Average of averages

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

ANGLE AVERAGING IS USED TO ACCOUNT FOR THE EFFECT OF POLARITY DUE TO A large DIPOLE ON THE ENERGY AND SIZE PARAMETERS. 

Also included in this table are some average execution times for the thermodynamic subroutines measured for the CDC 6400 of the Computer Center, University of California, Berkeley. 

Multiple reactions in parallel producing byproducts. Consider again the system of parallel reactions from Eqs. (2.16) and (2.17). A batch or plug-flow reactor maintains higher average concentrations of feed (Cfeed) than a continuous well-mixed reactor, in which the incoming feed is instantly diluted by the PRODUCT and... 

If the composition of the waste stream is known, then the theoretical oxygen demand can be calculated from the appropriate stoichiometric equations. As a first level of approximation, we can assume that this theoretical oxygen demand would be equal to the COD. Then, experience with domestic sewage indicates that the average ratio of COD to BOD will be on the order 1.5 to 2. The following example will help to clarify these relationships. 

Maintenance costs depend on whether processing fluids are solids on the one hand or gas and liquid on the other. Solids handling tends to increase maintenance costs. Highly corrosive process fluids increase maintenance costs. Average maintenance costs tend to be around 6 percent of the flxed capital investment. ... 

B = Average bond energy of P-CI bond. From the cycle it follows that ... 

Born-Haber cycle A thermodynamic cycle derived by application of Hess s law. Commonly used to calculate lattice energies of ionic solids and average bond energies of covalent compounds. E.g. NaCl ... 

If agitation and heating are not practical as in the cases of large volumes, it is better to withdraw samples from various levels in order to get an average sample. 

To extend the applicability of the characterization factor to the complex mixtures of hydrocarbons found in petroleum fractions, it was necessary to introduce the concept of a mean average boiling point temperature to a petroleum cut. This is calculated from the distillation curves, either ASTM or TBP. The volume average boiling point (VABP) is derived from the cut point temperatures for 10, 20, 50, 80 or 90% for the sample in question. In the above formula, VABP replaces the boiling point for the pure component. 

The measurement techniques most frequently used are derived from Raoult s and Van t Hoff s laws applied to cryometry, ebulliometry, osmometry, etc. They are not very accurate with errors on the order of ten per cent. Consequently, the molecular weight is often replaced by correlated properties. The mean average temperature or viscosity can thus replace molecular weight in methods derived from ndM. 

Although gas chromatography can give the concentration of each component in a petroleum gas or gasoline sample, the same cannot be said for heavier cuts and one has to be satisfied with analyses by chemical family, by carbon atom distribution, or by representing the sample as a whole by an average molecule. 

Using this concept, Burdett developed a method in 1955 to obtain the concentrations in mono-, di- and polynuclear aromatics in gas oils from the absorbances measured at 197, 220 and 260 nm, with the condition that sulfur content be less than 1%. Knowledge of the average molecular weight enables the calculation of weight per cent from mole per cent. As with all methods based on statistical sampling from a population, this method is applicable only in the region used in the study extrapolation is not advised and usually leads to erroneous results. 

One has seen that the number of individual components in a hydrocarbon cut increases rapidly with its boiling point. It is thereby out of the question to resolve such a cut to its individual components instead of the analysis by family given by mass spectrometry, one may prefer a distribution by type of carbon. This can be done by infrared absorption spectrometry which also has other applications in the petroleum industry. Another distribution is possible which describes a cut in tei ns of a set of structural patterns using nuclear magnetic resonance of hydrogen (or carbon) this can thus describe the average molecule in the fraction under study. 

The rapidity with which information is received one second instead of an average 10 minutes for dispersive spectrometry. 

The parameter giving the ratio of the number of substitutable aromatic carbon atoms to the total number of aromatic carbons which gives a measure of the average condensation of aromatic rings. 

The average length of the chains attached to aromatic rings (Brown and Ladner, 1960). 

Figure 3.12 shows the spectrum of carbon 13 obtained from a distillation residue and Table 3.10 gives average parameters for two FCC feedstocks as measured by NMR. 

This method follows the ASTM D 1159 and D 2710 procedures and the AFNOR M 07-017 standard. It exploits the capacity of the double olefinic bond to attach two bromine atoms by the addition reaction. Expressed as grams of fixed bromine per hundred grams of sample, the bromine number, BrN, enables the calculation of olefinic hydrocarbons to be made if the average molecular weight of a sufficiently narrow cut is known. 

It is common that a mixture of hydrocarbons whose boiling points are far enough apart petroleum cut) is characterized by a distillation curve and an average standard specific gravity. It is then necessary to calculate the standard specific gravity of each fraction composing the cut by using the relation below [4.8] ... 

The accuracy of the conversion depends on the smoothness of the D 86 curve. Errors affect essentially the points in the low % distilled ranges. Average error is on the order of 5°C for conversion of a smooth curve. 

Table 4.5 shows the results for an example. These results differ significantly from those obtained by the method of Riazi for the initial and 10% distilled points. The reported average error for this method is about 3°C, except for the initial point where it reaches 12°C. 

This relation should not be applied for temperatures less than 0°C. Its average accuracy is on the order of 5%. For a Watson factor ot 11.8, the C j can be obtained from the curve shown in Figure 4.4. For different K, values, the following correction is used (... 

This relation is used only for temperatures greater than 0°C. The average error is about 5 kJ/kg. Figure 4.5 gives the enthalpy for petroleum fractions whose is 11.8 as a function of temperature. For K, factors different from 11.8, a correction identical to that used for Cpi is used (to... 

The average error is about 30%. The relation can be used only if the reduced density is less than 2.5 and the reduced temperature of the mixture is greater than 0.80. 

The average error of this method is about 10%. The method is applicable... 

The method has an average error of 5% for all mixtures of hydrocarbons whose conductivities of its components are known. 

The effects of pressure are especially sensitive at high temperatures. The analytical expression [4.71] given by the API is limited to reduced temperatures less than 0.8. Its average error is about 5%. 

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