The Compressibility Factor

The PVT behavior of pure fluids is typically expressed in terms of the compressibility faaor z  

Flgure 8.3 Compressibility factors for i-butane (Goodwin and Haynes). 

It has been observed that the compressibility factor Z, which is defined as 

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This method utilizes essentially the concept developed by Fitzer in 1955. According to the principle of three-parameter corresponding states, the compressibility factor z, for a fluid of acentric factor w, is obtained by interpolating between the compressibilities Zj and Z2 for the two fluids having acentric factors w, and (p -... 

The compressibility factors Zy are obtained after solving the following equations in for i = 1 and i = 2 ... 

There have been several equations of state proposed to express the compressibility factor. Remarkable accuracy has been obtained when specific equations for certain components are used however, the multitude of their coefficients makes their extension to mixtures complicated. 

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 confinement term is unique because it alone causes a dependence of the binding free energy on the choice of unit concentration in the standard state the volume available per ligand molecule in the free state, and hence the compression factor, depend on the unit concentration. 

To correlate to the acentric factor a quadratic Taylor series in terms of the compressibility factor was formulated. This equation is represented as... 

FIG. 2-14 MoUier diagram for nitrous oxide. (Fig. 9, Cfniv. Texas Rep., Cont. DAI-23-072-ORD-685, June 1, 1956, hy Couch and Kobe. Reproduced hy permission.) Some irregularity in the compressibility factors from 80 to 160 atm, 50 to 100 C exists (Couch, private communication, 1.967). See Couch et al.,y, Chem. Eng. Data, 6, (1961) for P -T data. 

Example Many equations of state involve solving ciihic equations for the compressibility factor Z. For example, the Redlich-Kwong-Soave equation of state requires solving... 

Pitzer s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. Thennodynamic.s, 3ded., App. 3, McGraw-HiU, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor ... 

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

Pipecalc 2.0, Gulf Publishing Company, Houston, Texas. Note Pipecalc 2.0 will calculate the compressibility factor, minimum pipe ID, upstream pressure, downstream pressure, and flow rate for Panhandle A, Panhandle B, Weymouth, AGA, and Colebrook-White equations. The flow rates calculated in the above sample calculations will differ slightly from those calculated with Pipecalc 2.0 since the viscosity used in the examples was extracted from Figure 5, p. 147. Pipecalc uses the Dranchuk et al. method for calculating gas compressibility. 

A quick estimate of the compressibility factor Z can be made from the nomograph " shown as Figure 1. 

Figure 2 and Table 1 illustrate a sample problem (Z, the compressibility factor, is assumed to be 1.0). 

Since non-ideal gases do not obey the ideal gas law (i.e., PV = nRT), corrections for nonideality must be made using an equation of state such as the Van der Waals or Redlich-Kwong equations. This process involves complex analytical expressions. Another method for a nonideal gas situation is the use of the compressibility factor Z, where Z equals PV/nRT. Of the analytical methods available for calculation of Z, the most compact one is obtained from the Redlich-Kwong equation of state. The working equations are listed below ... 

The hard-sphere excess chemical potential p that appears on the left-hand side of the hierarchy is related to the compressibility factor Z = pV/pkT,... 

Determine inlet flow as Q, taking into account the compressibility factor, Z. 

Meters are accurate within close limits as legislation demands. However, gas is metered on a volume basis rather than a mass basis and is thus subject to variation with temperature and pressure. The Imperial Standard Conditions are 60°F, 30inHg, saturated (15.56°C, 1913.7405 mbar, saturated). Gas Tariff sales are not normally corrected, but sales on a contract basis are. Correction may be for pressure only on a fixed factor basis based on Boyle s Law or, for larger loads, over 190,000 therms per annum for both temperature and pressure using electronic (formerly mechanical) correctors. For high pressures, the compressibility factor Z may also be relevant. The current generation of correctors corrects for pressure on an absolute basis taking into account barometric pressure. 

Compressibility is experimentally derived from data about the actual behavior of a particular gas under pVT changes. The compressibility factor, Z, is a multiplier in the basic formula. It is the ratio of the actual volume at a given pT condition to ideal volume at the same pT condition. The ideal gas equation is therefore modified to ... 

An alternate to equation (6.15) is an equation that relates In

compressibility factor of the gas, that is defined as... 

Equations of state that are cubic in volume are often employed, since they, at least qualitatively, reproduce the dependence of the compressibility factor on p and T. Four commonly used cubic equations of state are the van der Waals, Redlich-Kwong, Soave, and Peng-Robinson. All four can be expressed in a reduced form that eliminates the constants a and b. However, the reduced equations for the last two still include the acentric factor u> that is specific for the substance. In writing the reduced equations, coefficients can be combined to simplify the expression. For example, the reduced form of the Redlich-Kwong equation is... 

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.

At very low pressures, deviations from the ideal gas law are caused mainly by the attractive forces between the molecules and the compressibility factor has a value less than unity. At higher pressures, deviations are caused mainly by the fact that the volume of the molecules themselves, which can be regarded as incompressible, becomes significant compared with the total volume of the gas. 

FIGURE 4.28 A plot of the compression factor, Z, as a function of pressure for a variety of gases. An ideal gas has Z = 1 for all pressures. For a few real gases with very weak intermolecular attractions, such as H2, Z is always greater than 1. For most gases, at low pressures the attractive forces are dominant and Z 1 (see inset). At high pressures, repulsive forces become dominant and Z 1 for all gases. 

We can assess the effect of intermolecular forces quantitatively by comparing the behavior of real gases with that expected of an ideal gas. One of the best ways of exhibiting these deviations is to measure the compression factor, Z, the ratio of the actual molar volume of the gas to the molar volume of an ideal gas under the same conditions ... 

The compression factor of an ideal gas is 1, and so deviations from Z = I are a sign of nonideality. Figure 4.28 shows the experimental variation of Z for a number of gases. We see that all gases deviate from Z = 1 as the pressure is raised. Our model of gases must account for these deviations. 

The presence of intermolecular forces also accounts for the variation in the compression factor. Thus, for gases under conditions of pressure and temperature such that Z > 1, the repulsions are more important than the attractions. Their molar volumes are greater than expected for an ideal gas because repulsions tend to drive the molecules apart. For example, a hydrogen molecule has so few electrons that the its molecules are only very weakly attracted to one another. For gases under conditions of pressure and temperature such that Z < 1, the attractions are more important than the repulsions, and the molar volume is smaller than for an ideal gas because attractions tend to draw molecules together. To improve our model of a gas, we need to add to it that the molecules of a real gas exert attractive and repulsive forces on one another. 

Real gases consist of atoms or molecules with intermolecular attractions and repulsions. Attractions have a longer range than repulsions. The compression factor is a measure of the strength and type of intermolecular forces. When Z > 1, intermolecular repulsions are dominant when Z < 2, attractions are dominant. 

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