In ref. 11, a reactivity of 170 /s is injected in a critical core (Wo = 1 GW), or in the same core made sub-critical at -1 , - 2 , - 3 . The results show that prompt criticality is reached in the critical core after 6 ms with a first power peak of 700 GW at 8.5 ms and a second peak of 500 GW at 13.2 ms. In the sub-critical mode, the peaks are respectively of 530 GW at - 1 , 6 GW at - 2 and 2.2 GW at - 3 (t = 16 ms) (see Figure 2).
In the case of loss-of-coolant-flow accidents, references 10 and 11 give simple examples, which show that, in the case of no shut-down of the source, the behavior of a -10 sub-critical system is less favourable, since in a critical system the increase of the coolant temperature is slower and lower due to the feed-back effects. Again, the choice of the level of sub-criticality is relevant, if one takes into account the potentially beneficial effects of the intrinsic characteristics of the core.
Report to Congress DOE/RW-0519, October 1999.
J. Phys. IV France 9, p. 17-33 (1999).
NEACRP-U-75.
Mito, Japan, October 1998, OECD-NEA.
CERN/AT/95-44 (ET), 1995.
(e.g. SNR-300 S/A with MOX fuel)
“the physics of subcritical multiplying systems” | |||
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The Impact of Nuclear Science on Life ScienceThe Physics of Sub-critical Multiplying SystemsM. Salvatores CEA - DEN/DDIN - Cadarache1. IntroductionThe role of accelerator-driven systems has been recognized world-wide [1, 2, 3] as a potential highly relevant tool in order to transmute very large amounts of radioactive wastes and, consequently, to lower the burden on a deep geological storage. Both the two major variants of the strategies of transmutation which make use of ADS [4], call for sub-critical cores with a fast neutron spectrum and a fuel dominated by mixtures of Pu and Minor Actinides (MA), essentially fertile-free. These cores are characterised by a very low fraction of delayed neutrons and by a low (even near zero) Doppler reactivity coefficient. In principle, the sub-criticality will help to reduce (or to eliminate) the negative consequences of these characteristics on the safety of the multiplying medium. This is one of the point that will be discussed in the present article. In fact, the physics of the ADS and of its sub-critical core is well understood, and there are several publications which deal extensively with the subject [see for example refs 5 and 6, among many others]. However, several concepts are new and their understanding requires experimental validation. In the present article, we will focus on a description of the basic physics phenomena in the sub-critical multiplying core, with reference to the coupling phenomena and their impact on the sub-critical core (SC), when needed. We will also indicate the areas that need particular care for experimental validation and we will quote some ongoing experimental programs and preliminary results. Finally, the inspection of some “visual” images of SC as they are presently studied in several laboratories, will be the occasion to point out some relevant design-oriented problems of sub-critical cores and their integration in an ADS. 2. The sub-critical multiplying core in stationary regime2.1 The flux distributionIn a critical system, the condition of balance of neutron production and neutron disappearing at each point of the phase space (E, r,) is expressed by the Boltzman equation, which can be expressed in matrix form :(1) where A is the “disappearance” and P the “production” operators, and the flux. In the same system, made sub-critical, the condition to have a stationary system is to have an external source S(E r,) such that, e.g, the equation (1) can be written as :(2) is the solution of the inhomogeneous equation (2). The distribution in space, energy, angle of is obviously different from that of . Of course, approaches as the level of sub-critically becomes smaller and smaller, approaching the critical configuration. For an ADS, once defined the material properties, the geometry of the system, the relevant cross-sections and the source intensity (in units of neutrons/sec), the distribution of the inhomogeneous flux is fully determined by equation (2). Relevant integral parameters, characterising the sub-critical core (SC), such as reaction rates can be easily calculated. This allows to evaluate the power deposited in each point of the system, the damage rate, the breeding ratio etc. This is done exactly as in critical systems, characterized by . 2.2 The reactivity of the sub-critical coreIt is formally possible to describe a sub-critical system with the introduction of a parameter keff which allows to “restore” the balance equation (2) : (3) Since has the same distribution as the “critical” flux, this equation is obviously an approximation of the real case, as described by eq. 2. In order to improve the definition of sub-criticality and to take into account the change in shape of the flux, it has been proposed a different definition of the sub-criticality, by means of a “K-source” KS. The procedure is to apply the formal balance condition (3) to the inhomogeneous flux equation (2) : (4) Integrating and recalling that one obtains : (5) 2.3 The * parameterIn a sub-critical system, it is of relevance to the understanding of the behaviour of the source-driven sub-critical core, the evaluation of the relative importance of the source neutrons with respect to the fission neutrons generated in the SC. One introduces a parameter *, which is the ratio of the average “importance”(a) of the source neutrons and of the average “importance” of fission neutrons. It can be shown that this parameter * is related to keff through the following relation : (b) (6) where is the average number of prompt neutrons per fission, and the average source neutrons per fission. Relation (6) is given in reference [7], where the experimental determination of * is discussed. The * parameter plays an important role in the ADS performance parameters assessment. In fact in ref. [8], it is shown that the relation among the proton beam current ip, the power in the SC and its sub-criticality is given by : (A) (7) where W is the power of the SC in watts, f the energy per fission (MeV) and Z is the number of neutrons per incident proton. It can be seen from (7) that a value of * higher than 1, can reduce proportionally the proton beam current requirement, for a given sub-criticality level. Measurements of * are made in the CEA facility MASURCA in Cadarache, in the frame of the MUSE program [11], which will be described shortly in a successive paragraph. 3. The kinetics of a sub-critical system3.1 The asymptotic behaviourThe equations which give the kinetic behaviour of a system driven by an external source, are of the type (ref. 9) : (8) where Ci are the precursors of delayed neutrons with decay constant i. i is the fraction of the total number of delayed neutrons emitted per fission ( i = ) due to precursors Ci. eff is the neutron generation time and is the reactivity . In steady state (i.e. if and ), we have : (9) A decrease by a factor h of the reactivity (’ = /h) or an increase by a factor h of the source (S’ = hS), induces an instantaneous increase of the power W’ = hW. For example, if, the system is sub-critical corresponding to - 10, a reactivity insertion of + 5, produces a doubling of the power (see Fig. 1). This of course is totally different from the behaviour of a critical system (which becomes prompt critical). In more general terms, the kinetic behavior of a critical system is characterized by delayed neutrons and their time constants (about 10 s), while the kinetic behaviour of a SC is determined by the time constants related to the external source, in the sense that an instantaneous variation of source has an effect on the time scale of the prompt neutron lifetime (typically of the order of microseconds). The evolution of the power with time and the related variation of the temperature is associated to the variation of the reactivity (Doppler reactivity effect, fuel expansion reactivity, reactivity due to the material concentrations in the core, including the coolant etc.). These reactivity effects (feed-back reactivity effects) are essential for the safety of a critical reactor. In a sub-critical core, the feedback reactivity effects are of different relevance according to the level of sub-criticality. In fact for a core deeply sub-critical, the dynamic behaviour is dominated by the external source and its variation in time. Closer to criticality, the feedback effects become more important and the behaviour of the core is approaching that of the corresponding critical core. In a very simplified way, if the core is sub-critical by -10 , a feedback reactivity equal to 1 , induces a 10 % variation of power and a 50 % variation of power if the system is sub-critical by - 2 . In a critical reactor + 1 reactivity insertion makes the reactor prompt critical and - 1 stops the chain reaction. In view of the definition of an “optimal” level of sub-criticality, it is of high relevance to verify the transition of the behavior of the SC from a “source-dominated” to a “feed-back dominated” regime. 3.2 Reactivity and loss-of-flow accidentsFast external reactivity insertions give rise to different consequences in critical or sub-critical core. Examples have been given in Refs [10, 11]. In ref. 10 a 0.55 /s reactivity insertion in a PHENIX fast reactor type core, critical or sub-critical at Keff = 0.95, gives rise (at constant external source level) to the following power and average temperature evolutions : | |||
| Critical core | Sub-critical core (k=0.95) | |
Delay before fuel fusion | 2 s | 12 s | |
Inserted reactivity | 1.1 | 6.6 | |
Power increase W’/W | 2.2 | 1.5 | |
| Type of | Range of | Diffusing buffer around the source |
MUSE-1 (1995) | Cf-252 spontaneous fission neutron source | - 1.5 % | None |
MUSE-2 (1996) | Cf-252 spontaneous fission neutron source | - 3.0 3.5 % K/K | - Sodium - Steel |
MUSE-3 (1998) | Pulsed neutron source from (d,t) | - 0.5 - 6. % % | - Sodium - Steel |
MUSE-4 (2000-2001) | Pulsed neutron source from (d,d) and (d,t) | - 1 .4 % | lead |
Configuration | Calculated * | Measured * | |
Cf-252 source at core center | 1.25 | 1.19 4 % | |
Cf-252 source at core/blanket axial interface | 0.91 | 0.90 4 % | |
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