Conduction, heat boundary-layer methods

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

The theoretical estimate of Slack has predicted k = 1.7 W/(cm K) for room temperature thermal conductivity of GaN [12], The thermal conductivity of GaN layers grown on sapphire substrates by the HVPE method [23] was measured by Sichel and Pankove using the heat flow method [24], The room temperature thermal conductivity was k = 1.3 W/(cm K). Sichel and Pankove attributed the smaller value to high impurity content, at least 1018 cm 3, and the presence of small angle grain boundaries. 

Conduction is treated from both the analytical and the numerical viewpoint, so that the reader is afforded the insight which is gained from analytical solutions as well as the important tools of numerical analysis which must often be used in practice. A similar procedure is followed in the presentation of convection heat transfer. An integral analysis of both free- and forced-convection boundary layers is used to present a physical picture of the convection process. From this physical description inferences may be drawn which naturally lead to the presentation of empirical and practical relations for calculating convection heat-transfer coefficients. Because it provides an easier instruction vehicle than other methods, the radiation-network method is used extensively in the introduction of analysis of radiation systems, while a more generalized formulation is given later. 

The semianalytical method developed earlier can be used to solve partial differential equations in composite domains also. Mass or heat transfer in composite domains involves two different diffusion coefficients or thermal conductivities in the two layers of the composite material.[6] In addition, even in case of solids with a single domain and constant physical properties, the reaction may take place mainly near the surface. This leads to the formation of boundary layer near one of the boundaries. In this section, the semianalytical method developed earlier is extended to composite domains. 

Summarizing, we should note that the methods presented in the present section can be applied without any modifications to heat exchange problems, because temperature distribution is described by an equation similar to the diffusion equation. The boundary conditions are also formulated in a similar way. One only has to replace D by the coefficient of thermal diffusivity, and the number Peo - by Pej-. The corresponding boundary layer is known as the thermal layer. Detailed solutions of heat conductivity problems can be found in [6]. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

Taking account of the boundary conditions, this equation can be integrated by elementary methods at each given instant and in a given layer. This determines the function late stage the plane field may be represented in the form H2 = curl (n ), where n = (0,0,1). After this the function (which now coincides with the vector potential component Az) is also subject to an equation of the heat conduction type. Consequently, H2 decays asymptotically. 

---