Upper subcritical limit

Note that subcritical means that the maximum neutron multiplication, adjusted appropriately by including a calculational bias, uncertainties and a subcritical margin, should be less than 1.0. See Appendix VII for specific advice on the assessment procedure and advice on determining an upper subcritical limit. 

V11.36. The maximum upper subcritical limit (USL) that should be used for a package evaluation is given by... 

The result of this situation is an extremely conservative biasing of criticality safety analyses toward the assumption of criticality. The most obvious manifestation of this is that the end result of fhe (very involved and formal) process of validating an analysis method is a determination of an upper subcritical limit (USL), which represents the value of the -effective prediction of the particular method that is going to be considered to represent a critical state. It sometimes comes as a surprise to reactor analysts to know that a A -effective of 1.000 is not where criticality is defined in a criticality safety analysis, but a much lower value—sometimes as low as 0.80 and seldom higher than 0.95 (except for extremely well-understood processes). 

Molecular Nature of Steam. The molecular stmcture of steam is not as weU known as that of ice or water. During the water—steam phase change, rotation of molecules and vibration of atoms within the water molecules do not change considerably, but translation movement increases, accounting for the volume increase when water is evaporated at subcritical pressures. There are indications that even in the steam phase some H2O molecules are associated in small clusters of two or more molecules (4). Values for the dimerization enthalpy and entropy of water have been deterrnined from measurements of the pressure dependence of the thermal conductivity of water vapor at 358—386 K (85—112°C) and 13.3—133.3 kPa (100—1000 torr). These measurements yield the estimated upper limits of equiUbrium constants, for cluster formation in steam, where n is the number of molecules in a cluster. 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

FlO. 5.4. The birth and growth of oscillatory solutions for the thermokinetic model with the full Arrhenius temperature dependence, (a) The Hopf bifurcations /x and ft are both supercritical, with [12 < 0, and the stable limit cycle born at one dies at the other, (b) The upper Hopf bifurcation is subcritical, with fl2 > 0. An unstable limit cycle emerges and grows as the dimensionless reactant concentration ft increases—at /rsu this merges with the stable limit cycle born at the lower supercritical Hopf bifurcation point ft. ... 

Of the fast neutrons produced in fission, some of them will be moderated to thermal energies and will induce other fission reactions while others will be lost. The ratio of the number of neutrons in the next generation to that in the previous generation is called the multiplication factor k. If the value of k is less than 1, then the reactor is subcritical and the fission process is not self-sustaining. If the value of k is greater than 1, then the number of fissions will accelerate with time and the reactor is supercritical. The goal of reactor operation is to maintain the system in a critical state with k exactly equal to 1. The extreme upper limit for the multiplication factor would correspond to the mean number of neutrons per fission ( 2.5 for 235U(n,f)) if each neutron produces a secondary fission. 

If such a zone exists on Jupiter, it is narrow. Where the temperature is 300 K (clearly suitable for organic molecules), the pressure (about 8 atm) is still subcritical. At about 200 km down, where the jovian pressure is supercritical, the temperature rises above 500 K, approaching the upper limit where carbon-carbon bonds are stable.24... 

The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can join up the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig. 

The behaviour spectrum of a homogeneous population as a function of parameter (fig. 6.3) reveals a rich variety of dynamic behavioural modes of the cAMP signalling system. Starting from a low initial value of fee (fig- 6.3a), the system evolves toward a stable steady state, represented by the value of the extracellular cAMP concentration, yo-Around k = 2.4 min (fig. 6.3b), a subcritical Hopf bifurcation occurs beyond which the steady state becomes unstable (dashed line) in a range roughly extending from k - 2.2 to 2.4 min in the conditions of fig. 6.3, the system thus admits a coexistence between a stable steady state and a stable limit cycle represented by the upper solid line showing the maximum cAMP level in the course of oscillations, y these two stable solutions are separated by an imstable limit cycle (dashed line). 

Of these three issues, the first two are the most serious, with the first severely limiting the systems that can be studied to those that are stable in the presence of hydrogen, and the second hmiting the upper temperature. The third constraint is not a major issue in high subcritical systems, because the transference numbers of the ions of most, if not all, binary electrolytes tend toward 0.5 with increasing temperature however, at temperatures above the critical temperature the solubility of a salt is severely restricted and it may not be possible to attain a sufficiently high concentration to suppress the liquid junction potential. Note that the isothermal liquid junction is most effectively suppressed if the transference numbers of the cation and the anion of the background electrolyte are equal, a condition that is fulfilled by KCl at ambient temperature (and hence the reason for the choice of KCl in ambient temperature studies). 

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