Neutron balance

For the present, we will not attempt a detailed description of all the processes that occur in an actual system, but instead will limit our study of the reactor system to a considerably simplified model. The model we select is admittedly an unrealistic one however, it will serve well to display some of the basic features of all reactors. The reactor model we propose has the following characteristics  

Number of neutrons gained per unit volume, per unit time 

This statement may be written in terms of the concepts developed in Sec. 3.1. If n t) denotes the number of neutrons per unit volume at time then by introducing the appropriate collision densities for absorption and fission [see Eq. (3.5)], we have 

2a s -1- 2 = total macroscopic-absorption cross section of reactor material 

Nr = fuel nuclei per unit volume No — nonfuel nuclei per unit volume 2/ = macroscopic-absorption cross section of fuel 2J, = macroscopic-absorption cross section of all nonfuel materials 

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The fission product Xe has the largest absorption cross section of all the nuclides in a thermal-neutron flux, and its buildup is especially important in affecting the neutron balance in a thermal reactor. The fission-product decay chain involving the production and decay of Xe is... 

In reactor control problems and in reactor neutron balances, the quantity of interest is the poisoning ratio r, which is the ratio of neutrons absorbed by the poison to neutrons absorbed in fission. Assuming for simplicity that the neutron flux is constant throughout the reactor, the xenon poisoning ratio at steady state is... 

For this fuel-cycle analysis, it is convenient to specify a reactor design that satisfies a neutron balance and contains a char of fresh fuel distributed uniformly throughout the reactor cote. 

We shall assume that a representative unit volume of this design contains atoms of a single fissile species (e.g., U) with absorption cross section ajjf and Ng atoms of a single fertile material (e.g., with absorption cross section Og. The unit volume is assumed also to contain steady-state amounts of Xe, Sm, and other fission products with cross sections above 10,000 b, which build up to equilibrium concentration in a few days at the neutron fluxes typical of power reactors. It is assumed that no other fission products are present to an extent sufficient to affect the neutron balance. The items that affect the thermal-neutron... 

The thermal-neutron balance equation for the reference design condition states that the rate of production of thermal neutrons equals the rate of consumption ... 

For an operating reactor, the terms of the neutron balance will differ from Table 3.9 and the neutron balance equation will differ from Eq. (3.25) because the fuel charged to the reactor may differ from the reference fuel and because the composition of fuel will change as a result of reaction with neutrons. The terms in the neutron balance of an operating reactor are listed in Table 3.10. 

Table 3.9 Theimal-neutron balance for reference deagn condition...

These terms determine the rate at which plutonium builds up during irradiation. One reason for giving the neutron balance in such detail is to be able to evaluate these terms. 

We are now in position to derive equations that will give the degree uf bumup nuclear fuel can experience before it ceases to be critical. First, we must determine how the concentration of each nuclide that affects the neutron balance changes with time. We consider fuel that at time zero contains N s atoms of U per cubic centimeter, atoms of U, and no other uranium isotopes, plutonium, or fission products. This fuel is then exposed to a thermal-neutron flux 0(0, which may be a function of time. The variation in concentration of each nuclide in this fuel with time is obtained as follows. 

Dimensions, material content, and nuclear parameters of the Douglas Point CANDU heavy-water, natural uranium reactor are summarized in Table 3.17. Effective properties of nuclides in this reactor are listed in Table 3.18. The neutron balance for this reactor when charged with natural uranium fuel is given in Table 3.19. 

The adjoint source in Eq. (259) represents some detector distribution which, when integrated over the reactor space (after weighting with the neutron distribution), yields a single measure of the reactor behavior. Physically, there is no reason why such a detector distribution should be positive definite. The existence condition [Eq. (260)] for a solution to Eq. (259) in a critical system, however, as in the neutron balance equation, requires the orthogonality of and V and since is (physically) positive definite, must be part positive, part negative. Physically, this orthogonality condition, therefore, expresses the idea that the acceptable virtual adjoint sources are detector distributions which, when reacting to the distribution of neutrons in the fundamental mode, lead to no total or net effect. 

If the alternate equation (6 ) is used in parts of a calculation, the neutron balance will not be complete, but the discrepancy may be used as a measure of whether a finer mesh spacing is called for or not. 

Reactor modes. The equations expressing the neutron balance in a reactor may be written in the form... 

A more general problem can be posed in reactor statics if the system is maintained in steady state by means of an independent source, with source density or distribution 5(x). The neutron balance for such a system can then be written as... 

Although many models of critical reactor theory take the form of a neutron balance such as... 

In changing the neutron balance equation, we transform the adjoint equation. For example, if we have transformed the critical equation by means of an operator, B, say, we compare... 

Neutron balance in fast reactor and thermal neutron reactor (Fuji-ie 1998)... 

Two general classes of methods are considered. The first is semlempirical models in which equations that describe the data on either critical or safe arrays in terms of some selected set of array parameters are written. Most of these methods (as, for example, the density analog technique) are directly based on comparing arrays to single units, to addition to the density analog, these models include the surface density and equilateral hyperbola techniques. The second Is array unit interaction models, which are based on equations written to describe the neutron balance for each unit. The solution... 

For a uniform reactor lattice with and as the only fissionable materials, the two-energy-group, neutron-balance condition. at critical (or steady-state sub-critical) can be written... 

As an example of the resulting spectrum, a simple approximation of the slow-ing-down region neutron balance (through the narrow resonance approximation) and a simple spatial approximation of a lump of resonance absorber surrounded by an infinite "sea" of moderator results in a simplified neutron balance relation of... 

Difference 2 The neutron balance relationship is the same, but the relative importance of loss mechanisms is different. 

As laid out in the Equation 21.1 definition of /c-effective, the basic neutron balance relationship is between fission neutron production and neutron loss mechanisms, the two neutron loss mechanisms being neutron absorption and neutron leakage. Nuclear reactors, for which criticality must be created and maintained for extended periods, tend to be large devices—both for neutron balance reasons and because of the large fuel inventory required to deliver substantial amounts of power for long periods of time. The result of this, from the point of view of the A -effective equation, is that neutron leakage is a relatively unimportant neutron... 

For example, consider a producing HFF based on fission-suppressed blanket using beryllium for neutron multiplication and moderation. Table VI compares the neutron balance and fuel production-ability of a simplified blanket of this type when driven by three different fusion neutron sources[39,42]. Without lithium (the SCD case), a large fraction of the neutrons are parasitically absorbed. By adding only a thin layer of lithium adjacent to the first wall it is possible to convert most of these neutrons to tritium. Using this tritium to assist the SCD fuel cycle, the resulting SCD-T HFF yields 30% more fissile fuel than the SCD-HFF and 70% more than the DT-HFF (all having the same total thermal capacity). Moreover, for the same fissile yield, the SCD-T plasma requires only about one-fourth of the volume of the SCD plasma. 

Th potentiml for power axcurilons in %hf NFR, their consequences and the effectiveness of remedial measures are discussed in this section Power excursions are directly related to nuclear excur-sions They are induced by an upset in the reactivity balance In the reactor ich can be brought about by either esnrors or accidents involving the control systems or possibly by other accidents ich can alter the neutron balance. 

Neutron-physical parameters of the salt coolants make it possible to use them effectively both as neutron moderators and reflectors thermo-physical and neutron-physical properties of NaF-BeF2 salt are slightly worse than those of LiF-BeF2, but tritium production using this salt in the reactor is considerably smaller to improve the neutron balance and reduce the tritium production, the initial enrichment by Xi shall be at least 99.999%. 

The general field of problems described above, except in some special areas, may be treated by the well-known methods and analytical models of mathematical physics. It has already been noted that the most general description of the neutron population usually starts with a neutron-balance relation of the Boltzmann type. The Boltzmann equation was developed in connection with the study of nonuniform gas mixtures, and the application to the neutron problem represents a considerable simplification of the general gas problem. (Whereas in gas problems all the particles are in motion, in reactor problems only the neutrons are in motion. ) The fundamental equation of reactor physics, then, is already a familiar one from the kinetic theory. Further, many of the most useful neutron models obtained from approximations to the Boltzmann equation reduce to familiar forms, such as the heat-conduction, Helmholtz, and telegraphist s equations. These simplifications result from the elimination of various independent variables in the... 

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. 

In our treatment of the infinite homogeneous reactor model, we argued that the neutron density was a spatially invariant function. Thus, if we determine the neutron-balance conditions in any one volume element of this infinite system, these conditions will apply to all such elements throughout the space. Let us write, then, the quantities involved in our definition of the multiplication constant in terms of the nuclear events which occur in a unit volume of the reactor. It is convenient here to write Eq. (3.16) in the form... 

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