AGA 8 1994 - Cesar Delgado

Vista previa del material en texto

'*'
Transmission Measurement Committee Report No 8
ip
American Petroleum Institute MPMS Chapter 14.2
Gas Research Institute
American Gas
Association
REPORT NO. 8 SOFIWARE
The methods preseoted in Report No. 8 have beeo incorporated into efficieot computer code and software. 80th
the executable and the FORTRAN 77 code can be purchased from A.GA. 00 diskette. The easy-to-use program
provides tabular output of compreSSloility, supercompressibility and density for applications 00 PC's, flow
computers aod mainframes.
System specifications:
Language: FORTRAN 77
Requirements: IBM Compatible PC (Minimum 512KB RAM)
Compiler: Microsoft 5.0 (Adaptable for other compilers)
The purchasing company is granted unlimited use of the program and subroutines witbin the purchasing
company. The purchasing company can incorporate subroutines provided on tbis diskette into executable
programs for sale, but cannot sell source code provided on this diskette.
To order a program diskette, complete tbis below and mail to:
A.G.A.
Director, Engineering Services
1515 Wilson Boulevard
Arlington, VA 22209
~- -- ~- , PIcase check to appropriate metering applications where yon plan to appiy tbe compnter programs.
Transmission Metering
Production/Gathering Metering
~TIMATED NUMBER OF COPI~ OF SOFI'WARE TO BE MADE IN YOURCOMPANY:
D ET AlLXZ GROSSXZAGA8PROG
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(AGA. Members $400 per copy). Diskette size - 3 1/2 inch, - 5 1/4 inch.
NOTE: Computer program includes a copy of Report No. 8
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TABLE OF CONTENTS
1 INTRODUCTION . 1
1
2
2
2
3
4
5
5
6
7
7
7
1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Background. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 FieldofApplication 1.3.1TypesofProperties 1.3.2TypesofGases 1.3.3TypesofConditions 1.4 Overview 01 Compressibility Factor Calculation Methods 1.4.1 DETAlL CHARACTERIZA TION METHOD . . . . . . . . . . . . . . . . . .
1.4.2 GROSS CHARACTERIZATION METHOD . . . . . . . . . . . . .. . . . .
1.5 Uncertalnty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .
1.5.1 DETAlL CHARACTERIZATION METHOD Uncertainty . . . . . . . . . .
1.5.2 GROSS CHARACTERIZA TION METHOD Uncertainty 1.6 Recommendations
2 SYMBOLS 7
3 DEFINITIONS 8
4 GENERAL EQUA TIONS 8, 5 NATURAL GAS CHARACTERIZA TION METHODS 10
6 REFERENCE CONDITIONS .. 11
6.1 Mass Density at Contract Reference Conditions . . . . . . . . . . . . . . . . .. 11
6.2 Supercompressibility Factor Reference Condition 11
6.3 Relative Density Reference Condition . . . . . . . . . . . . . . . . . . . . . . . . .. 12
6.4 Heating Value Reference Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
7 UNITS, CONVERSIONS, PRECISION AND ACCURACY . . 13
8 EQUATIONS FOR COMPRESSIBILITY FACTORS . . . . 15
~'"
~c
!~
="1"': t , .-
8.1 DETAlL CHARACTERIZATION METHOD Equation of State . . . . . . . . . .. 15
8.1.1 Nomenclature 15
8.1.2 DETAlL CHARACTERIZATION METHOD Equation of State for
CompressibilityFactor 16
8.1.3 DETAlL CH.6;.RACTERIZATION METHOD Equation 01 State 10r
Pressure 21
8.2 GROSS CHARACTERIZATION METHOD Equation of State . . . . . . . . ... 27
8.2.1 Nomenclature 27
8.2.2 GROSS CHARACTERIZA TION METHOD Equation 01 State 10r
CompressibilityFactor 28
C' A T E O. A.C.
CENTRO DE INFORMACION
vii
8.2.3 Interaction Virial Coefficient Terms for Nitrogen and Carbon
Dioxide 8.2.4 Interaction Virial Coefficient Terms for the Equivalent
Hydrocarbon,CH
..30
. 31
9 PROCEDURES FOR COMPUTATIONS OF COMPRESSIBILlTV
FACTORS .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10 COMPUTER PROGRAMS FOR COMPUTATIONS OF
COMPRESSIBILlTV FACTORS, SUPERCOMPRESSIBILlTV
FACTORS,ANDDENSITIES
..-
.. 33
...34
..34
10.1 Documentation for Computer Programa . . . . . . . .
10.2 Computer Program Code listings
11 TABLES OF COMPUTED COMPRESSIBILlTV FACTORS ANO
SUPERCOMPRESSIBILlTV FACTORS . . . . . . . . . . . . . . . . . . . . . . . . . .. 34
12 UNCERTAINTIES IN COMPUTEO COMPRESSIBILlTY FACTORS ANO
SUPERCOMPRESSIBILlTYFACTORS 34
APPENDIX A - DETAlL CHARACTERIZATION METHOD .. . . . . . . .. 35
viii
APPENDIX A.4 - DETAlL CHARACTERIZATION METHOD FORTRAN
CaCE LlSTING 49
FUNCTIONDDETAIL 50
FUNCTIONPDETAIL 54
FUNCTIONZDETAIL 55
SUBROUTINE B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .~ . . . . . . . .. 57
SUBROUTINEBRAKET 58
SUBROUTINECHARDL 61
SUBROUTINEPARAMDL 65
SUBROUTINETEMP 67
BLOCK DATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
APPENDIX A.S - DETAlL CHARACTERIZATlON METHOD COMPUTER
PROGRAMCALCULATIONS 74
APPENDIX A.6 - DETAlL CHARACTERIZA TION METHOD. CALCULA TION
UNCERTAINTIES .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
A.6.1 Uncertainties in the DETAlL CHARACTERIZATION METHOD
Computed Compressibility Factors . . . . . . . . . . . . . . . . . . . . . .. 76
A.6.2 Uncertainties in DETAlL CHARACTERIZATION METHOD
Computed Supercompressibility Factors . . . . . . . . . . . . . . . . . .. 78
APPENDIX A.7 - DETAlL CHARACTERIZATION METHOD REFERENCES . .. 79
APPENDIX B - GROSS CHARACTERIZA TION METHOD 81
.APPENDIX8.1-NOMENCLATURE 83
APPENDIX 8.2 - GROSS CHARACTERIZATION METHOD COMPUTATION
PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
8.2.1 Determination 01 the Molar Gross Heating Value 01 the
EquivalentHydrocarbon(HcH) 84
8.2.2 METHOD 1. Input Parameters: Volumetric Gross Heating
Value, Relative Density, Mole Fraction CO2 84
8.2.3 METHOD 2. Input Parameters: Relative Density, Mole Fractions
01 N2 and CO2 87
8.2.4 SGERG Method Equation 01 State 10r Pressure 90
APPENDIX 8.3 - GROSS CHARACTERIZATION METHOD COMPUTATION
PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
8.3.1 Computations 01 Compressibility Factors 92
8.3.2 Computations 01 Other Quantities 92
APPENDIX 8.4 - GROSS CHARACTERIZATION METHOD COMPUTER
PROGRAMDOCUMENTATION 93
8.4.1 Scope 93
8.4.2SummaryFlowDiagram 93
B.4.3 Driver Block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
8.4.4 Component Dependent Quantities Block. . . . . . . . . . . . . . . . . .. 93
8.4.5 Characterization Dependent Quantities Block. . . . . . . . . . . . . . .. 95
8.4.6 Temperature Dependent Quantities Block. . . . . . . . . . . . . . . . .. 95
B.4.7 Density Dependent Quantities Block. . . . . . . . . . . . . . . . . . , . .. 95
8.4.8 Descriptions 01 Functions and Subroutines . . . . . . . . . . . . . . . . .. 95
8.4.8.1 SubroutineCHARGS 95
8.4.8.2FunctionDGROSS 96
ix
1
COMPRESSIBILlTY FACTORS FOR NATURAL GAS
AND OTHER RELA TED HYDROCARBON GASES
1 INTRODUCTION
1. 1 Scope
This report presents detailed inforrnation for precise computations of compressibility
factors and densities of natural gas and other hydrocarbon gases, calculation uncertainty
estimations and FORTRAN computer program listings. Applications for computations of other
properties are summarized but are beyond fue scope of this reporto
1.2 Background
Research in 1928 and 1929 under the direction of Mr. Howard S. Bean of the National
Bureau of Standards provided the natural gas industry with its initial compressibility factor data
covering pressures up to 600 psia (4 MPa). However, it was not until 1954 that extensive
tablas of natural gas supercompressibility factors were published, based on tests supervised
by Professor Samuel R. Beitler of Ohio State University. The natural gas supercompressibility
factor tablas were extended and an equation of state was developed in 1956-1962 under the
direction of Mr. R. H. Zimmermanof Ohio State University. The results of this project,
designated PAR Project NX-19, appear in A.G.A.'s .Manual for Determination of
Supercompressibility Factors for Natural Gas,. published in 1962.
The research leading to the present report was initiated in 1981 under the sponsorship
of GRI in clase liaison with the A.G.A. Transmission Measurement Committee. This research,
carried out under the direction of Professor Kenneth E. Starling of the University of Oklahoma,
was aimed at extending capabilities for accurate computation of compressibility factors beyond
the temperatura, pressure and composition ranges of PAR Project NX-19. The results for
pipeline quality natural gases, which were completad in 1984, provided the basis for the 1985
reporto
The initial 1981-1984 research used data ranging in pressures up to approximately 900
psia (6 MPa) obtained from the literature and provided by GERG. However, the GERG data
bank was extended considerably in the period 1985-1990. The new data showed that the
original equation of sta te , developed in the period 1981-1984, needed to be improved. In
addition, velocity of sound data obtained under GRI sponsorship during 1985-1989 showed
calculation for rich gases were not sufficiently accurate for critical flow applications. The new
equations of state presented in this revision include the most recent GRI and GERG
compressibility factor data for natural gas mixtures. In addition, the revised method has
applied an improved correlation methodology developed by researchers under the direction of
Professor Richard T Jacobsen at the University of Idaho.
B.4.8.3 Subroutine PARAMGS 96
B.4.8.4FunctionPGROSS 96
B.4.8.5 Subroutine VIRGS . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96
B.4.8.6FunctionZGROSS 96
B.4.9 Example Calculations 96
APPENDIX 8.5 - GROSS CHARACTERIZATION METHOD FORTRAN
CODELlSTING(GROSSXZ) FUNCTION DGROSS FUNCTIONPGROSS FUNCTIONZGROSS "0 SUBROUTINE CHARGS .. . . . . . . . . . . . . . . . . . . . . . . . . . .
SUBROUTINEPARAMGS SUBROUTINE VIRGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - - -
APPENDIX 8.6 - GROSS CHARACTERIZATION METHOD COMPUTER
PROGRAMCALCULATIONS APPENDIX 8.7 - GROSS CHARACTERIZATION METHOD CALCULATION
UNCERTAINTIES .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. 7 .1 Uncertainties in Computed Compressibility Factors . . . . . . . . . . .
B.7.2 Uncertainties in Computed Supercompressibility Factors . . . . . . .
APPENDIX 8.8 - GROSS CHARACTERIZATION METHOD REFERENCES . .
1
1
1
. . . . . .~
1
. . . . .. 1
. . . . .. 1
116
120
120
122
123
APPENDIX C - REFERENCE CONDITIONS ANO CONVERSIONS FOR
HEATING VALUE AND RELATIVE OENSITY .. . . . . . .. . . . . . . . 125
130
132
132
133
135
142
143
APPENDIXC.1-NOMENCLATURE APPENDIX C.2 - DISCUSSION OF REFERENCE AND STANDARD
CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX C.3 - DENSITY, COMPRESSIBILITY FACTOR AND
SUPERCOMPRESSIBILITY FACTOR. . . . . . . . . . . . . . . . . . . . . . . .
C.3.1 Mass Density at Contract Reference Conditions . . . . . . . . . . . . .
C.3.2 Supercompressibility Factor Reference Condition C.3.3 Relative Density Reference Condition APPENDIX C.4 - MOLAR HEATING VALUE FROM COMPOSITION APPENDIX C.5 - VOLUMETRIC HEATING VALUE . . . . . . . . . . . . . . . . . . . .
APPENDIXC.6-REFERENCES
APPENDIX D - UNIT CONVERSIONS
APPENDIX Do1 - UNIT CONVERSION PROGRAM DOCUMENTA TION .
APPENDIX Do2 - UNIT CONVERSION PROGRAM FORTRAN SOURCE
CaCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENCIX Co3 - REFERENCES o . o o . o . . . . . . o . o
151
156
x
03
04
06
07
08
12
14
APPENDIX E - UTILITV PROGRAM (AGA8PROG) . .
APPENDIX E.1 - UTILITY PROGRAM DOCUMENTATION 159
APPENDIX E.2 - UTILITY PROGRAM FORTRAN SOURCE CODE LISTING . 164
PROGRAMAGA8PROG 165
SUBROUTINECOMPST 168
SUBROUTINECONFIG 169
SUBROUTINE INPUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 172
SUBROUTINEMETHOD 173
SUBROUTINEOUTPUT 179
SUBROUTINEPROPS 180
SUBROUTINESETUP 181
SUBROUTINESTATUS 183
SUBROUTINETABLES ~ 186
SUBROUTINETABLES2 190
SUBROUTINESTATUS2 194
SUBROUTINEUNITSG 196
FUNCTIONDCALC 199
FUNCTION PCALC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 200
FUNCTION ZCALC . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. 201
INDEX
xi
2
1.3 Field 01 Application
1.3. 1 Types 01 Properties
This report presents detaired information needed to compute gas phase compressibirity
factors, supercompressibirity factors and densities for naturar gas and other rerated
hydrocarbon gases. /~
The detailed information in this report on the subject of computations of compressibility
factorS, supercompressibility factors and densities can be applied directly in calculations of gas
volume and gas flow rateo These computations also can be used in other instances where the
relationship between temperature, pressure and volume of gas is important. A pertinent
example is gas reservoir calculations. The derived thermodynamic properties which can be
calculated using the information in this report include the heat capacity, enthalpy, entropy,
sonic velocity, critical flow factor and component chemical potentials. Applications using these
properties include sonic nozzle calculations, compressor calculations, heat exchanger
calculations, gas mixture reaction equilibrium calculations and gas mixture component fugacit}'
calculations (for use in vapor-liquid equilibrium calculations).
1.3.2 Types 01 Gases
This report is intended for natural gases and other related hydrocarbon gases. The
information in this report also can be used for calculations of compressibility factors anc
densities for pure methane, ethane, nitrogen, carbon dioxide, hydrogen and hydrogen sulfide
and gas mixtures of up to twenty-one compounds. Table 1 identifies the ranges of gas
characteristics for which this report can be used. The normal range column gives the range
of gas characteristics for which the average expected uncertainty corresponds to the
uncertainties identified in Figure 1. The expanded range of gas characteristics has an average
uncertainty which is expected to be higher, especially outside of Region 1 of Figure 1. The
use of this report for computations of the physical properties of gases with component mole
percentages outside the ranges given in Table 1 is not recommended.
An accepted database for water, heavy hydrocarbons and hydrogen sulfide in natura
gases is not presently available for determinations of uncertainties of calculated gas properties
Therefore, as a practical matter, the only limitation is that the calculation is for the gas phase
Thus, the limits are the water dew point for mole percent water, the hydrocarbon dew point fo
mole percent heavy hydrocarbons, and pure hydrogen sulfide. The presentation of method~
for calculations using the various heavy hydrocarbon fraction characterization methods use(
in the hydrocarbon industry is beyond the scope of this report; these methods will be presente(
subsequently in the technical literature.
f
"1
1
~
!
!
3
Table 1
Ranges of Gas Mixture Characteristics
Consistent with this Report
Ouantity
Relative Density*
Normal Range Expanded Range
554 to .87 0.07 to 1.52
Gross Heating Value** 477 to 1150 Btu/scf o to 1800 Btu/scf
18.7 to 45.1 MJ/m3 o to 66 MJ/m3Gross Heating Value***
Mole Percent Methane 45.0 to 100.0 Oto 100.0
Mole Percent Nitrogen o to 50.0 Oto 100.0
Mole Percent Carbon Dioxide o to 30.0 Oto 100.0
Mole Percent Ethane Oto 10.0 Oto 100.0
Mole Percent Propane o fa 4.0 Oto 12.0
Mole Percent Total Butanes o to 1.0 o to 6.0
Mole Percent Total Pentanes o to 0.3 o to 4.0
Mole Percent Hexanes Plus o to 0.2 o to Dew Point
Mole Percent Helium o to 0.2 o to 3.0
Mole Percent Hydrogen Oto 10.0 Oto 100.0
Mole Percent Carbon Monoxide o to 3.0 o to 3.0
Mole Percent Argon # o to 1.0
Mole Percent Oxygen # Oto 21.0Mole Percent Water o ~o 0.05 o to Daw Point
Mole Percent Hydrogen Sulfide
- - - o to 0.02 Oto 100.0
* Reference Condition: Relative density at 60°F, 14.73 psia
** Reference Conditions: Combustion at 60°F,14.73 psia; density at 60°F,14.73 psia.
*** Reference Conditions: Combustion at 25°C,O.1 01325 MPa; density at 0°C,O.1 01325 MPa.
, The normal range is considered to be zero for these compounds.
1.3.3 Types of Conditions
This report is only valid for the gas phase. It can be applied for temperatures from
-200°F to 760°F (-130°C to 400°C) at pressures up to 40,000 psia (280 MPa). Application at
extreme conditions should be verified by other means (e.g., experimental verification). Use
of the calculation methods is not recommended within the vicinity of the critical point. For
pipeline quality natural gas, this is usually not a constraint because operating conditions near
the critical point gene rally are not encountered.
5
The two methods are differentiated by the input parameters needed for the equation
of state calculations. One method applies a detail knowledge of natural gas composition to
compute the compressibility factor (i.e., using the gas analysis), Reference 1, Appendix A. 7.
This method is referred to herein as the -DETAlL CHARACTERIZATION METHOD.- It can
be applied over the entire temperature, pressure and composition regime referred to in Section
1.3. The second method applies an aggregate or gross knowledge of natural gas composition
(given by heating value and/or relative density and diluent content information) to compute the
compressibility factor, References 2 and 3, Appendix B.8. This method is referred to herein
as the -GROSS CHARACTERIZATION METHOD.- The GROSS CHARACTERIZATION
METHOD can be applied in a limited temperature, pressure region for natural gas
compositions shown in fue normal range column of Table 1. Both methods require the use
of temperature and pressure in absolute units and a gas analysis to initially determine the
applicable method.
1.4. 1 DETAlL CHARACTERIZATION METHOD
The DETAlL CHARACTERIZA TION METHOD was developed to accurately describe
the gas phase pressure-temperature-density behavior of natural gas mixtures over a wide
range of conditions. It can also accurately describe the gas phase pressure-temperature-
density behavior of the pure components methane, ethane, carbon dioxide, nitrogen and
hydrogen and mixtures of these components. In addition, a low density correlation was
developed for propane and heavier hydrocarbons and binary mixtures of these components
with methane, ethane, nitrogen and carbon dioxide. This method reduces the uncertainty of
compressibility factor and density calculations for natural gases from production separators,
which can contain mole percentages of hexanes plus heavier hydrocarbons greater than 1 %.
Correlations of the density behavior of pure hydrogen sulfide and binary mixtures of hydrogen
sulfide with methane, ethane, nitrogen and carbon dioxide were developed to reduce the
calculation uncertainty for natural gases containing hydrogen sulfide (sour gas). Finally,
second virial correlations were developed for water and binary mixtures of water with methane,
ethane, nitrogen and carbon dioxide to reduce the calculation uncertainty for natural gases
containing water vapor (wet gas). Section 8.1 and Appendix A present detailed information
for the DETAlL CHARACTERIZATION METHOD.
1.4.2 GROSS CHARACTERIZA TION METHOD
The GROSS CHARACTERIZATION METHOD was developed using the natural gas
characteristics database listed in Table 2 to accurately calculate compressibility factors for dry,
sweet natural gases. Section 8.2 and Appendix B present detailed information for fue GROSS
CHARACTERIZATION METHOD.
6
Tabla 2
Nominal Ranges of Natural Gas Characteristics from Available Experimental
Compressibility Factor Data Used to Test
the DETAlL and GROSS CHARACTERIZA TION METHODS
Quantity Range
Relative Density*
Gross Heating Value** 477 to 1150 Btu/scf
Gross Heating Value." 18.7 to 45.1 MJ/m3
Mole Percent Methane 45.2 ta 98.3
Mole Percent Nitrogen -", 0.3 to 53.6
Moje Percent Garbon Dioxide 0.04 to 28.94
Mole Percent Ethane 0.24 to 9.53
Mole Percent Propane 0.02 to 3.57
Mole Percent Total Butanes 0.01 to 1.08
Mole Percent Total Pentanes 0.002 to 0.279
Mole Percent Hexanes Plus 0.0005 to 0.1004
Mole Percent Helium o to 0.158
* Reference Condition: Relative density at 60°F,14.73 psia
** Reference Conditions: Combustion at 60°F,14.73 psia; density at 60°F,14.73 pala.
*** Reference Conditions: Combustion at 25°C,0.1 01325 MPa; density at 0°C,0.1 01325 MPa
1.5 Uncertainty
The uncertainties of compressibility factors calculated using either the DETAlL
CHARACTERIZATION METHOD or the GROSS CHARACTERIZATION METHOD depend
upon the natural gas composition and the temperature-pressure condition of the calculation.
Evaluations of the uncertainties of calculated compressibility factors for natural gases were
made by comparing them to the GRI and GERG compressibility factor reference databases.
The reference databases have the natural gas physical characteristics identified in Table 2.
Comparisons were also made with data for pure components and binary mixtures. Lastly,
comparisons were also made with experimental speed of sound data to access derived
therrnodynamic property capabilities of the equations. Detailed uncertainty analysis is
available in Reference 2, Appendix A. 7 .
7
1.5.1 DETAlL CHARACTERIZATION METHOD Uncertainty
In general, the expected uncertainty of the DETAlL CHARACTERlZATION METHOD
is within the targets given in Figure 1 for natural gases having the normal range of gas
characteristics identified in Table 1. For the expanded range of gas characteristics in Table
1, the method has average uncertainty which may be higher, especially outside of Region 1
of Figure 1. The GRI and GERG compressibility factor reference databases have verified the
expected uncertainty in the normal range of gas characteristics within Region 1 and parts of
Regions 2, 3, and 4 of Figure 1.
1.5.2 GROSS CHARACTERIZATION METHOD Uncertainty
In general, the expected uncertainty of the GROSS CHARACTERIZA TION METHOD
is within the targets given for Region 1 in Figure 1 for natural gases having the characteristics
identified in the normal ranga in Table 1. The GRI and GERG compressibility factor reference
databases haya verified the expected uncertainty in the normal range of gas characteristics
in Table 1 within Region 1. The equatíon was not desígned fo, and should not be used
outside of these limíts.
1.6 Recommendations
The GROSS CHARACTERIZATlON METHOD, because of its simplicity, is
recommended for calculations of natural gas compressibility factors and densities for
temperatures from 32°F to 130°F (O°C to 55°C) for pressures up to 1200 psia (8.3 MPa),
provided the natural gas characteristics are within fue normal ranga defined in Table 1. For
al! other conditions and natural gas compositions, fue DETAll CHARACTERIZATION
METHOD is recommended. In those cases when the operating conditions exceed fue
pressure, temperatura or compositionaJ limits of fue GROSS CHARACTERIZA TION METHOD,
the DETAll CHARACTERIZA TION METHOD is recommended.
2 SYMBOLS
The symbols used generally are specific to thermodynamics, although symbols specific
to flow measurement have been used as a convenience to the reader. Symbols used in
Sections 4 through 6, which do not refer to a specific equation of state, are given below.
Symbols specific to the equations of state used to compute compressibility factors are given
in Sections 8. 1. 1 and 8.2. 1.
Symbol = Represented Ouantity
d = molar density (moles per unit volume)
d(T d'P J = molar density at reference condition T d' P d
F pv = supercompressibility factor
Gr(T ~P gJ = relativa density (specific gravity) of gas mixtura at T '1" P gr
HN°(T h,PJ = molar ideal gross heating value at reference condition T h' Ph
HV(T h'Ph, T d'P J = volumetric gross heating value, reference conditions T h'Ph,T d'P d
Mr = molar mass (molecular weight)
8
Mr(air) = molar mass of air
Mr. = molar mass of ith component
In = number of moles of gas
N = number of components in gas mixtura
P = absolute pressure
P b = absoluta pressure at base conditions
P d = reference pressure for molar density
P v = reference pressure for relative density
P h = reference pressure for heating value
R = gas constant
T = absoluta temperatura of gas
T b = absoluta temperature at base conditions
T d = reference temperature for molar density
~r = reference temperatura for relative density (specific gravity)
T h = reference temperatura for heating value
V = gas volume
XI = mole fraction of component i in gas mixture
Z = compressibility factor at conditions of interest
~ = compressibility factor at base conditions
Z(T,P) = compressibility factor at T, P
Z(T ~ P~) = compressibility factor of gas mixture at T gr' P ~
Z(air, T gr' P gr) = compressibility factor of air at T gr' P gr
Z(60°F, 14.73 psia) = compressibility factor at 60°F, 14.73 psia
p = mass density (mass par unit volume)
p(T gr' P~) = mass density of gas mixtura at T gr' P gr
p(air, T gr' P gr) = mass density of air at T gr' P gr
Pb = mass density at contract reference base condition T b' P b
3 DEFINITIONS
The quantities used in the equations in this document gene rally are defined with the
specific discussion of the relations.
4 GENERAL EQUATIONS
This section contains general equations involving the compressibility factor. the molar
density and mass density of natural gas and other hydrocarbon gases.
The compressibility factor Z is defined by the equation:
z = PV
ñR'f
(1)
Where:
P = absolute static pressure of gas
V = gas volume
n = number of moles of gas
Z = compressibility factor of gas
R = gas constant
T = absolute temperature of gas-
9
Both the DETAlL CHARACTERIZATION METHOD and the GROSS
CHARACTERlZA TION METHOD express the compressibility factor Z in terms of the molar
density d:
d - n
V
(2)
Where:
d = molar density (moles per unit volume)
n = number 01 moles 01 gas
V = gas volume
The gas mixture molar mass M, is calculated from the composition using the relation:
N
Mr = ¿x,Mr,
1-1
(3)
Where:
Mr = molar mass (molecular weight)
x. = mole fraction of component i in gas mixture
Mrt = molar mass of ith component
N = number of components in gas mixture
The summation in Equation 3 is ayer all components in the gas mixture.
The mass density p is relatad to the molar density d by the relation:
p-Md r (4)
Where:
p = mass density (mass per unit volume)
d = molar density (moles per unit volume)
Mr = molar mass (mass per mole)
~
10
Thus, using Equations 2 and 4, in Equation 1, the following equations for the molar
density d and the mass density p in terms of the compressibility factor result:
(5)P
d=ZRf
MrPc)\¡')v
ZR'f - ,-
(6)p =
Where:
Z = compressibility factor at conditions of interest
P = absolute pressure
d = molar density (moles per unit volume)
R = gas constant
T = absolute temperature of gas
Mr = molar mass (mass per mole)
p = mass density (mass per unit volume)
5 NATURAL GAS CHARACTERIZATION METHODS
The natural gas characterization inforrnation required for calculations using the DETAlL
CHARACTERIZATION METHOD is the natural gas composition. i.e., the mole fractions or
mole percentages of the components in the natural gas mixture.
The natural gas characterization information required for calculations using the GROSS
CHARACTERIZA TION METHOD is the molar ideal gross heating value HCH of the mixtura of
hydrocarbon components present in the natural gas along with the compositions (i.e., mole
percentages) of the nonhydrocarbon components in the natural gas mixtura.
When a gas analysis is available, HCH can be calculated from the known molar ideal
gross heating values of the hydrocarbon components in the natural gas mixture for direct use
in the correlation. When the composition of the natural gas is not known, HCH and one
nonhydrocarbon composition can be estimated from limited natural gas characterization
information. With the natural gas relative density, volumetric gross heating value and carbon
dioxide composition as inputs, methods in Appendix B can be used to determine HCH and the
mole fraction of nitrogen consistent with these inputs. Similarly, with the natural gas relative
density, carbon dioxide composition and nitrogen composition as inputs, methods in Appendix
B can be used to determine the value ~f HCH consistent with these inputs.
11
6 REFERENCE CONDITIONS
This section contains a brief summary of reference condition relations for natural gas
density, supercompressibility factor, relative density and heating value. Oetailed equations are
given in Appendix C.
6.1 Mass Density at Contract Reference Conditions
The mass density Pb at the contract reference condition (i.e., base condition) T b' Pb can
be calculated using the following relation:
MrPb
-z;:~
(7)Pb 8
Where:
Pb = mass density at contract reference condition T b' P b
Mr = molar mass
R = gas constant
T b = contract reference absoluta temperatura
Pb = contract reference absoluta pressure
~ = compressibility factor at contract reference condition T b' Pb
This relation is discussed in detail in Appendix C.
6.2 Supercompressibility Factor Reference Condition
Tabulations of the supercompressibility factor F pv presented by the A.G.A. are defined
by the following relation:
F2 = Z(60°F, 14.73psia)
pv
Z(T,P)
(8)
Where:
F pv = supercompressibility factor at T, P
Z(600F, 14.73 psia) = compressibility factor at 60°F, 14.73 psia
Z(T ,P) = compressibility factor at T, P
In this equation, the condition 60°F, 14.73 psia is a specific reference condition. Note that the
contract reference condition equals the reference condition for supercompressibility factor only
when the contract reference condition is 60°F, 14.73 psia.
12
6.3 Relative Density Reference Condition
The gas relativa density (specific gravity) at the reference condition T~. P ~ is defined
by the relation:
p(T gr'P gr)
p(air, T gr' P gr)
(9)G,{T ~.P~) -
Where:
G,{T grt P gr) = relative density of gas mixture at T~, P ~
p{T ~ P~) = mass density of gas mixture at T~, P gr
p(air, T grt P~) = mass density of air at T~, P ~
T ~ = reference temperature for relative density
P ~ = reference pressure for relative density
In this relation for Gr (T gr' P gr), both the gas mixtura density p(T , P gr) and the air density p(air,
T~. P~) must be at the same temperature-pressure condition T~. P~. Thus, using Equation
6 for the mass density of both the gas mixtura and air, the following relation results from
Equation 9:
MrZ(air, T gr'P gr)
G,{T gr' P gr) = Mr(air)Z{T grtP w)
Where:
G,{T g" P gr) = relative density of gas mixture at T gr' P gr
M, = molar mass for the gas mixtura
M,(air) = molar mass of air
Z(T gr' P gr) = compressibility factor of gas mixtura
Z(air, T gr' P gr) = compressibility factor of sir at T~. P gr
Note that the compressibility factors of both the gas mixture and air must be at the reference
condition T gr' P gr" Also note that the relative densíty is not a constant but varíes with both T gr
and P gr because both Z{T gr' P gr) and Z(air, T gr' P gr) vary with T gr and P gr" The calculation of the
compressibility factor of sir is discussed in detail in Appendix C.
6.4 Heating Value Reference Conditions
The volumetric gross heating value is the product of the molar ideal gross heating value
and the molar density of the gas mixture:
HV(T h,Ph,T d'P J = HN °(Th,Ph)d(T d'P J (11)
Where:
HV(T h'Ph, T d'P J = volumetric gross heating value, reference conditions T h'Ph, T d'P d
H~(Th'Ph) = molar ideal gross heating value at Th' Pt¡
d(T d'P d) = molar density at T d' P d
,
13
Note that the reference condition T h' P h for the molar real gross heating value may differ from
the reference condition T d' P d for the molar density. This situation occurs in the use of the
GROSS CHARACTERIZATION METHOD. 80th T h' Ph and T d' P d must be specified to specify
the referenceconditions for the volumetric heating value. Relations for the heating value are
presented in detail in Appendix C.
7 UNITS, CONVERSIONS, PRECISION AND ACCURACY
Most equations are presented here without speci1ic units so that the equations are valid
provided consistent units are used. In cases where quantities require a speci1ic set 01 units,
the units are specified.
The computar program subroutines provided in this document uses metric units, which
in most cases correspond to SI units. For this reason, tabulations of dimensional constants
usad in the computar programs are given in metric units. A units conversion FORTRAN
subroutine is provided for conversions to other units which are more convenient to the user.
The computar program subroutines in this document use the following units for the
principal dimensional quantities; absoluta temperatura in kelvins (K), pressure in Megapascals
(MPa), molar density in moles par cubic decimeter (moVdm3), the real gas volumetric heating
value in kilojoules par cubic decimeter (kJ/dm3), and the molar ideal gross heating value in
kilojoules par mole (kJ/mol). Conversion factors are required for conversions to and from other
units. Soma conversion factors used in this document are given in Table 3. When possible,
the conversion factors given in Table 3 correspond to intemational standards (ISO 6976 and
GPA 2172-94) or anticipated consensus values (e.g., the value for the gas constant R is the
anticipated consensus value). Temperatures are based on the IPTS-68 temperature scale.
Other conversion factors are provided in Appendix D.
The quantities in Table 3 have been rounded to seven significant figures. Most of the
conversions are not accurate to more than seven significant figures. Thus, the accuracy of
computed compressibility factors is not improved by the use of more than seven significant
figures for the conversions in Table 3.
I
~
14
Table 3
Units Conversions
Quantity Units Conversion
1.0 in = 0.0254 m
I Length
12,0 in = 1.0 ft: Length
I 1.0 Ibm = 0.4535924 kg
Mass
I Moles ¡ 1.0 Ib-mole = 0.4535924 kg-mole
I Temperature (in °R) = Temperature (in °F) + 459.67I TemperatureI
I Temperature (in K) = Temperature (in °C) + 273.15
1.8 ~ = 1.0 K
1..1.0 psia = 0.006894757 MPa
T emperature
Temperature
Pressure
Pressure 11.0 bar = 0.10 MPa
132.17405 ft/sec = 9.806650 m/secI Standard Acceleration
01 Gravity
10.73164 psia fi'nbmol-R = 8.314510 J/mol-KI Gas Constant
I Gas Constant 1.985886 BtUI1bmol-R = 8.314510 J/mol-K
: Energy 1.0 Btu = 1.055056 kJ
Note that a calculation precision of five significant figures is approximately 1 part in
100,000 for the compressibility factor. Appendices A and B present information showing that
the expected uncertainties of computad natural gas compressibility factors in the custody
transfer region are 0.048% (one standard deviation) for both the DETAlL
CHARACTERIZATION METHOD and the GROSS CHARACTERIZATION METHOD. The
uncertainty of 0.048% corresponds to 48 parts in 100,000, so fue calculation precision of 1 part
in 100,000 is more than an order of magnitude less than the uncertainty in the computad
compressibility factor. The computed compressibility factor generally agrees with the best
available compressibility factor data for natural gases with difterences approaching the
experimental uncertainty of 0.1%. Thus, calculated compressibility factors are accurate to
about one part in 10,000, or if less than one (Z<1.0) to about tour significant figures. When
more than tour significant figures are quoted in this document, the purpose is for computer
program calculation verification only.
15
8 EQUATIONS FOR COMPRESSIBILlTV FACTORS
f
Two different equations for the compressibility factor Z, the DETAlL
CHARACTERIZATION METHOD and the GROSS CHARACTERIZATION METHOD are
provided. The choice of a particular method is dependent upon natural gas characteristics and
the region of application; see Section 1.6, Recommendations.
In the DETAlL CHARACTERIZATION METHOD, a natural gas is characterized by its
composition, that is the mole 1ractions or mole percentages 01 the components in the natural
gas. Procedures to apply the DETAlL CHARACTERIZA TION METHOD are given in Appendix
A. Computer implementation is given in Appendix A and Appendix E.
In fue GROSS CHARACTERIZA TION METHOD a natural gas is charactenzed either
by the compositional analysis or by using combinations of three of the following tour quantities:
(1) the real gas relative density (specific gravity) Gr. (2) fue real gas gross heating value per
unit volume. HV. (3) the mole fraction of carbon dioxide. Xc02 and (4) the mole fraction of
nitrogen, XN2- The input for fue GROSS CHARACTERIZATION METHOD can be either
measured values of these quantities or calculated values from a gas analysis. Procedures to
apply the GROSS CHARACTERIZATION METHOD for calculations of compressibility factors
are given in Appendix B and Appendix C. Computer implementation is given in Appendix B
and Appendix E.
8.1 DETAlL CHARACTERIZATION METHOD Equation of State
8.1. 1 Nomenclature
Symbol = Represented Ouantity
2n = constant in Table 4
B = second virial coefficient
B~q = binary characterization coefficient
bn = constant in Table 4
C~ = coefficients which are functions of composition
cn = constant in Table 4
O = reduced density of gas
d = molar density (moles per unit volume)
El = characteristic energy parameter for ith component (Table 5)
El = characteristic energy parameter for jth component (T able 5)
E, = second virial coefficient binary energy parameter
E; = second virial coefficiem energy binary interaction parameter (Table 6)
F = mixture high temperatura parameter
F, = high temperature parameter for ith component (Table 5)
Fj = high temperature parameter for jth component (Table 5)
fn = constant in Table 4
G = orientation parameter
G¡ = orientation parameter for ith component (T able 5)
GJ = orientation parameter for jth component (Table 5)
16
G. = binary orientation parameter
G; = binary interaction parameter for orientation (Table 6)
gn = constant in Table 4
K = size parameter
K. = size parameter for ith component (Table 5)
K¡ = size parameter for jth component (Table 5)
~ = binary interaction parameter for size (Table 6)
~ = constant in Table 4
N = number of components in gas mixture
P = absolute pressure
O = quadrupole parameter
O. = quadrupole parameter for ith component
01 = quadrupole parameter for jth component
qn = constant (n=1, 2, etc.) in Table 4
R = gas constant
SI = dipole parameter for ith component (Table 5)
SI = dipole parameter for jth component (Table 5)
sn = constant (n = 1, 2, etc.) in Table 4
T = absolute temperature
U = mixture energy parameter
U, = binary interaction parameter for conformal energy (Table 6)
un = constant in Table 4
WI = association parameter for ith component (Table 5)
W¡ = association parameter for jth component (Table 5)
wn = constant (n=1, 2, etc.) in Table 4
X¡ = mole fraction of component i in the gas mixture
X¡ = mole fraction of component j in the gas mixture
Z = compressibility factor
8.1.2 DETAlL CHARACTERIZA TION METHOD Equation of State for
Compressibility Factor
The equations, constants, and parameters needed to calculate compressibility factors
for natural gas mixtures using the DETAlL CHARACTERIZA TION METHOD are given here.
Computation procedures, FORTRAN programs, uncertainty discussion and references are
provided in Appendix A.
The equation of state used here is a hybrid formulation. The method is based on the
work of Starling et al., Reference 1, Appendix A.7. It combines features of the virial equation
of state (a power series in density) for low density conditions and exponential functions for
applications at high density conditions (extended Benedict-Webb-Rubin equation). This
formulation provides both high accuracy, broad temperature-pressure-composition application
ranga, and derived thermodynamic property capabilities. A detailed description of the
peñormance of the equation is available in Reference2, Appendix A. 7 .
17
The equation of state for the compressibility factor Z for the DETAll
CHARACTERIZA TION METHOD is given in its condensed form by the following equation:
:12)
ug
.~Z-1i
K'
Where
Z = compressibility factor
B = second virial coefficient
K = mixtura size parameter
D = reduced density
C: = coefficients which are functions of composition
T = absolute temperatura
Un,bn,Cn'~ = constants (n=13, 14, etc.) in Table 4
The reduced density D is related to the molar density d by the equation:
D.K~ (13)
Where:
D = reduced density
d = molar density (moles par unit volume)
K = mixtura size parameter
; ] a
!
,[lE1.1
N-1 N 5
+ 2E E X¡X¡(K.~-1)(K.K¡)'2'
1-1 1-1+1
K5 =
Where:
I
~
The subscript i refers to the ith component in the gas mixtura and the subscript j refers to the
jth component in the mixtura. In the single sum, i ranges ovar the integer values from 1 to N.
For example, for a mixtura of 12 components, N=12 and fuera would be 12 terms in the single
sumo In the double sum, i ranges from 1 to N-1 and, for each value of i, j ranges from i+ 1 to
N. For example, for a mixture of 12 components, there would be 66 terms in the double sum
18
i1 sIl values 01 Kq differed 1rom one. However, because many 01 the values 01 Kq are one, the
number 01 nonzero terms in the double sum is small 10r many natural gas mixtures. Note that
sIl values 01 Kq are one except 10r the values in Table 6.
The second virial coefficient B is given by the following equations. where N is the
number of components in the gas mixture and the values of i and j both range from 1 to N:
18 N N 3
8 = E 8n T -11. E E x,xIE¡¡u'(KtKJ '2'8':'
n-1 1-1 1-1
(15)
(16)
Where:
B = second virial coefficient
B~ = binary characterization coefficient
8n'Un = constants (n=1,2, etc.) in Table 4
gn,qn,fn,sn,wn = constants (n=1,2, etc.) in Table 4
T = absolute temperature
XI = mole fraction of component i in the gas mixture
X¡ = mole fraction of component j in the gas mixture
Gij = binary orientation parameter
01 = quadrupole para meter for ith component (Table 5)
O, = quadrupole parameter for jth component (Table 5)
F, = high temperature parameter for ith component (Table 5)
F, = high temperature parameter for jth component (Table 5)
S. = dipole parameter for ith component (Table 5)
SI = dipole parameter for jth component (Table 5)
W¡ = association parameter for ith component (Table 5)
Wj = association parameter for jth component (Table 5)
E. = second virial coefficient binary energy parameter
K¡ = size parameter for ith component (Table 5)
Kj = size parameter for jth component (Table 5)
N = number of components in gas mixture
It should be noted that W¡ is zero for all components except water and that F, is zero for all
components except hydrogen.
The binary parameters Eij and G, are calculated using the following equations:
19
(17)
(18)
Where:
~ = second virial coefficient binary energy parameter
e. = characteristic energy parameter for ith component (Table 5)
E, = characteristic energy parameter for jth component (Table 5)
~ = second virial coefficient energy binary interaction parameter (Table 6)
G, = binary orientation parameter
G, = orientation parameter for ith component (Table 5)
Gl = orientation parameter for jth component (Table 5)
G, = binary interaction parameter for orientation (Table 6)
Note that all values of the binary interaction parameters e; and G; are one except for
fue values in Table 6.
The coefficients ~ (n=13 to 58) are given by the equation:
c; = 8n(G+1 -g,JD.(Q2+1 -q,Jq.(F+1-f,J'oUu. (19)
Where:
~ = coefficients which are functions 01 composition
8n,9n,~,un,1n = constants (n=13, 14, etc.) in Table 4
T = absolute temperature
G = orientation parameter
a = quadrupole parameter
F = mixture high temperature para meter
U = mixture energy parameter
The mixture parameters U, G, Q, and F are calculated using the following equations,
where in the double sums, i ranges from 1 to N-1 and, for each value of i, j ranges from i+1
to N:
20
~ N 5
U5 = Ex.E,'"
'-1
N-1 N 5
+ 2 E E x,x¡(u.f -1 )(E,E¡)"2'
1-1 ¡-1+1
N N-1 N
G = EX.G. + E E ~(G,.-1)(G1+G~
1-1 1-1 1-1+1
N
Q = Ex.Q,
'-1
N
F = Ex.2F.
1-1
Where:
Xi = mole fraction for the ith component
Xi = mole fraction for the jth component
N = number of components in gas mixture
U = mixture energy parameter
El = energy parameter for ith component (Table 5)
E¡ = energy parameter for jth component (Table 5)
U~ = binary interaction parameter for conformal energy (Table 6)
G = orientation parameter
G1 = orientation parameter for ith component (Table 5)
GJ = orientation parameter for jth component (Table 5)
Gq = binary interaction parameter for orientation (Table 6)
a = quadrupole parameter
al = quadrupole parameter for ith component (Table 5)
a¡ = quadrupole parameter for jth component (Table 5)
F = mixture high temperature parameter
FI = high temperature parameter for ith component (Table 5)
F¡ = high temperature parameter for jth component (Table 5)
It should be noted that al! values of the binary interaction parameters ~. E;. G;. and
U. are one except for the values given in Table 6. Also note that F1 is zero for al! components
except hydrogen.
21
8.1.3 DETAlL CHARACTERIZA TION METHOD Equation of State for Pressure
In the computation of the compressibility factor Z using the DETAlL
CHARACTERIZA TION METHOD, the composition of the gas is known, the absolute
temperature T of the gas is known, and the absolute pressure P is known. The problem then
is to compute the molar density d using the equation of state expression for the pressure P.
For this purpose, Equation 12 is substituted into Equation 5 to obtain an equation for the
pressure:
18 58
P = dRT [1 + Bd - DE Cn ¡-u. + E Cn ¡-U.(bn -cn~D k.)D b-exp( -cnD k.)]
n-13 n-13
(24)
Where:
P = absolute pressure
R = gas constant
T = absolute temperature
d = molar density 01 gas
B = second viriaJ coefficient
D = reduced density
~ = coefficients which are 1unctions 01 composition
un,bn,Cn'~ = constants (n=13, 14, etc.) in Table 4
When the temperature, pressure and composition of a gas are known, the only
unknown quantity in Equation 24 is the molar density d. The density is determined using
appropriate iterative procedures. It should be noted that before the computation of d can be
performed, the coefficients B and ~ in Equation 24 must be calculated from the composition
and temperature of the gas.
22
Table 4
DETAlL CHARACTERIZA TION METHOD Equation of State Parameters
b.
1
1
1
-
1
1
1
1
1
-
1
-
1
--
1
-
1
i
1
1
1
f"
o
o
o
o
o
o
~
o
o
o
-
o
o
-
o
1
O
O
O
O
O
O
O
O
O
O
O
O
O
1
O
-
O
1
O
O
O
k.
o
o
~
o
~
o
~
.o
o
o
o
o
o
o
3
2
2
2
4
4
O
O
2
2
2
4
4
~
4
4
O
1
~
1
2
2
3
n
1-
2
---
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Cn
o
o
o
o
o
o
o
o
o
o
o
o
1
1
1
qft
o
o
o
o
o
o
1
O
O-
O
-
O
-
O
-
O
-
O
O
-
1
o
o
~
o
o
o
o
o
o
1
O
1
O
O
O
O
O
s,.
o
o
o
o
o
o
-
o
1
-
1-
O
O
O
-
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
Wft
o
o
_o.
o
o-
o.
o
o
o
~ 9..
o
o
o
o
1
10
--'
O
O
O
~
O
O~'
O
-
O
O
O
O
O
O
O
O
O
O
O
1
O
O
-
O
1
O
O
-
1
1
8n
0.153832600
1.341953000
-2.998583000
-0.048312280
0.375796500
~ I-1.589575000
-0.053588470
0.886594630
-0.710237040
-1.471722000
1.321850350
-0.786659250
2.291290E-9
0.157672400
-0.436386400
-0.044081590
-0.003433888
0.032059050
0.024873550
0.073322790
-0.001600573
0.642470600
-0.416260100
-0.066899570
0.279179500
-0.696605100
-0.002860589
-0.008098836
3.150547000
0.007224479
-0.705752900
0.534979200
-0.079314910
0.0
0.5
1.0
3.5
-0.5
4.5
0.5
7.5
9.5
6.0
12.0
12.5
-6.0
2.0
3.0
2.0
2.0
11.0
-0.5
0.5
0.0
4.0
6.0
21.0
23.0
22.0
-1.0
-0.5
7.0
-1.0
6.0
4.0
1.0
1
o-
o
o
o
o
o
~
o
o
o
o
o
o
o
o
o
o
-
o
o
o
o
o
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
1
O
~
O
'-
1
1
1
~
1
~
t
~
1
~
1
~
O
~
1
~
1
~
1
~
1
~
1
24
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25
Table 6
Binary Interaction Parameter Values for Use in the
OET AIL CHARACTERIZA TION METHOO*
CIO(l)
1
Identiflcatk)n
Number
~
u. G,
MeI1an8
"' '.'-"".'
E,
0.971640
c
0.960644
'0.886106
l' 0:963827
~
1 .003630
~ 0.995933 ;0.807653 '
~i 0.990877 1.0076190.994635
0.708218
0.931484
I ~ r? '_"_M 0.736833
1. 000080
~~=::~_:
1.170520
0.990126
1.156390 1 . 023260 1.957310
Component Palr
Nitrogen
-.. ~
Dioxide
Ethane
Propane
Water
Hydrogen
SUfide
~. Hydlogen
_.~ ~ ~
Monoxide
Oxygen
~._. ~ i-BuIane
n-Butane
w ~. i-Pentane
w...~ n-Pentane
~...~ n-Hex.¡e
~ n-Heptane
n-Octane
n-Nonane
n-Decane
Carbon
DIoxide
Et¡ane
Propane
Water
HydrogQl\
SI.Af!de
HydrogM
w CaIbon
Monoxlde
Oxygen
~ I-Butane
n-Butane
I-Pentane
n-Pentane
l~_~
[11
0.992291 0.997596:12, 13
14
15
~:=::=
117
1.019530
0.989844
1.002350
0.999268
1.107274
0.880880
0.880973
0.881067
0.881161
1.022740
1.003670
1.302576
1.191904
1.205769
1.219634
1.233498
0.835058
"..'.._M '
1.002529
0.982962
0.983565
M
0.982707
0.981849
0.980991
0.982381
l~
119-32 Nitrogen 0.982746
4
5
""'
6
7
1.0079600.970120
0.945939
0.746954
0.902271 0.993476 0.942596
~
0.408838
~ ~~
1.032270 '
'
8
9
1.086320
! 1.005710
10
11
12
13
14
c
1.021000
0.946914
0.973384
0.959340
0.945520
" 0.993556
. Note: Values of one should be used for al! binary interaction parameters except for the
entries in this table.
1
26
CIO(I)
3
Identlflcation
Number
~.:=¡~§ E,0.925053
U,
0.969870
~
1.008510
_s
0.370296
-,
5
6
17-
Propane
Water
c
Hydrogen
SWfide
c
Hydrogen
I 0.955052
1.281790
1.500000
1.045290 1.007790
l! '9 I~Carbon
Ma1Oxide
10 I
11
n-Nonane,
n-Decane
l4 Propane
Water
Hydrogen
SUfide
1"HYd~;~-'._.~ iCarbon
,'Monoxide
Oxygen
i-Butane
'...'-'.'-"--"'-
n-Butane
'-'.'..""'-"-"
i-Pent8ne
n-Pentane
ro-:971~- 0.999969
...H H
1.020340
... H.H'
8
9
r '.-.'.--'.-¡
-~-!-~ I
: 1.013060
1 . 005320
1.034787
1 . 004900
LP:~_. Hydrogen
n-Butane
ó
7 Hydrogen
Sulfide n-Hexane
n-Heptane
n-Octane
1.028973
1.033754
' N.._' '
1.038338
1 . 042735
1 . 046966
:16
17
18
19
n-Nonane
n-Decane
8 Catbon
Monoxide
IO~~ r¡:B_.' In-~' ':
9 Hydrogen
1.008692
t.010126
, 1.011501
1.012821
,
1.014089
1.100000
10
11
12
1.300000
1.300000
Note: Values of one should be used for al! binary interaction parameters except for the
entries in this table.
8.2 GROSS CHARACTERIZATION METHOD Equation 01 State
8.2.1 Nomenclature
Syrnbol = Represented Quantity
BCH-CH = binary CH-CH interaction coefficient
BCQ2oCH = binary CO2-CH interaction coefficient
B~.(:02 = binary CO2-CO2 interaction coefficient
B~-N2 = binary CO2-N2 interaction coefficient
~ = binary interaction coefficient
Bmlx = second virial coefficient for the mixture
~-cH = binary N2-CH interaction coefficient
~-N2 = binary N2-N2 interaction coefficient
Bo,B1,B2 = coefficients in second virial equation for CH-CH interaction
bo,b1,b2 = constants in Table 7
buJb,1,~ = constants in Table 8
CCH-cH-CH = temary CH-CH-CH interaction coefficient
CCQ2.af.cH = ternary CO2-CH-CH interaction coefficient
C~-CH = temary CO2-CO2-CH interactjon coefficient
C~.(:02 = temary CO2-CO2-CO2 interaction coefficient
C~-N2 = temary CO2-CO2-N2 interaction coefficient
C~-fQ-CH = temary CO2-N2-CH interactjon coefficient
C~-N2 = temary CO2-N2-N2 interaction coefficient
CN2OCH-CH = temary N2-CH-CH interaction coefficient
CN2-N2-cH = ternary N2-N2-CH interaction coefficient
CN2-N2-N2 = temary N2-N2-N2 interaction coefficient
Cmlx = third virial coefficient for the mixture
CO,C1,C2 = coefficients in third virial equation for CH-CH interaction
Co,c1,~ = constants in Table 7
c."G1'~ = constants in Table 8
C'k = temary interaction coefficient
d = molar density, moVdm3
HCH = molar gross heating value of the equivalent hydrocarbon
N = number of components in gas mixture
P = pressure
R = gas constant
T = temperature
XcH = mole fraction of equivalent hydrocarbon
Xco2 = mole fraction of CO2
~ = mole fraction of N2
XI = mole fraction of i" component in the natural gas
XI = mole fraction of j" component in the natural gas
~ = mole fraction of k" component in the natural gas
Z = compressibility factor
28
8.2.2 GROSS CHARACTERIZA TION METHOD Equation of State
for Compressibility Factor
Equations, constants and parameters needed to calculate compressibility factors for
natural gas mixtures using the GROSS CHARACTERIZA TION METHOD are presented in this
section. Computation procedures, FORTRAN programs, uncertainty discussion and references
are provided in Appendix B. The equation of state applied is a virial type modelo It is based
on the work of Schouten et al. (1990); see Reference 2, Appendix B.8. It is alsoknown as,the
SGERG model; see Reference 3, Appendix B.8. The method uses a range of natural gas
heating value, relative density (specific gravity) and diluent content data as gross
characterization parameters in place of the detailed composition of a natural gas. The model
predicts highly accurate compressibility factors for natural gases which contain component
concentrations in the ranges given in Table 2, with less than 0.1 mole percent water and 0.05
mole percent hydrogen sulfide.
!~I~'
A virial equation of state is a polynomial expansion in density. Each density term is
preceded by a viñal coefficient. The viñal coefficients are functions of temperatureand
composition. The application of the viñal equation of state vis the SGERG model truncates
the viñal equation for the compressibility factor alter the third virial coefficient termo This
truncation provides high accuracy for normal gas pipeline transmission and distribution
conditions. but limits applications of the SGERG model to moderate den sities and pressures.
The GROSS CHARACTERIZATION METHOD approximates a natural gas mixtura by
treating it as a mixture 01 three components: an equivalent hydrocarbon component (i.e.
pseudo-hydrocarbon component), nitrogen and carbon dioxide. The equivalent hydrocarbon,
CH, is used to collectively represent sIl the hydrocarbons found in the gas mixture. Nitrogen
and carbon dioxide are the diluent components. The original formulation of this method
contained the capability to include hydrogen and carbon monoxide as gas components at a
fixed ratio for coke/oven gas applications. These gases rarely occur in North American
pipeline applications and are not included in the description of the method in this documento
However, the programs in Appendix 6.5 include them as possible components if desired by
the user.
The equations and model are identical to the original work of Schouten et al. (1990),
Reference 2, Appendix B.8. The computation procedures provided in Appendix B have been
optimized for electronic flow measurement applications. The SGERG model was developed
using SI units. These units are used in the description of the method for the purpose of
maintaining continuity with the original work. Convenient unit conversion routines are provided
in the FORTRAN program listed in Appendix D.
29
The GROSS CHARACTERIZATION METHOD predicts the compressibility factor of
natural gas at a given temperature and pressure using three 01 the following four
characterization parameters:
the volumetric gross heating value with reference conditions of 7rF, 14.696 psia
(25°C, 0.101325 MPa) for molar (or mass) ideal gross heating value and 3~F,
14.696 psia (OOC, 0.101325 MPa) for molar (or mass) density,
the relativa density (specific gravity) with reference conditions of 3~F, 14.696
psia (Oac, 0.101325 MPa),
the mole fraction of carbon dioxide, and
the mole fraction of nitrogen
Two methods for using these characterization parameters are given in Appendix B. Method
1 uses heating value, relative density and carbon dioxide content as input. Method 2 uses
relativa density. carbon dioxide content and nitrogen content as input.
The SGERG model expresses the compressibility factor in terms of the molar density
(d), the mixture second virial coefficient (Bmix)' and the mixture third virial coefficient (CIYU):
Z - 1 + Bmlx d + Cmix d 2 (25)
(26)
N N N
Cna z E E E C"kx.X¡~
1-1 '-1 k-1
(27)
Where:
Z = compressibility factor
BmIx = second virial coefficient for the mixtura
Cmlx = third virial coefficient for the mixtura
d = molar density (moles par unit volume)
B, = the individual component(s) interaction second virial coefficient
C'k = the individual component(s) interaction third virial coefficient
X¡.X¡.~ = mole fractions of gas components
N = number of components in gas mixture
The indices i, j, and k represent molecular components in a natural gas mixture. The SGERG
model treatment for natural gas is basad on a simple 3 component mixtura where CO2 and N2
are the only diluents. AII hydrocarbon components are collected into a single equivalent
hYdrocarbon component, CH.
~
30
The Bij and Cilk terms in Equations 26 and 27 are the interaction virial coefficient terms.
They are temperature dependent functions. For natural gas applications within the specified
limits of this equation, the temperature dependence for fuese terms is a quadratic function.
Expanding Equations 26 and 27 for Bmlx and Cmlx identifies al! the terms needed to salve the
SGERG modelo Expanding Equations 26 and 27 gives:
BIIÜ
- B 2 2 2- co.-co. x COI + ~-N. X Ha + BQi-cH X Qi
+ 2. Bco.-Ha Xco.~ + 2 BCo.-c..Xco.XcH + 2 BHa.c..xHaXCH
and:
I Cco.-co.-co. X3co. + CHo-Ha-Ha X3Ho + CCH-CH-CH X3CH
+ 3 Cco.-co.-Ha X2co.~ + 3 Cco.-co.-CH X2co.~
+ 3 C co. -Ha -Ha X 2 Ho Xco. + 3 C co. -cH -CH X 2 CH Xco.
+ 3 CHo-Ha-CH X2Ho ~ + 3 CHo-cH-cH X2CH XHo
+ 6 Cco.-Ha-cH Xco.~~
Cmlx =
(29)
The following sections provide the equations and methods needed to compute sIl the second
and third virial interaction terms for Equations 28 and 29.
8.2.3 Interaction Virial Coefficient Terms for Nitrogen and Carbon Dioxide
The B, values for the terms involving only nitrogen and carbon dioxide are expressed
in (dm3/mol) and are given by:
BI¡ = bo + b1 T + b2 T2
(30)
Where values for boJ b1J and b2 are given in Table 7 and T is the temperature in kelvins.
Similarly, the C~k values for the terms involving only nitrogen and carbon dioxide are
expressed in (dm6/mol-) and are given by:
Cijk = Co + C1 T + ~ T2
Where values for coI C1J and ~ are sIso given in Table 7.
31
Table 7
Interaction Virial Coefficient Terms for Nitrogen and Carbon Dioxide
Fluid for~!I- bo (dm3/mol) b, (dm3/mol K) b2 (dm3/mol K2)
0.74091 Ox10.3N2-N2
CO2-CO2
CO2-N2
-O.911950x10-8-0.144600
,0.40376Ox10-2-0.868340 -O.516570x10.S
IO.161176x10-2 .0.204429x10.S1-0.339693
Co (dm6/moI2) Elldml/mol2 K) c2_(~~_'/mof K~
O. 784980x1 0.2~~~~~!:.~~I Nz-Nz-Nz! - - -
COz-COz-COz
COz-Nz-Nz
CO2-COz-~~ -
-O.398950x10" 0.611870x10-7
0.205130x10-2 0.348880x10" 1-0.83703Ox10.7
0.552066x10-2 -O.168609x10" O. 157169x1 0.7
0.358783x10-2 O.806674x10.S -0.325798x10.7
8.2.4 Interaction Virial Coefficient Terms for the Equivalent Hydrocarbon, CH
The only remaining virial coefficient terms and mole fraction needed to compute the
compressibility factor of a natural gas from Equations 28 and 29 are the quantities involving
the equivalent hydrocarbon CH. The second and third interaction virial coefficients for the
equivalent hydrocarbon (CH) must be calculated from the molar ideal gross heating value of
the equivalent hydrocarbon (HCH in kJ/mol at 25°C and 0.101325 MPa). The molar gross
heating value HCH can be determinad by one of two methods. These methods are summarized
in Appendix B.
The equations for the second and third interaction virial coefficients for fue equivalent
hydrocarbon are:
2- Bo + B, HCH + B2 HCH (32)BCH-cH
and
= Co + C, HCH + C2 H~ (33)c CH -cH -cH
Where Bol B" B21 COI C, and C21 are temperature dependent functions defined as:
B, (34)= b,o + bit T + b'2T2, i = 0,1,2
32
and:
= CK) + CI1 T + c2T2,c, = 0,1,2
The constants in Equations 34 and 35 are given in Table 8 and the temperature is in kelvins.
Table 8
Virial Coefficient Terrns for the Equivalent Hydrocarbon
b'1 bl2blO
0.286500x10-2 -o.462073x10.SI Bo (~~~~!!,~Il- o 1-0.425468
-O.556281x10-5 0.88151 Ox10"81 (dm3/kJ) O.877118x10-31
-0.608319x10-1182 (dm3moI/kJ2) 0.431436x10-82 I-O.824747x10-8
i CI2CIO Cl1
Ico (dm'/mol~ -
0.195861 x1 0-2 I-O.316302x10-5o 1-0.302488
O.688157x10-80.646422x1 0.3
-0.422876x10-5IC1 (dm6/mol-kJ) 1
1-0.367713x10-11'C2 (dm6JkJ2)- 0.223160x10-82 1-0.332805x10'6
The interaction second virial coefficient term for the equivalent hydrocarbon CH with
nitrogen N2 is calculated using the relation:
~-CH
For the equivalent hydrocarbon CH with carbon dioxide CO21 theWhere T is in kelvins.
relation is:
B -<:H
~
.
The interaction third virial coefficient terms for carbon dioxide and nitrogen with the equivalent
hydrocarbon are calculated by Equations 38 through 42:
CN.-CH-CH
33
1
= (0.92 + 0.0013 (T -270» (C~~~ CCH-cH.oJ'"
CNa~-cH (39) f
(40)c COa .(:H.(:H
(41)
Cm, -COa.ot
t
1.10 (Cco.-co.-co. CN.-N.-N. CCH-CH-CH )'1' (42)Cco,-fta-cH -
The equation for calculating the pressure using the GROSS CHARACTERIZATlON METHOD
is obtained by substituting Equation 25 into Equation 5:
P = dRT(1 + B.d + C.d ~ (43)
9 PROCEDURES FOR COMPUTATIONS OF COMPRESSIBILlTV FACTORS
Aow rate calculations for gas metering applications typically require values of both fue
compressibility factor Z at the flowing temperature and pressure and the compressibilityfactor
at base conditions~. Procedures for computations of Z and ~ are given in Appendix A 10r
the DETAlL CHARACTERIZATION METHOD and in Appendix B for the GROSS
CHARACTERlZA TION METHOD.
10 COMPUTER PROGRAMS FOR COMPUTATIONS OF COMPRESSIBILlTV
FACTORS, SUPERCOMPRESSIBILlTV FACTORS, AND DENSITIES
The two equations of state, the DETAlL CHARACTERIZA TION METHOD and the
GROSS CHARACTERIZA TION METHOD, have been incorporated into efficient computer
programs to compute the compressibility factor, Z, the molar density, d, the mass density, p,
and the supercompressibility factor, F pv.
The computer programs provided in this document can be used for the following
purposes: (1) direct use for computation of Z, d, p, and F pv; (2) as a guide for the development
of subroutines for computations of Z, d, p, and F pv for incorporation in other computer codes,
such as orífice flow computer programs; (3) for verification purposes when new flow programs
are developed; and (4) for utility purposes, such as modification to produce tabulations of Z,
p, or F pv for particular gas mixture compositions. The programs are documented and are
consistent with the equations and methods presented in this documento The programs can be
34
used for example computation purposes and for development of specific application
subroutines or programs.
10.1 Documentation for Computer Programs
Computer program documentation is given for the DETAlL CHARACTERlZA TION
METHOD in Appendix A and for the GROSS CHARACTERlZATION METHOD in Appendix B.
The documentation includes descriptions of the main programs. all subroutines. and calculation
flow diagrams. Brief documentation siso is given for a units conversion program in Appendix
D. A utility program which allows calculations using both the DETAlL CHARACTERlZATION
METHOD and the GROSS CHARACTERlZATION METHOD is documented in Appendix E.
10.2 Computer program Code Listings
Code listings of the computer programs in FORTRAN are given for the DETAlL
CHARACTERIZATION METHOD in Appendix A and for the GROSS CHARACTERIZATION
METHOD in Appendix B. The computer program listings provided have a one-to-one
correspondence with the program documentation. A multipurpose utility driver computer
program using the units conversion program listing in Appendix D has been provided in
Appendix E. The utility driver program calls subroutines from both the DETAlL
CHARACTERIZA TION METHOD program in Appendix A and the GROSS
CHARACTERIZA TION METHOD program in Appendix B and provides a convenient means
for inputting and outputting data and provides calculations using either the DETAlL
CHARACTERIZATION METHOD or the GROSS CHARACTERIZATION METHOD.
T ABLES OF COMPUTED COMPRESSIBILITV
SUPERCOMPRESSIBILlTV FACTORS
FACTORS AND11
Tables of computed compressibility factors and supercompressibility factors are
provided for the DETAlL CHARACTERIZATION METHOD in Appendix A and for the GROSS
CHARACTERIZA TION METHOD in Appendix B. These tables can be usad to verify computar
programs. The tablas cover a wide range of gas types.
12 UNCERTAINTIES IN COMPUTEO COMPRESSIBILlTV FACTORS ANO
SUPERCOMPRESSIBILlTV FACTORS
The expected uncertainty limits for natural gas compressibility factors computad using
the methods in this manual are summarized in Section 1.5, Uncertainty. More detailed
information regarding uncertainties in compressibility factors at typical custody transfer
conditions is presentad for the DETAlL CHARACTERIZA TION METHOD in Appendix A and
for the GROSS CHARACTERIZA TION METHOD in Appendix B.
35
APPENDIX A
DETAlL CHARACTERIZATION METHOD
APPENDIXA.1-NOMENCLATURE '~ 37
APPENDIX A.2 - COMPUTA TION PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
APPENDIX A.3 - COMPUTER PROGRAM DOCUMENTATION '.: 40
APPENDIXA.4-FORTRANCODELlSTING : 49
APPENDIX A.S - COMPUTER PROGRAM CALCULATIONS . . . . . . .. . . . . .. 74
APPENDIX A.6 - CALCULATION UNCERTAINTIES 76
APPENDIXA.7-REFERENCES 79
.
..
37
APPENDIX A.1
NOMENCLATURE
Symbol = Represented Quantity
AAD = average absolute deviation
B = second virial coefficient
BIAS = bias
Cn = coefficients which are function of composition
d = molar density
F pv = supercompressibility factor
Max.Dev. = maximum value of ~III
N = number of data points
P = absolute pressure
A = gas constant
RMS = root mean squared deviation
Std.Dev. = standard deviation
T = absolute temperature
XI = components of the mixture
Z = compressibility factor
~ = compressibility factor at base conditions
2caIc = calculated compressibility factor
~III = relative percentage difference between calculated and experimental
compressibility factors
~.I = ~ for ith data point
(~)max = maximum value of ~iIf
~ = experimental compressibility factor
p = mass density
1-
l'
ClArEa. A.C.
CENTRO DE INFORMACION
1
38
APPENDIX A.2
DETAlL CHARACTERIZA TION METHOD COMPUTA TION PROCEDURES
A.2.1 Computations 01 Compressibility Factors and Densities
Figure A.2.1 gives the procedure used for computation of the compressibility factor Z
using the DETAlL CHARACTERIZA TION METHOD in Section 8.1.
Input is made of the mole fractions of the components in the gas.
m ixtu re molar mass is calculated using Equation 3, Section 4.
The gas1
Input is made for the temperature and pressure for which the computation of
the compressibility factor is desired.
2.
Computation is made 01 the coefficients B and C~ in the equation 01 state.
These coefficients are 1unctions 01 absolute temperature and the mole 1ractions
01 the components in the gas mixture. Constants required 10r this calculation
are given in Section 8.1 in Tables 3, 4 and 5. Equations 15 through 23, Section
8.1.2, are required 10r the computations 01 B and C~.
3.
The molar density d is computed using the equation of state relation for the
pressure P given in Equation 24, Section 8.1.3. The computation procedure
used is referred to as Brent's method and is given in Reference 5 in Appendix
A.7. In the iterative procedure for solving for the molar density d, the
convergence criterion is either agreement of the pressure calculated using
Equation 24 with the specified pressure within an absolute relative deviation of
1x10-6 or agreement of successive iterative values of d within an absolute
relative deviation of 1 x1 0-6. The mass density is calculated using Equation 4.
4.
The compressibility factor Z is computed using Equation 12, Section 8.1.2.5.
A.2.2 Computations 01 Other Quantities
Other quantities of interest such as fue supercompressibility factor. the density at
contract conditions and fue volumetric gross heating value for specified reference conditions
can be calculated using relations in Equations 1-11 in Sections 4 and 6 and Appendix C.
39
ri~l
L~~~
!
IZ=1+Bd+...1
!
Figure A.2.1. Procedure for Calculation of Z, F pv' d, and p
'..
40
APPENDIX A.3
DETAlL CHARACTERIZA TION METHOD
COMPUTER PROGRAM DOCUMENTA TION
A.3.1 Scope
This appendix provides summary documentation for the FORTRAN language computer
program code listing which is given in Appendix A.4 for the DETAlL CHARACTERIZATION
METHOD. This documentation includes a summary flow diagram of the computer program
and descriptions of the main program and each 'subroutine. Additional documentation is
provided by comment statements within the computer program listing.
A.3.2 Summary Flow Diagram
Figure A.3-1 is a summary flow diagram for use of fue computer program in Appendix
A.4. This computer program was developed for efficient computations of natural gas
compressibility factors using the DETAlL CHARACTERIZA TION METHOD, in which the mole
percentages of gas mixture components are known.
One program driver block (Application Program) and tour computation blocks are shown
in Figure A.3-1. The principal subroutine usad by a given computation block is shown to the
left 01 that computation block. The computation blocks and the principal subroutines usad are,
(1) calculation of component dependent quantities (PARAMDL),
(2) calculation 01 composition dependent quantities(CHAROL),
(3) calculation 01 temperatura dependent quantities (TEMP),
(4) calculation of density dependent quantities (DDETAIL).
A.3.3 Driver Block
For efficient computations, the application program should call the principal subroutines
in a sequence corresponding to the sequence of the tour computation blocks. This
computation sequence is efficient because it minimizas unnecessary repeated calculations.
An example of an unnecessary repeated calculation is the calculation of a temperatura
dependent quantity each time the density is calculated at a series of pressures, even though
the temperatura may be the sama for some of these density calculations. Thus, if the
computation sequence in Figure A.3-1 is incorporated in any other driver program, the
efficiency of this sequence will be maintained.
42
A.3.4 Component Dependent Quantities Block
The computer program structure shown in Figure A.3-1 corresponds to the tour levels
of information which are required in the DETAlL CHARACTERIZATION METHOD equation of
sta te. First, the components of which the gas mixture is composed must be identified. The
number of components, NCC, and the component identification numbers, CID(1), CID(2), ...,
CID(NCC), are required by subroutine PARAMDL. PARAMDL obtains fixed parameters from
BLOCK DATA and calculates other parameters required subsequently. This completes the first
computation block. Note that for most efficient computation, subroutine PARAMDL is never
called again until the components in the gas mixture change.
A.3.5 Composition Dependent Quantities Block
For computations in the second computation block, the composition of the gas mixtura
must be identified. The mole percentages of each component, XI(1), XI(2), ..., XI(NCC), are
required by subroutine CHAROL. CHAROL calculates composition dependent quantities which
are required subsequently. Note that for most efficient computation, CHAROL subroutine is
never called again until there is a change in the composition of the gas mixture.
A.3.6 Temperature Dependent Quantities Block
For computations in fue third computation block, the temperature must be identified.
Subroutine TEMP requires fue absolute temperature T. TEMP calls subroutine B, which
calculates the second virial coefficient B for the gas mixture at fue temperature T. TEMP then
calculates fue other coefficients in fue equation of state for the gas mixture, C~, for fue
absolute temperature T. Note that for most efficient computation, subroutine TEMP and
subroutine B are never called again until there is a change in the temperature of the gas
mixture.
A.3.7 Density Dependent Quantities Block
For computations of density and density dependent quantities in the fourth computation
block, the pressure must be identified. Function DDETAIL requires the absolute pressure P.
DDET AIL uses Brent's method to solve for the density of the gas mixture of composition XI(1),
XI(2), ..., XI(NCC) at absolute temperature T and specified absolute pressure P. Function
DDET AIL calls subroutine BRAKET, which determines two values of density, RHOL and
RHOH, which, when used in the equation of state, yield calculated absolute pressures PRHOL
and PRHOH which bracket the specified pressure P. The densities RHOL and RHOH, which
bracket the solution for density at the specified pressure P, are then used in function DDETAIL
to solve for the density corresponding to the specified pressure P. Both function DDET AIL and
subroutine BRAKET call function PDETAIL, which calculates the pressure for a specified
density, RHO. Function PDETAIL calls function ZDETAIL, which calculates the compressibility
factor 10r a specified temperature and density.
43
A.3.8 Descriptions 01 Functions and Subroutines
FORTRAN functions and subroutines used to calculate density related properties from
the DETAlL CHARACTERIZATION MODEL are listed below. These routines are available in
the module DETAILXZ.FOR, listed in Appendix A.4.
FUNCTION DDETAIL (P. T)
FUNCTION PDETAIL (D. T)
FUNCTION ZDET AIL (D. T)
SUBROUTINE B (T, BMIX)
SUBROUTINE BRAKET (CODE, T, P, RHO, RHOL, RHOH, PRHOL, PRHOH)
SUBROUTlNE CHARDL (NCC, XI, ZB, DB)
SUBROUTINE PARAMDL (NCC, CID)
SUBROUTINE TEMP (T)
BLOCK DATA
A.3.S.1 BLOCK DATA
BLOCK DATA is a block data section which contains the fixed constants used in the
equation of state computations of the compressibility factor using the DETAlL
CHARACTERIZATION METHOD. These fixed constants include the gas constant, RGAS, the
equation of state constants, A(1) through A(58), compound characterization parameters, that
¡s, the molecular masses CMWB(J) for each compound, energy, size and orientation
parameters EIB(J), RKIB(J), and WIB(J) for each compound, and the binary interaction
parameters BEIJB(I,J), BKIJB(I,J), BWIJB(I,J), and BUIJB(I,J).
A.3.8.2 Subroutine PARAMDL
Subroutine P ARAMDL assigns component and binary parameters for a mixtura
containing NCC components with component identification numbers CID(1), CID(2), ...,
CID(NCC) from Tables 5 and 6. Pura component and binary parameters in BLOCK DATA
which are required by PARAMDL are accessed through the common block
COMMON/P ARAM/.
A.3.8.3 Subroutine CHARDL
Subroutine CHAROL calculates a number of composition dependent quantities using
the mole percent values for the components in fue mixture, XI(1), XI(2), ..., XI(NCC). The mole
percent values are used to calculate normalized mole fractions by dividing each component
mole percent by the sum of the component mole percents. The gas mixtura average molar
mass is calculated as the molar average of the component molar masses. Composition
dependent coefficients of temperature functions in the second virial coefficient are calculated
for subsequent use in subroutine B. Also calculated are values of the mixture Biza parameter
(K3 in the text and RK3PO in the program), the mixture quadrupole parameter (~ in the text
and 02PO in the program), the mixture energy parameter (U in the text and U in the program),
and the mixture orientation parameter (W in the text and W in the program).
44
A.3.8.4 Subroutine B
Subroutine B calculates the second virial coefficient for the gas mixture at the absolute
temperature T. The required composition dependent coefficients are accessed through
COMMONNIR/. The computed second virial coefficient BMIX is made accessible through the
subroutine argumento
A.3.8.5 Subroutine TEMP
Subroutine TEMP calculates a number of quantities which are functions of temperature
but flot density. These quantities are B1 through B12 and FN(13) through FN(58), which are
related to the coefficients of the density functions in the equation of state. Required mixture
parameters are accessed through COMMON/AGAVARIABLES/. Required equation of state
parameters are accessed from BLOCK DATA through COMMON/EOSAI/. The required
absolute temperature T is accessed through the subroutine TEMP argumento Subroutine
TEMP cal/s subroutine B for the calculation of the second viñal coefficient at fue absolute
temperature T. The quantities calculated in subroutine TEMP are made accessible through
COMMON/AGA V ARIABLES/ and COMMON/AGAEOS/.
A.3.8.6 Function Subprogram DDET AIL
Function subprogram DDETAIL uses Brenfs method to calculate the molar density for
the specific temperatura and pressure in argument of DDETAll (P, T).
DDETAIL calls subroutine BRAKET, which brackets the solution for the density.
BRAKET determines two values of density, RHOL and RHOH, which, when used in the
equation of state, yield calculated absolute pressures PRHOL and PRHOH which bracket the
specified pressure P. The densities RHOL and RHOH, which bracket the solution for density
at the specified pressure P, are then used by DDET AIL as initial values for vse in Brent's
method.
The first iteration of Brenfs method uses inverse linear interpolation between RHOL
and RHOH to estimate a third triar root. Subsequent iterations of Brenfs method make the
most efficient choice between inverse quadratic interpolation, inverse linear interpolation and