Velocity, phase

From Equation 1.67, the following relafionship is satisfied as phase angles are equal  

Voltage waveforms along x-axis at different times. 

The phase velocity c is found from the earlier equation as 

The earlier equation shows that the phase velocity is foimd from w and p and is independent of the location and time. 

As a result, for a lossless line, the phase velocity is found from Equations 1.68 and 1.70 as Equation 1.55. 

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Second corner reflection The first corner reflection appears as usual when the transducer is coupled to the probe at a certain distance from the V-butt weld. The second corner reflection appears if the transducer is positioned well above the V-hutt weld. If the weld is made of isotropic material the wavefront will miss (pass) the notch without causing any reflection or diffraction (see Fig. 3(a)) for this particular transducer position. In the anisotropic case, the direction of the phase velocity vector will differ from the 45° direction in the isotropic case. Moreover, the direction of the group velocity vector will no longer be the same as the direction of the phase velocity vector (see Fig. 3(b), 3(c)). This can be explained by comparing the corresponding slowness and group velocity diagrams. 

Figure 3 Snapshots of the shear wavefronts for different types of weld material to show the direction of the wave unit vector k (a phase velocity unit vector Cp ) and the group velocity unit vector...

The phase velocity is directly linked to the angles of incidence by the Snell-Descartes law ... 

The University developed a method of determination of the material residual strength, based on measurement of the change of phase velocity of ultrasonic waves, as well as an ultrasonic flaw detector-tomograph with multi-element transducers of the type of phased acoustic array. It enables control of the internal structure of materials and products of up to 300 mm thickness, with the resolution of up to 0.5 mm. In the same university, work on NDT is also carried out in the welding and electro-acoustic departments. 

Now t[i is a minimum when k = 2, that is, when = 3t . There is little increase in analysis time when k lies between 1 and 10. A twofold increase in the mobile-phase velocity roughly halves the analysis time (actually it is the ratio Wu which influences the analysis time). The ratio Wu can be obtained from the experimental plate height/velocity graph. 

Plot of the height of a theoretical plate as a function of mobile-phase velocity using the van Deemter equation. The contributions to the terms A B/u, and Cu also are shown. 

There is some disagreement on the correct equation for describing the relationship between plate height and mobile-phase velocity. In addition to the van Deemter equation (equation 12.28), another equation is that proposed by Hawkes... 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

The most common mobile phases for GC are He, Ar, and N2, which have the advantage of being chemically inert toward both the sample and the stationary phase. The choice of which carrier gas to use is often determined by the instrument s detector. With packed columns the mobile-phase velocity is usually within the range of 25-150 mF/min, whereas flow rates for capillary columns are 1-25 mF/min. Actual flow rates are determined with a flow meter placed at the column outlet. 

The settling velocity, is relative to the continuous Hquid phase where the particle or drop is suspended. If the Hquid medium exhibits a motion other than the rotational velocity, CO, the vector representing the Hquid-phase velocity should be combined with the settling velocity (eq. 2) to obtain a complete description of the motion of the particle (or drop). 

Use U.S. customary units only in this equation. In sizing the column diameter, it is ususlly assumed that the continuous phase velocity will set at 40 percent of this value, and therefore the column diameter is calculated by ... 

Qc = volumetric flow rate of the continuous phase = velocity of the continuous phase of flooding... 

For design, the slip velocity is derated to 70-80 percent of the calculated value to give some margin of safety this sets the design value of the continuous phase velocity (V ). The column cross sectional area (and therefore diameter) is set by QJVc- With the diameter set, the other dimensions can be set using the ratios given above. 

When a small quantity of a second liquid phase is present, a drawoff pot (commonly called a bootleg) is provided to make separation of the heavy liquid (frequently water) easier. The pot diameter is ordinarily determined for heavy phase velocities of 0.5ft/min. Minimum length is 3 ft for level controller connections. Minimum pot diameter for a 4 to 8 foot diameter reflux drum is 16 inches. For... 

Reflection and refraction of X-rays follow the laws of optics (Fig. 4.2) The glancing angles of incidence (i) and reflection ( ) are equal. The respective refraction angles follow Snell s law for different phase velocities v in mediums 1 and 2 ... 

As already mentioned, there are two so called "dead volumes" that are important in both theoretical studies and practical chromatographic measurements, namely, the kinetic dead volume and the thermodynamic dead volume. The kinetic dead volume is used to calculate linear mobUe phase velocities and capacity ratios in studies of peak variance. The thermodynamic dead volume is relevant in the collection of retention data and, in particular, data for constructing vant Hoff curves. 

It is also seen that, at very low velocities, where u E, the first term tends to zero, thus meeting the logical requirement that there is no multipath dispersion at zero mobile phase velocity. Giddings also introduced a coupling term that accounted for an increase in the effective diffusion of the solute between the particles. The increased diffusion has already been discussed and it was suggested that a form of microscopic turbulence induced rapid solute transfer in the interparticulate spaces. 

It is seen that the first term differs from the Giddings equation and now contains the mobile phase velocity to the power of one-half. However, when the first... 

The reduced velocity compares the mobile phase velocity with the velocity of the solute diffusion through the pores of the particle. In fact, the mobile phase velocity is measured in units of the intraparticle diffusion velocity. As the reduced velocity is a ratio of velocities then, like the reduced plate height, it also is dimensionless. Employing the reduced parameters, the equation of Knox takes the following form... 

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