Density compressibility factors

Let us assume for purposes of this paper that we can know the composition of a gas from some suitable measurement. It is then possible, in principle, to calculate pertinent properties such as heating value, relative density, compressibility factor. Unfortunately, as often happens in practice, these supposedly unambiguous calculations become clouded by accepted procedirres, misconceptions, misguided regulations. The discussion in this paper attempts to dispel the misconceptions and to clarify the accepted procedures. As for the regulations, it is only possible to wish that they would not always choose the path of maximum irrationality and to try to perform the calculations in the least offensive (technically) marmer possible. The calculations described in this paper reflect those suggested in GPA Standard 2172-85. However, this paper contains considerable amplification and discussion of the techniques. 

W. M. Haynes and R. D. Goodwin, Thermophysical Properties of Normal Butane from 135 to 700 K at Pressures to 70 MPa, U.S. Dept, of Commerce, National Bureau of Standards Monograph 169, 1982, 192 pp. Tabulated data include densities, compressibility factors, internal energies, enthalpies, entropies, heat capacities, fugacities and more. Equations are given for calculating vapor pressures, liquid and vapor densities, ideal gas properties, second virial coefficients, heats of vaporization, liquid specific heats, enthalpies and entropies. 

The virial equation of state, first advocated by Kamerlingh Oimes in 1901, expresses the compressibility factor of a gas as a power series in die number density ... 

The critical pressure, critical molar volume, and critical temperature are the values of the pressure, molar volume, and thermodynamic temperature at which the densities of coexisting liquid and gaseous phases just become identical. At this critical point, the critical compressibility factor, Z, is ... 

For pure organic vapors, the Lydersen et al. corresponding states method is the most accurate technique for predicting compressibility factors and, hence, vapor densities. Critical temperature, critical pressure, and critical compressibility factor defined by Eq. (2-21) are used as input parameters. Figure 2-37 is used to predict the compressibihty factor at = 0.27, and the result is corrected to the Z of the desired fluid using Eq. (2-83). 

No specific mixing rules have been tested for predicting compressibility factors for denned organie mixtures. However, the Lydersen method using pseudocritical properties as defined in Eqs. (2-80), (2-81), and (2-82) in place of true critical properties will give a reasonable estimate of the compressibihty faclor and hence the vapor density. 

Ideal gas obeys the equation of state PV = MRT or P/p = MRT, where P denotes the pressure, V the volume, p the density, M the mass, T the temperature of the gas, and R the gas constant per unit mass independent of pressure and temperature. In most cases the ideal gas laws are sufficient to describe the flow within 5% of actual conditions. When the perfect gas laws do not apply, the gas compressibility factor Z can be introduced ... 

Figure 8 Compressibility factor P/fiksT versus density p = pa3 of the hard-sphere system as calculated from both free-volume information (Eq. [8]) and the collision rate measured in molecular dynamics simulations. The empirically successful Camahan-Starling84 equation of state for the hard-sphere fluid is also shown for comparison. (Adapted from Ref. 71).

Estimating the critical density IE, ln(Pc7Pc) No consistency test is available for ln(pc7Pc), but for the original Van der Waals equation and the modified VdW equations discussed in this chapter the critical compressibility factors, Zc(VdW) = Pc/(pcRTc), are equal to 3/8 and (a2 - l)/(4a), respectively. In the latter case,... 

Figure 1 shows the Rule Sheet for a TKISolver model REALGAS.TK (12. The first rule is the van der Waals equation of state. The second defines the gas constant, and the third rule defines Ae number density. The fourth defines the compressibility factor z, a dimensionless variable which measures the amount of... 

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibility factor Z, defined as Z PV/RTy having either molar density, p( = V-1), or pressure, P, as the independent variable of expansion ... 

Figure 3.2 illustrates the relatively complex nature of the compressibility factor s dependence on temperature and pressure. It is evident that there can be very substantial departures from ideal-gas behavior. Whenever possible, it is useful to represent the equation of state as an algebraic relationship of pressure, temperature, and volume (density). Certainly, when applied in computational modeling, the benefits of a compact equation-of-state representation are evident. There are many ways that are used to accomplish this objective [332], most of which are beyond our scope here. 

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). 

Physical properties are termed either intensive or extensive. Intensive properties are independent of the quantity of material present. Density, specific volume, and compressibility factor are examples. Properties such as volume and mass are termed extensive their values are determined by the total quantity of matter present. 

Water invariably occurs with petroleum deposits. Thus, a knowledge of the properties of this connate, or interstitial, or formation water is important to petroleum engineers. In this chapter, we examine the composition of oilfield water water density, compressibility, formation volume factor and viscosity solubility of hydrocarbons in water and solubility of water in both liquid and gaseous hydrocarbons and, finally, water-hydrocarbon interfacial tension. An unusual process called hydrate formation in which water and natural gas combine to form a solid at temperatures above the freezing point of water is discussed in Chapter 17. 

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). 

The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the liquid region. The compressibility factor can be used to accurately determine gas volumes when used in conjunction with a vitial expansion or an equation such as equation 53 (77). However, the prediction of saturated liquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibility factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

Equations (8-111) to (8-115) are restricted to incompressible fluids. For gases and vapors, the fluid density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal gas law model. Deviations from ideal behavior are corrected for, to first order, with nonunity values of compressibility factor Z (see Sec. 2, Physical and Chemical Data, for definitions and data for common fluids). For compressible fluids... 

This quantity is often called the compressibility factor, a name that is easy to confuse with the isothermal compressibility, defined in Eq. (9).] Z is obviously unity for an ideal gas. At the other extreme, when the pressure or density is very large, excluded volume effects become dominant, V > Vig, and Z > 1.0. 

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... 

The GvdW equation of state contains a hard repulsive term, a van der Waals attractive term linear in density, and a correction term for medium and low densities. The compressibility factor Z is written as... 

From the density, we can calculate the compressibility factor using Equation (2.5) ... 

Virial Equations of State The virial equation in density is an infinite-series representation of the compressibility factor Z in powers of molar density p (or reciprocal molar volume V ) about the real-gas state at zero density (zero pressure) ... 

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