Normalized phase velocity

To indicate the advantages of the conventional (2, M) FDTD method, Figure 2.1(a) presents the relative error of the normalized phase velocity versus spatial frequency k Ah, where Ah corresponds to a uniform grid. Evidently, the second-order operator has a limited spectral... 

Relative error of normalized phase velocity Maximum Lo error norm... 

FIGURE 2.1 (a) Relative error of the normalized phase velocity and (b) maximum Lj error norm for... 

FIGURE 2.10 Relative error of the normalized phase velocity (a) for a 2-D angle-optimized FDTD scheme, optimized at 22.5° without filtering and (b) for a 3-D Chebyshev angle-optimized FDTD scheme at 9 = 90°,

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In fact, it is the careful choice of parameters Y and lZfn / in (3.75) that leads to this serious improvement. To illustrate these issues, consider Figures 3.10(a) and 3.10(b) that present the variation of the maximum Li error norm versus kAh (Ah denotes a uniform Au = Av = Aw mesh) and the relative error of the temporal evolution normalized phase velocity, respectively. 

FIGURE 3.10 (a) Maximum L2 error norm versus spatial frequency kAh and (b) relative error of the normalized phase velocity versus time for different higher order nonstandard FDTD arrangements... 

FIGURE5.2 (a) Normalized phase velocity and (b) reflection coefficient versus frequency for a second-order Lorentz medium... 

FIGURE 6.3 (a) A skewed mesh with localized discontinuities and (b) normalized phase velocity versus angle 6 for various second-order and higher order (HO) ADI-FDTD implementations... 

FIGURE 6.4 (a) Maximum dispersion error as a function of CFLN and lattice resolution and (b) normalized phase velocity versus frequency for diverse ADI-FDTD configurations... 

ADI-FDTD realizations is displayed. Recall that the CFLN is defined as CFLN = At/A/max, with A/max the maximum time-step denoted via the stability criterion of the second-order or higher order nonstandard FDTD method. As can be deduced, the superiority of (6.35) is more prominent when time-steps exceed the Courant limit at a great extent (CFLN > 20). On the other hand, the dispersion error of the improved algorithm is very small even for very coarse grids, whereas its normalized phase velocity is notably close to the free-space one, despite the fact that the Courant limit is greatly surpassed. 

Fig. 3.78. Normalized phase velocity versus size parameter computed with the EFMED routine and the Matlab program QCAMIE...

The basically correct equation appears to be that of Giddings but, over the range of mobile phase velocities normally employed i.e., velocities in the neighborhood of the optimum velocity), the Van Deem ter equation is the simplest and most appropriate to use. 

In extraction column design, the model equations are normally expressed in terms of superficial phase velocities, L and G, based on unit cross-sectional area. The volume of any stage in the column is then A H, where A is the cross-sectional area of the column. Thus the volume occupied by the total dispersed phase is h A H, where h is the fractional holdup of dispersed phase, i.e., the droplet volume in the stage, divided by the total volume of the stage and the volume occupied by the continuous phase, in the stage, is (1-h) A H. 

Equation (I.IS) Is valid for open tubular columns under all normal conditions and for packed columns at low mobile phase velocities. The average carrier gas velocity is calculated from the outlet velocity by correcting the latter for the pressure drop across the column, and is simply given by u - ju, where j is the gas compressibility correction factor, defined In equation (1.2). 

Depending on its subtraction from or addition to the buoyancy force, the continuous phase velocity can either increase or decrease the bubble volume. Normally, this velocity is such that the bubble detaches prematurely from the nozzle tip. Maier (M2) has shown that the shear force experienced by the bubble, which causes its premature detachment, is a maximum when the continuous phase flows at right angles to the nozzle axis. 

Expressed formally, the wave must be matched in amplitude at the surface and in phase velocity parallel to the crystal strrface. This implies that the tangential components of D and H must be continuous across the strrface, and the components in the crystal strrface of the wavevectors inside and outside the strrface must be the same. If n is a rmit vector normal to the crystal strrface, whatever the values of k o or the resttlting Bloch wave inside the crystal, then... 

The band profiles are obtained as the change of concentration against time at the column outlet. To better understand the effect of mobile phase dispersion, column length, mobile phase velocity, and other parameters, we introduce some normalization. We can rewrite Equation 10.8, using... 

Samples thus analyzed had two phase-velocity regions, the normal high-order detonation and an intermediate velocity region. The two abrupt changes in phase velocity are read from the streak records to give the distance to the intermediate region and the distance to detonation. 

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