Neutron flux spatial distribution

For an actual system having the nuclear and geometric properties assumed above, only one of the solutions (8.283) is physically acceptable. This solution is the fundamental mode o(r). It is clear that, for n > 0, the steady-state spatial distribution would require that i4(r, ) < 0 in certain regions of the reactor. The determination of the neutron-flux spatial distribution in a finite system by means of the integral-equation formulation was demonstrated in Sec. 5.7c for the case of an infinite slab. Those results may be applied directly to the multiplying medium problem. In the case of the infinite-slab reactor, of width 2a, Eq. (8.283) takes the form... 

The 0 weighting term involves the reaction rates and the spatial importance of the neutrons in question. From this expression it can be seen that the sensitivity of a reactor is inversely proportional to the reactor volume. A reactivity comparison of a standard absorber and a material of unknown cross section will yield a ratio of the total absorption area of the absorbers (N/pOj ) providing the neutron-flux spatial distribution remains constant. Flux depression in the vicinity of the absorber could be undesirable since this might change the distribution. 

Most reliable are the data on neutron fluxes which are determined by the plasma power density, reactor geometry and structural parameters. Most of the conceptual power reactor design studies contain a detailed neutronics analysis giving the energy as well as the spatial distribution of neutron fluxes. 

Spatial distributions of thermal and. epithermal neutron flux were measured in both the square reactor and the thin-slab reactor. The measurements were made with indium foils, used alternately bare and cadmium covered. Figure A2.J shows the distributions of thermal and epithermal neutron flux along a perpendicular bisector of the side of the square reactor. Figure A2.K shows the same distributions along a diagonal, of the square assembly. All these foil measurements were taken at mid-heights of the assemblies. 

Spatial distri But ions of thermal, indium resonance, and fast (E > 1 Mev) neutron flux in the core and in the reflector, particularly the distributions of thermal and indium resonance neutrons in the experimental holes. 

The spatial distributions of thermal and epithermal neutron flux in the 2.94-kg pile, the 3.95-kg pile, and in the various experimental holes of the mock-up assembly, as indicated in the listing below, were measured. These measurements were made with indium foils, used alternately bare and cadmium covered, as described previously. A calibration. of foils in the standard (sigma)reactor indicated that the absolute flux (nv) is obtained from the measured saturated activities, shown in the attached figures by the following relations, ... 

The analysis of quickly varying time dependence of fc-effective and neutron flux distribution, which gives us the needed information to control reactor total power and power spatial distribution changes with time... 

Calculation of the fission power distribution in Step 1 involves the handling of a tremendous amount of geometric details of the reactor because the reactor might contain tens of thousands of fuel elements, control elements, instrument ports, coolant flow channels, structural components, etc. In addition to the spatial detail, the designer needs to know the neutron flux distributions in energy and direction as well. This is an overwhelming amount of detail. 

The directed flux in an infinite medium wherein the spatial distribution of neutrons is given by a single variable x, and therefore by a single angle dy has been given by Wilson. In this special case Xyd) is symmetric in X and 6y and the elementary solution for 4> XyB) may be written in the form... 

This function should be a satisfactory solution to Eq. (8.88) at points far removed from the boundaries of the medium because in these regions the angular distribution of the neutron flux is nearly isotropic, and the diffusion-theory results will give an adequate representation of the spatial distribution of the neutrons. Thus sin Br/r is sometimes called the asymptotic solution to the transport equation. 

For many practical applications, such as preliminary studies, the primary information required is a reliable estimate of the critical mass. Since the two-group model is well suited to this purpose, it is clear that a formulation of the method which eliminates a large portion of the computational labor would be most useful. Such a method has been devised by R. P. Feynman and T. A. Welton. The Fe5mman-Welton method draws directly from the results of the one-velocity integral theory. The method effectively separates the space and energy dependence of the neutron-flux function and describes the spatial distribution of each... 

The eigenfunctions n(r) represent the spatial distributions of the flux according to these modes, and the coefficients n of Eq. (8.250) give the weighting function for each mode, i.e., the fraction of the neutron population distributed in each space form... 

The unit-cell method offers a relatively simple computational procedure for determining the various factors in /c . As already demonstrated, the calculation of these factors in each case reduces ultimately to the determination of the neutron spatial distributions for the entire energy range. For the thermal utilization we require the thermal-flux distribution and in particular the thermal-flux depression (i.e., the thermal disadvantage factor). The resonance-escape probability, on the other hand, requires a knowledge of the spatial distribution of resonance neutrons, and finally, of course, the calculation of the fast effect involves, essentially, the determination of the spatial distribution of successive (cascading) generations of fission neutrons. 

We have used here Eq. (10.131) for Pe., and Mr Vrpy When these expressions are compared with (10.38) and (10.39), it is seen that the NR approximation for leads rather easily to a statement in the standard form. Note, however, that because of the flat flux assumption the present derivation does not contain the resonance disadvantage factor fr. This quantity is customarily computed using a one-velocity model to represent the entire fast-neutron population. It is well recognized that this point of view is crude and somewhat unclear. On the one hand, when used with the NR approximation, it may be observed that only one collision is required to remove a resonance neutron from the vicinity of a resonance thus the spatial distribution would be very nearly uniform and isotropic. On the other hand, if the NRIA approximation is valid, then the absorptions are necessarily very strong and the use of a disadvantage factor based on diffusion theory is not well justified. For these reasons it has been omitted in this treatment as an unwarranted refinement not in keeping with the precision of the over-all model. ... 

The TWINKLE code is used to predict the kinetic behaviour of a reactor for transients that cause a major perturbation in the spatial neutron flux distribution. TWINKLE was used in the analysis performed in support of the Sizewell B PCSR (Reference 5.7). There is therefore a high degree of confidence that an acceptable verification statement can be made in the context of the UK regulatory regime. 

The traditional way to calculate the physical characteristics of a fast reactor is to carry out the following steps (1) preparation of the effective cross sections for regions of the reactor (2) a three-dimensional calculation to obtain k-eff, and real and adjoint fluxes (3) edit the results of the previous steps to estimate the power and reaction rate distributions, neutron kinetics parameters, control rod effectiveness, etc., and (4) a bumup analysis, calculating the variation of the isotopic composition with time, and then recalculating the results obtained in the previous steps for particular bumup states. This scheme has been implemented, for example, in the TRIGEX code [4.49]. This code calculates k-eff, few group real and adjoint fluxes, power spatial distribution, dose factor and reaction rates distributions, breeding parameters, bumup effects, and kinetics parameters (effective delayed neutron Auction, etc.). 

There are a number of techniques for meastuing subcritical reactivity relative to a calibrated reference control rod, in addition to soxuce multiplication. These include rod drop, rod jerk, source jerk, pulsed source and reactor power noise. Account must be taken of spatial flux transients (either by calculating them or measuring them with arrays of covmters) and of the spatial distribution of natural neutron sources due to spontaneous fission and (a,n) reactions and any fixed sources introduced to increase the subcritical flux level. The different methods have been reviewed and intercompared, for example, at the 1976 Specialists Meeting [4.87]. 

At low powers and for start-up, control neutron flux Is measured at a beam hole facility In the side of the calandrla. However, the errors due to spatial distributions become too large for adequate control to be based on such a measurement at or near full power. Consequently above 50 power, steam flow is used as a measure of reactor power, although a rate of change of flux signal Is still retained for stabilizing the control. 

The ASMBURN assembly bumup calculation code is based on the neutron flux calculation by the collision probability method and the bumup calculation by interpolations of macro-cross sections [14]. As the bumup proceeds, the compositions of the fuel rods in the assembly start to differ from each other depending on the spatial distribution of the neutron flux. Therefore, a precise modeling would require production and decay calculations for each fuel rod constituting the fuel assembly. However, when the fuel rods are aligned in a regular lattice, the differences in the... 

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