Fractionation model

Rhodes, and Scott Can. j. Chem. Eng., 47,445 53 [1969]) and Aka-gawa, Sakaguchi, and Ueda Bull JSME, 14, 564-571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles Can. ]. Chem. Eng., 52, 25-35 [1974]) and Barnea, Shoham, and Taitel Chem. Eng. Sci, 37, 741-744 [1982]). Use of drift flux theoiy for void fraction modeling in downflow is presented by Clark anci Flemmer Chem. Eng. Set., 39, 170-173 [1984]). Downward inclined two-phase flow data and modeling are given by Barnea, Shoham, and Taitel Chem. Eng. Set., 37, 735-740 [1982]). Data for downflow in helically coiled tubes are presented by Casper Chem. Ins. Tech., 42, 349-354 [1970]). 

Kase and Horiuchi (1996) obtained a large number of analytical data on sphalerites from sixteen Besshi-type deposits, mainly at Besshi and its vicinity, Hitachi, and Shimokawa. They revealed that (1) the Mn/Zn and Co/Zn ratios of sphalerite may have markedly increased during contact metamorphism, while the Cd/Zn ratios remained unchanged (2) the Emco/ lwzn (2/n total dissolved concentration in ore fluids) and Emco/S/wzn ratios in the initial ore solutions responsible for the mineralizations at Besshi which was calculated ba.sed on the equilibrium fractionation model between hydrothermal solution and sphalerite and analytical data on sphalerites are quite similar to the ratios of hydrothermal solutions at EPR 21 °N (3) however, these ratios for the Hitachi solutions are very low and different from those of the Besshi-subtype solution. 

Chexal, G., J. Horowitz, and G. Lellouche, 1987, An Assessment of Eight Void Fraction Models, in Heat Transfer—Pittsburgh, AIChE Symp. Ser. 83 (257) 249-254 Also 1986, EPRI, Nuclear Safety Analysis Center, Rep. NSAC-107. (3)... 

The results of the fractionation model (Fig. 24.9) differ from the equilibrium model in two principal ways. First, the mineral masses can only increase in the fractionation model, since they are protected from resorption into the fluid. Therefore, the lines in Figure 24.9 do not assume negative slopes. Second, in the equi-... 

The fractionation calculation is notable in that it predicts the formation of two minerals (bloedite and kainite) that did not precipitate in the equilibrium model. As well, hexahydrite, which appeared briefly in the equilibrium model, does not form in the fractionation model. The two classes of models, therefore, represent qualitatively distinct pathways by which an evaporating water can evolve. 

Procedures 1 to4describedinSections7.2.1 through 7.2.4 are applied in this example for determination of wastewater COD fractions, model parameters and a corresponding calibration/validation of the sewer process model under aerobic and dry-weather conditions. The number of repeated tests — a total of 29 during different seasons — demonstrates not just the validity of the sewer process model depicted in Table 5.3 but also the validity of the concept behind the model formulated in Section 5.2. 

Figure 4. Illustration of mass-dependent fractionation of Mg isotopes, cast in terms of 5 values. 5 Mg and 5 Mg values based on Mg/ Mg and Mg/ Mg ratios, respectively. A common equilibrium fractionation model, as defined by exponential relations between a values (fractionation factors) for different isotope ratios, is shown in the gray line. A simple linear relation, where the slope is proportional to the mass difference of the isotope pair, is shown in the black line. Additional mass-dependent fractionation laws may be defined, and all are closely convergent over small ranges (a few per mil) in isotope compositions at 5 values that are close to zero.

Mossbauer spectroscopy involves the measurement of minute frequency shifts in the resonant gamma-ray absorption cross-section of a target nucleus (most commonly Fe occasionally Sn, Au, and a few others) embedded in a solid material. Because Mossbauer spectroscopy directly probes the chemical properties of the target nucleus, it is ideally suited to studies of complex materials and Fe-poor solid solutions. Mossbauer studies are commonly used to infer properties like oxidation states and coordination number at the site occupied by the target atom (Flawthome 1988). Mossbauer-based fractionation models are based on an extension of Equations (4) and (5) (Bigeleisen and Mayer 1947), which relate a to either sums of squares of vibrational frequencies or a sum of force constants. In the Polyakov (1997)... 

CAI s that were once molten (type B and compact type A) apparently crystallized under conditions where both partial pressures and total pressures were low because they exhibit marked fractionation of Mg isotopes relative to chondritic isotope ratios. But much remains to be learned from the distribution of this fractionation. Models and laboratory experiments indicate that Mg, O, and Si should fractionate to different degrees in a CAI (Davis et al. 1990 Richter et al. 2002) commensurate with the different equilibrium vapor pressures of Mg, SiO and other O-bearing species. Only now, with the advent of more precise mass spectrometry and sampling techniques, is it possible to search for these differences. Also, models prediet that there should be variations in isotope ratios with growth direction and Mg/Al content in minerals like melilite. Identification of such trends would verify the validity of the theory. Conversely, if no correlations between position, mineral composition, and Mg, Si, and O isotopic composition are found in once molten CAIs, it implies that the objects acquired their isotopic signals prior to final crystallization. Evidence of this nature could be used to determine which objects were melted more than once. 

Figure 4.5 Cr as a function of the amount of Cr(VI) remaining in a batch slurry experiments with estuarine sediment. Line gives a Rayleigh fractionation model, with s = 3.4%o. Data from Ellis et al. (2002).

Biogenic particles which comprise primary (fungal spores, bacteria, viruses, plant debris) and secondary organic aerosol (SOA) from biogenic non-methane VOCs are part of the commonly measured organic carbon fraction. Model results [51] indicate a... 

For large molecules Orozco and Luque also suggested a practical procedure to calculate EP atomic charges [73]. They divided the molecule into several parts and these parts were treated independently concerning the calculation of EP atomic charges (fractional model). The influence exerted by the other parts is taken into account with the help of small groups of atoms. 

Akagawa, Sakaguchi, and Ueda (Bull JSME, 14, 564-571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles (Can. J. Chem. Eng., 52,25-35 [1974]) and Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 741—744 [1982]). Use of drift flux theory for void fraction modeling in downflow is... 

Mao, H.-K., Bell, P. M. Yagi, T. (1982) Iron-magnesium fractionation model for the Earth. In High-Pressure Research in Geophysics. (S. Akimoto M. H. Manghnani, eds Acad. Publ., Tokyo, Japan), pp. 319-25. 

Kehat and Shacham ( 6) used split fraction models to estimate the Jacobian when the Newton-Raphson method is used to solve Equation (1). The authors concluded that their method is very efficient for systems with more than one tear stream and when there is only a weak interaction between variables in the tear stream. 

Vuthandam et al. (1995) considered the top 2x2 subsystem of the heavy oil fractionator modeled in the Shell Standard Process Control Problem (Prett and Garcia, 1988) as... 

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