Infinite-medium model

It was remarked in Sec. 3.26 that the scattering cross section did not play a role in the analysis of the infinite-medium model, and indeed the result (3.22) is independent of 2,. This is entirely a consequence of the assumptions of an infinite medium and no change of speed in scattering. 

It is proposed that a reactor of pure UFe be constructed, the fluorine to provide the moderation and the U in the uranium to serve as the nuclear fuel. A first estimate of the critical fuel concentration for such a system can be obtained from the one-velocity infinite-medium model. 

In this treatment we examine the nuclear reactor with the aid of an infinite- medium model which describes the neutron-density distributions in terms of the kinetic energy of the neutrons. This model is the immediate extension of the one-velocity approximation introduced in Chap. 3. In the present analysis we discard the one-velocity restriction and attempt to describe the trajectory of the neutron in energy space as it is slowed down by elastic-scattering collisions with the nuclei of the medium. In... 

The Monte Carlo calculations carried out for the infinite-medium model described above were used to determine the distribution of neutrons around thermal energy. These distributions w ere obtained by tracing the individual life histories of many neutrons. The density of neutrons and the various collision densities at each energy were computed from the statistics compiled from these individual histories. Some of the... 

An important variation of the self-consistent model is the three-phase model, introduced by Kerner 20), according to which the inclusion is enveloped by a matrix annulus, which in turn is embedded in an infinite medium with the unknown macroscopic properties of the composite. 

Diffusion in Matrix. The transport equation for a semi-infinite medium of uniform initial concentration of mobile species, with the surface concentration equal to zero for time greater than zero, is given by Crank (13). The rate of mass transfer at the surface for this model is ... 

This model is a modification of the model developed by Kumar and Kuloor (K18) for bubble formation in inviscid fluids in the absence of surface-tension effects. The need for modification arises because the bubble forming nozzles actually used to collect data on bubble formation in fluidized beds differ from the orifice plates in that they do not have a flat base. Under such conditions the bubble must be assumed to be moving in an infinite medium and the value of 1/2 is more justified than the value 11/16. 

The model system is a cube of glassy polymer with 3D periodic boundaries, filled with chain segments at a density corresponding to the experimental value for the considered polymer. The entire contents of the cube are formed from a single parent chain with the chemical structure of the polymer. The cube can thus be considered as part of an infinite medium, consisting of displaced images of the same chain, as shown on Fig. 58. 

On the other hand, electrostatic models regard the ligands or the whole crystal as polarizable units and thereby lead to weaker Coulomb and spin-orbit interactions. In a dielectric screening model (DSM) from Morrison et al. (1967) the f element is placed within an empty sphere with radius Rs which is embedded into an infinite medium with dielectric constant e. This leads to a reduction AFk of the Slater parameters (Newman, 1973) ... 

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-infinite medium. We give here a brief account of... 

The problem is inherently a finite-size problem. Results that otherwise would be considered as finite-size effects and should be neglected are in this case essential. At the limit of infinite volume there will be no release at all. Bunde et al. [84] found a power law also for the case of trapping in a model with a trap in the middle of the system, i.e., a classical trapping problem. In such a case, which is different from the model examined here, it is meaningful to talk about finite-size effects. In contrast, release from the surface of an infinite medium is impossible. 

Assumptions Carbon penetrates into a very thin layer beneath the surface of the component, and thus the component can be modeled as a semi-infinite medium regardless of its thickness or shape. 

The concentration within a semi-infinite medium (C2) into which the substance diffuses out of the lower layer (initial concentration C0) may be calculated by the following relationship (24, p. 91 coordinates were changed to conform to the coordinates of the model discussed here)... 

Experiments are typically designed so that resulting concentration profiles can be modeled by the analytical constant surface concentration solution to the diffusion equation for a semi-infinite medium... 

The geological conditions, parameters and stress conditions have been presented elsewhere in this proceeding, see Andersson et. al (2003). Two kinds of models have been established to simulate the fracturing process (See Figure 4.). The Far-field model" simulates the effect of excavations in infinite medium without external boundaries. It is used to investigate parameter sensitivity and the effect of pre-existing fractures. The Pillar model is used to simulate real heating process and 3D excavation effects, with the aid of the stress reconstruction technique described above. 

Another simple model, the surface renewal model [31], predicts a dependence. In this model, the interface between the air and water is renewed periodically by turbulence eddies, a process that mixes gas that has diffused into the water surface down into the bulk phase (Fig. 3). Jacobs [32] gives the quantity of substance diffusing across a plane between two semi-infinite mediums as ... 

Equation (6.20) is for a noninsulated half-ellipsoid inserted from the surfaee. When the needle shaft is insulated so that when the needle is inserted from the surfaee and penetrates deeper into the tissue, the model may he regarded as a gradual ehange towards a full-length ellipsoid in an infinite medium. The eonductance given hy Eq. (6.20) would then be doubled. A slightly different model is to eonsider the half-ellipsoid positioned at the end of an insulating rod. 

The modelling of an isolated drop involves the calculation of the complete flow field around the drop in order to determine the forces acting on it. A series of recent papers by Subramanian (and collaborators) [10] have addressed the complete calculations of flow around drops in an infinite medium, taking into account the heat transport, for different ranges in the dimensionless parameters. 

It is assumed that the particle environment changes often such that a steady stale concentration boundary layer is never established thus, the convection terms are modeled by renewal of fresh gas at the particle surface. The solution of the diffusion equations for this model is the same as in the case of diffusion from a semi-infinite medium to a sphere. 

The neutron moderati properties ot polyetl lene (CHa) have been staled experimentally and ana ically at General Atomic ai. bther laboratories. Measurements have been, made of the neutron spectra with various 1/v absorber concentrations and temperatures as an over-till check idf the scattering model for the polyethylene molecule. Since the scattering model-for polyethylene should be very similar to that for paraffin, a homogeneous mixture of (UF 4-paraffin) was obtained from ORNL for quasi-Infinite-medium lieutron-spectrum measurements. This material has been studied extensively in critical experiments. 

Model 1 (Semi-Infinite Medium). For linear diffusion into a semi-infinite medium where the concentration at X = 0 is of the constant value Co for aU times t, we find the solution for the concentration field to be... 

No mention is made of the scattering cross section 2, these reactions have no effect on the neutron population in the present model. Both of these facts result from the assumption of an infinite medium and the consequent uniform neutron density. In the more general case of a finite reactor, the scattering cross section will affect the neutron balance, and migration (or transport) losses and gains must be considered. 

As noted above, the system is at steady state when k = In this case, each generation reproduces itself, and the neutron population remains fixed. If fc > 1, the subsequent generations of neutrons increase in population if /c < 1, the subsequent generations decrease in population. The case k = 1 corresponds exactly to the condition given by Eq. (3,13), i.e., the criticality condition. This is easily demonstrated for the infinite-medium reactor model defined by the assumptions (3.7). 

It is important to note that this result is based on a very elementary model of a chain-reacting system and should not be taken a priori as a representative value for the infinite-medium multiplication constant. Some of these limitations were mentioned in the discussion following Eq. (3.19). Better estimates for the infinite-medium multiplication constant than that given by Eq. (3.22) are developed in Chap. 4. 

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