Figure 1-1 The law of corresponding states applied to the PVT properties of methane and nitrogen. Experimental values [2]: o methane, • nitrogen.

of PVT data has encouraged similar correlations of other properties which depend primarily on intermolecular forces. Many of these have proved valuable to the practicing engineer. Modifications of the law are common to improve accuracy or ease of use. Good correlations of high-pressure gas viscosity have been obtained by expressing r)/t]c as a function of Pr and Tr. But since r/c is seldom known and not easily estimated, this quantity has been replaced in other correlations by rj°, 7]t, or the group M1/2PC2/3TC1/6, where ri° is the viscosity at Tc and low pressure, tjj is the viscosity at the temperature of interest, again at low pressure, and the group containing M, Pc, and Tc is suggested by dimensional analysis. Other alternatives to the use of t]c might be proposed, each modeled on the law of corresponding states but essentially empirical as applied to transport properties.

The law of corresponding states can be derived from statistical mechanics when severe simplifications are introduced into the partition function. Sometimes other useful results can be obtained by introducing less severe simplifications into statistical mechanics toward providing a framework for the development of estimation methods. Fundamental equations describing various properties (including transport properties) can sometimes be derived, provided that an expression is available for the poten-tial-energy function for molecular interactions. This function may be, at least in part, empirical; but the fundamental equations for properties are often insensitive to details in the potential function from which they stem, and two-constant potential functions frequently serve remarkably well for some systems. Statistical mechanics may at present be far removed from engineering practice, but there is good reason to believe that it will become increasingly useful, especially when combined with computer simulations.

Spherically symmetric molecules (for example, CH4) are well fitted by a two-constant law of corresponding states. Nonspherical and weakly polar molecules do not fit poorly, but deviations are often great enough to encourage the development of correlations using a third parameter, e.g., the acentric factor. The acentric factor is obtained from the deviation of the experimental vapor pressure-temperature function from that which might be expected for a similar substance consisting of spherically symmetric molecules. Typical corresponding states correlations express the dimensionless property as a function of Pr, T„ and the chosen third parameter.

Unfortunately, the properties of strongly polar molecules are often not satisfactorily represented by the two- or three-constant correlations which do so well for nonpolar molecules. An additional parameter based on the dipole moment has often been suggested but with limited success, since polar molecules are not easily characterized by using only the dipole moment and critical constants. As a result, although good correlations exist for properties of nonpolar fluids, similar correlations for polar fluids are often not available or else are of restricted reliability.

All macroscopic properties are related to molecular structure, which determines the magnitude and predominant type of the intermolecular forces. For example, structure determines the energy storage capacity of a molecule and thus the molecule's heat capacity.

The concept of structure suggests that a macroscopic property can be calculated from group contributions. The relevant characteristics of structure are related to the atoms, atomic groups, bond type, etc.; to them we assign weighting factors and then determine the property, usually by an algebraic operation which sums the contributions from the molecule's parts. Sometimes the calculated sum of the contributions is not for the property itself but instead is for a correction to the property as calculated by some simplified theory or empirical rule. For example, Lydersen's method for estimating Tc starts with the loose rule that the ratio of the normal boiling temperature to the critical temperature is about 2:3. Additive structural increments based on bond types are then used to obtain empirical corrections to that ratio.

Some of the better correlations of ideal-gas heat capacities employ theoretical values of C°p (which are intimately related to structure) to obtain a polynomial expressing C°p as a function of temperature; the constants in the polynomial are determined by contributions from the constituent atoms, atomic groups, and types of bonds.

1-4 Organization of the Book

Reliable experimental data are always to be preferred over values obtained by estimation methods. But all too often reliable data are not available.

In this book, the various estimation methods are correlations of experimental data. The best are based on theory, with empirical corrections for the theory's defects. Others, including those stemming from the law of corresponding states, are based on generalizations which are partly empirical but which nevertheless have application to a remarkably wide range of properties. Totally empirical correlations are useful only when applied to situations very similar to those used to establish the correlations.

The text includes a large number of numerical examples to illustrate the estimation methods, especially those which are recommended. Almost all of them are designed to explain the calculation procedure for a single property. However, most engineering design problems require estimation of several properties; the error in each contributes to the overall result, but some individual errors are more important than others. Fortunately, the result is often found adequate for engineering purposes, in spite of the large measure of empiricism incorporated in so many of the estimation procedures.

As an example, consider the case of a chemist who has synthesized a new compound which has the chemical formula CC12F2 and boils at —20.5°C at atmospheric pressure. Using only this information, is it possible to obtain a useful prediction of whether or not the substance has the thermodynamic properties which might make it a practical refrigerant?

Figure 1-2 shows portions of a Mollier diagram developed by the prediction methods described in later chapters. The dashed curves and points are obtained from estimates of liquid and vapor heat capacities, critical properties, vapor pressure, enthalpy of vaporization, and pressure corrections to ideal-gas enthalpies and entropies. The substance is, of course, a well-known refrigerant, and its known properties are shown by the solid curves.

For a standard refrigeration cycle operating between 48.9 and —6.7°C, the evaporator and condenser pressures are estimated to be 2.4 and 12.4 bar, vs. the known values 2.4 and 11.9 bar. The estimate of the heat absorption in the evaporator checks closely, and the estimated volumetric vapor rate to the compressor also shows good agreement: 2.39 versus 2.45 m3/hr per kW of refrigeration. (This number indicates the size of the compressor.) Constant-entropy lines are not shown in Fig. 1-2, but it is found that the constant-entropy line through the point for the low-pressure vapor essentially coincides with the saturated vapor curve. The estimated coefficient of performance (ratio of refrigeration rate to isentropic compression power) is estimated to be 3.8; the value obtained from the

Figure 1-2 Mollier diagram for dichlorodifluoromethane. The solid lines represent measured data. Dashed lines and points represent results obtained by estimation methods when only the chemical formula and the normal boiling temperature are known.

data is 3.5. This is not a very good check, but it is nevertheless remarkable because the only data used for the estimate were the normal boiling point and the chemical formula.

Most estimation methods require parameters which are characteristic of single pure components or of constituents of a mixture of interest. The more important of these are considered in Chap. 2, and tables of values for common substances are provided in Appendix A.

Thermodynamic properties (such as enthalpy and heat capacity) are discussed in Chaps. 3 to 6. Although the more accurate equations of state are employed, the basic thermodynamic relations are developed in a general way so that other equations of state can be introduced whenever they are more applicable for a particular purpose.

Chapters 5 and 6 discuss heat capacities; Chap. 6 discusses Gibbs energy and enthalpy of formation; and Chap. 7 discusses vapor pressures and enthalpies of vaporization of pure substances. Chapter 8 presents techniques for estimation and correlation of phase equilibria in mixtures. Chapters 9 to 11 describe estimation methods for viscosity, thermal conductivity, and diffusion coefficients. Surface tension is considered briefly in Chap. 12.

The literature searched was voluminous, and the lists of references following each chapter represent but a fraction of the material examined. Of the many estimation methods available, only a few were selected for detailed discussion. These were selected on the basis of their generality, accuracy, and availability of required input data. Tests of all methods i i Isotherms-

Figure 1-2 Mollier diagram for dichlorodifluoromethane. The solid lines represent measured data. Dashed lines and points represent results obtained by estimation methods when only the chemical formula and the normal boiling temperature are known.

40 80 120 160 200 240 Enthalpy, J/a t-55"c

40 80 120 160 200 240 Enthalpy, J/a

Was this article helpful?

## Post a comment