Empirical correlations

For systems containing four components, most previous attempts for calculating LLE use geometrical correlations of ternary data (Branckner, 1940), interpolation of ternary data (Chang and Moulton, 1953), or empirical correlations of ternary data (Prince, 1954 Henty, 1964). These methods all have two... 

The structural unit associated with an electronic transition m UV VIS spectroscopy IS called a chromophore Chemists often refer to model compounds to help interpret UV VIS spectra An appropriate model is a simple compound of known structure that mcor porates the chromophore suspected of being present m the sample Because remote sub stituents do not affect Xmax of the chromophore a strong similarity between the spectrum of the model compound and that of the unknown can serve to identify the kind of rr electron system present m the sample There is a substantial body of data concerning the UV VIS spectra of a great many chromophores as well as empirical correlations of sub stituent effects on k Such data are helpful when using UV VIS spectroscopy as a tool for structure determination... 

The values of d and n are given in Table 3 typical values for can be found in Table 4. The exponent of 0.5 on the Schmidt number (l-L /PiLj) supports the penetration theory. Further examples of empirical correlations provide partial experimental confirmation of equation 78 (3,64—68). The correlation reflecting what is probably the most comprehensive experimental basis, the Monsanto Model, also falls in this category (68,69). It is based on 545 observations from 13 different sources and may be summarized as... 

To estimate the number of transfer units for design, the following empirical correlations which were derived from efficiency measurements employing a variety of trays and operating conditions under the aforementioned assumptions are recommended (111) ... 

Another approximation, one of the most enduring empirical correlations in multiphase systems, is the Richardson-Zaki correlation for a single particle in a suspension (3) ... 

Dow Fire and Explosion Index. The Dow Eire and Explosion Index (3) is a procedure usehil for determining the relative degree of hazard related to flammable and explosive materials. This Index form works essentially the same way as an income tax form. Penalties are provided for inventory, extended temperatures and pressures, reactivity, etc, and credits are appHed for fire protection systems, process control (qv), and material isolation. The complete procedure is capable of estimating a doUar amount for the maximum probable property damage and the business intermptionloss based on an empirical correlation provided with the Index. 

Likewise, the microscopic heat-transfer term takes accepted empirical correlations for pure-component pool boiling and adds corrections for mass-transfer and convection effects on the driving forces present in pool boiling. In addition to dependence on the usual physical properties, the extent of superheat, the saturation pressure change related to the superheat, and a suppression factor relating mixture behavior to equivalent pure-component heat-transfer coefficients are correlating functions. 

Spray Correlations. One of the most important aspects of spray characterization is the development of meaningful correlations between spray parameters and atomizer performance. The parameters can be presented as mathematical expressions that involve Hquid properties, physical dimensions of the atomizer, as well as operating and ambient conditions that are likely to affect the nature of the dispersion. Empirical correlations provide useful information for designing and assessing the performance of atomizers. Dimensional analysis has been widely used to determine nondimensional parameters that are useful in describing sprays. The most common variables affecting spray characteristics include a characteristic dimension of atomizer, d Hquid density, Pjj Hquid dynamic viscosity, ]ljj, surface tension. O pressure, AP Hquid velocity, V gas density, p and gas velocity, V. ... 

Many empirical correlations have been pubHshed in the Hterature for various types of Hquid atomizers, eg, one book (2) provides an extensive coUection of empirical equations. Unfortunately, most of the correlations share some common problems. Eor example, they are only vaHd for a specific type of atomizer, thereby imposing strict limitations on thein use. They do not represent any specific physical processes and seldom relate to the design of the atomizer. More important, they do not reveal the effect of interactions among key variables. This indicates the difficulty of finding a universal expression that can cover a wide range of operating conditions and atomizer designs. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

Generalized Correlations. Generalized correlations are often the only recourse when a property value cannot be determined from empirical correlations or by other means. Several powerful correlating techniques fall under this category, including the principle of corresponding states (3,17), reduced property models (1), and the Polanyi-type characteristic curve for microporous adsorbents (14). 

Over 25 years ago the coking factor of the radiant coil was empirically correlated to operating conditions (48). It has been assumed that the mass transfer of coke precursors from the bulk of the gas to the walls was controlling the rate of deposition (39). Kinetic models (24,49,50) were developed based on the chemical reaction at the wall as a controlling step. Bench-scale data (51—53) appear to indicate that a chemical reaction controls. However, flow regimes of bench-scale reactors are so different from the commercial furnaces that scale-up of bench-scale results caimot be confidently appHed to commercial furnaces. For example. Figure 3 shows the coke deposited on a controlled cylindrical specimen in a continuous stirred tank reactor (CSTR) and the rate of coke deposition. The deposition rate decreases with time and attains a pseudo steady value. Though this is achieved in a matter of rninutes in bench-scale reactors, it takes a few days in a commercial furnace. 

Empirical correlations for i< -values found in the older literature have httle relation to thermodynamics. Their proper evaluation comes directly from Eq. (4-277) ... 

Vigne.s empirically correlated mixture diffusivity data for 12 binary mixtures. Later Ertl et al. evaluated 122 binary systems, which showed an average absolute deviation of only 7 percent. None of the latter systems, however, was veiy nonideal. 

The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surface-renewal models. Thus, in view of its relative simphcity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for maldug a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. 

Semiempirical correlations are often preferred to purely empirical or purely theoretical correlations. Purely empirical correlations are dangerous to use for extrapolation. Purely theoretical correlations may predict trends accurately but they can be several orders of magnitude off in the value of k. 

Probably Fl is a function of particle Reynolds number and concentration, but Fig. 6-33 gives Durand s empirical correlation for Fl as a function of particle diameter and the input, feed volume fraction solids, Cs = QsKQs + Ql)- The form of Eq. (6-145) may be derived from turbulence theory, as shown by Davies (Chem. Eng. Sci., 42, 1667-1670 [1987]). 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

On the basis of experimental studies, Mathur and Gishler derived an empirical correlation to describe the minimum fluid flow necessaiy for spouting, in 3- to 12-in-diameter columns ... 

Feed analyses in terms of component concentrations are usually not available for complex hydrocarbon mixtures with a final normal boihng point above about 38°C (100°F) (/i-pentane). One method of haudhug such a feed is to break it down into pseudo components (narrow-boihng fractions) and then estimate the mole fraction and value for each such component. Edmister [2nd. Eng. Chem., 47,1685 (1955)] and Maxwell (Data Book on Hydrocarbons, Van Nostrand, Princeton, N.J., 1958) give charts that are useful for this estimation. Once values are available, the calculation proceeds as described above for multicomponent mixtures. Another approach to complex mixtures is to obtain an American Society for Testing and Materials (ASTM) or true-boihng point (TBP) cui ve for the mixture and then use empirical correlations to con-strucl the atmospheric-pressure eqiiihbrium-flash cui ve (EF 0, which can then be corrected to the desired operating pressure. A discussion of this method and the necessary charts are presented in a later subsection entitled Tetroleum and Complex-Mixture Distillation. ... 

Because of the time and expense involved in conducting laboratoiy distillation tests of all three basic types, it has become increasingly common to use empirical correlations to estimate the other two distillation curves when either the ASTM, TBP, or EFN- curve is available. Preferred correlations given in the API Technical Data Book—Petroleum Refining (op. dt.) are based on the work of Edmister and Pollock [Chem. Eng. Prog., 44, 905 (1948)], Edmister and Okamoto [Pet. Refiner, 38(8), 117 (1959) 38(9), 271 (1959)], Maxwell Data Book on... 

Empirical Predictive Methods Two empirical correlations which have found wide use are the one of Drickamer and Bradford... 

As shown by Table 14-12, empirical correlations for two-fluid atomization show dependence on high gas velocity to supply atomizing energy, usually to a power dependence close to that for turbulent breakup. In addition, the correlations show a dependence on the ratio of gas to liquid and system dimension. 

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