Corresponding-states principle, extended

At low and moderate pressures, the viscosity of a gas is nearly independent of pressure and can be correlated for engineering purposes as a function of temperatnre only. Eqnations have been proposed based on kinetic theory and on corresponding-states principles these are reviewed in The Properties of Gases and Liquids [15], which also inclndes methods for extending the calculations to higher pressures. Most methods contain molecular parameters that may be fitted to data where available. If data are not available, the parameters can be estimated from better-known quantities such as the critical parameters, acentric factor, and dipole moment. The predictive accuracy for gas viscosities is typically within 5%, at least for the sorts of small- and medinm-sized, mostly organic, molecules used to develop the correlations. 

Pai-Panandiker, R. S., Nieto de Castro, C. A., Marrucho, I. M., and Ely, J. E, Development of an extended corresponding states principle method for volumetric property predictions based on Lee-Kesler reference fluid, Int. J. Themwphys., 23, 771-785 (2002). 

For nonassociating molecules, the methods of Tsonopoulos [14] and Hayden and O Connell [15] yield pretty good results for the estimation of the second virial coefficient. The first method works with the three-parameter corresponding-states principle (Section 2.4.4) the second method takes into account several molecular effects. In case of vapor phase association (Section 13.2), the model of Hayden-O Connell is extended by a chemical equilibrium term, whose parameters have to be fitted to experimental data. 

As discussed in Section 2.5.4, the simple two-parameter corresponding states principle indicates that a generalized equation of state for all substances can be created using only two specific parameters, for example, T, and P. The success of this approach is restricted to simple, spherical molecules like Ar, Kr, Xe, or CH4, where vapor pressure and compressibility factor can be reasonably described. For other molecules, the simple two-parameter corresponding states principle leads to significant errors. A large improvement has been achieved with the introduction of a third parameter which describes the vapor pressure curve (extended three-parameter principle of corresponding states). The most common parameter of this kind is the so-called acentric factor, which is defined as... 

An equation of state has been developed by Kosinski and Anderko (2001) for the representation of the phase behavior of high-temperature and supercritical aqueous systems containing salts. They improved the EOS by Anderko and Pitzer (1993a) to enhance the predictive capability of the EOS using the three-parameter corresponding-states principle. The model was successfully apphed to the H2O + NaCl solutions up to 573 K, and correctly predicts the p VTx properties of H2O + KCl solution up to 773 K. This EOS also considerably extended the validity range. The EOS is also applicable to water + nonelectrolyte solutions such as water + methane and water + n-decane systems. 

The second approach is to extend the simple two-parameter corresponding-states principle at its molecular origin. This is accomplished by making the intermolecular potential parameters functions of the additional characterization parameters /I, and the thermodynamic state, for example, the density p and temperature T. This can be justified theoretically on the basis of results obtained by performing angle averaging on a non-spherical model potential and by apparent three-body effects in the intermolecular pair potential. The net result of this substitution is a corresponding-states model that has the same mathematical form as the simple two-parameter model, but the definitions of the dimensionless volume and temperature are more complex. In particular the... 

In order to extend the simple molecular corresponding-states principle to non-spherical fluids, two approaches are possible. The first simply amounts to introducing models for the non-spherical interactions into the intermolecular potential. For example, the intermolecular potential between two axially symmetric molecules whose electrostatic interactions can be represented as point dipoles and quadrupoles can be modeled as ... 

The second approach to extending the molecular corresponding-states principle to non-spherical molecules was suggested by the work on angle-averaged potentials by Rushbrooke, Pople and Cook and Rowlinson. For example, if the spherical portion of the potential of an axial dipolar molecule can be represented by the Lennard-Jones (12-6) model with... 

Lee and Kesler (1975) extended the Benidict- Webb-Rubin equation to a wider variety of substances, using the principle of corresponding states. The method was modified further by Plocker et al. (1978). 

The PR eos has been modified by Stryjek and Vera to extend to polar substances that do not follow the three-parameter principle of corresponding states. The modified eos is fitted to the vapor pressure of polar substances with additional substance-specific parameters. The PRSV equation has been described in Equation (4.163) et seq. The free-energy-matched mixture eos parameters are given in Equations (4.436) and (4.438) the fugacity coefficients are given in Equation (4.439). PRSV eos using the UNIEAC activity coefficient predicts the vie data for both ethanol/water mixtures at 423-623°K and acetone/water mixtures at 373-523°K from low to high pressure. 

Either to or (Z - 0.293) can be used as a third parameter with which to extend the principle of corresponding states to substances which depart from strict agreement with equations (3) to (6). The first was used by Pitzer and his colleagues, (18-20) and the second by Hougen, Watson, and Ragatz (21). The first, has the practical advantage that the vapour pressure at (T/T ) = 0.7 is known for most substances since this temperature is not far above the normal boiling point. 

The principle of corresponding states, extended as above to mixtures of acentric molecules, has been applied to the calculation of many of the properties needed for the design of separation equipment. The examples reviewed briefly here are taken from our own work on cryogenic fluids, liquified natural gas (LNG), mixtures of hydrocarbons, and mixtures of carbon dioxide with hydrocarbons. In all this work methane was used as the reference substance. 

When comparative tests have been made, the accuracy of the BWR equation and of the principle of corresponding states has been about the same (29), but the extended BWR equations are more accurate for the systems to which they have been fitted. The use of the principle of corresponding states is now little more expensive in computing time. 

The extended principle of corresponding state methods are being further developed. Wider experience with this approach might produce earlier adoption of new methods of prediction because most practicing engineers have a better intuitive understanding of these methods. 

Turning now to transport properties, it is obvious that it is possible to extend the principle of corresponding states from the equation of state to the transport coefficients. This step was soon taken by engineers, using critical constants (Hirschfelder et al. 1964). For example, a dimensionless reduced viscosity could be defined as... 

The theoretical basis, naturally, is the kinetic theory as summarized in Chapter 4. The explicit formulas of the Chapman-Enskog theory given there are all that is needed to formulate a principle of corresponding states, and were the basis for the two-parameter correlation for the noble gases given in 1972 (Kestin et al. 1972a,b), and for the extended correlation for the noble gases (Najafi et al. 1983). 

It is a virtue of the principle of corresponding states that most of the details given in several textbooks and summarized in Chapter 4 are not needed. None at all are needed for the two-parameter correlation, and only a few for the extended correlation. 

For molecular gases the extended principle of corresponding states holds only in a limited form, for the reasons discussed above. In particular, it holds only in the temperature range in which the repulsive wall of an effective potential is dominant, and so fails at low temperatures k T/e < 1) this eliminates the need for one of the three additional parameters introduced by the extended principle for the noble gases. However, the extension to higher temperatures is successful. The second virial coefficients can also be included, provided that corrections for specific contributions of nonspherical components of the potential are made. The overall accuracy of the correlation for molecular gases is estimated to be somewhat less than for the noble gases, but viscosity, diffusion and thermal diffusion are all included. 

There are two reasons for discussing the two-parameter correlation, even though it has been superseded to a large extent by correlations based on the extended principle of corresponding states. First, the two-parameter principle exhibits the major features involved in a correlation, without some of the complexity introduced by the extended principle. Second, and more important, the two-parameter correlation is available in its more restricted temperature range for a number of molecular gases for which an extended correlation has not yet been developed. However, an explicit discussion of mixtures is omitted, since these are included under the extended principle of corresponding states. 

The extended principle of corresponding states developed by Najafi et al. (1983) is thus based on the following form for the potential... 

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