Thermal wave propagation

Equation 17.102 was derived on the assumption that concentration and thermal waves propagated at the same velocity. Amundson et al.<4y> showed that it was possible for the temperatures generated in the bed to propagate as a pure thermal wave leading the concentration wave. A simplified criterion for this to occur can be obtained from equations 17.75 and 17.101. Since there is no adsorption term associated with a pure thermal wave, and if changes within the bed voids are small, then ... 

For a bed initially free of adsorbate, the thermal wave propagates more quickly than the concentration wave if ... 

Operation of thermal swing systems for separation in the traveling wave mode (thermal wave propagation through the adsorbent bed) seems to be restricted to gas/dense gas regimes, because of the inability to adjust the velocity of the thermal wave in liquid systems. Liquid systems generally require the use of fixed temperature zones. 

Aldushin, A. P., Filtration combustion of metals. In Thermal Wave Propagation in Heterogeneous Media (Russ.). Nauka, Novosibirsk, 1988, p. 52. 

One important application of PAS is the depth-resolved measurement of layered samples. As the thermal waves propagate from the point where absorption oc-... 

The use of phase shifts for PA/FT-IR depth profiling was first discussed by Dittmar et al. [8], who correlated the depth from which a spectral feature originated with its phase spectrum, 9, as calculated by Eq. 20.3. The photoacoustic phase shift, 0 can be calculated for thermal wave propagation across a layer of thickness, t, from the velocity of the thermal wave [9,10] ... 

The distance that a thermal wave travels in one cycle is V /f. Since L = the phase shift of the PA signal emanating from a distance t below the surface, 0, is tIL radians, or 3(>Qtl2nL degrees. McClelland et al. [10] have listed the thermal wave propagation parameters with different modulation frequencies for a material with D = 0.001 cm s these parameters are hsted in Table 20.3. 

Table 20.3. Thermal Wave Propagation Parameters for Various Modulation Frequencies with D = 10 cm s ...

Equation (21) represents a series of thermal waves propagating into the surface and attenuating with distance. The higher harmonics are attenuated more rapidly than the lower harmonics since increases as the square root of the harmonic number n. The attenuation and phase of each harmonic depend on the quantity Pi, which is termed the thermal absorption coefficient of the planetary material. The reciprocal of(Lt = l/ i) is termed the thermal skin... 

The pressure is to be identified as the component of stress in the direction of wave propagation if the stress tensor is anisotropic (nonhydrostatic). Through application of Eqs. (2.1) for various experiments, high pressure stress-volume states are directly determined, and, with assumptions on thermal properties and temperature, equations of state can be determined from data analysis. As shown in Fig. 2.3, determination of individual stress-volume states for shock-compressed solids results in a set of single end state points characterized by a line connecting the shock state to the unshocked state. Thus, the observed stress-volume points, the Hugoniot, determined do not represent a stress-volume path for a continuous loading. 

No theoretical criterion for flammability limits is obtained from the steady-state equation of the combustion wave. On the basis of a model of the thermally propagating combustion wave it is shown that the limit is due to instability of the wave toward perturbation of the temperature profile. Such perturbation causes a transient increase of the volume of the medium reacting per unit wave area and decrease of the temperature levels throughout the wave. If the gain in over-all reaction rate due to this increase in volume exceeds the decrease in over-all reaction rate due to temperature decrease, the wave is stable otherwise, it degenerates to a temperature wave. Above some critical dilution of the mixture, the latter condition is always fulfilled. It is concluded that the existence of excess enthalpy in the wave is a prerequisite of all aspects of combustion wave propagation. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

Three groups of phenomena affect the frequency-dependence of ultrasonic wave propagation classical processes, relaxation, and scattering, of which scattering is likely to dominate in foodstuffs due to their particulate nature. The two classical thermal processes are radiation and conduction of heat away from regions of the material, which are locally compressed due to the passage of a wave they can lead to attenuation but the effect is negligible in liquid materials (Herzfield and Litovitz, 1959 Bhatia, 1967). The third classical process is due to shear and bulk viscosity effects. Attenuation in water approximates to a dependence on the square of the frequency and because of this it is common to express the attenuation in more complex liquids as a()/o or a(f)jf2 in order to detect, or differentiate from, water-like properties. 

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