Solution of the Transport Equations

Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

Aquifers which are closely connected to a river (as for the case of Groundwater System S) may be influenced by abrupt input variations that are driven by the corresponding concentration changes in the river. In order to analyze the resulting concentrations in the aquifer, the solution of the transport equation (Eq. 25-10) will be discussed for different time-dependent input concentrations at x = 0, Cin(t). In the river typical time scales of change are of the order of minutes (in case of an accidental spill) to days, but seasonal variations also exist for some chemicals. In contrast, transport within the aquifer is slower than most riverine concentration variations. The question arises how the river dynamics are transmitted to the groundwater. Three different cases are discussed ... 

Solution of the transport equation including the chemical reactions (one equation system for each species to be solved)... 

Fig. 26 Numeric dispersion and oscillation effects for the numeric solution of the transport equation (after Kovarik, 2000)...

The definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds e = Dg eI may be obtained from Maxwell s equation... 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

Given the difficulties facing the analytical approach, we must turn to approximate methods for the solution of the transport equation. Retaining, for the moment, the idea of a first-order recombination, we find, as above, for < W... 

Trying to overcome the limitations of the Gaussian plume assumptions, Lamb presented a Green s function approach to the solution of the transport equation with both area sources at the ground and volumetric sources caused by reactions (17), Two other features included in the... 

We must emphasise that the combination of the boundary condition of type (2.36) with the solution of the transport equation (diffusion equation) needed to calculate the subsurface concentration, is not trivial. In Section 4.4. the complexity of this type of models is demonstrated in detail. 

Finally it should be mentioned that a direct numerical solution of the transport equations will allow us to obtain more exact quantitative results. This way of solving the complex partial differential equation system is not trivial. First results are obtained by MacLeod Radke... 

We shall restrict the comparison to the flux and birth-rate density formulations of integral transport theory and to two sets of distribution functions one consists of the flux and source importance function, and the other set consists of the solutions of the transport equations in the formulation under consideration. 

Breit felt that the work on chain reactions should be extended. For one, he asked Teller (who was at Colmnbia for that year) and me to visit him in Washington and we discussed theoretical questions relating to the chain reaction. It was during this time that Teller conceived the idea of the fissionability of rare earths. Breit was interested in accurate solutions of the transport equation and I began to be interested in resonance absorption. Much of my acquaintance with the subject was acquired during this time (Dec. 1940-Jan. 1941). 

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. 

In the moderator the neutron density should satisfy (17) with the i determined from (15). The expression for Ki represents a higher order of approximation to the true solution of the transport equation than is justified by the simple slowing down picture of the previous section. We therefore shall use the simplified, first-order expression for ki (not involving the 2/5 aa/o- correction) in our comparison of the two calculations. In this approximation /cf = 3iVi crti<7i i/ri, and we have in the moderator. 

The model representing diffusion/reaction involved solution of the transport equations for each single pore simultaneously to give concentration profile in the pore network. The calculations related to capillary condensation were performed in the same way as for the Fickian model, described in Section lll.C. 

IV. Spatially dependent velocity distributions. When the spectrum is independent of position, the central problem is the determination of the energy-transfer cross sections. The calculation of the spectrum once these cross sections are known is a straightforward procedure. The cross-section aspect of the problem is both more difficult from a physics point of view, and more time consuming from the point of view of machine computation. This situation is reversed when we come to consider the spatial dependence of the slow neutron spectrum. The cross sections needed are the same ones that already have been computed for the infinite medium spectrum problem. The transport equation must now be solved in at least two variables, and in a form for which the existing approximate techniques are not very well adapted. The focus of the problem therefore shifts to the development of appropriate techniques for solution of the transport equation when the energy and position variables are coupled in such a way that neutrons can both gain and lose energy in a collision. 

Solutions of the transport equation by expansion in functions which have overlapping regions of energy. 

The most profitable alternative is probably a combination of Items Nos. 4 and 5 above. Below a certain energy of the order of five or ten times kT, an essentially new approximate method for solution of the transport equation should be developed. Above this energy, studies of the asymptotic behavior of the spectrum can be used to determine correction terms to be used in the ordinary multigroup formulation. The treatment of the asymptotic behavior of the spectrum has been studied extensively by Corngold [29]. The development of new techniques for the energy region below 10 kT is still at a very preliminary stage. At this point it is clear only that the flux must be expanded in a form... 

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. 

B. G. Carlson, Solution of the transport equation by approximations, Los Alamos Scientific Laboratory Report, LA-1891, 1955. 

The fact that the flux density of the electroinactive species, i 1, is zero simplifies significantly the solution of the transport equations. These species are in electrochemical equilibrium within the DEL, and hence their concentrations c, are related to the electric potential

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N.C. Francis et al., Variational Solutions of the Transport Equation, p/627 Proc. 2nd UN Conf. on Atomic Energy Geneva (1958) 16, 517 United Nations, New York. 

The spherical critical radii were computed by means of the DTF transport-theory code using an S approximation in the solution of the transport equation . Multigioup constants (16 energy groups) used in the DTF code were Obtained from the GAMTEC-n computer code. ... 

The measurement of mobilities usually requires fitting transient experimental data with formulas based on anal3 ical solutions of the transport equations. Such formulas are much easier to derive with flux-exphcit transport laws. 

CFD providers treat gas-phase combustion by using a mixture fraction model (Wang et al., 2006). The model is based on the solution of the transport equations for the fuel and oxidant mixture fractions as scalars and their variances. The combustion chemistry of the mixture fractions is modeled by using the equilibrium model through the minimization of the Gibbs free energy, which assumes that the chemistry is rapid enough to assure chemical equilibrium at the molecular level. Therefore, individual component concentrations for the species of interest are derived from the predicted mixture fraction distribution. 

For the numerical solutions of the transport equations developed above, KT involved in Equation 5.116 should be rewritten using the equilibrium constant K of the salt given in Equation 5.125. The derivation is too lengthy to be presented in detail. The readers who are interested in this subject should refer ... 

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