numerical examples are presented to illustrate the computational procedures. a,A m ,b n ,C n B 0 c D n /"erfcz I(a) n,m,q t T x z 6 X P A Subscripts i s n 1 2 Nomenclature = general constants or coefficients = constant for surface temperature behavior = specific heat = constant, Eq. (23) = repeated error integral of variable z = mean square difference integral = number of equal parts or degree of polynomial approximation = (' Jo (6,0 "dt -time variable = temperature = space variable = similarity variable, Eq. (12) = modified thermal conductivity slope = conventional thermal conductivity slope = thermal diffusivity = thermal conductivity = density = distance between thermocouples, x 2 -x { , or unspecified length = initial value = surface value = A?th valuê value of first location, or first iteration value at second location, or second iteration
SynopsisAn examination of the theory of thermal conductivity of aniorphous dielectrics as applied to polymeric materials indicates that it is reasonable to expect that the conductivity is stress dependent. An experimental investigation was undertaken to determine the validity of this hypothesis for a number of plastics at temperatures below their respective glass transition points. Poly(methy1 methacrylate), nylon, and Delrin were chosen as representative of a wide range of percentage crystallinity and were tested a t compressive stresses up to 140 kg./cm.2 and temperatures between 4 and 38°C. The results indicate that the conductivity may increase as much as 20% and that the dependence on stress is a function of temperature and the type of polymer examined.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.