# Determination of the stationary thermal state of simple geometry layered structureswith themperature dependent heat conductivity factors

## Date

2018-06-18

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## Abstract

An analytical-numerical method to determine the
one-dimensional stationary thermal state of simple geometry
multilayer structures for arbitrary dependences of heat-conductivity
factors on temperature is proposed (the multilayer bodies of
thermosensitive materials, referred to one of the classical orthogonal
coordinate systems (a,b,g ) are considered, the thermal state
caused by thermal load is characterized by a one-dimensional
stationary temperature field t (a) ).
The method is based on:
· utilization of elements of generalized functions algebra;
· approximation of temperature dependences of heatconductivity
factors of materials by piecewise constant
temperature functions –( ) ( ) ( ) ( ( ) ( ) ) ( ) 1 11() ( )mi i i i it j j ijl t t + S+ t t=» L = L +å L -L - ,where t0 = 0<t1 <t2 <...<tm <tk =tm+1; (i)Lj – with the given
accuracy corresponds to the value of the heat-conductivity factor
of the corresponding layer in the interval t j-1 < t < t j ,S+ (a ) = {0, a £ 0; 1, a > 0} ;· introduction into consideration the function of Kirchhoff
function type –( ) ( )( ) ( )0 =1= L å òtniiiJ t x N a dx ,where ( ) ( ) ( ) Ni a = S+ a -ai-1 - S+ a -ai .Therefore, the temperature field is determined by therelation( ) ( ) ( ) ( ) ( )1 1n nii i ii it J F J N a J N a= =é ù é ù= ê + ú ê L úêë úû êë úûå å ,where ( ) ( ) ( )11 21ink i i iiCf C K Q S-+== + +å + - a J a J a a isthe solution of the partially degenerate equation derived from
the heat equation in accordance with generalized functions
algebra, taking into account the perfect thermal contact of the
layers; С1 , С2 are the constants of integration, in the general
case determined from the system of two nonlinear algebraic
equations obtained from the boundary conditions; fk (a) , Ki ,
Qi are the functions and constants, determined by the
recurrence relations obtained in the work.
Approbation of the methodology by studying the stationary
thermal state of a two-layer cylinder is realized. The cases of
existence of a closed-form analytic solutions for the nonlinear heat
conduction problem are considered.

## Description

## Keywords

multilayer structures, solids of simple geometry, steady thermal state, temperature-dependent heat conduction factors, generalized functions

## Citation

Makhorkin M. Determination of the stationary thermal state of simple geometry layered structureswith themperature dependent heat conductivity factors / M. Makhorkin, I. Makhorkin, T. Makhorkina // Econtechmod. — Lublin, 2018. — Vol 7. — No 3. — P. 23–28.