Page 32 - Dynamics and Control of Nuclear Reactors

3.3 Development of the point reactor kinetics equations 23

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dn ð ρ βÞ X
¼ n + λ i C i (3.12)
dt Λ
i¼1
β
dC i i
¼ n λ i C i (3.13)
dt Λ
An alternate form of the kinetics equations may also be developed. The development
begins by re-writing Eqs. (3.4) and (3.5) as follows:
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dn P T X
¼ L a + L 1 βð Þ 1+ λ i C i (3.14)
dt L a + L
i¼1
dC i P T
¼ L a + L β λ i C i (3.15)
i
dt L a + L
Using the same procedure and definitions as above gives
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dn k eff 1 β X
¼ n + λ i C i (3.16)
dt l
i¼1
β
dC i i
¼ k eff n λ i C i (3.17)
dt l
This is called the lifetime formulation, with l¼1/{(Σ a +leakage operator)v}.
Note that the generation time and lifetime are equal for a critical reactor. Upon a
change in the reactor’s absorption cross section (as in motion of a control rod) the
generation time is expected to stay unchanged while the lifetime would undergo a
small (essentially negligible) change. The generation time formulation will be used
in subsequent sections of this book.
The simple derivations provided above are rigorous and independent of reliance
on specific quantities from reactor physics other than the generation time or lifetime.
Of course, reactivity or k eff could be computed from first principles for simulating a
specific situation. But, that is not how transient simulations are normally done. Tran-
sients are calculated for specified changes in reactivity or k eff , not for some change in
a fundamental nuclear property.
It should be noted that there are several different ways to express the neutron mul-
tiplication factor, k eff or reactivity, ρ. Different authors chose among the following
measures:

k eff
ρ (reactivity)
mk (¼ 0.001 k eff )
per cent mill or pcm (¼ 0.00001 k eff )
Δk/k
%Δk/k¼(0.01*Δk/k)