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Neutronics for critical fission reactors and sub-critical fission in hybrids Massimo Salvatores (CEA, Cadarache, France). WORKSHOP ON FUSION FOR NEUTRONS AND SUB-CRITICAL NUCLEAR FISSION FUNFI Villa Monastero, Varenna, Italy September 12 - 15, 2011. Outline.
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Neutronics for critical fission reactors and sub-critical fission in hybrids
Massimo Salvatores (CEA, Cadarache, France)
FUSION FOR NEUTRONS AND
SUB-CRITICAL NUCLEAR FISSION
Villa Monastero, Varenna, Italy
September 12 - 15, 2011
(A+1) +γ: (n, γ) capture reaction
(n+nucleus with A nucleons)= (A+1)excited
A+n+γ: (n,n‘) inelastic scattering
(A-1) +2n: (n,2n) scattering
The radius of a nucleus is given by:
where r0=1.2x10-15m and A is the nucleon number.
The following units are generally used:
The scattering cross section for the interaction of a neutron with a nucleus considered as as
spherical target, would be:
σ ~ 4πR2
For a nucleus with A~200, σ~10barn.
k is the factor by which the neutron population is multiplied going from one generation to the next
The time interval between two generations is the mean lifetime of neutrons in the medium, i.e. the time interval between their birth by fission and their desappearence by absorption or leakage out of the system.
Its order of magnitude is 10-5-10-7 sec
The evolution of the neutron population n(t) is given by:
The power from the fission is proportional to the fission reaction rate, i.e. proportional to n, according to the same exponential behaviour:
P(t) = P(0).exp[(k - 1) t/]. (P in Watt)
If t = 1 s, = 2.5x10-5 s and, respectively, k = 1.0001 and k = 0.9999 we could expect:
P(1) = 55 P(0) and P(1) = 0,018 P(0)
However, things are different due to the presence of a fraction of neutrons that are emitted at fission with some « delay » (delayed neutrons). For example in the case of U-235, 99.32 % of the neutrons are « prompt » and 0.68 % are « delayed ».
These delayed neutrons come from the radioactive decay of some fission products, with periods of ~0.2 sec- 1 min.
And this is enough to change the kinetic behavior of a reactor!
If is the fraction of delayed neutrons, their average emission delay, and the mean neutron lifetime, as before
now the average generation time separating two fissions is equal to :
= (1 - ) x + x ( + )= + .
In the case of U-235:
= 2.5 x 10-5 + 0.00679 x 11.31 = 0.077 s
The second term is dominating, and delayed neutrons change completely the overall generation time.
In the previous examples, the power increase/second is ~0.1% (and not a factor 55!) and similarly the power decrease is ~(-0.13%) and not a factor ~50.
The most general neutron transport equation (Boltzmann equation) is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows:
Minimum energy of a neutron after elastic collision is determined by the parameter α: Emin= αE where:
The fission/absorption ratios are consistently higher for the fast spectrum SFR. Thus, in a fast spectrum, actinides are preferentially fissioned, not transmuted into higher actinides
How cross section characteristics translate into neutron balance: the concept of neutron consumption/fission.
The total number of neutrons Dj consumed by the given J-family can be calculated according to the following scheme:
where PJNr J(N +1)s is the probability of transmutation of the nuclide JJNr (belonging to
the Nth generation within family J) into nuclide J(N +1)s (belonging to the (N+1)th
Generation within family J). All these nuclides are the members of the J-family.
is the number of neutrons consumed during transition AB and depends on the nuclear reaction type
Positive D means “consumption” and negative D means “production”.
In practice, starting with the “father” isotope in family J, one evaluates the probability of each nuclear reaction, calculating at the same time the number of neutrons consumed as a result of each reaction.
This procedure is systematically repeated up to (almost) complete disappearance of the J- family.
The following figure presents the U-235 chain (family structure) as a simplified example:
D (neutron consumption/fission) value for different isotopes in different systems
NScore values for a typical Light Water (Thermal) Reactors, LWRs and Sodium Cooled Fast Reactors, SFRs
As for « transmutation », several features characterize the transmutation potential of a specific neutron field for each isotope, e.g.:
Fission of isotope A should be favoured against (n,γ) and (n,xn) reactions. I.e., starting from isotope A reactions giving rise to A+1, A+2 etc should be minimized.
The isotopes that successively lead towards full fission should, as much as possible, be “neutron producers” rather than “neutron consumers”, in order to allow a viable core neutron balance and surplus
As for point 1, we have seen that the fission to absorption ratio is more favorable (i.e. larger) in a fast neutron system.
This means that a fast neutron spectrum reactor leads to fewer high mass isotopes compared to a thermal reactor, since the transuranics (TRU) isotopes are more likely to fission
As for point 2, the discussion on the neutron surplus indicates that, again, fast neutron spectra systems should be preferred
For the purpose of transmutation, both critical and sub-critical fast neutron systems can be used.
The transmutation performances of three fast neutron systems, i.e. a critical fast reactor, a Fusion-Fission Hybrid and an ADS will be shortly compared (for a specific fuel cycle scenario) at the end of the next Tutorial.
(1-f) : to the grid
For a sub-critical system (Keff < 1), the condition to have a stationarysystem is heuristically written as follows:
For an ADS:
is the number of neutrons which come back to the subcritical core if all the fission energy Ef is transformed into proton current:
f : fraction of Ef
used to feed the accelerator.
If Ep~ 0.5 - 1.5 GeV and for targets such that:
Z = neutrons/proton ~ 20 - 50
= = neutrons/fission
(with e~ 0.4, p~ 0.5)
The stationary system condition:
The fraction f of energy produced in the subcritical core used for feeding the accelerator depends on the subcriticality level:
If (average number of prompt fission neutrons per fission) = 2.8, one has:
f = 2.6 % if 1 - Keff = 0.01
f = 5.3 % if 1 - Keff = 0.02
f = 13 % if 1 - Keff = 0.05
Current of the proton beam ip
If Ef (energy released/fission) ~ 200 MeV, and W is the core power:
which correspond, respectively to the following power in the proton beam:
The current ip will be increasingly small, when the system will be closer to criticality and when the power of the sub-critical reactor will be low.