Isothermal compressibility factor

A virial equation is a power-series expansion of the compressibility factor isotherm in molar density from the zero-density ideal gas, z = 1 at p = 0, which is the common point of intersection of aU z isotherms. 

Obviously, this procedure for calculating the compressibility factor is applicable only in the regions where the compressibility factor isotherms are nearly linear. 

For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived ... 

Equations 4.55 and 4.57 are the most convenient for the calculation of gas flowrate as a function of Pi and Pi under isothermal conditions. Some additional refinement can be added if a compressibility factor is introduced as defined by the relation Pv -= ZRT/M, for conditions where there are significant deviations from the ideal gas law (equation 2.15). 

Since the derivation of governing equations may be found elsewhere (S6, W6), we need only state them briefly in order to establish our notation and to point out some further simplifications. Let c be the isothermal sonic velocity, then the equation of state in terms of compressibility factor Z is... 

Numerous representations have been used to describe the isotherms in Figure 5.5. Some representations, such as the Van der Waals equation, are semi-empirical, with the form suggested by theoretical considerations, whereas others, like the virial equation, are simply empirical power series expansions. Whatever the description, a good measure of the deviation from ideality is given by the value of the compressibility factor, Z= PV /iRT), which equals 1 for an ideal gas. 

Let us first examine the pressure dependence of the compressibility factor under isothermal conditions. Table 2.2 and Fig. 2.2 exhibit some representative P-dependent values of Z for gaseous C02 (at fixed temperature 40°C = 313K) in tabular and graphical form. 

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... 

This coefficient normally is referred to simply as compressibility or gas compressibility. You must understand that the term compressibility is used to designate the coefficient of isothermal compressibility whereas, the term compressibility factor refers to z-factor, the coefficient in the compressibility equation of state. Although both are related to the effect of pressure on the volume of a gas, the two are distinctly not equivalent. 

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. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

The flow of compressible fluids (e.g., gases and vapors) through pipelines and other restrictions is often affected by changing conditions of pressure, temperature, and physical properties. The densities of gases and vapors vary with temperature and pressure. During isothermal flow, i.e., constant temperature (PV = constant) density varies with pressure. Conversely, in adiabatic flow, i.e., no heat loss (PV " = constant), a decrease in temperature occurs when pressure decreases, resulting in a density increase. At high pressures and temperatures, the compressibility factor can be less than unity, which results in an increase in the fluid density. 

Table II. Comparison of Experimental and Calculated Compressibility Factors along the Critical Isotherm...

The applicability of the proposed equation was tested in terms of its predicted values of the compressibility factors, liquid fugacity coefficients, and isothermal enthalpy departures of pure compounds. 

In Figure 5, we present a comparison of our preferred values for the ethylene second virial coefficient with comparable state of the art results obtained by Douslin and Harrison (2). The experimental method and data analysis used by Douslin are independent from ours. In Douslin s experiment, all of the variables required for the calculation of the compressibility factor are measured, whereas in the Burnett method only two variables are measured. Aslo, in this experiment the same sample of gas is retained for the entire experiment in the Burnett isothermal method, the sample is changed for each sequence of measurements. Furthermore,... 

Here, p is the amount-of-substanee density, equal to the reeiproeal molar volume. To carry out the numerical integration of these equations, initial values are required for two of the three quantities Cym,Z and (dZldT)p . For example, values for Z and (dZldT)p can be determined at evenly spaced densities along an initial isotherm when accurate gas-density data are known. Then, (dZldp )T can be calculated and combined with speed of sound data to give values for Cy at each density from the first of the equations above, and then f < Z/dT )p can be determined from the second equation. A simple predictor-corrector method is then used to determine Z and dZ/dT)p at a temperature AT from the reference isotherm. This process is then repeated to cover the range of thermodynamic states of the speed of sound measurements. Accurate coefficients in the virial equation of state can then be derived from the compression factors. 

Many physical and chemical properties are related to the density of the fluid. Properties such as the isothermal compressibility factor, /Cj, and the coeflicient of thermal expansion, a, are simply derivatives of density (molar volume, V) with temperature or pressure [Eq. (2)]. 

Eq. (8.15) that F2 also varies linearly with P/T for finite and small P, P 2 P/T. For isotherms T = const and sufficiently low reducec/pressure, we expect linear variation of the compressibility factor PVsIRT with P/T. 

Figure 4.3 Effects of pressure on residual volumes and compressibility factors along three supercritical isotherms for pure ethane. Broken horizontal lines represent values for the ideal gas. The ethane critical point occurs at = 305.3 K and = 48.7 bar. Note that Z —> 1 as P —> 0,...

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