108
2 Critical Mass, Efficiency, and Yield
where the lower limit of integration of the time integral was dropped on the rationale
that it will be small compared to the upper limit. Setting G shut = t shut /τ and computing
the energy E f N G liberated to shutdown with the help of (2.138) gives the yield as
E shut ∼
E f N O
ν − 1
ν
(tshut / τ ) ∼
M tot
8
R ln ν
τ
2
,
(2.142)
where the same exponential manipulation following (2.139) has been used. As with
the approach of Sect. 2.5, the initial number of neutrons cancels out. This result
possesses the same dependence of the yield on M core and τ as does the more sophisticated analysis which led to (2.95). The dependence on ΔR is different, however:
Calculations with the CriticalityAnalytic spreadsheet for untamped cores of about
one to two critical masses show that < ρr > , α, and ΔR are roughly linearly proportional to each other, which gives the numerator in (2.95) an overall dependence on ΔR
as being roughly proportional to (ΔR)
4 . But this comparison should not be stretched
too far: (2.95) applies to an untamped core, while the model in this section is for a
tamped core. In (2.95), there is no explicit dependence on ν, which is implicit in α.
The question remains of what value of τ to use in (2.142). In the numerical
simulation program, τ was adjusted at each timestep according as the core density.
In the results related below, I use the value of τ at the time of second criticality on the
rationale that the vast majority of the energy liberation occurs in the last few fission
generations.
For the Little Boy model of Sect. 2.6 with a 53 kg core and 550 kg WC tamper,
this model predicts a yield of 29.6 kt. Because the same threshold constraint was
imposed in both models, they should both give the same expansion distance ΔR, and
they do (2.30 cm). That the yield predictions are different is to be expected as they
take very different approaches to calculating that quantity; for example, in the present
model, there is no gas/radiation pressure constant γ . In this respect, time-dependent
simulation surely does a better job of encoding the underlying physics, so we should
probably not be too alarmed by the high yield indicated here. Again, predicted yields
are very sensitive to input parameters: For the present model, leaving the core mass
at 53 kg but reducing ν to 2.3 is by itself sufficient to reduce the predicted yield to
~14 kt. For Fat Man, taking a 6.3 kg core and 230 kg DU tamper with an initial
compression ratio of 2.5, the yield emerges as 27 kt, again high compared with the
numerical simulation, but not outrageously so.
Spreadsheets are available for performing the calculations involved in this “toy”
model: ToyYield(LB).xls and ToyYield(FM).xls.
Figures 2.22 and 2.23 show some results from this model for uncompressed cores
of
235 U with WC tampers. The curves in Fig. 2.22 show the expansion distances
ΔR to second criticality for given core/tamper mass combinations. For example, a
56 kg core plus 600 kg tamper will expand by ~2.5 cm before criticality shutdown.
Figure 2.23 shows corresponding yields; for this example, Y ~ 35 kt. The numerical
simulation of Sect. 2.6 indicates Y ~ 15.3 kt for this arrangement. You should be able
to verify the ~30 kt yield predicted for the 53 kg core/550 kg tamper combination
described above.
2 Critical Mass, Efficiency, and Yield
where the lower limit of integration of the time integral was dropped on the rationale
that it will be small compared to the upper limit. Setting G shut = t shut /τ and computing
the energy E f N G liberated to shutdown with the help of (2.138) gives the yield as
E shut ∼
E f N O
ν − 1
ν
(tshut / τ ) ∼
M tot
8
R ln ν
τ
2
,
(2.142)
where the same exponential manipulation following (2.139) has been used. As with
the approach of Sect. 2.5, the initial number of neutrons cancels out. This result
possesses the same dependence of the yield on M core and τ as does the more sophisticated analysis which led to (2.95). The dependence on ΔR is different, however:
Calculations with the CriticalityAnalytic spreadsheet for untamped cores of about
one to two critical masses show that < ρr > , α, and ΔR are roughly linearly proportional to each other, which gives the numerator in (2.95) an overall dependence on ΔR
as being roughly proportional to (ΔR)
4 . But this comparison should not be stretched
too far: (2.95) applies to an untamped core, while the model in this section is for a
tamped core. In (2.95), there is no explicit dependence on ν, which is implicit in α.
The question remains of what value of τ to use in (2.142). In the numerical
simulation program, τ was adjusted at each timestep according as the core density.
In the results related below, I use the value of τ at the time of second criticality on the
rationale that the vast majority of the energy liberation occurs in the last few fission
generations.
For the Little Boy model of Sect. 2.6 with a 53 kg core and 550 kg WC tamper,
this model predicts a yield of 29.6 kt. Because the same threshold constraint was
imposed in both models, they should both give the same expansion distance ΔR, and
they do (2.30 cm). That the yield predictions are different is to be expected as they
take very different approaches to calculating that quantity; for example, in the present
model, there is no gas/radiation pressure constant γ . In this respect, time-dependent
simulation surely does a better job of encoding the underlying physics, so we should
probably not be too alarmed by the high yield indicated here. Again, predicted yields
are very sensitive to input parameters: For the present model, leaving the core mass
at 53 kg but reducing ν to 2.3 is by itself sufficient to reduce the predicted yield to
~14 kt. For Fat Man, taking a 6.3 kg core and 230 kg DU tamper with an initial
compression ratio of 2.5, the yield emerges as 27 kt, again high compared with the
numerical simulation, but not outrageously so.
Spreadsheets are available for performing the calculations involved in this “toy”
model: ToyYield(LB).xls and ToyYield(FM).xls.
Figures 2.22 and 2.23 show some results from this model for uncompressed cores
of
235 U with WC tampers. The curves in Fig. 2.22 show the expansion distances
ΔR to second criticality for given core/tamper mass combinations. For example, a
56 kg core plus 600 kg tamper will expand by ~2.5 cm before criticality shutdown.
Figure 2.23 shows corresponding yields; for this example, Y ~ 35 kt. The numerical
simulation of Sect. 2.6 indicates Y ~ 15.3 kt for this arrangement. You should be able
to verify the ~30 kt yield predicted for the 53 kg core/550 kg tamper combination
described above.