In the most frequently used form of the kinetics equation the energy and space — dependence is eliminated averaging over all energies (one-group approximation) and assuming that the flux, source and delayed neutron precursor concentrations are separable in space and time:

Ф(г, t) = fair)<f(t),

C(r, t) = fo(r)C,(t),

S(r, t) = /„(r)S(t),

where /u(r) is the fundamental mode of the flux, first eigenfunction of the equation

v2/ + b2/ = o.

This means that the flux shape during the transient is supposed to remain the same as in the steady state. This approximation is sufficient in many cases. Higher modes of the flux distribution are excited during the transient, but they decay very rapidly. As the heat capacity of HTR fuel is very high, these short-lived higher modes do not last long enough to produce appreciable temperature increases and can therefore be neglected. The assumption of space-time separability is then valid in all cases in which the fundamental mode of the flux does not appreciably change during the transient.

After introducing the above expressions for ф(г, t), Ci(r, t) and S(r, t) into eqn.

(12.1) and substituting V2/o by ~B2fa the following equations are obtained:

-^~ = C-p)i’^it) + fJ ,C(t) + S(t) — ОВ2ф«) — Хаф(1),

V dt (12.2)

= Р, сЪ! ф(1) — ЛіСі(0 (f = l,…m)

In most cases S(t) = 0 and then, neglecting the delayed neutrons eqn. (12.2) reads:

(12.6)

If we eliminate all sources (including fission) from the first eqn. (12.2) we have

— фП is then the decay rate of the neutron flux in absence of all sources, whence the name “neutron lifetime”.

Equation (12.5) is the general equation describing a population growth where kctt is the gain between two generations. Considering also the delayed neutrons the zero­dimensional one-group equation takes the form

Equation (12.7), which can be easily obtained from (12.2), (12.3) and (12.4), is used for most space-independent dynamic calculations and is a simple equation if A к is independent of feedback effects. As A к is usually a function of temperature, the problem is complicated by the equations describing the temperature feedback, and eqn. (12.7) is no longer linear. In the most general case A к can represent also the whole control system feedback. Defining

kcff~ 1 А к. . л оч

p=— ——- = -— reactivity, (12.8)

kctt kctt

A = generation time. (12.9)

kctt vvZf

Equation (12.7) can take the form

Фф p ~ (3 , , — Vі, c,

-cr = — 7- Ф + V 2j AiCi + vS,

ut а і = і

dCt /3,

—77 = 77- ф — A, C, (i = 1,. . . m ).

dt Av

In a critical reactor l = Л, and in most cases the difference between l and Л is negligible compared to the uncertainty with which l is known. Equations (12.7a) can be still further simplified defining

Д = p — /3 prompt reactivity.

If ДзїО the reactor is critical on the prompt neutrons alone (prompt critical).

The method we have used here to obtain the one-group space-independent kinetics eqns. (12.7), while giving a clear picture of the approximations to be made, does not show explicitly the way of obtaining the energy — and space-independent constants appearing in these equations, starting from the energy — and space-dependent reactor parameters.

Because of this reason we show here also a different method of obtaining eqn. (12.7), based on perturbation theory (see for details ref. 4 or ref. 1, p. 179), starting from the time-dependent Boltzmann equation [or simply from eqn. (12.1)] and from the time-independent adjoint equation of a just-critical reference system. The flux is expressed as the product of an amplitude <f>(f) multiplied by a shape function/(r, E, t)

ф(г, Е,П = 4>(t)f(r, E,t). (12.10)

In order to obtain the space-independent equations, the time dependence of / is ignored.

Substituting (12.10) in (12.1), multiplying (12.1) by the time-independent adjoint flux Фо(г, E), multiplying the adjoint equation by ф(г, E, t), subtracting the resulting equations and integrating over volume, angle and energy it is possible to obtain eqns. (12.7a).

In this case the coefficients of (12.7a) have the following definitions:

Л(0 = 4: f —f(r, E, f)<^o(r, E) dV dE, (12.11)

r J V

0.(0 = -^ f x<(E)Pii’Zf(r, E’)f(r, E’,tWo(r, E)dVdEdE’ (12.12)

with

E = J x(E)vi,(r, E’)f(r, E’, t )фо{г, E) dV dE dE’.

It can easily be seen that eqn. (12.11) represents an energy and space weighted version of eqn. (12.9).