Fission Reactors Physics 1

Formule a quattro e sei fattori nella reazione a catena di fissione nucleare

If we now insert these new definitions into our earlier expressions for the infinite medium multiplication factor we get k_fp = η ε_∞. This is known as the four-factor formula. Moreover, one now writes k_fp P_P = η ε FNL TNL, which is known, surprisingly enough, as the six-factor formula. The values of each of the factors in the six-factor formula, for a typical thermal reactor, are:

Of course, the calculation of these factors is quite difficult in general. Nevertheless, the four-factor and six-factor formulas are quite useful because they provide insight into the various mechanisms involved in nuclear fission chain reactions.

One can also vary the non-leakage probabilities by simply making the reactor core larger or surrounding the reactor by a material with a large scattering cross-section so that some of the neutrons leaking out will be scattered back into the reactor. Actually, when leakage is changed, there will be some change in the parameters in the four-factor formula.

as well, since these are actually averages over the various neutron energies in the reactor, and this distribution of energies will vary with the amount of leakage. Such considerations have given rise to a different notation for the multiplication factor characterizing a finite system, which is occasionally referred to as the effective multiplication factor and denoted by k_eff.

How to adjust k

There are several ways to adjust k in the initial design of the reactor. One could first regard the size of the reactor as the design variable, since the ratio of surface area to volume decreases as the reactor geometry is enlarged and one can control the relative importance of the leakage factors by adjusting the reactor size. Most thermal reactor cores are so large that leakage represents a rather small loss mechanism (typically about 3% of the neutrons leak out from the core in large thermal reactors); usually the core size and geometry for a power reactor are dictated by thermal considerations. An

An alternative way to achieve the same reduction in leakage is to surround the reactor with a scattering material that acts as a neutron reflector.

The primary design variable at the disposal of the nuclear engineer is the core composition. In particular, he can vary the composition (enrichment) and shape of the fuel, the ratio of fuel to moderator density, the type of moderator, coolant, and structural materials used, or the manner in which reactor multiplication is controlled.

One would refer to the amount of fuel required to achieve a critical chain reaction as the critical mass of fuel.

In reality, however, a nuclear reactor is always loaded with much more fuel than is required merely to achieve criticality. This extra multiplication is required for several reasons: first, if the reactor is to operate at power for a period of time, one must provide enough excess fuel to compensate for those fuel nuclei destroyed in fission reactions during the power production - the so-called burnup; second,

one must include enough extramultiplication to allow for reactor power level changes. Of course when this excess multiplication is not beingused, some mechanism has to be provided to cancel it out to achieve reactor criticality; this is the function ofreactor control mechanisms.

C 4: N R DHAPTER UCLEAR EACTOR YNAMICSFor a nuclear reactor to operate at a constant power level, the rate of neutron production via fission reactionsshould be exactly balanced by neutron loss via absorption and leakage. Changes from the stationary conditionmay occur for a number of reasons: for example, the reactor power level may change via control rodadjustments; or there may be longer term changes in core multiplication due to fuel depletion and isotopicbuildup.

It is important that one be able to predict the time behavior of the neutron population in a reactor core inducedby changes in reactor multiplication. Such a topic is known as nuclear reactor kinetics. However, we shouldrecognize that the core

multiplication is never completely under the control of the reactor operator. Indeed, since multiplication will depend on the core composition, it will also depend on other variables not directly accessible to control (such as the fuel temperature or coolant density distribution throughout the reactor), but these variables depend, in turn, on the reactor power level and hence the neutron flux itself. The study of the time-dependence of the related processes involved in determining the core multiplication as a function of the power level of the reactor is known as nuclear reactor dynamics and usually involves a detailed modeling of the entire nuclear steam supply system, either of its neutronics and its thermohydraulic.

The principal applications of such an analysis are not only to the study of operating transients in reactors, but also to the prediction of the consequences of accidents involving changes in core multiplication, and to the interpretation of experimental techniques measuring.

reactor parameters by inducing time-dependent changes in the neutron flux. However, what is important when talking about nuclear reactor dynamics is to keep in mind~ 38 ~the time scale of the possible events; in particular, the problems of dynamics can be classified in:

• fast transients (μs to s): nuclear explosives, pulsed reactors, reactor accidents;
• short-term operating transients (minutes to hours): are the operation related to control, such as start-up,
• effects of power level on the reactivity of the reactor, fluctuations of power demand dictated by the external electrical grid;
• long-term operating transients (days to months): fuel consumption (or burn-up), breeding and
• conversion of the fuel, effects of radiation on materials.

4.1 POINT REACTOR KINETICS PROBLEM

We already introduced the concept of precursors, those fission products that emit neutrons with a considerable time delay after the fission event (some seconds). Such neutrons typically account to less than

6β= Σ β ii=1 235

1 6T T= Σ β1/ 2 i 1β /2,ii =1

1 1 16= Σ βiλ β λi=1 i

The neutron flux in a simple bare slab reactor could be written as a superposition of spatial modes (eigenfunctions) characteristic of the reactor geometry, each weighted with an exponentially varying time-dependence:

Φ(r, t) = Σ A exp(-λ t)Φ(r) ψ(r)

Here the spatial eigenfunctions were determined as the solution to the eigenvalue problem:

2 2 r∇ψ + B ψ(r) = 0, ψ(∼) = 0

while the time eigenvalues λ were given by:

λ = 2D B v(ν + ν Σ - ν Σ)

These eigenvalues are ordered as -λ > -λ > ... . Hence, for long times, the flux approaches an asymptotic form:

Φ(r, t) ≈ A exp(-λ t) Φ(r) ψ(r)

where we identify the mean lifetime of a neutron in the reactor:

τ = [(ν - 1) + L BΣ()]ag

and the multiplication factor:

ν = Σkf/(Σka)

In this sense, n(t) can be interpreted as the total number of neutrons in the reactor at time t. Actually, since the normalization of n(t) is arbitrary, we could also scale the dependent variable n(t) to represent the total instantaneous power P(t) being generated in the reactor core at any particular time. That is, we could let nPv n(t) → P(t) = wΣ(t)fff where w is the usable energy released per fission event. Since the reactor power level is usually a more convenient variable to monitor, we will frequently express the reactor time-dependence in terms of P(t). In this sense we have derived a "lumped parameter" description of the reactor in which the neutron flux time behavior is of the form of the product of a shape factor Ψ(r) and a time-dependent amplitude factor n(t), 1 k-1, t) = vnne^tΦ(r(t)Ψ(r) = v[( )]Ψ(r)10l characterized by a time constant called reactor period lT = k-1. However, there is, of course, something very important.