General Solution of the Transport Equation

The N 1 eigenvectors of the matrix A form a complete set, and A has A - - 1 positive eigenvalues, which forms a canonical matrix under similarity transformation, and is consequent nonsingular. Thus, Ct t) and Q t) can be expanded as a linear combi- 

Substituting (12) and (13) into (lb), and using the orthogonality relation (6), one obtains 

One can, therefore, solve the time-dependent random-walk problems if the transition probabilities ior every possible configuration are known. As long as N is not laige, the eigenvectors are, therefore, obtainable by using electronic computers. However, the particular solutions with a priori transition probabilities for large N are not, in general, obtainable. 

We shall present here some potential profiles which give analytic expressions of ip s and s for any N, and shall generalize these results to include broader multi-barrier problems. Nearest-neighbor transition probabilities only are involved throughout the following considerations except for the perturbation theory treatment the inclusion of the next-nearest neighbor may be introduced by the perturbation method as discussed later. 

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It is probably unnecessary to emphasize for anyone who has considered, for instance, Milne s problem that the knowledge of the general solution of the transport equations in all the distinct media contained in the reactor does not yet constitute a solution of the reactor equations. The adjustment of the boundary conditions at the interfaces between media can be, and often is, more difficult than the determination of the general solution in the homogeneous media. It is the adjustment of boundary conditions which is rendered much simpler if the homogeneous parts of the reactor have shapes of high symmetry. 

Conclusions. The discrete Sn method represents, from the point of view of the numerical analyst, a general solution of the transport equation. It includes as special cases the earlier methods which have been referred to, also based on discrete ordinates but of limited applicability. In practice the Sn method has proved both accurate and versatile, especially in the formulation given here, and has been tested extensively in geometries of one dimension and recently also in the case of finite cylinders (two space variables). With a better mathematical understanding of the processes involved, one can undoubtedly make further progress here and perhaps solve more comphcated particle flow problems than those represented by the transport equation. 

To preview the results somewhat, it will be shown that the general form of the transport equations contains expressions for the property flux variables (momentum flux P, energy flux q, and entropy flux s) involving integrals over lower-order density functions. In this form, the transport equations are referred to as general equations of change since virtually no assumptions are made in their derivation. In order to finally resolve the transport equations, expressions for specific lower-ordered distribution functions must be determined. These are, in turn, obtained from solutions to reduced forms of the Liouville equation, and this is where critical approximations are usually made. For example, the Euler and Navier-Stokes equations of motion derived in the next chapter have flux expressions based on certain approximate solutions to reduced forms of the Liouville equation. Let s first look, however, at the most general forms of the transport equations. 

The above system of equations admits a similarity solution in terms of erf function. The growth rate and the position of the interface are usually calculated from the equation obtained from the solution of the transport equations coupled with the boundary conditions. The resulting relation is generally a complex transcendental equation whose eigen values A are related to the interface location as h(t) = IX-y/Dt. 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The concentrations of the reactants and reaction prodncts are determined in general by the solution of the transport diffusion-migration equations. If the ionic distribution is not disturbed by the electrochemical reaction, the problem simplifies and the concentrations can be found through equilibrium statistical mechanics. The main task of the microscopic theory of electrochemical reactions is the description of the mechanism of the elementary reaction act and calculation of the corresponding transition probabilities. 

Solutions of the combined equations of mass transfer, kinetics and electrochemical transport expressed in terms of the limiting current, i generally are of the form... 

The general solution of the system of transport equations for electrons and holes permits the photopotential of an open circuit to be calculated. The assumption that the total potential change due to illumination occurs in the space-charge region of a semiconductor, i.e., equilibrium value of , and that the exchange currents and... 

In the foregoing, the expressions needed to account for mass transport of O and R, e.g. eqns. (23), (27), (46), and (61c), were introduced as special solutions of the integral equations (22), giving the general relationship between the surface concentrations cG (0, t), cR (0, t) and the faradaic current in the case where mass transport occurs via semi-infinite linear diffusion. It is worth emphasizing that eqns. (22) hold irrespective of the relaxation method applied. Of course, other types of mass transport (e.g. bounded diffusion, semi-infinite spherical diffusion, and convection) may be involved, leading to expressions different from eqns. (22). 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Owing to the nature of thin films generated by both spinning disks and spinning cones, transport by diffusion can become a significant method of transport of both heat and mass within the film. In the case of heat transfer this is especially true and will be the dominant mode of transport within the film. The equations for transport by diffusion have been studied and solved for many simple systems. One of the most comprehensive studies of this process was given by Crank in which the equations for diffusion are examined for several generalized systems. In the solution of the diffusion equation one parameter is of paramount importance that is, the Fourier number, Fo. This can be expressed as... 

Quantitative optimization or prediction of the performance of photoelectrochemical cell configurations requires solution of the macroscopic transport equations for the bulk phases coupled with the equations associated with the microscopic models of the interfacial regions. Coupled phenomena govern the system, and the equations describing their interaction cannot, in general, be solved analytically. Two approaches have been taken in developing a mathematical model of the liquid-junction photovoltaic cell approximate analytic solution of the governing equations and numerical solution. 

As usual, the eigenvalue B is to be computed from (7.254) this may be demonstrated by substituting (7.258) into (7.250). The above expression, then, is an exact solution to the transport equation. However, it has no direct physical application for two reasons (1) it applies only to the infinite medium and (2) not all the solutions (7.258), for the various values of B from (7.254), are positive. In spite of these facts, (7.258) can be used to construct physically useful solutions. The technique is to replace the actual finite body by an infinite medium for which (7.258) gives mathematically exact solutions and so construct the solution that the regions of the infinite space, wherein < 0, are outside the boundaries of the real system. This approach is analogous, then, to the method of images which was used for the Fermi age solution to the slab problem (see Sec. 6.4b). In general, this method cannot yield an exact solution to the medium with a finite dimension since it is not possible to satisfy... 

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