On the tolerance modelling of heat conduction in functionally graded laminated media

Abstract

The object of analysis is a heat conduction problem within the framework of tolerance modelling in laminated media with a functional gradation of effective properties. In contrast to the known asymptotic models (based on the homogenization technique), the characteristic feature of the tolerance model equations is that they make it possible to analyze the effect of the microstructure size on the overall behavior of the laminate. The proposed model equations describe heat conduction in laminates by mean of partial differential equations with smooth and slowly varying functions. This paper describes a certain extension of the unified tolerance modelling procedure, which makes it possible to analyze specific problems of heat conduction in laminates with a transversal gradation of effective properties.

Теоретичний Та Експериментальний Аналіз Розподілу Температур У Багатошаровій Структурі

“…W pracy wybrano wariant asymptotyczny modelowania tolerancyjnego cechujący się największą prostotą równań spośród wszystkich modeli tolerancyjnych (por, [4], [5], [7], [8]). Dla dwuwymiarowego stacjonarnego zagadnienia przewodnictwa ciepła w dwuskładniowej warstwowej periodycznej strukturze rozkład temperatury wyznacza się w ramach tego modelu ze wzoru 3 Równanie (6) z dekompozycją pola temperatury (3) wraz z warunkami brzegowymi reprezentują wariant asymptotyczny modelowania tolerancyjnego, opisujący dwuwymiarowe przewodnictwo ciepła w analizowanej strukturze.…”

On Determination of the Effective Thermal Conductivity of a Bundle of Steel Bars Using the Krischer Model and Considering Thermal Radiation

“…Due to its cellular structure, a bundle of bars (an example of which is presented in Figure 1 ) is a granular two-phase medium, where one phase is steel and the other phase is gas, which fills in spaces that occur between individual bars. For this reason, the basic thermal property of a charge is the effective thermal conductivity k ef , which is commonly used in the theory of porous [ 2 , 3 ] and nonhomogeneous [ 4 , 5 , 6 ] media. Knowledge of the value of this coefficient is necessary to determine the heating time of the heat-treated bars [ 7 , 8 ].…”

Analysing the Contact Conduction Influence on the Heat Transfer Intensity in the Rectangular Steel Bars Bundle

“…It is possible to simplify this description considerably by introducing the notion of the effective thermal conductivity kef. This parameter is commonly used in the theory of porous [6][7][8] and nonhomogeneous [9][10][11] media. By using the effective thermal conductivity it is possible to describe a transient heat transfer within a bundle of bars [12].…”

“…where: h-convection heat transfer coefficient, ε-bundle surface emissivity (ε = 0.7), σ-Stefan-Boltzmann constant, TF and T0-thermodynamic temperatures of the furnace and bundle surface. It was assumed that the furnace temperature is rising up to the final value as the function of time described by Equation (11) = 20 + 0.173 (11) where time τ is expressed in seconds. Upon reaching the final value, for the rest of the process temperature, tF is kept constantly at this level.…”