Solid governing equations

The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

Another model consisting of elements in series and parallel is that attributed to Zener. It is known as the Standard Linear Solid and is illustrated in Fig. 2.41. The governing equation may be derived as follows. 

It may be observed that the governing equation of the standard linear solid has the form... 

As shown in the previous question, the governing equation for this type of Standard Linear Solid is given by ... 

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

Equation (A9) accounts for the change in the amount of daughter within the solid while Equation (A 10) accounts for that in the fluid. The transfer of the nuclide from the solid to the fluid phase is governed by the first term on the right side of both equations. Note that in this formulation, as given in Spiegelman and Elliott (1993), the time of melt transport is accounted for and depends on the physically based transport velocity. Like Equation (A8), both equations can be solved simultaneously by numerical integration or more sophisticated numerical method. 

Anderson and Jackson (1967, 1968, 1969) and Ishii (1975) have separately derived the governing equations for TFMs from first principles. Although the details of constructing the averaged equations are different, the final equations are essentially the same. The TFMs differ significantly from each other as different closures for the solid stress tensor are used. 

This equality will allow us to eliminate the a terms from the governing equations. We will carry these variables through the mathematical development, however, so that the results can be readily extended to account for solid solutions, even though we will not apply them in this manner. 

It is important to emphasize that the similarity behavior is not the result of approximations or assumptions where certain physical effects have been neglected. Instead, these are situations where the full two-dimensional behavior can be completely represented by a one-dimensional description for special sets of boundary conditions. Of course, in all finite-dimensional systems there are edge effects that violate that similarity behavior. By way of contrast, however, one may consider the difference between the governing equations used here and the boundary-layer equations for flow parallel to a solid surface. The boundary-layer equations are approximations in which certain terms are neglected because they are small compared to other terms. Thus terms are dropped even though they are not exactly zero, whereas here the mathematical reduction is accomplished because certain terms vanish naturally over nearly all of the domain (excluding edge effects). 

This chapter sets out the basic formulation and governing equations of mass-action kinetics. These equations describe the time evolution of chemical species due to chemical reactions in the gas phase. Chapter 11 is an analogous treatment of heterogeneous chemical reactions at a gas-solid interface. A discussion of the underlying theories of gas-phase chemical reaction rates is given in Chapter 10. 

This chapter describes the fundamental principles of heat and mass transfer in gas-solid flows. For most gas-solid flow situations, the temperature inside the solid particle can be approximated to be uniform. The theoretical basis and relevant restrictions of this approximation are briefly presented. The conductive heat transfer due to an elastic collision is introduced. A simple convective heat transfer model, based on the pseudocontinuum assumption for the gas-solid mixture, as well as the limitations of the model applications are discussed. The chapter also describes heat transfer due to radiation of the particulate phase. Specifically, thermal radiation from a single particle, radiation from a particle cloud with multiple scattering effects, and the basic governing equation for general multiparticle radiations are discussed. The discussion of gas phase radiation is, however, excluded because of its complexity, as it is affected by the type of gas components, concentrations, and gas temperatures. Interested readers may refer to Ozisik (1973) for the absorption (or emission) of radiation by gases. The last part of this chapter presents the fundamental principles of mass transfer in gas-solid flows. 

Note that /ep in Eq. (5.238) is replaced with /Ep for Eq. (5.240), where /Ep is the heat generated by thermal radiation per unit volume and Qap is the heat transferred through the interface between gas and particles. Thus, once the gas velocity field is solved, the particle velocity, particle trajectory, particle concentration, and particle temperature can all be obtained directly by integrating Eqs. (5.235), (5.237), (5.231), and (5.240), respectively. Since the equations for the gas phase are coupled with those for the solid phase, final solutions of the governing equations may have to be obtained through iterations between those for the gas and solid phases. 

Two- and Three-Dimensional Conduction The one-dimensional solutions discussed above can be used to construct solutions to multidimensional problems. The unsteady temperature of a rectangular, solid box of height, length, and width 2H, 2L, and 2 W, respectively, with governing equations in each direction as in (5-18), is... 

There is an inherent coupling of the behavior of the micro-scale variables to the behavior of macro-scale variables. This in itself presents complications when simrrlating these models. A few researchers have tried to address this problem of couphng of scales in these models. The solid state concentration term defined by the micro scale diffusion equation need to be coupled with the governing equations for the macro-scale to predict electrochemical behavior. Wang and co-workers used volume averaged equations and a parabolic profile approximation for solid-phase concentration. Subramanian et al. developed approximations assuming that the solid-state concentration inside the spherical electrode particle can be expressed as a polynomial in the spatial direction. 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

Consider a solid sphere of radius a, which is surrounded by another concentric spherical liquid envelope of radius yS, whose thickness is adjusted so that the porosity of the medium is equal to that of the model. The governing equation for the steady state mass transport in the fluid phase within the porous medium can be written in spherical coordinates as... 

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