Heat conduction problem

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

The computation time for calculations of energy losses to the ground can be quite significant because of the three-dimensional heat conduction problem. Simplified methods are given in ISO/FDIS 13370 1998. ... 

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

An one-point heat source. Of special interest is the nonstationary heat conduction problem in the situation when a heat source is located only at a single point x = under the agreement that at this point the solution of problem (l)-(3) satisfies the condition of conjugation... 

Cylindrically symmetric and spherically symmetric heat conduction problems. In explorations of many physical processes such as diffusion or heat conduction it may happen that the shape of available bodies is cylindrical. In this view, it seems reasonable to introduce a cylindrical system of coordinates (r, ip, z) and write down the heat conduction equation with respect to these variables (here x = r) ... 

The cylindrically symmetric heat conduction problem is reproduced by... 

Being concerned with the heat conduction problem in the case of a spherical symmetry, we are now in a position to produce on the same grounds the difference scheme associated with problem (8T)-(82) ... 

The stationary problem. To avoid misunderstanding, we concentrate primarily on the simplest problem, the statement of which is related to the stationary heat conduction problem with nonlinear sources ... 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

The following example, taken from Welty et al. ( 1976), illustrates the solution approach to a steady-state, one-dimensional, diffusional or heat conduction problem. 

Figure 5.243. The temperature profile is symmetrical for this heat conduction problem.

From Eq. (17) it is easy to see that the heat flux is proportional to thermal conductivity and temperature gradient and inversely related to film thickness. Not all heat conduction problems lead to such a simple solution. 

This chapter demonstrated the computational simplification that is possible in systems consisting of a one-dimensional chain of coupled reservoirs, which arise in diffusion and heat conduction problems. In such systems each equation is coupled just to its immediate neighbors, so that much of the work involved in Gaussian elimination and back substitution can be avoided. I presented here two subroutines, GAUSSD and SLOPERD, that deal efficiently with this kind of system. 

In Chapter 7 I showed how much computational effort could be avoided in a system consisting of a chain of identical equations each coupled just to its neighboring equations. Such systems arise in linear diffusion and heat conduction problems. It is possible to save computational effort because the sleq array that describes the system of simultaneous linear algebraic equations that must be solved has elements different from zero on and immediately adjacent to the diagonal only. 

Let us return to our discussion of the prediction of ignition time by thermal conduction models. The problem reduces to the prediction of a heat conduction problem for which many have been analytically solved (e.g. see Reference [13]). Therefore, we will not dwell on these multitudinous solutions, especially since more can be generated by finite difference analysis using digital computers and available software. Instead, we will illustrate the basic theory to relatively simple problems to show the exact nature of their solution and its applicability to data. 

In heat conduction problems solutions can often be expressed in terms of the error-function... 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

Unsteady state diffusion processes are of considerable importance in chemical engineering problems such as the rate of drying of a solid (H14), the rate of absorption or desorption from a liquid, and the rate of diffusion into or out of a catalyst pellet. Most of these problems are attacked by means of Fick s second law [Eq. (52)] even though the latter may not be strictly applicable as mentioned previously, these problems may generally be solved simply by looking up the solution to the analogous heat-conduction problem in Carslaw and Jaeger (C2). Hence not much space is devoted to these problems here. 

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

Sect. 5.4), the heat transfer process can be modeled using classical unsteady state heat conduction theory [142-144]. From the mathematical solutions to heat conduction problems, a thermal diffusivity can be extracted from measurements of temperatures vs. time at a position inside a gel sample of well-defined geometry. 

Instead of starting with a rigorous and mathematical development of the finite element technique, we proceed to present the finite element method through a solution of onedimensional applications. To illustrate the technique, we will first find a numerical solution to a heat conduction problem with a volumetric heat source... 

Transient Heat Conduction Problem Using Constant Strain Triangle... 

Three-Dimensional Transient Heat Conduction Problem With Convection... 

What would the constant strain finite element equations look like for the transient heat conduction problem with internal heat generation if you were to use a Crank-Nicholson time stepping scheme ... 

Green s identities for a 2D Laplace s equation (heat conduction) Here, we will demonstrate how to develop Green s identities for a two-dimensional heat conduction problem, which for a material with constant properties is described by the Laplace equation for the temperature, i.e.,... 

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential equation... 

Example 5.4 Melting of a Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature The Stefan-Neumann Problem The previous example investigated the heat conduction problem in a semi-infinite solid with constant and variable thermophysical properties. The present Example analyzes the same conduction problem with a change in phase. 

The preceding examples discuss the heat-conduction problem without melt removal in a semi-infinite solid, using different assumptions in each case regarding the thermophysical properties of the solid. These solutions form useful approximations to problems encountered in everyday engineering practice. A vast collection of analytical solutions on such problems can be found in classic texts on heat transfer in solids (10,11). Table 5.1 lists a few well-known and commonly applied solutions, and Figs. 5.5-5.8 graphically illustrate some of these and other solutions. 

As sintering proceeds and coalescence and densification occur, the overall heat conduction problem does not remain unaffected. Clearly, the effective thermophysical properties change, thereby influencing the overall temperature distribution and the local sintering problem as well. 

The last term on the right-hand side can be obtained by solving the temperature profile in the solid bed. Consider a small, x-direction portion of the film and solid [Fig. 5.12(b)], We assume the solid occupies the region y > 5 (where 3 is the local film thickness) and moves into the interface with constant velocity vs.v. The problem thus reduces to a onedimensional steady heat-conduction problem with convection. In the solid, a steady, exponentially dropping temperature profile develops. The problem is similar to that in Section 5.4. The equation of energy reduces to... 

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