Friedel model

The first part of the chapter is devoted to an analysis of these correlations, as well as to the presentation of the most important experimental results. In a second part the following stage of development is reviewed, i.e. the introduction of more quantitative theories mostly based on bond structure calculations. These theories are given a thermodynamic form (equation of states at zero temperature), and explain the typical behaviour of such ground state properties as cohesive energies, atomic volumes, and bulk moduli across the series. They employ in their simplest form the Friedel model extended from the d- to the 5f-itinerant state. The Mott transition (between plutonium and americium metals) finds a good justification within this frame. 

Those in part (b) are from the Friedel model with parameters taken from the Solid State Table. 

The energies in the figure exhibit a parabolic-like behavior as a function of the atomic number. Such a behavior is explained in terms of the occupation of the valence d-band by the Friedel model [29,30] in which the surface energies follow the same trend as the corresponding cohesive energies and can be estimated from... 

The density of states in hexagonal close-packed osmium, with Fermi energy indicated by the dashed line. The monotonically rising line is the integral over the density of states, with scale given to the right. The curves in part (a) are after Jepsen, Andersen, and Mackintosh (1975). Those in part (b) are from the Friedel model with parameters taken from the Solid State Table. 

The R-Fe system (fig. 14.77) shows the anomalous Tc increase as the Fe concentration decreases even for the Y-Fe system. As we shall see in the following sections Fe tends to show more localized character than Ni and Co. The application of the Friedel model (subsection 3.1.6), with couplings between semi-localized 3d-moments within a band, could in principle account for the increased strength of exchange on diluting Fe with Y since the 3d polarization around an impurity centre oscillates in space. 

In the present investigation, this methodology has been applied to check the accuracy of the Lockhart-Martinelli and Friedel models also used for the straight sections. The acceleration terms have been calculated as explained in the previous section. 

In the case of the homogeneous (no slip) Friedel model, the acceleration term of the total pressure variation between two pipe cross-sections can be directly calculated from Equation 2, with the homogeneous specific volume estimated from Equation 4. This allows the direct calculation of the friction loss by Equation 1. 

In Eigure 4, the pressure drop per unit length (bar/m) due to friction losses calculated with the Friedel model is reported against the experimental values relative to the same pipe sections. In order to check the possible influence of some other parameters involved in the flow, the vapor quality at the entrance of each section has been classified in a discrete number of intervals, and the measurements belonging to each class have been reported with different marks in the figure (see attached key). It can be seen that no significant effect is associated with the vapor quality. 

Despite the Friedel model is considered a general one, independent of the flow pattern, in order to assess its possible influence, the flow pattern has been characterized for all the runs. The flow pattern map proposed by Moreno-Quiben Thome (2007) has been adopted, and it was found that, based on that classification, the flow pattern in all the runs could be associated with either the intermittent or dispersed bubble regimes. In Figure 5, the ratio between the calculated and experimental data is reported, as a function of the vapor quality, with the flow pattern as a parameter. 

Figure 4. Pressure drop in straight pipes. Comparison between Friedel model s predictions and experimental data.

As already done for the Friedel model, the possible influence of the specific flow pattern has been checked also for the Lockhart-Martinelli model. However, since the influence of the vapor quality has been already found negligible from the previous results, the ratio of the predicted pressure drop over the experimental one, has been plotted as a function of the inlet pressure (Fig. 7). It can be seen that, apart from the expected marked dispersion of the data (based on the RMS value already reported in Fig. 6), a general increase of the ratio is observed at increasing inlet pressures. Again, an apparent dependency of the predictions with the flow pattern is found, with the intermittent flow regime always characterized by larger ratios of the pressure drop, compared with the dispersed bubble. 

In Figure 8 the experimental pressure drops and those calculated with the Friedel model are reported for all the runs, along with the corresponding statistical indicators. The indicators present slightly higher values than in the case of straight pipelines, but the average error is still quite low. 

If the data corresponding to Re < 10 only are taken into consideration (see Fig. 12), the above results are more apparent, with the statistical parameters worse than those obtained for the Friedel model, when the single phase equivalent length is adopted, and definitely the worst in absolute when adopting the measured two-phase equivalent length (cr34.2% and RMS 46.4%). 

In conclusion, based on the present experimentation, the Friedel model resulted to be characterized by a higher level of accuracy in the calculation of the friction losses for two-phase vapor-liquid flows. However, further work is needed for the characterization of the pressure losses connected with fittings, where the particular method adopted for the calculation of the equivalent pipe length can play a significant role. 

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