Scaled energy

Figure 5. Variation of n ace(u) with scaled energy, E/htii, derived from Eq. (A.13) with Eq — 0.5/t j>, The Roquet bands in Figure cover energy ranges such that trace( ) < 2.

In Figure 6.35, lines have been added for a sphere bursting into 2 or 100 pieces for pi/po = 50 and 10, in accordance with Figure 6.33. Obviously, the simple relations proposed by Brode (1959) and Baum (1984) predict the highest velocity. Differences between models become significant for small values of scaled energy E, in the following equation ... 

In most industrial applications, scaled energy will be between 0.1 and 0.4 (Baum 1984), so under normal conditions, few fragments are expected, and Figure 6.33 can be applied. However, if an operation or process is not under control and pressure rises dramatically, higher scaled-energy values can be reached. 

In the relationships proposed by Brode (1959) and in Figure 6.33, velocity has no upper limit, although Figure 6.33 is approximately bounded by scaled pressures of 0.05 and 0.2 (scaled energies of approximately 0.1 and 0.7). Baum (1984) states, however, that there is an upper limit to velocity, as follows The maximum velocity... 

Baum (1984) states that the scaled energy, which is determined by ... 

Thus, is 20% of the energy calculated for nonideal gases or for flash-vaporization situations. For scaled energies ( ) larger than about 0.8 as calculated by Eq. (9.3.5), the calculated velocity is too high, so method 3 should be applied. 

Separate regions in the figure account for the scatter of velocities for spheres and cylinders separating into 2, 10 or 100 fragments. The number of fragments must first be chosen, usually on the basis of scaled energy. 

It is inadvisable to extrapolate outside the regions given in Figure 9.7. For high scaled-pressure values (i.e., scaled energy larger than 0.8), method 3 should be used. 

Moore s equation was derived from fragments accelerated from high explosives packed in a casing. Baum (1984) showed, in comparing different models, that the Moore equation tends to follow the theoretical upper-velocity limit for high scaled energies. 

In order to determine which method should be applied for the calculation of initial velocity, the scaled energy should first be determined (see Section 9.3.2.5). With Eq. (9.3.5) ... 

Since the scaled energy is lower than 0.8 and nitrogen can be considered to be an ideal gas, both methods 1 and 2 can be applied. 

Because the scaled energy is higher than 0.8, method 3 has to be applied for both cases. Method 3 [Eq. (9.3.11)] gives... 

Thus, Segura replaced ct with a scaled energy change, obtained by a molecular orbital calculation, for this proton transfer ... 

Davos, 22nd-26th March 1993, paper 6/5. 8(13) LARGE SCALE ENERGY RECOVERY TRIALS ON POLYURETHANE, PET, ACRYLIC AND NYLON... 

Figure 3.4 shows a more correctly scaled energy level diagram that results for the hydrogen molecule. Note that the energy for the Is atomic orbital of a hydrogen atom is at — 1312 kJ moT1 because the... 

The heat of combustion of a stoichiometric hydrocarbon-air mixture is approximately 3.5 MJ/m3, and by multiplying by the confined volume, the resulting total energy is (2094 m3)(3.5 MJ/m3) = 7329 MJ. To apply the TNO multi-energy method, a blast strength of 7 is chosen. The Sachs-scaled energy is determined using Equation 6-25. The result is... 

Eq. (5) shows that the classical dynamics depends on the scaled energy e = E Y-1/2. As it is clear from Eq. (5) the Hamiltonian has the singularity at f = 0. This singularity can be removed by performing the following transformations... 

The equations of motion obtained from the Hamiltonian (5) at a fixed value of the scaled energy are equivalent to ones obtained from the Hamiltonian (8). 

In Figs. l(a)-l(c) the Lyapunov functions are shown for Z = 50, m = 1 and different scaled energies. Fig. 1(a) shows results for v = 0, p = 0, e = 10. Al( ) tends to zero indicating that this trajectory is regular. This figure has the same shape as that for the nonrelativistic hydrogen atom in a uniform magnetic field (Schweizer et.al., 1988). In Fig. 1(b) the Lyapunov function for v = 0, p = 0, e = 50 is shown. It tends to some positive value, which means that this trajectory is chaotic. While for v = 0, p = 0, e = 100 (Fig. 1(c)) we find that the trajectory is unstable. 

As follows from the above treatment for higher values of the scaled energy e which includes both the energy and magnetic field strength the behaviour of the system becomes more chaotic, while for the smaller values regular behaviour is observed. 

Fig. 8.13 Power spectra of measured Na photoexcitation cross sections from the 3p states vs the classical action S in atomic units (a) o polarization and (c) ic polarization and calculated power spectra of H (b) from the 2p m = 1 state with o polarization and (d) from the 2p m = 1 state with jt polarization. All are for fixed scaled energy W = W rE = —2.5...

While the success of scaled energy spectroscopy suggests that the behavior of atoms does become classically chaotic near the ionization limit, higher resolution reveals a surprisingly orderly structure. Iu et alhave studied the odd parity Li m = 0 states in a beam travelling in the direction of the magnetic field. They... 

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