Residual compressibility factor

The original, two-parameter corresponding-states principle leads to an equation of state which expresses the residual compressibility factor (or compression factor) in terms of a universal function of the dimensionless temperature and molar volume (or density) ... 

The shape factors are weak functions of temperature and, in principle, density and can be visualized as distorting scales that force the two fluids to conformality. Although there is no direct theoretical evidence for the density dependence of the shape factors, mathematical solutions for exact shape factors found by equating the dimensionless residual compressibility factor and Helmholtz energy of two pure-fluids exhibit weak density dependence. 

The first attempt to find exact shape factors is due to Leach,who equated the residual compressibility factor and fugacity coefficient of two fluids, with... 

The parameters Qa and are universal for this equation of state and m (o) is a simple quadratic function of the acentric factor. The residual compressibility factor and dimensionless residual Helmholtz energy are given by... 

Factors affecting the apparent fracture toughness in a laminate aa is the distribution of applied stress resulted from bending o>is the residual compressive stress a is the crack length. 

Since V = ZRT/ P, the residual volume and the compressibility factor are related ... 

The compressibility factor is by definition Z = PV/RT values of Z and of (dZ/BT)P are calculated directly from experimental PVT data, and the two integrals in Eqs. (6.40) through (6.42) are evaluated by numerical or graphical methods. Alternatively, the two integrals are evaluated analytically when Z is expressed by an equation of state. Thus, given PVT data or an appropriate equation of state, we can evaluate HR and SR and hence all other residual properties. It is this direct connection with experiment that makes residual properties essential to the practical application of thermodynamics. 

Of the two kinds of data needed for evaluation of thermodynamic properties, heat capacities and PVT data, the latter are most frequently missing. Fortunately, the generalized methods developed in Sec. 3.6 for the compressibility factor are also applicable to residual properties. 

Figure 3.16, drawn specifically for the compressibility-factor correlation, is also used as a guide to the reliability of the correlations of residual properties based on generalized second virial coefficients. However, all residual-property correlations are less precise than the compressibility-factor correlations on which they are based and are, of course, least reliable for strongly polar and associating ... 

As with the generalized compressibility-factor correlation, the complexity ol the functions (H f/RZ. H Y/RZ. (S /R, and S Y/R preclude then general representation by simple equations. However, the correlation for Z basec on generalized virial coefficients and valid at low pressures can be extended U the residual properties. The equation relating Z to the functions and ia derived in Sec. 3.6 from Eqs. (3.46) and (3.47) ... 

The graphs are based on the Peng-Robinson equation of state (1) as improved by Stryjek and Vera (2, 3). The equations for thermodynamic properties using the Peng-Robinson equation of state are given in the appendix for volume, compressibility factor, fugacity coefficient, residual enthalpy, and residual entropy. Critical constants and ideal gas heat capacities for use in the equations are from the data compilations of DIPPR (8) and Yaws (28, 29, 30). 

As with the generalized compressibility-factor correlation, tire complexity of tire fuirc-tioirs (H f/RTc, (H f/RTc, S f/R, and (Sy/R precludes tlreir geireral representation by simple equations. However, tire generalized secoird-virial-coefficientcorrelationvalid at low pressures fonrrs the basis for airalytical correlations of tire residual properties. The equation relating B to the functions aird 5 is derived iir Sec. 3.6 ... 

Certainly 117 lb is a more realistic figure than 100 lb, and it is easily possible to be in error by 8 lb if the residual weight of NH3 in the tank is determined by difference. As a matter of interest you might look up the specific volume of NHa at the conditions in the tank in a handbook—you would find that V = 0.973 ft /lb, and hence the compressibility factor calculation yielded a volume with an error of only about 4%. 

The residual properties of gases and vapors depend on their PVT behavior. This is often expressed through correlations for the compressibility factor Z, defined by Eq. (4-36). Analytical expressions for Z as functions of T and P or T and V are known as equations of state. They may also be reformulated to give P as a function of T and V or V as a function of T and P. 

Given an expression for the compressibility factor Z as a function of temperature and volume for a fluid (an equation of state), we can determine the residual Helmholtz free energy of the system. 

Table 1. Compression factor, fugacity coefficient, and residual thermodynamic functions of real fluid Refrigerant 500...

In the present investigation, the compression factor, fugacity coefficient, and residual thermodynamic functions of the real fluid R-500 have been calculated using the BACK equation of state. For this purpose, the experimental data, i.e., critical constants, saturated liquid density, saturation vapor pressure, and the PVT data have been utilized. 

---