Page 67 - Dynamics and Control of Nuclear Reactors

6.2 Dynamics of xenon-135 59

dX=N f
¼ γ σ f Φ + λ I I=N f X=N f σ aX Φ λ X X=N f (6.4)
X
dt
or
dI 0
¼ γ σ f Φ λ I I 0 (6.5)
I
dt
dX 0
0
¼ γ σ f Φ + λ I I X σ aX Φ λ X X 0 (6.6)
0
X
dt
where
0
I/N f ¼ I (the number of I-135 atoms per fissile atom in the reactor)
0
X/N f ¼ X (the number of Xe-135 atoms per fissile atom in the reactor)
6.2.4 Steady state Xe-135
The steady state quantities of I-135 and Xe-135 are obtained by setting the derivative
terms in the equations equal to zero. The result is
γ σ f Φ
0 I
I ¼ (6.7)
ss
λ Ι
ð γ + γ Þσ f Φ
0 X I
X ¼ (6.8)
ss
ð λ X + σ aX ΦÞ
Note that I ss increases in proportion to the neutron flux, while X ss increases in pro-
0
0
portion to the neutron flux at low flux levels (when λ X » σ aX Φ is small) and reaches
saturation at high flux when λ X « σ aX Φ is large.
Since I-135 becomes Xe-135 upon radioactive decay, its steady state value is a
reservoir of future Xe-135. Therefore, the ratio of steady state I-135 to steady state
Xe-135 serves as an indicator of future Xe-135. The ratio is
I 0 γ λ X + σ aX ΦÞ
ð
ss ¼ I (6.9)
X 0 ss λ I γ + γð X I Þ
Inserting values for fission yields and decay constants gives (using a moderator tem-
perature of 300°C in evaluating the Xe-135 absorption cross section resulting in
6
2.22 10 b and using fission yields and decay constants for fission in U-235)
I ss 0 13 Φ
X 0 ss ¼ 0:695 + 0:738 10 (6.10)
Fig. 6.1 shows the ratio of steady-state I-135 concentration to the steady-state con-
centration of Xe-135.
0
Note that X ss is slightly greater than I ss for small flux levels, but I ss exceeds Xe-
0
0
135 at higher flux levels.