Thermal wave velocity

It has been assumed that the gas and solid have the same temperature at any point, and that the fluid concentration is constant throughout a pellet at a value equal to that immediately outside the pellet. Within the limits of these assumptions, the thermal wave velocity up is independent of temperature. As discussed in Section 17.8.4, the velocity of the thermal wave relative to that of the concentration wave can be positive, as it normally is in liquids, negative or zero. 

In Figure 1, a cold feed is the influent for a hot column and the thermal wave (dashed line) is seen to have a higher velocity than either solute. Figure 2 describes the breakthrough profiles for an initially cold column with an intermediate feed temperature, where the thermal wave velocity is faster than only one solute velocity. In Figure 3, the thermal wave is shown to exit the column more slowly than either solute, for a column initially at an intermediate temperature with a hot influent. 

Changing the temperature of the feed to a sorption column will cause a thermal wave to pass through the column. The velocity of this thermal wave can be calculated by a procedure analogous to that used for solute waves. The thermal wave velocity will be the fraction of the change in thermal energy in the mobile phase multiplied by the interstitial velocity. 

As a first approximation, and solute movement theory is a first approximation, the thermal wave velocity is independent of concentration and tenperature. Tenperature is constant along the lines with a slope equal to the numerical value of 11. ... 

C. Plan. Since mass fractions are used in the equilibrium expression, we use Eq. (18-15cl to calculate the velocity of the solute at both 0 and 80°C. The thermal wave velocity is determined from Eq. fl8-21i with W = 0. The effect of the ten erature change on the fluid concentration can be determined either from a mass balance over one cycle or fromEq. (18=24). 

W ().379[Pg.821]

This frequency is a measure of the vibration rate of the electrons relative to the ions which are considered stationary. Eor tme plasma behavior, plasma frequency, COp, must exceed the particle-coUision rate, This plays a central role in the interactions of electromagnetic waves with plasmas. The frequencies of electron plasma waves depend on the plasma frequency and the thermal electron velocity. They propagate in plasmas because the presence of the plasma oscillation at any one point is communicated to nearby regions by the thermal motion. The frequencies of ion plasma waves, also called ion acoustic or plasma sound waves, depend on the electron and ion temperatures as well as on the ion mass. Both electron and ion waves, ie, electrostatic waves, are longitudinal in nature that is, they consist of compressions and rarefactions (areas of lower density, eg, the area between two compression waves) along the direction of motion. 

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

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 ... 

The velocity with which a pure thermal wave travels through an insulated packed bed may be obtained from equation 17.100 by putting Uq = 0 and (d/dT)(CsAH) = 0 to give ... 

Figure 17.37 shows a thermal wave plotted as a dotted line of distance against time. The velocity uc of the concentration wave depends on where it is in relation to the thermal wave, as can be seen by comparison with the full line in the Figure 17.37. 

Uj lower/higher temperatures Velocity of a point on a thermal wave m/s LT ... 

Furthermore, comparison of the thermal vT and the concentration vc wave velocities as defined by Gould (1969)... 

Table VII shows that, for the methanation reactor model, the dynamic response of the gas temperatures and CO and C02 concentrations should be much faster (by two orders of magnitude) than the response of the catalyst and thermal well temperatures. This prediction is verified in the dynamic responses shown in Figs. 18 and 19 and the previous analysis of the thermal and concentration wave velocities.

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. 

Heinrich Hertz long ago solved the problem of a thermal wave in front of a heated surface which is moving with constant velocity. This solution, first applied to a flame by Michelsohn, has the form... 

The slower autowave process is similar in some respects to classical combustion, despite the differences in their physical nature. The wave velocity shows the same dependence on thermal conductivity as in the case of flame propagation. Analogously to combustion, the reaction zone is near the maximum temperature Tm [it is near Tm that the critical gradient (dT/dx) switching on the reaction is realized], whereas the greater part of the front... 

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. 

In an early work by Kottke and Niiler (1988), a cellular model was used to simulate the combustion wave initiation and propagation for the TH-C model system. The interactions between neighboring cells were described by the electrical circuit analogy to heat conduction. At the reaction initiation temperature (i.e., melting point of titanium), the cell is instantly converted to the product, TiC, at the adiabatic combustion temperature. The cell size was chosen to be twice as large as the Ti particles (44 /xm). Experimentally determined values for the green mixture thermal conductivity as a function of density were used in the simulations. As a result, the effects of thermal conductivity of the reactant mixture on combustion wave velocity were determined (see Fig. 21). Advani et al. (1991) used the same model, and also computed the effects of adding TiC as a diluent on the combustion velocity. 

Aluminum nitride has received attention as an alternative to Si02 dielectric layers in microelectronic circuits because of its high dielectric strength. Being a refractory ceramic with high thermal conductivity, AIN is useful for electronics packaging. There are also uses for AIN as a piezoelectric material because of its high surface acoustic wave velocity. 

The results obtained for lead azide are summarized in Table IX and in Figure 30. The highest temperature achieved at the aluminum-lead azide interface was less than 120°C, which is significantly below the lowest value (297°C) for the thermal initiation of lead azide. The data indicated a stress initiation threshold of 3.6 kbar for the lead azide, assuming a sound velocity of 2.5 km/sec for the explosive. The stress pulse-width was approximately 0.2 psec. If the wave velocity (shock velocity) of 1.23 km/sec is assumed for dextrinated lead azide, then a lower bound of 2.2 kbar can be placed on the threshold for RD1333 lead azide. 

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