Reduced pressure %
Figure 3-1 Generalized compressibility chart. Vri is V/(RTJPC). (From Ref. 73.)
Reduced pressure Pr
Figure 3-2 Generalized compressibility chart. Vri is V/(RTC/Pr). (From Ref. 73.)
Figure 3
Figure 3
Reduced pressure Pr -3 Generalized compressibility chart. Vri is V/(RTC/P,). (From Ref. 73.)
Vn is not defined in the usual manner, that is, V/Vc, but instead is an "ideal reduced volume" given by
Equation (3-2.2) is an example of the law of corresponding states. This law, though not exact, suggests that reduced properties of all fluids are essentially the same if compared at equal reduced temperatures and pressures. For PVT properties, this law gives
Except for monatomic gases, highly polar fluids, and fluids composed of large molecules, values of Zc for most organic compounds range from 0.27 to 0.29. If it is assumed to be a constant, Eq. (3-2.4) reduces to Eq. (3-2.2). In Sec. 3-3, Zc is introduced as a third correlating parameter (in addition to Tc and Pc) to estimate Z, but not in the form of Eq. (3-2.4).
In Eq. (3-2.2), Tc and Pc are scaling factors to reduce T and P; i.e., to make them nondimensional. Other scaling factors have been proposed, but none have been widely accepted. A tabulation of Tc and Pc for a number of elements and compounds is given in Appendix A, and methods for estimating them are described in Sec. 2-2.
Equation (3-2.2) is a two-parameter equation of state, the two parameters being Tc and Pc. That is, by knowing Tc and Pc for a given fluid, it is possible to estimate the volumetric properties at various temperatures and pressures. The calculation may involve the use of Figs. 3-1 to 3-3, or one may employ an analytical function for /( ) in Eq. (3-2.2). Both methods are only approximate. Many suggestions which retain the general concept yet allow an increase in accuracy and applicability have been offered. In general, the more successful modifications have involved the inclusion of an additional third parameter into the function expressed by Eq. (3-2.2). Most often, this third parameter is related to the reduced vapor pressure at some specified reduced temperature or to some volumetric property at or near the critical point, although one correlation employs the molar polarizability as the third parameter [93]. Two common and well-tested three-parameter correlations are described below.
Assume that there are different, but unique, functions Z = /(Tr, Pr) for each group of pure substances with the same Zc. Then, for each Zc we have a different set of Figs. 3-1 to 3-3. All fluids with the same Zc values then follow the Z-Tr-Pr behavior shown on charts drawn for that particular Zc. Such a structuring indeed leads to a significant increase in accuracy. This is exactly what was done in the development of the Lydersen-Greenkorn-Hougen tables, which first appeared in 1955 [53] and were later modified [37]. There Z is tabulated as a function of Tr and Pr with separate tables for various values of Zc. Edwards and Thodos [20] have also utilized Zc in a correlation to estimate saturated vapor densities of nonpolar compounds.
An alternate third parameter is the Pitzer acentric factor [86-89], defined in Sec. 2-3. This factor is an indicator of the nonsphericity of a molecule's force field; e.g., a value of co = 0 denotes rare-gas spherical symmetry. Deviations from simple-fluid behavior are evident when co > 0. Within the context of the present discussion, it is assumed that all molecules with equal acentric factors have identical Z = /(Tr, Pr) functions, as in Eq. (3-2.2). However, rather than prepare separate Z, Tn Pr tables for different values of co, it was suggested that a linear expansion could be employed:
Thus, the Zi0) function would apply to spherical molecules, and the Z(1) term is a deviation function.
Pitzer et al. tabulated Z{0} and Z{1} as functions of Tr and Pr [89], and Edmister has shown the same values graphically [18]. Several modifications as well as extensions to wider ranges of Tr and Pr have been published [38, 51, 99]. Tables 3-2 and 3-3 list those prepared by Lee and Kes-ler [47]. The method of calculation is described in Sec. 3-7. With Tables 3-2 and 3-3, Z can be determined for both gases and liquids.* The Z{0) table agrees well with that presented originally by Pitzer et al. over the range of Tr and Pr common to both. The deviation function table of Lee and Kesler (Table 3-3) differs somewhat from that of Pitzer and Curl, but extensive testing [47, 114] indicates the new table is the more accurate.
Tables 3-2 and 3-3 were not intended to be applicable to strongly polar fluids, though they are often so used with surprising accuracy except at low temperatures near the saturated vapor region. Though none have been widely adopted, special techniques have been suggested to modify Eq. (3-3.1) for polar materials [21, 31, 50, 75, 111, 120].
Considerable emphasis has been placed on the Pitzer-Curl generalized relation. It has proved to be accurate and general when applied to pure gases. Only the acentric factor and critical temperature and pressure need
*For mixtures, see Table 4.3.
be known. It is probably the most successful and useful result of corresponding states theory [48, 109, 110].
Example 3-1 Estimate the specific volume of dichlorodifluoromethane vapor at 20.67 bar and 366.5 K.
solution From Appendix A, Tc = 385.0 K, Pc = 41.4 bar, and o> = 0.204.
366.5 20.67
P 20.67
The value reported in the literature is 1109 cmVmol [4].
If the Pitzer-Curl method were to be used, from Tables 3-2 and 3-3, Z(0) = 0.761 and Zw = -0.082. From Eq. (3-3.1),
3-4 Analytical Equations of State
An analytical equation of state is an algebraic relation between pressure, temperature, and molar volume. Three classes of equations of state are presented in the next three sections. The virial equation is discussed in Sec. 3-5. In its truncated form, it is a simple equation, and it can represent only modest deviations in the vapor phase from ideal-gas behavior. In Sec. 3-6, equations which are cubic in volume are discussed. These equations can represent both liquid and vapor behavior of nonpolar molecules over limited ranges of temperature and pressure, and they remain relatively simple from a computational point of view. Section 3-7 describes the Lee-Kesler generalized version of the Benedict-Webb-Rubin equation, which is applicable over broader ranges of temperatures and pressure than are the cubic equations. But it is also computationally more complex.
3-5 Virial Equation
The virial equation of state is a polynomial series in inverse volume which is explicit in pressure and can be derived from statistical mechanics:
The parameters B, C,... are called the second, third,... virial coefficients and are functions only of temperature for a pure fluid. Much has been written about this particular equation, and several reviews have been pub-
36 Properties of Gases and Liquids TABLE 3-2 Values of Zm
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