Velocity, ultrasonic equations

The ultrasonic relaxation loss may involve a thermally activated stmctural relaxation associated with a shifting of bridging oxygen atoms between two equihbrium positions (169). The velocity, O, of ultrasonic waves in an infinite medium is given by the following equation, where M is the appropriate elastic modulus, and density, d, is 2.20 g/cm. ... 

For ideal mixtures there is a simple relationship between the measurable ultrasonic parameters and the concentration of the component phases. Thus ultrasound can be used to determine their composition once the properties of the component phases are known. Mixtures of triglyceride oils behave approximately as ideal mixtures and their ultrasonic properties can be modeled by the above equations [19]. Emulsions and suspensions where scattering is not appreciable can also be described using this approach [20]. In these systems the adiabatic compressibility of particles suspended in a liquid can be determined by measuring the ultrasonic velocity and the density. This is particularly useful for materials where it is difficult to determine the adiabatic compressibility directly, e.g., powders, biopolymer or granular materials. Deviations from equations 11 - 13 in non-ideal mixtures can be used to provide information about the non-ideality of a system. 

In non-ideal mixtures, or systems where scattering of ultrasound is significant, the above equations are no longer applicable. In these systems the ultrasonic properties depend on the microstructure of the system, and the interactions between the various components, as well as the concentration. Mathematical descriptions of ultrasonic propagation in emulsions and suspensions have been derived which take into account the scattering of ultrasound by particles [20-21]. These theories relate the velocity and attenuation to the size (r), shape (x) and concentration (0) of the particles, as well as the ultrasonic frequency (co) and thermophysical properties of the component phases (TP). 

This equation is also a consequence of Eqs. 9.19 and 9.20. The distance between the piezotransducers, d, can be determined from measurements of (f - f i) for the resonator filled with a liquid of known ultrasonic velocity, c. Based on the simple model for an ideal resonator, Eqs. 9.21 and 9.22 work surprisingly well in real resonators at frequencies that are close to the resonance frequency of the piezotransducer. These equations have been widely used with high-resolution measurements of ultrasonic velocity in liquids. The scope of these equations has been expanded by including empirical correction factors [70]. 

As with any analytical technique, it is important for US spectrometry users to have a thorough understanding of its underlying physical principles and of potential sources of errors adversely affecting measurements. The basis of ultrasonic analyses in a number of fields (particularly in food analysis) is the relationship between the measurable ultrasonic properties (velocity, attenuation and impedance, mainly) and the physicochemical properties of the sample (e.g. composition, structure, physical state). Such a relationship can be established empirically from a calibration curve that relates the property of interest to the measured ultrasonic property, or theoretically from equations describing the propagation of ultrasound through materials. 

The changes undergone by US on interacting with a solid, and the information that can thus be obtained from the solid, have been measured mainly through the US velocity and attenuation under resonant conditions. The ultrasonic parameters to be used and their processing are dictated by the final information required. Thus, the resistance to deformation is obtained by calculating the elastic moduli, the number and nature of which are a function of the nature of the solid (e.g. isotropic, anisotropic). However, equations relating the acoustic measurement to sample density, dimensions, material microstructure and thickness can usually be derived from simple parameters such as US velocity... 

CSIRO Minerals has developed a particle size analyzer (UltraPS) based on ultrasonic attenuation and velocity spectrometry for particle size determination [269]. A gamma-ray transmission gauge corrects for variations in the density of the slurry. UltraPS is applicable to the measurement of particles in the size range 0.1 to 1000 pm in highly concentrated slurries without dilution. The method involves making measurements of the transit time (and hence velocity) and amplitude (attenuation) of pulsed multiple frequency ultrasonic waves that have passed through a concentrated slurry. From the measured ultrasonic velocity and attenuation particle size can be inferred either by using mathematical inversion techniques to provide a full size distribution or by correlation of the data with particle size cut points determined by laboratory analyses to provide a calibration equation. 

Membrane Formation. In earlier work. 2.) it was found that fumed silica particles could be dispersed in aqueous suspension with the aid of ultrasonic sound. Observations under the electron microscope showed that the dispersion contained disc-like particles, approximately 150-200 1 in diameter and 70-80 1 in height. Filtration experiments carried out in the "dead-end" mode (i.e., zero crossflow velocity) on 0.2 urn membrane support showed typical Class II cake formation kinetics, i.e., the permeation rate decreased according to equation (12). However, as may be seen from Figure 7, the decrease in the permeation rate observed during formation in the crossflow module is only t 1, considerably slower than the t 5 dependence predicted and observed earlier. This difference may be expected due to the presence of lift forces created by turbulence in the crossflow device, and models for the hydrodynamics in such cases have been proposed. 

Using the Brock-Bird relation (Toxvaerd, 1975), the Flory theory (see Pandey et al., 1999, for a discussion), and the well-known Auerbach relation (Auerbach, 1948), the following equation can be formulated to predict ultrasonic velocity ... 

Another method used to evaluate the ultrasonic velocity is the Flory statistical theory, which, as it happens, has no direct link with ultrasonic velocity. Patterson and Rastogi (1970) have used this theory to calculate a characteristic surface tension using appropriately defined characteristic pressure and temperature. Equation 3.47 can then be used to calculate s. 

The main developments in experimental techniques for measuring high pressure to obtain reliable pressure sensors are extensively discussed by Decker et al. [42]. These include (1) the establishment of a primary pressure scale using a free piston gauge (2) the selection and precise measurement of identifiable phase transitions as fixed pressure points and (3) the use of interpolation and extrapolation techniques for continuous-pressure calibration based on changes in resistance, volume, or optical spectra (based on an equation of state). An alternative method of estimating absolute pressure in isotropically compressed materials is based on measurements of ultrasonic velocity [43, 44]. 

Properties.—A polyimide from succcinic acid has been fractionated and Mark-Houwink equation parameters determined, Several studies have appeared devoted to the investigation of the flexibility of polyimide molecules. Theoretical and experimental data indicate that these polymers have rather a high flexibility. " The Kuhn segment has been reported to consist of 2.5 monomer units. Also used to examine polyesterimides have been ultrasonic velocity and Rao formulism. ... 

Elastic Coefficients The elastic stiffness coefficients Cy can be calculated from the measured velocity of propagation of bulk acoustic ultrasonic waves, according to the Papadakis method (quartz transducer with center frequency of 20MHz) (Papadakis, 1967), on differently oriented bar-shaped samples using the equations given by Truell et al. (1969) and corrected for the piezoelectric contributions (Ljamov, 1983 Ikeda, 1990). The samples were oriented in axial directions XYZ, and 45° rotated against the X- and Y-axes, respectively. In order to obtain optimized values for the elastic materials parameters, the elastic stiffness coefficients Cy were used to calculate and critically compare the results of surface acoustic wave (SAW) measurements. 

Figure 3 displays the hypersonic speed dispersion curves of CCl, at four different temperatures. The complete dispersion curve contains additionally an experimental point corresponding to the ultrasonic velocity, i.e. lying at a comparatively low frequency which is located on the ordinate axis. The fit has been made with the classical dispersion relation with the relaxation time, and the amplitude of the dispersion curve as free fit parameters according to the equation ... 

Extensive measurements of ultrasonic velocity and absorption have been made on many polymer solutions at concentrations of the order of 0.01 to 0.1 g/cc. " Results for acoustic absorption are usually expressed as a/v, where p is the frequency in Hz (= cji/lir), which from equation 15 is given by... 

The velocity of elastic ultrasonic waves in solution is strongly influenced by solute-solvent and solute-solute interactions which, in turn, are determined by the chemical structure of the solute and solvent molecules. Adiabatic compressibility P for a system having the density p, is related to the measured ultrasonic velocity u by the following equation ... 

An ultrasonic field applied parallel to the helical axis k (k ) can cause a square gridlike pattern deformation of the planar texture [17, 58, 62]. This can be observed in nematic/cholesteric mixtures with a helix of large pitch. According to experimental data, the spatial period of distortion follows the equation L= p dy and tends to decrease slightly with frequency. The threshold particle velocity in the wave i is practically... 

Subsequent ultrasonic relaxation studies - also involved the PVP/SDS system. The reported results are very different from those on the same system given in Reference 225, in the sense that in the binding range the value of 1/Xx is claimed to be nearly independent of the surfactant concentration. This is apparently true for the data in References 226 and 227 (PVP/SDS systems) but certainly not for the results reported in Reference 228 for several other polymer/surfactant systems. There it is clearly seen that 1/Xi is nearly constant over a very short range of concentration but increases linearly with C in the binding range determined from ultrasonic velocity measurements, as predicted by Equation 3.9. Besides in Reference 226 to Reference 228, the range where 1/Xi is nearly... 

The velocity v of the ultrasonic wave in a material is given by the following equation ... 

Table 1 shows relations between the elastic constants of materials of a hexagonal symmetry and the velocity,direction of propagation and plane of polarisation of the ultrasonic waves. The relations between the material constants (Young"s modulus, stiffness modulus and Poisson number) and the elastic constants, derived from the equations quoted in Table 1 are as follows ... 

V is the ultrasonic frequency and r is the equilibrium resonant radius of the gas bubble. Once the maximum velocity gradient is known, it is possible to calculate the maximum stretching force Fmax, which a molecule will experience in the flow fieM. According to their validity any equation from Section 6.3 may be used. 

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