The semi-infinite solid

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 5.3 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. 

Consider the semi-infinite solid shown in Fig. 4-3 maintained at some initial temperature T,. The surface temperature is suddenly lowered and maintained at a temperature T0, and we seek an expression for the temperature distribution in the solid as a function of time. This temperature distribution may subsequently be used to calculate heat flow at any x position in the solid as a function of time. For constant properties, the differential equation for the temperature distribution T(x, r) is... 

Fig. 4-5 Temperature distribution in the semi-infinite solid with convection boundary condition.

Using the slug-flow model, show that the boundary-layer energy equation reduces to the same form as the transient-conduction equation for the semi-infinite solid of Sec. 4-3. Solve this equation and compare the solution with the integral analysis of Sec. 6-5. 

If the surface of the semi-infinite solid is heated with constant heat flux % (Case b), the temperature distribution, from No. 7 in Table 2.3 is found to be... 

The detailed approach would require a meticulous summation of all molecule-molecule interactions. Consider the situation, shown schematically in Figure 3, of a deep layer of adsorbate, j molecules thick. The particular molecule shown in the ith layer experiences a potential given by the sum of interactions with the semi-infinite solid and the interactions with other adsorbate molecules, infinite in two dimensions, but finite in the third. The last was approximated in Equation 16 by treating the adsorbed film as normal liquid in state. The detailed analysis would involve complete potential functions, radial distributions, and structural information, and would then lead to some total expression for the free energy of an adsorbed film as a function of its thickness. From the corresponding partial molar free energy, a relationship to kT In P °/P could in principle be obtained. 

In order to overcome these problems, simplified systems are studied theoreti-eally. One such corresponds to approximating the semi-infinite solid through a finite cluster and then studying the interactions between this and the reactants. In this approximation a number of bonds that are present in the infinite solid have been cut and the resulting dangling bonds have therefore to be saturated through, e.g. hydrogen atoms. Nevertheless, finite-size effects as well as effects due to the saturated bonds may obscure the results of such calculations. 

Solutions of the Fick law for diffusion in the a -direction may be divided into those for infinite, semi-infinite, and finite solids. The infinite solid extends to infinity in -h and —x directions, the semi-infinite solid extends from a bounding plane at x = 0 to a = +cx), and the finite solid is bounded by planes a,t x = 0, x = I and sometimes x = l + h). All these solutions are exemplified by physical systems, which will be illustrated in the text. 

Figure 8.5 shows the normalized temperature distribution inside the semi-infinite solid after imposition of the temperature pulse from both Fourier s conduction and hyperbolic conduction... 

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