Материал: Апсе ENVIRONMENTAL PROTECTION 2014
4.2. Energy yield of NED with reactor-grade plutonium
There are two different, in principle, types of NED.
One of them is based on the implosion effect and can be named as the implosive NED. In the implosive NED a spherical charge (metal δ- plutonium, for instance) is surrounded by annular layer of high chemical explosive acting, in addition to its main mission, as a neutron reflector and reducing the critical mass by a factor of two, or so, below the bare critical mass. The ingoing shock wave from the high explosives compresses the central plutonium charge, the charge-reflector system becomes supercritical, and nuclear explosion can occur. So, the inwarddirected shock wave from the high explosives creates the conditions for initiation and fast propagation of the chain fission reaction (CFR) accompanied by release of huge energy amount (energy yield).
Another type of NED, so-called the gun-type NED, is based on fast joining of two sub-critical masses into one supercritical system where the CFR could be initiated and propagated.
These two NED types, in addition to their evident constructive peculiarities, are characterized by substantially different times needed to transform the NED into the state with maximal super-criticality. In this respect the implosive NED demonstrates an obvious superiority over the gun-type NED. The implosive NED can transform itself into the maximal supercriticality state for a substantially shorter time interval, by one order of magnitude, as compared with the gun-type NED. In the implosive NED the inward-directed shock wave from the high explosives moves with mean velocity of 5 km/s and compresses the plutonium charge (~5 cm in diameter) for about 10-5 s. In the gun-type NED two sub-critical masses can move to meet with mean velocity of 300 m/s, and maximal supercriticality, after completion of the assemblage process, can be achieved for about 10-4 s, i.e. ten times slower. The slower actuation of the guntype NED can produce a negative effect on the CFR initiation and propagation.
Mathematical model for the CFR initiation and propagation
Let assume that time dependency of the NED reactivity can be described by a simple linear function. Initially, the NED is a sub-critical
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system with effective neutron multiplication factor |
КEFF is near but |
less than unity ( КEFF » 0.99 ). When the NED |
is activated (high |
chemical explosive begins compressing the central plutonium charge or two sub-critical masses begins moving to join), the system gradually becomes super-critical. Let t = t1 be the time moment when the system
becomes critical ( КEFF = 1), and t = t0 be the time moment when the system reaches maximal super-criticality ( КEFF = 2 ): the shock wave came to the center, or two sub-critical masses formed a single mass. Since initial state is very close to criticality, i.e. t1 » 0 and
КEFF (t) = 1 + t . t0
At t ³ 0 the CFR can be initiated by the appearance of some neutrons. In principle, the CFR can not be initiated at all, if no neutrons appeared in the system during the time interval [0, t0 ]. At t ³ t0 the
system would be destructed and flown away.
The flying-away stage can begin when the fissile material overheats, vaporizes and creates so high internal pressure that can stop the further compression or mutual rapprochement. Numerical evaluations have
demonstrated that e42 fission reactions could produce дthermal energy that is large enough to overheat and vaporize fissile materials. Unfortunately, these evaluations could give only qualitative results. That is why the further analysis was oriented on somewhat larger value of e45 fission reactions as a threshold that defines transition of the system from the compression stage to the flying-away stage, i.e.
tF
e45 = ∫ N f (t)dt;
ti
where ti - time moment of the CFR initiation; t f - time moment when
the CFR finishes because e45 fission reactions already occurred; N f (t) - fission rate.
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Time dependency of the fission rate in a nuclear charge can be written as follows:
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N f ( t ) = N f ( 0 )×exp( ∫ |
КЭФ( t¢ ) - 1 |
dt¢ ) = N f |
( 0 )×exp( |
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lprompt |
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where N f (0) = S f |
× F(0); |
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a nuclear charge; lprompt - prompt neutron lifetime ( 10-8 s). |
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Then: |
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tF |
t2 |
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e45 |
= Σ f ×Φ ( 0 )× ∫ exp( |
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t |
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If the integral in the right part is replaced by its approximate value, then:
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e45 = ( t f |
- ti )×Σ f |
×Φ ( 0 )×exp |
t f |
- ti |
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2lprompt ×t2 |
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By using the following identity:
(t f - ti ) × S f × F(0) º exp{ln[(t f - ti ) × S f × F(0)]},
the balance equation for the required quantity of fission reactions can be re-written:
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t f |
- ti |
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= exp ln |
( t f |
- ti )× |
Σ f ×Φ ( 0 ) |
+ |
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2lprompt ×t2 |
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The summand |
ln[(t f - ti ) × S f |
× F(0)] in the exponential function |
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defines the contribution given by neutrons-initiators of the CFR into
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total quantity of fission reactions. It seems evident that this contribution is negligibly small in comparison with total contribution from all fission reactions taking place during the CFR propagation. If the summand is removed, then we can derive the equation that links the time moments related with the CFR initiation and completion:
t 2f |
- t 2i |
= 90. |
(4.1) |
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Maximal and minimal energy yields
Energy yield of a nuclear explosion could achieve its maximal value only if the initiated CFR would not be able to prevent the nuclear charge from reaching maximal super-criticality. In other words, if e45 fission reactions can occur only by the time moment t f ³ t0 , then the
CFR can not stop the compression stage. As it follows from equation (4.1), the CFR must not be initiated too early, i.e.
t |
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×( 1 - |
90 ×lprompt |
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t0 |
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In the implosive NED t |
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s. It is easy to calculate that, in or- |
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der to produce maximal energy yield, the CFR must be initiated at 0.954·10-5 s. In this case, e45 fission reactions can occur only for the remaining 0.046·10-5 s, and effective neutron multiplication factor КEFF can reach its maximal value.
Minimal energy yield (“fizzle”) could be produced i f the CFR is initiated just at the time moment when the system became critical, i.e. at ti = 0 . This is the worst moment for neutrons-initiators to appear in the system. As it follows from equation (4.1), in this case the CFR lasts about 3·10-6 s. The necessary quantity of fission reactions ( e45 ) occurs in the system, and effective neutron multiplication factor increases up to
КEFF = 1,3 .
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Similar evaluations can be carried out for the gun-type NED where t0 = 10−4 s . In order to produce maximal energy yield, the CFR must be initiated at 0.995·10-4 s. Then, e45 fission reactions can occur for the
remaining 0.005·10-4 s, |
and effective neutron multiplication factor |
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can reach its maximal value. Minimal energy yield could be pro- |
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duced if the CFR is initiated at ti = 0 . Then, the CFR lasts 9,5·10-6 |
s, |
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multiplication factor increases up |
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= 1,095 only. |
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Premature initiation of the CFR by neutrons that appeared in the system before an optimal time moment is called as the pre-detonation regime. It is evident that energy yield from the pre-detonation regime is lower than maximal (nominal) value of energy yield, and minimal energy yield (“fizzle yield”) is produced if the CFR is initiated at the very inappropriate time moment (at ti = 0 ).
As is known, main fraction of energy yield Y releases at the flyingaway stage, and Y is defined by maximal value of effective neutron multiplication factor achieved at the time moment when the CFR
propagation finishes, |
i.e. energy yield is directly proportional toα 3 |
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where |
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pre-detonation regime YFIZZLE and nominal |
energy yield YNOM |
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linked by the following relationship: |
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Y |
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= ( |
КEFF ( t f |
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− 1 |
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FIZZLE |
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where |
КEFF ( t f )NOM |
= КEFF ( t2 ) = 2 , |
КEFF ( t f )FIZZLE = 1.3 for |
the |
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implosive NED and |
КEFF ( t f )FIZZLE = 1.095 |
for the gun-type NED. It |
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