Basic Equations and Boundary Conditions

To yield a constitutive relation for the mass flux of particles, it is convenient to begin with the equation of motion of a single particle, which is expressed by 

For a fully developed flow, dUp/dt is zero and, thus, Eq. (11.60) becomes 

Since a fully developed motion excludes deposition of the solid particle phase under field forces (i.e., zero deposition rate), the flux due to diffusion must be equal to the flux due to relaxation under the field force, i.e., 

It is seen that Eqs. (11.65) and (11.66) are not independent because both can be integrated to the form 

Therefore, we only need to investigate the axial component of the momentum equations. Equation (11.59) indicates that the electrostatic force does not contribute to the momentum transfer in the axial direction. For high Reynolds number and Froude number of the gas 

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The basic equation and boundary conditions for the symmetrical fluctuations are the same as those for the asymmetrical fluctuations except for the superscript s. The diffusion equation is written in the form... 

The fluid dynamics of bubble column reactors is very complex and several different CFD models may have to be used to address critical reactor engineering issues. The application of various approaches to modeling dispersed multiphase flows, namely, Eulerian-Eulerian, Eulerian-Lagrangian and VOF approaches to simulate flow in a loop reactor, is discussed in Chapter 9 (Section 9.4). In this chapter, some examples of the application of these three approaches to simulating gas-liquid flow bubble columns are discussed. Before that, basic equations and boundary conditions used to simulate flow in bubble columns are briefly discussed. 

A basic waveguide structure, which is sketched in Fig. 1, is composed of a guiding layer surrounded by two semi-infinite media of lower refractive indices. The optical properties of the stmcture are described by the waveguiding layer refractive index Hsf, and thickness t, and by the refractive indices of the two surrounding semi-infinite media, here called (for cover) and (for substrate). Application of Maxwell s equations and boundary conditions leads to the well-known waveguide dispersion equation [6] ... 

The basic governing equations (2.1 to 2.10) along with appropriate constitutive equations and boundary conditions govern the flow of fluids, provided the continuum assumption is valid. To obtain analytical solutions, the governing equations are often simplified by assuming constant physical properties and by discarding unimportant... 

Traditionally, a simulation is constructed as follows (1) the real world problem is analysed and cast in terms of mathematical expressions (e.g. the differential equations for mass transport and the initial and boundary conditions for what happens at the electrode surface and in the bulk) (2) the expressions are then rewritten in dimensionless form (3) the continuous variables (typically, concentration, space and time) are discretised (4) the differential equations and boundary conditions are disaetised (5) an algorithm is chosen and a program written in Fortran, Pascal, Basic or C (6) the simulation is tested with conditions yielding a known solution (7) finally, the simulation is applied to the conditions of interest. 

Equations describing the CCL impedance are Equations 5.91 and 5.92 (the section Basic Equations ). However, boundary conditions to these equations now differ from those discussed in the section Basic Equations. Here, the solution to the system (5.91) and (5.92) is subjected to more general boundary conditions ... 

Equation 3-10 is the most basic diffusion equation to be solved, and has been solved analytically for many different initial and boundary conditions. Many other more complicated diffusion problems (such as three-dimensional diffusion with spherical symmetry, diffusion for time-dependent diffusivity, and... 

Note that neither initial nor boundary conditions have been applied yet. The above equation is the general solution for infinite and semi-infinite diffusion medium obtained from Boltzmann transformation. The parameters a and b can be determined by initial and boundary conditions as long as initial and boundary conditions are consistent with the assumption that C depends only on q (or ). Readers who are not familiar with the error function and related functions are encouraged to study Appendix 2 to gain a basic understanding. 

The preceding discussion thus indicates that the basic equations, with their initial and boundary conditions, will determine whether a ray of comparatively large transverse dimensions is to be represented by a plane wave or by a beam of particle-like photon wavepackets. Thus two general cases should be observed ... 

The basic set of differential material-balance equations for the various species in the column can also be written in terms of the h instead of Xi and t/i. This new set of differential equations reveals a particular property of the H-function roots no root values other than those appearing in the initial and boundary conditions can arise anywhere in the column. The column behavior thus is completely described by the trajectories of the initial and boundary values of the roots. (The same is not true for concentrations, as the behavior of species 2 in the example in Figure 4 has shown.)... 

To analyze a physical problem analytically, we must obtain the governing equations that model the phenomenon adequately. Additionally, if the auxiliary equations pertaining to initial and boundary conditions are prescribed those are also well-posed, then conceptually getting the solution of the problem is straightforward. Mathematicians are justifiably always concerned with the existence and uniqueness of the solution. Yet not every solution of the equation of motion, even if it is exact, is observable in nature. This is at the core of many physical phenomena where ohservahility of solution is of fundamental importance. If the solutions are not observable, then the corresponding basic flow is not stable. Here, the implication of stability is in the context of the solution with respect to infinitesimally small perturbations. 

The conditions specified by Eq. (6.206) provide the conditions required to design the model, also called similarity requirements or modeling laws. The same analysis could be carried out for the governing differential equations or the partial differential equation system that characterize the evolution of the phenomenon (the conservation and transfer equations for the momentum). In this case the basic theorem of the similitude can be stipulated as A phenomenon or a group of phenomena which characterizes one process evolution, presents the same time and spatial state for all different scales of the plant only if, in the case of identical dimensionless initial state and boundary conditions, the solution of the dimensionless characteristic equations shows the same values for the internal dimensionless parameters as well as for the dimensionless process exits . 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

The original Danckwerts model has later been extended to relate the renewal rate s to many flow parameters, and to account for the existence of the micro scale flow of the fluid within the individual eddies. Further modifications relate to the fact that not all the penetrating eddies reach the whole way to the interface. Many model extensions have thus been developed based on the basic surface-renewal concept. A review of these models is given by Sideman and Pinczewski [135]. The various extensions are motivated by the inherent assumptions regarding, the governing equations, the boundary conditions, the age distribution function and/or the mean contact time. 

The basic solutions for the plate and the cylinder can be used to obtain solutions within rectangular plates, cuboids, and finite circular cylinders. The equations and the initial and boundary conditions are well known [4,11, 23, 28, 29, 38, 49, 56, 80, 87]. The solutions presented below follow the recent review of Yovanovich [151]. The Heisler [36] cooling charts for dimensionless temperature are obtained from the series solution ... 

An attempt to combine in a single dimensionless equation the numerous variables involved in extraction from a drop was reported recently. Lileeva and Smirnov (LI4) derived the dimensionless groups from the four basic differential equations (motion, continuity, kinetic diffusion, and convection) and the initial and boundary conditions. The eleven groups derived were then reduced to the following equation ... 

For the basic equations of coupled stress-flow analysis mentioned above, it is very difficult to solve them in closed-form. The transposition method of progression and integration can only be applied for problems of boundary value problems of simple geometry and boundary conditions. Therefore the finite element method (FEM) is used to solve the coupled partial differential equations in this paper. 

Basic differential equations for the dynamic analysis, together with the initial and boundary conditions, are summarized as follows ... 

Consider conservation laws for mass and momentum of viscous Newtonian liquids. In case of isothermal flow of incompressible liquid, it is necessary to add to these laws the basic equation (4.13) together with appropriate initial and boundary conditions. It is sufficient to determine the velocity distribution and stresses at any point of space filled with the liquid, and at any moment of time. If the flow is not isothermal, then in order to And the temperature distribution in liquid, we need to use the energy conservation. If, besides, the liquid is compressible, it is necessary to add the equation of state. 

Solutions of the differential equation systems including initial and boundary conditions as discussed in the previous sections are generally provided by simulation. Mathematical models describing the basic physico-chemical processes are numerically resolved [25]. LSV and CV are among the main methods treated in electrochemical simulations. Further information on this subject can be found in Chapter 1.3 of this Volume and (with mechanistic background) in Chapter 1 of Volume 8. 

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