Axial correlation

In the preceding chapters the concept of the random coil was developed, where axial correlation of mers is only of very short distance see Figure 5.1. Now a number of polymers have been synthesized which are substantially rodshaped. An example is poly(p-phenylene terephthalamide), PPD-T, with the chemical structure... 

NMR relaxation and small-angle X-ray scattering indicated that the intramolecular orientation is only slightly affected by the presence of other chains in the glassy amorphous state and that axial correlation is of the order of 1 nm or less. 

The recommended rapid design procedure consists of the following steps (/) The apparent is calculated using equation 56. (2) The extent of axial dispersion is estimated from Hterature correlations for each phase in terms of Pe numbers and transformed into values. (3) The correction... 

The servo voltage is a function of mass-flow rate. Axial-flow angular-momentum meters are sometimes used in measuring jet engine fuel flow as the fuel energy content correlates much mote closely with mass than volume. 

Numerous studies for the discharge coefficient have been pubHshed to account for the effect of Hquid properties (12), operating conditions (13), atomizer geometry (14), vortex flow pattern (15), and conservation of axial momentum (16). From one analysis (17), the foUowiag empirical equation appears to correlate weU with the actual data obtained for swid atomizers over a wide range of parameters, where the discharge coefficient is defined as — QKA (2g/ P/) typical values of range between 0.3 and 0.5. 

The recommended correlation for the gas-phase axial-dispersion coefficient is given by Field and Davidson (loc. cit.) ... 

Heat Transfer Heat-transfer rates are gener ly large despite severe axial dispersion, with Ua. frequently observed in the range 18.6 to 74.5 and even to 130 kW/(m K) [1000 to 4000 and even to 7000 Btu/(h fF °F)][see Bauerle and Ahlert, Ind. Eng. Chem. Process Des. Dev., 4, 225 (1965) and Greskovich et al.. Am. Tn.st. Chem. Eng. J., 13,1160 (1967) Sideman, in Drewet al. (eds.). Advances in Chemical Engineering, vol. 6, Academic, New York, 1966, p. 207, reviewed earlier work]. In the absence of specific heat-transfer correlations, it is suggested that rates be estimated from mass-transfer correlations via the heat-mass-transfer analogy. 

Axial Dispersion Vermeiilen et al. [Chem. Eng. Prog., 62(9), 95 (1966)] summarized many of the data for packings. Their correlation for the continuous phase is shown in Fig. 15-36. For the dispersed phase, their correlation is given by... 

Correlations for axial dispersion in beds packed with very small particles (<50 Im) that take into account the holdup of liquid in a bed are discussed by Horvath and Lin [J. Chromatogr, 126, 401 (1976)]. 

TABLE 16-10 Coefficients for Axial Dispersion Correlations in Packed Beds Based on Eq (16-79)... 

FIG. 16 11 Axial dispersion coefficient correlations for well-packed beds for e = 0.4. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

Two complementai y reviews of this subject are by Shah et al. AIChE Journal, 28, 353-379 [1982]) and Deckwer (in de Lasa, ed.. Chemical Reactor Design andTechnology, Martinus Nijhoff, 1985, pp. 411-461). Useful comments are made by Doraiswamy and Sharma (Heterogeneous Reactions, Wiley, 1984). Charpentier (in Gianetto and Silveston, eds.. Multiphase Chemical Reactors, Hemisphere, 1986, pp. 104—151) emphasizes parameters of trickle bed and stirred tank reactors. Recommendations based on the literature are made for several design parameters namely, bubble diameter and velocity of rise, gas holdup, interfacial area, mass-transfer coefficients k a and /cl but not /cg, axial liquid-phase dispersion coefficient, and heat-transfer coefficient to the wall. The effect of vessel diameter on these parameters is insignificant when D > 0.15 m (0.49 ft), except for the dispersion coefficient. Application of these correlations is to (1) chlorination of toluene in the presence of FeCl,3 catalyst, (2) absorption of SO9 in aqueous potassium carbonate with arsenite catalyst, and (3) reaction of butene with sulfuric acid to butanol. 

Axial Dispersion and the Peclet Number Peclet numbers are measures or deviation from phig flow. They may be calculated from residence time distributions found by tracer tests. Their values in trickle beds are fA to Ve, those of flow of liquid alone at the same Reynolds numbers. A correlation by Michell and Furzer (Chem. Eng. /., 4, 53 [1972]) is... 

For the preliminary estimate of the expected efficiency of expansion turbines, in most cases it is sufficient to neglect Reynolds number effects (Rg > 10 ) and use the efficiency and specific speed correlations shown in Figure 2-12 for partial admission axial impulse, reaction radial inflow and full admission impulse and reaction axial turbines. Due to the economic advantage of the radial turbine, die radial inflow turbine is die best selection when operating in die specific speed range 20 < Nj < 140, whereby die optimum efficiency will be achieved at N, = 80. 

Similar behaviour is found in many other carboxylates of Cu as well as their adducts in which axial water is replaced by other O-or A -donor ligands. In spite of a continuous flow of work on these compounds there is still no general agreement as to the actual mechanism of the interaction nor on possible correlations of its magnitude with relevant... 

The power of the new spectrometer to reveal configurations of difficult cyclitols or sugars was first tested with mt/o-inositol (2), using deuterium oxide as solvent. At 60 or 100 MHz. the one equatorial and five axial protons appear to have different chemical shifts as shown by Lemieux in 1956 with a 40 MHz. instrument (14,15). However, since the five-proton axial signal could not be resolved, one could probably not have assigned the configuration 2 (which was already known from laborious chemical correlations extending over many years.)... 

Mullis made measurements of the heat flux for axially directed jets and for jets directed 30° to the motor axis. For the axially directed jets, Mullis found that the flux 4 to 10 diameters downstream of the contact point could be correlated by the boundary layer relations... 

For the region near the attachment point, Mullis found a strong effect of axial position on flux, but no satisfactory general correlation for this effect. In addition, he found no quantitative relation for the heat-transfer characteristics of jets directed toward the propellant surface. Under most conditions studied by Mullis, the radiation contribution is approximately 10% of the convective flux. The effects of solid-particle impingement were not investigated. 

It seems probable that a fruitful approach to a simplified, general description of gas-liquid-particle operation can be based upon the film (or boundary-resistance) theory of transport processes in combination with theories of backmixing or axial diffusion. Most previously described models of gas-liquid-particle operation are of this type, and practically all experimental data reported in the literature are correlated in terms of such conventional chemical engineering concepts. In view of the so far rather limited success of more advanced concepts (such as those based on turbulence theory) for even the description of single-phase and two-phase chemical engineering systems, it appears unlikely that they should, in the near future, become of great practical importance in the description of the considerably more complex three-phase systems that are the subject of the present review. 

Dunn et al. (D7) measured axial dispersion in the gas phase in the system referred to in Section V,A,4, using helium as tracer. The data were correlated reasonably well by the random-walk model, and reproducibility was good, characterized by a mean deviation of 10%. The degree of axial mixing increases with both gas flow rate (from 300 to 1100 lb/ft2-hr) and liquid flow rate (from 0 to 11,000 lb/ft2-hr), the following empirical correlations being proposed ... 

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