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Prediction of water vapor sorption in the polymeric membranes using PHSC equation of state
Journal of Natural Gas Science and Engineering 21 (2014) 757e763
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Prediction of water vapor sorption in the polymeric membranes using PHSC equation of state Fatemeh Sabzi*, Mohammad Reza Talaghat, Abbas Hosseini Department of Chemical Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 August 2014 Received in revised form 30 September 2014 Accepted 7 October 2014 Available online
Dehydration of gas streams such as natural gas, flue gas and compressed air is a major process which has applications in the industry. The most common method for the dehydration of gas streams is the sorption method. Recently attention has been given to the use of membranes for the adsorption of water vapor from gas streams. In this study, water vapor sorption in the polymeric membranes including PEBAX-1074, SPEEK, PEO-PBT, NTDA and Nafion-117 has been modeled by using a thermodynamic model named perturbed-hard-sphere-chain (PHSC) equation of state (EoS) at temperatures 20e70 C. PHSC equation of state is the sum of a hard-sphere-chain term as a reference system and a van der Waals attractive term as a perturbation term. PHSC equation of state for the simple fluids and spherical molecules contains two parameters which are ε, the well depth in molecular potential energy function, and s, separation distance between segment centers of molecules. While this equation is used for the chain molecules such as polymers, the parameter r, the number of effective hard spheres per molecule is also required. The equilibrium calculations have been performed and results have been compared with the experimental data showing an overall average absolute deviation equal 0.53. © 2014 Elsevier B.V. All rights reserved.
Keywords: Dehydration Sorption Polymeric membrane PHSC equation of state
1. Introduction Dehydration of gas streams is very important in industrial processes. Applications can include: dehydration of natural gas (Ohlrogge et al., 2001; Ohlrogge and Brinkmann, 2003), dehydration of flue gas (Sijbesma et al., 2008) and drying of compressed air (Balik, 1996). The raw natural gas compositions extracted from different underground sources cover a wide range of components. The raw natural gas contains undesirable impurities, such as water, carbon dioxide, nitrogen, and hydrogen sulfide. Although the composition of raw natural gas varies widely, but the composition of the gas delivered to commercial pipeline network needs to be controlled. Natural gas requires treatment before its delivery to the pipeline. Water vapor removal of natural gas is crucial for corrosion control, hydrate formation and frost formation prevention and also to adjust the dew point of produced gas to a desirable and permissible limit for delivery to the pipeline network (Baker and Lokhandwala, 2008). Flue gas dehydration is necessary to prevent corrosion in industrial units, due to the condensation of water vapor in the stack. Pre-heating is required for preventing water vapor condensation in the flue gas, but this will increase the energy * Corresponding author. E-mail address: [email protected] (F. Sabzi). http://dx.doi.org/10.1016/j.jngse.2014.10.003 1875-5100/© 2014 Elsevier B.V. All rights reserved.
consumption as well as the costs (Sijbesma et al., 2008; Bikson et al., 2003). The production of compressed air in the industry is a difficult process because of its costs. With regard to the likelihood of disturbance in the controlling equipment, corrosion will be increased severely if there will be water in the compressed air, so drying of the compressed air is significant (Balik, 1996; Jansen, 2007). Sorption is the most common method for dehydration of gas streams. In the recent decades, the use of membranes has been noticed due to the low energy consumption, low maintenance costs, reduction in the size and weight of facilities and also the lack of moving parts. The other advantages of the membrane technology are easy installation, easy operation, energy saving, the lack of the environmental pollution and its consistency whit the environment. Due to the advantages of membrane separation, this technique could be a viable alternative for the absorption (Saidur et al., 2010). Nowadays, the commercial membranes are mainly fabricated from polymeric materials. Thermal resistance in a wide temperature range along with mechanical and chemical stability is the properties of polymers which makes them suitable for separation applications (Li et al., 2008; Takht Ravanchi et al., 2009). In this study, as the separation of water vapor from the gas streams has been spotted, the use of hydrophilic membranes is
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Nomenclature
List of symbols AAD Average Absolute Deviation A Helmholtz free energy (J mol1) A* Characteristic surface area (cm2 mol1) av Activity of water vapor a Attractive forces between two non-bonded segments (bar cm6 mol2) b van der Waals co-volume (cm3 mol1) d Hard-sphere diameter (Å) E* Characteristic interaction energy (bar cm3 mol1) Fa, Fb Universal functions of reduced temperature g(dþ) Pair radial distribution function k Boltzmann's constant (J K1) Kij Binary interaction parameter NA Avogadro's number Np Number of equilibrium data N Number of molecules Nr Total number of segments OF Objective function P Pressure (bar) Q Parameter in Eq. (22)
more efficient. PEBAX-1074, SPEEK, PEO-PBT, NTDA and Nafion-117 are polymeric membrane candidates to be considered here. PEBAX-1074 has a unique composition of 45 wt% of hard phase (poly amide, PA) and 55 wt% of soft phase (poly ethylene oxide, PEO). Permeation of gas and water vapor occurs through the soft, hydrophilic PEO phase preferentially, while the hard PA phase provides the mechanical stability of the membrane material (Shangguan, 2011; Potreck, 2009; Potreck et al., 2009a). SPEEK (Sulfonated poly ether ether ketone) has been made by the sulfonation of PEEK (poly ether ether ketone). SPEEK is a glassy hydrophilic polymer, and is similar to the commercially available PEBAX-1074. This polymer shows excellent permeability and high selectivity of water vapor (Potreck, 2009; Potreck et al., 2009b). The PEOePBT (poly (Ethylene Oxide) Poly (Butylene Terephthalate)) multi-block copolymer consists of two segments: a hard hydrophobic rigid crystalline PBT segment (x) and a soft hydrophilic rubbery PEO segment (y). The following notation classifies the various block copolymers: mPEOyPBTx, where m is the molecular weight of the PEO segment and y and x are the weight percentages of PEO and PBT, respectively (Metz et al., 2005). Finally, sulfonated polymers such as 1,4,5,8-naphthalene tetracarboxylic dianhydride (NTDA) and Nafion-117, are used for membrane separation due to the high mechanical strength and good chemical stability (Watari et al., 2003; Kim et al., 2006; Elfring, 2005). PHSC equation of state can be used for both small and large molecules such as polymers (Song et al., 1994a; Hino and Prausnitz, 1997). This equation has been also applied for the vapor-liquid equilibrium in copolymer solutions demonstrating the acceptable results (Chen et al., 2004). For example the sorption of CO2, C2H2 and C2H4 in HOF-1a has been investigated through this equation of state and good agreement between the model and experimental data has been observed (Sabzi and Molaei, 2013). The water vapor sorption in selected polymeric membranes has not been modeled previously with any equation of state. In this study, PHCS model has been used for the first time in order to model the phase equilibrium in the system of water vapor and the
r T V* V W X x
Number of segments per molecule Absolute temperature (K) Characteristic volume (cm3 mol1) Volume (cm3 mol1) Parameter in Eq. (22) Parameter in Eq. (22) Mole fraction
Greek Symbols ε Well depth of molecular potential energy function (J mol1) s Separation distance between segment centers (Å) h Packing fraction for pure substances m Chemical potential (J mol1) n Number of groups x Packing fraction for mixtures r Density (mol cm3) S Summation 4 Volume fraction Superscripts and subscripts cal Calculated exp Experiment i, j Type of component
above-mentioned polymeric materials. By applying PHSC equation of state, storage properties can be achieved in vast classes of polymers and fluids. By combining this equation with the method of group contribution and knowing the structure of molecules, characteristic parameters and equilibrium conditions have been calculated well. 2. Theory 2.1. PHSC equation of state Perturbed-hard-sphere-chain equation of state is the modified and extended version of Chiew equation (Chiew, 1990). This equation was proposed by Song et al. for a pure substance, considering the hard sphere chain term as the reference state and a van der Waals attractive term as the perturbation term as below (Song et al., 1994a; Hino and Prausnitz, 1997; Prausnitz and Gupta, 1996):
P rKT
h i r 2 ar ¼ 1 þ r 2 brg dþ ðr 1Þ g dþ 1 KT
(1)
where P is the pressure, T is the absolute temperature, r ¼ N/V is the number density, N is the number of molecules and V is the volume of system, K is Boltzmann's constant, d is the hard sphere diameter and g(dþ) is the radial distribution function of hard spheres in the interface. The best analytical expression for g is Carnahan-Starling equation (Carnahan and Starling, 1972):
1 h=2 g dþ ¼ ð1 hÞ3
(2)
where packing fraction h is:
h ¼ rbr=4
(3)
Three parameters r, a and b appeared in Eq. (1) have a clear physical meaning: The first parameter, r, is the number of effective hard spheres per molecule, the second parameter a, is the attractive
F. Sabzi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 757e763
forces between the two non-bonded segments and the third parameter b, stands for van der Waals co-volume per segment. a and b are temperature-dependent functions given by Song-Mason method (Song and Mason, 1991):
a ¼ ð2p=3Þs3 εFa ðKT=εÞ
(4)
b ¼ ð2p=3Þs3 εFb ðKT=εÞ
(5)
where ε is the depth of minimum in the pair potential energy function, and s is the separation distance between segment centers at this minimum. Fa and Fb are universal functions determined from thermodynamic properties of argon and methane over a wide range of temperature and pressure (Favari et al., 2000):
Fa
KT ε
KT ¼ 1:8681 exp 0:0619 ε " 3 # KT 2 þ 0:6715 exp 1:7317 ε
(6)
759
1 si þ sj 2
(12)
1 εij ¼ εij εij 2 1 Kij
(13)
sij ¼
Parameters aij and bij are obtained by Bertucco et al., 1999:
aij ¼ ð2p=3Þs3ij εij Fa KT εij
(14)
bij ¼ ð2p=3Þs3ij εij Fb KT εij
(15)
Kij is an adjustable binary interaction parameter which is calculated by using an optimization method for each pair of components in the mixture. Because of the existence of polymers in mixture, it is more convenient to write Eq. (8) on the basis of the segment fraction as (Song et al., 1994a):
m m X X P 1 ¼ 1 þ rr g dþ 4i 4j bij gij dþ 4i 1 ij rKT h ij ij ij
Fb
KT ε
KT ¼ 0:7303 exp 0:1649 ε " 3 # KT 2 þ 0:2697 exp 2:3973 ε
i
m r X 44a r KT ij i j ij
(7)
The mixture extension of PHSC EoS is obtained by applying the one-fluid-mixing rules to the EoS parameters leading to the following equation (Song et al., 1994b):
(16)
New parameter of 4i represents the segment or volume fraction and the relationship between xi and 4i is:
xi ¼
Ni 4 =r ¼ m i i P N 4j rj
(17)
j m m X i h X P ¼1þr 1 xi xj ri rj bij gij dþ xi ðri 1Þ gij dþ ij ij rKT ij
i
m r X xxrra KT ij i j i j ij
where xi ¼ Ni/N is the number fraction of molecules of species i, ri is the number of segments for the ith component, and gij ðdþ Þ is the i, j ij pair radial-distribution function of the hard-sphere mixtures in the interface. The exact statistical mechanical expression for gij ðdþ Þ is ij given by Boublik (1970): 2
xij 1 3 1 xij þ þ 1 h 2 ð1 hÞ2 2 ð1 hÞ
(9)
Values of h and xij is the packing fraction of pure substance and mixture, respectively, obtainable from following relations:
h¼
m rX xrb 4 i i i i
xij ¼
bi bj bij
!1=3
(10)
m rX 2=3 xk rk bk 4
Ni ri xr ¼ mi i P Nr xj r j
(18)
j
(8)
gij h; xij ¼
4i ¼
(11)
k
For pure components, xij is equal to h. Mixture parameters aij and bij have similar relations to Eqs. (4) and (5) for pure substances, with an exception that they have used s and ε instead of sij and εij in the equations. The combining rules for sij and εij are defined by Bertucco et al., 1999:
rr, segment density, and Nr, total segment number, in Eqs. (16) and (18) are defined as following: Nr ¼
m X
Ni ri
rr ¼ rr ¼ Nr =V
(19)
i
In all systems, equilibrium occurs when there is the equality of the chemical potential of each component in each phase. Chemical potential can be obtained from Helmholtz free energy by Prausnitz and Gupta, 1996:
mk ¼
vA vNk
(20) T;V;Nisk
where A is Helmholtz free energy and the general equation for its calculation from a pressure-explicit equation of state is (Song et al., 1994a):
AðT; V; Ni Þ ¼
m X i
Z∞ A0i ðTÞ þ
r t
m X NKT N KT dv þ KT Ni ln i V V i (21)
The substitution of Eq. (16) into the Eq. (21) and taking the partial derivation from Helmholtz free energy function, gives the chemical potential of each component for pure and mixture systems:
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m0 mk vW vQ þ ðr 1ÞQ þ ðr 1Þh ¼ k þ rW þ rh vh vh KT KT 4 ε Fa vX þ ln h Xþh r KT Fb vh
Table 1 Calculation parameters for polymers.
(22)
m m X X m0 vWij mk ¼ k þ 2rk rr 4i bik Wik þ rr 4i 4j bij Nr KT KT vNk i¼1 ij¼1 m m X vQij 1 2r r X Nr k r 4i 1 4a ðrk 1ÞQkk ri vNk KT i¼1 i ik i¼1 4 þ ln k rk KT þ 1 rk (23) The expression for W, Q and X in Eq. (22) is given in the works of Kim and Bae (Kim and Bae, 2000). The calculation procedure and the expression for W and Q in Eq. (23) are also discussed in the works of Song et al. (1994a).
PHSC
r
ε/k (K)
s (A )
PEBAX-1074 SPEEK-%59 SPEEK-%75 3000PEO52PBT48 2000PEO52PBT48 1000PEO52PBT48 1000PEO64PBT36 600PEO41PBT59 300PEO32PBT68 1000PEO40PBT60 1000PEO56PBT44 1000PEO75PBT25 NTDA-ODADS-ODA(1-1) NTDA-ODADS-ODA(3-1) NTDA-DMBDSA-BAPF(1-1) NTDA-DMBDSA-BAPF(9-1) NAFION-117
5.28 13.32 14.04 102.10 96.36 38.38 45.47 22.32 13.64 31.56 40.80 51.93 29.70 32.47 31.91 33.35 24.38
221.19 171.88 175.30 159.90 162.79 111.50 106.77 175.70 183.84 117.22 109.56 103.72 175.54 174.67 171.54 175.21 137.21
3.66 4.41 4.35 3.19 3.25 3.36 3.32 3.53 3.74 3.45 3.34 3.29 3.96 3.85 4.14 3.87 3.56
3. Results and discussion 2.2. Parameters determination PHSC equation of state for the simple fluids and spherical molecules contains two parameters ε and s. When the equation is used for the chain molecules such as polymers, the parameter r is also required. An appropriate method to obtain the above parameters is such that, these parameters are defined by characteristic volume (V*), characteristic surface Area (A*) and characteristic interaction energy (E*) parameters. The resulted relations will be (Elvassore et al., 2002):
V* ¼
p 6
rs3 NA
A* ¼ prs2 NA E* ¼ r
ε Rg ¼ rεNA K
(24) (25) (26)
where NA and Rg are the Avogadro's number and the universal gas constants, respectively. To calculate r, ε and s from the above equations, first of all, each molecule is broken down into their constituting groups and V*,A* and E* are identified for each group in accordance with the table provided by Elvassore et al. (2002). Then the characteristic parameters are obtained through:
Vi* ¼
n X
vk Vk
(27)
vk Ak
(28)
vk Ek
(29)
The binary interaction parameter (Kij) has been estimated by fitting the experimental data of water vapor sorption on the polymeric membrane. The minimized objective function used for fitting the experimental data is given as following (Lee and Kim, 2007):
OF ¼
NP Pical Piexp 1 X exp NP i¼1 Pi
!2 (30)
where NP is the number of equilibrium data for a given system and exp Pi and Pical are the experimental and calculated pressure, respectively. In this study, water vapor sorption on the polymeric membrane has been studied by using only one adjustable parameter. This is one of the advantages of this model which does not need more experimental data. The measured equilibrium uptake values which are on the basis of the volume or mass of water vapor sorbed in a determined volume or mass of polymer have been converted to volume fraction for using in the calculation of the equilibrium. Fig. 1 shows the isotherms of water vapor sorption by PEBAX1074 or concentration of water vapor in the polymer membrane as a function of the water vapor activity at 20, 30, 50 and 70 C.
k¼1
A*i ¼
n X k¼1
Ei* ¼
n X k¼1
After obtainingV*, A* and E*, for each substance or molecule, r, ε and s are gained from Eqs. (24)e(26) with a simple mathematical calculation. These parameters have been reported in Table 1 to be placed directly in the equation of state for the equilibrium calculations. r, ε and s for water used in this study have been adopted from Lee and Kim work (Lee and Kim, 2007).
Fig. 1. Prediction of water vapor sorption by PEBAX-1074 at 20, 30, 50 and 70 C, experimental data (Potreck et al., 2009a).
F. Sabzi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 757e763
Fig. 2. Prediction of water vapor sorption by SPEEK-%59 at 20, 30, 50 and 70 C. experimental data (Potreck, 2009).
It is evident from Fig. 1 that at low pressures or activities (avi ¼ Pi =Pis ), absorption does not show significant variations by increasing the pressure. But at higher pressures, there is a sharp uptake. One of the reasons for this phenomena can be the flexible structure of PEBAX-1074. This trend has been predicted by the model perfectly. Also at higher water vapor activities (av > 0.6), temperature has a significant effect on the amount of water vapor absorbed into the polymer. The concentration of water vapor inside the polymer decreases with increasing temperature. It is observed from PHSC model that water vapor sorption by PEBAX-1074 has been predicted at different temperatures satisfactorily. Fig. 2 shows the isotherms of water vapor sorption by SPEEK with sulfonation degree of %59 as a function of the water vapor activity at 20, 30, 50 and 70 C. Behavior of this polymer in water vapor sorption is completely different from PEBAX-1074. Sorption isotherms for the glassy polymer SPEEK show a concave increase towards the x-axis at low water vapor activities (av < 0.5) and at higher water vapor activities (av > 0.5) the isotherms show a convex increase towards the x-axis. The reason of this behavior is the inherent properties of the polymer. This trend has been predicted perfectly by PHSC equation of
Fig. 3. Prediction of water vapor sorption by SPEEK (sulfonation degrees of %59 and % 75) at 20 C, experimental data (Potreck et al., 2009b).
761
Fig. 4. Prediction of water vapor sorption by various species PEO-PBT with PHSC EoS at 20 c, experimental data (Metz et al., 2005).
state and model shows a good precision in predicting the experimental data. The concentration of water vapor inside the polymer increases by increasing the temperature especially at higher activities. Fig. 3 shows the isotherms of water vapor sorption by SPEEK with sulfonation degrees of %59 and %75 at 20 C. Water vapor sorption increases by increasing the Sulfonation degree. Sulfonation degree has significant impact on water vapor sorption inside the polymer especially in the higher activities. This trend has been predicted by PHSC model very well. Figs. 4 and 5 show the isotherms of water vapor sorption by the various species of PEO-PBT at temperatures 20 and 30 C. Behavior of this polymer in water vapor sorption is much like PEBAX-1074, and concentration of water vapor inside PEO-PBT increases exponentially with increasing water vapor activity. Generally, the water vapor concentration in polymer increases with increasing the molecular weight of the hydrophilic segment of polymer (PEO) and also with increasing the weight percent of hydrophilic segment (PEO) ratio to hydrophobic segment (PBT). This trend has been predicted perfectly by PHSC equation of state. Figs. 6 and 7 show the concentration of water vapor inside NTDA-ODADS-ODA and NTDA-DMBDSA-BAPF as a function of the water vapor activity at constant temperatures 50 C. Behavior of
Fig. 5. Prediction of water vapor sorption by various species PEO-PBT with PHSC EoS at 30 c, experimental data (Metz et al., 2005).
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Fig. 6. Prediction of water vapor sorption by NTDA-ODADS-ODA (1-1) and NTDAODADS-ODA (3-1) with PHSC EoS at 50 C, experimental data (Watari et al., 2003).
this polymer in water vapor sorption is the same as SPEEK, and concentration of water vapor inside polymer increases with increasing water vapor activity. Generally, the water vapor concentration in polymer increases with increasing sulfonation to nonsulfonation segment ratio. This trend has been predicted perfectly by PHSC equation of state. Fig. 8 shows the isotherms of water vapor sorption by NAFION117 as a function of the water vapor activity at 27 and 30 C. The concentration of water vapor inside the polymer increases with increasing the temperature and with water vapor activity. As it is observed, PHSC model has predicted the water vapor sorption by NAFION-117 at different temperatures, satisfactorily. Table 2 reports the optimized binary interaction parameters and the percent of average absolute deviation (%AAD) in activity of water vapor calculated by the following equation:
%AAD ¼
Np 100 X cal exp avi avi Np i¼1
(31)
As it is evident from Table 2, the percentage of overall average absolute deviation equal 0.53 has been obtained. This result
Fig. 7. Prediction of water vapor sorption by NTDA-DMBDSA-BAPF (1-1) and NTDADMBDSA-BAPF (9-1) with PHSC EoS at 50 c, experimental data (Watari et al., 2003).
Fig. 8. Prediction of water vapor sorption by Nafion-117 with PHSC EoS at 27 and 30 C, experimental data (Kim et al., 2006; Elfring, 2005).
represents the high precision of PHSC model in predicting equilibrium data of water vaporepolymer systems. 4. Conclusions In this work, perturbed hard sphere chain (PHSC) equation of state has been used for thermodynamic modeling of water vapor sorption in PEBAX-1074, SPEEK, PEO-PBT, NTDA and Nafion-117. The parameters of PHSC model such as r, ε and s have been calculated by group contribution approach. One binary interaction parameter (Kij) has been optimized by fitting the experimental data for modeling of each system in this study. PHSC equation of state has proved to be an appropriate approach for modeling of phase equilibrium conditions in water vaporepolymer systems. This model is a suitable method for predicting the equilibrium in systems including complex molecules with high molecular weight where other models are less powerful.
Table 2 Optimized binary interaction parameters and average absolute deviations for water vaporepolymer systems.
H2OePEBAX 1074 H2OePEBAX 1074 H2OePEBAX 1074 H2OePEBAX 1074 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%75 H2Oe3000PEO52PBT48 H2Oe2000PEO52PBT48 H2Oe1000PEO52PBT48 H2Oe1000PEO64PBT36 H2Oe600PEO41PBT59 H2Oe300PEO32PBT68 H2Oe1000PEO40PBT60 H2Oe1000PEO56PBT44 H2Oe1000PEO75PBT25 H2OeNTDA-ODADS-ODA(1-1) H2OeNTDA-ODADS-ODA(3-1) H2OeNTDA-DMBDSA-BAPF(1-1) H2OeNTDA-DMBDSA-BAPF(9-1) H2OeNAFION-117 H2OeNAFION-117
T(K)
Kij
%AAD
293 303 323 343 293 303 323 343 293 293 293 293 293 293 293 293 303 303 303 323 323 323 323 300 303
0.05 0.05 0.05 0.05 0.15 0.14 0.17 0.15 0.16 0.14 0.12 0.11 0.11 0.11 0.10 0.12 0.06 0.05 0.05 0.96 0.96 0.63 0.73 0.36 0.37
0.29 0.57 0.91 0.59 0.69 0.39 0.54 0.74 0.57 0.50 0.15 0.26 0.23 0.09 0.25 0.37 0.34 0.52 0.60 0.67 0.56 0.47 0.39 1.17 1.05
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Predictions by PHSC model are in good agreement with the experimental data. For mixtures, excellent results have been obtained with one single value of binary interaction parameter. However, it is worth to mention that PHSC equation of state in the presented form has some inadequacy to model the vaporeliquid or liquideliquid equilibrium in systems with polar or hydrogen bonding interactions especially in higher temperatures and pressures. Acknowledgments The authors wish to thank the computer facilities provided by Shiraz University of Technology. References Baker, R.W., Lokhandwala, K., 2008. Natural gas processing with membranes: an overview. Ind. Eng. Chem. Res. 47, 2109e2121. Balik, C.M., 1996. On the extraction of diffusion coefficients from gravimetric data for sorption of small molecules by polymer thin films. Macromolecules 29, 3025e3029. Bertucco, A., Elvassore, N., Fermeglia, M., Prausnitz, J.M., 1999. A perturbed-hardsphere-chain equation of state for phase equilibria of mixtures containing a refrigerant and a lubricant oil. Fluid Phase Equilibria 158e160, 183e191. Bikson, B., Giglia, S., Hao, J., 2003. Novel Composite Membranes and Process for Natural Gas Upgrading. In: Technical Report to Department of Energy, United States. FC26e99FT40497. Boublik, T.J., 1970. Equation of state of linear fused hard-sphere models. Chem. Phys. 53, 471e483. Carnahan, N.F., Starling, K.E., 1972. Intermolecular repulsions and the equation of states for fluids. AIChE J. 18, 1184e1189. Chen, X., Sato, Y., Takishima, S., Masuoka, H., 2004. Chain reference system and scaling factor algorithm for perturbed hard- sphere- chain equation of state. Fluid Phase Equilibria 226, 129e139. Chiew, Y.C., 1990. Percus- Yevick integral- equation theory for athermal hardsphere chain Part I: equation of state. Mol. Phys. 70, 129e143. Elfring, G.J., 2005. Thermodynamics of Sorption and Distribution of Water in Nafion (Master of applied science thesis). University of Victoria. Elvassore, N., Fermeglia, M., Bertucco, A., 2002. Phase-equilibria calculation by group- contribution perturbed-hard-sphere-chain equation of state. AIChE J. 48, 359e368. Favari, F., Bertucco, A., Elvassore, N., Fermeglia, M., 2000. Multiphase multicomponent equlibria for mixtures containing polymers by the perturbation theory. Chem. Eng. Sci. 55, 2379e2392. Hino, T., Prausnitz, J.M., 1997. A perturbed hard-sphere-chain equation of state for normal fluids and polymers using the square-well potential of variable width. Fluid Phase Equilibria 138, 105e130. Jansen, J., 2007. Gas and Water Permeation through Highly Hydrophilic Membranes (Bachelor thesis). Membrane Technology Group, University of Twente.
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Kim, H.I., Bae, Y.C., 2000. Phase equilibria of polymer solutions: modified perturbed hard-sphere-chain equation of state. Polymer 41, 875e881. Kim, M.H., Glinka, C.J., Grot, S.A., Grot, W.G., 2006. SANS study of the effects of water vapor sorption on the nano scale structure of perfluorinated sulfonic acid (NAFION) membranes. Macromolecules 39, 4775e4787. Lee, B.S., Kim, K.C., 2007. Phase equilibria of associating fluid mixtures using the perturbed-hard-sphere-chain equation of state combined with the association model. Korean J. Chem. Eng. 24, 133e147. Li, G.M., Feng, C., Li, J.F., Liu, J.Z., Wu, Y.L., 2008. Water vapor permeation and compressed air dehydration performances of modified polyimide membrane. Sep. Purif. Technol. 60, 330e334. Metz, S.J., van de Ven, W.J.C., Mulder, M.H.V., Wessling, M., 2005. Mixed gas water vapor/N2 transport in poly(ethylene oxide) poly(butylene terephthalate) block copolymers. J. Membr. Sci. 266, 51e61. Ohlrogge, K., Keil, B., Wind, J., 2001. Dehydration and Hydrocarbon Dew Pointing of Natural Gas by Membrane Technology. Symposium on Membrane Technology in Petroleum. Petrochemical and Gas Processing, San Diego, pp. 138e141. Ohlrogge, K., Brinkmann, T., 2003. Natural gas cleanup by means of membranes. Adv. Mem. Tech. 984, 306e317. Potreck, J., 2009. Transport Behavior of Water and Gas in Hydrophilic Polymer Membranes (Doctoral dissertation). University of Twente. Potreck, J., Nijmeijer, K., Kosinski, T., Wessling, M., 2009a. Mixed water vapor gas transport through the rubbery polymer PEBAX® 1074. J. Membr. Sci. 338, 11e16. Potreck, J., Uyar, F., Sijbesma, H., Nijmeijer, K., Stamatialis, D., Wessling, M., 2009b. Sorption induced relaxations during water diffusion in S-PEEK. Phys. Chem. Chem. Phys. 11, 298e308. Prausnitz, J.M., Gupta, R.B., 1996. Vapor-liquid equilibria for solvent-polymer system from a perturbed hard-sphere-chain equation of state. Ind. Eng. Chem. Res. 35, 1225e1230. Sabzi, F., Molaei, H., 2013. Sorption of CO2, C2H2 and C2H4 in HOF-1a studied by PHSC equation of state. Fluid Phase Equilibria 360, 23e28. Saidur, R., Rahim, N.A., Hasanuzzaman, M., 2010. A review on compressed-air energy use and energy savings. Renew. Sustain. Energy Rev. 14, 1135e1153. Shangguan, Y., 2011. Intrinsic Properties of Poly(Ether-B-Amide) (Pebax®1074) for Gas Permeation and Pervaporation (Master of applied science thesis). Waterloo University. Sijbesma, H., Nijmeijer, K., van Marwijk, R., Heijboer, R., Potreck, J., Wessling, M., 2008. Flue gas dehydration using polymer membranes. J. Membr. Sci. 313, 263e276. Song, Y., Mason, E.A., 1991. Statistical-mechanical theory of a new analytical equation of state. J. Chem. Phys. 12, 7840e7853. Song, Y., Lambert, S.M., Prausnitz, J.M., 1994a. Liquid-liquid phase diagrams for binary polymer solutions from a perturbecl hard-sphere equation of state. Chem. Eng. Sci. 49, 2765e2775. Song, Y., Lambert, S.M., Prausnitz, J.M., 1994b. A perturbed hard-sphere-chain equation of state for normal fluids and polymers. Ind. Eng. Chem. Res. 33, 1047e1057. Takht Ravanchi, M., Kaghazchi, T., Kargari, A., 2009. Application of membrane separation processes in petrochemical industry: a review. Desalination 235, 199e244. Watari, T., Wang, H., Kuwahara, K., Tanaka, K., Kita, H., Okamoto, K.I., 2003. Water vapor sorption and diffusion properties of sulfonated polyimide membranes. J. Membr. Sci. 219, 137e147.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Prediction of water vapor sorption in the polymeric membranes using PHSC equation of state Fatemeh Sabzi*, Mohammad Reza Talaghat, Abbas Hosseini Department of Chemical Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 August 2014 Received in revised form 30 September 2014 Accepted 7 October 2014 Available online
Dehydration of gas streams such as natural gas, flue gas and compressed air is a major process which has applications in the industry. The most common method for the dehydration of gas streams is the sorption method. Recently attention has been given to the use of membranes for the adsorption of water vapor from gas streams. In this study, water vapor sorption in the polymeric membranes including PEBAX-1074, SPEEK, PEO-PBT, NTDA and Nafion-117 has been modeled by using a thermodynamic model named perturbed-hard-sphere-chain (PHSC) equation of state (EoS) at temperatures 20e70 C. PHSC equation of state is the sum of a hard-sphere-chain term as a reference system and a van der Waals attractive term as a perturbation term. PHSC equation of state for the simple fluids and spherical molecules contains two parameters which are ε, the well depth in molecular potential energy function, and s, separation distance between segment centers of molecules. While this equation is used for the chain molecules such as polymers, the parameter r, the number of effective hard spheres per molecule is also required. The equilibrium calculations have been performed and results have been compared with the experimental data showing an overall average absolute deviation equal 0.53. © 2014 Elsevier B.V. All rights reserved.
Keywords: Dehydration Sorption Polymeric membrane PHSC equation of state
1. Introduction Dehydration of gas streams is very important in industrial processes. Applications can include: dehydration of natural gas (Ohlrogge et al., 2001; Ohlrogge and Brinkmann, 2003), dehydration of flue gas (Sijbesma et al., 2008) and drying of compressed air (Balik, 1996). The raw natural gas compositions extracted from different underground sources cover a wide range of components. The raw natural gas contains undesirable impurities, such as water, carbon dioxide, nitrogen, and hydrogen sulfide. Although the composition of raw natural gas varies widely, but the composition of the gas delivered to commercial pipeline network needs to be controlled. Natural gas requires treatment before its delivery to the pipeline. Water vapor removal of natural gas is crucial for corrosion control, hydrate formation and frost formation prevention and also to adjust the dew point of produced gas to a desirable and permissible limit for delivery to the pipeline network (Baker and Lokhandwala, 2008). Flue gas dehydration is necessary to prevent corrosion in industrial units, due to the condensation of water vapor in the stack. Pre-heating is required for preventing water vapor condensation in the flue gas, but this will increase the energy * Corresponding author. E-mail address: [email protected] (F. Sabzi). http://dx.doi.org/10.1016/j.jngse.2014.10.003 1875-5100/© 2014 Elsevier B.V. All rights reserved.
consumption as well as the costs (Sijbesma et al., 2008; Bikson et al., 2003). The production of compressed air in the industry is a difficult process because of its costs. With regard to the likelihood of disturbance in the controlling equipment, corrosion will be increased severely if there will be water in the compressed air, so drying of the compressed air is significant (Balik, 1996; Jansen, 2007). Sorption is the most common method for dehydration of gas streams. In the recent decades, the use of membranes has been noticed due to the low energy consumption, low maintenance costs, reduction in the size and weight of facilities and also the lack of moving parts. The other advantages of the membrane technology are easy installation, easy operation, energy saving, the lack of the environmental pollution and its consistency whit the environment. Due to the advantages of membrane separation, this technique could be a viable alternative for the absorption (Saidur et al., 2010). Nowadays, the commercial membranes are mainly fabricated from polymeric materials. Thermal resistance in a wide temperature range along with mechanical and chemical stability is the properties of polymers which makes them suitable for separation applications (Li et al., 2008; Takht Ravanchi et al., 2009). In this study, as the separation of water vapor from the gas streams has been spotted, the use of hydrophilic membranes is
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Nomenclature
List of symbols AAD Average Absolute Deviation A Helmholtz free energy (J mol1) A* Characteristic surface area (cm2 mol1) av Activity of water vapor a Attractive forces between two non-bonded segments (bar cm6 mol2) b van der Waals co-volume (cm3 mol1) d Hard-sphere diameter (Å) E* Characteristic interaction energy (bar cm3 mol1) Fa, Fb Universal functions of reduced temperature g(dþ) Pair radial distribution function k Boltzmann's constant (J K1) Kij Binary interaction parameter NA Avogadro's number Np Number of equilibrium data N Number of molecules Nr Total number of segments OF Objective function P Pressure (bar) Q Parameter in Eq. (22)
more efficient. PEBAX-1074, SPEEK, PEO-PBT, NTDA and Nafion-117 are polymeric membrane candidates to be considered here. PEBAX-1074 has a unique composition of 45 wt% of hard phase (poly amide, PA) and 55 wt% of soft phase (poly ethylene oxide, PEO). Permeation of gas and water vapor occurs through the soft, hydrophilic PEO phase preferentially, while the hard PA phase provides the mechanical stability of the membrane material (Shangguan, 2011; Potreck, 2009; Potreck et al., 2009a). SPEEK (Sulfonated poly ether ether ketone) has been made by the sulfonation of PEEK (poly ether ether ketone). SPEEK is a glassy hydrophilic polymer, and is similar to the commercially available PEBAX-1074. This polymer shows excellent permeability and high selectivity of water vapor (Potreck, 2009; Potreck et al., 2009b). The PEOePBT (poly (Ethylene Oxide) Poly (Butylene Terephthalate)) multi-block copolymer consists of two segments: a hard hydrophobic rigid crystalline PBT segment (x) and a soft hydrophilic rubbery PEO segment (y). The following notation classifies the various block copolymers: mPEOyPBTx, where m is the molecular weight of the PEO segment and y and x are the weight percentages of PEO and PBT, respectively (Metz et al., 2005). Finally, sulfonated polymers such as 1,4,5,8-naphthalene tetracarboxylic dianhydride (NTDA) and Nafion-117, are used for membrane separation due to the high mechanical strength and good chemical stability (Watari et al., 2003; Kim et al., 2006; Elfring, 2005). PHSC equation of state can be used for both small and large molecules such as polymers (Song et al., 1994a; Hino and Prausnitz, 1997). This equation has been also applied for the vapor-liquid equilibrium in copolymer solutions demonstrating the acceptable results (Chen et al., 2004). For example the sorption of CO2, C2H2 and C2H4 in HOF-1a has been investigated through this equation of state and good agreement between the model and experimental data has been observed (Sabzi and Molaei, 2013). The water vapor sorption in selected polymeric membranes has not been modeled previously with any equation of state. In this study, PHCS model has been used for the first time in order to model the phase equilibrium in the system of water vapor and the
r T V* V W X x
Number of segments per molecule Absolute temperature (K) Characteristic volume (cm3 mol1) Volume (cm3 mol1) Parameter in Eq. (22) Parameter in Eq. (22) Mole fraction
Greek Symbols ε Well depth of molecular potential energy function (J mol1) s Separation distance between segment centers (Å) h Packing fraction for pure substances m Chemical potential (J mol1) n Number of groups x Packing fraction for mixtures r Density (mol cm3) S Summation 4 Volume fraction Superscripts and subscripts cal Calculated exp Experiment i, j Type of component
above-mentioned polymeric materials. By applying PHSC equation of state, storage properties can be achieved in vast classes of polymers and fluids. By combining this equation with the method of group contribution and knowing the structure of molecules, characteristic parameters and equilibrium conditions have been calculated well. 2. Theory 2.1. PHSC equation of state Perturbed-hard-sphere-chain equation of state is the modified and extended version of Chiew equation (Chiew, 1990). This equation was proposed by Song et al. for a pure substance, considering the hard sphere chain term as the reference state and a van der Waals attractive term as the perturbation term as below (Song et al., 1994a; Hino and Prausnitz, 1997; Prausnitz and Gupta, 1996):
P rKT
h i r 2 ar ¼ 1 þ r 2 brg dþ ðr 1Þ g dþ 1 KT
(1)
where P is the pressure, T is the absolute temperature, r ¼ N/V is the number density, N is the number of molecules and V is the volume of system, K is Boltzmann's constant, d is the hard sphere diameter and g(dþ) is the radial distribution function of hard spheres in the interface. The best analytical expression for g is Carnahan-Starling equation (Carnahan and Starling, 1972):
1 h=2 g dþ ¼ ð1 hÞ3
(2)
where packing fraction h is:
h ¼ rbr=4
(3)
Three parameters r, a and b appeared in Eq. (1) have a clear physical meaning: The first parameter, r, is the number of effective hard spheres per molecule, the second parameter a, is the attractive
F. Sabzi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 757e763
forces between the two non-bonded segments and the third parameter b, stands for van der Waals co-volume per segment. a and b are temperature-dependent functions given by Song-Mason method (Song and Mason, 1991):
a ¼ ð2p=3Þs3 εFa ðKT=εÞ
(4)
b ¼ ð2p=3Þs3 εFb ðKT=εÞ
(5)
where ε is the depth of minimum in the pair potential energy function, and s is the separation distance between segment centers at this minimum. Fa and Fb are universal functions determined from thermodynamic properties of argon and methane over a wide range of temperature and pressure (Favari et al., 2000):
Fa
KT ε
KT ¼ 1:8681 exp 0:0619 ε " 3 # KT 2 þ 0:6715 exp 1:7317 ε
(6)
759
1 si þ sj 2
(12)
1 εij ¼ εij εij 2 1 Kij
(13)
sij ¼
Parameters aij and bij are obtained by Bertucco et al., 1999:
aij ¼ ð2p=3Þs3ij εij Fa KT εij
(14)
bij ¼ ð2p=3Þs3ij εij Fb KT εij
(15)
Kij is an adjustable binary interaction parameter which is calculated by using an optimization method for each pair of components in the mixture. Because of the existence of polymers in mixture, it is more convenient to write Eq. (8) on the basis of the segment fraction as (Song et al., 1994a):
m m X X P 1 ¼ 1 þ rr g dþ 4i 4j bij gij dþ 4i 1 ij rKT h ij ij ij
Fb
KT ε
KT ¼ 0:7303 exp 0:1649 ε " 3 # KT 2 þ 0:2697 exp 2:3973 ε
i
m r X 44a r KT ij i j ij
(7)
The mixture extension of PHSC EoS is obtained by applying the one-fluid-mixing rules to the EoS parameters leading to the following equation (Song et al., 1994b):
(16)
New parameter of 4i represents the segment or volume fraction and the relationship between xi and 4i is:
xi ¼
Ni 4 =r ¼ m i i P N 4j rj
(17)
j m m X i h X P ¼1þr 1 xi xj ri rj bij gij dþ xi ðri 1Þ gij dþ ij ij rKT ij
i
m r X xxrra KT ij i j i j ij
where xi ¼ Ni/N is the number fraction of molecules of species i, ri is the number of segments for the ith component, and gij ðdþ Þ is the i, j ij pair radial-distribution function of the hard-sphere mixtures in the interface. The exact statistical mechanical expression for gij ðdþ Þ is ij given by Boublik (1970): 2
xij 1 3 1 xij þ þ 1 h 2 ð1 hÞ2 2 ð1 hÞ
(9)
Values of h and xij is the packing fraction of pure substance and mixture, respectively, obtainable from following relations:
h¼
m rX xrb 4 i i i i
xij ¼
bi bj bij
!1=3
(10)
m rX 2=3 xk rk bk 4
Ni ri xr ¼ mi i P Nr xj r j
(18)
j
(8)
gij h; xij ¼
4i ¼
(11)
k
For pure components, xij is equal to h. Mixture parameters aij and bij have similar relations to Eqs. (4) and (5) for pure substances, with an exception that they have used s and ε instead of sij and εij in the equations. The combining rules for sij and εij are defined by Bertucco et al., 1999:
rr, segment density, and Nr, total segment number, in Eqs. (16) and (18) are defined as following: Nr ¼
m X
Ni ri
rr ¼ rr ¼ Nr =V
(19)
i
In all systems, equilibrium occurs when there is the equality of the chemical potential of each component in each phase. Chemical potential can be obtained from Helmholtz free energy by Prausnitz and Gupta, 1996:
mk ¼
vA vNk
(20) T;V;Nisk
where A is Helmholtz free energy and the general equation for its calculation from a pressure-explicit equation of state is (Song et al., 1994a):
AðT; V; Ni Þ ¼
m X i
Z∞ A0i ðTÞ þ
r t
m X NKT N KT dv þ KT Ni ln i V V i (21)
The substitution of Eq. (16) into the Eq. (21) and taking the partial derivation from Helmholtz free energy function, gives the chemical potential of each component for pure and mixture systems:
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m0 mk vW vQ þ ðr 1ÞQ þ ðr 1Þh ¼ k þ rW þ rh vh vh KT KT 4 ε Fa vX þ ln h Xþh r KT Fb vh
Table 1 Calculation parameters for polymers.
(22)
m m X X m0 vWij mk ¼ k þ 2rk rr 4i bik Wik þ rr 4i 4j bij Nr KT KT vNk i¼1 ij¼1 m m X vQij 1 2r r X Nr k r 4i 1 4a ðrk 1ÞQkk ri vNk KT i¼1 i ik i¼1 4 þ ln k rk KT þ 1 rk (23) The expression for W, Q and X in Eq. (22) is given in the works of Kim and Bae (Kim and Bae, 2000). The calculation procedure and the expression for W and Q in Eq. (23) are also discussed in the works of Song et al. (1994a).
PHSC
r
ε/k (K)
s (A )
PEBAX-1074 SPEEK-%59 SPEEK-%75 3000PEO52PBT48 2000PEO52PBT48 1000PEO52PBT48 1000PEO64PBT36 600PEO41PBT59 300PEO32PBT68 1000PEO40PBT60 1000PEO56PBT44 1000PEO75PBT25 NTDA-ODADS-ODA(1-1) NTDA-ODADS-ODA(3-1) NTDA-DMBDSA-BAPF(1-1) NTDA-DMBDSA-BAPF(9-1) NAFION-117
5.28 13.32 14.04 102.10 96.36 38.38 45.47 22.32 13.64 31.56 40.80 51.93 29.70 32.47 31.91 33.35 24.38
221.19 171.88 175.30 159.90 162.79 111.50 106.77 175.70 183.84 117.22 109.56 103.72 175.54 174.67 171.54 175.21 137.21
3.66 4.41 4.35 3.19 3.25 3.36 3.32 3.53 3.74 3.45 3.34 3.29 3.96 3.85 4.14 3.87 3.56
3. Results and discussion 2.2. Parameters determination PHSC equation of state for the simple fluids and spherical molecules contains two parameters ε and s. When the equation is used for the chain molecules such as polymers, the parameter r is also required. An appropriate method to obtain the above parameters is such that, these parameters are defined by characteristic volume (V*), characteristic surface Area (A*) and characteristic interaction energy (E*) parameters. The resulted relations will be (Elvassore et al., 2002):
V* ¼
p 6
rs3 NA
A* ¼ prs2 NA E* ¼ r
ε Rg ¼ rεNA K
(24) (25) (26)
where NA and Rg are the Avogadro's number and the universal gas constants, respectively. To calculate r, ε and s from the above equations, first of all, each molecule is broken down into their constituting groups and V*,A* and E* are identified for each group in accordance with the table provided by Elvassore et al. (2002). Then the characteristic parameters are obtained through:
Vi* ¼
n X
vk Vk
(27)
vk Ak
(28)
vk Ek
(29)
The binary interaction parameter (Kij) has been estimated by fitting the experimental data of water vapor sorption on the polymeric membrane. The minimized objective function used for fitting the experimental data is given as following (Lee and Kim, 2007):
OF ¼
NP Pical Piexp 1 X exp NP i¼1 Pi
!2 (30)
where NP is the number of equilibrium data for a given system and exp Pi and Pical are the experimental and calculated pressure, respectively. In this study, water vapor sorption on the polymeric membrane has been studied by using only one adjustable parameter. This is one of the advantages of this model which does not need more experimental data. The measured equilibrium uptake values which are on the basis of the volume or mass of water vapor sorbed in a determined volume or mass of polymer have been converted to volume fraction for using in the calculation of the equilibrium. Fig. 1 shows the isotherms of water vapor sorption by PEBAX1074 or concentration of water vapor in the polymer membrane as a function of the water vapor activity at 20, 30, 50 and 70 C.
k¼1
A*i ¼
n X k¼1
Ei* ¼
n X k¼1
After obtainingV*, A* and E*, for each substance or molecule, r, ε and s are gained from Eqs. (24)e(26) with a simple mathematical calculation. These parameters have been reported in Table 1 to be placed directly in the equation of state for the equilibrium calculations. r, ε and s for water used in this study have been adopted from Lee and Kim work (Lee and Kim, 2007).
Fig. 1. Prediction of water vapor sorption by PEBAX-1074 at 20, 30, 50 and 70 C, experimental data (Potreck et al., 2009a).
F. Sabzi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 757e763
Fig. 2. Prediction of water vapor sorption by SPEEK-%59 at 20, 30, 50 and 70 C. experimental data (Potreck, 2009).
It is evident from Fig. 1 that at low pressures or activities (avi ¼ Pi =Pis ), absorption does not show significant variations by increasing the pressure. But at higher pressures, there is a sharp uptake. One of the reasons for this phenomena can be the flexible structure of PEBAX-1074. This trend has been predicted by the model perfectly. Also at higher water vapor activities (av > 0.6), temperature has a significant effect on the amount of water vapor absorbed into the polymer. The concentration of water vapor inside the polymer decreases with increasing temperature. It is observed from PHSC model that water vapor sorption by PEBAX-1074 has been predicted at different temperatures satisfactorily. Fig. 2 shows the isotherms of water vapor sorption by SPEEK with sulfonation degree of %59 as a function of the water vapor activity at 20, 30, 50 and 70 C. Behavior of this polymer in water vapor sorption is completely different from PEBAX-1074. Sorption isotherms for the glassy polymer SPEEK show a concave increase towards the x-axis at low water vapor activities (av < 0.5) and at higher water vapor activities (av > 0.5) the isotherms show a convex increase towards the x-axis. The reason of this behavior is the inherent properties of the polymer. This trend has been predicted perfectly by PHSC equation of
Fig. 3. Prediction of water vapor sorption by SPEEK (sulfonation degrees of %59 and % 75) at 20 C, experimental data (Potreck et al., 2009b).
761
Fig. 4. Prediction of water vapor sorption by various species PEO-PBT with PHSC EoS at 20 c, experimental data (Metz et al., 2005).
state and model shows a good precision in predicting the experimental data. The concentration of water vapor inside the polymer increases by increasing the temperature especially at higher activities. Fig. 3 shows the isotherms of water vapor sorption by SPEEK with sulfonation degrees of %59 and %75 at 20 C. Water vapor sorption increases by increasing the Sulfonation degree. Sulfonation degree has significant impact on water vapor sorption inside the polymer especially in the higher activities. This trend has been predicted by PHSC model very well. Figs. 4 and 5 show the isotherms of water vapor sorption by the various species of PEO-PBT at temperatures 20 and 30 C. Behavior of this polymer in water vapor sorption is much like PEBAX-1074, and concentration of water vapor inside PEO-PBT increases exponentially with increasing water vapor activity. Generally, the water vapor concentration in polymer increases with increasing the molecular weight of the hydrophilic segment of polymer (PEO) and also with increasing the weight percent of hydrophilic segment (PEO) ratio to hydrophobic segment (PBT). This trend has been predicted perfectly by PHSC equation of state. Figs. 6 and 7 show the concentration of water vapor inside NTDA-ODADS-ODA and NTDA-DMBDSA-BAPF as a function of the water vapor activity at constant temperatures 50 C. Behavior of
Fig. 5. Prediction of water vapor sorption by various species PEO-PBT with PHSC EoS at 30 c, experimental data (Metz et al., 2005).
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F. Sabzi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 757e763
Fig. 6. Prediction of water vapor sorption by NTDA-ODADS-ODA (1-1) and NTDAODADS-ODA (3-1) with PHSC EoS at 50 C, experimental data (Watari et al., 2003).
this polymer in water vapor sorption is the same as SPEEK, and concentration of water vapor inside polymer increases with increasing water vapor activity. Generally, the water vapor concentration in polymer increases with increasing sulfonation to nonsulfonation segment ratio. This trend has been predicted perfectly by PHSC equation of state. Fig. 8 shows the isotherms of water vapor sorption by NAFION117 as a function of the water vapor activity at 27 and 30 C. The concentration of water vapor inside the polymer increases with increasing the temperature and with water vapor activity. As it is observed, PHSC model has predicted the water vapor sorption by NAFION-117 at different temperatures, satisfactorily. Table 2 reports the optimized binary interaction parameters and the percent of average absolute deviation (%AAD) in activity of water vapor calculated by the following equation:
%AAD ¼
Np 100 X cal exp avi avi Np i¼1
(31)
As it is evident from Table 2, the percentage of overall average absolute deviation equal 0.53 has been obtained. This result
Fig. 7. Prediction of water vapor sorption by NTDA-DMBDSA-BAPF (1-1) and NTDADMBDSA-BAPF (9-1) with PHSC EoS at 50 c, experimental data (Watari et al., 2003).
Fig. 8. Prediction of water vapor sorption by Nafion-117 with PHSC EoS at 27 and 30 C, experimental data (Kim et al., 2006; Elfring, 2005).
represents the high precision of PHSC model in predicting equilibrium data of water vaporepolymer systems. 4. Conclusions In this work, perturbed hard sphere chain (PHSC) equation of state has been used for thermodynamic modeling of water vapor sorption in PEBAX-1074, SPEEK, PEO-PBT, NTDA and Nafion-117. The parameters of PHSC model such as r, ε and s have been calculated by group contribution approach. One binary interaction parameter (Kij) has been optimized by fitting the experimental data for modeling of each system in this study. PHSC equation of state has proved to be an appropriate approach for modeling of phase equilibrium conditions in water vaporepolymer systems. This model is a suitable method for predicting the equilibrium in systems including complex molecules with high molecular weight where other models are less powerful.
Table 2 Optimized binary interaction parameters and average absolute deviations for water vaporepolymer systems.
H2OePEBAX 1074 H2OePEBAX 1074 H2OePEBAX 1074 H2OePEBAX 1074 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%59 H2OeSPEEK-%75 H2Oe3000PEO52PBT48 H2Oe2000PEO52PBT48 H2Oe1000PEO52PBT48 H2Oe1000PEO64PBT36 H2Oe600PEO41PBT59 H2Oe300PEO32PBT68 H2Oe1000PEO40PBT60 H2Oe1000PEO56PBT44 H2Oe1000PEO75PBT25 H2OeNTDA-ODADS-ODA(1-1) H2OeNTDA-ODADS-ODA(3-1) H2OeNTDA-DMBDSA-BAPF(1-1) H2OeNTDA-DMBDSA-BAPF(9-1) H2OeNAFION-117 H2OeNAFION-117
T(K)
Kij
%AAD
293 303 323 343 293 303 323 343 293 293 293 293 293 293 293 293 303 303 303 323 323 323 323 300 303
0.05 0.05 0.05 0.05 0.15 0.14 0.17 0.15 0.16 0.14 0.12 0.11 0.11 0.11 0.10 0.12 0.06 0.05 0.05 0.96 0.96 0.63 0.73 0.36 0.37
0.29 0.57 0.91 0.59 0.69 0.39 0.54 0.74 0.57 0.50 0.15 0.26 0.23 0.09 0.25 0.37 0.34 0.52 0.60 0.67 0.56 0.47 0.39 1.17 1.05
F. Sabzi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 757e763
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