Section 2.4 describes how power production changes the effective multiplication factor of a nuclear reactor so that

K = K (N) (2.31)

As a result equation (2.26) is non-linear. However, for control analysis it is sufficient to examine this equation for small perturbations about an operating point for which the effective multiplication factor is a constant derived from neutron diffusion and possibly thermal-hydraulic calculations. Effecting the Laplace transformation of equation (2.26) and involving the definition

j

b = b

j=1

(2.33)

For analytical purposes, equation (2.33) is more conveniently expressed in terms of the core reactivity

r, (K — 1)/k

to give

The algebraic roots of

^ bi’Tj’S

F(s) = Is — r(1s + 1)^^ = 0 (2.36)

j=f rjs + 1

define the poles of the neutron population’s kinetics, and the real root closest to the origin is termed dominant whose reciprocal T* defines the reactor period. As reactivity is increased from zero, the dominant pole moves into the right half s-plane causing the neutron population to diverge exponentially in the form Noexp (tT*). The location of the dominant pole s* can be determined iteratively using the Newton — Raphson algorithm

where

s—an estimated location of the dominant pole and

For small enough reactivities, the reactor period is estimated from the above equations as

which from Table 2.2 further approximates to

so that the radioactive decay periods of the precursor groups govern the growth of a neutron population. On the other hand for large reactivities the dominant root of equation (2.36) is far removed from the {1/tj}, and under these conditions it behaves as

1s — p(1s + 1)+ b = 0 (2.41)

The corresponding reactor period is then asymptotic to

1(1 — p)/(p — b) (2.42)

which is dominated by the lifetime of prompt neutrons, because as seen from Figure 2.4 these alone constitute a self-sustaining subpopulation. Typical data in Table 2.2 and the above analysis enable a simple digital computation of the stable reactor period as a function of normalized reactivity (p/b). Reactivity are often quantified in this normalized form for operational purposes, and the so-called prompt critical value of unity is ascribed a magnitude of one dollar ($1) with corresponding subdivisions of cents. Once a reactor enters the super-prompt critical regime, the reactor period is seen from Figure 2.5 to decrease dramati­cally: especially for fast reactors.[34] Bearing in mind the typical rate

constraints associated with induced thermal stresses in a power plant, and the interval necessary for emergency intervention, then reactivity changes in normal operation must be restricted to a few cents (1/100th of 1$) to achieve a stable reactor period of no less than about 30 seconds [80,117]. After circumspect increases in reactor power by restricting withdrawal of control rods, the negative reactivity feedbacks described in Section 2.4 restablize a neutron population. In terms of equation (2.40) such increases in reactor power or neutron population correspond to an infinite reactor period T* with K = 1 or p = 0. An alternative viewpoint from equation (2.36) is that a neutron population (reactor power) in equilibrium corresponds to the dominant pole at the origin with p = 0.