The heat-balance integral: 1. How to calibrate the parabolic profile?

Abstract

The Heat-Balance Integral Method (HBIM) of Goodman under classic prescribed temperature boundary conditions has been studied towards it optimization. Because the parabolic profile satisfies both the boundary conditions and the heat-balance integral at any value of the exponent the calibration is of a primary importance in generation of the approximate solution. The simple 1-D heat conduction problem, enabling one to demonstrate the HBIM performance with the entropy generation minimization (EGM) concept in calibration of a parabolic temperature profile with unspecified exponents, has been developed. The EGM concept provides constraints that impose addition boundary conditions at the approximate parabolic profile. Additionally, entire domain optimizations based on the mean-squared error concept has been performed in two versions -the method Myers and through a similarity transformed diffusion equation.

A note on the integral approach to non-linear heat conduction with Jeffrey’s fading memory

“…Hence, a parabolic profile with either n M T = 2.235 or n T h = 1.5047 is applicable, depending on the required actuary of approximation (see details in ref. [10]). …”

“…The optimal exponent of the parabolic profile when k 2 = 0 (Fourier law) is discussed in the literature [9][10][11] and the problem at issue refers to adequate function F n = [2n(n + 1)] 1/2 in eq. (10).…”

“…, when the approximate profile is calibrated at x = 0 [8,10,11]. This approach avoids the condition (11); (b) n M T = 2.235, obtained after minimization of the L 2 norm [9]; (c) n T h = 1.5047, when the minimization of the L 2 norm is performed through a preliminary transform of the governing equation by the similarity variable h = x/2(a 1 t) 1/2 [10] . Hence, a parabolic profile with either n M T = 2.235 or n T h = 1.5047 is applicable, depending on the required actuary of approximation (see details in ref.…”

“…With this approach we have several cases (Dirichlet boundary condition) of optimal exponents, namely (a) n T 0 175 » . , when the approximate profile is calibrated at x = 0 [8,10,11]. This approach avoids the condition (11); (b) n M T = 2.235, obtained after minimization of the L 2 norm [9]; (c) n T h = 1.5047, when the minimization of the L 2 norm is performed through a preliminary transform of the governing equation by the similarity variable h = x/2(a 1 t) 1/2 [10] .…”

“…The introduction of the concept of the final penetration depth d(t) is, in fact, ad hoc, definition of a damping function in the approximate profile. Thus, from mathematical point of view, the inexact approximate profile satisfies the diffusion equation in average, i. e. satisfying the energy balance, but on other hand, this is a profile with a relaxation function.Because the approximate profile T a = T 0 (1 -x/d) n satisfies the Goodman's conditions T(d, t) = 0 and ¶T(d, t)/ ¶x = 0 with any value of the exponent n[8] the accuracy of approximation depends on the choice of the exponent n and refers to consequent optimizations of the parabolic profile[8][9][10][11] . Therefore, the dimensionless approximate profile is:…”