Diffusion theory criticisms

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

Several authors have made restricted comparisons between experiment and calculations of diffusion theory. Thus, Turner et al. (1983, 1988) considered G(Fe3+) in the Fricke dosimeter as a function of electron energy, and Zaider and Brenner (1984) dealt with the shape of the decay curve of eh (vide supra). These comparisons are not very rigorous, since many other determining experiments were left out. Subsequently, more critical examinations have been made by La Verne and Pimblott (1991), Pimblott and Green (1995), Pimblott et al. (1996), and Pimblott and LaVeme (1997). These authors have compared their... 

S A study of turbulent diffusion of gas clouds over several terrains, Report OSRD No. 6185, 84 pages, by Harold Johnston, Robert Merrill, and Robert Mills. 1945. Part I. An Empirical Approach to the Effect of Turbulent Diffusion on Gas Clouds over Several Terrains, pages 1-23. Part II. A Critical Examination of the British Statistical Diffusion Theories, pages 24-46. 

The value of critical mass, M., calculated in this way is, however, considerably overestimated by the elementary diffusion theory. The more exact diffusion theory, allowing for the long free path, drops R. by a factor of about 2/3 giving... 

The simplest scheme which might be autocatalytic is indicated in the sketch where the active material is disposed in a hollow shell as indicated in Fig. 8.3a. Suppose that when the firing plug is in place one has just the critical mass for this configuration. If as the reaction proceeds the expansion were to proceed only inward it is easy to see from diffusion theory that v would increase. Of course in actual fact it will proceed outward (tending to decrease v ) as well as inward and the outward expansion would in reality give the dominant effect. However, even if the outward expansion were very small compared to the inward expansion, it has been calculated that this method gives very low efficiency with 12 an efficiency of only about 10 was calculated. 

Despite these criticisms, however, we are not saying that diffusion-created interfaces or interphases do not exist or do not influence mechanical behavior of joints. They do. But Voyutskii s diffusion theory simply is not, and cannot be, the basis for a general understanding of the mechanical behavior of, e.g., adhesive joints or other adhering systems. 

The Shape Factor analyses utilized three-energy-group diffusion theory in spherical and two-dimensional r,z geometry. Table 11 is a summary of the analyses. It may be seen that the optimum geometry Shape Factor is a function of the reactor system. It is clearly not an invariant quantity. Furthermore, the variation of critical mass with core geometry also is a function of the specific reactor system. 

The fuel compacts were in the form of 2.01-in. cubes. Each cube was spray-painted with one-mil-thick coat of aluminum paint and 11-mil-thick rubberized plastic. The approach-to-critical technique was used to assemble the critical arrays utilizing a remotely operated split-table device. A summary of the experimental data is given in Table I. Also given are the criticality data for an equivalent Pu/water mixture fully reflected by water and having an H/Pu atomic ratio of 15 and a plutonium density of 1.52 g/cm. The conversion factors were obtained from a 12-group diffusion-theory calculation. 

Some experimental data have also been obtained for mixed Pu(N03)4, VOa(NOb)a solutions in the United Kingdom. An analysis of those data Indicates that the critical parameters of mixed, neutronlcally thermal, solutions can be predicted quite accurately using the KFK diffusion theory code. 

Critical thicknesses for clean slabs of the above-mentioned plutonium nitrate solutions were derived from the iperimental data. These values compare quite favor ly with the values computed using the HFN multigroup diffusion theory code, with multigroup constants obtained from the GAMTEC-II code, as shown in Table n. 

In preparation for the experiment, extensive calculations and selected critical facility measurements were made. The basic analytical approach is the multigroup method in the Sn and diffusion theory approximations to the transport equation. The fast and epithermal spectra used in computing the multigroup cross sections are calculated using HRO. The thermal spectra are calculated using Battelle Revised THERMOS. ) The NeUdn kernel is used to describe the thermal-neutron scattering in water. 

A two-dimensional diffusion theory code in four enef . groups" was used to predict the critical loadings. These critical predictions were documented previous to the experiment. When the critical loading was achieved it was with the predicted loading of 7 fixed fuel elements,... 

The diipensions of Assembly 2 are given in Table I, as calculated using MC -generated ENDF/B cross sections and two-dimensional diffusion theory. Initial criticidity hsiS just been achieved, and toe critical mass, uncor-rected for various minor effects, is 1066 e 5 kg fissile ( Pu + Pu + " U). 

The analytic study was di ded into three phases. In the first phase, homogeneous two-region, systems were examined using the diffusion theory WANDA Code. The configuration studied consisted of a spherical core surrounded by a homogeneous miJ reof natural uranium and HiO. The H-to- U ratio was varied in the central core and reflector region. The results indicate a minimum critical mass of 451 g U occurs for a H/ U equal to 450 in both regions. ... 

The approximate methods used frequently for criticality calculations are classified into several broad categories Monte Carlo, discrete ordinates, integral transport, and diffusion theory. Within each category there exist several computer programs, each with a different treatment of spatial detail and energy group structure. Several of these methods are topics for later papers in this session. In the remainder of this paper we will describe some general aspects of each method and indicate the types of situations where each is used. 

Few data cvirrently exist on the ellectlveness of boron and cadmium for criticality prevention In operations external to reactors that may involve fuel elements under conditions of water Immersion. Material bucklings and extrapolation distances have previously been measured and reported for 25.2 wt% Pu02-U(Nat)0a fuel pins in water. These experimental results have also teen compared with one-dimensional diffusion theory calculations using ENDF/B version n cross-section data. The previous measurements have now been extended to include criticality data On these same fuel pins positioned in lattices with boron- and cadmium-poisoned water. 

Experimentally determined critical sizes and masses obtained from Plexiglas-reflected parallelepipeds, constructed from 2- X 2- x 2-in. PuOi-UOa-polystyrene fuel compacts, are shown in Table n. Calculated values of k ff have been obtained for these critical systems utilizing ENDF/B-n cross-section data with diffusion theory, transport ttieory, and Monte Carlo-type calculations. Typical resiHts obtained were 1.027 a 0.007, with the Monte Carlo code KENO-I, for the reflected 40.72-X 40.64- X 36.42-cm assembly of 7.89 wt% Pu fuel and 0.991 with the diffusion theory code HFN, for a reflected infinite slab of this same material. 

Two other techniques were correlated with the same critical eiqieriments with water-reflected spheres. The ffrst of toese involved a 12-groiq > Bo calculation to Obtain the buckling and two-group parameters for a subsequent diffusion theory calculation of extrapolation distance. ... 

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. 

A series of benchmark calculations of critical experiments has been performed to assess the effects that recent chanp es in the ENDF/B data files have had on calculated LMFBR parameters. Three well-documented critical assemblies were studied using standard methods of fast reactor analysis [two-dimensional (2-D) multi-group diffusion theory] with both Versions III and IV of ENOF/B. A review of the changes in the principal cross sections incorporated in the latest evaluation has been made and was used to interpret the changes in calculated integral parameters. 

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