Evaluation of the Performance of the Equations of State Available in
CFTurbo for Air
VASILEIOS MOUTSIOS, DIONISSIOS MARGARIS, NICHOLAS PITTAS
Department of Mechanical Engineering and Aeronautics,
University of Patras,
University Campus Rio-Patras 26504,
GREECE
Abstract: - In this publication, we evaluate the performance of equations of state Perfect Gas, Redlich-Kwong,
Aungier/ Redlich-Kwong, Soave/ Redlich-Kwong, Peng-Robinson, and Cool Prop in calculating the properties
of air at temperatures between 250 K to 800_K and pressure between 1 bar to 250 bar. Firstly, we research the
available literature to find the real density (or molar volume or specific volume) at each point. Then we create
these equations and use the temperature and density (or molar volume or specific volume) to calculate the
pressure, which is then compared to the pressure of the real values. This comparison evaluates the performance
of each equation. Finally, we compare our calculations with online calculators to verify our results and
comment on the performance of each equation.
Key-Words: - Thermodynamics, Air properties, Equation of State, Perfect Gas, Redlich-Kwong, Aungier/
Redlich-Kwong, Soave/ Redlich-Kwong, Peng-Robinson, Cool Prop.
Received: May 9, 2023. Revised: November 14, 2023. Accepted: December 13, 2023. Published: December 31, 2023.
1 Introduction
During the lifetime of a fluid dynamics analyst, they
would face many times the challenge of choosing a
gas equation of state for their analysis. There are
indeed a lot of equations of state and new ones are
developed. Each one has its one advantages and
disadvantages.
We also found ourselves in the same position.
Specifically, we were designing a high-pressure air
compressor using the turbomachinery designing
software CFTurbo. CFTurbo gives the option of
using the following six equations of state to
calculate air properties: Perfect Gas, Redlich-
Kwong, Aungier/ Redlich-Kwong, Soave/ Redlich-
Kwong, Peng-Robinson, and Cool Prop (we must
note that Cool Prop is a library with several
equations). The question is which of the above
equations of state is more suitable to our case as
each equation may perform better under different
conditions.
Unfortunately, we searched the available
literature, and we did not find any publication that
would cover our needs. What we need is studies
showing the performance of the above models for
air from low to high temperature and pressure. What
we found is studies using the above equations to
evaluate unrelated to our interest phenomena and
properties, such as vapor-liquid equilibrium, [1],
throttle reduction efficiency, [2], efficiency of air-
conditioning, [3], evaluation using different acentric
factors, [4], combustion gases properties, [5]. Even
academic and specialized literature has limited
information about these equations. We found
contemporary literature just mentioning some of
those equations, [6], and literature presenting only
theoretical background and focusing on more
advanced topics, [7], [8]. So, we concluded that
there is no available literature, at least in public, that
provides this information. This is strange as there
are high-pressure systems using air and such studies
should have been performed. For these reasons, we
decided to perform the analysis presented in this
article and give public information about the
performance of these equations in predicting air
properties.
In this publication, we study which of the above
equations calculates better air properties (78.08%
Ν2, 20.95% Ο2, 0.93% Ar, and 0.04% CO2) in the
range of temperature values between 250 K to
800_K and the range of pressure values between 1
bar to 250 bar.
2 Problem Formulation
To determine which is the best equation of state to
calculate air properties in the above range we
searched to find the real properties of air in these
conditions. This was achieved by finding the
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1284
Volume 19, 2023
compressibility factor, Z, of air in the ranges. As
you will see in the following paragraphs, most of the
equations of state are solved to pressure and are
complex to be solved to other variables. So, to avoid
round-up and iteration errors we calculate the
pressure in the specified ranges using the
corresponding density (or molar volume or specific
volume) and temperature, and we compare the
calculated pressure value with the real pressure
value corresponding to the used density (or molar
volume or specific volume) and temperature.
2.1 Compressibility Factor of Air
The process of finding the compressibility factor of
air was challenging as the required range goes far
beyond the needs of conventional applications.
Also, we must note that after extensive research we
did not find contemporary literature providing the
compressibility factor of air in the requested ranges.
So, our only option is to base our analysis on older
literature that provides the wanted data.
We found in total four sources that agree and
one with a small deviation from the other four.
Specifically, we found a maximum deviation of 1 %
between the values in, [9], (approximately 300
pressure-temperature points in the required region)
and the values found in [10] and in [11], which
provide fewer points (approximately 90). This
deviation is considered significant as we aim to
achieve the highest accuracy possible for our
analysis. The deviation found in the literature is
normal to occur as the compressibility factor is
calculated using both experimental and
computational methods. By carrying out further
literature search we found, [12], and, [13], which
agree with [10] and [11]. Having found four sources
proposing the same values we will use them for our
analysis.
The values for the compressibility factor Z in the
required range are presented in Table_1. Z values
are used to calculate the density (or molar volume or
specific volume) of air and together with the related
temperature are used in the equations of state to
calculate the pressure, which then is compared with
the pressure of the corresponding point of Z value.
Table 1. Compressibility factor Z of air.
kkk
kkk
kk
Compressibility factor Z
Pressure (bar)
Temperature
(K)
1
10
20
40
60
80
100
150
200
250
250
0.9
99
2
0.9
91
1
0.9
82
2
0.9
67
1
0.9
54
9
0.9
46
3
0.9
41
1
0.9
45
0.9
71
3
1.0
15
2
300
0.9
99
9
0.9
97
4
0.9
95
0.9
91
7
0.9
90
1
0.9
90
3
0.9
93
1.0
07
4
1.0
32
6
1.0
66
9
350
1.0
00
0
1.0
00
4
1.0
01
4
1.0
03
8
1.0
07
5
1.0
12
1
1.0
18
3
1.0
37
7
1.0
63
5
1.0
94
7
400
1.0
00
2
1.0
02
5
1.0
04
6
1.0
1
1.0
15
9
1.0
22
9
1.0
31
2
1.0
53
3
1.0
79
5
1.1
08
7
450
1.0
00
3
1.0
03
4
1.0
06
3
1.0
13
3
1.0
21
1.0
28
7
1.0
37
4
1.0
61
4
1.0
91
3
1.1
18
3
500
1.0
00
3
1.0
03
4
1.0
07
4
1.0
15
1
1.0
23
4
1.0
32
3
1.0
41
1.0
65
1.0
91
3
1.1
18
3
600
1.0
00
4
1.0
03
9
1.0
08
1
1.0
16
4
1.0
25
3
1.0
34
1.0
43
4
1.0
67
8
1.0
92
1.1
17
2
800
1.0
00
4
1.0
03
8
1.0
07
7
1.0
15
7
1.0
24
1.0
32
1
1.0
40
8
1.0
62
1
1.0
84
4
1.1
06
1
2.2 Equations of State
We recreated every available equation of state in
CFTurbo in Excel software, except Cool Prop, to
calculate the pressure of air at each point of Table 1
as described above. The equations are described
below.
2.2.1 Perfect Gas
Perfect Gas equation of state is the simplest of all
and was used in the following form:
(1)
Where P – pressure [Pa], ρ – density [kg/m3], R – air
gas constant = 287
, and T – temperature [K].
2.2.2 Redlich-Kwong
The Redlich-Kwong equation was taken from a
practice engineering book, [14]:
󰇛󰇜
(
(2)
Where P – pressure [Pa], R – air gas constant =
287_
, T – temperature [K], Vm – molar volume
[m3/mol], b - is a constant that corrects for volume
and, α - is a constant that corrects for the attractive
potential of molecules.
Vm is calculated with the following equation:
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1285
Volume 19, 2023
(3)
Where Mw – the molar mass of air = 0.029 kg/mol.
Mw was calculated using the formula calculating gas
mixture properties:

(4)
Where Mwi is the molar mass of each component of
the air mixture, and xi is the percentage of each
component, which are provided in Table 2.
Table 2. Essential properties of the components
of air for the used equations. x – the percentage of
each component of air, Mw – molar mass, Tc – the
temperature at the critical point, Pc – the pressure at
the critical point, Vc - the volume at the critical
point, ω – acentric factor.
N2
O2
Ar
CO2
x (%)
78.08%
20.95%
0.93%
0.04%
Mw
(kg/mol)
0.02802
0.03200
0.03995
0.04401
Tc (K)
126.2
154.6
150.8
304.13
Pc (bar)
33.90
50.50
48.65
73.97
Vc
(m3/kg)
0.00318
0.00250
0.00186
0.00214
ω
0.040
0.022
0.001
0.228
α is calculated using Eq. 4 by replacing Μwi with
αi values. αi of each component of air is calculated
by the following equation:
󰇛
󰇜


(5)
Where Tci and Pci of each component of air are given
in Table 2.
b is calculated using Eq. 4 by replacing Μwi with
bi values. bi of each component of air is calculated
by the following equation:


(6)
Where Tci and Pci of each component of air are
given in Table 2.
2.2.3 Aungier/ Redlich-Kwong
The Aungier/ Redlich-Kwong equation was taken
from the ANSYS documentation, [15]:

󰇛󰇜
(7)
Where P – pressure [Pa], R – air gas constant =
287_
, T – temperature [K], V – specific volume
[m3/kg], b and c - are constants that correct for
volume and α0 - is a constant that corrects for the
attractive potential of molecules. Tr and n are
presented below.
V is calculated using the values of Table 1 and
the following equation:
󰇛󰇜
(8)
α0 is calculated using Eq. 4 by replacing Μwi with
α0i values. α0i of each component of air is calculated
by the following equation:
  


(9)
Where Tci and Pci of each component of air are
given in Table 2.
b is calculated using Eq. 4 by replacing Μwi with
bi values. bi of each component of air is calculated
by the following equation:
 

(10)
Where Tci and Pci of each component of air are given
in Table 2.
c is calculated using Eq. 4 by replacing Μwi with
ci values. ci of each component of air is calculated
by the following equation:



󰇛󰇜 
(11)
Where Tci, Pci, and Vci of each component of air are
given in Table 2.
Tr is calculated using the following equation:
(12)
Where Tc is the temperature at the critical point of
air and is calculated using Eq. 4 by replacing Μwi
with Tci values provided in Table 2.
n is calculated using the following equation:

(13)
Where ω – the acentric factor = 0.0360 which is
calculated using Eq. 4 by replacing Μwi with ωi
values of each component of air. ωi are provided in
Table 2.
2.2.4 Soave/ Redlich-Kwong
The Soave/ Redlich-Kwong equation was taken
from the publication of the author himself for this
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1286
Volume 19, 2023
equation, [16]:
󰇛󰇜
(14)
Where P – pressure [Pa], R – air gas constant =
287_
, T – temperature [K], Vm – molar volume
[m3/mol], b - is a constant that corrects for volume,
α - is a constant that corrects for the attractive
potential of molecules and Α – is a modification to
the attractive term.
Vm is calculated using Eq. 3.
α is calculated using Eq. 4 by replacing Μwi with
αi values. αi of each component of air is calculated
by the following equation:
󰇛
󰇜

(15)
Where Tci and Pci of each component of air are given
in Table 2.
A is calculated following equation:
󰇛󰇛
󰇜󰇛󰇜󰇜
(16)
Where Τr and ω are calculated the same way as
described in Eq. 12 and 13 respectively.
b is calculated using Eq. 4 by replacing Μwi with
bi values. bi of each component of air which are
calculated using an equation using Eq. 6.
2.2.5 Peng-Robinson
The Peng-Robinson equation was taken from the
publication of the authors themselves for this
equation, [17]:
󰇛󰇜 󰇛󰇜
(17)
Where P – pressure [Pa], R – air gas constant =
287_
, T – temperature [K], Vm – molar volume
[m3/mol], b - is a constant that corrects for volume,
α - is a constant that corrects for the attractive
potential of molecules and Α – is a modification to
the attractive term.
Vm is calculated using Eq. 3.
α is calculated using Eq. 4 by replacing Μwi with
αi values. αi of each component of air is calculated
by the following equation:


(18)
Where Tci and Pci of each component of air are
given in Table 2.
A is calculated following equation:
󰇛 󰇛   󰇜
󰇛󰇜󰇜
(19)
Where Τr and ω are calculated the same way as
described in Eq. 12 and 13 respectively.
b is calculated in Eq. 4 by replacing Μwi with αi
values. bi of each component of air which is
calculated by the following equation:


(20)
Where Tci and Pci of each component of air are given
in Table 2.
2.2.6 Cool Prop
CoolProp is a C++ library that implements several
equations for the calculation of properties of
substances and mixtures, as described by the Cool
Prop creators on their website coolprop. Its
construction would require a lot of effort. For this
reason, we used the online calculator created by
Cool Prop creators themselves, available on their
website, which also ensures correct calculations.
3 Problem Solution
In this section, we present the calculations
performed to figure in which area the performance
of each equation of state of Section 2 is optimized.
As we described in Section 2, we use pressure
values to evaluate the performance of each equation
of state to avoid computational errors as all the
equations are solved to pressure. The error of each
point Pi and Tj of the range of the analysis, Table 1,
is calculated using the following equation:

󰇛󰇜 󰇛󰇜
󰇛󰇜
(21)
This calculation is used to evaluate the
performance of each equation of state. To calculate
the pressure of each point in Table 1 we apply the
following process. For each point Pi and Tj of
Table_1 we calculate their density ρij (or molar
volume Vmij or specific volume Vij) by using the
corresponding compressibility factor Zij of Table 1.
Then both Tj and ρij (or Vmij or Vij) are inserted in
the equations to calculate the pressure
󰇛󰇜, which is used together
with the 󰇛󰇜 in equation (21) to
calculate the .
For example, we want to calculate the error of
the equations at the point 100 bar and 500 K. At this
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1287
Volume 19, 2023
point the compressibility factor is Z = 1.041. This Z
corresponds to density ρ = 66.94 kg/m3, specific
molar Vm = 4.33 10-4 m3/mol, and specific volume
V = 1.493 10-2 m3/kg. The calculated pressure at
this point using the Soave/ Redlich-Kwong equation
is 100.43 bar. By using equation (21) we find that
the error is 0.43%
The results are presented in the following Tables.
The tables present the error of each equation on
predicting the properties of air in different
combinations of temperature and pressure in the
ranges 250 K to 800 K and 1 bar to 250 bar. To
facilitate the evaluation, error values equal to or
smaller than 0.1_% are colored with blue, error
values between 0.1_% and 1 % and equal to 1 % are
colored with green, error values between 1 % and 2
% and equal to 2 % are colored with orange, and
error values bigger than 2 % are colored with red.
Table 3. The error of Perfect Gas equation
calculations from real values
kkk
kkk
kk
Error
Pressure (bar)
Temperature
(K)
1
5
10
20
40
60
80
100
150
200
250
250
0.0
8
%
0.4
3
%
0.9
0
%
1.8
1
%
3.4
0
%
4.7
2
%
5.6
7
%
6.2
6
%
5.8
2
%
2.9
5
%
1.5
0%
300
0.0
1
%
0.1
3
%
0.2
6
%
0.5
0
%
0.8
4
%
1.0
0
%
0.9
8
%
0.7
0
%
0.7
3
%
3.1
6
%
6.2
7%
350
0.0
0
%
0.0
2
%
0.0
4
%
0.1
4
%
0.3
8
%
0.7
4
%
1.2
0
%
1.8
0
%
3.6
3
%
5.9
7
%
8.6
5%
400
0.0
2
%
0.1
2
%
0.2
5
%
0.4
6
%
0.9
9
%
1.5
7
%
2.2
4
%
3.0
3
%
5.0
6
%
7.3
6
%
9.8
0%
450
0.0
3
%
0.1
6
%
0.3
4
%
0.6
3
%
1.3
1
%
2.0
6
%
2.7
9
%
3.6
1
%
5.7
8
%
8.3
7
%
10.
58
%
500
0.0
3
%
0.2
0
%
0.3
4
%
0.7
3
%
1.4
9
%
2.2
9
%
3.1
3
%
3.9
4
%
6.1
0
%
8.3
7
%
10.
58
%
600
0.0
4
%
0.2
2
%
0.3
9
%
0.8
0
%
1.6
1
%
2.4
7
%
3.2
9
%
4.1
6
%
6.3
5
%
8.4
2
%
10.
49
%
800
0.0
4
%
0.2
0
%
0.3
8
%
0.7
6
%
1.5
5
%
2.3
4
%
3.1
1
%
3.9
2
%
5.8
5
%
7.7
8
%
9.5
9%
From Table 3 we observe that the Perfect Gas
equation provides excellent calculations (error equal
to or less than 0.1 %) only near the atmospheric
pressure of 1 bar while its calculations worsen as the
pressure increases. However, for applications that
require less than 1 % error this model can be used
up to 10 bar and some points between 20 bar and
150 bar and temperatures between 300 K and 400 K.
All the other areas have more than 1 % error with
more than 2 % error to appear mainly after 60 bar
and 400 K.
Table 4. The error of Redlich–Kwong equation
calculations from real values
kkkk
kkkk
Error
Pressure (bar)
Temperature
(K)
1
5
10
20
40
60
80
100
150
200
250
250
0.0
3
%
0.1
1
%
0.1
8
%
0.2
9
%
0.5
4
%
0.7
3
%
0.8
9
%
0.9
9
%
1.1
2
%
1.3
0
%
1.7
2
%
300
0.0
3
%
0.0
9
%
0.1
7
%
0.3
0
%
0.5
8
%
0.8
1
%
1.0
0
%
1.2
4
%
1.7
1
%
2.1
3
%
2.5
8
%
350
0.0
1
%
0.0
8
%
0.1
5
%
0.3
2
%
0.6
1
%
0.8
9
%
1.1
3
%
1.4
0
%
1.9
6
%
2.4
9
%
3.0
1
%
400
0.0
1
%
0.0
9
%
0.1
9
%
0.3
1
%
0.6
1
%
0.8
6
%
1.1
5
%
1.4
7
%
2.0
9
%
2.6
9
%
3.2
3
%
450
0.0
1
%
0.0
8
%
0.1
8
%
0.3
0
%
0.6
0
%
0.9
1
%
1.1
5
%
1.4
4
%
2.1
6
%
3.1
6
%
3.6
3
%
500
0.0
1
%
0.1
0
%
0.1
3
%
0.3
0
%
0.5
9
%
0.8
9
%
1.2
1
%
1.4
6
%
2.1
6
%
2.8
6
%
3.4
6
%
600
0.0
1
%
0.0
9
%
0.1
3
%
0.2
9
%
0.5
7
%
0.8
8
%
1.1
5
%
1.4
6
%
2.2
4
%
2.8
7
%
3.5
0
%
800
0.0
1
%
0.0
7
%
0.1
2
%
0.2
4
%
0.5
1
%
0.7
9
%
1.0
4
%
1.3
4
%
2.0
1
%
2.7
1
%
3.3
2
%
From Table 4 we observe that the Redlich–
Kwong equation provides excellent calculations
almost up to 5 bar while again its calculations
worsen as the pressure increases. Generally, it
performs better than the Perfect Gas equation and
provides less than 1 % error up to and around 60
bar. The area with errors of more than 2% is limited
between 150 bar and 250 bar, and between 300 K
and 800 K.
Aungier/ Redlich-Kwong, Table 5, is an even
better equation as it provides excellent calculations
up to 5 bar, almost in the same area as Redlich–
Kwong while again its calculations worsen as the
pressure increases. Generally, it performs better than
the Redlich–Kwong as it provides less and 1 % error
up to 80 bar, including a triangular area between
100 bar and 250 bar and between 250 K and 350 K.
The area with errors of more than 2% is even
smaller in this case between 200 bar and 250 bar,
and between 400 K and 800 K. So, regarding high-
pressure analyses, Aungier/ Redlich-Kwong
equation performs well at low temperatures between
250K and 350 K.
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1288
Volume 19, 2023
Table 5. The error of Aungier/ Redlich-Kwong
equation calculations from real values
kkkk
kkkk
Error
Pressure (bar)
Temperature
(K)
1
5
10
20
40
60
80
100
150
200
250
250
0.0
2
%
0.0
8
%
0.1
2
%
0.1
6
%
0.2
7
%
0.3
1
%
0.3
1
%
0.2
3
%
0.0
8
%
0.3
3
%
0.3
0
%
300
0.0
3
%
0.0
6
%
0.1
1
%
0.1
9
%
0.3
5
%
0.4
6
%
0.5
4
%
0.6
6
%
0.8
2
%
0.9
6
%
1.1
4
%
350
0.0
1
%
0.0
5
%
0.1
0
%
0.2
3
%
0.4
2
%
0.6
0
%
0.7
5
%
0.9
3
%
1.2
6
%
1.5
7
%
1.8
9
%
400
0.0
1
%
0.0
7
%
0.1
5
%
0.2
3
%
0.4
5
%
0.6
3
%
0.8
3
%
1.0
8
%
1.5
1
%
1.9
4
%
2.3
1
%
450
0.0
1
%
0.0
7
%
0.1
5
%
0.2
3
%
0.4
6
%
0.7
0
%
0.8
8
%
1.1
1
%
1.6
7
%
2.5
3
%
2.8
7
%
500
0.0
1
%
0.0
8
%
0.1
0
%
0.2
4
%
0.4
7
%
0.7
1
%
0.9
7
%
1.1
7
%
1.7
3
%
2.3
2
%
2.8
0
%
600
0.0
1
%
0.0
8
%
0.1
1
%
0.2
4
%
0.4
7
%
0.7
4
%
0.9
7
%
1.2
4
%
1.9
1
%
2.4
5
%
2.9
9
%
800
0.0
1
%
0.0
6
%
0.1
0
%
0.2
1
%
0.4
5
%
0.7
0
%
0.9
2
%
1.1
9
%
1.7
9
%
2.4
3
%
2.9
7
%
Table 6. The error of Soave/ Redlich-Kwong
equation calculations from real values
kkkk
kkkk
Error
Pressure (bar)
Temperature
(K)
1
5
10
20
40
60
80
100
150
200
250
250
0.0
0
%
0.0
3
%
0.1
1
%
0.2
9
%
0.6
2
%
1.0
1
%
1.4
1
%
1.8
5
%
2.8
7
%
3.5
0
%
3.5
7
%
300
0.0
1
%
0.0
6
%
0.1
3
%
0.2
8
%
0.5
8
%
0.9
0
%
1.2
3
%
1.4
8
%
2.0
9
%
2.5
0
%
2.6
6
%
350
0.0
2
%
0.0
6
%
0.1
3
%
0.2
4
%
0.4
9
%
0.7
1
%
0.9
5
%
1.1
2
%
1.5
6
%
1.8
3
%
1.9
3
%
400
0.0
1
%
0.0
4
%
0.0
7
%
0.2
1
%
0.4
0
%
0.6
0
%
0.7
6
%
0.8
4
%
1.1
3
%
1.2
8
%
1.3
5
%
450
0.0
1
%
0.0
4
%
0.0
6
%
0.1
7
%
0.3
2
%
0.4
3
%
0.5
7
%
0.6
5
%
0.7
6
%
0.4
3
%
0.5
4
%
500
0.0
1
%
0.0
1
%
0.0
9
%
0.1
2
%
0.2
3
%
0.3
1
%
0.3
5
%
0.4
3
%
0.4
9
%
0.4
2
%
0.3
6
%
600
0.0
0
%
0.0
0
%
0.0
4
%
0.0
5
%
0.1
0
%
0.0
9
%
0.1
1
%
0.0
7
%
0.0
9
%
0.1
9
%
0.3
5
%
800
0.0
0
%
0.0
1
%
0.0
1
%
0.0
2
%
0.0
8
%
0.1
6
%
0.2
2
%
0.3
4
%
0.5
8
%
0.9
2
%
1.1
9
%
Table 7. The error of Peng–Robinson calculations
from real values
kkkk
kkkk
Error
Pressure (bar)
Temperature
(K)
1
5
10
20
40
60
80
100
150
200
250
250
0.0
5
%
0.2
1
%
0.3
7
%
0.6
4
%
1.1
3
%
1.4
7
%
1.7
2
%
1.8
9
%
2.2
8
%
3.0
0
%
4.2
3
%
300
0.0
4
%
0.1
3
%
0.2
5
%
0.4
4
%
0.8
0
%
1.0
7
%
1.2
9
%
1.5
4
%
2.0
6
%
2.6
6
%
3.4
4
%
350
0.0
1
%
0.0
9
%
0.1
6
%
0.3
4
%
0.6
3
%
0.8
9
%
1.1
1
%
1.3
5
%
1.8
7
%
2.4
4
%
3.1
1
%
400
0.0
1
%
0.0
8
%
0.1
7
%
0.2
7
%
0.5
2
%
0.7
2
%
0.9
5
%
1.2
3
%
1.7
5
%
2.3
3
%
2.9
1
%
450
0.0
1
%
0.0
6
%
0.1
4
%
0.2
2
%
0.4
5
%
0.6
8
%
0.8
6
%
1.0
9
%
1.6
9
%
2.6
3
%
3.1
0
%
500
0.0
0
%
0.0
7
%
0.0
8
%
0.2
1
%
0.4
1
%
0.6
3
%
0.8
7
%
1.0
6
%
1.6
1
%
2.2
4
%
2.8
0
%
600
0.0
1
%
0.0
7
%
0.0
8
%
0.1
9
%
0.3
8
%
0.6
0
%
0.8
0
%
1.0
4
%
1.6
7
%
2.2
1
%
2.7
8
%
800
0.0
1
%
0.0
5
%
0.0
8
%
0.1
7
%
0.3
5
%
0.5
7
%
0.7
6
%
1.0
0
%
1.5
5
%
2.1
7
%
2.7
1
%
According to Table 6, the Soave/ Redlich-
Kwong equation has almost similar performance to
the Aungier/ Redlich-Kwong equation with the
difference that it performs better at high
temperatures. Soave/ Redlich-Kwong equation has a
bigger area of low error (less than 0.1 %) between
1_bar and 10 bar which extends also mainly up to
40_bar but only for high temperatures between 600
K and 800 K. Its area with less than 1 % is also
bigger and covers all the area of analysis excluding
a triangular area between 60 bar and 250 bar, and
between 250 K and 400 K.
Peng–Robinson, Table 7, has almost similar
performance to Redlich–Kwong with the difference
that it performs better at high temperatures. Its
tendency to perform better in high temperatures is
clear even in the low error area (error less than 0.1
%) covering a wide triangular area between 1 bar
and 10_bar. Its area with less than 1 % extends
almost up to 80 bar, excluding a small triangular
area between 250 K and 350 K. Its area with almost
and more than 2 % error is between 150 bar and 250
bar.
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1289
Volume 19, 2023
Table 8. The error of Cool Prop calculations from
real values
kkkk
kkkk
Error
Pressure (bar)
Temperature
(K)
1
5
10
20
40
60
80
100
150
200
250
250
0.0
0
%
0.0
2
%
0.0
1
%
0.0
4
%
0.0
6
%
0.0
7
%
0.0
4
%
0.0
3
%
0.0
1
%
0.0
1
%
0.0
5
%
300
0.0
0
%
0.0
0
%
0.0
0
%
0.0
0
%
0.0
2
%
0.0
4
%
0.0
7
%
0.0
3
%
0.0
2
%
0.0
3
%
0.0
1
%
350
0.0
2
%
0.0
1
%
0.0
1
%
0.0
2
%
0.0
3
%
0.0
4
%
0.0
3
%
0.0
6
%
0.0
8
%
0.1
0
%
0.1
0
%
400
0.0
2
%
0.0
1
%
0.0
3
%
0.0
2
%
0.0
5
%
0.0
4
%
0.0
6
%
0.1
3
%
0.1
6
%
0.2
1
%
0.2
1
%
450
0.0
2
%
0.0
0
%
0.0
4
%
0.0
2
%
0.0
6
%
0.1
0
%
0.1
0
%
0.1
4
%
0.2
5
%
0.6
7
%
0.6
0
%
500
0.0
2
%
0.0
2
%
0.0
1
%
0.0
4
%
0.0
7
%
0.1
2
%
0.1
9
%
0.2
0
%
0.3
0
%
0.4
4
%
0.5
0
%
600
0.0
2
%
0.0
2
%
0.0
0
%
0.0
5
%
0.1
0
%
0.1
8
%
0.2
2
%
0.3
1
%
0.5
3
%
0.6
3
%
0.7
3
%
800
0.0
1
%
0.0
1
%
0.0
1
%
0.0
4
%
0.1
2
%
0.2
2
%
0.2
8
%
0.3
9
%
0.5
9
%
0.8
4
%
1.0
0
%
Finally, the most accurate calculations were
achieved by Cool Prop, Table 8, which is a library
with several equations. The maximum error is 1 %
at the point 250 bar and 800 K. Its low error area
(less than 0.1 %) covers almost all the range of the
analysis, excluding a triangular area between 60 bar
and 250 bar and between 350 K and 800 K. And this
triangular area has a maximum error of 1 %. And
thus Cool Prop is the best model for predicting air
properties in the range of 1 bar to 250 bar and 250 K
to 800 K.
The fact that many of our calculations are close
to the real values is a good sign that our calculations
are correct. However, we must check if this is the
case or if we have underestimated an equation. So,
we will compare our equations with existing
calculators from other sources. The Ideal Gas was
not verified as it is simple enough to not require
validation. Also, Cool Prop does not require
validation as we used the original calculator of Cool
Prop creators.
A comparison with equations is presented in
Table 9. To compare them we calculated the
properties of air in fewer points which cover the
ranges of analysis, focusing on the points where our
equations presented the highest errors because our
equations and the equations from other sources
presented the same error pattern. So, we compared
the maximum errors found.
For the Redlich-Kwong equation, we used the
online calculator found on the website of vCalc
which is an open calculator, equation, and dataset
library. Its calculations are slightly better than ours.
At their worst point, our equation has a 3.634 %
error while the online calculator has a 3.628 %
error. Their difference is 0.006 % which is very
small.
For the Soave/ Redlich-Kwong equation, we
used the vCalc website. Its calculations are a lot
worse than ours. At their worst point, our equation
has a 3.57 % error while the online calculator has a
27.38 % error. Their difference is 23.81 % which is
significant. This may be the result of the different
ways these equations can be used for mixtures, such
as air. One way is that the constants of the
equations, α, α0, b, c, can be calculated by
considering air as a mixture and using equation (4),
as we described. Another way is that these constants
can be calculated by considering air as a pure
substance and using equations (5), (6), (9), (10),
(11), (15), (18), (19), and (20), ignoring the i index
and calculating them by using the critical
temperature and critical pressure of air instead of
each component. Both methods can be applied, and
this may be the reason for the high difference that
appeared.
For the Peng-Robinson equation, we used the
vCalc website. Its calculations are worse than ours.
At their worst point, our equation has a 4.23 % error
while the online calculator has a 7.64 % error. Their
difference is 3.41_%.
For the Aungier/ Redlich-Kwong equation, we
did not find an online calculator, so we used
CFTurbo for the calculations. Its calculations are
slightly worse than ours. At their worst point, our
equation has a 2.99 % error while the CFTurbo has
a 3.32 % error. Their difference is 0.33 % which is
very small.
With the above comparison, we showed that the
performance of our equations is close to the
performance of available calculators, excluding the
Soave/ Redlich-Kwong equation which was
outperformed by our approach. So, we can trust that
our calculations and the results of this analysis do
not underestimate the performance of any equation.
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1290
Volume 19, 2023
Table 9. Comparison of our equations with
equations from other sources (*calculated using
CFTurbo)
Maximum Error (%)
Difference (Ours-
Online)
Best
results
(Ours/
Online)
Equation
Our
Equations
Online
Calculators
Perfect
Gas
10.58 %
-
-
-
Redlich
–Kwong
3.634 %
3.628 %
0.006 %
Other
Aungier/
Redlich-
Kwong
2.99 %
*3.32 %
- 0.33 %
Ours
Soave/
Redlich-
Kwong
3.57 %
27.38 %
- 23.81
%
Ours
Peng–
Robinso
n
4.23 %
7.64 %
- 3.41 %
Ours
Cool
Prop
-
1.00 %
-
-
4 Conclusion
After the above calculations, we have studied the
available models in a wide range of conditions. The
percentages presented in Table 3, Table 4, Table 5,
Table 6, Table 7 and Table 8 describes the error of
each model at the specific temperature and pressure
condition.
It is very interesting to observe how the
performance of each model varies. Specifically, we
notice that as the complexity of a model increases so
does its accuracy.
Table 3, Table 4, Table 5, Table 6, Table 7 and
Table 8 provide valuable data for an analyst to
choose, based on their intentions, the best model for
their analysis to achieve the desired accuracy and
save computational cost.
Also, if they need the same analysis for a
different substance, they can follow the steps
described in this article to realize it.
According to the results of Section 3, we come to
the following conclusions about each equation of
state.
According to Table 3, the calculations of the
Perfect Gas equation have a maximum of 1 % error
mainly at low pressures between 1 bar and 20 bar.
According to Table 4, the calculations of
Redlich–Kwong have a maximum of 1 % error from
1 bar to 60 bar of pressure.
According to Table 5, the calculations of the
Aungier/ Redlich-Kwong equation are even better
by having a maximum 1 % error from 1 bar to
80_bar including a small triangular area between
100 bar to 250 bar and 250 K to 350 K.
According to Table 6, the calculations of the
Soave/ Redlich-Kwong equation have a maximum
of 1 % error inside all the ranges of the analysis
excluding a triangular area between 60 bar to 250
bar and 250 K to 400 K and the point of 250 bar and
800 K.
According to Table 7, the calculations of Peng–
Robinson have a maximum of 1 % error from 1 bar
to 80 bar excluding a small triangular area between
40 bar to 80 bar and 250 K to 350 K.
Finally, according to Table 8, the calculations of
Cool Prop have a maximum of 1 % error inside all
the ranges of the analysis.
So, if someone needs to calculate the properties
of air and has available all the above equations the
best option is Cool Prop. However, there is no
available Cool Prop setup for all the substances in
every conventional software, especially for custom
mixtures.
The next best options for air are the Aungier/
Redlich-Kwong equation and Soave/ Redlich-
Kwong equation as they cover a larger area than the
other equations. Also, they perform well in different
areas which means that combined cover almost all
the areas of analysis. Both have calculations with a
maximum 1% error from pressures greater than 100
bar. Above this pressure, Aungier/ Redlich-Kwong
equation has a maximum 1% error in a triangular
area between 250 K and 350 K, while Soave/
Redlich-Kwong equation in the area between 450 K
and 800 K, excluding point 250 bar and 800 K. We
notice that the other part of the triangular area
between 250 K and 350 K and the point 250 bar and
800 K have more than 1% errors. However, we
notice that if we use each model where it performs
better, calculations with a maximum of 2% error can
be achieved. This is an exceptionally good
performance considering the relative simplicity of
these two equations.
An interesting and useful future study would be
to evaluate the performance of these equations using
other substances in the same area and to observe if
each equation performs better for different
substances.
Even more interesting and useful future studies
would be to collect analyses of different substances
and search for a relationship between the
performance of the equations and key properties of
the substances, e.g. the performance of the equations
to Pr and Tr.
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1291
Volume 19, 2023
References:
[1] H. Li, J. Yan, Evaluating cubic equations of
state for calculation of vapor-liquid
equilibrium of CO2 and CO2-mixtures for CO2
capture and storage processes, Applied Energy
Journal, Vol. 86, No. 6, 2009, pp. 826–836.
[2] M. M. Rashidi, N. Rahimzadeh, N. Laraqi,
Evaluation of the equations of state for air,
nitrogen, and, oxygen on throttle reduction
efficiency by using exergy analysis,
International Journal of Exergy, Vol. 9, No.
3, 2011.
[3] X. Jin, D. Xu, X. Zhou, Performance
Evaluation of Air-Conditioning System Using
NARM by Equation of State, International
Refrigeration and Air Conditioning
Conference, 1994, pp. 261.
[4] Kouremenos, D.A., Antonopoulos, K.A.
Compressibility charts for gases with different
acentric factors and evaluation of the Redlich-
Kwong-Soave equation of state. Forsch Ing-
Wes, Vol. 57, 1991, pp. 158-161.
[5] Yousef S. H. Najjar, Awad R. Mansour,
Evaluation of Peng Robinson Equation of
State in Calculating Thermophysical
Properties of Combustion Gases, Chemical
Engineering Communications, Vol. 61, No. 1-
6, 1987, pp. 327-435.
[6] Y. A. Cengel, M. A. Boles, M. Kanoglu,
Thermodynamics: An Engineering Approach,
10th Ed., Mc Graw Hill, 2023.
[7] Sh. Eliezer, A. Ghatak, H. Hora,
Fundamentals of Equations of State, World
Scientific, 2002.
[8] R. Span, Multiparameter Equations of State,
Springer, 2000.
[9] R. Humphrey, C. A. Neel, Tables of
Thermodynamic Properties of Air from 90 to
1500 K, Arnold Engineering Development
Center Air Force Systems Command USAF,
1961.
[10] J. H. Perry, Perry's chemical engineers'
handbook (6 ed.), MCGraw-Hill, 1984.
[11] Vasserman and K. A. Rabinovich,
Thermophysical Properties of Air and Air
Components, U.S. Department of Commerce,
National Technical Information Service,
Springfield, 1971 (Translation from Russian,
first published 1961).
[12] J. Hilsenrath, C. W. Beckett, W. S. Benedict,
L. Fano, H. J. Hoge, J. F. Masi, R. L. Nuttall,
Y. S. Touloukian, H. W. Woolley, Tables of
Thermal Properties of Gases, National Bureau
of Standards, 1955.
[13] N. B. Vargaftic, Handbook Thermophysical
Properties of Gases and Liquids, Nauka,
1972.
[14] Murdock, James W., Fundamental fluid
mechanics for the practicing engineer, CRC
Press, 1993.
[15] ANSYS, ANSYS FLUENT 12.0/12.1
Documentation, ANSYS Inc., 2009, [Online].
https://www.afs.enea.it/project/neptunius/docs
/fluent/html/ug/main_pre.htm (Accessed Date:
January 17, 2024).
[16] G. Soave, Equilibrium constants from a
modified Redlich–Kwong equation of state,
Chemical Engineering Science, 1972.
[17] D. Peng, D. Robinson, A New Two-Constant
Equation of State, Industrial & Engineering
Chemistry Fundamentals, Ind. Eng. Chem.
Fundamen. 1976, 15, 1, 59–64
https://doi.org/10.1021/i160057a011.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Vasileos Moutsios performed research of
resources, formal analysis, and the writing -
original draft.
- P. Dionissios Margaris performed supervision and
validation.
- Nicholas Pittas performed supervision.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The present work was financially supported by the
“Andreas Mentzelopoulos Foundation” and by the
“CFTurbo”.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT
DOI: 10.37394/232015.2023.19.116
Vasileios Moutsios,
Dionissios Margaris, Nicholas Pittas
E-ISSN: 2224-3496
1292
Volume 19, 2023