Equations For Heavy Gases In Centrifugals

Equations For Heavy Gases in Centrifugals

Lee Chong Jin (Team leader)

Mohd Zakiyuddin Mohd Zahari

Cheah Cang To

James Bryan

- Rotating Equipment Department,

Technip Geoproduction Malaysia

August 12, 2014

Abstract

Centrifugal compressor performance prediction relies heavily on accurate modelling of

thermodynamic properties using Equations of State (EOS); In particular, the gas compressibility factor

(Z) and ratio of specific heat (k). There have been efforts to develop more generalised EOS such as

GERG, but the challenge remains on identifying the best EOS fit for specific duties.

More recent EOS including AGA8 and REFPROP’s NIST EOS haven been explored in this paper, along

with some earlier ones. The boundary limits of the various EOS are herein described with comparison

of the results of all of these equations on various gas mixtures encountered in real applications.

The purpose of this work is to explore the more thermodynamically challenging heavy gas and

mixtures. Operating points are selected to cover typical duties that are commonly encountered in

LNG and offshore compression. Z and k derived from the EOS are then compared with REFPROP’s

EOS as a reference and the deviations are tabulated.

More specifically, Mixed Refrigerant gases are typically used for LNG liquefaction applications while

CO2 gas are common in sour gas fields, hence relevant for the intended investigation.

Discharge temperature is not calculated and compared between EOS in this paper; a reliable model

for calculating polytropic exponents is open for further research.

Nomenclature

Symbols

( )

( )

Abbreviations

The following are abbreviations of the different EOS names used throughout the report:

( )

Eq. 1

Eq. 2

Fundamentals

The Compressibility Factor, Z, is the fundamental thermodynamic property for modifying the ideal

gas law to account for the real gas behavior. Z is introduced into the Ideal Gas equation [1]:

Due to the various factors involved such as having infinite possible combinations of ratios between

each component in a gas mixture, it is infeasible to develop an EOS that will accurately calculate Z

across a wide combination of operating conditions (in terms of gas compositions, temperatures and

pressures).

With the specific heat ratio, k, polytropic exponents can be obtained and in turn gas compression can

then be expressed in terms of pressure and temperature variation [2]:

( )

The power required to compress a gas is directly proportional to the gas compressibility factor, Z. For

an ideal gas, Z=1 regardless of the gas’ state. Since in practice Z changes depending on the gas

conditions P and T, power calculation will deviate between a real gas and ideal gas calculation by as

much as the Z deviates. Similarly, k affects the accuracy of the head and power equations.

Therefore, it is worth investigating the different EOS that can be used to obtain Z and k for a specified

mixture and operating condition. The EOS that are investigated in this report are tabulated in Table

1:

Equation of State General Form of Equation

Redlich-Kwong (RK) [1]

( )√

Soave-Redlich-Kwong (SRK) [1]

( )

Peng-Robinson (PR) [1]

( ) ( )

Benedict-Webb-Rubin-Starling & Han modified by Nishiumi & Saito (BWRS-NS) [3][4]

(

)

(

)

(

)

(

) ( ) ( )

Lee-Kesler-Plöcker (LKP) [1]

(

) (

)

AGA8 [5] ∑

∑ (

) (

)

GERG (2004) [6]

(∑ (

)

∑ ∑ (

)

)

NIST [7] Based on GERG2008 which is an updated GERG2004 EOS

Eq. 3

Eq. 4

Table 1: Equations of States analysed and the general form of the equation.

REFPROP, a commercially available program developed by the National Institute of Standards and

Technology (NIST), performs estimation of real gas thermodynamic properties based on three

models for the thermodynamic properties of pure fluids: EOS explicit in Helmholtz energy, the

modified Benedict-Webb-Rubin equation of state, and an extended corresponding states (ECS) model

[7]. Equation of state modules available from the REFPROP package are:-

1. AGA 8 (for pipelines)

2. GERG 2008

3. Peng-Robinson (PR)

4. NIST

The NIST EOS is primarily based on the GERG 2008 EOS (which is used in [8;9;10;11]), in turn

expanded from GERG 2004 to include additional fluids (e.g. ethylene, propylene, methanol, etc.).

NIST's database is widely recognised as a reliable source of reference in terms of real gas behaviour,

as can be traced in both the academic and turbo-machinery industry [12;13;14;15]. Thus, with the

established database in REFPROP software, the default NIST EOS will be the benchmark EOS which

other EOS will be referred to for the purpose of this paper.

The standalone EOS (not included in REFPROP) compiled by the authors for the purpose of this

discussion are as follows:-

1. Redlich-Kwong (RK)

2. Lee-Kesler-Plӧcker (LKP)

3. Modified Benedict, Webb, Rubin, Starling and Han by Nishiumi and Saito (BWRS-NS)

4. Soave-Redlich-Kwong (SRK)

RK and SRK EOS are relatively straightforward to model as they are cubic EOS. Virial EOS such as

BWRS-NS and LKP are developed as an improvement to the former; These are EOS which represents

a power series of density with temperature coefficients [1]:

The roots in these virial EOS are evaluated using the Newton-Raphson method where the initial guess

for compressibility factor is set to be 0.8 for the vapor phase [1]. AGA8 which is an extended virial

equation is even lengthier where it contains summations of 58 polynomial terms.

BWRS-NS is selected over the standard BWRS model for its wider range of operations; specifically in

the cryogenic range [3]. Nevertheless, BWRS would still suffice for noncryogenic CO2 duties.

In the absence of REFPROP, the Multiparameter EOS such as GERG could also be modelled. GERG is

represented in the Helmholtz Free Energy form in terms of reduced density and inverse reduced

temperature [6]:

( ) ( ) ( )

Comparison of Z & k between different EOS

For a given pressure, temperature and gas mixture, different EOS will yield different values of Z and

k. The goal would be to tabulate and understand the differences between each EOS for various gas

compositions. One way to demonstrate the differences is by plotting graphs of Z and k versus

pressure (at a specific temperature) for different EOS. This plot shows how the different EOS varies

with each other over the range of pressures. Note that we can also choose to plot Z versus

temperature for specific pressures instead, however for convention sake plots of Z versus pressure

will be used (i.e. in the standard form of Nelson-Obert compressibility charts). [16]

By investigating the different EOS, if results are very similar for specific EOS for certain common

gasses then the results can be interchangeable for future comparisons. In general, at lower pressure

and higher temperature it is expected that the different EOS should corroborate better among each

other as the conditions are approaching that of an Ideal Gas, thus yielding a more accurate and

predicable model.

To ensure accurate and consistent results, the operating ranges for a given gas mixture are selected

to ensure the fluid is in a pure gas state, and not as a multi-phase, liquid-phase fluid or near the

critical point. This is verified by plotting a Phase Map using REFPROP NIST (i.e. vapor-liquid

equilibrium curves) and ensuring the operating points chosen are in the gas phase region. Operating

points are selected based on typical compressor applications of each gas, such as refrigerant

compressors and high-pressure CO2 reinjection duties.

As stated previously, NIST’s EOS from the REFPROP software is widely recognized as a reliable EOS

and thus will be used as the reference for a comparison datum for other EOS. REFPROP software is

used to calculate Z and k for NIST, GERG, AGA8 and PR EOS. RK, SRK, BWRS-NS and LKP are not

available in REFPROP and thus are compiled individually by the authors. Deviations will be quoted in

terms of % deviation from NIST. Since NIST is an updated form of the GERG (2004) EOS, it is expected

that the NIST and GERG2004 will have negligible deviation.

The different EOS are utilized for the calculation of Z and k for Mixed Refrigerant Gas, Pure CO2 gas

and a CO2 gas mixture. For reference, Pure Methane and a Natural Gas Mixture is also investigated as

the properties of methane are well-established.

Note that for mixture comparisons, RK is not used due to lack of interaction parameters on-hand.

Instead, AGA8 is used. AGA8 is not used for pure fluids because in REFPROP, AGA8 is only used for

mixtures and will revert back to NIST EOS when calculating pure fluids.

Compressibility and Cp/Cv vs Pressure Graphs

Figure 1: Compressibility Z vs Pressure for Methane gas at T=210K and T=300K

Figure 1 illustrates a typical compressibility graph of Z vs P for pure Methane using the various EOS.

On the left side of the graph close to P=0 bara, it can be seen that all the EOS converges to 1. This

signifies the point where the gas behaves closest to an ideal gas; the lower the P, the less collisions

and force interactions between the gas molecules. As P increases, gas intermolecular forces become

prevalent. This causes the gas molecules to occupy a denser space (Z reduces) than predicted by the

ideal gas model. Each EOS notably branches off from each other; the different EOS models have their

own set of parameters to estimate Z with varying degrees of accuracy. As P is increased even further,

Z slowly increases due to the physical size of the molecules (Ideal gas model neglects gas molecule

size).

At lower temperatures, the gas molecules’ kinetic energy is low enough that the interaction forces

between molecules are prevalent. Thus, there is a huge variation in Z as P increases. As T is increased

however, the kinetic energy of the gas molecules renders the interaction forces to be less significant

which approaches the ideal gas model. Thus the curve starts to approach a flatter, straight line closer

to Z=1 throughout the range of P. This trend is visible in Figure 1 by comparing the two curves at

T=210K and T=300K.

The Critical Point of Methane is at P=46bara and T=190.6k. Thus, data to the right of the critical P line

in Figure 1 in this case are within the Supercritical Region.

Figure 2: Specific Heat Ratio (k=Cp/Cv) vs Pressure for Methane gas at T=210K and T=300K

Similarly, the Specific Heat Ratios, k can be plotted versus P, as seen in Figure 2. As the ratio k=Cp/Cv

and Cp>Cv in all cases, the graph for k is always above k=1. A notable feature is that near the critical

point, the EOS spikes to infinity yielding erroneous results. This issue is not apparent on the Z graph,

thus it is possible to obtain operating points near the critical point where values of Z appear sensible

while k becomes overly sensitive. The curve slowly flattens out as Temperature is increased beyond

Tc. Deviations of k also increase when P increases as the gas deviates from the Ideal Gas model.

It is not practical to present the entire range of data on this paper as this requires a 3D graph to

effectively plot Z for various P and T; even then, it will be difficult to compare multiple 3d surfaces

representing each EOS. Therefore, selected operating points applicable for the gas examined will be

used to compare Z and k to evaluate how the EOS differ from each other.

Pure Methane

Methane is the simplest alkane molecule and the main constituent in natural gas, serving as a

reference to establish the comparisons between EOS. Selected points are chosen rather than

presenting all the data on a graph here as it is impractical to overlay all the data of the various EOS.

Operating Point 1 represents boil-off gas conditions. Operating points 2-7 represent typical values of

a natural gas compressor.

Operating Point 1 2 3 4 5 6 7

Pressure, bar a 1.01 13.82 24.41 24.6 105.66 56.42 128.94

Temp, K 115 320 320 350 350 380 380

Table 2: Selected Operating Points for Pure Methane

Note that the critical point of methane is at 46bara, 191K. None of the operating points are selected

near this value.

-0.5

0

0.5

1

1.5

2

1 2 3 4 5 6 7

%D

evi

atio

n Z

Operating Points

Deviation of Z for Pure Methane

Z GERG

Z LKP

Z BWRSNS

Z SRK

Z PR

Z RK

Figure 3 & 4: Deviation of Z and k for Pure Methane

Z: With reference to NIST, it can be seen that for Z most of the EOS agree with each other

within these ranges; exhibiting deviations <0.5% except for SRK at higher P and T. LKP appears to be

the most consistent throughout the range with the lowest deviations. SRK on the other hand appears

to deviate more as P and T increases by up to nearly 2%. What appears promising is that even at low

T of 115K (i.e. near boil-off gas conditions), the EOS all fall within 0.4%< deviation.

k: Specific heat ratios (k) on the other hand do not follow the same trend even with the same

operating conditions. It is apparent that SRK has generally the least deviations this time, however the

trend indicates that k for SRK gradually shifts to negative deviation as P increase. For BWRS-NS the

opposite is observed; k gradually shifts to positive deviation as P increase. LKP and PR demonstrates

stable deviations throughout the range. Again, at 115K, the EOS are within 0.5%< deviation.

Hence for Z, all the EOS throughout the range (even for low T) fall within +- 0.5% deviation except for

SRK at higher P (>50bara). For k, all the EOS throughout the range (even for low T) fall within +- 0.5%

deviation except for RK at higher P (>50bara).

-2

-1.5

-1

-0.5

0

0.5

1

1 2 3 4 5 6 7

%D

evi

atio

n k

Operating Points

Deviation of k for Pure Methane

k GERG

k LKP

k BWRSNS

k SRK

k PR

k RK

Natural Gas Mixture

The difficulty of modeling mixtures arises from the interactions between the different components.

Thus, more reliable EOS consider the interactions by using binary interaction parameters. By

considering the effects between each pair of compounds in a mixture and taking an average, a more

accurate result for the calculation of EOS parameters is obtained. Hence, it is expected that the EOS

will result in higher deviations for mixtures than pure components.

With the analysis of EOS for pure Methane, it can be predicted that a natural gas composition will

exhibit similar trends for Z and k. A sample typical natural gas mixture consisting of 82% methane is

analysed using similar operating points to pure methane gas. The exact composition can be found in

Appendix A.

Operating Point 1 2 3 4 5 6

Pressure, bar a 10.91 21.42 24.3423 102.8026 55.93 126.52

Temp, K 320 320 350 350 380 380

Table 3: Selected Operating Points for Natural Gas Mixture

-1

-0.5

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6

%D

evi

atio

n Z

Operating Points

Deviation of Z for Natural Gas

Z GERG

Z LKP

Z BWRSNS

Z SRK

Z PR

Z AGA8

Figure 5 & 6: Deviation of Z and k for Natural Gas Mixture

As predicted, the results of the graphs (Figures 1 & 3, 2 & 4) for natural gas are similar to the pure

Methane gas graph except with slightly more deviations.

Z: For Natural Gas data, it is observed that LKP gives stable results; with less than 0.5%

deviation for each data. However, the results show that AGA8 show the least deviation throughout

the range; demonstrating great correlation with NIST for natural gas mixtures. As predicted, SRK

again exhibits large deviations up to 2% at high P and T. PR has low deviations in this range but may

overshoot at higher P and T; same goes for BWRS-NS.

k: AGA8 represents the closest EOS to NIST but deviates more significantly at the higher

P>100bara (around 0.5%). SRK still models k well comparatively (around 0.25% at most) despite poor

comparison with Z, however k continues to deviate more negatively as P and T increase. PR basically

demonstrates to be a worse SRK in this mixture. LKP remains fairly consistent with deviations

throughout the range around 0.25%.

Overall, it can be concluded that for the ranges above, AGA8 resembles closest to NIST for this

methane-predominant mixture. Otherwise, LKP is also a strong contender for Z, and LKP/SRK for k.

However within the P and T ranges analysed above, most EOS do agree well with each other as

deviations are at most 1%. Therefore for a natural gas mixture with similar composition to the above,

any of the above EOS can be used and a deviation of not more than 1% for Z and k can be expected

(except for RK and SRK at higher P).

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6

%D

evi

atio

n k

Operating Points

Deviation of k for Natural Gas

k GERG

k LKP

k BWRSNS

k SRK

k PR

k AGA8

Mixed Refrigerant Gas

Mixed refrigerant gas consists of a wider range of heavier hydrocarbons compared to a typical

natural gas mixture and therefore has a significantly heavier molecular weight. Since the longer chain

hydrocarbon molecules come into the picture, their size and interaction are important to consider. A

sample gas consisting primarily of Methane, Ethylene and Butane is investigated for the following

operating conditions:

Operating Point 1 2 3 4 5 6

Pressure, bar a 3.35 16.73 16.73 43.48 43.48 56.86

Temp, K 300 310 390 360 400 400

Table 4: Selected Operating Points for Mixed Refrigerant Gas

Note that the critical point of this mixture is at 103bara, 334K. The operating P typically do not

exceed the critical P. GERG2004 does not contain parameters for ethylene and propylene, therefore

is excluded from the analysis of this mixed refrigerant composition.

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1 2 3 4 5 6

%D

evi

atio

n Z

Operating Points

Deviation of Z for Mixed Refrigerant Gas

Z LKP

Z BWRSNS

Z SRK

Z PR

Z AGA8

Figure 7 & 8: Deviation of Z and k for Mixed Refrigerant Gas

Z: Compared to the primarily Methane mixture, some larger deviations can be observed. PR

and BWRS-NS appears to be the more stable EOS; with a deviations below 1%. The other EOS

demonstrates inconsistent trends even at lower P; AGA8 and LKP exceed 1% deviation at higher P to

T ratios.

K: Again, trends appear inconsistent especially for AGA8. The most stable EOS appears to be

BWRS-NS, however there is excellent correlation between EOS at Operating Points 1 and 3 (i.e. at

relatively lower P values with sufficient T).

Overall, BWRS-NS seems like a safer option for Z and k to compare with NIST for this heavier

hydrocarbon mixture especially at higher pressure and temperature conditions. However, PR

performs relatively well too for computing Z. The other EOS are expected to deviate at least 1% at

higher P.

-1.5

-1

-0.5

0

0.5

1

1.5

1 2 3 4 5 6

%D

evi

atio

n k

Operating Points

Deviation of k for Mixed Refrigerant Gas

k LKP

k BWRSNS

k SRK

k PR

k AGA8

Pure Carbon Dioxide

Carbon Dioxide (CO2) is more difficult to model accurately, as its properties are far from an ideal gas.

As it is a linear molecule, it has a high acentric factor (i.e. highly non-spherical). This affects the

interaction between the molecules in the gas, yielding inaccurate values of Z and k if not taken into

account. CO2 is of interest in oil & gas for CO2 reinjection, thus high pressure ranges are investigated

for comparison.

Operating Point 1 2 3 4 5 6 7* 8 9 10

Pressure, bar a 32 48.1 54.1 62.9 79.6 95.6 100.6 416.8 481 520

Temp, K 310 350 320 370 400 420 320 400 400 430

Table 5: Selected Operating Points for Pure Carbon Dioxide

*The critical point of CO2 is at 73.77bara, 304.1K. Operating point 7 is reasonably close and therefore

may result in anomalous results.

Figure 9 & 10: Deviation of Z and k for Pure CO2 Gas

-4

-2

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10

%D

evi

atio

n Z

Operating Points

Deviation of Z for Pure CO2

Z GERG

Z LKP

Z BWRSNS

Z SRK

Z PR

Z RK

-20

-15

-10

-5

0

5

10

1 2 3 4 5 6 7 8 9 10

%D

evi

atio

n k

Operating Points

Deviation of k for Pure CO2

k GERG

k LKP

k BWRSNS

k SRK

k PR

k RK

Z: Below the critical point, LKP and BWRS-NS have the lowest deviations to NIST (<1%). At

P=100.57bara and T=320K which is close to the critical point of CO2, most EOS have massive

deviations with NIST. At this point, only LKP demonstrates astonishingly high correlation with NIST,

while the other EOS deviates by at least 7%. RK performs surprisingly well throughout these

conditions, deviating at most 2% throughout the range. RK does not take acentric factor into account

when calculating Z and k, thus for CO2 it was expected that RK will have large deviations from NIST.

Beyond the critical point - at supercritical conditions (Operating Points 8, 9, 10), some EOS exhibits

larger inconsistencies to NIST; with SRK having up to 8% deviations. In general however, LKP and PR

shows the most consistent deviation with NIST at around 1% even in supercritical regions.

k: Massive deviations can be seen throughout the range, with average deviations at least 2%

among the EOS. This is because k is more sensitive than Z - especially near the critical point; k

theoretically shoots to infinity while Z is not affected. Below the critical point, BWRS-NS and

surprisingly RK demonstrate excellent correlation with NIST. In the supercritical region, each EOS

deviates by large amounts with each other and thus, it is unsafe to draw a general conclusion as to

the validity of the EOS models.

Thus, for Z it is generally safe to use LKP as the EOS with the lowest deviation to NIST. PR may be

used for Z in the supercritical region (about 1% Deviation). However for k, it is advised there will be

deviations between EOS of at least 2% in the supercritical region. Otherwise, below the critical point

BWRS-NS and RK does comparatively well (<1.5% Deviation).

CO2 Gas mixture

For CO2 reinjection purposes, a 95% CO2 gas mixture is analysed. Again, similar operating points are

selected inline with the pure CO2 gas for comparison purpose; It is expected that the trends of both

graphs should be similar.

Operating Point 1 2 3 4 5 6 7** 8 9 10

Pressure, bar a 30.99 46.29 52.84 60.36 75.9 90.75 100.85 415.61 475.1 536.18

Temp, K 310 350 320 370 400 420 320 400 400 430

Table 6: Selected Operating Points for CO2 Gas Mixture

**The critical point of CO2 is at 73.77bara, 304.1K. Since this gas mixture consists of 95% CO2, the

mixture’s critical point is expected to be very similar to pure CO2. Operating point 7 is reasonably close

and therefore may result in anomalous results.

Figure 11 & 12: Deviation of Z and k for CO2 Gas Mixture

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10

%D

evi

atio

n Z

Operating Points

Deviation of Z for CO2 Gas Mixture

Z GERG

Z LKP

Z BWRSNS

Z SRK

Z PR

Z AGA8

-10

-8

-6

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10

%D

evi

atio

n k

Operating Points

Deviation of k for CO2 Gas Mixture

k GERG

k LKP

k BWRSNS

k SRK

k PR

k AGA8

Z: Similar to the pure CO2 graph, LKP demonstrates consistently low deviations throughout the

range. However for mixtures, the correlation between AGA8 and NIST is unrivalled; only deviating

notably near the critical point. At P higher than Tc, LKP and PR compares reasonably well with NIST at

1% deviation but AGA8 correlates much better.

k: Similar to pure CO2, BWRS-NS correlates well with NIST below the critical point (<1%

Deviation). Near the critical point, all of the EOS examined tend to deviate significantly; PR, SRK and

BWRS-NS deviates around 8-10%. In the supercritical region, again it is not possible to establish with

confidence the validity of the results due to the large deviations between EOS.

A similar conclusion for CO2 mixture can be drawn; LKP is fairly reliable for Z across the range, and

BWRS-NS correlates well with NIST below the critical point for k. However, AGA8 demonstrates the

best comparison with NIST for both Z and k for CO2.

Conclusion

The selection of a reliable EOS ensures more accurate calculation of a compressor’s power and

discharge temperature. By establishing NIST as a datum, results of Z and k of different EOS for Mixed

Refrigerant and CO2 duties were compared.

For predominantly methane based mixtures, most EOS agree well with each other as the properties

of methane are well established (0.5% average deviation for Z and k). However, heavier hydrocarbon

mixtures such as Mixed Refrigerants and CO2 gas demonstrate larger deviations among EOS. The

results are summarised in Table 7. The following EOS are therefore recommended (with some

caution):

Mixture Recommended EOS Remarks (% Deviation with respect to NIST EOS) Z k

Mixed Refrigerants BWRS-NS, PR BWRS-NS <0.5% for Z and k (BWRS-NS), <1% for Z (PR)

Pure CO2 (gas) LKP/BWRS-NS

BWRS-NS <1% for Z and <0.5% for k

Pure CO2 (supercritical)

LKP/PR - 1% for Z, k inconclusive

CO2 Gas Mixtures (gas)

AGA8 AGA8 <0.2% for Z and <1% for k

LKP/BWRS-NS

BWRS-NS <1% for Z and k

CO2 Gas Mixtures (supercritical)

LKP/PR - 1% for Z, k inconclusive

Table 7: Summary of EOS comparisons with NIST EOS

The findings are generally in agreement with those of authorities such as Sandberg [17] and Lüdtke

[18], where BWRS and LKP are already shown to be reliable EOS models.

This also emphasizes the need for vendors to justify their EOS selection when providing quotes to

consultant engineers - especially for mixed refrigerants and CO2. Typically, datasheets of compressors

provided by vendors do not clarify the EOS used in calculation of the Z and k values. Therefore, it is

not possible to verify the values as different EOS will have deviations between each other.

Clarification will ensure consistency between both parties, and consequently better confidence to

the operator/end-user of the compressor.

References

[1] Marc J. Assael, J. P. M. Trusler, Thomas F. Tsolakis (1996) Thermophysical Properties of Fluids, Imperial College Press

[2] Heinz P. Bloch (2006) A Practical Guide to Compressor Technology 2nd Edition, John Wiley & Sons, Inc.

[3] Nishiumi H., Saito S. (1975) An Improved Generalized BWR Equation of State Applicable to Low Reduced Temperature, Journal of Chemical Engineering of Japan

[4] Nishiumi H., Saito S. (1977) Correlation of the Binary Interaction Parameter of The Modified Generalized BWR Equation of State, Journal of Chemical Engineering of Japan

[5] ISO 12213-2 (2006) Natural Gas-Calculation of compression factor (Part 2: Calculation using molar composition analysis)

[6] Kunz O., Klimeck R., Wagner W., Jaeschke M. (2007) The GERG-2004 Wide-Range Equation of State for Natural Gases and Other Mixtures, Lehrstuhl für Thermodynamik Ruhr-Universität Bochum Germany

[7] Lemmon, E.W., Huber, M.L., McLinden, M.O. (2013) NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg

[8] Nimtza M., Klatta M., Joachim H. Krautza (2011) Evaluation of the GERG-2008 Equation of State for the Simulation of Oxyfuel Systems, 2nd Oxyfuel Combustion Conference

[9] Raimondi L. (2010) Rigorous calculation of LNG flow reliefs using the GERG-2004

equation of state, 4th International Conference on Safety & Environment in Process

Industry

[10] Yildiz T. (1996) Analytical Gas Pipeline Design Method Using The GERG Equation of

State, European Petroleum Conference 22-24 October, Milan, Italy

[11] Mark R. Sandberg, Gary M. Colby (2014) Limitations of ASME PTC 10 in Accurately

Evaluating Centrifugal Compressor Thermodynamic Performance, 42nd Turbomachinery

Symposium

[12] Aicher W. (1993) Test of Process Turbocompressors Without CFC Gases, 22nd Turbomachinery Symposium

[13] Moore J., Lerche A., Delgado H., Allison T., Pacheco J. (2011) Development of Advanced Centrifugal Compressors and Pumps for Carbon Capture and Sequestration Applications, 40th Turbomachinery Symposium

[14] F-Chart Software (2014) Engineering Equation Solver REFPROP Interface, <http://www.fchart.com/ees/ees-refprop.php>

[15] AspenTech (2014) Aspen Properties®, <http://www.aspentech.com/products/aspen-properties.aspx>

[16] Yunus A. Cengel, Michael A. Boles (2005) Thermodynamics: An Engineering Approach 5th Edition, McGraw-Hill College, Boston, MA

[17] Mark R. Sandberg (2005) Equation of State Influences on Compressor Performance Determination, 34th Turbomachinery Symposium

[18] H. K. Lüdtke (2004) Process Centrifugal Compressor: Basics, Function, Operation, Design, Application, 1st Edition, Springer

Appendix A – Gas Mixture Compositions

Gas composition Natural Gas Mixture Mixed Refrigerant CO2 Gas Mixture

Mole fraction

Methane 0.829700 0.257800 0.041018

Nitrogen 0.060810 0.079780 0.001157

Carbon dioxide 0.029040 0.000000 0.947150

Ethane 0.042890 0.002530 0.005496

Propane 0.017600 0.016730 0.002540

n-Butane 0.005328 0.047480 0.000393

i-Butane 0.003198 0.215113 0.000558

n-Pentane 0.001838 0.000078 0.000131

i-Pentane 0.001929 0.001319 0.000210

n-Hexane 0.001500 0.000000 0.000245

n-Heptane 0.004157 0.000000 0.001103

Water 0.002010 0.000000 0.000000

Ethylene 0.000000 0.378900 0.000000

Propylene 0.000000 0.000270 0.000000

Appendix B – Tabulated values of Z and k of the gas compositions for various

EOS

NIST, GERG, AGA8 and PR EOS calculations are done via REFPROP software. RK, SRK,

LKP and BWRS-NS calculations are compiled by the authors independantly.

Pure Methane

Pressure, bar a Temp, K Z NIST Z GERG Z LKP Z BWRS-NS Z SRK Z PR Z RK

1.01 115 0.9671 0.9674 0.9667 0.9685 0.9704 0.9699 0.9691

%Deviation to NIST 0.00 0.03 -0.05 0.14 0.34 0.29 0.20

13.82 320 0.9820 0.9820 0.9826 0.9806 0.9830 0.9790 0.9808

%Deviation to NIST 0.00 0.00 0.06 -0.14 0.10 -0.30 -0.12

24.41 320 0.9687 0.9687 0.9697 0.9665 0.9706 0.9639 0.9667


24.59628 350 0.9786 0.9786 0.9793 0.9763 0.9809 0.9750 0.9766


105.6605 350 0.9280 0.9280 0.9315 0.9306 0.9429 0.9260 0.9248

%Deviation to NIST 0.00 0.00 0.38 0.28 1.61 -0.21 -0.34

56.42 380 0.9706 0.9706 0.9718 0.9675 0.9772 0.9667 0.9668


128.94 380 0.9525 0.9525 0.9563 0.9583 0.9700 0.9532 0.9479

%Deviation to NIST 0.00 0.01 0.40 0.61 1.84 0.07 -0.48

Pure Methane

Pressure, bar a Temp, K k NIST k GERG k LKP k BWRS-NS k SRK k PR k RK

1.01 115 1.3693 1.3697 1.3760 1.3733 1.3680 1.3627 1.3709

%Deviation to NIST 0.00 0.03 0.49 0.29 -0.10 -0.48 0.12

13.82 320 1.3250 1.3250 1.3221 1.3222 1.3266 1.3293 1.3229

%Deviation to NIST 0.00 0.00 -0.21 -0.21 0.12 0.32 -0.16

24.41 320 1.3504 1.3503 1.3464 1.3469 1.3539 1.3579 1.3472

%Deviation to NIST 0.00 -0.01 -0.30 -0.26 0.26 0.56 -0.24

24.60 350 1.3232 1.3231 1.3197 1.3206 1.3252 1.3288 1.3201


105.66 350 1.4822 1.4818 1.4761 1.4844 1.4803 1.4905 1.4597

%Deviation to NIST 0.00 -0.02 -0.41 0.15 -0.13 0.56 -1.52

56.42 380 1.3445 1.3443 1.3401 1.3433 1.3465 1.3521 1.3380


128.94 380 1.4462 1.4456 1.4412 1.4502 1.4398 1.4490 1.4223

%Deviation to NIST 0.00 -0.04 -0.35 0.28 -0.45 0.19 -1.66

Natural Gas Mixture

Pressure, bar a Temp, K Z NIST Z GERG Z LKP Z BWRS-NS Z SRK Z PR Z AGA8

10.91 320 0.9811 0.9811 0.9819 0.9802 0.9817 0.9781 0.9811

%Deviation to NIST 0.00 0.00 0.08 -0.09 0.06 -0.31 0.00

21.42 320 0.9632 0.9632 0.9647 0.9617 0.9647 0.9578 0.9634


24.3423 350 0.9711 0.9711 0.9722 0.9693 0.9734 0.9665 0.9713


102.8026 350 0.9003 0.9004 0.9032 0.9049 0.9164 0.8972 0.9025

%Deviation to NIST 0.00 0.01 0.32 0.51 1.78 -0.35 0.24

55.93 380 0.9579 0.9579 0.9596 0.9558 0.9650 0.9528 0.9585


126.52 380 0.9286 0.9287 0.9315 0.9365 0.9480 0.9285 0.9305

%Deviation to NIST 0.00 0.01 0.32 0.85 2.09 -0.01 0.21

Natural Gas Mixture

Pressure, bar a Temp, K k NIST k GERG k LKP k BWRS-NS k SRK k PR k AGA8

10.91 320 1.2921 1.2921 1.2898 1.2899 1.2938 1.2958 1.2921

%Deviation to NIST 0.00 -0.01 -0.18 -0.18 0.13 0.28 0.00

21.42 320 1.3206 1.3205 1.3168 1.3172 1.3247 1.3279 1.3204


24.3423 350 1.2991 1.2990 1.2958 1.2964 1.3023 1.3053 1.2986


102.8026 350 1.4820 1.4817 1.4783 1.4798 1.4840 1.4923 1.4743


55.93 380 1.3258 1.3256 1.3215 1.3235 1.3296 1.3345 1.3241


126.52 380 1.4414 1.4410 1.4387 1.4411 1.4374 1.4448 1.4342

%Deviation to NIST 0.00 -0.03 -0.19 -0.02 -0.27 0.24 -0.50

Mixed Refrigerant

Pressure, bar a Temp, K Z NIST Z LKP Z BWRS-NS Z SRK Z PR Z AGA8

3.35 300 0.9762 0.9781 0.9754 0.9765 0.9747 0.9722

%Deviation to NIST 0.00 0.19 -0.08 0.03 -0.16 -0.42

16.73 310 0.8870 0.8963 0.8835 0.8890 0.8808 0.8664


16.73 390 0.9504 0.9552 0.9487 0.9526 0.9457 0.9440


43.48 360 0.8225 0.8371 0.8180 0.8302 0.8149 0.7985


43.48 400 0.8838 0.8945 0.8806 0.8919 0.8773 0.8705


56.86 400 0.8498 0.8630 0.8470 0.8621 0.8450 0.8346


Mixed Refrigerant

Pressure, bar a Temp, K k NIST k LKP k BWRS-NS k SRK k PR k AGA8

3.35 300 1.1959 1.1943 1.1959 1.1955 1.1960 1.1859

%Deviation to NIST 0.00 -0.13 0.00 -0.03 0.01 -0.84

16.73 310 1.2778 1.2692 1.2798 1.2799 1.2796 1.2908

%Deviation to NIST 0.00 -0.67 0.16 0.17 0.14 1.02

16.73 390 1.1823 1.1801 1.1827 1.1859 1.1854 1.1750

%Deviation to NIST 0.00 -0.18 0.03 0.30 0.26 -0.62

43.48 360 1.3471 1.3331 1.3517 1.3606 1.3577 1.3659

%Deviation to NIST 0.00 -1.04 0.34 1.00 0.78 1.40

43.48 400 1.2513 1.2446 1.2530 1.2598 1.2589 1.2496


56.86 400 1.2989 1.2901 1.3023 1.3096 1.3074 1.2988


Pure Carbon Dioxide


31.96 310 0.8448 0.8448 0.8406 0.8418 0.8469 0.8329 0.8454

%Deviation to NIST 0.00 0.00 -0.49 -0.35 0.25 -1.41 0.07

48.07 350 0.8503 0.8503 0.8487 0.8476 0.8554 0.8377 0.8418

%Deviation to NIST 0.00 0.00 -0.18 -0.31 0.60 -1.48 -1.00

54.05 320 0.7487 0.7487 0.7418 0.7437 0.7542 0.7329 0.7459


62.86 370 0.8415 0.8415 0.8413 0.8390 0.8508 0.8300 0.8280


79.61 400 0.8551 0.8550 0.8575 0.8535 0.8702 0.8473 0.8366

%Deviation to NIST 0.00 -0.01 0.28 -0.19 1.77 -0.90 -2.16

95.62 420 0.8605 0.8603 0.8647 0.8598 0.8808 0.8561 0.8386

%Deviation to NIST 0 -0.01 0.49 -0.07 2.37 -0.51 -2.54

100.57 320 0.3642 0.3644 0.3638 0.3924 0.4137 0.3910 0.3730

%Deviation to NIST 0.00 0.05 -0.11 7.73 13.58 7.36 2.40

416.84 400 0.8037 0.8036 0.8119 0.8301 0.8757 0.8210 0.8152

%Deviation to NIST 0.00 -0.01 1.03 3.30 8.97 2.16 1.44

480.90 400 0.8678 0.8677 0.8766 0.8910 0.9366 0.8763 0.8804


520.02 430 0.9357 0.9355 0.9471 0.9671 1.0063 0.9454 0.9334

%Deviation to NIST 0.00 -0.02 1.21 3.36 7.55 1.03 -0.25

Pure Carbon Dioxide

Pressure, bar a Temp, K k NIST k GERG k LKP k BWRS-NS k SRK k PR k RK

31.96 310 1.5324 1.5327 1.5608 1.5408 1.5688 1.5702 1.5105


48.07 350 1.5125 1.5123 1.5374 1.5196 1.5551 1.5586 1.5056


54.05 320 1.8137 1.8142 1.8838 1.8396 1.9101 1.9177 1.7882


62.86 370 1.5344 1.5341 1.5611 1.5421 1.5801 1.5856 1.5330


79.61 400 1.5093 1.5089 1.5321 1.5146 1.5451 1.5525 1.5111


95.62 420 1.5059 1.5055 1.5278 1.5102 1.5347 1.5434 1.5082


100.57 320 7.1967 7.1328 6.9563 5.8332 7.4018 7.4032 5.8767


416.84 400 1.9801 1.9762 2.0692 2.1368 1.8982 1.9317 1.6844

%Deviation to NIST 0.00 -0.20 4.50 7.91 -4.14 -2.45 -14.93

480.90 400 1.9129 1.9087 2.0017 2.0962 1.8230 1.8595 1.6220


520.02 430 1.8221 1.8178 1.9000 1.9570 1.7314 1.7594 1.5766


CO2 Mixture


30.99 310 0.8551 0.8551 0.8529 0.8524 0.8575 0.8439 0.8556

%Deviation to NIST 0 0.00 -0.26 -0.31 0.29 -1.30 0.06

46.29 350 0.8613 0.8613 0.8613 0.8589 0.8666 0.8496 0.8617

%Deviation to NIST 0 0.00 0.01 -0.27 0.62 -1.35 0.05

52.84 320 0.7655 0.7654 0.7619 0.7612 0.7716 0.7508 0.7665

%Deviation to NIST 0 0.00 -0.47 -0.56 0.80 -1.91 0.13

60.36 370 0.8538 0.8537 0.8551 0.8515 0.8629 0.8431 0.8542

%Deviation to NIST 0 0.00 0.16 -0.26 1.08 -1.25 0.06

75.90 400 0.8673 0.8672 0.8708 0.8659 0.8817 0.8600 0.8680

%Deviation to NIST 0 -0.01 0.41 -0.16 1.66 -0.84 0.08

90.75 420 0.8728 0.8727 0.8780 0.8723 0.8919 0.8684 0.8738

%Deviation to NIST 0 -0.01 0.59 -0.06 2.19 -0.50 0.11

100.85 320 0.4357 0.4357 0.4281 0.4510 0.4743 0.4505 0.4434

%Deviation to NIST 0 -0.01 -1.74 3.52 8.87 3.40 1.77

415.61 400 0.8243 0.8242 0.8325 0.8533 0.8948 0.8419 0.8243

%Deviation to NIST 0 -0.01 1.00 3.52 8.55 2.13 0.01

475.10 400 0.8838 0.8837 0.8922 0.9098 0.9513 0.8934 0.8829

%Deviation to NIST 0 -0.01 0.95 2.94 7.64 1.09 -0.10

536.18 430 0.9712 0.9709 0.9827 1.0044 1.0388 0.9787 0.9703

%Deviation to NIST 0 -0.03 1.18 3.42 6.97 0.78 -0.09

CO2 Mixture

Pressure, bar a Temp, K k NIST k GERG k LKP k BWRS-NS k SRK k PR k AGA8

30.99 310 1.5054 1.5056 1.5263 1.5119 1.5362 1.5377 1.5052

%Deviation to NIST 0 0.01 1.39 0.44 2.05 2.15 -0.01

46.29 350 1.4840 1.4838 1.5028 1.4894 1.5200 1.5233 1.4856


52.84 320 1.7469 1.7471 1.7961 1.7661 1.8252 1.8319 1.7463

%Deviation to NIST 0 0.01 2.82 1.10 4.48 4.87 -0.03

60.36 370 1.5012 1.5010 1.5214 1.5069 1.5397 1.5448 1.5043


75.90 400 1.4764 1.4761 1.4938 1.4801 1.5068 1.5135 1.4813


90.75 420 1.4718 1.4714 1.4885 1.4746 1.4964 1.5043 1.4782


100.85 320 5.0262 5.0257 5.2633 4.5261 5.4992 5.4315 4.7783

%Deviation to NIST 0 -0.01 4.72 -9.95 9.41 8.06 -4.93

415.61 400 1.9311 1.9274 2.0184 2.0756 1.8556 1.8852 1.9485

%Deviation to NIST 0 -0.19 4.52 7.48 -3.91 -2.37 0.90

475.10 400 1.8743 1.8701 1.9616 2.0466 1.7921 1.8248 1.8853


536.18 430 1.7721 1.7681 1.8483 1.9062 1.6873 1.7131 1.7843


Equations For Heavy Gases In Centrifugals

Engineering

gas state

co2 gas

gas compressibility

ideal gas calculation

ideal gas equation

various eos

ideal gas law

various gas mixtures