Heat conduction approximate solutions

The general heat-conduction equation, along with the familiar diffusion equation, are both consequences of energy conservation and, like we have just seen for the Navier-Stokes equation, require a first-order approximation to the solution of Boltz-man s equation. 

Homogeneous through execution schemes are quite applicable in the cases where the diffusion coefficient is found as an approximate solution of other equations. For instance, such schemes are aimed at solving the equations of gas dynamics in a heat conducting gas when the diffusion coefficient depends on the density and has discontinuities on the shock waves. 

Only a finite difference numerical solution can give exact results for conduction. However, often the following approximation can serve as a suitable estimation. For the unsteady case, assuming a semi-infinite solid under a constant heat flux, the exact solution for the rate of heat conduction is... 

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. 

For mi > 0 the time derivative was approximated by a forward difference ratio and the space derivatives by central difference ratios. It is known (E4, F2) that the solution to the difference approximation to the heat conduction equation will not be stable unless the ratio (Ay/Az 2) < 5. Stability implies that small perturbations introduced by rounding off or truncation are damped out instead of being magnified. Taking this ratio to be 5, the difference approximation to Eq. (99)... 

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

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. 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

Assumptions 1 The egg is spherical in shape with a radius of r, = 2.5 cm. 2 Heat conduction in the egg is one dimensional because or thermal symmetry about the midpoint. 3 The thermal properties of the egg and the heat transfer coefficient are constant, 4 The Fourier number is t > 0,2 so that the one-term approximate solutions are applicable. 

CoWsider a plane wall of thickness 2L initially at a uniform temperature of T , as shown in Fig. 4—1 In. At lime t = 0, the wall is immersed in a fluid at temperature 7 and is subjected to convection heal transfer from both sides with a convection coefficient of h. The height and the widlh of the wall are large relative to its thickness, and thus heat conduction in the wall can be approximated to be one-dimensional. Also, there is thermal symmetry about the inidplane passing through.x = 0, and thus the temperature distribution must be symmetrical about tlie midplane. Therefore, the value of temperature at any -.T value in - A "S. t 0 at any time t must be equal to the value at f-.r in 0 X Z, at the same time. This means we can formulate and solve the heat conduction problem in the positive half domain O x L, and then apply the solution to the other half. 

Coefficients used in the one-term approximate solution of transient one-dimensional heat conduction in plane walls, cylinders, and spheres (B = hUk for a plane wall of thickness ZL, and Bi = hrjkfor a cylinder or sphere of radius r )... 

SOLUTION A long cylindrical shaft is allowed to cool slowly. The center temperature and the heat transfer per unit length are to be determined. Assumptions 1 Heat conduction in the shaft is one-dimensional since it is long and it has thermal symmetry about the centerline. 2 The thermal properties of the shaft and the heat transfer coefficient are constant. 3 The Fourier number is t > 0.2 so that the one-term approximate solutions are applicable. Properties The properties of stainless steel 304 at room temperature are k - 14,9 W/m °C, p = 7900 kg/m r. = 477 J/kg X, and a = 3.95 X 10 mVs (Table A-3). More accurate results can be oblained by using properties at average temperature. 

Using the one-term approximation, the solutions of onedimensional transient heat conduction problems are expressed analytically as... 

So far we Kave mostly considered relatively simple heat conduction problems Involving simple geoineiries with simple boundary conditions because only such simple problems can be solved analytically. But many problems encountered in practice involve complicated geometries with complex boundary conditions or variable properties, and cannot be solved analytically. In such cases, sufficiently accurate approximate solutions can be obtained by computers using a numerical method. 

In Section S-3 we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Many heat transfer problems encountered in practice can be approximated as being one-dimensional, but this is not always the case. Sometimes we need to consider heat transfer in other directions as well when the variation of temperature in other directions is significani. In this section we consider the numerical formulation and solution of two-dimensional steady lieat conduclion in rectangular coordinates using the finite difference method. The approach presented below can be extended to three-dimensional cases. 

In the finite difference method, the derivatives dfl/dt, dfl/dx and d2d/dx2, which appear in the heat conduction equation and the boundary conditions, are replaced by difference quotients. This discretisation transforms the differential equation into a finite difference equation whose solution approximates the solution of the differential equation at discrete points which form a grid in space and time. A reduction in the mesh size increases the number of grid points and therefore the accuracy of the approximation, although this does of course increase the computation demands. Applying a finite difference method one has therefore to make a compromise between accuracy and computation time. 

What is the ratio of heat flows by conduction in a cube of side length a in the three coordinate directions As an approximate solution of this part of the question, presume that the heat flows in the three coordinate directions are independent of each other. 

---