Transient heat conduction numerical methods

In the following sections we will discuss simple solutions, which are also important for practical applications, of the transient heat conduction equation. The problems in the foreground of our considerations will be those where the temperature field depends on time and only one geometrical coordinate. We will discuss the most important mathematical methods for the solution of the equation. The solution of transient heat conduction problems using numerical methods will be dealt with in section 2.4. 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the time variable and numerical in the spatial dimension) for linear parabolic partial differential equations using Maple, the method of lines and the matrix exponential. 

Wood WL, Lewis RW (1975) A comparison of time marching schemes for the transient heat conduction equation. Int J Numer Methods Eng 9 679-689... 

In this chapter we will deal with steady-state and transient (or non steady-state) heat conduction in quiescent media, which occurs mostly in solid bodies. In the first section the basic differential equations for the temperature field will be derived, by combining the law of energy conservation with Fourier s law. The subsequent sections deal with steady-state and transient temperature fields with many practical applications as well as the numerical methods for solving heat conduction problems, which through the use of computers have been made easier to apply and more widespread. 

The problem was solved by using a numerical method based mainly on the Dusinberre generalization of the increment method [9] applied to one-dimensional transient conduction. In fact, the heat transfer coefficient at the steel-rubber interface is very large, the surface rubber temperature changed very quickly, and consequently the initial temperature was taken as the arithmetic mean of the original surface temperatures of the mold and rubber. 

The analysis results in Figure 1 are from a numerical transient thermal and structural collapse analysis, where heat conduction is included and where the structural stiffness (and hence the load path) is computed for every time step in the solution process. This analysis is more accurate than the former because it includes the composite action of the three dimensional imevenly heated structure, where the load path is shed from the hot to the cold members. The method therefore represents the structural redimdancy and gives (more accurately) longer times to collapse than the lumped thermal mass method. 

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