Fluids Handling
Working with Non-Ideal Gases Here are two proven methods for predicting gas compressibility factors.
Jimmy Peress, P.E., Tritech Consulting Engineers
N
UMEROUS METHODS HAVE BEEN proposed to model pressure-volume-temperature (PVT) relationships of non-ideal gases (1). The approaches described in literature are generally classified into three groups. The first includes virial equations that are derived from statistical mechanics. The methods in the second group are represented by analytical equations, such as the Redlich-Kwong equation. The third group includes corresponding-state correlations that rely on a dimensional analysis of the gas system to identify the key modeling parameters. This article focuses on two proven and relatively simple methods to predict the compressibility factor, Z. One method is based on generalized compressibility charts. It is easy to use, but requires a graphical interpolation that introduces uncertainty in the estimated value. The second method is derived from the Redlich-Kwong equation. It is somewhat tedious, but Z can be obtained by mathematical manipulation.
Compressibility factor defined An ideal gas is represented by this equation of state: PV = nRT
(1)
For a non-ideal gas, Z is defined as: Z = PV/nRT
(2)
The compressibility factor can be viewed as that which corrects for the non-ideality of the gas. Thus, a gas for which Z = 0.90 will occupy only 90% of the volume occupied by an ideal gas at the same temperature and pressure. The values of Z range from about 0.2 to a little over 1.0 for pressures and temperatures of up to 10 times the critical values. Z itself is a complex function of the reduced temperature Tr, reduced pressure Pr and one or more other parameters, such as the critical compressibility factor, Zc, or acentric factor, ω. No simple analytical equations have yet been offered to express this complex relationship.
Experimental compressibility factor Where available, experimental Z values should be used for PVT calculations. Compilations of Z factors for some common gases and compounds as a function of T and P are
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available in standard handbooks (2). Thermodynamic tables for many common compounds often include PVT data in addition to enthalpy data. When such data exist, Z can be determined by direct substitution of the tabulated values into Eq. 2. Product brochures may also provide a good source of PVT data for specific gases or vapors. The following example illustrates how Z can be determined from tabulated thermodynamic data. Example 1. A relief valve must be sized to relieve saturated ammonia at 317.2 psia and 127.4°F. Thermodynamic data for ammonia indicate that the specific volume, v, of ammonia vapors at 307.8 psia (20.94 atm) and 125°F (324.8 K) is 0.973 ft3/lb. The molecular weight, MW, of ammonia is 17.03 lb/lbmol. Calculate Z at relieving conditions. Solution. The tabulated values are close to the relieving conditions, so the calculated value of Z will be applicable for the relieving conditions. Z is obtained by substituting the PVT data into Eq. 2 and using R = 1.314 atm-ft3/lbmolK and MW = 17.03 lb/lbmol. Z = (20.94 × 0.973 × 17.03)/(1 × 1.314 × 324.8) = 0.813.
Estimated compressibility factor Z factor from generalized compressibility charts: Nelson and Obert (3–4) developed a set of three generalized charts representing, respectively, the low-pressure region (0 atm < Pr < 1 atm), the medium-pressure region (1 atm < Pr < 10 atm) and the high-pressure region (10 atm < Pr < 40 atm). In these charts, Z is plotted as a function of Pr and Tr. The charts are also available in standard references (1–2). Estimation of Z involves the following steps: 1. Set the operating temperature and pressure, T and P 2. Obtain critical temperature and pressure, Tc and Pc 3. Calculate reduced temperature and pressure, Tr and Pr 4. From the appropriate chart, locate Pr and Tr and read the value of Z. Example 2. Determine the Z factor for saturated ammonia vapors at P = 307.8 psia (20.94 atm) and T = 125°F (324.8 K) from the generalized compressibility charts. Tc is 405.7 K and Pc is 111.3 atm. Solution. From the data above, Tr = 0.801 and Pr = 0.188. From the generalized compressibility charts in Ref. 2, Z = 0.84. The deviation from the experimental value determined in Example 1 is +3.3%.
Fluids Handling
Finding Z from Redlich-Kwong equation of state: The Redlich-Kwong equation of state (5) has constants, a and b: P = RT/(V – b) – a/( T1/2V(V + b))
(3)
Eq. 3 can be combined with Eq. 2 to give: Z = 1/(1 – b/V) – a/(RT3/2V(1 + b/V))
(4)
Eq. 4 can be rearranged to define Z as: Z = 1/(1 – h) – (a*2/b*) h/(1 + h)
(5)
where a*2 = a/R2T2.5, b* = b/RT and h = b/V. From thermodynamic considerations, a and b in Eq. 4 can be expressed in terms of the critical properties. A detailed derivation of these terms is beyond the scope of this article. Redlich and Kwong have shown that substitution of the critical properties Tc and Pc, and rearrangement, lead to the following equalities: a*2 = a/R2T2.5 = 0.4278Tc2.5/PcT2.5 atm–1
(6)
b* = b/RT = 0.0867Tc/PcT atm–1
(7)
h = b/V = b*P/Z
(8)
Z can be determined by first substituting the values for T, P, Tc and Pc and solving the resulting expression by trial and error. The built-in “goal-seek” capability of spreadsheets such as Microsoft Excel can be used to find the correct Z value that satisfies the equation. It should be noted
Nomenclature a = constant in Redlich-Kwong equation of state a*2 = term defined in Eq. 5 b = constant in Redlich-Kwong equation of state b* = term defined in Eq. 5 h = term defined in Eq. 5 MW = molecular weight of gas, lb/lbmol n = moles, lbmol P = pressure, atm Pc = critical pressure, atm Pr = reduced pressure, P/Pc, dimensionless R = universal gas constant = 1.314 atm-ft3/lbmol-K T = operating temperature, K Tc = critical temperature, K Tr = reduced temperature, T/Tc, dimensionless V = volume, ft3 w = weight, lb x = mole fraction of gas component in mixture Z = compressibility factor, dimensionless Zav = average compressibility factor as defined in Eq. 9, dimensionless Zc = critical compressibility factor, dimensionless v = specific volume, ft3/lb ω = acentric factor
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that since the equation is cubic, three roots are possible, but only one of the solutions will be physically meaningful and will represent the correct value. Example 3. Determine the Z factor for boron trifluoride (BF3) at 68°F (293.15 K) and 1,500 psig (103.04 atm) using the Redlich-Kwong equation of state. For BF3, Tc = 260.9 K and Pc = 49.20 atm. The Z factor reported by the manufacturer at these conditions is 0.468. Solution. From the data given above and using Eqs. 6–8: a*2 = 0.4278 × 260.92.5/49.20 × 293.152.5 atm–1 = 6.497 × 10–3 atm–1; b* = 0.0867 × 260.9/49.2 × 293.15 atm–1 = 1.568 × 10–3 atm–1; h = 1.568 × 10–3 × 103.04/Z = 0.1616/Z. Substituting the values of a*2, b* and h into Eq. 5 gives Z = 1/(1 – 0.1616/Z) – [(6.497 × 10–3)/(1.568 × 10–3)] × [(0.1616/Z)/(1 + 0.1616/Z)]. Rearranging and simplifying gives: Z = Z/(Z – 0.1616) – (0.6695)/(Z + 0.1616). Then, by trial and error, Z = 0.464. The deviation from the reported value is –0.9%. Example 4. BF3 is shipped in cylinders that have a net volume of 43.6 L (1.54 ft3). The temperature of the gas in the cylinders is 293.15 K and the pressure in the cylinders is 103.04 atm. The Z factor reported by the manufacturer at these conditions is 0.468. Estimate the weight of BF3 contained in the cylinder. Solution. Rearranging Eq. 2 and substituting the appropriate values gives: w = MW × n = (67.80 × 103.04 × 1.540)/(0.468 × 1.314 × 293.15) = 59.7 lb.
Compressibility factor of gas mixtures Gas mixtures that display significant deviations from ideal gas behavior can be modeled by using an average Z factor that is defined by the following relationship: Zav = Z1x1 + Z2x2 + Znxn
(9)
Z1, Z2, and Zn represent the Z factors for each component contained in the mixture, and x1, x2, and xn represent the respective mole fraction of each gas in the mixture. Example 5. Compressed air is stored at 200 K under a pressure of 98.7 atm. At these conditions, Z = 0.6997 for O2 and Z = 0.8455 for N2. Determine the Z factor for air containing 20.9% (by volume) O2. The tabulated Z factor for air at these conditions is 0.8142. Solution. The estimated Z factor is obtained by substituting the values into Eq. 9: Zav = 0.6977 × 0.209 + 0.8455 × 0.791 = 0.8146.
Comparing methods The table lists Z factors estimated by the two methods reviewed in this article and compares these values with experimental Z factors (where available) at the same T and P. The conditions selected are somewhat arbitrary, but generally reflect conditions encountered in the chemical industries. The analysis reveals that the two methods predict Z factors that are in good agreement with each other and with the
Table. Comparison between predicted and actual compressibility factors.* Actual Conditions
Critical Data
Reduced Values
Case Compound Molecular Weight T, K
P, atm V, ft3/lb
Tc, K
Pc, atm
Tr
Pr
Actual Z Factors
Compressibility Factors Compressibility Redlich-Kwong Charts Equation
1 2 3 4 5 6 7 8 9 10 11
Acetic acid 60.05 Acetone 58.08 Ammonia 17.03 Boron trifluoride 67.80 Bromine 159.82 n-Butane 58.12 1-Butene 56.10 CFC-114 170.92 Carbon dioxide 44.01 Carbon monoxide 28.01 Chlorine 70.91
533.15 463.15 324.80 293.00 380.00 388.80 522.00 344.10 323.00 300.00 299.80
21.27 22.47 20.94 103.04 3.96 20.41 40.82 7.46 59.21 100.00 7.93
NA NA 0.9730 0.0258 0.7640 0.2761 0.2460 0.2934 0.1183 0.1398 0.6357
594.80 509.10 405.70 260.90 584.00 425.20 419.60 418.90 304.20 133.00 417.00
57.10 47.00 111.32 49.20 101.65 37.50 39.70 32.20 72.90 34.50 76.10
0.896 0.910 0.801 1.123 0.651 0.914 1.244 0.821 1.062 2.256 0.719
0.373 0.478 0.188 2.094 0.039 0.544 1.028 0.232 0.812 2.899 0.104
NA NA 0.8130 0.4682 0.9684 0.6411 0.8213 0.8274 0.7263 0.9934 0.9074
Z % Error 0.80 NA 0.75 NA 0.84 3.3 0.45 –3.9 0.94 –2.9 0.66 3.0 0.82 –0.2 0.82 –0.9 0.72 –0.9 0.99 –0.3 0.88 –3.0
Z % Error 0.795 NA 0.734 NA 0.866 6.5 0.464 –0.9 0.955 –1.4 0.687 7.2 0.808 –1.6 0.843 1.9 0.722 –0.6 0.980 –1.3 0.903 –0.5
12 13 14 15 16 17 18 19 20 21 22
Ethanol Hydrogen Methane Methanol Methyl chloride Nitrogen Oxygen n-Octane Sulfur dioxide Toluene Water
427.15 300.00 300.00 487.15 349.80 300.00 300.00 563.15 366.30 553.15 502.40
6.80 100.00 100.00 39.25 19.31 98.70 98.70 21.34 23.61 24.07 27.21
1.6630 2.0802 0.2101 0.3691 0.3613 0.1433 0.1191 NA 0.2460 NA 11610
516.35 33.20 190.70 513.20 416.30 126.20 154.80 569.00 430.70 592.00 647.00
63.00 12.96 45.50 78.50 65.90 33.50 50.10 24.50 77.70 41.60 218.30
0.827 9.036 1.573 0.949 0.840 2.377 1.938 0.990 0.850 0.934 0.777
0.108 7.716 2.198 0.500 0.293 2.946 1.970 0.871 0.304 0.579 0.125
0.9282 1.0607 0.8549 0.7251 0.7664 1.0050 0.9543 NA 0.7730 NA 0.8618
0.92 1.05 0.86 0.73 0.78 0.99 0.95 0.55 0.80 0.68 0.87
0.934 1.063 0.855 0.762 0.808 0.990 0.947 0.551 0.807 0.690 0.907
46.07 2.01 16.04 32.04 50.49 28.01 32.00 114.23 64.06 92.13 18.01
–0.9 –1.0 0.6 0.7 1.8 –1.5 –0.4 NA 3.5 NA 0.9
0.6 0.2 0.0 5.1 5.4 –1.5 –0.8 NA 4.4 NA 5.2
* Compounds for which the specific volume was not readily available in standard references are marked with “NA.” For these compounds, the actual Z value could not be calculated.
Z factors reported in literature. For the cases evaluated, the Z factors interpolated from the generalized compressibility chart show an absolute average deviation of about 1.5% and a maximum deviation of about 4% for BF3. It should be noted that some regions of the charts are quite crowded and the interpolation may be somewhat subjective. The Redlich-Kwong method yields Z factors that show an average absolute deviation of about 2.5%, with a maximum deviation of about 7% for ammonia and n-butane. Its main advantage is that it allows the user to obtain Z factors by mathematical computation without the need for graphical interpolation. The use of a spreadsheet program with a built-in “goal seek” capability can prevent computational errors and overcome the tedium of repetitive calculations.
Literature Cited 1. Reid, R. and T. Sherwood, “The Properties of Gases and Liquids — Their Estimation and Correlation,” 2nd Ed., McGraw-Hill Co., New York (1966). 2. Perry, R. (Editor), “Chemical Engineers Handbook,” 5th Ed., McGraw-Hill Book Co., New York. 3. Nelson, L. and E. Obert, Trans. ASME, 76, p. 1057 (1954) as cited in Ref. 1. 4. Nelson, L. and E. Obert, “How to Use the New Generalized Compressibility Charts,” Chem. Eng., 61, (7) pp. 203–208 (July 1954). 5. Redlich, O. and J. Kwong, “On the Thermodynamics of Solutions,” Chem. Rev., 44, pp. 233–244 (1949).
The generalized compressibility charts are based on experimental data for 30 gases. Typical deviations are reportedly 1–2%, but some gases such as methane, water and fluoromethane display deviations of up to 4% in different regions of the charts. The Redlich-Kwong equation is a well-established method to model PVT relationships, but information on its accuracy was not readily available to the author. It reportedly yields Z factors that are in good agreement with experimental values over a wide range of conditions, and is particularly recommended for high-pressure conditions. However, this equation is not accurate at or near critical conditions.
Program available A Microsoft Excel-based template is available to calculate the compressibility factor using the Redlich-Kwong equation of state. Readers who are interested in obtaining the template should contact the author via e-mail. CEP JIMMY PERESS is director of Tritech Consulting Engineers, Inc. (Jamaica, NY; Phone: (718) 454-3920; E-mail:
[email protected]). He has over 30 years of experience in process engineering, process troubleshooting, software development and regulatory compliance (emission inventories and air permitting). Prior to starting his consulting practice in 1983, Peress was a process development manager and a senior consultant for Chem Systems, Inc. (Tarrytown, NY). He earned a BS in chemical engineering from the Israel Institute of Technology and an MS in chemical engineering from the City Univ. of New York. He is a professional engineer registered in New York, New Jersey and Delaware, holds seven U.S. patents and is a member of AIChE.
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