Equilibrium Core Calculations

The R-Z two-dimensional core calculation model, as described by Fig. 2.30, may be a good first approximatiOTi to calculate a fast reactor core with a relatively simple loading pattern of hexagonal fuel assemblies (a tight fuel lattice). In such a configuration, the spatial dependence of the fast neutron flux is small and the rough estimation by the R-Z two-dimensional model may be applicable. 

However, when calculating a thermal-spectrum core with large heterogeneities, the R-Z two-dimensional model is inadequate for design purposes. In a thermal-spectrum core, the spatial dependence of the thermal neutron flux is large. The fuel assemblies are loaded with a relatively complex pattern to flatten the neutron flux distributions. Hence, the calculation of such a core requires the modeling of each fuel assembly with a three-dimensional model as shown in Fig. 2.31. To conserve computational power, symmetric boundary conditions can be applied. 

In the case of the Super LWR core, design, the X-Y-Z three-dimensional core calculation model is essential. It is a thermal-spectrum core with large heterogeneities. Not only the neutron flux but also the special dependences of the coolant temperature and density are large. These parameters may also be largely affected by the local insertions of control rods. The core characteristics also depend on the bumup distributions, which ultimately depend on the core power distributions, control rod patterns and fuel replacement patterns. In order to consider these parameters in the design, the three-dimensional core calculation model is required. 

The coupling of neutronic and thermal-hydraulic calculations is especially important for designing the Super LWR core. The density change of the coolant (and moderator) is large and sensitive to the enthalpy rise of the coolant as it flows from the core inlet to the outlet. On the other hand, the core neutronic characteristics strongly depend on the coolant and moderator density distributions. 

The COREBN code does not have the coupling function. Hence, the bumup calculations for one cycle of the core operation is divided into a number of bumup steps. Within each bumup step, the neutronic and thermal-hydraulic calculations are coupled by the core power and density distributions (within each bumup step, the coolant density distribution is assumed to be constant). These calculations are repeated until the core power distribution and the density distributions are converged. Once the convergence is obtained, the bumup step proceeds to the next step. For the coupling calculations, the macro-cross section sets of the fuel assemblies are prepared for different coolant and moderator densities and these are interpolated by bumups. 

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Fig. 2.32 Outline of the equilibrium core calculations. (Taken from doctoral thesis of A. Yamaji, the University of Tokyo (2005) [9])...

Neutronic Thermal-Hydraulic Coupled Equilibrium Core Calculation Method... 

Nominal Reactivity Control Worths The calculation of control rod and reserve shutdown control (RSC) worths under both hot and cold conditions have been performed for both the initial cycle BOG conditions and the equilibrium cycle EOC condition. In addition, the worth of all 30 control rods has been calculated for other times in cycle for both the initial core and an equilibrium reload cycle to determine how the total control rod bank worth is expected to change over the cycle. Other specific rod pattern control worths for hot conditions for the selected withdrawal of groups of three rods each in the outer bank of control rods were analyzed to define the maximum group worth for use in the transients analyzed in Chapter 15. These calculations were only performed for the EOC equilibrium core loading since that cycle condition yields the minimum temperature coefficient of reactivity and the maximum rod group reactivity worth for a rod group withdrawal transient. No reduction in control rod poison worth due to burnup has been assumed in this or other EOC rod worth calculations discussed below, although this effect would be minimal. 

Equilibrium core releases of cesium, strontium, and silver were calculated by using the results for the fuel and graphite temperature and fuel particle... 

For a certain fast breeder reactor, the average neutron flux in the core is 9 X 10 n cm" s . The average capture cross section of Na in the reactor spectrum is 2 mb. If the sodium spends 2 % of the time in-core, calculate the activity per cm of the coolant under equilibrium conditions. (Density of liquid sodium at operating temperature of reactor is 0.81 gem. )... 

In addition to its atomic structure, it is also of interest to calculate the effective size and energy associated with the equilibrium core, as done previously for Ta at... 

In practice, the core contribution is usually included as an additive correction. First, the equilibrium geometry is determined in a sufficiently accurate frozen-core cc-pVXZ calculation. Next, the core correction is estimated by comparing all-electron and frozen-core calculations in the cc-pCVFZ and cc-pVFZ sets where F < X. This approach is acceptable since the cote correction is small and need only be determined to a low accuracy. 

The bumup profile of the density reactivity coefficient of the equilibrium core is shown in Fig. 2.61 [9]. Although the calculation methods used in this chapter are not accurate enough to state the precise density coefficient values, the tendency of the density reactivity coefficient to decrease with the cycle bumup exposure can be seen. This decreasing trend is due to the increase in the core average density with the bumup from about 0.50 g/cm at the BOC to about 0.57 g/cm at the EOC. The gradual increase of the core average water density can be explained by the gradual shift of the axial core power distribution from the bottom peak to the top peak towards the EOC. As the axial power distribution shifts to the top peak, the axial... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

The active space used for both systems in these calculations is sufficiently large to incorporate important core-core, core-valence, and valence-valence electron correlation, and hence should be capable of providing a reliable estimate of Wj- In addition to the P,T-odd interaction constant Wd, we also compute ground to excited state transition energies, the ionization potential, dipole moment (pe), ground state equilibrium bond length and vibrational frequency (ov) for the YbF and pe for the BaF molecule. 

Water decompression may be assumed to be an isentropic equilibrium process for unheated blowdown or limited heating of fluid in the core region in order to calculate the transient pressure-time relationships with reasonably good accuracy for the full duration of the blowdown. 

The slopes of the peaks in the dynamic adsorption experiment is influenced by dispersion. The 1% acidified brine and the surfactant (dissolved in that brine) are miscible. Use of a core sample that is much longer than its diameter is intended to minimize the relative length of the transition zone produced by dispersion because excessive dispersion would make it more difficult to measure peak parameters accurately. Also, the underlying assumption of a simple theory is that adsorption occurs instantly on contact with the rock. The fraction that is classified as "permanent" in the above calculation depends on the flow rate of the experiment. It is the fraction that is not desorbed in the time available. The rest of the adsorption occurs reversibly and equilibrium is effectively maintained with the surfactant in the solution which is in contact with the pore walls. The inlet flow rate is the same as the outlet rate, since the brine and the surfactant are incompressible. Therefore, it can be clearly seen that the dynamic adsorption depends on the concentration, the flow rate, and the rock. The two parameters... 

Most of the calculations have been done for Cu since it has the least number of electrons of the metals of interest. The clusters represent the Cu(100) surface and the positions of the metal atoms are fixed by bulk fee geometry. The adsorption site metal atom is usually treated with all its electrons while the rest are treated with one 4s electron and a pseudopotential for the core electrons. Higher z metals can be studied by using pseudopotentials for all the metals in the cluster. The adsorbed molecule is treated with all its electrons and the equilibrium positions are determined by minimizing the SCF energy. The positions of the adsorbate atoms are varied around the equilibrium position and SCF energies at several points are fitted to a potential surface to obtain the interatomic force constants and the vibrational frequency. 

Models of hot isentropic neutron stars have been calculated by Bisnovatyi-Kogan (1968), where equilibrium between iron, protons and neutrons was calculated, and the ratio of protons and neutrons was taken in the approximation of zero chemical potential of neutrino. The stability was checked using a variational principle in full GR (Chandrasekhar, 1964) with a linear trial function. The results of calculations, showing the stability region of hot neutron stars are given in Fig. 7. Such stars may be called neutron only by convention, because they consist mainly of nucleons with almost equal number of neutrons and protons. The maximum of the mass is about 70M , but from comparison of the total energies of hot neutron stars with presupemova cores we may conclude, that only collapsing cores with masses less that 15 M have... 

Abstract Hybrid stars with extremely high central energy density in their core are natural laboratories to investigate the appearance and the properties of compactihed extra dimensions with small compactification radius - if these extra dimensions exist at all. We introduce the necessary formahsm to describe quantitatively these objects and the properties of the formed hydrostatic equilibrium. Different scenarios of the extra dimensions are discussed and the characteristic features of these hybrid stars are calculated. 

First-order phase transition from a nucleonic matter to the strange quark state with a transition parameter A > 3/2 that occurs in superdense nuclear matter generally gives rise to a toothlike kink on the stable branch of the dependence of stellar mass on central pressure. Based on the extensive set of calculated realistic equations of state of superdense matter, we revealed a new stable branch of superdense configurations. The new branch emerges for some of our models with the transition parameter A > 3/2 and a small quark core (.Mcore 0.004 A- 0.03M ) on the M(PC) curve, with Mmax 0.08M and A 0.82M for different equations of state. Stable equilibrium layered... 

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