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Experimental and modeling investigation on surface tension and surface properties of (CH4 + H2O), (C2H6 + H2O), (CO2 + H2O) and (C3H8 + H2O) from 284.15 K to 312.15 K and pressures up to 60 bar
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
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Experimental and modeling investigation on surface tension and surface properties of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) from 284.15 K to 312.15 K and pressures up to 60 bar Shahin Khosharay, Farshad Varaminian* School of Chemical, Gas and Petroleum Engineering, Semnan University, Semnan, Iran
article info
abstract
Article history:
In this work, the experimental surface tensions for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O)
Received 29 June 2014
and (C3H8 þ H2O) binary systems have been measured with the pendant drop tensiometer
Received in revised form
at temperatures ranged (284.15e312.15) K and pressures ranged (1e60) bar. Subsequently,
2 August 2014
the combination of gradient theory (GT) and CheneKreglewski Statistical Association Fluid
Accepted 8 August 2014
Theory equation of state (CK-SAFT EOS) is applied for modeling these experimental surface
Available online 16 August 2014
tensions and determining the density profiles of interface. Also, a proposed influence parameter as a function of the densities of the bulk phases is applied for these systems.
Keywords:
The binary interaction parameters for these systems are determined according to the bulk
Surface tension
densities and surface tensions. Subsequently, the density profiles for the surface layer are
Pendant drop
determined. The surface tensions determined with the gradient theory (GT) agree well with
Gradient theory
experimental surface tensions for these systems (overall AAD ~ 1.36 and 1.37 for constant
CK-SAFT EOS
and the proposed influence parameters, respectively). © 2014 Elsevier Ltd and IIR.
rimentale et par mode lisation de la tension Etude expe te s superficielles de (CH4 þ H2O), superficielle et des proprie (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) pour des ratures allant de 284,15 K a 312,5 K et des pressions tempe 60 bar allant jusqu'a orie du gradient ; CK-SAFT EOS Mots cles : Tension superficielle ; Goutte suspendue ; The
* Corresponding author. Tel./fax: þ98 231 3354280. E-mail address: [email protected] (F. Varaminian). http://dx.doi.org/10.1016/j.ijrefrig.2014.08.003 0140-7007/© 2014 Elsevier Ltd and IIR.
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Nomenclature A 0 A ares aseg achain aassoc AAD B 0 B c calc d EOS exp f f0 g k kij L
1.
constant coefficient of influence parameter constant coefficient of binary interaction parameter reduced Helmholtz Energy Helmholtz energy of segments Helmholtz energy of chain-forming bonds among the segments Helmholtz energy of association contribution average absolute deviation constant coefficient of influence parameter constant coefficient of binary interaction parameter influence parameter calculated result diameter equation of state experimental fugacity Helmholtz free energy density gravitational constant Boltzmann constant binary interaction parameter (BIP) liquid
Introduction
Because of increasing the need for the properties of refrigerants, several studies focus on the modeling the properties of the (refrigerant gases þ water) systems. The surface tension of (refrigerant gas þ water) system is also an important parameter for the conditions with hydrate formation. Furthermore, the interfacial phenomenon strongly influences the nucleation of gas hydrate; therefore, interfacial properties such as the surface tension of gas/water systems may greatly affect the hydrate formation kinetics. On the other hand, the surface tension relates to the nature of molecular forces at the interface phase so that knowledge of interfacial properties such as interfacial profiles and thickness of surface layer is of importance. It is known that the direct experimental measurement of surface layer profile is quite difficult and it was investigated for few systems. One method to achieve this aim is using equations of states (EOSs) for determining the interfacial properties (Abraham et al., 2001; Enders and Kahl, 2008; Tjahjono and Garland, 2010). The gradient theory of inhomogeneous fluid (Miqueu et al., 2003, 2011; Li et al., 2008; Khosharay and Varaminian, 2013) is one of the models in which the properties of bulk phases can be related to the properties surface layer by using a suitable equation of state (EOS). Furthermore, the gradient theory requires two inputs: (1) the Helmholtz free energy density that belongs to the homogeneous fluid; (2) the influence parameter that belongs to the inhomogeneous fluid. The Helmholtz free energy density for the homogeneous fluid can be calculated by using a suitable thermodynamic model. The influence parameter is a molecular theory based input of this theory, but
27
m effective number of segments within the molecule n mole density R ideal gas constant P pressure ref reference fluid CK-SAFT CheneKreglewski statistical associating fluid theory T temperature reduced temperature Tr temperature-dependent dispersion energy of u0/k interaction between segments V vapor temperature-independent segment volume voo x mole fraction z position in the interface Greek letters g surface tension m chemical potential molar density of component i ri 3 /k association energy k volume of association 4 fugacity coefficient U grand thermodynamic potential
it is almost impossible to use this parameter based on molecular theory in practice; therefore, applying a semiempirical expression is one of the practical ways of using this parameter in practice (Lin et al., 2007; Lafitte et al., 2010). In recent years, the experimental and modeling investigations are conducted on the interfacial properties of gasewater systems. Yan et al. (2001) measured the surface tension of N2 þ H2O, (CH4 þ N2) þ H2O and (CO2 þ N2) þ H2O mixtures with pendant drop method at different pressures, temperatures and compositions. Subsequently they applied linear gradient theory (LGT) in combination with SRK EOS for N2 þ H2O and (CH4 þ N2) þ H2O systems. They concluded that modeling results are in a good agreement with the experimental surface tensions. Ren et al. (2000) measured the high pressure surface tensions for CH4 þ H2O and (CH4 þ CO2) þ H2O mixtures by using pendant drop method under various pressure, temperature and composition conditions. Georgiadis et al. (2010) measured the surface tension of CO2 þ H2O system at pressure ranged (1e60) MPa and temperatures ranged (298e374) K. They also applied the pendant drop method for surface tension measurement. Kvamme et al. (2007) applied a novel apparatus and the quasi-static pendant drop method to measure measured the surface tension of CO2 þ H2O system at pressure ranged (0.1e20) MPa and temperatures ranged (278e335) K. They also performed molecular dynamics simulations for this system. Chiquet et al. (2007) reported pendant drop measurements of surface tensions for water/CO2 system at temperatures ranged (308e383) K and pressures ranged (5e45) MPa. They also measured phase densities simultaneously. Khosharay and Varaminian (2013) applied the linear gradient theory (LGT) in combination with the cubic-plus-association equation of state (CPA EOS) to
28
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
calculate the surface tensions of (CH4 þ N2) þ H2O and (N2 þ CO2) þ H2O systems. The estimations of this model agreed well with experimental surface tensions. Schmidt et al. (2007) applied linear gradient theory (LGT) in combination with the SoaveeRedlicheKwong (SRK) and the PengeRobinson (PR) equations of state to correlate the surface tensions of the binary system of methane/water. They also determined the temperature dependant influence parameters for pure component and binary interaction coefficient of the influence parameter. Several other investigations are done in (Wiegand and Franck, 1994; Jho et al., 1978; Chun and Wilkinson, 1995; Wesch et al., 1997; Hebach et al., 2002; Park et al., 2005; Bachu and Bennion, 2009; Chalbaud et al., 2009; Bikkina et al., 2011). In this study, the surface tensions of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems are measured by using pendant drop technique at temperatures ranged from 284.15 K to 312.15 K and pressures up to 60 bar. Subsequently, the gradient theory of inhomogeneous fluid in combination with the CheneKreglewski Statistical Association Fluid Theory equation of state (CK-SAFT EOS) (Kontogeporgis and Folas, 2010; Yarrison and Chapman, 2004; Chapman et al., 1990; Huang and Radosz, 1990) is applied to determine the surface tension and the surface layer density profiles for these binary systems. Additionally, the proposed expression on the basis of the densities of bulk coexist phases (Khosharay et al., 2013, 2014) is used for the influence parameters of the gradient theory. To identify the advantages of the proposed influence parameter for these binary systems, the proposed influence parameter is applied and compared with the constant influence parameter for these binary systems.
2.
Experimental
2.1.
The applied materials
The analytical grade methane (CH4), ethane (C2H6), propane (C3H8) and carbon dioxide (CO2) are used with a purity of 99.99% that were supplied by Technical Gas Services. The double distilled water is also applied. The information of these materials has been shown in Table 1.
Table 1 e Materials used for experiments. Component
Chemical formula
Purity
Carbon dioxide Methane
CO2
99.99%
CH4
99.99%
Ethane
C2H6
99.99%
Propane
C3H8
99.99%
Water
H 2O
Deionizeddistilled
Supplier Technical Gas Services Technical Gas Services Technical Gas Services Technical Gas Services Bahrezolal, Iran
Fig. 1 e Schematic of experimental apparatus for surface tension measurements (pendant drop), 1. Handle pump; 2. Drop needle; 3. High pressure cell; 4. Digital camera; 5. Light source; 6. Temperature transmitter; 7. Pressure transmitter; 8. Temperature indicator; 9. Pressure indicator; 10. Computer; 11. Gas tank; 12. To vacuum trap; 13. Temperature controlled air bath.
2.2.
Apparatus
The schematic of experimental high-pressure pendant drop set-up has been shown in Fig. 1. The experiments are conducted in the cylindrical and high-pressure stainless steel cell with a total capacity of 21 mL which can operate pressures up to 10 MPa and temperatures ranging from 283.15 K to 323.15 K. The cell has two polished Pyrex-glass which allows observing inside the cell from a horizontal axis; therefore, the visual observations of the droplet shape is possible. The cell is equipped with the stainless steel capillary tube (o.d. 1.22 mm and i.d. 1.00 mm) for hanging the drop of liquid. A handle pump is also applied for feeding the liquid into the cell through the capillary tube which can operate pressures up to 40 MPa. The temperature is measured by using a PT100 thermometer (Pro-Temp Controls, Santa Ana, California, United States) with ±0.1 K accuracy. A BD-Sensors-Str.1 pressure transmitter with ±0.1 bar accuracy is utilized to measure the pressure of the cell. The apparatus contains a coolant bath with controllable circulator that is utilized to circulate the air. The cell is immersed in a constant temperature air bath whose temperature is controlled within 0.1 K by using a PID controller. In order to capture the images of the drop and measure the surface tension, the system is also equipped with the digital camera which is connected to a personal computer.
2.3.
Experimental procedure
Prior to conducting each experiment and measurement, the pendant drop cell and its connections were soaked in acetone for 30 min. Subsequently, the cell and its connections were washed and rinsed with double distilled water for three times and dried with compressed air. Then the cell was evacuated with a vacuum pump and charged with the gas up to the
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
specified pressure. By using a handle pump, the liquid water was introduced slowly into the pendant drop cell through the stainless steel capillary tube. The water pendant drop formed at the tip of the stainless steel capillary tube which was vertically inserted in the pendant drop cell. The pendant drop cell was left more than 1 h to reach the specified temperature and make sure that the pendant drop was in the thermal equilibrated with its surroundings and the adsorption of gas molecules on the drop surface reached equilibrium. During the process, for measuring the surface tension, the images of the drop were captured with the digital camera and light source.
2.4.
Calculation of the surface tension
In this work, the surface tension values are calculated by using equations of Andreas et al. (1938). g¼
Drd2e g H
1 ds ¼f H de
3.
Brief description of the model
3.1.
Gradient theory
Z g¼
rV ref
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X X dni dnj u t2ðUðnÞ UB Þ cij dnref dnref dnref i j
UðnÞ ¼ f0 ðnÞ
in which UB ¼ P; P denotes the equilibrium pressure; r the mole density; g the surface tension. rL and rV show the mole
ni mi;b
(4)
in which mi,b shows the chemical potential of component i belonging to the bulk phase. As it is mentioned in the introduction part, to apply the gradient theory, it is necessary to calculate two inputs for determining surface tension: the Helmholtz energy density of the homogeneous fluid and the influence parameters of the inhomogeneous fluid. In order to calculate the density profile in the surface layer, the following equation is applied: ðzÞ
rref
Z
(2)
(3)
X i
z ¼ z0 þ
The extensive description of gradient theory of fluid interfaces (Miqueu et al., 2003, 2011; Li et al., 2008; Khosharay and Varaminian, 2013) exists in several investigations; therefore, in this work, the description of important aspects of gradient theory is considered. The detailed description of gradient theory can be found in Miqueu et al. (2003, 2011), Li et al. (2008), Khosharay and Varaminian (2013), Lafitte et al. (2010), Khosharay et al. (2013, 2014). Based on the gradient theory of interfaces, without presence of any external potential and assuming the planar interface, the surface tension of a mixture containing N components between the liquid (L) and vapor (V) phases is expressed as function of the local gradients in density as follows: rLref
densities belonging to the liquid and vapor bulk phases, respectively. Subscript ref represents the reference component. dri/drref is the mole density profile belonging to component i and cij shows the influence parameter. U is the grand thermodynamic potential and it is expressed as follows:
(1)
in which Dr denotes the density difference between liquid and gas phases, g shows the gravitational constant, ds represents the pendant drop diameter at height that is equal to the maximum diameter of the pendant drop (de). The relation between 1/H and ds/de is given from the work of Drelich et al. (2002). The uncertainty of the experimental data is in the range of (0.2e0.7) mN m1. The CK-SAFT EOS is applied to compute the density of gas phase. Except the carbon dioxide, the influence of the solubility of gases on the aqueous phase density is neglected because it is very low. The densities of aqueous phase for (CO2 þ H2O) system are given from Hebach et al. (2004).
29
r0ref
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP P dr drj u cij drrefi drref u t i j drref 2ðUðrÞ UB Þ
(5)
in which z shows the coordinate and subscript 0 shows the arbitrarily chosen origin.
3.2.
Free energy density and phase equilibrium
The CheneKreglewski-Statistical-Association-Fluid-Theory equation of state (CK-SAFT EOS) (Kontogeporgis and Folas, 2010; Yarrison and Chapman, 2004; Chapman et al., 1990; Huang and Radosz, 1990) is a version of the SAFT EOS. The CK-SAFT EOS is expressed in the terms of the residual molar Helmholtz energy as follows: ares aseg achain aassoc ¼ þ þ RT RT RT RT
(6)
where aseg belongs to the Helmholtz energy of segments, achain belongs to the chain-forming bonds among the segments and aassoc belongs to the association term. The detailed descriptions of these terms are given in earlier publications (Kontogeporgis and Folas, 2010; Yarrison and Chapman, 2004; Chapman et al., 1990; Huang and Radosz, 1990). Additionally, the temperature dependent of binary interaction parameter has been used for the dispersion term of CKSAFT EOS in which A and B show the constants and T shows the temperature in K. kij ¼ A0 þ B0 T
(7)
In order to calculate the phase equilibrium for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems, the fugacity (fi) of each component i should be equal in the coexisting equilibrium phases. fiI ¼ fiII
(8)
where superscripts I and II show the coexisting equilibrium phases. Eq. (8) can be expressed as follows: P4Ii xIi ¼ P4IIi xIIi
(9)
30
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where xIi and xIIi denote the mole fraction of components in the coexisting equilibrium phases. P shows the pressure and 4 denotes the fugacity coefficient. Based on the work of Li et al. (2008), the binary interaction parameters (kij) are fitted according to the experimental liquid bulk densities and surface tension data.
3.3.
The influence parameter
Since the theoretical based influence parameter is not applicable for many systems, the influence parameters for the pure fluids (c) should be regressed according to the experimental surface tensions of the pure fluids. 2
32
6 7 6 7 6 7 6 7 gexp 16 c ¼ 6Z rV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 26 7 6 f0 ðrÞ rmðrÞ þ Pdr7 4 rV 5
(10)
The deviations of the surface tensions are given as follows:
AADg ¼
N exp gcalc 1 X i gi 100 exp N i¼1 gi
(11)
In this work, two different forms of influence parameter are applied: (1) constant value of influence parameter and (2) the influence parameter based on the liquid and vapor densities (Khosharay et al., 2013, 2014). For pure fluids, the second form of influence parameter is expressed as follows: L r rV c ¼ V Ar þ BrL
(12)
To test the applicability of the proposed influence parameter for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems, the influence parameters of component i for the mixture are determined on the basis of the mole densities of component i in the bulk phases. The binary interaction coefficients (lij) are also set to be zero. I I r x rII xIIi ci ¼ I iI II II Ai r xi þ Bi r xi
(13)
in which and xIi are xIIi are the mole fractions of component i in the bulk phases.
3.4.
Molar density gradients
In this study, according to the geometric combing rule for the influence parameters (lij ¼ 0), the molar density gradients of
components through the surface layer are calculated by solving algebraic equations. This equation derived on the basis of the minimum Helmholtz free energy criteria (Lin et al., 2007): pffiffiffiffiffi vm1 pffiffiffiffiffi 2 c1 vm c2 vr2 vr2 dr1 T;P;r1 T;P;r1 ¼ pffiffiffiffiffi 1 dr2 pffiffiffiffiffi vm2 c2 vm c1 vr1 vr1 T;P;r2
(14)
T;P;r2
4.
Results and discussion
4.1.
Pure fluids
Prior to determining any surface layer properties, the influence parameters of each pure component should be obtained. In this work, the influence parameters for methane (CH4), ethane (C2H6), propane (C3H8), carbon dioxide (CO2) and water (H2O) are obtained based on the fitting of their surface tensions to the experimental data. The pure component parameters for CK-SAFT EOS are given from Huang and Radosz (1990). The pure fluid parameters and the gradient theory influence parameters are given in Tables 2 and 3, respectively.
4.2.
The binary systems
Firstly, the experimental surface tensions for (CH4 þ H2O) and (CO2 þ H2O) are reported in Table 4. In order to check the validity of the apparatus and measured surface tensions, the experimental surface tensions are compared with the literature reported by Ren et al. (2000), Georgiadis et al. (2010), Sachs and Meyn (1995). In Fig. 2, one can see good agreements with the literature data (Ren et al., 2000; Georgiadis et al., 2010; Sachs and Meyn, 1995). As it is shown in Table 4 and Fig. 2, at constant temperature, with increasing pressure, the steep decrease in surface tension is seen both for (CH4 þ H2O) and (CO2 þ H2O) systems due to enrichment of interface layer with the gaseous component. At constant temperature, for (CH4 þ H2O) system, increasing the temperature results in the decreasing surface tension. The temperature dependence for (CO2 þ H2O) system is rather complex. At the pressure range of (10e30) bar, the increase of the temperature results in lowering surface tension. At higher pressures, the slopes of isotherms are different; therefore, these isotherms intersect each other. These behaviors are also seen in the literature data of Georgiadis et al. (2010), Kvamme et al. (2007), Chiquet et al. (2007). Subsequently, the gradient theory of interface was utilized for modeling the surface tension of (CH4 þ H2O) and (CO2 þ H2O) systems. In this study, to test the advantages of
Table 2 e The pure component parameters of CK-SAFT EOS with their average absolute deviations. Fluid
m
yoo (mL mol1)
u0/k (K)
CO2 CH4 C2H6 C3H8 H2O
1.417 1.000 1.941 2.696 1.179
13.578 21.556 14.460 13.457 10.000
216.08 190.29 191.44 193.09 528.17
3 /k
(K)
e e e e 1809
102kAB
T range
AAD vliq
AAD P
e e e e 1.593
218e288 92e180 160e300 190e360 283e613
0.86 0.35 1.60 1.80 3.20
2.80 1.40 1.80 2.10 1.30
Ref. Huang Huang Huang Huang Huang
and and and and and
Radosz Radosz Radosz Radosz Radosz
(1990) (1990) (1990) (1990) (1990)
31
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Table 3 e The influence parameters for gradient theory with the average absolute deviations. Fluid
Density dependant influence parameter
AADg
A (mol2 J1 m5) 1020 B (mol2 J1 m5) 1020
Constant influence AADg T range parameter C (J m5 mol2) 1020
CO2
1.02859
3.65019
0.17
2.32500
2.81
218e288
CH4
0.55932
0.42956
3.89
6.75000
5.28
92e180
C2H6
0.19646
0.18233
1.49
5.20312
1.53
160e300
C3H8
0.066749
0.091551
0.94
10.62500
2.08
190e360
H 2O
5.32781
0.90123
4.17
1.20321
4.82
283e613
using the proposed expression of the influence parameters, the gradient theory (GT) is reapplied. To determine the surface tension of the (CH4 þ H2O) and (CO2 þ H2O) systems at temperatures ranged (284.15e312.15) K and pressures ranged (10e60) bar. To improve the description of the equilibrium densities that belongs the bulk phases and subsequently surface tensions, the binary interaction parameters of CKSAFT EOS is determined based on the experimental bulk densities and surface tensions. For constant influence parameter, the temperature dependent binary interaction parameters can be expressed as follows, respectively for (CH4 þ H2O) and (CO2 þ H2O) systems: kij ¼ 0:8343 þ 0:002578T
(15)
kij ¼ 0:1419 0:0004644T
(16)
For the proposed influence parameter, the temperature dependent binary interaction parameters are written as follows, respectively for (CH4 þ H2O) and (CO2 þ H2O) systems: kij ¼ 0:9393 þ 0:02821T
(17)
kij ¼ 0:1328 0:0004718T
(18)
As it is illustrated in Fig. 3 and Table 4, the combination of gradient theory (GT) and CK-SAFT EOS is capable of reproducing the surface tension for (CH4 þ H2O) and (CO2 þ H2O) systems. The deviations of the surface tensions for (CH4 þ H2O) and (CO2 þ H2O) systems are listed in Table 4. Furthermore, in Table 4, the results of the present model with
gexp Ref.
NIST Chemistry WebBook Rathjen, and Straub (1977) Vargaftik (1975) NIST Chemistry WebBook Miqueu et al. (2000) Holcomb and Zollweg (1992) NIST Chemistry WebBook Miqueu et al. (2000) Maass and Wright (1921) NIST Chemistry WebBook Miqueu et al. (2000) Maass and Wright (1921) NIST Chemistry WebBook Fu et al. (2000) Goncalves et al. (1991)
constant influence parameter are compared with the proposed influence parameter based on the densities of bulk coexist phases. Figs. 4 and 5 show the density profiles of the interface for (CH4 þ H2O) and (CO2 þ H2O) systems at T ¼ 298.15 K and P ¼ 30 bar. It is seen that the water partial density increases monotonically from the CH4-rich phase to the water-rich phase with the shape of tanh. Figs. 4 and 5 also illustrate the enhancement of CH4 and CO2 densities which corresponds to the adsorption of CH4 and CO2 molecules at the surface phase. In this study, the surface tensions of (C2H6 þ H2O) and (C3H8 þ H2O) systems are measured. The dependence of the surface tension of (C2H6 þ H2O) and (C3H8 þ H2O) systems on temperature and pressure are the same as (CO2 þ H2O) system. The combination of the gradient theory and CK-SAFT EOS is applied to model the surface tensions of these systems for the first time. Similar to (CH4 þ H2O) and (CO2 þ H2O) systems, the binary interaction parameters are determined on the basis of the experimental bulk densities and surface tensions. For constant influence parameter, the dependence of binary interaction parameters on the temperature is expressed as follows, respectively for (C2H6 þ H2O) and (C3H8 þ H2O) systems: kij ¼ 0:9421 0:00312T
(19)
kij ¼ 0:4022 þ 0:001523T
(20)
By using the proposed influence parameter for (C3H8 þ H2O) system, the density profiles and surface tensions cannot be determined. The temperature dependent binary interaction
32
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Table 4 e Measured and modeled surface tensions for (CH4 + H2O), (C2H6 + H2O), (CO2 + H2O) and (C3H8 + H2O) binary systems. T (K)
P (bar)
gexp (mN m1)
gcalc (mN m1) GT + density dependant influence parameter
GT + constant influence parameter
74.12 73.31 71.27 69.11 67.03 65.07 72.43 72.23 70.43 68.33 66.25 64.30 71.12 70.51 68.96
74.81 73.59 71.31 69.1 67.1 65.3 73.04 72.56 70.50 68.31 66.25 64.39 70.82 70.19 68.84
T (K)
P (bar)
CH4 + H2O 284.15 284.15 284.15 284.15 284.15 284.15 291.15 291.15 291.15 291.15 291.15 291.15 298.15 298.15 298.15
10 20 30 40 50 60 10 20 30 40 50 60 10 20 30
AAD
Constant influence parameter: 0.82; Proposed influence parameter: 0.79
CO2 + H2O 284.15 284.15 284.15 284.15 284.15 284.15 291.15 291.15 291.15 291.15 291.15 291.15 298.15 298.15 298.15
10 20 30 40 50 60 10 20 30 40 50 60 10 20 30
AAD
Constant influence parameter: 2.59; Proposed influence parameter: 2.54
C 2H6 + H2O 284.15 6 284.15 12 284.15 18 284.15 24 284.15 30 291.15 6 291.15 12 291.15 18 291.15 24 291.15 30 291.15 36 298.15 8 298.15 16 298.15 24 AAD
73.25 71.84 70.55 69.12 67.55 65.56 72.69 70.93 69.65 68.32 66.84 64.75 70.63 69.25 68.15
66.98 58.66 51.39 42.85 30.00 29.02 66.07 58.23 51.05 45.02 35.00 34.86 65.55 58.00 50.86
71.04 67.01 62.50 58.28 54.37 70.26 66.15 61.23 57.18 52.87 47.34 68.27 64.78 60.57
66.21 56.9 52.15 42.88 34.06 33.79 66.08 55.46 49.74 46.39 35.23 34.78 65.55 55.89 49.42
71.18 66.61 62.06 57.60 54.55 70.64 65.98 60.99 56.34 51.05 46.94 68.36 63.87 59.76
68.72 58.66 52.18 42.4 30.72 29.81 67.59 55.46 49.35 45.03 35.5 33.63 67.16 56.44 49.84
71.50 66.47 62.38 57.20 54.39 70.79 65.85 60.95 56.54 51.32 47.00 68.56 63.54 59.37
298.15 298.15 298.15 305.15 305.15 305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
298.15 298.15 298.15 305.15 305.15 305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
298.15 298.15 305.15 305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
gexp (mN m1)
gcalc (mN m1) GT + density dependant influence parameter
GT + constant influence parameter
40 50 60 10 20 30 40 50 60 10 20 30 40 50 60
67.02 65.65 63.68 69.36 68.61 67.53 66.34 65.17 62.74 68.58 67.67 66.42 65.45 64.25 61.29
67.04 65.13 63.29 69.09 68.58 67.80 66.34 64.69 63.03 67.22 66.65 66.44 65.44 64.12 62.70
67.02 65.10 63.30 69.03 68.68 67.71 66.13 64.42 62.75 67.14 66.91 66.59 65.45 64.04 62.58
40 50 60 10 20 30 40 50 60 10 20 30 40 50 60
46.65 41.85 36.28 64.86 57.72 50.5 46.24 42.67 37.79 63.55 57.45 50.15 45.94 42.85 39.31
46.19 41.96 37.11 64.83 56.3 50.19 45.85 43.27 39.77 62.59 55.59 50.13 45.92 43.05 40.67
46.65 39.67 37.81 66.12 56.78 50.17 46.38 42.69 37.76 62.87 55.69 49.96 45.93 42.91 39.31
32 40 8 16 24 32 40 10 20 30 40 50 60
56.36 52.28 67.33 63.57 59.16 55.89 51.25 66.46 62.15 58.01 53.85 48.15 43.04
56.01 51.98 67.52 62.86 58.77 54.97 50.51 66.50 62.00 57.81 53.03 47.99 42.63
55.65 52.38 67.48 62.50 58.36 54.94 50.08 66.88 61.84 57.34 52.28 47.86 42.56
7.5 9.0 2
63.38 61.56 70.09
e e e
62.50 60.95 69.28
Constant influence parameter: 0.81; Proposed influence parameter: 0.79
C 3H8 + H2O 284.15 1.5 284.15 3.0 284.15 4.5
73.08 71.28 69.50
e e e
72.69 70.96 68.87
298.15 298.15 305.15
33
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Table 4 e (continued ) T (K)
P (bar)
gexp (mN m1)
gcalc (mN m1) GT + density dependant influence parameter
284.15 291.15 291.15 291.15 291.15 291.15 298.15 298.15 298.15 298.15
6.0 1.5 3.0 4.5 6.0 7.5 1.5 3.0 4.5 6.0
67.36 72.67 70.57 68.48 66.45 64.73 71.32 69.18 67.32 65.64
e e e e e e e e e e
T (K)
P (bar)
gexp (mN m1)
GT + constant influence parameter 66.57 71.80 69.53 67.80 65.70 63.54 70.92 68.81 66.63 64.78
gcalc (mN m1) GT + density dependant influence parameter
305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
4 6 8 10 2 4 6 8 10 12
68.55 66.48 64.26 62.55 69.09 67.45 65.24 63.19 61.33 59.75
e e e e e e e e e e
GT + constant influence parameter 67.18 65.89 63.76 61.60 68.45 66.09 64.08 62.16 60.23 58.70
Constant influence parameter: 1.22; Proposed influence parameter: e
parameters for the proposed influence parameter are determined, respectively for (C2H6 þ H2O) system: kij ¼ 0:7394 0:00253T
(21)
Fig. 6 and Table 4 indicate the experimental and calculated surface tensions for (C2H6 þ H2O) and (C3H8 þ H2O) systems. Similar to (CH4 þ H2O) and (CO2 þ H2O) systems, for (C2H6 þ H2O) and (C3H8 þ H2O) systems, it is concluded that the applied model can reproduce the surface tensions for (C2H6 þ H2O) and (C3H8 þ H2O) systems. Also, it is seen that the combination of present model with constant influence parameter performs better than the proposed influence parameter for (C2H6 þ H2O) and (C3H8 þ H2O) systems. The very good performance of the applied model in combination the constant influence parameter allows us to determine the reliable interfacial density profiles. Figs. 7 and 8 show the partial density profiles of interface for (C2H6 þ H2O)
Fig. 2 e Surface tension vs. pressure for (CH4 þ H2O) and (CO2 þ H2O) systems. (A) (CH4 þ H2O) experimental data at 298.15 K from Yan et al. (2001); (-) (CH4 þ H2O) experimental data at 298.15 K from Sachs and Meyn (1995); (C) (CH4 þ H2O) experimental data at 298.15 K (this work); (:) (CO2 þ H2O) experimental data at 298.15 K from Ren et al. (2000); and (þ) (CO2 þ H2O) experimental data at 298.15 K (this work).
Fig. 3 e Surface tension vs. pressure for the (CH4 þ H2O) and (CO2 þ H2O) systems at 298.15 K; (-) (CH4 þ H2O) experimental data (this work); (C) (CO2 þ H2O) experimental data (this work); (e) gradient theory þ proposed influence parameter for (CH4 þ H2O); and ( … ) gradient theory þ proposed influence parameter for (CO2 þ H2O).
Fig. 4 e Determined density profiles of CH4 and H2O in the interface at T ¼ 298.15 K and P ¼ 30 bar; (e) CH4 and ( … ) H2O.
34
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Fig. 5 e Determined density profiles of CO2 and H2O in the interface at T ¼ 298.15 K and P ¼ 30 bar; (e) CO2 and ( … ) H2O.
Fig. 8 e Determined density profiles of C3H8 and H2O in the interface at T ¼ 284.15 K and P ¼ 1.5 bar; (e) C3H8 and ( … ) H2O.
5.
Fig. 6 e Surface tension vs. pressure for the (C2H6 þ H2O) system at 284.15 K; (-) (C2H6 þ H2O) experimental data (this work); (e) gradient theory þ proposed influence parameter.
and (C3H8 þ H2O) systems at T ¼ 284.15 K and P ¼ 6 bar and P ¼ 1.5 bar, respectively. It is concluded that the interfacial behavior of (C2H6 þ H2O) and (C3H8 þ H2O) systems is the same as (CH4 þ H2O) and (CO2 þ H2O) systems.
Fig. 7 e Determined density profiles of C2H6 and H2O in the interface at T ¼ 284.15 K and P ¼ 6 bar; (e) C2H6 and ( … ) H2O.
Conclusions
The surface tensions of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems are measured with the pendant drop technique at temperatures ranged (284.15e312.15) K and pressures ranged (1e60) bar. The gradient theory in combination with CK-SAFT EOS is utilized for modeling the surface tensions of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems. A proposed influence parameter correlation of the gradient theory is also used for the surface tension of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) systems. The binary interaction parameters of these four systems are fitted according to the bulk densities and surface tensions. The density profiles of the interface phase are also determined. Surface tensions could be reproduced well for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) systems by the proposed influence parameter and combination of the GT with the CK-SAFT EOS.
references
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chain statistical associating fluid theory combined with density-gradient theory. Ind. Eng. Chem. Res. 47, 8911e8917. Lin, H., Yuan, Y.-Y., Min, Q., 2007. Gradient theory modeling of surface tension for pure fluids and binary mixtures. Fluid Phase Equilib. 254, 75e90. Maass, O., Wright, C.H., 1921. Some physical properties of hydrocarbons containing two and three carbon atoms. J. Am. Chem. Soc. 43, 1098e1111. Miqueu, C., Broseta, D., Satherley, J., Mendiboure, B., Lachaise, J., Graciaa, A., 2000. An extended scaled equation for the temperature dependence of the surface tension of pure compounds inferred from an analysis of experimental data. Fluid Phase Equilib. 172, 169e182. Miqueu, C., Mendiboure, B., Graciaa, A., Lachaise, J., 2003. Modelling of the surface tension of pure components with the gradient theory of fluid interfaces: a simple and accurate expression for the influence parameters. Fluid Phase Equilib. 207, 225e246. Miqueu, C., Migues, J.M., Pineiro, M.M., Lafitte, T., Mendiboure, B., 2011. Simultaneous application of the gradient theory and Monte Carlo molecular simulation for the investigation of methane/ water interfacial properties. J. Phys. Chem. B 115, 9618e9625. NIST Chemistry WebBook. http://www.webbook.nist.gov/ chemistry/fluid/. Park, J.-Y., Lim, J.S., Yoon, C.H., Lee, C.H., Park, K.P., 2005. Effect of a fluorinated sodium bis(2-ethylhexyl) sulfosuccinate (aerosol-OT, AOT) analogue surfactant on the interfacial tension of CO2 þ water and CO2 þ Ni-plating solution in nearand supercritical CO2. J. Chem. Eng. Data 50, 299e308. Rathjen, W., Straub, J., 1977. Surface tension and refractive index of six refrigerants from the triple point to the critical point. In: Proceedings of the 7th Symposium for Thermophysical Properties, New York. Ren, Q.-Y., Chen, G.-J., Yan, W., Guo, T.-M., 2000. Interfacial tension of (CO2 þ CH4) þ water from 298 K to 373 K and pressures up to 30 MPa. J. Chem. Eng. Data 45, 610e612. Sachs, W., Meyn, V., 1995. Pressure and temperature dependence of the surface tension in the system natural gas/water principles of investigation and the first precise experimental data for pure methane/water at 25 C up to 46.8 MPa. Colloids Surf. A: Physicochem. Eng. Asp. 94, 291e301. Schmidt, K.A.G., Folas, G.K., Kvamme, B., 2007. Calculation of the interfacial tension of the methaneewater system with the linear gradient theory. J. Fluid Phase Equilib. 261, 230e237. Tjahjono, M., Garland, M., 2010. A new modified parachor model for predicting surface compositions of binary liquid mixtures. On the importance of surface volume representation. J. Colloid Interf. Sci. 345, 528e537. Vargaftik, N.B., 1975. Tables on Thermophysical Properties of Liquids and Gases. Hemisphere Publishing Corporation, John Wiley & Sons Inc., New York, USA. € n, J., 1997. GrenzflachensWesch, A., Dahmen, N., Ebert, K., Scho € ben und Kontaktwinkel im pannungen, Tropfengro Zweiphasensystem H2O/CO2 bei Temperaturen von 298 bis 333 K und Dru¨cken bis 30 MPa. Chem. Ing. Tech. 69, 942e946. Wiegand, G., Franck, E.U., 1994. Interfacial tension between water and non-polar fluids up to 473 K and 2800 bar. Ber. BunsenGes. 98, 809e817. Yan, W., Zhao, G.-Y., Chen, G.-J., Guo, T.-M., 2001. Interfacial tension of (methane þ nitrogen) þ water and (carbon dioxide þ nitrogen) þ water systems. J. Chem. Eng. Data 46, 1544e1548. Yarrison, M., Chapman, W.G., 2004. A systematic study of methanol þ n-alkane vaporeliquid and liquideliquid equilibria using the CK-SAFT and PC-SAFT equations of state. Fluid Phase Equilib. 226, 195e205.
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Experimental and modeling investigation on surface tension and surface properties of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) from 284.15 K to 312.15 K and pressures up to 60 bar Shahin Khosharay, Farshad Varaminian* School of Chemical, Gas and Petroleum Engineering, Semnan University, Semnan, Iran
article info
abstract
Article history:
In this work, the experimental surface tensions for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O)
Received 29 June 2014
and (C3H8 þ H2O) binary systems have been measured with the pendant drop tensiometer
Received in revised form
at temperatures ranged (284.15e312.15) K and pressures ranged (1e60) bar. Subsequently,
2 August 2014
the combination of gradient theory (GT) and CheneKreglewski Statistical Association Fluid
Accepted 8 August 2014
Theory equation of state (CK-SAFT EOS) is applied for modeling these experimental surface
Available online 16 August 2014
tensions and determining the density profiles of interface. Also, a proposed influence parameter as a function of the densities of the bulk phases is applied for these systems.
Keywords:
The binary interaction parameters for these systems are determined according to the bulk
Surface tension
densities and surface tensions. Subsequently, the density profiles for the surface layer are
Pendant drop
determined. The surface tensions determined with the gradient theory (GT) agree well with
Gradient theory
experimental surface tensions for these systems (overall AAD ~ 1.36 and 1.37 for constant
CK-SAFT EOS
and the proposed influence parameters, respectively). © 2014 Elsevier Ltd and IIR.
rimentale et par mode lisation de la tension Etude expe te s superficielles de (CH4 þ H2O), superficielle et des proprie (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) pour des ratures allant de 284,15 K a 312,5 K et des pressions tempe 60 bar allant jusqu'a orie du gradient ; CK-SAFT EOS Mots cles : Tension superficielle ; Goutte suspendue ; The
* Corresponding author. Tel./fax: þ98 231 3354280. E-mail address: [email protected] (F. Varaminian). http://dx.doi.org/10.1016/j.ijrefrig.2014.08.003 0140-7007/© 2014 Elsevier Ltd and IIR.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Nomenclature A 0 A ares aseg achain aassoc AAD B 0 B c calc d EOS exp f f0 g k kij L
1.
constant coefficient of influence parameter constant coefficient of binary interaction parameter reduced Helmholtz Energy Helmholtz energy of segments Helmholtz energy of chain-forming bonds among the segments Helmholtz energy of association contribution average absolute deviation constant coefficient of influence parameter constant coefficient of binary interaction parameter influence parameter calculated result diameter equation of state experimental fugacity Helmholtz free energy density gravitational constant Boltzmann constant binary interaction parameter (BIP) liquid
Introduction
Because of increasing the need for the properties of refrigerants, several studies focus on the modeling the properties of the (refrigerant gases þ water) systems. The surface tension of (refrigerant gas þ water) system is also an important parameter for the conditions with hydrate formation. Furthermore, the interfacial phenomenon strongly influences the nucleation of gas hydrate; therefore, interfacial properties such as the surface tension of gas/water systems may greatly affect the hydrate formation kinetics. On the other hand, the surface tension relates to the nature of molecular forces at the interface phase so that knowledge of interfacial properties such as interfacial profiles and thickness of surface layer is of importance. It is known that the direct experimental measurement of surface layer profile is quite difficult and it was investigated for few systems. One method to achieve this aim is using equations of states (EOSs) for determining the interfacial properties (Abraham et al., 2001; Enders and Kahl, 2008; Tjahjono and Garland, 2010). The gradient theory of inhomogeneous fluid (Miqueu et al., 2003, 2011; Li et al., 2008; Khosharay and Varaminian, 2013) is one of the models in which the properties of bulk phases can be related to the properties surface layer by using a suitable equation of state (EOS). Furthermore, the gradient theory requires two inputs: (1) the Helmholtz free energy density that belongs to the homogeneous fluid; (2) the influence parameter that belongs to the inhomogeneous fluid. The Helmholtz free energy density for the homogeneous fluid can be calculated by using a suitable thermodynamic model. The influence parameter is a molecular theory based input of this theory, but
27
m effective number of segments within the molecule n mole density R ideal gas constant P pressure ref reference fluid CK-SAFT CheneKreglewski statistical associating fluid theory T temperature reduced temperature Tr temperature-dependent dispersion energy of u0/k interaction between segments V vapor temperature-independent segment volume voo x mole fraction z position in the interface Greek letters g surface tension m chemical potential molar density of component i ri 3 /k association energy k volume of association 4 fugacity coefficient U grand thermodynamic potential
it is almost impossible to use this parameter based on molecular theory in practice; therefore, applying a semiempirical expression is one of the practical ways of using this parameter in practice (Lin et al., 2007; Lafitte et al., 2010). In recent years, the experimental and modeling investigations are conducted on the interfacial properties of gasewater systems. Yan et al. (2001) measured the surface tension of N2 þ H2O, (CH4 þ N2) þ H2O and (CO2 þ N2) þ H2O mixtures with pendant drop method at different pressures, temperatures and compositions. Subsequently they applied linear gradient theory (LGT) in combination with SRK EOS for N2 þ H2O and (CH4 þ N2) þ H2O systems. They concluded that modeling results are in a good agreement with the experimental surface tensions. Ren et al. (2000) measured the high pressure surface tensions for CH4 þ H2O and (CH4 þ CO2) þ H2O mixtures by using pendant drop method under various pressure, temperature and composition conditions. Georgiadis et al. (2010) measured the surface tension of CO2 þ H2O system at pressure ranged (1e60) MPa and temperatures ranged (298e374) K. They also applied the pendant drop method for surface tension measurement. Kvamme et al. (2007) applied a novel apparatus and the quasi-static pendant drop method to measure measured the surface tension of CO2 þ H2O system at pressure ranged (0.1e20) MPa and temperatures ranged (278e335) K. They also performed molecular dynamics simulations for this system. Chiquet et al. (2007) reported pendant drop measurements of surface tensions for water/CO2 system at temperatures ranged (308e383) K and pressures ranged (5e45) MPa. They also measured phase densities simultaneously. Khosharay and Varaminian (2013) applied the linear gradient theory (LGT) in combination with the cubic-plus-association equation of state (CPA EOS) to
28
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
calculate the surface tensions of (CH4 þ N2) þ H2O and (N2 þ CO2) þ H2O systems. The estimations of this model agreed well with experimental surface tensions. Schmidt et al. (2007) applied linear gradient theory (LGT) in combination with the SoaveeRedlicheKwong (SRK) and the PengeRobinson (PR) equations of state to correlate the surface tensions of the binary system of methane/water. They also determined the temperature dependant influence parameters for pure component and binary interaction coefficient of the influence parameter. Several other investigations are done in (Wiegand and Franck, 1994; Jho et al., 1978; Chun and Wilkinson, 1995; Wesch et al., 1997; Hebach et al., 2002; Park et al., 2005; Bachu and Bennion, 2009; Chalbaud et al., 2009; Bikkina et al., 2011). In this study, the surface tensions of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems are measured by using pendant drop technique at temperatures ranged from 284.15 K to 312.15 K and pressures up to 60 bar. Subsequently, the gradient theory of inhomogeneous fluid in combination with the CheneKreglewski Statistical Association Fluid Theory equation of state (CK-SAFT EOS) (Kontogeporgis and Folas, 2010; Yarrison and Chapman, 2004; Chapman et al., 1990; Huang and Radosz, 1990) is applied to determine the surface tension and the surface layer density profiles for these binary systems. Additionally, the proposed expression on the basis of the densities of bulk coexist phases (Khosharay et al., 2013, 2014) is used for the influence parameters of the gradient theory. To identify the advantages of the proposed influence parameter for these binary systems, the proposed influence parameter is applied and compared with the constant influence parameter for these binary systems.
2.
Experimental
2.1.
The applied materials
The analytical grade methane (CH4), ethane (C2H6), propane (C3H8) and carbon dioxide (CO2) are used with a purity of 99.99% that were supplied by Technical Gas Services. The double distilled water is also applied. The information of these materials has been shown in Table 1.
Table 1 e Materials used for experiments. Component
Chemical formula
Purity
Carbon dioxide Methane
CO2
99.99%
CH4
99.99%
Ethane
C2H6
99.99%
Propane
C3H8
99.99%
Water
H 2O
Deionizeddistilled
Supplier Technical Gas Services Technical Gas Services Technical Gas Services Technical Gas Services Bahrezolal, Iran
Fig. 1 e Schematic of experimental apparatus for surface tension measurements (pendant drop), 1. Handle pump; 2. Drop needle; 3. High pressure cell; 4. Digital camera; 5. Light source; 6. Temperature transmitter; 7. Pressure transmitter; 8. Temperature indicator; 9. Pressure indicator; 10. Computer; 11. Gas tank; 12. To vacuum trap; 13. Temperature controlled air bath.
2.2.
Apparatus
The schematic of experimental high-pressure pendant drop set-up has been shown in Fig. 1. The experiments are conducted in the cylindrical and high-pressure stainless steel cell with a total capacity of 21 mL which can operate pressures up to 10 MPa and temperatures ranging from 283.15 K to 323.15 K. The cell has two polished Pyrex-glass which allows observing inside the cell from a horizontal axis; therefore, the visual observations of the droplet shape is possible. The cell is equipped with the stainless steel capillary tube (o.d. 1.22 mm and i.d. 1.00 mm) for hanging the drop of liquid. A handle pump is also applied for feeding the liquid into the cell through the capillary tube which can operate pressures up to 40 MPa. The temperature is measured by using a PT100 thermometer (Pro-Temp Controls, Santa Ana, California, United States) with ±0.1 K accuracy. A BD-Sensors-Str.1 pressure transmitter with ±0.1 bar accuracy is utilized to measure the pressure of the cell. The apparatus contains a coolant bath with controllable circulator that is utilized to circulate the air. The cell is immersed in a constant temperature air bath whose temperature is controlled within 0.1 K by using a PID controller. In order to capture the images of the drop and measure the surface tension, the system is also equipped with the digital camera which is connected to a personal computer.
2.3.
Experimental procedure
Prior to conducting each experiment and measurement, the pendant drop cell and its connections were soaked in acetone for 30 min. Subsequently, the cell and its connections were washed and rinsed with double distilled water for three times and dried with compressed air. Then the cell was evacuated with a vacuum pump and charged with the gas up to the
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
specified pressure. By using a handle pump, the liquid water was introduced slowly into the pendant drop cell through the stainless steel capillary tube. The water pendant drop formed at the tip of the stainless steel capillary tube which was vertically inserted in the pendant drop cell. The pendant drop cell was left more than 1 h to reach the specified temperature and make sure that the pendant drop was in the thermal equilibrated with its surroundings and the adsorption of gas molecules on the drop surface reached equilibrium. During the process, for measuring the surface tension, the images of the drop were captured with the digital camera and light source.
2.4.
Calculation of the surface tension
In this work, the surface tension values are calculated by using equations of Andreas et al. (1938). g¼
Drd2e g H
1 ds ¼f H de
3.
Brief description of the model
3.1.
Gradient theory
Z g¼
rV ref
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X X dni dnj u t2ðUðnÞ UB Þ cij dnref dnref dnref i j
UðnÞ ¼ f0 ðnÞ
in which UB ¼ P; P denotes the equilibrium pressure; r the mole density; g the surface tension. rL and rV show the mole
ni mi;b
(4)
in which mi,b shows the chemical potential of component i belonging to the bulk phase. As it is mentioned in the introduction part, to apply the gradient theory, it is necessary to calculate two inputs for determining surface tension: the Helmholtz energy density of the homogeneous fluid and the influence parameters of the inhomogeneous fluid. In order to calculate the density profile in the surface layer, the following equation is applied: ðzÞ
rref
Z
(2)
(3)
X i
z ¼ z0 þ
The extensive description of gradient theory of fluid interfaces (Miqueu et al., 2003, 2011; Li et al., 2008; Khosharay and Varaminian, 2013) exists in several investigations; therefore, in this work, the description of important aspects of gradient theory is considered. The detailed description of gradient theory can be found in Miqueu et al. (2003, 2011), Li et al. (2008), Khosharay and Varaminian (2013), Lafitte et al. (2010), Khosharay et al. (2013, 2014). Based on the gradient theory of interfaces, without presence of any external potential and assuming the planar interface, the surface tension of a mixture containing N components between the liquid (L) and vapor (V) phases is expressed as function of the local gradients in density as follows: rLref
densities belonging to the liquid and vapor bulk phases, respectively. Subscript ref represents the reference component. dri/drref is the mole density profile belonging to component i and cij shows the influence parameter. U is the grand thermodynamic potential and it is expressed as follows:
(1)
in which Dr denotes the density difference between liquid and gas phases, g shows the gravitational constant, ds represents the pendant drop diameter at height that is equal to the maximum diameter of the pendant drop (de). The relation between 1/H and ds/de is given from the work of Drelich et al. (2002). The uncertainty of the experimental data is in the range of (0.2e0.7) mN m1. The CK-SAFT EOS is applied to compute the density of gas phase. Except the carbon dioxide, the influence of the solubility of gases on the aqueous phase density is neglected because it is very low. The densities of aqueous phase for (CO2 þ H2O) system are given from Hebach et al. (2004).
29
r0ref
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP P dr drj u cij drrefi drref u t i j drref 2ðUðrÞ UB Þ
(5)
in which z shows the coordinate and subscript 0 shows the arbitrarily chosen origin.
3.2.
Free energy density and phase equilibrium
The CheneKreglewski-Statistical-Association-Fluid-Theory equation of state (CK-SAFT EOS) (Kontogeporgis and Folas, 2010; Yarrison and Chapman, 2004; Chapman et al., 1990; Huang and Radosz, 1990) is a version of the SAFT EOS. The CK-SAFT EOS is expressed in the terms of the residual molar Helmholtz energy as follows: ares aseg achain aassoc ¼ þ þ RT RT RT RT
(6)
where aseg belongs to the Helmholtz energy of segments, achain belongs to the chain-forming bonds among the segments and aassoc belongs to the association term. The detailed descriptions of these terms are given in earlier publications (Kontogeporgis and Folas, 2010; Yarrison and Chapman, 2004; Chapman et al., 1990; Huang and Radosz, 1990). Additionally, the temperature dependent of binary interaction parameter has been used for the dispersion term of CKSAFT EOS in which A and B show the constants and T shows the temperature in K. kij ¼ A0 þ B0 T
(7)
In order to calculate the phase equilibrium for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems, the fugacity (fi) of each component i should be equal in the coexisting equilibrium phases. fiI ¼ fiII
(8)
where superscripts I and II show the coexisting equilibrium phases. Eq. (8) can be expressed as follows: P4Ii xIi ¼ P4IIi xIIi
(9)
30
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
where xIi and xIIi denote the mole fraction of components in the coexisting equilibrium phases. P shows the pressure and 4 denotes the fugacity coefficient. Based on the work of Li et al. (2008), the binary interaction parameters (kij) are fitted according to the experimental liquid bulk densities and surface tension data.
3.3.
The influence parameter
Since the theoretical based influence parameter is not applicable for many systems, the influence parameters for the pure fluids (c) should be regressed according to the experimental surface tensions of the pure fluids. 2
32
6 7 6 7 6 7 6 7 gexp 16 c ¼ 6Z rV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 26 7 6 f0 ðrÞ rmðrÞ þ Pdr7 4 rV 5
(10)
The deviations of the surface tensions are given as follows:
AADg ¼
N exp gcalc 1 X i gi 100 exp N i¼1 gi
(11)
In this work, two different forms of influence parameter are applied: (1) constant value of influence parameter and (2) the influence parameter based on the liquid and vapor densities (Khosharay et al., 2013, 2014). For pure fluids, the second form of influence parameter is expressed as follows: L r rV c ¼ V Ar þ BrL
(12)
To test the applicability of the proposed influence parameter for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems, the influence parameters of component i for the mixture are determined on the basis of the mole densities of component i in the bulk phases. The binary interaction coefficients (lij) are also set to be zero. I I r x rII xIIi ci ¼ I iI II II Ai r xi þ Bi r xi
(13)
in which and xIi are xIIi are the mole fractions of component i in the bulk phases.
3.4.
Molar density gradients
In this study, according to the geometric combing rule for the influence parameters (lij ¼ 0), the molar density gradients of
components through the surface layer are calculated by solving algebraic equations. This equation derived on the basis of the minimum Helmholtz free energy criteria (Lin et al., 2007): pffiffiffiffiffi vm1 pffiffiffiffiffi 2 c1 vm c2 vr2 vr2 dr1 T;P;r1 T;P;r1 ¼ pffiffiffiffiffi 1 dr2 pffiffiffiffiffi vm2 c2 vm c1 vr1 vr1 T;P;r2
(14)
T;P;r2
4.
Results and discussion
4.1.
Pure fluids
Prior to determining any surface layer properties, the influence parameters of each pure component should be obtained. In this work, the influence parameters for methane (CH4), ethane (C2H6), propane (C3H8), carbon dioxide (CO2) and water (H2O) are obtained based on the fitting of their surface tensions to the experimental data. The pure component parameters for CK-SAFT EOS are given from Huang and Radosz (1990). The pure fluid parameters and the gradient theory influence parameters are given in Tables 2 and 3, respectively.
4.2.
The binary systems
Firstly, the experimental surface tensions for (CH4 þ H2O) and (CO2 þ H2O) are reported in Table 4. In order to check the validity of the apparatus and measured surface tensions, the experimental surface tensions are compared with the literature reported by Ren et al. (2000), Georgiadis et al. (2010), Sachs and Meyn (1995). In Fig. 2, one can see good agreements with the literature data (Ren et al., 2000; Georgiadis et al., 2010; Sachs and Meyn, 1995). As it is shown in Table 4 and Fig. 2, at constant temperature, with increasing pressure, the steep decrease in surface tension is seen both for (CH4 þ H2O) and (CO2 þ H2O) systems due to enrichment of interface layer with the gaseous component. At constant temperature, for (CH4 þ H2O) system, increasing the temperature results in the decreasing surface tension. The temperature dependence for (CO2 þ H2O) system is rather complex. At the pressure range of (10e30) bar, the increase of the temperature results in lowering surface tension. At higher pressures, the slopes of isotherms are different; therefore, these isotherms intersect each other. These behaviors are also seen in the literature data of Georgiadis et al. (2010), Kvamme et al. (2007), Chiquet et al. (2007). Subsequently, the gradient theory of interface was utilized for modeling the surface tension of (CH4 þ H2O) and (CO2 þ H2O) systems. In this study, to test the advantages of
Table 2 e The pure component parameters of CK-SAFT EOS with their average absolute deviations. Fluid
m
yoo (mL mol1)
u0/k (K)
CO2 CH4 C2H6 C3H8 H2O
1.417 1.000 1.941 2.696 1.179
13.578 21.556 14.460 13.457 10.000
216.08 190.29 191.44 193.09 528.17
3 /k
(K)
e e e e 1809
102kAB
T range
AAD vliq
AAD P
e e e e 1.593
218e288 92e180 160e300 190e360 283e613
0.86 0.35 1.60 1.80 3.20
2.80 1.40 1.80 2.10 1.30
Ref. Huang Huang Huang Huang Huang
and and and and and
Radosz Radosz Radosz Radosz Radosz
(1990) (1990) (1990) (1990) (1990)
31
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Table 3 e The influence parameters for gradient theory with the average absolute deviations. Fluid
Density dependant influence parameter
AADg
A (mol2 J1 m5) 1020 B (mol2 J1 m5) 1020
Constant influence AADg T range parameter C (J m5 mol2) 1020
CO2
1.02859
3.65019
0.17
2.32500
2.81
218e288
CH4
0.55932
0.42956
3.89
6.75000
5.28
92e180
C2H6
0.19646
0.18233
1.49
5.20312
1.53
160e300
C3H8
0.066749
0.091551
0.94
10.62500
2.08
190e360
H 2O
5.32781
0.90123
4.17
1.20321
4.82
283e613
using the proposed expression of the influence parameters, the gradient theory (GT) is reapplied. To determine the surface tension of the (CH4 þ H2O) and (CO2 þ H2O) systems at temperatures ranged (284.15e312.15) K and pressures ranged (10e60) bar. To improve the description of the equilibrium densities that belongs the bulk phases and subsequently surface tensions, the binary interaction parameters of CKSAFT EOS is determined based on the experimental bulk densities and surface tensions. For constant influence parameter, the temperature dependent binary interaction parameters can be expressed as follows, respectively for (CH4 þ H2O) and (CO2 þ H2O) systems: kij ¼ 0:8343 þ 0:002578T
(15)
kij ¼ 0:1419 0:0004644T
(16)
For the proposed influence parameter, the temperature dependent binary interaction parameters are written as follows, respectively for (CH4 þ H2O) and (CO2 þ H2O) systems: kij ¼ 0:9393 þ 0:02821T
(17)
kij ¼ 0:1328 0:0004718T
(18)
As it is illustrated in Fig. 3 and Table 4, the combination of gradient theory (GT) and CK-SAFT EOS is capable of reproducing the surface tension for (CH4 þ H2O) and (CO2 þ H2O) systems. The deviations of the surface tensions for (CH4 þ H2O) and (CO2 þ H2O) systems are listed in Table 4. Furthermore, in Table 4, the results of the present model with
gexp Ref.
NIST Chemistry WebBook Rathjen, and Straub (1977) Vargaftik (1975) NIST Chemistry WebBook Miqueu et al. (2000) Holcomb and Zollweg (1992) NIST Chemistry WebBook Miqueu et al. (2000) Maass and Wright (1921) NIST Chemistry WebBook Miqueu et al. (2000) Maass and Wright (1921) NIST Chemistry WebBook Fu et al. (2000) Goncalves et al. (1991)
constant influence parameter are compared with the proposed influence parameter based on the densities of bulk coexist phases. Figs. 4 and 5 show the density profiles of the interface for (CH4 þ H2O) and (CO2 þ H2O) systems at T ¼ 298.15 K and P ¼ 30 bar. It is seen that the water partial density increases monotonically from the CH4-rich phase to the water-rich phase with the shape of tanh. Figs. 4 and 5 also illustrate the enhancement of CH4 and CO2 densities which corresponds to the adsorption of CH4 and CO2 molecules at the surface phase. In this study, the surface tensions of (C2H6 þ H2O) and (C3H8 þ H2O) systems are measured. The dependence of the surface tension of (C2H6 þ H2O) and (C3H8 þ H2O) systems on temperature and pressure are the same as (CO2 þ H2O) system. The combination of the gradient theory and CK-SAFT EOS is applied to model the surface tensions of these systems for the first time. Similar to (CH4 þ H2O) and (CO2 þ H2O) systems, the binary interaction parameters are determined on the basis of the experimental bulk densities and surface tensions. For constant influence parameter, the dependence of binary interaction parameters on the temperature is expressed as follows, respectively for (C2H6 þ H2O) and (C3H8 þ H2O) systems: kij ¼ 0:9421 0:00312T
(19)
kij ¼ 0:4022 þ 0:001523T
(20)
By using the proposed influence parameter for (C3H8 þ H2O) system, the density profiles and surface tensions cannot be determined. The temperature dependent binary interaction
32
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Table 4 e Measured and modeled surface tensions for (CH4 + H2O), (C2H6 + H2O), (CO2 + H2O) and (C3H8 + H2O) binary systems. T (K)
P (bar)
gexp (mN m1)
gcalc (mN m1) GT + density dependant influence parameter
GT + constant influence parameter
74.12 73.31 71.27 69.11 67.03 65.07 72.43 72.23 70.43 68.33 66.25 64.30 71.12 70.51 68.96
74.81 73.59 71.31 69.1 67.1 65.3 73.04 72.56 70.50 68.31 66.25 64.39 70.82 70.19 68.84
T (K)
P (bar)
CH4 + H2O 284.15 284.15 284.15 284.15 284.15 284.15 291.15 291.15 291.15 291.15 291.15 291.15 298.15 298.15 298.15
10 20 30 40 50 60 10 20 30 40 50 60 10 20 30
AAD
Constant influence parameter: 0.82; Proposed influence parameter: 0.79
CO2 + H2O 284.15 284.15 284.15 284.15 284.15 284.15 291.15 291.15 291.15 291.15 291.15 291.15 298.15 298.15 298.15
10 20 30 40 50 60 10 20 30 40 50 60 10 20 30
AAD
Constant influence parameter: 2.59; Proposed influence parameter: 2.54
C 2H6 + H2O 284.15 6 284.15 12 284.15 18 284.15 24 284.15 30 291.15 6 291.15 12 291.15 18 291.15 24 291.15 30 291.15 36 298.15 8 298.15 16 298.15 24 AAD
73.25 71.84 70.55 69.12 67.55 65.56 72.69 70.93 69.65 68.32 66.84 64.75 70.63 69.25 68.15
66.98 58.66 51.39 42.85 30.00 29.02 66.07 58.23 51.05 45.02 35.00 34.86 65.55 58.00 50.86
71.04 67.01 62.50 58.28 54.37 70.26 66.15 61.23 57.18 52.87 47.34 68.27 64.78 60.57
66.21 56.9 52.15 42.88 34.06 33.79 66.08 55.46 49.74 46.39 35.23 34.78 65.55 55.89 49.42
71.18 66.61 62.06 57.60 54.55 70.64 65.98 60.99 56.34 51.05 46.94 68.36 63.87 59.76
68.72 58.66 52.18 42.4 30.72 29.81 67.59 55.46 49.35 45.03 35.5 33.63 67.16 56.44 49.84
71.50 66.47 62.38 57.20 54.39 70.79 65.85 60.95 56.54 51.32 47.00 68.56 63.54 59.37
298.15 298.15 298.15 305.15 305.15 305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
298.15 298.15 298.15 305.15 305.15 305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
298.15 298.15 305.15 305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
gexp (mN m1)
gcalc (mN m1) GT + density dependant influence parameter
GT + constant influence parameter
40 50 60 10 20 30 40 50 60 10 20 30 40 50 60
67.02 65.65 63.68 69.36 68.61 67.53 66.34 65.17 62.74 68.58 67.67 66.42 65.45 64.25 61.29
67.04 65.13 63.29 69.09 68.58 67.80 66.34 64.69 63.03 67.22 66.65 66.44 65.44 64.12 62.70
67.02 65.10 63.30 69.03 68.68 67.71 66.13 64.42 62.75 67.14 66.91 66.59 65.45 64.04 62.58
40 50 60 10 20 30 40 50 60 10 20 30 40 50 60
46.65 41.85 36.28 64.86 57.72 50.5 46.24 42.67 37.79 63.55 57.45 50.15 45.94 42.85 39.31
46.19 41.96 37.11 64.83 56.3 50.19 45.85 43.27 39.77 62.59 55.59 50.13 45.92 43.05 40.67
46.65 39.67 37.81 66.12 56.78 50.17 46.38 42.69 37.76 62.87 55.69 49.96 45.93 42.91 39.31
32 40 8 16 24 32 40 10 20 30 40 50 60
56.36 52.28 67.33 63.57 59.16 55.89 51.25 66.46 62.15 58.01 53.85 48.15 43.04
56.01 51.98 67.52 62.86 58.77 54.97 50.51 66.50 62.00 57.81 53.03 47.99 42.63
55.65 52.38 67.48 62.50 58.36 54.94 50.08 66.88 61.84 57.34 52.28 47.86 42.56
7.5 9.0 2
63.38 61.56 70.09
e e e
62.50 60.95 69.28
Constant influence parameter: 0.81; Proposed influence parameter: 0.79
C 3H8 + H2O 284.15 1.5 284.15 3.0 284.15 4.5
73.08 71.28 69.50
e e e
72.69 70.96 68.87
298.15 298.15 305.15
33
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Table 4 e (continued ) T (K)
P (bar)
gexp (mN m1)
gcalc (mN m1) GT + density dependant influence parameter
284.15 291.15 291.15 291.15 291.15 291.15 298.15 298.15 298.15 298.15
6.0 1.5 3.0 4.5 6.0 7.5 1.5 3.0 4.5 6.0
67.36 72.67 70.57 68.48 66.45 64.73 71.32 69.18 67.32 65.64
e e e e e e e e e e
T (K)
P (bar)
gexp (mN m1)
GT + constant influence parameter 66.57 71.80 69.53 67.80 65.70 63.54 70.92 68.81 66.63 64.78
gcalc (mN m1) GT + density dependant influence parameter
305.15 305.15 305.15 305.15 312.15 312.15 312.15 312.15 312.15 312.15
4 6 8 10 2 4 6 8 10 12
68.55 66.48 64.26 62.55 69.09 67.45 65.24 63.19 61.33 59.75
e e e e e e e e e e
GT + constant influence parameter 67.18 65.89 63.76 61.60 68.45 66.09 64.08 62.16 60.23 58.70
Constant influence parameter: 1.22; Proposed influence parameter: e
parameters for the proposed influence parameter are determined, respectively for (C2H6 þ H2O) system: kij ¼ 0:7394 0:00253T
(21)
Fig. 6 and Table 4 indicate the experimental and calculated surface tensions for (C2H6 þ H2O) and (C3H8 þ H2O) systems. Similar to (CH4 þ H2O) and (CO2 þ H2O) systems, for (C2H6 þ H2O) and (C3H8 þ H2O) systems, it is concluded that the applied model can reproduce the surface tensions for (C2H6 þ H2O) and (C3H8 þ H2O) systems. Also, it is seen that the combination of present model with constant influence parameter performs better than the proposed influence parameter for (C2H6 þ H2O) and (C3H8 þ H2O) systems. The very good performance of the applied model in combination the constant influence parameter allows us to determine the reliable interfacial density profiles. Figs. 7 and 8 show the partial density profiles of interface for (C2H6 þ H2O)
Fig. 2 e Surface tension vs. pressure for (CH4 þ H2O) and (CO2 þ H2O) systems. (A) (CH4 þ H2O) experimental data at 298.15 K from Yan et al. (2001); (-) (CH4 þ H2O) experimental data at 298.15 K from Sachs and Meyn (1995); (C) (CH4 þ H2O) experimental data at 298.15 K (this work); (:) (CO2 þ H2O) experimental data at 298.15 K from Ren et al. (2000); and (þ) (CO2 þ H2O) experimental data at 298.15 K (this work).
Fig. 3 e Surface tension vs. pressure for the (CH4 þ H2O) and (CO2 þ H2O) systems at 298.15 K; (-) (CH4 þ H2O) experimental data (this work); (C) (CO2 þ H2O) experimental data (this work); (e) gradient theory þ proposed influence parameter for (CH4 þ H2O); and ( … ) gradient theory þ proposed influence parameter for (CO2 þ H2O).
Fig. 4 e Determined density profiles of CH4 and H2O in the interface at T ¼ 298.15 K and P ¼ 30 bar; (e) CH4 and ( … ) H2O.
34
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
Fig. 5 e Determined density profiles of CO2 and H2O in the interface at T ¼ 298.15 K and P ¼ 30 bar; (e) CO2 and ( … ) H2O.
Fig. 8 e Determined density profiles of C3H8 and H2O in the interface at T ¼ 284.15 K and P ¼ 1.5 bar; (e) C3H8 and ( … ) H2O.
5.
Fig. 6 e Surface tension vs. pressure for the (C2H6 þ H2O) system at 284.15 K; (-) (C2H6 þ H2O) experimental data (this work); (e) gradient theory þ proposed influence parameter.
and (C3H8 þ H2O) systems at T ¼ 284.15 K and P ¼ 6 bar and P ¼ 1.5 bar, respectively. It is concluded that the interfacial behavior of (C2H6 þ H2O) and (C3H8 þ H2O) systems is the same as (CH4 þ H2O) and (CO2 þ H2O) systems.
Fig. 7 e Determined density profiles of C2H6 and H2O in the interface at T ¼ 284.15 K and P ¼ 6 bar; (e) C2H6 and ( … ) H2O.
Conclusions
The surface tensions of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems are measured with the pendant drop technique at temperatures ranged (284.15e312.15) K and pressures ranged (1e60) bar. The gradient theory in combination with CK-SAFT EOS is utilized for modeling the surface tensions of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) binary systems. A proposed influence parameter correlation of the gradient theory is also used for the surface tension of (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) systems. The binary interaction parameters of these four systems are fitted according to the bulk densities and surface tensions. The density profiles of the interface phase are also determined. Surface tensions could be reproduced well for (CH4 þ H2O), (C2H6 þ H2O), (CO2 þ H2O) and (C3H8 þ H2O) systems by the proposed influence parameter and combination of the GT with the CK-SAFT EOS.
references
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 2 6 e3 5
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