Reactor multiplication factor

Hence we may regard the multiplication factor, k, as composed of a prompt part, kp, and a delayed part, kj  

Since the total number of neutrons in the next generation will be proportional to k and the number of next-generation prompt neutrons will be proportional to kp, it follows that the fraction of prompt neutrons in the next generation will be kp/k. Similarly the fraction of next-generation delayed neutrons will be kdi/k, for I = 1 to 6. The delayed neutron fraction for group i is given the symbol ft, so that 

Substituting from equation (21.6) back into equation 

We may also note that, since each delayed neutron is preceded by a precursor nucleus, it follows that 

The control engineer will be able to obtain data on the items above from the reactor designers in the first instance and then from reactor physicists employed to work on the reactor, who can be expected to refine the match to the operational reactor by collecting extensive operating data and performing experiments where necessary. For example, it is customary to perform experiments to calibrate the worth of each control rod. In this case the data is usually given in the form of a graph of control rod insertion distance 

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In general, each of the phenomena described fuel burnup, moderator temperature increase, fuel temperature increase, and poison addition, result in a decrease in the reactor multiplication factor, Kgff, if means are not taken to compensate. 

The nuclear chain reaction can be modeled mathematically by considering the probable fates of a typical fast neutron released in the system. This neutron may make one or more coUisions, which result in scattering or absorption, either in fuel or nonfuel materials. If the neutron is absorbed in fuel and fission occurs, new neutrons are produced. A neutron may also escape from the core in free flight, a process called leakage. The state of the reactor can be defined by the multiplication factor, k, the net number of neutrons produced in one cycle. If k is exactly 1, the reactor is said to be critical if / < 1, it is subcritical if / > 1, it is supercritical. The neutron population and the reactor power depend on the difference between k and 1, ie, bk = k — K closely related quantity is the reactivity, p = bk jk. i the reactivity is negative, the number of neutrons declines with time if p = 0, the number remains constant if p is positive, there is a growth in population. 

Neutron cycle is die average life history of a neutron m a nuclear reactor. The gain in the number of neutrons in a reactor during any individual neutron cycle ts given by n(k-1). where n is the number of neutrons in the reactor of the beginning of the cycle and k is the multiplication factor. 

Of the fast neutrons produced in fission, some of them will be moderated to thermal energies and will induce other fission reactions while others will be lost. The ratio of the number of neutrons in the next generation to that in the previous generation is called the multiplication factor k. If the value of k is less than 1, then the reactor is subcritical and the fission process is not self-sustaining. If the value of k is greater than 1, then the number of fissions will accelerate with time and the reactor is supercritical. The goal of reactor operation is to maintain the system in a critical state with k exactly equal to 1. The extreme upper limit for the multiplication factor would correspond to the mean number of neutrons per fission ( 2.5 for 235U(n,f)) if each neutron produces a secondary fission. 

The multiplication factor for an infinite sized reactor core is given by the four-factor formula ... 

For operation of nuclear reactors, the delayed neutrons (section 8.9) play an important role, because they cause an increase in the time available for control. The multiplication factor due to the prompt neutrons alone is kesi - P), P being the contribution of the delayed neutrons, and as long as / eff(l P) < 1, the delayed neutrons are necessary to keep the chain reaction going. In the fission of 0.65% of the fission neutrons are emitted as delayed neutrons from some neutron-rich fission fragments such as or Xe. 

For use in nuclear weapons, the concentration of °Pu in the plutonium should be low, because the presence of this nuclide leads to the production of appreciable amounts of neutrons by spontaneous fission if the concentration of °Pu is too high the neutron multiplication would start too early with a relatively small multiplication factor, and the energy release would be relatively low. Higher concentrations of " Pu also interfere, because of its transmutation into " Am with a half-life of only 14.35 y. To minimize the formation of " °Pu and " Pu, Pu for use in weapons is, in general, produced in special reactors by low bum-up (<20 000 MWth d per ton). 

In any neutron generation the neutrons experience a variety of fates (Fig. 19.6). Some neutrons escape the reactor and some are absorbed in the reactor structural materials and shielding (i.e. fuel cans, control rods, moderator, coolant, etc.). To take this loss into accoimt, two different multiplication factors are used k refers to a reactor of infinite dimensions (i.e. no leakage) while k f refers to a reactor of a finite size ... 

This report describes the data and procedures used to predict the multiplication factor for several measured critical core configurations using a select set of APR analysis codes. The analyses were performed for precise state points at beginning of cycle (BOC) (mixture of fresh and burned fuel) and at measured state points throughout the cycle depletion (all burned fuel). Self-consistency among the reactor criticals in the prediction of k will allow the determination of the bias of the approach taken in representing the effect of those materials not present in fi esh fuel. 

This section will cover some simple calculations related to the reactor. The reactor had a cold, beginning of life, neutron multiplication factor of 1.037 0.001, which corresponds to an excess positive reactivity of 5.7 based on a delayed neutron fraction of 0.0065. The burnup for the reactor was determined using a fairly simple set of equations. The consumption over 10 years at a power level of 200 kWth was 0.8 kgs of and at 400 kWth, 1.6 kgs of would be consumed. Given that the fuel loading is 186 kgs of the burnup is -0.86% for the upper end of the uranium consumed. This burnup results in a loss of 1 reactivity. 

The primary goal of this study was to ensure that there was sufficient excess reactivity in the neutron multiplication factor to keep the reactor critical for the 10 year lifespan while ensuring that the reactor would be subcritical during major accident scenarios. The position of the reflector can be used to set the multiplication factor of the reactor. Burnup in the reactor causes a proliferation of additional materials to absorb neutrons and reduces the density of fissile materials, lowering the neutron multiplication of the reactor. This can be offset by closing the reflector, which decreases the neutron leakage of the system. This is shown in Figure 5-1. 

The change in neutron multiplication vs reflector position is nearly linear for an extended region. Beyond 22 cm it starts curving, asymptotically approaching the neutron multiplication factor of the core without reflectors. The combined worth of the moveable reflectors is roughly 26.5 from full open to full closed position. Figure 5-2 shows how the reflectors open up on the core and where the centerline of the reactor is. It also indicates the length of the reflectors. 

The runs show that between 6 and 7 of the 20 runs done fall outside the one sigma deviation, exactly what would be expected. This give an average multiplication factor of 1.037 0.001. The second set of runs was focused on what the spectrum of the neutrons causing fissions in the reactor during accident cases. The results are shown in Figure 5-17... 

Therefore, the problems which faced the would-be designers of chain reactors early in 1941 were (1) the choice of the proper moderator to uranium ratio, and (2) the size and shape of the uranium lumps which would most likely lead to a self-sustaining chain reaction, i.e., give the highest multiplication factor. In order to solve these problems, one had to understand the behavior of the fast, of the resonance, and of the thermal neutrons. We were concerned with the second problem which itself consisted of two parts. The first was the measurement of the characteristics of the resonance lines of isolated uranium atoms, the second, the composite effect of this absorption on the neutron spectrum and total resulting absorption. One can liken the first task to the measurement of atomic constants, such as molecular diameter, the second one, to the task of kinetic gas theory which obtains the viscosity and other properties of the gas from the properties of the molecules. The first task was largely accomplished by Anderson and was fully available to us when we did our work. Anderson s and Fermi s work on the absorption of uranium, and on neutron absorption in general, also acquainted us with a number of technics which will be mentioned in the third and fourth of the reports of this series. Finally, Fermi, Anderson, and Zinn carried out, in collaboration with us in Princeton, one measurement of the resonance absorption. This will be discussed in the third article of this series. 

The evaluated multiplication factor of fuel in storage racks under normal and AOEs should be equal to or less than an estabIi shed max i mum mu 11 i pIi cat i on factor. Procedures for determining the limiting multiplication factor are given in detail in ANSI/ANS-8.1-1983, American National Criticality Safety in Operation with Fissionable Materials Outside Reactors. 

The aim of this talk is to indicate how the concept of positivity can be used systematically, to provide a mathematical basis for the concepts of criticality, multiplication factor, period, principal distribution, and importance function in nuclear reactor theory. Thus, its aim is to provide rigorous existence theorems for these related concepts, the concepts themselves being already familiar to reactor physicists. Though not all cases have been worked through as yet, the general facts seem to be clear. 

Work of Perron and Frobenius. It was shown in [3] that the concepts of criticality, multiplication factor, period, and importance function could be deduced in finite models from the theories of positive and non-negative matrices, developed a half-century ago by Perron and Frobenius. From the heuristic assumption that an arbitrarily close approximation to the behavior of any reactor can be obtained, by dividing the neutron phase-space 2 into sufficiently small cells , one would expect the behavior of continuous reactors to follow similar lines. As explained in [3], digital computations can also be interpreted as referring to finite-dimensional multiplicative processes, of the type (7)-(7a) or (8)-(8a). It therefore seems appropriate... 

A simple but significant example arises when we wish to consider the effect of heterogeneity in the infinite cell of a fast reactor. When the cell is uniform or homogeneous, the calculation is straightforward. It is desirable to have a relatively simple procedure for estimating the change of effective multiplication factor due to redistribution of the materials in an inhomogeneous cell when the structure and coolant are considered. 

While the actinide region has achieved much recent attention, (n,y) reactions throughout the periodic table are important for the analytic tool of neutron activation. Cd is also important for reactor control as the reaction Cd(n,y), with cr = 2.0 x 10 b, is used to control the reactor neutron flux and hence the multiplication factor k in reactor design. The reaction Xe(n,y) with cr = 2.6 x 10 b is a prominent fission-product poison that creates problems in the operation of nuclear reactors by consuming neutrons unproductively. 

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