On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

Abstract

relationship (Eqs. 9, 10) (m 2 /s) bCoefficient in Eq. (24b) which should be defined trough the initial condition δ (t = 0) = 0 C p Specific heat capacity (J/kg) E L (n, β, t) Squared-error function in accordance with the Langford criterion (Eq. 34) E LT (n, β, t) Squared-error function in accordance with the Langford criterion (Eq. 35) E Mq (p, β, t) Squared-error function in accordance with the Langford criterion and fixed flux BC problem (Eq. 76) e LT (n, β, t) Squared-error sub-function in accordance with the Langford criterion (Eq. 35) e Lq (p, β) Squared-error sub-function in accordance with the Langford criterion and the fixed flux BC problem k Thermal conductivity (W/mK) k (T) Temperature-dependent thermal conductivity (W/mK) k 0 Thermal conductivity of the linear problem (β = 0) (W/mK) m Dimensionless parameter of the nonlinearity (power-law diffusivity) n Dimensionless exponent of the parabolic profile p Dimensionless exponent of the parabolic profile of the assumed profile used to solve Eq. (48) (m) Abstract Closed form approximate solutions to nonlinear transient heat conduction with linearly temperaturedependent thermal diffusivity have been developed by the integral-balance integral method under transient conditions. The solutions uses improved direct approaches of the integral method and avoid the commonly used linearization by the Kirchhoff transformation. The main steps in the new solutions are improvements in the integration technique of the double-integration technique and the optimization of the exponent of the approximate parabolic profile with unspecified exponent. Solutions to Dirichlet and Neumann boundary condition problems have been developed as examples by the classical Heat-balance integral method (HBIM) and the Double-integration method (DIM). Additional examples with HBIM and DIM solutions to cases when the Kirchhoff transform is initially applied have been developed. List of symbolsThermal diffusivity (m 2 /s) a 0 Thermal diffusivity of the linear problem (β = 0) (m 2 /s) a p Thermal diffusivity coefficient in the case of power-law non-linear

The radiation diffusion equation: Explicit analytical solutions by improved integral-balance method

“…Equations (8) and (9) 12,13,16,17]. The principle drawback of (8) is that the right-side depends on gradient expressed through the type of the assumed profile.…”

“…Equations (8) and (9) are the principle relationships of the simplest version of the integralbalance method known as Heat-balance Integral Method (HBIM) [16]. After this first step, replacing T by an assumed profile a T (expressed as a function of the relative space co- 12,13,16,17]. The principle drawback of (8) is that the right-side depends on gradient expressed through the type of the assumed profile.…”

“…The integral relation (12) allows to work with either integer-order time-derivatives (as in the present case) or with time-fractional derivatives [19] where the Leibniz rule is inapplicable If the thermal diffusivity is non-linear and expressed as a power-law m p a a T  ( 0 m  ), corresponding to degenerate diffusion problems) then equation (12) takes the forms [12,13]   The integral relation (13) will be used further in this work in the solutions of the problems at issue.…”

“…The aim of the present work is to present an approximate closed form analytical solution allowing to estimate the penetration depth of the heat wave and the spatial temperature distribution applying an improved integral-balance approach [12,13] already successfully applied to nonlinear heat conduction problems modelled by degenerate parabolic equations (see further in the text).…”

“…The principle drawback of (8) is that the right-side depends on gradient expressed through the type of the assumed profile. An improvement, avoiding the drawback of HBIM is the double integration method (DIM) [12,13]. The first step of DIM is integration of from 0 to x , namely…”