Semi-infinite solid, heat conduction

The application of Fourier s conduction equation and hyperbolic heat conduction equation to the transient heating of semi-infinite solid is discussed in the following section. 

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. 

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... 

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... 

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. 

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. 

Transient Heat Conduction in Semi-Infinite Solids 240 Contact of Two Semi-Infinite Solids 245... 

TRANSIENT HEAT CONDUCTION IN SEMI-INFINITE SOLIDS... 

Heat conduction in a semi-infinite solid is governed by the thermal conditions imposed on the exposed surface, and thus the solution depends strongly on the boundary condition at x = 0. Below we present a detailed analytical solution for the case of constant temperature T, on the surface, and give the results for other more complicated boundary conditions. When the surface temperature is changed to at f = 0 and held constant at that value at all tirne.s, tli formulation of the problem can be expressed as... 

Transient Heat Conduction in Semi-Infinite Solids... 

Yuen, W. W. and Lee, S. C. (1989) Non-Fourier Heat Conduction in a Semi-Infinite Solid Subjected to Oscillatory Surface Thermal Disturb ices, Journal of Heat Transfer, Vol. Ill, pp. 178-181. 

Zubair, A.S., and Aslam Chaudhry, M. (1996) Heat Conduction in a Semi-Infinite Solid due to Time-Dependent Laser Source, Int. J. Heat Mass Transfer, Vol. 39, pp. 3067-3074. 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

The viscous liquid is passed at low velocity through the central tube. Portions of this liquid adjacent to the heat-transfer surface are essentially stagnant, except when disturbed by the passage of the scraper blade. Heat is transferred to the viscous liquid by unsteady-state conduction. If the time between disturbances is short, as it usually is, the heat penetrates only a small distance into the stagnant liquid, and the process is exactly analogous to unsteady-state heat transfer to a semi-infinite solid. 

The particular solution of Eq. (21.41) is the same as that for transient heat conduction to a semi-infinite solid, Eq. (10.26). 

M. F. Modest and H. Abakians, Heat Conduction in a Moving Semi-Infinite Solid Subjected to Pulsed Laser Irradiation, J. Heal Transfer, 108, pp. 597-601,1986. 

Here, the concentration at x = 0 always remains constant contrary to the previous example, where a fixed concentration is introduced once. This problem is analogous to transient conduction in semi-infinite solid with constant surface temperature boundary condition. The detailed solution procedure can be found in regular heat transfer book. The solution for the above problem can be obtained by using f The governing equation in partial differential form... 

The application of hyperbolic heat conduction equation is shown here using a pulse heating example of semi-infinite solid. The wall at x = 0 is impulsively stepped to a temperature (Figure 8.4). 

A convective boundary condition was assumed at the mold-melt interface to take the contact resistance into account. Thus, heat from the polymo- must first be transferred by convection across the into-face and then conducted through the mold. The equation for transient conduction in a semi-infinite solid is given by [3] d T 1 dT dz a dt... 

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. 

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... 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

---