Simple Methods of Engineering Calculation for Solving Heat Transfer Problems

Abstract

There are well-known numerical methods for solving the initial‐boundary value problems for partial differential equations. We mention only some of them: finite difference method (FDM), finite element method (FEM), boundary element method (BEM), Galerkin type methods and others. In the given work FDM and BEM are considered for determination a distribution of heat in the multilayer media. These methods were used for the reduction of the 1D heat transfer problem described by a partial differential equation to an initial‐value problem for a system of ordinary differential equations (ODEs). Such a procedure allows us to obtain a simple engineering algorithm for solving heat transfer equation in multilayered domain. In a stationary case the exact finite difference scheme is obtained. An inverse problem is also solved. The heat transfer coefficients are found and temperatures in the interior layers depending on the given temperatures inside and outside of a domain are obtained.

Increasing of Accuracy for Engineering Calculation of Heat Transfer Problems in Two Layer Media

“…In different coordinates it is possible to enlarge the accuracy of the given algorithm, when second order derivatives are used instead of first order derivatives. 4. Such formulations have a big practical importance as compared to Cartesian coordinates, e.g.…”

“…If p(0) = 0 (cylindrical and spherical coordinates), then a 1 = α 0 = 0 and the difference equation (2.1) is defined as 4) and the equation (22) for k = 1 is given as…”

Simple Methods of Engineering Calculation for Solving Stationary 2 –D Heat Transfer Problems in Multilayer Media

“…Using a method of finite volumes [1][2][3][4], we would find the finite-difference scheme of 1  N equations on a joint of layers in the form…”

Simple Methods of Engineering Calculations for Solving Transfer Problems of Multi – Substances in Horizontal Layer

“…used, then we have the following system of ODE -s of first order with error term ) ( 4 h O [7,8] ( ) ( ) ( )…”

“…The calculation of heat transfer in a ground layer and in an atmosphere [10] is possible. It is possible to find out the algorithm for describing the temperature changes depending on time for surfaces on fibbers glasses at its heating without radiation [3] and with radiation [4,5], the algorithm for calculation of a temperature mode in a wall of a building with linings taking into account an opportunity of there heating [6,7]. Here it is very important to find simple algorithms for engineering -technical calculations, which are characterized by the sufficient accuracy of calculations, by the simplicity of calculations (with an opportunity of use of mathematical systems of a high level, for example, Mathematica, Maple, Matlab, etc.…”