Isokinetic approximation

This equation can be used when the ratio between the nueleation rate and the growth rate is constant with time and, therefore, the athermal nueleation is negh-gible (isokinetic approximation). The drawback of Ziabicki s theory is that it can be apphed only in the range of temperatures where isothermal crystallization data are available. 

Obviously for this method to work the ratio T1IT2 must be appreciably smaller than unity. Provided this condition is met, this method is a simple and reliable way to test for an isokinetic relationship or to detect deviations from such a relationship. Exner shows examples of systems plotted both as log 2 vs. log and as AH vs. A5, demonstrating the inadequacy of the latter plot. Exner has also developed a statistical analysis of the Petersen method this analysis yields p and an uncertainty estimate of p. Exner has applied his statistical methods to 100 reaction series, finding that 78 of them follow approximately valid isokinetic relationships. 

Several doubts about the correctness of the usual statistical treatment were expressed already in the older literature (31), and later, attention was called to large experimental errors (142) in AH and AS and their mutual dependence (143-145). The possibility of an apparent correlation due only to experimental error also was recognized and discussed (1, 2, 4, 6, 115, 116, 119, 146). However, the full danger of an improper statistical treatment was shown only by this reviewer (147) and by Petersen (148). The first correct statistical treatment of a special case followed (149) and provoked a brisk discussion in which Malawski (150, 151), Leffler (152, 153), Palm (3, 154, 155) and others (156-161) took part. Recently, the necessary formulas for a statistical treatment in common cases have been derived (162-164). The heart of the problem lies not in experimental errors, but in the a priori dependence of the correlated quantities, AH and AS. It is to be stressed in advance that in most cases, the correct statistical treatment has not invalidated the existence of an approximate isokinetic relationship however, the slopes and especially the correlation coefficients reported previously are almost always wrong. 

The method outlined is quick and useful for testing isokinetic relationships described in the literature and for finding approximate values of j3 (149). It should replace the incorrect plotting of E versus log A, which gives fallacious results for the value of (3 and which usually simulates better correlations than in fact apply. Particularly, the values of correlation coefficients (1) in the E versus log A plane are meaningless. As shown objectively in Figures 9-12, the failure of this plotting is not caused by experimental errors only (3, 143, 153), nor is it confined to values of j5 near the error slope or within the interval of experimental temperatures (151). 

By repeating the calculation for various values of x, one can obtain y and Sx as functions of x and find the minimum of the latter by successive approximations. The value of x at this minimum (xo) gives the estimate of the isokinetic temperature Xo The corresponding values yo and So are obtained from eqs. (52) and (53) So has... 

The most fundamental thermodynamic approach of Rudakov (6) applies to all condensed systems. The actual linear relationship is argued to exist between enthalpy (AH) and entropy (AS) of intermolecular interaction, as reflected in an approximately linear relationship between the total enthalpy and entropy. Special attention has been given to hydrophobic interaction (89, 90) in water solutions, which makes the isokinetic behavior more pronounced and markedly changes its slope. 

For simplicity we assumed that the transition states are charged. However, it is not necessary to do so because the only requirement is that the difference in entropy of forming the transition states be offset by the difference in enthalpy of activation. The transition states could have different polarities and the same result be obtained. In fact, the transition states need not have high polarity. Forming a transition state in which there is a reduction in charge separation could result in more favorable solvation when the solvent is nonpolar. For there to be an isokinetic relationship for a series of reactions, it is required only that AH and AS be related in such a way that AG be approximately constant. 

Table XV lists the isokinetic temperatures of several reactions representing a wide variety of mechanisms, these examples having been chosen because the isokinetic temperature happened to fall in the popular experimental range between 0 and 100°. There are many other polar reactions that have isokinetic temperatures well outside of the accessible temperature range there are many whose variations in activation energy and entropy are not parallel and these, of course, do not have an isokinetic temperature even approximately. When one of a series of reactions deviates markedly from a parallel trend in activation energy and entropy established by the others, it is probable that it differs in mechanism from the others. This is a better indication of a change in mechanism than either marked differences in rate or in activation energy.

Some reactions are characterized by straight-line plots of TAS versus having a slope of approximately one, where this linearity results from compensatory, or off-setting, changes of AH and TAS. For this reason, the change in the Gibbs energy of activation, AG = AH - TAS, is a better description of the variation in the reaction than either AH or alone . See also Isokinetic Relationship... 

Mass balance measurements for 41 elements have been made around the Thomas A. Allen Steam Plant in Memphis, Tenn. For one of the three independent cyclone boilers at the plant, the concentration and flow rates of each element were determined for coal, slag tank effluent, fly ash in the precipitator inlet and outlet (collected isokinetically), and fly ash in the stack gases (collected isokinetically). Measurements by neutron activation analysis, spark source mass spectroscopy (with isotope dilution for some elements), and atomic adsorption spectroscopy yielded an approximate balance (closure to within 30% or less) for many elements. Exceptions were those elements such as mercury, which form volatile compounds. For most elements in the fly ash, the newly installed electrostatic precipitator was extremely efficient. 

Since reactions at comparable concentrations of surface species and involving an almost identical bond redistribution energy requirement Es must be expected to exhibit at least approximate isokinetic behavior, it follows that the observed values of A must compensate for changes in E — Es — t — 2). 

It is evident that a calculation of the rate constant neglecting AS will only be correct for a limited number of cases. Series of reactions are known in which AS really remains approximately constant. Other series obey the isokinetic relation in others the change in AS is independent of E [14]. Nevertheless, for the time being, our only choice is to consider only the energy, and compare the result with experiment. A procedure for calculating E at variable AS in a generally applicable form is not yet available. 

The existence of compensation behaviour can be accounted for as follows. All samples of calcite undergo dissociation within approximately the same temperature interval, many kinetic studies include the range 950 tolOOO K. The presence of COj (product) may decrease reactivity and a delay in heat flow into the reactant will decrease the reaction temperature. Thus, imder varied conditions, the reaction occurs close to a constant temperature. This is one of the conditions of isokinetic behaviour (groups of related reactions showing some variations of T within the set will nonetheless exhibit a well-defined compensation plot [61]). As already pointed out, values of A and E calculated for this reaction, studied under different conditions, show wide variation. This can be ascribed to temperature-dependent changes in the effective concentrations of reaction precursors, or in product removal [28] at the interface, and/or heat flow. The existence of the (close to) constant T, for the set of reactions, for which the Arrhenius parameters include wide variations, requires (by inversion of the argument presented above) that the magnitudes of A and E are related by equation (4.6). 

In the applications of gas-solid flows, measurements of particle mass fluxes, particle concentrations, gas and particle velocities, and particle aerodynamic size distributions are of utmost interest. The local particle mass flux is typically determined using the isokinetic sampling method as the first principle. With the particle velocity determined, the isokinetic sampling can also be used to directly measure the concentrations of airborne particles. For flows with extremely tiny particles such as aerosols, the particle velocity can be approximated as the same as the flow velocity. Otherwise, the particle velocity needs to be measured independently due to the slip effect between phases. In most applications of gas-solid flows, particles are polydispersed. Determination of particle size distribution hence becomes important. One typical instrument for the measurement of particle aerodynamic size distribution of particles is cascade impactor or cascade sampler. In this chapter, basic principles, applications, design and operation considerations of isokinetic sampling and cascade impaction are introduced. 

It is worth pointing out that the isokinetic sampling is not required when very small particles are sampled. A correlation for a better approximation of the aspiration efficiency was proposed by Belyaev and Levin (1972, 1974) as... 

From the results shown in Figure 2 and 3 one can infer that the Avrami isokinetic condition is not satisfied, i.e., N/G constant. However, over the temperature range from 200 to 140 C the number of nuclei is approximately constant, and Figure 2 shows that nuclei site saturation occurs... 

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