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

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.