5 Properties of Real Gases
5.4 COMPRESSIBILITY CHARTS .1 Compressibility Factor
The compressibility factor measures the deviation of a gas from ideal behaviour. It is defined as Z = PV/RT. It can be interpreted as the ratio of the actual volume of a gas (V) to the volume that the gas will occupy if it were an ideal gas (RT/P) at the given temperature and pressure. Compressibility charts are plots of the compressibility factor against pressure with temperature as parameter as shown in Figure 5.2. For a perfect gas, the compressibility factor is unity and
2.0 Ethane Methane Hydrogen
1.0 Ideal gas Ammonia 0.0
0 200 400 600 Pressure, bar 800 1000 Figure 5.2 Compressibility factor.
the curve of Z versus P is a straight horizontal line at Z = 1.0. The shape of the curve for a real gas depends on the nature of the gas. However, since a real gas behaves ideally as pressure is reduced, the compressibility factor of a real gas approaches unity as the pressure is reduced to zero. This is clear from Figure. 5.2. At other conditions, the compressibility factor may be less than or more than unity depending on the temperature and pressure of the gas.
5.4.2 Principle of Corresponding States
The principle of corresponding states can be stated as that all gases when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from the ideal behaviour to the same extent. The reduced variables are obtained by dividing the actual variables by the corresponding critical constants. Thus the reduced pressure Pr = P/Pc, the reduced temperature Tr = T/Tc, and reduced volume Vr = V/Vc.
According to the principle of corresponding states, the compressibility factor of any substance is a function of the reduced temperature and reduced pressure alone. For example, consider the
compressibility factors of methane with the critical properties Tc =191 K, Pc = 46 bar and nitrogen with the critical properties Tc =126 K, Pc = 34 bar. Methane at 382 K and 69 bar and nitrogen at 252 K and 51 bar have identical reduced properties Tr = 2, Pr =1.5. Then according to the law of
corresponding states, nitrogen at 252 K and 51 bar and methane at 382 K and 69 bar should have the same compressibility factors. Thus the principle of corresponding states enables one to coordinate the properties of a range of gases to a single diagram such as the generalised compressibility chart.
5.4.3 Generalized Compressibility Charts
These charts enable engineering calculations to be made with considerable ease and also permit the
development of thermodynamic functions of gases for which no experimental data are available. The generalised compressibility chart (Figure 5.3) is constructed based on the principle
Tr = 1.00 5.00 2.00 1.60 0.90 1.40 1.30 0.80 1.20 1.15 0.70 1.10 0.601.05 0.50 0.40
0.300.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Reduced pressure, Pr Figure 5.3(a) Generalized compressibility chart (lower pressures).
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Reduced pressure, Pr Figure 5.3(b) Generalized compressibility chart.
of corresponding states. Based on experimental P-V-T data on 30 gases, Nelson and Obert* prepared these charts which give the compressibility factors with an average deviation of less than 5% from the experimental results.
On these charts, the compressibility factor Z is plotted as a function of reduced temperature and reduced pressure. The generalized compressibility charts provide one of the best means of expressing the deviation from ideal behaviour. When precise equations of state are not available to represent the P-V-T behaviour, these charts provide a convenient method for the evaluation of compressibility factor.
5.4.4 Methods Based on the Generalized Compressibility Charts The principle of corresponding state can be mathematically expressed as
Z = f (Pr, Tr) (5.18) Though the generalized charts and correlations based on the above equations provide better
results than the ideal gas equation, significant deviations from experimental results exist for many fluids. Considerable improvement can be made when a third parameter is introduced into
PV
Eq. (5.18). Lyderson et al.† used the critical compressibility factor, Zc =cc as the thirdRTc parameter.
Pitzer‡ used the acentric factorwas the third parameter.
*Nelson, L.E. and E.F. Obert, Chem. Eng., 61(7), 203–208 (1954).
†See Hougen, O.A., K.M. Watson and R.A. Ragatz, Chemical Process Principles, vol. 2, Asia Publishing House (1963).
‡Pitzer, K.S., J. Am. Chem. Soc., 77, 3427 (1955).
Lydersen et al. method: The method assumes that Z = f(Tr, Pr, Zc). Assume that there are different, but unique, functions Z = f(Pr, Tr) for each group of pure substances with the same Zc. Then, for each
Zc we have a different set of compressibility charts such as the one shown in Figure 5.3. All fluids with the same Zc values then follow the Z-Tr-Pr behaviour shown on charts drawn for that particular Zc. The generalized charts were drawn for Zc = 0.27 as most materials fell in the range of 0.26 to 0.28 and correction tables were developed to correct for values of Zc different from 0.27. The generalized charts for compressibility is given in Figure 5.4.
0 0.2 0.4 0.6 0.8 1.0 Reduced pressure, Pr
Figure 5.4(a) Generalized compressibility factor, low pressure range, Zc = 0.27. 1.2 1.1
Figure 5.4(b) Generalized compressibility factor, Zc = 0.27. The values given in Figure 5.4 can be corrected for values of Zc different from 0.27 according to the following equation.
Z ¢ = Z + D(Zc – 0.27) (5.19) where Z¢ is the compressibility factor of the given fluid, Z is the
compressibility factor at Zc = 0.27 and D is the deviation term. Generalized charts for deviation of the compressibility factor are given in Figue 5.5.
Pitzer method: The acentric factors measure the deviation of the intermolecular potential function of a substance from that of simple spherical molecules. For simple fluids it has been
observed that at a temperature equal to 7/10 of the critical temperature, the reduced vapour pressure closely follows the following empirical result:
PS 1 at T=0.7P=
c 10 Tc
where PS is the vapour pressure. Pitzer defined the acentric factor (w) in terms of the reduced vapour pressure at a reduced temperature of 0.7 as
1.2 1.2
0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2 0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2 Reduced pressure, Pr Reduced pressure, Pr Figure 5.5 Generalized deviation D for compressibility factor at Zc = 0.27.
w =− − log
⎛⎞
⎜⎟⎜⎟⎝⎠
Tr = 0.7
(5.20)
For simple fluids the acentric factor = 0; for more complex fluids the acentric factor > 0. The acentric factors are listed in standard references. [See, for example, Prausnitz, J.M., Molecular
Thermodynamics of Fluid Phase Equilibria, Prentice-Hall Inc. (1986).] Table 5.2 gives the acentric factors of some common substances.
Table 5.2 Acentric factors Molecule w Molecule w CH4 0.008 n-C4H10 0.193 O2 0.021 C6H6 0.212 N2 0.040 CO2 0.225 CO 0.049 NH3 0.250 C2H4 0.085 n-C6H14 0.296 C2H6 0.098 H2O 0.344
Pitzer proposed the following functional form for the compressibility factor. Z = f(Tr, Pr, w) (5.21) It was suggested that instead of preparing separate Z-Tr-Pr charts for different values of w, a linear relation could be used
Z = Z(0) (Tr, Pr) + wZ(1) (Tr, Pr) (5.22)
Here, the function Z(0) would apply to spherical molecules, and the Z(1) term is a deviation function.
Pitzer presented tabular and graphical values of the function Z(0) and the Z(1). Figures 5.6 and 5.7 give these plots incorporating the Lee–Kessler* modifications.
*Lee, B.L. and M.G. Kessler, AIChE J., 21, 510 (1975).
1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.2 0.3 0.4 0.5 0.6 0.8 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0 Reduced pressure, Pr Figure 5.6 Generalized compressibility factor for simple fluid.
0.35 0.3 0.5 0.7 1.0 2.0 3.0 Tr = 1.50
0.25 1.80 2.00 2.50 3.00
0.15 3.50 4.00
0.05 0.00 –0.05 –0.15 –0.25
0.2 0.3 0.4 0.6 0.8 1.0 2.0 3.0 Figure 5.7 Generalized compressibility factors correction term.
EXAMPLE 5.6 Using the Lydersen method, determine the molar volume of n-butane at 510 K and 26.6 bar. The critical pressure and temperature of n-butane are 38 bar and 425.2 K respectively, and the critical compressibility factor is 0.274.
Solution The reduced properties of n-butane are:
P26.6 0.70r==
38
T510 1.20r==
425.2
From Figures 5.4 and 5.5 Z = 0.865 D = 0.15
Using Eq. (5.19),
Z¢ = 0.865 + 0.15(0.274 – 0.27) = 0.8656 The molar volume is
V
RTZ (8.314)(510)(0.8656) P== 26.6×105
= 1.3798 ¥ 10–3 m3/mol
EXAMPLE 5.7 Using the Pitzer correlation, determine the molar volume of n-butane at 510 K and 26.6 bar. The critical pressure and temperature of n-butane are 38 bar and 425.2 K, respectively and the acentric factor is 0.193.
Solution The reduced properties of n-butane are:
P 26.6
r
== 0.70 38 T 510
r
==1.20 425.2
From Figures 5.6 and 5.7 Z(0) = 0.855 Z(1) = 0.042
Using Eq. (5.22)
Z = 0.855 + 0.193(0.042) = 0.8631 The molar volume is
V
RTZ (8.314)(510)(0.8631) P== 26.6×105
= 1.3798 ¥ 10–3 m3/mol