Age-diffusion theory

Corrections have to be made in the situations mentioned in 3. The case of closely packed assemblies has already been treated in 7. For lattices, where the distances are not small compared to the slowing length and also for the case that lump dimensions are larger than the moderator mean free path, modifications have been introduced by Pershagen and Carivik [24]. The method used is a lattice theory in which the neutrons from each fuel rod are followed using age-diffusion theory and their densities superposed. The corrections are very small for most practical cases and great accuracy is not required for these calculations. 

Core. The standard LMFR is predominantly thermal, nearly homogeneous, and moderated by graphite. Thus age-diffusion theory is applicable, and therefore the following formula can be used for a critical system ... 

Critical mass. The results of age-diffusion theory are in good agreement with multigroup calculations for predominantly thermal LMFR reactors. At higher fuel concentrations, however, the age theory overestimates the critical mass, as shown in Table 19-5 [1]. The differences in critical mass estimates are large only for weakly moderated reactors. 

In the case of some measurements, it is possible to interpret the data in several different ways. For instance, measurements of migration area may be interpreted by age-diffusion, age, one group, or multigroup theory. Numerical values of M as computed by the different methods will vary coMiderably, although each will form a con-slstent set of reactor parameters in its own critical equation, hi addition, there appears to be agreement in values of M as measured by subcritical and by critical. methods. Thus, it is necessary to make any comparison between critical and subcritical experiments with a common theoretical interpretation. 

In the early days of criticality safety computations, when a two-group diffusion theory calculation in cylindrical or spherical coordinates 1 desk calculator was a tedious and somewhat formidable task and when cross-section data were more sparse, the selection of cross sections was perhaps a sinqiler task. The fbur-fitctor formula was widely used for moderated systems. A factor was used to indicate the deviation from 1/v behavior of an absorption cross section at thermal energy. - Thermal and epithermal cross sections were related to the Integral parameters, diffusion area, and neutron age. 

The analytical method employed in both of these presentations involves the use of expansions in spherical harmonics. In each case the original integrodifferential equation, which states the neutron-balance condition, is reduced to an infinite set of coupled equations in the various harmonics of the function . By truncating the series expansion for the flux, this set can be further reduced to finite size (depending on the accuracy desired), and it is shown that in first approximation the resulting equations yield the diffusion theory and Fermi age models. 

Apart from this limitation, the principal restriction in the use of the Fermi age model is due to the large number of conditions which must be satisfied by the physical system if the model is to be valid (see Sec. 7.3i). However, even with this shortcoming, the model is still very useful for preliminary studies and for gaining insight into the essential features of many reactor problems. The Fermi age or continuous-slowing-down diffusion theory, then, offers some generality without introducing excessive computational difficulties. 

The authors have attempted to present a discussion of all the principal topics of reactor analysis, with an entire chapter devoted to each. In many instances several analytical methods are presented in order to provide as wide a treatment as possible. These include Chap. 2 on probability concepts Chap. 3 on the neutron flux Chap. 4 on slowing down Chap. 5 on diffusion theory Chap. 6 on the Fermi age model Chap. 7 on transport theory Chap. 8 on reflected reactors Chap. 9 on reactor kinetics and Chap. 10 on heterogeneity. It is important to mention that the remaining chapters represent in main part extensions and applications of these general topics. 

In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average, mass Iransfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is VlD/(7rt, where D is the diffusivity. 

From the penetration theory, the mass transfer rate per unit area N, is given in terms of the concentration difference AC, between the interface and the bulk fluid, the molecular diffusivity D and the age t of the. surface clement by ... 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

The techniques of monomolecular rate theory easily transform measured reaction data into a form where we can analyze apparent kinetics and the effects of intracrystalline diffusion by the use of selectivity data. Time dependency has been eliminated. Since selectivity is extremely reproducible and is independent of short-term aging effects, the number of experimental runs is reduced while data reliability is maintained. For catalyst evaluation at any temperature, it is necessary to determine the equilibrium composition and the straight-line reaction path. With this information any catalyst can be evaluated at this temperature with simply the additional information from a curved-line reaction path. The approach used in the application of monomolecular rate theory to the xylene isomerization selectivity kinetics is as follows. Reference is made to the composition diagram, Figure 1. 

From these considerations, Cottrell demonstrated that the rate at which solute atoms diffuse to dislocations and subsequently pin them in place is proportional to time2/3 (this time dependence is derived by an approximate method in Exercise 3.9). This provided the first quantifiable theory for the strain aging caused by solute pinning of dislocations [22]. 

Information relating to the diffusion of metal-bearing compounds in catalytic materials at reaction conditions has been obtained indirectly through classic diffusion and reaction theory. Shah and Paraskos (1975) calculated effective diffusitivities of 7 x 10-8 and 3 x 10-8 cm2/sec for V and Ni compounds in reduced Kuwait crude at 760°F. These low values may be indicative of a small-pore HDS catalyst. In contrast, Sato et al. (1971) report that the effective diffusivity of vanadium compounds was one-tenth that of the nickel compounds on the basis of metal deposition profiles in aged catalysts. This large difference may be influenced by relative adsorption strengths not explicitly considered in their analysis. 

Instead, they proposed a time on stream theory to model the catalyst deactivation. However, in an earlier work by Voorhies (2), a linear correlation between conversion and coke on catalyst for fixed-bed catalytic cracking was derived. Rudershausen and Watson (3) also observed the similar behavior. Coke on catalyst can reduce the activity by covering the active sites and blocking the pores. The effects of pore size on catalyst performance during hydrotreating coal oils in trickle-bed reactors have been studied experimentally by Ahmed and Crynes (4) and by Sooter (5). The pore size effects in other studies are also reported 7, 8). Prasher et al. (9) observed that the effective diffusivities of oils in aged catalysts were severely reduced by coke deposition. 

There are approaches to analyses of turbulent combustion that, although not deductively based on the Navier-Stokes equations, nevertheless appeal to concepts of coherent structures [68], [69]. We shall not have space here to present these approaches and must refer instead to reviews [18], [27], [40]. These methods share some aspects in common with age theories of stirred reactors [19], theories that we also shall forego discussing for the sake of brevity. Instead, we shall consider a promising approach to the theoretical analysis of turbulent diffusion flames. 

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