Abstract

In this paper we study the simple algorithms for modelling the heat transfer problem in two layer media. The initial model which is based on a partial differential equation is reduced to ordinary differential equations (ODEs). The increase of accuracy is shown if instead of first order ODE initial value problem ([4, 5]) the second order differential equations is taken. Such a procedure allows us to obtain a simple engineering algorithm for solving heat transfer equations in two layered domain of Cartesian, cylindrical (with axial symmetry) and spherical coordinates (with radial symmetry). In a stationary case the exact finite difference scheme is obtained. Šiame straipsnyje yra nagrinejami paprasti dvisluoksnes srities šilumos laidumo problemos modeliavimo algoritmai, keičiant diferencialines lygtis dalinemis išvestinemis i paprastas diferencialines lygtis. Parodoma, kad didesnio tikslumo pasiekimui, vietoje pirmos eiles paprastu diferencialiniu lygčiu pradinio uždavinio nagrinejamos antros eiles diferencialines lygtys. Ši proced ura leidžia gauti paprasta inžinerini dvisluoksies srities šilumos laidumo lygties sprendini stačiakampeje, cilindrineje (su ašiu simetrija) ir sferineje (su spinduline simetrija) koordinačiu sistemoje. Tiksli baigtiniu skirtumu schema buvo sudaryta stacionariam atvejui.

Highlights

  • Kangro conductivity in every layer, t[s] is the time, qk = qk(x, t) is the function of thermal sources, x[m] is the space coordinate, p = p(x) is given function depending on the system of coordinates: p = 1 in the Cartesian coordinates, p = x in cylindrical coordinates with an axial symmetry, p = x2 in spherical coordinates with a radial symmetry

  • Where the constants of heat transfer parameters c, ρ, λ are normalizing magnitudes and cρ/λ is used as appropriate factor to the time t and function q

  • The following stiff system of three ordinary differential equations (ODEs) of the second order is obtained for finding u0(t), u1(t), u2(t):

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Summary

The Mathematical Model

We obtain the initial–boundary value problem (1.1-1.4) for the heat transfer equation. The nonlinear function f (uN ) in the boundary condition (1.3) describes the radiation from heaters and convection, for example f (uN (L, t)) = αL(TL − uN (L, t)) + σ(T∗4 − u4N (L, t)), where α0. If α0 = αL = ∞, we have the first kind boundary conditions in the form u1(0, t) = T0, uN (L, t) = TL. In the case of homogeneous media we consider the following partial differential equation. Where the constants of heat transfer parameters c, ρ, λ are normalizing magnitudes and cρ/λ is used as appropriate factor to the time t and function q. In every layer the heat equation (1.1) can be presented in the following form uk(x, ∂x t) where Fk

The Exact 3-Points Finite-Difference Scheme
The Cartesian Coordinates
The Cylindrical Coordinates
The Spherical Coordinates
Conclusions

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