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MIRARCO MINING INNOVATION RESEARCH REPORT
Solubility Calculations for Hydraulic Gas
Compressors
JULIA ANDRADE
21 August 2013
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Table of Contents
1 Introduction ................................................................................................................................ 5
2 Vapor-Liquid Equilibrium (VLE) Theory .................................................................................. 7
2.1 Henry’s Law Applied for Solubility Calculations ............................................................. 8
2.1.1 Henry’s Law .............................................................................................................. 8
2.1.2 Henry’s Constant ....................................................................................................... 9
2.1.3 Limitations of Henry’s Law .................................................................................... 10
2.2 Cubic Equations of State applied for solubility calculations ........................................... 10
3 Approximation of the vapour-liquid equilibrium of air in water .............................................. 14
3.1 Applying Henry’s Law - Estimating Henry’s Constant from van’t Hoff Expression ...... 14
3.2 Applying Henry’s Law - Estimating Henry’s Constant from Analytical Expressions .... 15
3.3 Comparison with Experimental Data from the Literature ............................................... 16
4 Approximation of the vapour-liquid equilibrium of combustion gases in water ...................... 19
4.1 Combustion gases from different fuels ............................................................................ 19
4.1.1 Estimation of the Solubility of Combustion Gases using Henry’s Law .................. 20
5 Approximation of the vapour-liquid equilibrium of carbon dioxide in water ........................... 22
5.1 Estimating the solubility of CO2 in water using Henry’s Law......................................... 22
5.2 Experimental Data of the System CO2 + Water ............................................................... 23
5.3 Estimating the solubility of CO2 in water using the VR Therm ...................................... 24
5.4 Limitations of Henry’s Law in predicting the solubility of the system CO2 + Water...... 25
5.5 Estimating the solubility of CO2 in water using the model of Diamond (2003) .............. 26
6 Conclusions ............................................................................................................................... 28
7 References ................................................................................................................................. 29
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List of Figures
Figure 1 – Schematic Diagram of a Hydraulic Air Compressor ......................................................... 6
Figure 2 - Mass of N2, O2 and Ar in the gas phase as the absolute pressure of the system increases 15
Figure 3 - Solubility of air in water at pressure ranging from 0 to nearly 1,200 meters of water:
Results from the Application of Henry's Law and Kolev's Correlations and Experimental Data from
[15] and [16] ...................................................................................................................................... 18
Figure 4 - Pressures at which all CO2 is dissolved in water for the fuel methane, diesel, anthracite
and HFSO .......................................................................................................................................... 21
Figure 5 - Solubility of CO2 in water at 50°C obtained applying Henry's Law ................................ 22
Figure 6 - Solubility of CO2 in water at pressure ranging from 0 to nearly 1,500 meters of water:
Results from the Application of Henry's Law and S-P-E [20] and P-CO2 [16] Experimental Data . 24
Figure 7 - Solubility of CO2 in water predicted from SKR EoS and PR EoS, using the VRTherm,
compared with the S-P-E Experimental Data.................................................................................... 25
Figure 8 - CO2 solubility in water at 50°C using the model from [22] referred as D-A Data in
comparison with S-P-E and P-CO2 experimental data ...................................................................... 27
List of Tables
Table 1: Values of molar fraction, Henry's Law Constants and Temperature Dependence .............. 10
Table 2: B-R-T Data of Air Solubility in Water Expressed as a Mass Fraction (gram of air per gram
of water) ............................................................................................................................................ 17
Table 3: Henry's Law Constants for the Solubility of Air in Water as atm....................................... 17
Table 4: Composition of the flue gases from the complete combustion of fuels with oxygen
contained in air in molar fraction ...................................................................................................... 20
Table 5 - Solubility of CO2 in water: Experimental Data at 50°C and from nearly 10 to 7,233 meters
of water.............................................................................................................................................. 23
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Nomenclature
Symbol Name
P Absolute Pressure
� Acentric factor
�� Activity coefficient of component � ��� Binary Interaction Parameter
� Compressibility Factor
P Critical pressure
T Critical temperature
�� Fugacity of component � in a multicomponent mixture
��� Fugacity of pure species � in phase �
���� Fugacity of species � in phase � in a multicomponent mixture
�� K-value of equilibrium
����� Henry’s Constant
�� Mole fraction of component � in the liquid phase
�� Mole fraction of component � in the gas phase
� Molar gas constant
�� Molecular weight
�� Partial pressure of a gas � ∆���� Solute enthalpy of solution
Specific volume
T Temperature
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1 Introduction
There are many studies of gas solubility in water, both experimental and theoretical, especially at
the low-pressure region, where partial pressures do not exceed a few atmospheres. If working in this
low-pressure region there are advantages, such as; the majority of the data are reliable and of high
precision, the thermodynamic relations are estimated by generally-accepted and well-defined
correlations, and the results are barely influenced by inaccuracies from semi-empirical data. High-
pressure equilibrium data encounters thermodynamic difficulties for analysis, because of the
complexity of the mathematical formalism, the absence of fundamental data, and the dependence on
empirical correlations for their estimation [1].
The reliable estimation of high-pressure vapour-liquid equilibrium is important for a variety of
applications, especially in the chemical engineering field. For instance, due to concerns with global
warming, equilibrium data is essential for the study of sequestration of carbon dioxide (CO2) when
capturing and injecting it into deep geological formations [2]. A more specific application is in the
analysis of air loss in hydraulic air compressors (HACs). These involve the pressurisation of air by
water hence determining the solubility of air in water during the compression process is vital to
determine the loss of gaseous species due to their solution and for the correct operation of the other
elements of the equipment, such as air water separation [3].
An hydraulic air compressor can produce compressed air by the means illustrated in Figure 1. The
hydraulic energy source is a head of water, and its downward moving flow entrains air. The
separation of the air from the water at depth produces compressed air.
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Figure 1 – Schematic Diagram of a Hydraulic Air Compressor
The applications for HACs range from wave energy recovery to the production of compressed air
for compressed air energy storage. The performance of the HAC can be adversely affected by the
absorption of air by water during the compression process, representing a loss of product for the
equipment. Consequently, the accurate estimation of the maximum impact of air solubility by
applying vapor-liquid equilibrium calculations is essential for the modeling of the two-phase flow
in HACs.
The range of applications for hydraulic compressors has recently been conceptually extended to the
sequestration of combustion gases, such as carbon dioxide, a so-called greenhouse that is important
in global warming. In this case, the more gas that is absorbed by water, the better the sequestration
performance by hydraulic gas compression. Therefore, again, accurate calculations for gas
solubility in water are essential for the proper evaluation of HAC equipment performance in this
context.
Taking into account the importance of predicting the amount of gases absorbed by water during
hydraulic gas compression, this work focus on the estimation of solubility of gaseous species in air
and combustion gases at elevated pressure. Firstly, a discussion of the thermodynamic fundamentals
of vapour-liquid equilibrium is presented, followed by solubility calculations and their comparison
with experimental data. The results provided by this study aim to be used to update the
hydrodynamic calculations of hydraulic gas compressors (as a generalisation of hydraulic air
compressors).
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2 Vapor-Liquid Equilibrium (VLE) Theory
According to [5], thermodynamically, the equilibrium state is defined as:
“The equilibrium state of a closed system at a given P and T is the one for which the Gibbs free
energy of the system is at a minimum with respect to all possible changes.”
As a consequence of this definition and as exposed in details by Smith and Van Ness [10], there is
equilibrium between phases at the same P and T when the fugacity of each species (��) is the same
in all phases. Systems involving a single liquid phase in equilibrium with its vapor phase are in
vapor-liquid equilibrium (VLE) and the criterion for equilibrium is written as:
��! = ��� (1)
Fugacity is a thermodynamic property that has dimensions of pressure and is given by:
�� = ���#$%��#$&�� '()*+,+(-./012 (2)
where the exponential component is known as the Poynting factor.
For the case of multi-component vapor-liquid equilibrium, Equation (1) becomes:
��!� = ���4 (�=1, 2, ..., N) (3)
where N is the number of components in the mixture.
The dimensionless ratio �/% is a mixture property called the fugacity coefficient and given the
symbol �. The fugacity coefficient of species � in solution is given the symbol �� and defined as:
�� ≡ 7847889� (4)
where ���:� represents the fugacity of a species in an ideal-gas mixture and is equal to the partial
pressure of the species:
���:� = ��% (5)
The activity coefficient of species � in solution is defined as follows and depends upon the
chemicals involved, particularly whether they are polar or not (i.e. have a dipole moment) [5].
�� ≡ 784;(7( (6)
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For species � in the vapor mixture, the combination of Equations (4) and (5) becomes:
��!� = ���� % (7)
and for species � in the liquid solution, Equation (6) is written as:
���4 = ������. (8)
As stated in Equation (3), at equilibrium the two above equations must be equal, therefore:,
���� % = ������ (9)
2.1 Henry’s Law Applied for Solubility Calculations
2.1.1 Henry’s Law
William Henry in 1803 pioneered work describing the solubility of gases in water. Henry’s Law
states that the fugacity of a component in the gas phase can be linearly related to its liquid
concentration, when the mixture is sufficiently diluted, and is expressed as: [6]
��! = ���������� → 0� (10)
Where ����� is a proportionality constant called Henry’s Constant.
For relatively low pressures (up to 2 MPa, typically) and dilute solutions (�� < 0.03, typically),
ideal gas phase behaviour can be assumed. Substituting ��! into Eq. (5) and solving for ��, Henry’s
law becomes: [6]
�� = A(B( ��� → 0, DEF%� (11)
where �� is the partial pressure of the gas.
In Eq. (11) gas solubility is expressed in terms of mole fractions; nevertheless, the solubility can be
expressed in terms of mass fractions, volume fractions, molarity (moles per litre of solution), and
molality (moles per kilogram of solvent). Consequently, Henry’s law can be expressed in a variety
of ways depending on the dimensional units employed. [7] For instance, if it is desirable to state the
solubility inG:9.-:-H) I, Eq. (11) can be modified to:
J� = ��. ��. K(L-H) (12)
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where J� is the solubility of the component � in G :(:-H)I, �� is its partial pressure in MNOPQ, �� is its
molecular weight inG :(R��(I, and S��� is the density of the solution G:-H)T-H)I. Taking into account the
units adopted, the dimensional unit of �� must be G�R��(/T-H)�#$R I.
2.1.2 Henry’s Constant
For pressures up to 5 MPa, the effects of pressure on Henry’s Constant are quite small and it can be
considered that �� is solely a function of temperature. Qualitatively, the significant non-linear
temperature dependence generally increases with temperature at low temperatures, reaches a
maximum, and then decreases at high temperatures. The temperature at which the maximum �� occurs depends on the specific solute-solvent pair. As a guideline, the maximum tends to increase
with increasing solute critical temperature for a given solvent and with increasing solvent critical
temperature for a given solute. Inaccuracies can result, if the temperature dependence is ignored.
The Henry’s Constant can be extrapolated from a single data point at a certain temperature using the
van’t Hoff expression [8]:
B(�2U�B(�2V� ≈ &�� GX∆B-H)
1 Y X Z2U − Z
2VYI (13)
where ∆���� is the solute enthalpy of solution and � is the molar gas constant. The temperature
dependence is:
∆B-H)1 = − \ ]^B(
\�Z 2⁄ � (14)
The van’t Hoff approach assumes that ∆���� remains constant with temperature; hence, the
expression is usually a fair approximation of Henry’s Constant for modest temperature differences
when independent values of ∆���� exist at the desired temperature, either from calorimetric data, or
from a reliable estimation techniques[8]. Sander [9] presents a compilation of Henry’s Constants at
�̀ =298.15 K, as well as the temperature dependence defined in Eq. (14). In this work, the most
frequent values of those constants were adopted for the components of interest and are shown in
Table 1.
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Table 1: Values of molar fraction, Henry's Law Constants and Temperature Dependence
Air component a N2 O2 Ar CO2 SO2
����̀ � XPED� �b. NOP�c Y 0.00061 0.0013 0.0014 0.034 1.2
− d ln��d�1 �⁄ ���� 1,300 1,500 1,100 2,400 3,200
2.1.3 Limitations of Henry’s Law
The Henry’s Law constants can be satisfactorily used to express solubility, but it must be
remembered from thermodynamics that it is a limiting law applicable only for dilute solutions. [7]
The larger the deviations from ideal behaviour of the system, the narrower the range of
concentrations suitable for the application of Henry’s Law. Moreover, Henry’s Law is restricted to
describe the equilibrium of a single species that does not react chemically with the solvent. Hence,
Henry’s Law cannot predict the actual equilibrium when the aqueous form of a substance is
partitioned within the aqueous phase; for instance, carbon dioxide (CO2) in water quickly forms
hydrated carbon dioxide and then carbonic acid (H2CO3). [8]
2.2 Cubic Equations of State applied for solubility calculations
In the case of mixtures that largely deviate from the ideal behaviour the equilibrium relation of Eq.
(3) can be represented by:
��%�h�! = ��%�h�� or simply:
���h�! = ���h�� �� = 1, 2, . . . , N� (15)
Eq. (15) may be written as
�� = ������ = 1, 2, . . . , N� (16)
where �� is the K-value given by:
�� = k ()k (l (17)
From [10], the general equation for calculating the fugacity coefficients �h�� and �h�! is:
ln�� = � − 1 − ln� − m nXo�pq�op( Y2,',pr − 1s \'''t (18)
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where � is the compressibility factor defined as the ratio % ��⁄ and calculated from Equations of
State (EoS) that relate pressure, temperature and specific volume ( ). For ideal gases, � equals one
and the EoS becomes:
% = �� (19)
EoS are the basis of thermodynamic models and are used to represent phase equilibria, as well as
the calculation of thermal and volumetric properties [11]. Under high pressures, the P-V-T
behaviour is not correctly represented by the ideal gas EoS. In 1873, the first generalization known
as van der Waals Equation of State was proposed: [12]
% = 12!,u − #
!v (20)
where N accounts for the interaction forces between two molecules and w accounts for the excluded
volume.
Numerous models based on van der Waals EoS were formulated since its advent. They are
conventionally called cubic equations of state, since they can be rearranged to be cubic in , as
shown in Eq. 21. [5]
x − Xw + 12+ Y z + #
+ − #u+ = 0 (21)
The cubic EoS provide great advantage over other complex EoS, such as: accuracy of the results,
computational efficiency, and ease of implementation [13]. One of the variations of the van der
Waals EOS is the Soave-Redlich-Kwong (SRK) EoS proposed in 1972, which is common in
process simulators: [12]
% = 12!,u − #
!�!{u� (22)
Where: (from [9]):
N = 0.42748X�v��v�� Yα;
w = 0.08664X12�+� Y;
�� = 22�;
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� = �1 + P*1 − ��`.�0�z, and;
P = 0.480 + 1.574 − 0.176�z.
The Peng-Robinson equation of state (PR) from 1976 is another variation given as:
% = 12!,u − #
!�!{u�{u�!,u� (23)
Where:
N = 0.45724X�v��v�� Yα,
w = 0.08664X12�+� Y,
� = �1 + P*1 − ��`.�0�z, and;
P = 0.37464 + 1.54226 − 0.26992�z.
The critical temperature (T), critical pressure (P), and ‘acentric’ factor (�) are tabulated pure-fluid
physical properties [12].
The extension of an EoS from pure gas to mixtures requires the employment of empirical mixing
rules that represent the composition dependence of the parameters. Soave (1972) suggested
determining N and w from:
N = ∑ ∑ ����N�����Z���Z (24)
w = ∑ ��w����Z (25)
where �� and �� are the mole fractions of component i and j, respectively, in the vapour or liquid
phase. The term w� is the b-parameter defined above for the ith component. The term N�� is
determined as:
N�� = �*N�N�0�1 − ���� (26)
The parameter ��� is a binary interaction parameter for each binary component pair. ��� is zero by
definition if � = � and it is close to zero for two different components of approximately the same
polarity. Otherwise, the binary interaction parameter is tabulated.
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For the SRK EoS, the expression for the fugacity coefficient of component � in a mixture takes the
form: [5]
ln�� =− ln�� − �� + �� − 1� u(u − �� GZ# *2�N� ∑ ���N�*1 − ���0���Z 0 − u(
u I ln X1 + �qY (27)
In comparison, the PR EoS expression for the fugacity is: [5]
ln�� =− ln�� − �� + �� − 1� u(u − �zU.�� GZ# *2�N� ∑ ���N�*1 − ���0���Z 0 − u(
u I ln Xq{*zV.�{Z0�
q,�zV.�,Z��Y (28)
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3 Approximation of the vapour-liquid equilibrium of air in water
In this work, the solubility of air in water was analysed for a temperature of 298.15K and for
pressures ranging from approximately 1 to 97 atm. The air was considered a mixture of nitrogen,
oxygen, and argon in a concentration expressed in molar fraction of 0.7557, 0.2316, and 0.012691,
respectively. These values were taken from the NIST Reference Fluid Thermodynamic and
Transport Properties Database Version 9.1 (REFPROP) and the other components of air were not
included due to their relative low concentration.
3.1 Applying Henry’s Law - Estimating Henry’s Constant from van’t Hoff Expression
Firstly, the solubility of air in water was approximated using Henry’s law. Since our temperature of
interest (298.15K) is the same as the temperature of reference of the Henry’s Law constants shown
in Table 1, it was not necessary recalculate Henry’s constants for the air components. The higher
the��, the higher the solubility of the component in water; therefore argon dissolves in water faster
than nitrogen and oxygen.
Knowing the value of ��, the solubility of air in water was estimated as G:.(�:-H)I , by applying Eq. (12)
in the pressure range of interest, using intervals of 0.5 atm. In this work we considered the flow of
water as 20,000 kg/s and the flow of air as 20 kg/s. The initial mass of component i in the air was
determined by multiplying the composition of air by its flow, and as the pressure increased, the
mass of a gaseous species in air was calculated by subtracting the amount transferred to the water.
The mass of each component in water at a certain pressure was estimated by multiplying its
solubility at that pressure for the flow of water (20,000 kg/s); and the mass transferred from the air
to the liquid phase was considered to be the difference of its mass in water at each pressure. Figure
2 illustrates the mass behaviour in the gas phase for each component of air analysed and, as
predicted, argon is the first component to become entirely removed from the gas phase, due to its
higher solubility in water, and nitrogen is the last one to be completely removed.
J. Andrade Solubility Calculations for Hydraulic Gas Compressors
Figure 2 - Mass of N2, O2 and Ar in the gas phase as the absolute pressure of the system increases
3.2 Applying Henry’s Law Expressions
In his book, Multiphase Flow Dynamics, Kolev
literature on the solubility of O
expressions in order to ease their application in computational s
following recommended expressions of
��v��, %� = �NZ + Nz� + Nx�
where:
NZ:� = −133424.80726−7.78632 ∗
��v��� = 10�&��MNZ + Nz�̅ + N�NZ` + NZZ�̅ + NZz�̅z� ln �Q where:
NZ:Zz = −2.10973
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and Ar in the gas phase as the absolute pressure of the system increases
Applying Henry’s Law - Estimating Henry’s Constant from Analytical
In his book, Multiphase Flow Dynamics, Kolev [14] collected experimental data available in the
literature on the solubility of O2, N2, H2 and CO2 in water and correlated them with analytical
expressions in order to ease their application in computational simulations. In the present work, the
following recommended expressions of �� for N2 and O2 in water are applied:
�z�10¡ + �N¢ + N�� + N¡�z�%10 + �N£ + N¤� + N
80726, 826.81456,−1.17389,−66.79008,0.4994610,¢, 0.02728,−2.03146 ∗ 10,¢, 3.28 ∗ 10,£
Nx�̅z + �N¢ + N��̅ + N¡�̅z�� + �N£ + N¤�̅ + N��̅z��Q (30)
∗ 10z, 2.32745,−1.19186 ∗ 10,z, −2.02733 ∗ 10,
21 August 2013
Estimating Henry’s Constant from Analytical
collected experimental data available in the
in water and correlated them with analytical
imulations. In the present work, the
N��z�%z10,¢
(29)
49946,
��z +
,Z,
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2.45925 ∗ 10,x, −1.21107 ∗ 10,�, 9.77301 ∗ 10,�, −1.43857 ∗ 10,¡, 6.84983 ∗ 10,�, 4.79875 ∗ 10Z, −5.14296 ∗ 10,Z, 2.61610 ∗ 10,x
And
�̅ = % 10�⁄ .
These analytical expressions give Henry’s Constant in Pa as a function of temperature in K and
pressure in Pa. The mean error of Eq. (29) and Eq. (30) is 0.3% and 7.1%, respectively, in
comparison with all the experimental data analysis in [14]. The molar concentration of each
component in water has been calculated by the application of Henry’s law, already stated in Eq.
(11) and displayed again here:
�� = ����
The equivalent mass concentration, J�, in M¥� ¥¦#$§�⁄ Q is:
J� = ;(K(;(K({�Z,;(�K¨v©
(31)
In this work, it was considered that the gases in air behave independently; consequently, the total
mass fraction of air in water was considered to be the sum of each component’s mass fraction. For
greater precision, direct measurements must be carried out.
3.3 Comparison with Experimental Data from the Literature
A review of the solubility of nitrogen and air in most liquids for which experimental data were
available was proposed in [15]. The authors present the smoothed values of the air solubility in
water at elevated pressures as cubic centimeters of gas at standard pressure and temperature per
gram of water (cm³ (STP) per gram of water), a unit referred as S. To express solubility as a mass
fraction (gram of air per gram of water), the values were multiplied by 4.462 × 10,�� , where �
is the molar mass of air, as stated in [14]. The results, referred in this work as “B-R-T Data”, are
presented in Table 2 below.
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Table 2: B-R-T Data of Air Solubility in Water Expressed as a Mass Fraction (gram of air per gram of water)
T, K 1 MPa 5 MPa 10 MPa 15 MPa 20 MPa 25 MPa
273.15 0.000267 0.001505 0.002959 0.004212 0.005246 0.006086
278.15 0.000242 0.001361 0.002675 0.003812 0.004742 0.005505
283.15 0.000221 0.001243 0.002442 0.003476 0.004342 0.005026
288.15 0.000204 0.001146 0.002248 0.003205 0.003993 0.004639
293.15 0.00019 0.001066 0.002093 0.002985 0.003721 0.004303
298.15 0.000177 0.000999 0.001964 0.002791 0.003489 0.004032
303.15 0.000168 0.000942 0.001848 0.002636 0.003282 0.003812
308.15 0.000159 0.000895 0.001757 0.002507 0.003127 0.003618
313.15 0.000152 0.000858 0.001693 0.002403 0.002998 0.003463
318.15 0.000147 0.000826 0.001628 0.002313 0.002881 0.003334
323.15 0.000142 0.0008 0.001576 0.002235 0.002791 0.00323
328.15 0.000138 0.000779 0.001538 0.002184 0.002714 0.003153
333.15 0.000136 0.000764 0.001499 0.002132 0.002662 0.003088
338.15 0.000133 0.000751 0.001473 0.002106 0.002623 0.003037
343.15 0.000132 0.000742 0.00146 0.00208 0.002584 0.002998
According to Perry’s Handbook of Chemical Engineering [16], which presents tabulated values of
Henry’s Law constants over a range of temperatures (Table 3), at 25°C (298.15K) Henry’s Constant
is equal to 72,000 atm. Applying Eq. (11) in conjunction with Eq. (31) and considering the partial
pressure in the vapor phase equals to the total pressure of the system, the mass fraction of air in
water was obtained (P-Air Results) for the range of pressures analysed.
Table 3: Henry's Law Constants for the Solubility of Air in Water as atm
T, °C 0 5 10 15 20 25 30 35
10⁻⁴ ·�� 4.32 4.88 5.49 6.07 6.64 7.20 7.71 8.23
T, °C 40 45 50 60 70 80 90 100
10⁻⁴ ·�� 8.70 9.11 9.46 10.1 10.5 10.7 10.8 10.7
The graph in Figure 3 presents the solubility of air in water calculated via the estimation of �� using
the van’t Hoff expression, as well as, via the application of Kolev’s correlations, as described in
Sections 3.1 and 3.2, respectively. In addition, the B-R-T Results and P-air Results are also plotted.
From the graph, up to pressures equivalent to 300 meters of water, the solubility calculated in this
work matches the values from the literature (B-R-T and P-air Results) with good accuracy. At
pressures higher than 300 meters of water, the results from Henry’s Law approximate more closely
with the P-air; whereas, the use of Kolev’s Correlations lead to values closer to the B-R-T Results.
Overall, from the comparison, the results from this work closely agree with the experimental data.
Hence, it can be said that Henry’s Law is able to predict the solubility of air in water at high
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pressures with reasonable precision. Consequently, for this system it was not necessary to use EoS
for the solubility calculation.
Figure 3 - Solubility of air in water at pressure ranging from 0 to nearly 1,200 meters of water: Results from the
Application of Henry's Law and Kolev's Correlations and Experimental Data from [15] and [16]
0
0.0005
0.001
0.0015
0.002
0.0025
0 200 400 600 800 1000 1200
So
lub
ilit
y [
g a
ir/
g w
ate
r]
Pressure [meters of water]
Henry Law
Kolev's Correlations
B-R-T Data
P-air Data
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4 Approximation of the vapour-liquid equilibrium of combustion gases in water
4.1 Combustion gases from different fuels
Combustion is defined as a “chemical process of oxidation that occurs at a rate fast enough to
produce temperature rise and usually light, either as a glow or flame” [17]. In this work, the
complete combustion of the fuels: methane, diesel, anthracite, and high-sulphur fuel oil (HSFO)
with oxygen was studied. The solubility of the flue gases in water, especially CO2, was also
analysed. Herein, the source of the oxidizing element (oxygen) was air with a composition of
78.12% N2, 20.96% O2 and 0.92% Ar in molar percentage.
The combustion reactions for methane and diesel with air are based on the general formulas of these
compounds, «�¢ and «Zz�zx, respectively, and are explicit in the equations:
1«�¢ + 2¬z + 7.45z + 0.09®¯ → 1«¬z + 7.45z + 0.09®¯ + 2�z¬
1«Zz�zx + 17.75¬z + 66.16z + 0.78®¯ → 12«¬z + 66.16z + 0.78®¯ + 11.5�z¬
The composition of the flue gases on a dry basis is determined by the stoichiometry of these
equations, associated with the formula mass of the components, and is shown in Table 4.
In the case of anthracite and HSFO, there are no general compound formulae available in the
literature; however, in [18] the authors propose a methodology to calculate the carbon to CO2 mass
conversion factor for HFSO and the result was 3.021 - read as: 3.021 mass units of CO2 emitted per
each mass unit of HFSO combusted. The HFSO analysis from [18] indicated 2.7 wt% sulphur (S)
content; hence, we calculated the Sulphur to SO2 mass conversion factor for HFSO as 0.054 using
the same methodology. In [19], the authors presented an ultimate analysis of a raw anthracite coal
from China and the mass percentage of C is 92.27%. In this work, we considered that all the other
components are inert, including sulphur, since its content of 0.79% is negligible. Based on the
methodology of [18], the carbon to CO2 mass conversion factor for anthracite was 3.384.
The general combustion equation of carbon and sulphur with oxygen contained in air is shown
below.
1« + 1¬z + 3.73z + 0.04®¯ → 1«¬z + 3.73z + 0.04®¯
1° + 1¬z + 3.73z + 0.04®¯ → 1°¬z + 3.73z + 0.04®¯
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Considering the stoichiometry of the combustion reactions and the formula mass of the compounds,
for each gram of CO2 emitted, 2.37 g of N2, and 0.04 g of Ar are emitted. For each gram of SO2,
1.63 g of N2 and 0.03 g of Ar are emitted. The dry compositions of the flue gases from the
combustion with air of anthracite and HFSO, in conjunction with methane and diesel, are shown in
Table 4 in molar and mass fractions.
Table 4: Composition of the flue gases from the complete combustion of fuels with oxygen contained in air in molar
fraction
SO2 CO2 N2 Ar
mol/mol g/g mol/mol g/g mol/mol g/g mol/mol g/g
methane - - 0.117069 0.17169 0.872654 0.814629 0.010277 0.013681
diesel - - 0.152024 0.21891 0.838106 0.768189 0.00987 0.012901
anthracite - - 0.2096 0.293064 0.7812 0.69526 0.0092 0.011676
HSFO 0.002527 0.005137 0.206163 0.287927 0.782099 0.69526 0.009211 0.011676
4.1.1 Estimation of the Solubility of Combustion Gases using Henry’s Law
In this work, the solubility of combustion gases from the fuels: methane, diesel, anthracite, and
HFSO in water at 25°C and high pressures was analysed. The solubility was estimated applying
Henry’s Law and the values of Henry’s Constant for SO2, CO2, N2 and Ar were directly taken from
Table 1, as the temperature of reference (T̀ ) is the same as our temperature of study. Using the
composition of the flue gases shown in Table 4, the solubility was estimated applying Eq. (12) at
absolute pressures increasing from atmospheric until nearly 60 atm, in intervals of 0.5 atm.
Additionally, it was considered that the flow of flue gas and the excess flow of water was
approximately 20 kg/s and 2,000 kg/s, respectively. The mass of each gas component � in water at a
given pressure was determined by the multiplication of its solubility in G :(:±./²�I for the water flow.
The mass transferred from the gas to the liquid phase as the system pressure increases was the
difference between the mass of component � at each pressure. The initial mass of � in the
combustion gas was calculated by multiplying its mass fraction in the flue gas from Table 4 for the
flue gas flow, and this mass diminishes as the gas is transferred between the phases due to
increasing pressure, until the component � is completely removed from the gas phase. Due to
interest in methods of sequestration of CO2, there is special concern for the estimation of the
pressure at which all CO2 is removed from the flue gas produced by the complete combustion of
methane, diesel, anthracite and HFSO. The mass transfer of CO2 from the gas phase to the liquid
phase for the combustion gas of different fuels is illustrated in Figure 4, which shows the total mass
of CO2 in the gas phase diminishing as the pressure increases until the total removal of CO2 from
the gas.
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Figure 4 - Pressures at which all CO2 is dissolved in water for the fuel methane, diesel, anthracite and HFSO
The initial mass of CO2 in the combustion gas from anthracite and HFSO is greater than the other
fuels due to the higher initial mass fraction of this component in the gas. Additionally, the analysis
of the graph shows that the pressure at which the mass of CO2 in the gas phase reaches zero is
practically the same for all of these fuels, that is, around 20 m H2O. This can be explained by the
fact that we do not take into account in this work the kinetics of the mass transfer, instead, we
consider that CO2 instantaneously transfers from the gas to the liquid phase.
0
1000
2000
3000
4000
5000
6000
7000
10 20 30 40 50 60
Ma
ss i
n t
he
ga
s p
ha
se [
g]
Pressure [m of water]
Methane
Diesel
Anthracite
HFSO
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5 Approximation of the vapour-liquid equilibrium of carbon dioxide in water
The solubility of carbon dioxide in water has been calculated at 50°C, because the experimental
data available in the literature are more abundant at this temperature, and the pressure range
analysed was from 1 atm until 60 atm.
5.1 Estimating the solubility of CO2 in water using Henry’s Law
Figure 5 - Solubility of CO2 in water at 50°C obtained applying Henry's Law
The solubility of CO2 in water has been estimated using Henry’s Law following the concepts
discussed in Section 2.2 and Figure 5 above correlates the solubility of CO2 with the system’s
absolute pressure. To predict Henry’s Constant at 50°C, the vant’t Hoff expression (Eq. 13) was
used with the parameters for CO2 presented in Table 1. The simulations were performed using
Excel, assuming infinite dilution, because the flow of H2O (20,000 kg/s) was much higher than the
gas flow (20 kg/s), and that the gas phase has initially the properties of pure CO2, because the
amount of H2O in the CO2-rich phase is reduced at temperatures below 100°C. [20] The solubility
of CO2 in water in mass concentration G :³©v:±./²�I was calculated using Eq. (12) and it could be
modified to molar concentration by using Equation (31).
0
0.005
0.01
0.015
0.02
0.025
0 10 20 30 40 50 60 70
So
lub
ilit
y [
mo
l C
O2
/m
ol
H2
O]
Pressure [atm]
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5.2 Experimental Data of the System CO2 + Water
Because of the importance of the CO2 + H2O system, there have been many experimental studies
conducted on its solubility. Carrol, Slupsky and Mather assembled experimental solubility data for
the carbon dioxide-water system limiting the investigation to the low pressure region (i.e. pressures
less than 1 MPa) due to the limitations of the Henry’s Law chosen to correlate the solubility [21]. In
Spycher et al. (2003), data from experimental work was summarized for a CO2-H2O mixture at
temperatures from 12°C to 110°C and pressures up to 60 MPa. At our temperature and range of
pressures of interest (50°C and 0.1 to 70.3 MPa) the authors in [20] reviewed at least nine published
studies, and their experimental data on CO2 solubility in water are tabulated in Table 5 and referred
here as “S-P-E Data”. Another source of experimental data considered in this work is Perry’s
Handbook of Chemical Engineering. The results of CO2 solubility in water at 50°C are referred here
as “P - CO2 Data” and are also presented in Table 5.
Table 5 - Solubility of CO2 in water: Experimental Data at 50°C and from nearly 10 to 7,233 meters of water
P (m H2O) x CO2 P (m H2O) x CO2 P (m H2O) x CO2 P (m H2O) x CO2
10.332 0.000342 ᵃ 695.4354 0.0161 ᵇ 1033.262 0.0198 ᵇ 2039.4 0.023 ᵇ
20.664 0.000683 ᵃ 721.9476 0.0176 ᵇ 1131.867 0.021 ᵇ 2049.597 0.02347 ᵇ
103.32 0.0033 ᵃ 767.8341 0.0175 ᵇ 1233.837 0.0214 ᵇ 2066.932 0.02289 ᵇ
206.64 0.00634 ᵃ 774.972 0.01779 ᵇ 1245.054 0.02096 ᵇ 3069.297 0.02514 ᵇ
257.9841 0.00774 ᵇ 823.9176 0.019 ᵇ 1291.96 0.02106 ᵇ 3099.888 0.02457 ᵇ
309.96 0.0091 ᵃ 889.1784 0.01768 ᵇ 1438.797 0.0217 ᵇ 4132.844 0.02606 ᵇ
371.952 0.01063 ᵃ 926.9073 0.02 ᵇ 1504.058 0.02215 ᵇ 5098.5 0.028 ᵇ
412.9785 0.0109 ᵇ 1028.877 0.0205 ᵇ 1504.058 0.02207 ᵇ 6199.776 0.02868 ᵇ
515.9682 0.0137 ᵇ 1029.897 0.02075 ᵇ 1549.944 0.02174 ᵇ 7232.732 0.02989 ᵇ
516.9879 0.01367 ᵇ 1032.956 0.02081 ᵇ 1549.944 0.021 ᵇ
617.9382 0.0161 ᵇ 1032.956 0.02018 ᵇ 1802.83 0.02262 ᵇ
ᵃ Values selected from [16]
ᵇ Values selected from [20]
The experimental data tabulated above are plotted in Figure 6 along with the results obtained by the
application of Henry’s Law from Section 5.1 for comparison purposes.
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Figure 6 - Solubility of CO2 in water at pressure ranging from 0 to nearly 1,500 meters of water: Results from the
Application of Henry's Law and S-P-E [20] and P-CO2 [16] Experimental Data
The graph shows that Henry’s Law predicts well the solubility of CO2 in water in the range of
pressures from atmospheric until approximately 300 m H2O. Considering that for hydraulic gas
compressors, from Section (4.1.1.), the estimated pressure at which all CO2 from combustion gases
would be completely dissolved in water was approximately 202.65kPa (20.65 m H2O), the
prediction of CO2 solubility in water using Henry’s Law seems acceptable.
5.3 Estimating the solubility of CO2 in water using the VR Therm
Due to the limitations of Henry’s law to predict equilibrium data at high pressure, the solubility of
CO2 in water was also predicted using cubic Equations of State for more precise results. Since the
application of these EoS requires interactive mathematical procedures, the software program,
VRTherm, was used as an add-in to Excel to determine the equilibrium properties of the system.
VRTherm is a library program that can predict thermodynamic properties and physical properties of
complex mixtures, and can be downloaded at <www.vrtech.com.br>. VRTherm can calculate the
fraction of vaporization, the composition of the liquid phase and vapor phase composition, given the
overall composition of the input stream, and the temperature and pressure of the equilibrium. In
VRTherm several Equations of State are available, including Soave-Redlich-Kwong and Peng-
Robinson, presented in this work as Eq. (22) and Eq. (23). Flash calculations were performed using
these two EoS and, in Figure 7, the composition of the liquid phase as a function of the absolute
0
0.005
0.01
0.015
0.02
0.025
0 200 400 600 800 1000 1200 1400 1600
So
lub
ilit
y [
mo
l C
O2
/m
ol
wa
ter]
Pressure [m H2O]
S-P-E Data
P-CO2 Data
Henry's Law Results
J. Andrade Solubility Calculations for Hydraulic Gas Compressors 21 August 2013
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pressure of the system is shown. The results from VRTherm were less accurate than the results
obtained from using Henry’s Law.
Figure 7 - Solubility of CO2 in water predicted from SKR EoS and PR EoS, using VRTherm software, compared
with the S-P-E Experimental Data
5.4 Limitations of Henry’s Law in predicting the solubility of the system CO2 + Water
The carbon dioxide-water system is of great scientific and technological importance. The solubility
of carbon dioxide (CO2) in water (H2O) is one of the most often studied phenomena in all physical
chemistry; however, the dissimilarity between the molecules makes the phase relations in the
system quite complex and the CO2 + water system is highly non-ideal. [4] This dissimilarity is
mainly due to the fact that H2O is small and dipolar whereas CO2 (in the gaseous state) is large and
non-polar; therefore, the miscibility of the components in the liquid phase is extremely low. At low
temperatures, the molecules do not mix, and the two liquids are independently stable– one rich in
CO2 and the other in H2O. The solubility rises with increasing temperature until 265°C and at
approximately 220 MPa when mutual solubility is complete.
CO2 is more volatile than H2O. CO2 has its triple point located at low pressure and temperature (0.5
MPa and -56.6°C) and its critical point at high pressure and approximately at room temperature (7.4
MPa and 31.1°C); whereas, the triple point of H2O is at 0.01°C and at very low pressure (0.0006
MPa) and its critical point is at high pressure and temperature (374°C and 22.1 MPa) [4].
0
0.005
0.01
0.015
0.02
0.025
0 200 400 600 800 1000
So
lub
ilit
y [
mo
l C
O2
/m
ol
wa
ter]
Pressure [m of water]
S-P-E
PR
SRK
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In addition, partitioning occurs, in which CO2 quickly forms hydrated carbon dioxide in water and
then dissociates as carbonic acid (H2CO3). However, the kinetics of these reactions are not taken
into account when using Henry’s Law.
Moreover, some errors in this study can arise due to the effects of electrolytes. It is known that
dissolved solids affect the phase portioning of H2O and CO2; therefore, further studies are necessary
to investigate the impact of dissolved solids on the mutual solubility of CO2 and H2O. [20]
5.5 Estimating the solubility of CO2 in water using the model of Diamond (2003)
Diamond and Akinfiev (2003) evaluated twenty five experimental measurements of the solubility of
CO2 in pure water from 0.1 to 100 MPa. The reliable experimental data were separated from the
unreliable data using the criteria presented in [22] and were correlated by a basic thermodynamic
model based on Henry’s Law and on new high-accuracy EoS. The performance of this model was
evaluated and it is reasonable to consider the activity coefficient of aqueous CO2 equal to one up to
solubilities of nearly 2 mol%. At higher solubilities, the activity coefficients are greater than unity
at low temperatures, and lower than unity at high temperatures. The authors applied an empirical
correlation function for this trend to the basic model and the resulting modified model describes the
accepted data with a 2% level of precision . The corrected model is available as a computer code at
< http://www.geo.unibe.ch/diamond> and, using it in this work, the solubility of CO2 in water at
50°C and from 1 atm to 60 atm was estimated. The results are referred as “D-A Data” and are
shown in Figure 8, in conjunction with the S-P-E and P-CO2 experimental data.
The application of the model from Diamond and Akinfiev is suitable for the estimation of the
solubility of CO2 in pure water from 1 to 60 atm, due to the increased precision of the results when
compared with experimental data.
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Figure 8 - CO2 solubility in water at 50°C using the model from [22] referred as D-A Data in comparison with S-P-
E and P-CO2 experimental data
0
0.005
0.01
0.015
0.02
0.025
0 200 400 600 800 1000
So
lub
ilit
y [
mo
l C
O2
/m
ol
wa
ter]
Pressure [m H2O]
S-P-E Data
P-CO2 Data
D-A Data
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6 Conclusions
Despite recent efforts to predict the solubility of gases at high pressures, there are insufficient
results in the literature, many of which are imprecise. This work focused on the estimation of air
and combustion gases solubility in pure water with the aim to update hydrodynamic calculations for
hydraulic gas compressors.
The solubility of air components in water was shown to be precisely estimated via the utilization of
Henry’s Law, either using either the van’t Hoff expression or Kolev’s correlations, when compared
with reliable experimental data from the literature. As expected, due to the higher solubility in
water, argon is the first component that is completely dissolved in water, followed by O2 and N2.
The solubility of combustion gases produced from anthracite, methane, HFSO and diesel were also
estimated using Henry’s Law, however, by neglecting the kinetics associated with the mass transfer
from the gas to the liquid phase, the pressure at which the combustion gases would be totally
dissolved was almost the same for all fuels studied.
However, the application of Henry’s Law to estimate the solubility of CO2 in water was shown to
be inadequate at pressures higher than 20 atm; moreover, the use of the program VRTherm led to
results far from the experimental measurements. The application of the model proposed by
Diamond and Akinfiev was the most suitable for the estimation of CO2 solubility in water from 1 to
60 atm.
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