Velocity, root

Ifgi(x) equals the screw root velocity Kz andgjW = Vbz" then ... 

A root velocity can be defined as the rate of advance of a given value of an H function root. For any given composition, roots with lower index numbers have lower velocities. An arbitrary initial noncoherent boundary involving variations of all roots thus is resolved, upon undisturbed development, into separate variations of the roots. This is shown by schematic trajectories of root values in a distance-time diagram in Figure 6. After resolution, each trajectory bundle involves variation of... 

This paper presents solutions of two different NDT problems which could not be solved using standard ultrasonic systems and methods. The first problem eoncems the eraek detection in the root of turbine blades in a specified critical zone. The second problem concerns an ultrasonie thiekness measurement for a case when the sound velocity varies along the object surface, thus not allowing to take a predetermined eonstant velocity into account. 

As m increases, At becomes progressively smaller (compare the difference between the square roots of 1 and 2 (= 0.4) with the difference between 100 and 101 (= 0.05). Thus, the difference in arrival times of ions arriving at the detector become increasingly smaller and more difficult to differentiate as mass increases. This inherent problem is a severe restriction even without the second difficulty, which is that not all ions of any one given m/z value reach the same velocity after acceleration nor are they all formed at exactly the same point in the ion source. Therefore, even for any one m/z value, ions at each m/z reach the detector over an interval of time instead of all at one time. Clearly, where separation of flight times is very short, as with TOF instruments, the spread for individual ion m/z values means there will be overlap in arrival times between ions of closely similar m/z values. This effect (Figure 26.2) decreases available (theoretical) resolution, but it can be ameliorated by modifying the instrument to include a reflectron. 

For stand-alone or hybrid TOF mass spectrometry, the ions examined must all start from some point at the same instant. From this zero time, the ions are accelerated through a short region by applying a short pulse of electric potential of several kilovolts. The acceleration gives the ions velocities that vary in proportion to the square root of their m/z values. 

The detached section of ions sets off along the TOP analyzer, with the ions having velocities proportional to the square roots of their m/z values. 

After the initial acceleration phase, the velocity reached by an ion is inversely proportional to its mass (strictly, inversely proportional to the square root of its m/z value). 

Since the distance from the source to the detector is fixed, the time taken for an ion to traverse the analyzer in a straight line is proportional to its velocity and hence its mass (strictly, proportional to the square root of its m/z value). Thus each m/z value has its characteristic time of flight from the source to the detector. 

A single bubble rises through a fluid bed at a velocity, proportional to the square root of its diameter, or more accurately, the diameter of a sphere of equivalent volume ... 

In a free jet the absence of a pressure gradient makes the momentum flux at any cross section equal to the momentum flux at the inlet, ie, equations 16 and 17 define jet velocity at all points. For a cylindrical jet this leads to a center-line velocity that varies inversely with (x — aig), whereas for slot jets it varies inversely with the square root of (x — Xq As the jet proceeds still further downstream the turbulent entrainment initiated by the jet is gradually subordinated to the turbulence level in the surrounding stream and the jet, as such, disappears. 

Time Delay and Velocity of Propagation. Time delay is direcdy proportional to the square root of the dielectric constant and describes the time that it takes for a signal to travel through a cable. The lower the dielectric constant, the less time required for a signal to travel through a cable. 

Velocity of propagation is the speed of transmission in a cable as compared to the speed of transmission in air and is therefore expressed as a percentage. Since the velocity of propagation is inversely proportional to the square root of the dielectric constant of the core, a lower dielectric constant results in higher transmission speed (3). 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Head meters with density compensation. Head meters such as orifices, venturis, or nozzles can be used with one of a variety of densitometers [e.g., based on (a) buoyant force on a float, (b) hydrauhc couphug, (c) voltage output from a piezoelectric ciystal, or (d) radiation absolution]. The signal from the head meter, which is proportional to pV" (where p = fluid density aud V = fluid velocity), is multiphed by p given by the densitometer. The square root of the produc t is proportional to the mass flow rate. 

The parameter used to design rapid mix and flocculation systems is the root mean square velocity gradient G, which is defined by equation... 

Equation (13) is the first important equation for open tubular column design. It is seen that the optimum radius, with which the column will operate at the optimum velocity for the given inlet pressure, increases rapidly as an inverse function of the separation ratio (cc-1) and inversely as the square root of the inlet pressure. Again it must be remembered that, when calculating (ropt)5 the dimensions of the applied pressure (P) must be appropriate for the dimensions in which the viscosity (r)) is measured. 

We see that the heat transfer coefficient is inversely proportional to the square root of the wire diameter, which is the reason for the development of fine wire heat exchangers after all. With an air velocity v of 0.5 m/s and a wire of 100 m, we have a=226 W/m K, which is around ten times the typical value of flat plate heat exchangers to air. 

Zone 2 is a transition zone, and its length depends upon the diffuser type. For a compact jet the transition zone typically extends to eight or ten diameters from the outlet. Within this zone, the maximum velocity may vary inversely with the square root of the distance from the outlet. Some researchers 3-5 suggest use of a simplified scheme of the jet (Fig. 7.20b) with a transition cross-section for practical purposes. 

For any vector, the magnitude is the square root of the sum of the squares of the components. Thus the magnitude of the velocity of point P would be... 

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