Pressure reduced, compressibility factor

The free volume 5v treatment of glass formation can be extended to a description that encompasses constant volume conditions by considering a generalized reduced compressibility factor, 5Z = PV — PV)f /PV, which reduces to Sv and 5P under constant pressure and constant volume conditions, respectively. The o subscript indicates that the product of the pressure P and the total volume V is be evaluated under the constraint of vanishing s. ... 

The terms on the right-hand sides of these equations depend only on the upper limit Pr of the integrals and on the reduced temperature at which they are evaluated. Thus, values of In ( ) and HVRTc may be determined once and for all at any reduced temperature and pressure from generalized compressibility factor data. 

Figure 3-1 shows the relationship between the compressibility factor and pressure and temperature, couched in terms of reduced pressure and temperature ... 

T = Reduced temperature of mixture P,n, = Reduced pressure of mixture = Compressibility factor of mixture... 

Figure A3.2 Graph of the compressibility factor r for a number of gases versus their reduced pressure at several reduced temperatures. Reprinted with permission, taken from Goug-Jen Su, Ind. Eng. Chem.. 38,803 (1946), the data illustrate the validity of the principle of corresponding states. The line is Goug-Jen Su s estimate of the average value for r.

A chart which correlates experimental P - V - T data for all gases is included as Figure 2.1 and this is known as the generalised compressibility-factor chart.(1) Use is made of reduced coordinates where the reduced temperature Tr, the reduced pressure Pr, and the reduced volume Vr are defined as the ratio of the actual temperature, pressure, and volume of the gas to the corresponding values of these properties at the critical state. It is found that, at a given value of Tr and Pr, nearly all gases have the same molar volume, compressibility factor, and other thermodynamic properties. This empirical relationship applies to within about 2 per cent for most gases the most important exception to the rule is ammonia. 

The compressibility factor can be estimated from a generalised compressibility plot, which gives z as a function of reduced pressure and temperature (Chapter 3, Figure 3.8). 

From the ideal gas equation, it is found that for 1 mole of gas, PV/KT = 1, which is known as the compressibility factor. For most real gases, there is a large deviation from the ideal value, especially at high pressure where the gas molecules are forced closer together. From the discussions in previous sections, it is apparent that the molecules of the gas do not exist independently from each other because of forces of attraction even between nonpolar molecules. Dipole-dipole, dipole-induced dipole, and London forces are sometimes collectively known as van der Waals forces because all of these types of forces result in deviations from ideal gas behavior. Because forces of attraction between molecules reduce the pressure that the gas exerts on the walls of the container, van der Waals included a correction to the pressure to compensate for the "lost" pressure. That term is written as w2a/V2, where n is the number of moles, a is a constant that depends on the nature of the gas, and V is the volume of the container. The resulting equation of state for a real gas, known as van der Waals equation, is written as... 

If a gas follows any two-parameter equation of state, such as the van der Waals or the Redlich-Kwong, it has been shown in Section 5.2 that Z, the compressibility factor, is a universal function of the reduced pressure P = P/Pc and the reduced temperature Pj = P/ Pe- Then if Z is plotted as a function of Pj, at a given reduced temperature P, all gases fit a single curve. At another reduced temperature Tj, a new curve is obtained for Z versus Pj, but it too fits all gases. Gases at equal reduced pressures and reduced temperatures are said to be in corresponding states. 

Finally, the use of additives is required to lower the operating conditions and reduce compression costs. This should be done in a way that does not compromise the separation factor. Use of tetrahydrofuran in the hydrate formation process has been suggested in the literature in order to reduce the hydrate formation pressure for the C02/N2 system (Kang and Lee, 2000 Seo et al., 2005). A suitable additive is also required to lower the hydrate forming conditions for the C02/H2 system. One such additive might be propane (Kumar et al., 2006). 

The assumption known as the law of corresponding states asserts that the compressibility factor Z should be a function only of the reduced temperature Ti and the reduced pressure Pi, which is approximately correct for many real gases. It is seen that, for a van der Waal gas, the minimum value of Vr = 1/3, which can be achieved only at infinite pressure. From the equation of state written for the reduced temperature and pressure, we can derive the equivalent formula of compressibility as... 

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. Tbe most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibility factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

For an ideal gas, Z = 1. In general, the law of corresponding states provides that the compressibility factor depend on the reduced temperature and pressure,... 

Fig. 3.2 Plot of the compressibility factor as a function of reduced pressure and parameterized by the reduced temperature. The reduced values are normalized by their corresponding values at the critical point. This plot is adapted from one originally prepared by Nelson and Obert [295,332],...

The above equations were obtained from twenty non-polar gases including inert gases, hydrocarbons and carbon dioxide (but not hydrogen and helium). Hence, possible errors can be as large as 20%. The maximum pressure corresponds to a reduced density of 2.8. In the above equations, Zc represents the critical compressibility factor. The value of gamma is calculated using Eqn. (3.4-26). 

Notice that the shapes of the isotherms of compressibility factors for the three gases given in Figures 3-2, 3-3, and 3-4 are very similar. The realization that this is true for nearly all real gases led to the development of the Law of Corresponding States and the definition of the terms reduced temperature and reduced pressure. Reduced temperature and reduced pressure are defined as... 

Generalized charts are applicable to a wide range of industrially important chemicals. Properties for which charts are available include all thermodynamic properties, eg, enthalpy, entropy, Gibbs energy, and PVT data, compressibility factors, liquid densities, fugacity coefficients, surface tensions, diffusivities, transport properties, and rate constants for chemical reactions. Charts and tables of compressibility factors vs reduced pressure and reduced temperature have been produced. Data is available in both tabular and graphical form (61—72). 

All fluids, when compared at the same reduced temperature and reduced pressure have approximately the same compressibility factor and deviate from ideal gas behavior to the same extent, giving... 

This equation is useful for gases above the critical point. Only reduced pressure, /J, and reduced temperature, T, are needed. In the form represented by equation 53, iteration quickly gives accurate values for the compressibility factor, Z. However, this two-parameter equation only gives accurate values for simple and nonpolar fluids. Unless the Redlich-Kwong equation (eq. 53) is explicidy solved for pressure in nonreduced variables, it does not give accurate liquid volumes. 

No actual gas follows the ideal gas equation exactly. Only at low pressures are the differences between the properties of a real gas and those of an ideal gas sufficiently small that they can be neglected. For precision work the differences should never be neglected. Even at pressures near 1 bar these differences may amount to several percent. Probably the best way to illustrate the deviations of real gases from the ideal gas law is to consider how the quantity PV/RT, called the compressibility factor, Z, for 1 mole of gas depends upon the pressure at various temperatures. This is shown in Figure 7.1, where the abscissa is actually the reduced pressure and the curves are for various reduced temperatures [9]. The behavior of the ideal gas is represented by the line where PV/RT = 1. For real gases at sufficiently low temperatures, the PV product is less than ideal at low pressures and, as the pressure increases, passes through a minimum, and finally becomes greater than ideal. At one temperature, called the Boyle temperature, this minimum... 

For use of the generalized Redlich/Kwong equation one needs only the critical temperature and critical pressure of the gas. This is the basis for the two-parameter theorem of corresponding states All gases, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor, and all deviate from ideal-gas behavior to about the same degree. 

Lydersen, Greenkom, and Hougenl developed a general method for estimation of liquid volumes, based on the principle of corresponding states. It applies to liquids just as the two-parameter compressibility-factor correlation applies to gases, but is based on a correlation of reduced density as a function of reduced temperature and pressure. Reduced density is defined as... 

Pig. The compressibility factors as a function of reduced pressures and reduced temperatures. 

Component (i) Yi Partial pressure (Pi =PYi) Reduced pressure (Pi /Pc.i) Reduced temperature (T/Tc,i) Compressibility factor (Zj) ZiYi... 

The observation that the properties could be expressed in terms of the reduced quantities had many important ramifications. These including the possibility that if you plotted the reduced vapor pressure as a function of reduced temperature, all substances would fall onto a single curve. Furthermore, if you plotted the compressibility factor versus the reduced pressure with the reduced temperature as a parameter, all fluids would lie on the same plot. 

If the gas is ideal, z = 1. In Figure 5.14, the compressibility factor is plotted as a function of reduce pressure and temperature. The compressibility factor in Equation 5.11 will vary as the temperature and pressure changes from the compressor inlet to the compressor outlet. 

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