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


Подпись: (2.32)b = b


Подпись: Is —Подпись:image052(2.33)

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

r, (K — 1)/k

Подпись: (2.34)

Подпись: yields Подпись: 1s T image056 Подпись: 1 X bjtj Подпись: N = 1CA/S
Подпись: (2.35)

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


=1 (tjs + 1 2

Подпись: dF ds Подпись: 1(1 — r) + Подпись: jj Подпись: (2.38)

s—an estimated location of the dominant pole and

image065 Подпись: p Подпись: (2.39)

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

image068 Подпись: p Подпись: (2.40)

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


Reactivity ($)

Figure 2.5 Typical Variations of Reactor Period as a Function of Reactivity

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.

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