The compressibility factor Z of natural hydrocarbon gases can be estimated by using the revised Awoseyin method32:
Z F
A P T F F F F
Sg
= +
( (
× ×) )
+ ×é ë êê ê
ù û úú ú
+ +
1
6 11 785 3 825 2 3 4 5
1 10. .
where
T is temperature (R) P is pressure (Kpsia)
F1=P
(
0 251. Sg-0 15.)
-0 202. Sg+1 106. F2=1 4. e-0 0054. (T-460)F3 A P1 5 A P A P A P A P
2 4 3 3
4 2
= + + + + 5
F S Pg e
Sg P
4
3 18 1 0 0 5
0 154 0 152 0 02
=
(
. - .)
(. -. ) -. - .F S eg P
5=0 35 0 6.
(
. -)
-1 039. ( -1 8.)2TABLE 3.5
Atomic Diffusion Volumes for Use in Fuller, Schettler, and Giddings’s Method
Atomic and Structural Diffusion Volume Increments, v
C 16.5 Cl 19.5
H 1.98 S 17.0
O 5.48 Aromatic ring −20.2
N 5.69 Heterocyclic ring −20.2
Diffusion Volumes for Simple Molecules, ∑v
H2 7.07 CO 18.9
D2 6.70 CO2 26.9
He 2.88 N2O 35.9
N2 17.9 NH3 14.9
O2 16.6 H2O 12.7
Air 20.1 CCl2F2 114.8
Ar 16.1 SF6 69.7
Kr 22.8 Cl2 37.7
Xe 37.9 Br2 67.2
SO2 41.1
Source: Coker, A.K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, 1995, p. 124.
and the values of the constants are
A1=0 001946. , A2= -0 027635. , A3=0 136315. , A4-0 23849. A5=0 105168. , A6=3 44 10. ´ 8
The specific gravity, Sg, of natural gas can be calculated from its density or molecular weight:
Sg=
( )
Sg(
Density of gas)
= °° =Density of air gas F or M
air F
r r
, , 60 60
oolecular weight of gas Molecular weight of air
gas air
= M M
w w , ,
The MATLAB function ngasZ estimates the compressibility factor Z of natural hydrocarbon gases. This function can be used as
nz = ngasZ(T,P,Sg)
where T is temperature (°F) (a scalar or a vector), P is pressure (psia) (a scalar or a vector), Sg is the specific gravity of the natural gas, and nz is the estimated compressibility factor of the natural gas at T and P. The units of T and P are automatically converted to R and Kpsia, respectively, within the function.
function nz = ngasZ(T,P,Sg)
% Estimates the compressibility factor Z of natural gases
% input
% T: temperature (F) (scalar or vector)
% P: pressure (psia)
% Sg: specific gravity of the natural gas
% output
% nz: estimated compressibility factor Z of natural gases P = P/1000; T = T + 460;
A1 = 0.001946; A2 = -0.027635; A3 = 0.136315;
A4 = -0.23849; A5 = 0.105168; A6 = 3.44e8;
F1 = P.*(0.251*Sg-0.15) - 0.202*Sg + 1.106;
den = 1 + A6*P.*10.^(1.785*Sg)./(T.^3.825);
F2 = 1.4*exp(-0.0054*(T-460));
F3 = A1*P.^5 + A2*P.^4 + A3*P.^3 + A4*P.^2 + A5*P;
F4 = (0.154-0.152*Sg).*P.^(3.18*Sg-1).*exp(-0.5*P) - 0.02;
F5 = 0.35*(0.6-Sg).*exp(-1.039*(P-1.8).^2);
nz = F1.*(1./den + F2.*F3) + F4 + F5;
end
Example 3.20: Compressibility Factor of Natural Gases33
Estimate compressibility factors of natural gases at 60°F and at the pressure range of 100 ≤ P ≤ 5000 (psia) when the specific gravity is 0.5, 0.6, 0.7, and 0.8. Plot the results as a function of pressure and specific gravity.
Solution
The following shows the calculation procedure and produces the curves shown in Figure 3.6.
>> T=60; P=100:10:5000; nz=[];
>> for k = 1:4, Sg = 0.5+(k-1)*0.1; nzv = ngasZ(T,P,Sg); nz = [nz nzv']; end
>> plot(P,nz(:,1),P,nz(:,2),':',P,nz(:,3),'.-',P,nz(:,4),'--')
>> axis([100 5000 0 1.1]), legend('Sg=0.5','Sg=0.6','Sg=0.7','Sg=0.8','locati on','best')
>> xlabel('P(psia)'), ylabel('Compressibility factor, Z')
PROBLEMS
3.1 Estimate the physical properties of saturated steam for 200 ≤ T ≤ 300 (°C) at 20°C intervals and tabulate the results.
3.2 Calculate the values of absolute humidity (g/m3) when the dry bulb temperature (Td) increases from 10°C to 40°C at 10°C intervals and the relative humidity varies from 20% to 80% at 20%
intervals.
3.3 In a storage room, the initial air temperature is 28°C and the relative humidity is 73%. After a while, the air temperature and the relative humidity decreased to 16°C and 55%, respectively.
Assuming that the door of the room was kept closed, calculate the amount of water condensed.
The volume of the air in the room is 150 m3.
3.4 Calculate the density of water for 0 ≤ T ≤ 350 (°C) at 1°C intervals and plot the results as a function of temperature.
3.5 Calculate the viscosity of water for 0 ≤ T ≤ 350 (°C) and plot the results as a function of temperature.
3.6 Estimate the viscosity of ethane for 25 ≤ T ≤ 900 (°C) at 1°C intervals and plot the results as a function of temperature.
3.7 Estimate the heat capacity of water for 0 ≤ T ≤ 350 (°C) and plot the results as a function of temperature.
3.8 Estimate the heat capacity of carbon dioxide for 300 ≤ T ≤ 1500 (K) at 1 K intervals and plot the results as a function of temperature.
3.9 Estimate the thermal conductivity of phenol for 0 ≤ T ≤ 350 (°C) at 1°C intervals and plot the results as a function of temperature.
3.10 Estimate the thermal conductivities of carbon dioxide and methane for 20 ≤ T ≤ 500 (°C) and plot the results as a function of temperature.
3.11 Calculate the surface tension of water (dynes/cm) for 0 ≤ T ≤ 350 (°C) and plot the results as a function of temperature.
500 1000 1500 2000 2500 3000 P (psia)
Sg = 0.5 Sg = 0.6 Sg = 0.7 Sg = 0.8
0 0.2 0.4
Compressibility factor, Z
0.6 0.8 1
3500 4000 4500 5000
FIGURE 3.6 Plot of compressibility factors of natural gases as a function of pressure and specific gravity.
3.12 Calculate the vapor pressures of benzene and phenol (mmHg) for 0 ≤ T ≤ 300 (°C) and plot the results as a function of temperature.
3.13 Estimate the vapor pressure of RE-218 (CF3–O–CF2–CF3) at 50.3°C and at −45°C by using the Rarey/Moller method. The normal boiling point is Tb = −23.7°C. Compare the results with experimental values (11.319 bar at 50.3°C and 0.372 bar at −45°C).34 Each group and frequency of the RE-218 molecule are shown in Table P3.13.
3.14 Estimate the enthalpy of vaporization of acetone at 273.15 K by using the derivative relation based on the Antoine equation. The parameters of the Antoine equation are A = 7.11714, B = 1210.595, and C = 229.664.
3.15 Estimate the standard heat of formation of CO and CO2 for 25 ≤ T ≤ 1200 (°C) and plot the results as a function of temperature.
3.16 Calculate the Gibbs free energy of formation (kcal/gmol) of ethane for 300 ≤ T ≤ 1500 (K) and plot the results as a function of temperature.
3.17 Determine the value of the infinite-dilution diffusion coefficient of propane (A) in chloroben- zene (B) at 0°C by using the Wilke–Chang relation. The molecular weight of chlorobenzene is 112.56, μ = 1.05 cP, and the molar volume of propane is 74.5 cm3/gmol, T = 273.15 K, and φ = 1.0.
3.18 Estimate the diffusion coefficient of allyl chloride in air at 25°C and 1 atm. Compare the result with the experimental value of 0.0975 cm2/s.35
REFERENCES
1. International Association for the Properties of Water and Steam, Revised Release of the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam, IAPWS, Lucerne, Switzerland, August 2007.
2. International Association for the Properties of Water and Steam, Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use, IAPWS, Lucerne, Switzerland, September 2016.
3. Smith, J. M., H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, New York, NY, p. 719, 2005.
4. Yaws, C. L. et al., Physical Properties, A Chemical Engineering Publication, McGraw-Hill, New York, NY, 1977.
5. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, pp. 95–96, 2012.
6. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, p. 96, 2012.
7. Andrade, E. N. d. C., Properties of dense gases and liquids, Philosophical Magazine, 17, 497, 698, 1934.
8. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 121, 1995.
9. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, pp. 9–21, 2001.
TABLE P3.13
Each Group and Frequency of the RE-218 Molecule
Group v (Frequency) ΔBi
C 3 −0.0896
F3 2 0.09402
F2 1 0.1054
O 1 0.15049
No H 1 −0.19373
10. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, pp. 9–22, 2001.
11. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 109, 1995.
12. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, p. 111, 2012.
13. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 110, 1995.
14. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 111, 1995.
15. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, pp. 12–13, 12–18, 2001.
16. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, pp. 12–18, 2001.
17. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 112, 1995.
18. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, p. 7, 2001.
19. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., pp. 84–85, 2012.
20. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., p. 86, 2012.
21. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., p. 89, 2012.
22. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., pp. 89–90, 2012.
23. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, pp. 7–24, 2001.
24. Pitzer, K. S., D. Lippman, R. F. Curl, C. M. Higgins, and D. E. Peterson, A new correlation method for enthalpies of vaporization, Journal of the American Chemical Society, 77, 3433, 1955.
25. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, pp. 101–102, 2012.
26. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, pp. 102–103, 2012.
27. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 115, 1995.
28. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, pp. 118–119, 1995.
29. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw- Hill, New York, NY, pp. 11.21–11.22, 2001.
30. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 123, 1995.
31. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 124, 1995.
32. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 125, 1995.
33. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, pp. 126–127, 1995.
34. Gmehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Chemical Thermodynamics for Process Simulation, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, p. 91, 2012.
35. Coker, A. K., Chemical Process Design, Analysis and Simulation, Gulf Publishing Company, Houston, TX, p. 127, 1995.
151
4 Thermodynamics
In chemical process modeling and simulation, the degree of accuracy of thermodynamic prop- erties, phase equilibria, mass and heat transfer, and chemical reactions mainly determines the quality of the model and simulation. Since these parameters are strongly influenced by thermo- dynamic relationships, a proper application of chemical engineering thermodynamic principles is an essential requirement of a successful process modeling and simulation. Chemical engi- neering thermodynamics deals with real substances whose properties are not entirely known from experiment at all possible temperatures and pressures and therefore are approximations described by model equations. These equations include volumetric equations of state that inter- relate pressure, volume, and temperature, and equations that relate activity coefficients to com- positions. Practicing engineers often need to solve problems even when descriptions of physical properties are imperfect, and a selection of equation of state or activity coefficient model must be made. Typically, the equations of state and activity coefficient models used in chemical engi- neering thermodynamics are not simple linear algebraic equations, so computations involving them may be difficult.
The main objective of this chapter is to provide readers with various thermodynamic models and corresponding MATLAB® programs that have already been found—or potentially would become—
useful in academic and industrial applications. The MATLAB programs in this chapter can be used in undergraduate or graduate courses on chemical engineering thermodynamics, provided that stu- dents have a prerequisite understanding of the fundamentals of chemical engineering. Researchers and practicing engineers in the field of chemical engineering can use these MATLAB programs in the modeling and simulation of chemical processes.
4.1 EQUATION OF STATE