Real coil

For a real coil of the type shown in Fig. 4, the magnetic field B can also be expressed by the following equation ... 

As was mentioned above, real coils in induction probes can very often be considered as magnetic dipoles. For this reason, let us consider the behavior of the magnetic field of a magnetic dipole in more detail. Suppose that a magnetic dipole is located at the origin of a spherical system of coordinates, as shown in Fig. 1.31a, with its moment oriented along the z-axis. Then, in accord with eq. 1.123, we have ... 

Real coiled molecules have an external and internal excluded volume. The external excluded volume results from the volume excluded to other molecules it is intermolecular and its influence disappears at infinite dilution. 

In equation (4-41), the expansion factor gives directly the average linear expansion of the coil. The expansion factor calculated in this way is, of course, fictitious, since the coil is not spherical in shape, and will probably have various expansion factors for various directions in space. The existence of unequal expansion factors over all directions in space means that the distribution of molecular segments of real coils is no longer the same as for ideal coils. 

Nevertheless, even the critical concentration obtained from absolute polymer coil dimensions, is only a relative value since the radii measured with scattering experiments are not equal to the hydrodynamic radii of the same polymer coils in solution. A detailed discussion on how to calculate a hydrodynamic radius is given in The critical concentration of a real coil in Chap. 8. 

The radius of gyration is often corrected with a factor that converts the radius of gyration Rq of a real coil with a density that decreases with the radius, into the... 

Because real coils will always exhibit ohmic losses, the phase shift will not be completely 90°, as shown in Equation 9.33 ... 

Fig. 2.2 Correcting function e for a real penetration depth, calculated by 3D-FEMfor different coils and frequencies.

Fia 5 Real Part of Calculated Field Distribution around Exciter Coil... 

Measurement of real- and imaginary part of a coil complex impedance... 

A common known method to get eddy-current informations about material flaws is the measurement of real- and imaginary part of the complex impedance of a coil in absolute circuit. The measurement, shown in this paper, are done with an impedance analyzer (HP4192A). The device measures the serial inductance L, and the serial resistance Rs of the complex impedance with an auto-balance bridge measurement circuit [5]. 

An ethylene plant contains more than 300 equipment items. Traditionally, operators were trained at the site alongside experienced co-workers. With the advent of modem computers, the plant operation can be simulated on a real-time basis, and the results displayed on monitors (107). Computers are used in a modem plant to control the entire operation, eg, they are used to control the heaters and the recovery section (108). A weU-controUed plant is much more profitable than a poorly controlled plant. For the heaters, a model-based control system is gaining importance (109). Instead of simply controlling the coil outlet temperature (COT), severity is actually controlled. The measurement of severity (either or C H /CH ratio) requires on-line effluent... 

The coiled cable is not acceptable for low-speed (i.e., less than 300 rpm) applications or where there is a strong electromagnetic field. Because of its natural tendency to return to its relaxed length, the coiled cable generates a low level frequency that corresponds to the oscillation rate of the cable. In low-speed applications, this oscillation frequency can mask real vibration that is generated by the machine. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

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