Nonlinear Heat Transfer in Planar Solids: Direct and Inverse Applications

Abstract

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

Two-dimensional linear transient inverse heat conduction problem - Boundary condition identification

“…Figure 6 gives the nine selected components of Ucl(t) to be identified, and the corresponding meshes interval of the direct model [Eqs. (8) and (9)]. In the same way, as for the thermocouples, the choice of the input localization is made to test the method ability to restore the two-dimensional effects.…”

Inverse Heat Transfer Problems

“…The need for a statement and solution for inverse heat transfer problems arises in various thermophysical investigations and also in designing and operating thermally loaded engineering units, [1,14,17,81,87,95,100,116,127,[160][161][162]184]. Below we will consider a number of possible applications of inverse problem techniques which could have been even greater.…”

ORMDIN: a finite element program for two-dimensional nonlinear inverse heat conduction analysis

“…Equations 14 This application of the finite element method to the inverse heat conduction problem considers a two-dimensional model of the (r,8) cross section of a circular cylinder. An isoparametric [14] discretization is employed, so •that the spatial coordinate.s are interpolated using the same functions NI as those used for Tin equation (6). The NI associated with the 4-to 8-noded two-dimensional isoparametric element are described in numerous references, including [14] and [15], and ~ill not be given here.…”

“…In the second numerical example, the inverse formulation is applied to actual thermocouple transients taken from a representative test of ORNL's single-rod test apparatus. 6 The heater power input to the rod during the period of the transient considered here is essentially 3 constant at Q = 5300 watts/cm • Figure 12 illustrates the time-history of the thermocouple temperatur.es recorded by the active BN-filled 6 The single~rod test facility [l] at ORNL is used primarily to qualify heaters for the large rod bundle loop and to obtain blowdown heat tr.ansfer results for a single rod in an annular geometry. The test data utilized in this example.were recorded during blowdown FCTF 79-1~6 at axial level D in the heater rod.…”