Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2021 / Module NE1 / 2

Solution

The point kinetics model replaces the spatially and spectrally distributed neutron field by a single amplitude

n (t)

, effectively assuming a fixed or sufficiently slowly varying flux shape. The source amplitude is

s

. A single effective delayed neutron precursor population

c (t)

, delayed neutron fraction

β

and decay rate

λ

stand in for many precursor species with different lifetimes.

Λ

is the prompt neutron generation time and

ρ

the specified reactor reactivity.

The displayed equations omit explicit thermal-hydraulic, fuel-temperature and isotope-composition feedback. For the stated constant-coefficient transients,

ρ, β, Λ, λ

are prescribed rather than being solved as evolving fields. The model is useful for an elementary timescale calculation; it does not establish the spatial stability or operating limits of an actual core.

c

Solution

At equilibrium, precursor production balances precursor decay:

0 = \frac{β}{Λ} n_{0} - λ c_{0} .

Therefore

\frac{c _{0}}{n _{0}} = \frac{β}{λ Λ} = \frac{0.007}{( 0.1 ) ( 0.5 \times 1 0 ^{- 3} )} = 140.

The delayed neutron precursor population can exceed the instantaneous neutron population even when the delayed neutron fraction is small: the delayed neutron precursor reservoir persists much longer than an individual prompt-neutron generation.

d

Solution

Multiply the supplied inhour equation by

ω + λ

and collect powers of the inverse period:

Λ ω^{2} + (Λ λ + β - ρ) ω - ρ λ = 0.

With the stipulated constants this becomes

ω^{2} + 12.1 ω - 0.2 = 0,

with roots

ω_{+} = 0.0165064 s^{- 1}

and

ω_{-} = - 12.1165 s^{- 1}

. The negative root describes a rapidly decaying component. The positive root governs the late-time growth, giving

T_{+} = ω_{+}^{- 1} = 60.58 s ≃ 60.6 s .

(3)

The growing-mode e-folding time is not the precursor mean lifetime, and the negative root is a legitimate transient mode rather than an unphysical negative time.

e

Solution

After the short prompt adjustment, the prompt jump approximation takes the neutron amplitude to remain nearly in balance with the slowly changing delayed neutron precursor reservoir. For the source-free model,

0 ≃ \frac{ρ - β}{Λ} n + λ c, n ≃ \frac{Λ λ}{β - ρ} c .

Substitute this quasi-steady relation into the precursor equation:

\frac{d c}{d t} = \frac{β λ}{β - ρ} c - λ c = \frac{ρ λ}{β - ρ} c .

For constant coefficients,

c (t) = c_{0} exp (\frac{ρ λ}{β - ρ} t) .

(3)

The approximation is meaningful in the stated delayed-neutron regime with

ρ < β

and a separated fast transient. The delayed neutron precursor reservoir remains continuous across an idealized instantaneous reactivity step; the approximation does not resolve the short prompt adjustment itself.

f

i

Solution

Differentiating the supplied relation gives

\frac{d ρ}{d ω} = Λ + \frac{β λ}{( ω + λ ) ^{2}}, \frac{d ρ}{d ω}_{0} = Λ + \frac{β}{λ} .

Thus the tangent approximation is

ρ ≃ ω (Λ + β / λ)

and

T_{tangent} = \frac{Λ + β / λ}{ρ} = 70.5 s .

It is about

16.4%

longer than

60.58 s

and therefore predicts slower growth. For this positive step it is not conservative as a growth-rate estimate. The local expansion needs

∣ ω ∣ ≪ λ

; here the actual ratio is approximately

0.165

, so its error is appreciable.

ii

Solution

The precursor exponential from part (e) has inverse period

ρ λ / (β - ρ)

. Hence

T_{prompt jump} = \frac{β - ρ}{ρ λ} = 60.0 s .

This is approximately

0.96%

shorter than the exact dominant period, so it slightly overestimates growth for these stipulated constants. Neglecting the small

Λ

contribution is more accurate here than replacing the complete frequency dependence by its tangent at the origin.

iii

Solution

With no delayed neutrons, set

β = 0

and omit the precursor equation. The source-free model reduces to

\overset{n}{˙} = (ρ /Λ) n

, giving

T_{prompt only} = \frac{Λ}{ρ} = 0.500 s .