Semi-infinite solid

Semi-infinite solid Semi-infinite solid... 

Figure 26.18 shows a theoretical calculation of the temperature rise along the contact area of a mbber strip under a constant load and speed over a semi-infinite dry mbber solid [30]. The figure... 

FIGURE 26.18 Theoretical temperature rise in the contact area of a pad sliding over a semi-infinite solid for different depths from the surface. Width 2b 2 mm, speed 3 m/s, pressure 2 Mp, p—l, heat conductivity 0.15 W/m/K, heat diffusivity 10 " m /s. 

The thermally thin case holds for d of about 1 mm. Let us examine when we might approximate the ignition of a solid by a semi-infinite medium. In other words, the backface boundary condition has a negligible effect on the solution. This case is termed thermally thick. To obtain an estimate of values of d that hold for this case we would want the ignition to occur before the thermal penetration depth, <5T reaches x d. Let us estimate this by... 

Hence, we might expect solids to behave as thermally thick during ignition and to be about 8 cm for t-lg = 300 s and about 2.5 cm for 30 s. Therefore, a semi-infinite solution might have practical utility, and reduce the need for more tedious finite-thickness solutions. However, where thickness and other geometric effects are important, such solutions must be addressed for more accuracy. 

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... 

The semiconductor support can be treated in a similar manner (Bose and Foo 1974), since the Greenians G2 and g2 of the infinite and semi-infinite solids, respectively, are also linked by the Dyson equation, i.e.,... 

Approach to the Simulation of Elastic, Semi-Infinite Solids. 

The diffusion coefficients for Rb, Cs and Sr in obsidian can be calculated from the aqueous rate data in Table 1 as well as from the XPS depth profiles. A simple single-component diffusion model (9j characterizes onedimensional transport into a semi-infinite solid where the diffusion coefficient (cm2-s 1) is defined by ... 

Assuming one-dimensional heat transfer is the mode of the solid bed heating due to the heating of the film by conduction and dissipation, the temperature will only change in the y direction. The same assumption that was made by Tadmor and Klein will be made here that the heat transfer model is a semi-infinite slab moving at a velocity Vsy c (melting velocity) with the boundary conditions T(0) = and j(-oo) = 7 , This assumption is not strictly correct because it will also be proposed that the other four surfaces are melting. The major error will occur at the corners of the solid bed. is the velocity of the solid bed surface adjacent to Film C as it moves toward the center of the solid bed in the y direction. 

Figure 14. Simple model demonstrating how adsorption and surface diffusion can co-Urnit overall reaction kinetics, as explained in the text, (a) A semi-infinite surface establishes a uniform surface coverage Cao of adsorbate A via equilibrium of surface diffusion and adsorption/desorption of A from/to the surrounding gas. (b) Concentration profile of adsorbed species following a step (drop) in surface coverage at the origin, (c) Surface flux of species at the origin (A 4i(t)) as a function of time. Points marked with a solid circle ( ) correspond to the concentration profiles in b. (d) Surface flux of species at the origin (A 4i(ft>)) resulting from a steady periodic sinusoidal oscillation at frequency 0) of the concentration at the origin.

Let us imagine that the inhnite periodic 3D solid discussed in Section 1.7 is separated into two halves, leading to two semi-inhnite 3D solids, preserving their 3D bulk periodicity but becoming aperiodic in the direction perpendicular to the generated surfaces. Because the translation symmetry is lost in this direction, the Bom-von Karman boundary conditions can no longer be applied, hence the apparent paradox that a semi-infinite problem becomes more complex than the infinite case. This fact inspired W. Pauli to formulate his famous sentence God made solids, but surfaces were the work of the Devil. 

The solution to this partial differential equation depends upon geometry, which imposes certain boundary conditions. Look np the solution to this equation for a semi-infinite solid in which the surface concentration is held constant, and the diffusion coefficient is assumed to be constant. The solution should contain the error function. Report the following the bonndary conditions, the resulting equation, and a table of the error function. 

The difference between a specimen that is effectively a semi-infinite halfspace and one that has a boundary at a finite depth is illustrated in the V(z) curves of two glass specimens in Fig. 10.7. The dashed curve is V(z) for a microscope slide 2 mm thick. The solid curve is V(z) for a glass cover-slip 0.11 mm thick. In a thin plate such as a cover-slip Lamb waves can be excited (Auld 1973). At low frequencies the lowest-order symmetric mode is like a longitudinal wave, except that, because the plate is thin, the modulus governing the velocity is cn — c2n/cn (Table 6.1). If the frequency is increased until... 

In the absence of an external field, electrons in the metal are confronted by a semi-infinite potential barrier (upper solid line in Fig. la), so that escape is possible only over the barrier. The process of thermionic emission consists of boiling electrons out of the Fermi sea with kinetic energy > x + M- The presence of a field F volts/cm. at and near the surface modifies the barrier as shown. It follows from elementary electrostatics that the potential V will not be noticed by electrons sufficiently far in the interior of the metal. However, electrons approaching the surface are now confronted by a finite potential barrier, so that tunneling can occur for sufficiently low and thin barriers. 

Fig. 6.3 Nondimensional axial and radial velocity profiles for the axisymmetric stagnation flow in the semi-infinite half plane above a solid surface. The flow is approaching the surface axially (i.e., u < 0) and flowing radially outward (i.e., V > 0). The temperature profile, which is the result of solving the thermal-energy equation, is discussed in Section 6.3.6.

For many applications, like chemical-vapor-deposition reactors, the semi-infinite outer flow is not an appropriate model. Reactors are often designed so that the incoming flow issues through a physical manifold that is parallel to the stagnation surface and separated by a fixed distance. Typically the manifolds (also called showerheads) are designed so that the axial velocity u is uniform, that is, independent of the radial position. Moreover, since the manifold is a solid material, the radial velocity at the manifold face is zero, due to the no-slip condition. One way to fabricate a showerhead manifold is to drill many small holes in a plate, thus causing a large pressure drop across the manifold relative to the pressure variations in the plenum upstream of the manifold and the reactor downstream of the manifold. A porous metal or ceramic plate would provide another way to fabricate the manifold. 

Lightfoot (L5, L6) considered the solidification of a semi-infinite steel mass in contact with a semi-infinite steel mold, where the thermal properties of the phases were taken to be identical. In a later paper, (L7) different thermal constants for the liquid and solid metal and for the mold were assumed. With uniform initial phase temperatures, it is seen that all boundaries of the system are immobilized in jj-space. Yang (Y2) rederived this result and further extended the application of the similarity transformation to three-region problems with induced motion. An example is the condensation of vapor, as a result of sudden pressurization, in a tank with relatively thick walls. 

Fig. 4. Rate of melting of semi-infinite solid for various values of mi (L4). Reproduced by permission of Quarterly of Applied Mathematics.

Citron (C4) generalizes Landau s derivation of the steady-state melting rate of a semi-infinite solid with instantaneous removal of the melt to temperature-dependent thermal conductivity and specific heat, expressible in the form... 

This leads to a powerful method when the densities, specific heat, and thermal conductivity can be assumed to have constant average values throughout the entire solid-liquid region. In this case, the moving boundary can be considered to be a source surface. As an example, consider the freezing of a semi-infinite region occupied by liquid with arbitrary initial and surface temperature and conditions. In this case Eq. (159) becomes... 

Gutowski Method. An alternative approach is given by Gutowski (G10), who considers the particular case of a radiation boundary condition for the solidification of a semi-infinite liquid mass. The ambient temperature is considered to be constant, and the liquid is initially at the fusion temperature. The heat equation in the solid phase is formulated in the integral form... 

As pointed out in the previous section, melting can often be modeled in terms of simple geometries. Here we analyze the transient conduction problem in a semi-infinite solid. We compare the solutions of this problem, assuming first (a) constant thermophysical properties, then (b) variable thermophysical properties and finally, and (c) a phase transition with constant thermophysical properties in each phase. These solutions, though useful by themselves, also help demonstrate the profound effect of the material properties on the mathematical complexities of the solution. 

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... 

Fig. E5.2 Temperature profiles in a semi-infinite solid with a step change in temperature at the boundary.

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