While a reactor is operating at steady power the chain reaction, described in Chapter 1, Section 4, is just sus­tained and the multiplication constant kerr is exactly equal to unity. If there is any change in any of the items which make up the value of keff, the value of keff will deviate from unity and the reactor power will change. For example, a control rod may be moved up or down; reactor gas temperature may change due to a change in feedwater flow into a boiler, or gas circulator speed may change thereby upsetting the balance of temperature within the reactor core, and changes in temperature lead to changes in keff via the temperature coefficients described in Section 3 of this chapter.

Reactor kinetics is concerned with deviations of keff from unity and the effect that the deviation has on the neutron population within the reactor; if the neutron population changes, the rate of fission will change, so the reactor power will also change. Reactor kinetics is concerned with the magnitude of such changes and the way in which the neutron population varies with time.

2.4 Prompt neutrons

In the fission process, most of the resultant neutrons appear within 10“ 14 s of the fission occurring; these neutrons are called ‘prompt neutrons’.

In Chapter 1, Section 6 the relationship between the multiplication constant к and neutron life cycle was described. Let us consider how the neutron po­pulation caries with time, by following the ecents as the neutrons go through their life cycle. At the be­ginning of the cycle, suppose the thermal neutron population in the reactor core amounts to nt neu­trons, let us call this the first generation. These П| neutrons will be absorbed in the various reactor materials (some will escape from the core), some which are absorbed in the fuel will cause fission and the fast neutrons produced in fission will be slowed down to thermal energies (some fast neutrons will escape from the core, some will be absorbed during the slowing- down phase). When one round of the neutron life cycle has been completed there will be a new thermal neutron population of ny neutrons, let us call this the second generation. The number of neutrons in the second generation may be expressed in terms of the number in the first generation and the multiplication constant:

ny = keff x щ

The change in neutron population from one generation to the next, fin, is then:

fin = П2 — щ = (kerr — 1) x щ (3.10)

The term (kelf — 1) is the deviation of kerr from unity, and is given the symbol Дк:

Дк = keff — 1

So for Equation (3.10) we can write:

fin = Дк x n] (3.11)

In a power reactor the value of kerr does not deviate markedly from the value unity. While the reactor is at power the value of к0rr lies in the range 0.998 to 1.002, so the deviation Дк is very close in value to the reactivity q which is defined as:

kef: ~ 1

e = —————

к ett

Therefore, for all practical purposes, Дк is a close approximation to reactivity, and it is common prac­tice to refer to л к as reactivity.

The timescale over which the change in neutron population on occurs is the time taken to go exactly once round the neutron lifecycle. This is called the ‘mean neutron lifetime’, 1*. and its value depends on the reactor system. The mean neutron lifetime is made up of two components, ‘slowing-down time’ and ‘diffusion time’:

• ‘Slowing-down time’ is the time from production of a fast neutron by fission to the neutron be­coming a thermal neutron.

• ‘Diffusion time’ is the time the thermal neutron spends diffusing through the reactor materials be­fore it is absorbed.

Typical values of slowing-down time and diffusion time in a graphite-moderated reactor are approximate­ly 0.33 ms and 0.66 ms respectively, adding to give a mean neutron lifetime 1* of approximately 1 ms. The time from absorption of a neutron in a fissile nucleus, through the fission process to the fast neu­trons being released, is several orders of magnitude smaller than the slowing-down and diffusion times, so is negligible.

The diffusion time of 0.66 ms may seem surpris­ingly long, since it makes up two-thirds of the neutron lifetime and is unproductive. It is a consequence of two main factors. First, there are many more mod­erator atoms in the core than fuel atoms, the ratio being approximately 100 : I (see Chapter 1, Section 6), so there are more opportunities for the neutrons to collide with the graphite atoms than with fuel atoms, Second, the neutron capture cross-section of graphite is very low (see Chapter 1, Section 5), so it is to be expected that the neutrons will be able to exist in the graphite for a relatively long time before being ab­sorbed. In a PWR, by comparison, the ratio of mod­erator atoms to fuel atoms is much lower, approxi­mately 4:1, and the neutron capture cross-section of water is higher than that of graphite, so the diffusion time in a PWR is shorter than in a magnox reactor or an AGR. The slowing-down time in a PWR is also shorter than in a magnox or an AGR, because it requires fewer collisions to slow down a neutron in a hydrogenous moderator than in a graphite moderator. Thus the mean neutron lifetime 1* in a PWR is shorter than in a magnox or an AGR, a typical value of 1* for a PWR is between 10-i and 10~4 s. It will be seen later that the time-dependent behaviour of the neutron population in a power reactor is most strongly influenced by factors other than mean neu­tron lifetime, so these differences between PWR and graphite-moderated reactors are of little consequence in practice.

The change in neutron population fin takes place over a time interval fit which, in the case w:e have considered in Equation (3.11), is the mean neutron lifetime I*. Thus the rate of change of neutron popu­lation, from Equation (3.11), is given by:

<5n Д к x n <5t 1*

Ak 1*

which give rise to delayed neutrons (in the case of Br-87, this is 2%) and divided by the total number of neutrons released per fission (because the fission yield is per fission, but delayed neutron yield is to be expressed as per neutron). Thus ue ha e:

2,6’ro x 2го

debased neutron yield = ———————- = 0.02 PP

2.438

• ‘Mean life’ is related to the halflife of the pre­cursor. Mean life is similar in concept to a time constant in other branches of science and engineer­ing which deal with exponential behaviour, see Fig 3.17, Mean life is related to halflife by the factor log,, 4 as follows:

halflife = mean life x logn 4 (3.14)

Br-87 has a halflife of 55 s, so the mean life of the delayed neutrons is 79 s. Note that the term ‘mean life’ does not refer to the life of the neutron, it refers to the delay between the fission occurring and the delayed neutron being formed.

There are over 50 fission products which can give rise to delayed neutrons in this way. For convenience they are grouped initially into six groups, each group characterised by a yield and a mean life (Table 3.1). Table 3.1 shows that Br-87 accounts for virtually all the delayed neutrons in group 1.

Tvbi. E 3.1 Delayed neutron yields from thermal fission of U-235 Delayed neutron group, і Mean life, г, seconds Yield, (3j 1 79 0.025 2 31 0.154 3 7.8 0.134 4 3.3 0.258 5 0.7 0.089 6 0.3 0.016

It is convenient to combine the six groups into one

average group as follows:

6

total yield, 3 = ^ d, = 0.68^0

і = і

The mean life is determined by weighted averaging:

6

З-, x ті

…. i = i

mean lire, г = ————— = P 9 s

6

J———— L Fig. 3.17 Relationship between mean lire and halt’ll(e The curve shows an exponential deea, tor example, the decay of a fission product giving rise to delated neutrons. Halflife is the time taken to reach half of its initial value, mean life is the time taken to reach l/e of its initial value where e is the base of natural logarithms.

Although delayed neutrons account for only 0.68 °"o of the total neutron population in the reactor, their mean life of 12.9 s is very much longer than the mean neutron lifetime of prompt neutrons, 1 ms, so the delayed neutrons have a very marked effect on reactor kinetics. An ‘effective neutron lifetime’, le, may be calculated:

U = 1* + (£ — Дк) x r (3.15)

In this calculation, care must be taken to express 3 and Дк as ‘per unit’ values. For example, calculate. the effective neutron lifetime in a reactor with a reactivity of +-50 mN:

le = 1* + (3 — дк) x t

= 0.001 + (0.0068 — 0.0005) x 12.9 = 0.082 s = 82 ms

The effective neutron lifetime is applied to the neu­tron kinetics equation as follows;

Compare the equation for prompt neutrons only, Equation (3.13). Equations (3.13) and (3.16) are shown in Fig 3.18 for a reactivity of +50 mN. The marked effect of the delayed neutrons is clearly demonstrated.

In reality, the time-dependent behaviour of the, neutron population is a combination of the prompt and delayed neutrons, Consider a reactor operating at steady power, then suddenly being subjected to a

step change in reactivity of +200 mN. The effect is as shown in Fig 3.19. The neutron population in­creases rapidly at first, as the prompt neutrons re­spond quickly to the change in reactivity. The prompt rise is followed by a steady increase, as the delayed neutrons respond slowly. Figure 3.19 also shows the effect of a -200 mN step in reactivity, namely a prompt drop followed by a steady decay.

An interesting result is obtained from Equation (3.15) if a reactivity of +680 mN is inserted. In this case the effective neutron lifetime le becomes equal to the mean neutron lifetime 1[32], i. e., 1 ms. This is known as ‘prompt criticality’, and the result is that the reactor will diverge as though the delayed neutrons did not exist. In power reactors it is of academic interest only, for the following reasons:

• There is no credible means by which a net reactivity

of ^680 mN can be created.

Fjg. 3.19 Effect on the reactor of a step change
in reactivity

of neutron population. Not only will this tell him how quickly the reactor power is changing, but it will also indicate the magnitude of the disturbing influence, i. e., the deviation of kerf front unity. We have seen above that the neutron population changes exponentially under the action of a given value of reactivity дк. Clearly the simple rate of change of neutron population 6n/St will change in value as the exponential proceeds, see Equation (3.12) and Fig 3.19, so a meter showing this parameter would be of limited value. In dealing with an exponential change it is convenient to take the logarithm of the func­tion, for example, take the logarithm of Equation (3.16):

logn = (n/no) = Дк/1е x t (3.17)

Then the differential of Equation (3.17) gives a rate
of change which is constant for a given value of Дк:

4.3 Practical operation

4.3.1 Rate of change instrument

It is useful for the reactor operator to have an in­dication on the control desk of the rate of change

o/6t [logn (n/no)] = Дк/Іе (3.18)

By this method, a useful rate of change indication can be presented to the operator. Note that it is directly proportional to the deviation Дк. The units

of this rate of change parameter are not immediately obvious, and a slightly different approach yields a more useful unit as follows.

Returning to Equation (3.16), consider the time taken for the neutron population to increase from its initial value n(> at time zero to a value e x n0 uhere e is the base of natural logarithms (e = 2." 18, etc.). In Equation (3.16). set n to the alue e x n, j and determine the time T to reach this value:

e x no = n0 exp( Д к 1L, x t) therefore T = le /Д к

This time period for the neutron population to change from some initial value to e times that initial value is called the ‘reactor period’;

reactor period, T = 1е/Дк (3.19)

Note that the reactor period is the inverse of the rate of change parameter given by Equation (3.18). Equation (3,16) can now be rewritten:

n = no exp(t/T) (3.20)

A Tate of change’ meter on the reactor control desk which derives its signal from the differential of the logarithm of the neutron population (more precisely, of the neutron flux, see Section 6 of this chapter) can now be calibrated in units of reactor period. A ariation on reactor period is ‘doubling time’, i. e., the time taken for the neutron population to double. From Equation (3.20), determine the time taken for the neutron population to double from its initial value n0 to 2 x no:

2 x no = no exp(t/T) (3.21)

therefore doubling time = reactor period x Iogn2

Thus the principle of our Tate of change’ meter is the same, but the scaling is different. Doubling time is a more convenient unit because it is easier to

understand.

One of the properties of a pure exponential is that us doubling time is constant for a given value of Дк, ‘■e., the reactor will take the same time to double from 1 MW to 2 MW, 2 MW to 4 MW, 4 MW to 8 MW, etc., so for a constant value of Дк the dou­bling time meter will settle to a steady reading during the transient. Figure 3.20 shows a scale for a meter to display doubling time or reactor period. For com­parison, Fig 3,20 also shows corresponding values ot the rate of change parameter k/le given by Equation (3.18), it also shows the corresponding val — aes of reactivity дк.

It should be noted that the numerical work neces­sary to produce the above results for the effect of delayed neutrons has been carried out using an aver­age group of delayed neutrons with a yield of 0.68ro and a mean life of 12.9 s. The true behaviour of a power reactor is more complex, with many delayed neutron groups as mentioned in Section 4.2 of this chapter. The method of combining the delayed neu­trons into one average group is adequate for many purposes, particularly when studies are confined to small values of the deviation Дк. Table 3.2 gives a comparison of doubling times derived from a ri­gorous treatment of delayed neutrons and doubling times derived from one average group using Equations (3.15), (3.19) and (3.21).

Comparison of the single average group model with a 6-group model (see Table 3.1) is given in Fig 3.21 for a step change in reactivity of — 100 mN, similar to that shown in Fig 3.19. It can be seen in Fig 3.21 that there is very little difference between the two models.

0 25 50

ГМЕ SECONDS

Fig. 3.21 Comparison of 6-group and single average
group models