Correlating equation

The theory and appHcation of SF BDV and COV have been studied in both uniform and nonuniform electric fields (37). The ionization potentials of SFg and electron attachment coefficients are the basis for one set of correlation equations. A critical field exists at 89 kV/ (cmkPa) above which coronas can appear. Relative field uniformity is characterized in terms of electrode radii of curvature. Peak voltages up to 100 kV can be sustained. A second BDV analysis (38) also uses electrode radii of curvature in rod-plane data at 60 Hz, and can be used to correlate results up to 150 kV. With d-c voltages (39), a similarity rule can be used to treat BDV in fields up to 500 kV/cm at pressures of 101—709 kPa (1—7 atm). It relates field strength, SF pressure, and electrode radii to coaxial electrodes having 2.5-cm gaps. At elevated pressures and large electrode areas, a faH-off from this rule appears. The BDV properties ofHquid SF are described in thehterature (40—41). 

Cetane number is difficult to measure experimentally. Therefore, various correlation equations have been developed to predict cetane number from fuel properties. One such equation may be found in ASTM D4737 to calculate a cetane index (Cl). ASTM D975 allows use of Cl as an approximation if cetane numbers are not available. 

Extensive tabulations of Antoine parameters are available for many chemicals of importance to engineers, chemists, and environmental scientists (9,19,20). Caution is in order when using tabulated Antoine constants because several forms of the correlating equation are found in the Hterature. In particular, there are variations in the sign before the second term, the units of temperature, and the use of natural or decimal logarithms of the vapor pressure. 

Mathematical Consistency Requirements. Theoretical equations provide a method by which a data set s internal consistency can be tested or missing data can be derived from known values of related properties. The abiUty of data to fit a proven model may also provide insight into whether that data behaves correctiy and follows expected trends. For example, poor fit of vapor pressure versus temperature data to a generally accepted correlating equation could indicate systematic data error or bias. A simple sermlogarithmic form, (eg, the Antoine equation, eq. 8), has been shown to apply to most organic Hquids, so substantial deviation from this model might indicate a problem. Many other simple thermodynamics relations can provide useful data tests (1—5,18,21). 

Fitting Simple Functions. In many engineering appHcations there may be no apparent theoretical basis for the relationship of two variables or the relationship may be too complex to apply. Thus the search for a correlating equation form may at first be along empirical lines. A simple plot of the data in ordinary Cartesian coordinates gives an immediate indication of the essential form of the data. 

The activity coefficient (y) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. 

Values of g given by the correlating equation, Eq. (4-286), are called derived values, and associated derivea values of the activity coefficients are given by specialization of Eqs. (4-58) ... 

The data-reduction procedure just desciiDed provides parameters in the correlating equation for g that make the 8g residuals scatter about zero. This is usually accomphshed by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y. Equation (4-282) is solved for yjP and written for species 1 and for species 2. Adding the two equations gives... 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Neither the principles of thermodynamics nor theories of reaction rates require that there should be such linear relationships. There are, in fact, numerous reaction series that fail to show such correlations. Some insight into the origin of die correlation can be gained by considering the relationship between the correlation equation and the free-energy changes involved in the two processes. The line in Fig 4.2 defines an equation in which m is the slope of the line ... 

Give a rationalization for the form of this correlation equation. What information does it give regarding involvement of the Ari ring in the rate-determining step ... 

Only a small amount of work has been done up to now concerning the prediction of bond strengths and other properties based on the results of the analysis of the resin. Ferg et al. [59] worked out correlation equations evaluating the chemical structures in various UF-resins with different F/U molar ratios and different types of preparation on the one hand and the achievable internal bond as well as the subsequent formaldehyde emission on the other hand. These equations are valid only for well defined series of resins. The basic aim of such experiments is the prediction of the properties of the wood-based panels based on the composition and the properties of the resins used. For this purpose various structural components are determined by means of - C NMR and their ratios related to board results. Various papers in the chemical literature describe examples of such correlations, in particular for UF, MF, MUF and PF resins [59-62]. For example one type of equation correlating the dry internal bond (IB) strength (tensile strength perpendicular to the plane of the panel) of a particleboard bonded with PF adhesive resins is as follows [17]... 

The nucleation rate is plotted versus the supersaturation for different stirrer speeds in a log-log diagram (Figure 6.21). The kinetic order n in the correlating equation... 

The work of Uhl [22] gave particular emphasis to viscous materials in Jacketed vessels, and the correlating equations are ... 

B = parameter in correlating equation In = natural logarithm log = logarithm to the base 10 N = actual theoretical stages required for a given separation... 

Silipo and Hansch 77) have developed correlation equations for the formation of a-cyclodextrin-substituted phenyl acetate complexes (Eq. 13), a-cyclodextrin-RCOO complexes (Eq. 14), and P-cyclodextrin-substituted phenylcyanoacetic acid anion complexes (Eq. 15). 

In defining a 7-factor (jd) for mass transfer there is therefore good experimental evidence for modifying the exponent of the Schmidt number in Gilliland and Sherwood s correlation (equation 10.225). Furthermore, there is no very strong case for maintaining the small differences in the exponent of Reynolds number. On this basis, the /-factor for mass transfer may be defined as follows ... 

The whole concept based on parameters, although used several times (3, 57, 155, 156, 201) and advocated particularly by Good and Stone (200), has a principal defect. The results are dependent not only on experimental rate constants, but also on the values of the parameter and on the form of the correlation equation used. Furthermore, the procedure does not give any idea of the possible error. Hence, it could be acceptable only in an unrealistic case, that in which the isokinetic relationship itself and the correlations with the parameter are very precise. 

For the transition state effect correlation equation (equation 7), the slopes (M) for the various reactions of this study are positive in sign (Tables II and V). When the negative AE(x) values (Table I) are multiplied by the slope, for a particular initiator at a given rate, a negative (reaction temperature-lowering) value is obtained. For the reactant state effect correlation equation (equation 8), the slopes (N) for the various reactions of this study are negative in... 

Further studies of the 2-substituted 5-nitroanilines, conducted by Kier and coworkers, searched for a linear combination of structural variables that describes a line, plane, or surface that separates the molecule classes in the optimum manner. They found that sweetness correlated very well with the substituent polarizability-constants for the 2-substituent, implicating the involvement of the 2-substituents in dispersive-binding interactions at the receptor. This is in agreement with the results of Hansch " and McFarland. The correlation equation was not, however, reported. 

Here,5 is a correlation equation which minimizes errors of p2 (PsiO P f(y)=y/, where n is between -0.2 and 1. 

G. J. Janz, J. Phys. Chem. Ref Data 17, Supplement (1988) Thermodynamic and Transport Properties for Molten Salts Correlation Equations for Critically Evaluated Density, Surface Tension, Eleetrieal Conduetance and Viseosity Data, American Chemical Society-American Institute of Physics-National Bureau of Standards, Washington, DC, 1988. 

This means that representing the equilibrium adsorbate mass (or volume) adsorbed versus concentration (or partial pressure) should be a straight line if plotted on logarithmic scales. Other theoretically based correlating equations are available for the adsorption of gases and vapors9, but all equations have their limitations. 

The following correlation equations were used in the estimation of partition coefficients used in this paper. 

Fluid Reference Range of NRe and/or eB where applicable Correlation Equation number... 

Several correlating equations for the friction factor have been proposed for both the laminar and turbulent flow regimes, and plots of fM (or functions thereof) versus Reynolds number are frequently presented in standard fluid flow or chemical engineering handbooks (e.g., 96, 97). Perhaps the most useful of the correlations is that represented by the Ergun equation (98)... 

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