1040
Ind. Eng. Chem. Res. 1994,33, 1040-1046
Prediction of Water Solubilities in Hydrocarbons and Polyethylene at Elevated Temperatures and Pressures Charles W. Haschets, Annette D. Shine,' and Robert M. Secor Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The solubility of water in n-alkanes from c6 t o for temperatures from 200 t o 300 OC and pressures from 9.8 t o 39.2 MPa was modeled using the lattice fluid (LF) theory of Sanchez and Rodgers. The theory quantitatively described the temperature, pressure, and hydrocarbon molecular weight dependence of the experimental data using a lattice fluid mixing parameter which was linearly dependent on temperature. The solubility of water in n-hexane or in n-octane along the three-phase line, which varied by over 3 orders of magnitude, could be semiquantitatively described for temperatures ranging from 38 to 265 "C, when values of the mixing parameter a t lower temperatures were estimated by extrapolation. The solubility of water in molten polyethylene at high temperatures and pressures was estimated based on the n-alkane results. T h e LF model predicts that the solubility of water in low-density polyethylene a t temperatures of 200-300 "C and pressures of 7.0-35.0 MPa is appreciable, of the order kg/kg, and warrants experimental verification. This information may be useful in modeling water stripping in polymer devolatilization extrusion processes.
Introduction High-temperature and -pressure polymer melt processing applications such as extrusion often use water as a stripping agent to remove volatile contaminants such as solvents and unreacted monomers in order to improve product quality, to recycle materials, and to comply with regulations on volatile contaminants. The addition of water enhances the removal of low molecular weight compounds. For example, the addition of 1.5 or 3% water to low-density polyethylene (LDPE) under typical extrusion conditions of temperatures from 200 to 300 "C and pressures of 7-35 MPa enhances the removal of contaminants by an order of magnitude during devolatilization (Welling, 1980). Several researchers have developed models for describing devolatilization in single-screw extruders (Roberts, 1970;Biesenberger and Kessidis, 1982) and in twin-screw extruders (Collins et al., 1985; Secor, 1986) which provide estimates for volatile removal rates. These models, however, do not account for the effect of stripping agents on devolatilization. An estimate for the solubility of water in the polymer-rich phase would be needed to develop such a model. This solubility is often presumed negligible. The solubility of water in solid polymers at ambient temperature and near-atmospheric pressures has been extensively studied, due to the importance of solubility in packaging and coating applications. For hydrocarbon polymers such as polyethylene and polypropylene, water solubility under such conditions is typically on the order of 10-6 kg of water/kg of polymer (Barrie, 1968). The solubility of water in molten polymers at high temperatures and pressures, however, has not been experimentally studied. These experiments are often unappealing because of the difficulty and expense associated with working at high pressures and temperatures. In lieu of experimental characterization, it is attractive to utilize a theoretical model to predict the experimental phase behavior of water-polymer solutions. Lattice fluid (LF) molecular models are often used in studying the phase behavior of high-pressure systems since they account for compressibility and can describe both vapor and liquid properties and lower consolute solution phase behavior.
* Author to whom correspondence should be addressed.
One main advantage of LF models over other analytical equations of state is that LF models encompass both low molecular weight compounds and high molecular weight polymers. The LF model of Sanchez and Lacombe has been successfullyused to describe the phase behavior of various polymer-supercritical fluid solutions for pressures to 65 MPa, temperatures to 187 "C,and molecular weights from 2100 to 420 000 using a single set of mixing parameters for each binary (Haschets and Shine, 1993). Meilchen et al. (1991) also used the LF theory to describe the phase behavior of acrylate polymers in chlorodifluoromethane or propane using temperature-dependent lattice fluid parameters. Sanchez and Rodgers have used a modified LF model to successfullydescribe the solubilities of several probe gases at near-ambient conditions in amorphous polymers including atactic polypropylene, poly(methy1 acrylate), and polystyrene (Sanchez and Rodgers, 1988; Rodgers and Sanchez, 1993)and in polydimethylsiloxane (Pope et al., 1991). Furthermore, the LF model has been modified to account for morphological features such as glass transition temperature and crystallization (Kalcepiros and Paulaitis, 19931,so it offers possibilities for completely describing complex polymer systems. These models have not yet been tested for water-hydrocarbon or waterpolymer systems. Kleintjens and Koningsveld have used a similar meanfield lattice gas model (MFLG) to describe the phase behavior of water-methane systems which exhibit gasgas demixing (1980a) and n-alkane-linear polyethylene systems (1980b). Beckman et al. (1990) have also used the MFLG model to describe the phase behavior of mixtures of supercritical COz and poly(methy1methacrylate) or polystyrene. The MFLG model requires four characteristic parameters for each pure component, one of which is temperature dependent, and two mixture parameters, one of which is often considered a function of temperature. The LF theory uses a simpler description requiring three temperature-independent parameters for each pure component and either one or two mixture parameters. Dee (1991)has used the cell model of Flory, Orwoll, and Vrij (FOV) to describe the phase behavior of watercyclohexane mixtures for temperatures to 370 "C and pressures to 150MPa assuming that water exists as clusters
OSSS-58S5~94~2633-1040$04.50/0 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 4,1994 1041 of five hydrogen-bonded water molecules. The phase behavior of the cyclohexane-water-polystyrene ternary systems at elevated temperatures and pressures was also estimated. Similar to the LF model, each pure component in the FOV model is characterized by three temperatureindependent parameters while the mixture is described by one adjustable parameter. The FOV model, however, was developedusing an often misinterpreted cell parameter to account for external degrees of freedom per molecular segment. Even though the solubility of water in liquid polymers at elevated temperatures and pressures has not been experimentally studied or modeled, several researchers have studied the solubility of water under these conditions in low molecular weight polymer analogs such as linear hydrocarbons. Tsonopoulos and Wilson (1985) experimentally examined the solubilities of water in c6 hydrocarbons, and Heidman et al. (1985) studied the solubilities of water in cg hydrocarbons up to the three-phase equilibrium end point of the binary. These groups also used a modified Redlich-Kwong equation of state with one or two adjustable parameters to model the solubilities of water in the hydrocarbons. Hooper et al. (1988) described the solubility of water in n-octane with a modified UNIFAC model using temperature-dependent interaction parameters, and Prausnitz and co-workers (Michel et al., 1989) used a perturbation of the Boublik and Mansoori et al. equation of state to describe the solubilities of water in octane. In eachof the above studies, the researchers only studied the solubilities at pressures along the three-phase curve. Skripka (1976)and Sultanov and Skripka (1973) have experimentally studied the solubility of water in n-alkanes from hexane to hexadecane from 200 to 300 "C for pressures starting near the threephase pressure to pressures of 78 MPa. A complete review of researchers who have studied the mutual solubilities of water and hydrocarbons is given in the Technical Data Book-Petroleum Refining of the American Petroleum Institute (1992) and in the IUPAC Solubility Data Series (Kertes, 1989a,b). In this research, a modified lattice fluid model of Sanchez and Rodgers (1990) is used to describe the high-pressure solubilities of water in low molecular weight polymer analogs from hexane to hexadecane for temperatures of 200-300 "C and pressures of 9.8-39.2 MPa (Kertes, 1989a,b). The solubility data of water in n-hexane (Tsonopoulosand Wilson, 1983)and in n-octane (Heidman et al., 1985) along the three-phase line is also described using the LF theory to verify the applicability of the theory to low-pressure and -temperature systems. Based on these results, the solubility of water in molten low-density polyethylene at high temperatures and pressures is estimated.
Theory In this section, we have adopted the development of Sanchez and Rodgers (1990) to derive an expression for the solubility of water in hydrocarbon liquids. The solubility of water in a hydrocarbon or polymer can be described by Henry's law provided that the solubility is linear in fugacity: C, = kH-l(T9)f l ( T 9 )
hydrocarbons have very low solubility in liquid water, e.g., the mole fraction of n-hexane and n-octane soluble in the water-rich phase is about 1O-e-lo" (Tsonopoulos and Wilson, 1983; Heidman et al., 1985), the fugacity of the water-rich phase is approximated here as that of pure water. This fugacity is given as the departure of the molar Gibbs free energy from the ideal gas value:
The ideal gas Gibbs free energy GIGwas calculated here using the method suggested by Sandler (1989), while G was determined from enthalpy and entropy data taken from the steam tables (Grigull et al., 1990). Sanchez and Rodgers derived an expression for the inverse Henry's law constant, kH-l for the solubility of a gas in liquid polymers (eq 17 of their paper) by equating the chemical potential of the gas a t temperature T and pressure P to the chemical potential of the gas absorbed in the liquid. Limiting conditions of dilute gas absorption and infinite polymer molecular weight were assumed. For dilute liquid or gas absorption in a host liquid of arbitrary molecular weight, kH-l thus differs slightly from that of Sanchez and Rodgers:
-
(
kH-l Y1* - RT exp -7,2~1*x- +
M1
Y2*
where subscripts 1and 2 refer to water and hydrocarbon, respectively. x is the bare water-hydrocarbon interaction parameter given as RTx = Pl*
+ P2*- 2r(P1*P2*)1'2
(4)
where t is a dimensionless adjustable parameter that accounts for deviations from the geometric mean. Equation 3 is identical in form to eq 17 of Sanchez and Rodgers except that their expression omits the v1*Iv2* term for high molecular weight because u2* approaches polymers. In deriving the inverse Henry's law constant, Sanchez and Rodgers neglected the h/(7,?'d term in their eq 10 for the chemical potential for water adsorbed in the hydrocarbon due to the infinite molecular weight assumption. In our derivation, the p1/(7,?'1) term for the hydrocarbon systems was at least an order of magnitude smaller than the other terms in the exponential expression and was therefore also neglected. Due to the assumption of dilute absorption, eqs 1and 3 are not expected to be valid above 0.1 kg/kg water solubility. The other variables in eqs 3 and 4 are related by the lattice fluid equation of state (Sanchez and Rodgers, 1990):
where p, p, and 7, are the reduced temperature, pressure, and density of the pure component defined by:
(1)
where C1is the concentration of water in the hydrocarbon, f1 is the fugacity of the water in the water-rich phase, and kH-l is the inverse Henry's law constant. In the limit of low pressures, the fugacity is replaced by the partial pressure of water above the hydrocarbon liquid. Because
where p* is given in terms of the close-packed volume, Y*, and the dimensionless size parameter, r: (7)
1042 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 Table 1. Pure Component Equation of State Parameters comDonent P (K) P (MPa) n-hexane' 476 298 487 309 n-heptane' 502 307 n-octane' 530 305 n-decane' 588 29 1 n-hexadecaneb 673 359 low-density polyethylenec 623 2690 wateP
P*
(kg/m3) 775 800 815 837 875 887 1105
r 8.37 9.57 10.34 11.75 15.39 1.012 per CH2 repeat unit 8.46
4 Sanchez and Lacombe, 1976. Interpolated from data reported by Sanchez and Lacombe (1976) for CtdHa and C1,HM. Sanchez and Lacombe, 1978.
h
1.25 1.20
1.15 1.10 1.05 1 .oo 0.95
180
200
220
240
260
280
300
320
Temperature ("C) Figure 2. Correlation for the lattice fluid parameter f as a function of temperature for hydrocarbons from C6 to CIS. fwas optimized for each case over a pressure range of 9.8-39.2 MPa. W)"C A
0.02
225'C
b
1 z
;
0.04
0.00
0
10
20
30
40
F'reasure ("a)
Figure 1. Solubility of water in n-decane at elevated temperatures and pressures. Symbols are data from Kertes (1989b);solid lines are LF isobar predictions with f given by eq 8.
Each pure component is characterized by three paramP, and p*, which are determined for a particular eters, P, fluid by fitting the equation of state to PVT or saturation data. Values for the pure components used in this work for the hydrocarbons and water (Sanchez and Lacombe, 1976) and low-density polyethylene (Sanchez and Lacombe, 1978) are shown in Table 1. Parameters for hexadecane were interpolated from those reported for C14H30 and C17H36. Other parameters for water (P= 614 K, PC = 2847 MPa, p* = 1151 kg/m3, r = 8.73) have been reported by Kilpatrick and Chang (1986). Solubilities calculated using the Kilpatrick and Chang values for water were 11-16% lower than those calculated using the water parameters of Table 1. The solubility (kg of water/kg of hydrocarbon) was calculated by dividing the concentration from eq 1 by the pure hydrocarbon density determined by the LF equation of state, eq 5. A sample calculation can be found in the Appendix. In the limit of dilute gas absorption, the assumption that the solution density is equal to the pure polymer density from eq 5 is presumed valid up to about 0.05-0.1 kg/kg gas solubility.
Comparison with Experiment Figure 1 shows the experimental solubility data of water in n-decane over the experimental temperature range of 225-290 OC. The solubilities decreased with increasing pressure and increased with increasing temperature. The solubilities were surprisingly high for such dissimilar components, ranging from about 0.015 kg/kg at the lowest temperature of 225 "C to as high as 0.175 kg/kg at 290 "C and a pressure of 9.8 MPa. Solubilities greater than 0.085 kg/kg were not fit to the LF model because of the limiting assumption of dilute concentration, so they are not included in Figure 1.
In order to quantitatively describe these data, as well as the water solubilities in hexane, octane, or hexadecane, it was necessary to allow for deviations from the geometric mean of the mixture x parameter, Le., to assume a value of { different from unity. However, using { = 1.0 resulted in a correct prediction of the order of magnitude of the solubilities ( kg/kg), but the values were systematically lower than the experimental data. Even for a single hydrocarbon, the water solubility data could not be quantitatively described at all temperatures and pressures using a single value for the LF parameter {. For example, the best-fit value of {found for the decane 250 OC isotherm gave an overprediction of the 225 "C isotherm by as much as 65%,while the 290 "C isotherm was underpredicted by as much as 40 5%. Similar results were also found for all the water-hydrocarbon systems. Thus it was necessary to assume that { is a function of temperature. In our previous work with polar/polar or nonpolar/nonpolar mixtures of polymer and supercritical fluid solvents, we demonstrated that the phase behavior of dilute polymer solutions could be described without using temperature-dependent parameters (Haschets and Shine, 1993). Meilchen et al. (1991), however, invoked temperature-dependent LF parameters to describe the phase behavior of acrylate polymers in supercritical chlorodifluoromethane or propane. The temperature dependence of { is shown in Figure 2 for the solubility of water in hydrocarbons from c6 to CIS over a temperature range of 200-300 "C. In each case the value of {was optimized to obtain the best least-squares fit of the experimental solubilities over a pressure range of 9.8-39.2 MPa for each temperature. It appears that the same value of the interaction parameter {can be used for each of the water-n-alkane systems. A straight line was used to correlate {over the temperature range for all four water-hydrocarbon systems: { = 0.433 799
+ 2.6645 X lo3 T ("C)
(8)
The solubility of water in decane using {given by eq 8 is shown in Figure 1. Agreement within 15%between the experimental and LF solubilities was found for the lower temperatures, but the LF model could not describe the high solubilities at the lower pressures for the 275 and 290 OC isotherms. The deviation of the model from the experimental data at these high solubilities is not unexpected since Henry's law is only valid in the limit of low
Ind. Eng. Chem. Res., Vol. 33,No. 4, 1994 1043
I
I
0.05
.01 7
,0017 0.03
/
,00017
0.02
0 Tsonopoulos and Wilson
- 6 given by eq 8
C16
Polyethylene 0.011 0.000
.
I
0.003
.
1
0.006
.
I
0.009
.
I
0.012
.
I
.00001
0.015
!
'
'
I
solubility. Similar results were observed using eq 8 to describe the other water-hydrocarbon systems studied. Figure 3 shows the experimental solubility data of Skripka and co-workers for water in n-octane, n-decane, and n-hexadecane at 250 "C for pressures of 9.8-39.2 MPa (Kertes, 1989b). Hexane data were unavailable at 250 "C, while the octane data were interpolated from solubility data at 240 and 265 "C. All the experimental data were above the three-phase pressure of each system, in the twophase liquid-liquid equilibrium region. At a given pressure, the solubility of water decreases with increasing hydrocarbon chain length. As was seen in Figure 1 for water in decane, the solubility of water in the other hydrocarbons decreases with increasing pressure. Figure 3 also shows the LF description of the experimental solubilities. Using the value of l =1.100 from eq 8for 250 "C, the LF model correctly described the pressure and molecular weight dependence of the solubilities within 9 % for all water-hydrocarbon systems studied except for the highest solubilities at 9.8 MPa, which were within 20%. One of the main advantages of using a molecular thermodynamic model to describe the experimental solubility data is that the model can be used as a tool for predicting solubilities under other circumstances, such as pressure, temperature, or carbon number, where experimental data are not available. Since reasonable agreement was achieved between experiment and LF predictions in Figure 3, the LF model could be used with some confidence to interpolate the solubilities of water in other n-alkanes such as Clz or CSor to estimate the solubility of water in lighter hydrocarbons such as n-heptane or n-hexane over the pressure and temperature ranges studied. As a test of this predictive capability, the LF model was used to estimate water solubility in hydrocarbons for temperatures along the three-phase liquid-liquid-vapor line. Figure 4 shows the experimental solubilities of water in n-hexane for temperatures from 40 to 200 "C and for three-phase pressures ranging from 0.045 to 3.52 MPa (Tsonopoulos and Wilson, 1983); these conditions are considerably milder than the 200-300 "C,9.8-39.2 MPa ranges treated earlier. Experimental solubilities varied over 3 orders of magnitude over the range of the threephase conditions. Figure 4 also shows the LF prediction of the experimental solubilities. Since l was found empirically to be a linear function of temperature from 200 to 300 "C,LF predictions in Figure 4 were also made
.
I
100
4
.
1
150
200
250
Temperature PC)
(MolecularWeight)" Figure 3. Solubility of water in n-alkanes from Ca to Cle and in polyethylene at 250 O C and elevated pressures. Symbols are data from Kertes (1989b); solid lines are LF isobar predictions with { given by eq 8.
.
I
50
0
Figure 4. Solubility of water in n-hexane along the three-phase line. 0 , experimental data of Tsonopoulos and Wilson (1983);solid curve is LF prediction with I; estimated from eq 8.
0
.I
0
7
.017
,001 7
0 Heidmanetal. 0 Criticalendpoint given by eq 8
.mol .00001! 0
'
'
8
50
.
1
100
'
1
150
.
b
'
200
1
250
'
3 0
Temperature ("C) Figure 5. Solubility of water in n-octane along the three-phase line. 0,experimental data; 0 , critical end point (Heidman et al., 1985); solid curve is LF prediction with { from eq 8.
using the linear relation of eq 8 at the low temperatures. The LF predictions are seen to describe the general trend of the data reasonably well, with large quantitative deviations only at the lowest temperature, where the extrapolation based on eq 8 is expected to be least reliable. Figure 5 shows the experimental and calculated solubilities of water in n-octane (Heidman et al., 1985) from 38 to 266 "C for three-phase pressures ranging from 0.01 to 7.0 MPa. The solid curve shows the LF prediction using values of l from eq 8. As was seen in Figure 4 for hexane, the model predicted the correct order of magnitude of the three-phase experimental water solubility in octane, except at the lowest temperature and near the critical end point, where the solubility is extremely large and increases rapidly with increasing temperature. A main objectiveof this study is to estimate the solubility of water in molten polymers such as low-density polyethylene (LDPE) at the high pressures and temperatures used in polymer extrusion processes. The solubility of water in solid polyethylene is quite small under near ambient conditions, e.g., 6.5 X 106 kg/kg at 25 "C and at the water vapor pressure of 0.0032 MPa (Barrie, 1968); however, experimental data are not readily available for water-polymer systems at high pressures and temperatures. Thus the solubility of water in molten LDPE has
1044 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
generally not been accounted for heretofore in models describing water-enhanced devolatilization in extrusion processes. As seen in Figures 4 and 5, the LF model predicted that water solubilitywas 104106kg/kg in hexane and in octane at near ambient temperatures of 38-40 "C and at the threephase pressures of 0.01-0.05 MPa. For water-hydrocarbon systems, the three-phase line lies slightly above the water vapor pressure curve (Kertes, 1989a). Thus, assuming the same temperature dependence of f as described in eq 8, the LF model predicts the solubility of water in amorphous polyethylene at 25 OC and 0.032 MPa to be 1.0 X 104. The LF prediction at this low temperature is significantly lower than the experimental value, as was also seen for the threephase lines, suggesting that the linear extrapolation of eq 8 predicts values of {which are too low. Nevertheless, the LF model correctly predicts that the solubility of polar water in nonpolar LDPE is quite low at ambient conditions. As a test of the capability of the LF model to describe high-temperature melts, experimental solubility data for ethylene in LDPE (Welling, 1980;Maloney and Prausnitz, 1976) were modeled. The experimental solubility of ethylene in LDPE at 230 "C and 0.1 MPa is about 5 X lo4 kg/kg. We have shown before that a different set of LF parameters were needed to describe the phase behavior of propane-polyethylene and n-pentane-polyethylene dilute solutions (Haschets and Shine, 1993); thus eq 8 is not appropriate to describe the ethylene-LDPE system. However, using a value of f = 1.0, an order of magnitude estimate of the solubility was determined to be lo4 kg/kg, in agreement with the experimental value. Since the LF model can describe solubility of water in LDPE under mild conditions, as well as the solubility of ethylene in LDPE at high temperatures, it is reasonable to assume that the solubility of water in the LDPE melt can also be described. The LF model was thus used to estimate the solubility of water in LDPE for number-average molecular weights ranging from 500 to 500 000. The same t function used for the n-alkane polymer analogs was used to describe the LDPE melts. This is not an overly restrictive assumption since typical values off needed to describe polymer-solvent binaries range from only 0.875 to 1.13 (Haschets and Shine, 1993;Meilchen et al., 1991;Sanchez and Lacombe, 1976). As seen in Figure 3, the solubility of water in LDPE at 250 "C for pressures varying from 9.8 to 39.2 MPa is predicted to be appreciable, about 0.02 kg/kg. The predicted solubility of water in other hydrocarbon polymers such as polypropylene and high-density polyethylene was also of the same order of magnitude, about 0.01-0.04 kg/kg. Figure 6 predicts the effect of temperature on the solubility of water in LDPE at typical extrusion conditions for temperatures of 200-300 "C and pressures of 7-35 MPa. The model predicts a substantial increase in the solubility as the temperature is increased a t a constant pressure and a decrease in solubility with increasing pressure at a constant temperature, as was found for the water-n-alkane systems. The solubility is relatively independent of the molecular weight for the molecular weight range of 500500 000. This information alone could be used to aid in the design of high-pressure processesor in the development of a model that accounts for stripping agents in devolatilization in screw extrusion. For example, in a typical extrusion process for LDPE, the polymer is first melted at temperatures between 200 and 300 OC at pressures from 7 to 35 MPa. The LF model predicts that the fraction of empty holes in the lattice as well as the isothermal compressibility
200T 0.00 . , . 0.0000 0.0005 ,
I
.
.
..,... . 0.0010
I
.
0.0015
.
..
I
.
0.0020
,
,
.
0.0025
( ~ o l e c ~ lweight) ar -* Figure 6. LF prediction of the solubility of water in polyethylene at elevated temperatures and pressures. Solid curves are 7.0 m a ; dashed curves are 35.0 MPa.
factor for the water-LDPE system increases over the temperature range 200-300 OC. This increase in vacant sites with increasing temperature increases the free volume of the system, which could account for the increased solubility with increasing temperature. The above modeling results suggest that this added water could be completely dissolved in the LDPE melt, provided the process is not mass transfer limited. Since the predicted solubilities of water in molten hydrocarbon polymers are appreciable, experimental verification of these solubilities is recommended. The above description of the solubility of water in liquid hydrocarbons and polymers at elevated temperatures and pressures assumes that the water molecules are randomly placed on the lattice and that there are no specific interactions such as hydrogen bonding. Water, however, is a hydrogen bonding liquid with possible aggregation or clustering of water molecules occurring in highly concentrated mixtures. The description of the solubility using a random statistical model does not give any insight on water aggregation in the liquid-rich phase of the mixture. Panayiotou and Sanchez (1991) have recently modified the LF theory to account for hydrogen bonding fluids. We have previously demonstrated (Haschets and Shine, 1993) for polymer-SCF systems which show cross-association hydrogen bonding that inclusion of the hydrogen bonding term in the chemical potentials improved the description of polymer solubility in the SCF for the case of no adjustable LF mixing parameters. However, adjustable mixing parameters were still needed in order to quantitatively describe the phase behavior. For the case considered here of water dissolved in a hydrocarbon fluid, only self-associationhydrogen-bonding of water molecules would be expected to occur. Although expressions for the chemical potential for pure water and for water absorbed in the hydrocarbon liquid can be derived using the theory of Panayiotou and Sanchez, the self-association hydrogen bonding contributions to the expression for kH-1 can be shown to approach zero in the limit of low gas density and in the limit of dilute liquid or gas absorption, which were both assumed in the derivation of eq 3. A more complex analysis which does not invoke the above assumptions would thus be required to account for self-association hydrogen bonding of water. Conclusions The Sanchez-Rodgers lattice fluid model was used to describe the solubilities of water in n-alkanes from Cg to
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 1045 Cl6 for temperatures from 200 to 300 O C and pressures from 9.8 to 39.2 MPa. Using a temperature-dependent lattice fluid parameter, the LF model quantitatively described the temperature, pressure, and molecular weight dependence of the experimental solubility data up to the maximum solubility of about 0.085 kg/kg. The LF model could also describe the solubilities of water in hexane and in octane for conditions along the three-phase line. A semiquantitative description of the experimental data was found for temperatures as low as 38 O C up to the critical end point of the system. The success of the model in describing water solubility in hydrocarbons over this wide range of temperatures and pressures suggests that the model might be used as a tool for estimating the solubility of water in polymers under extrusion conditions. LF predictions for low-density polyethylene using the same interaction parameter found for low molecular weight n-alkane analogs indicate that equilibrium water solubility under extrusion conditions is appreciable, of the order of kg/kg. This information can be used to develop models for devolatilization of polymers using stripping agents which require an estimate of the solubility of water in the molten polymers.
Acknowledgment This work was funded by the National Science Foundation under Grant No. DDM-9102435. Also, the s u g gestions of Professors Stanley I. Sandler and Michael E. Paulaitis of the University of Delaware are gratefully appreciated. Nomenclature
C1= concentration of water in hydrocarbon, kg/m3 = fugacity of water above hydrocarbon liquid, Pa G = molar Gibbs free energy, J/kmol = molar enthalpy, J/kmol S = molar entropy, J/(kmol K) C, = ideal gas heat capacity, J/(kmol K) kH-l = inverse Henry's law constant M = molecular weight P = pressure, Pa r = number of sites a molecule occupies in a lattice R = gas constant, J/(kmol K) T = temperature, K x = mole fraction
f1
E
Greek Letters
f = lattice fluid parameter = close-packed volume, m3/kmol
V*
p
x
= density, kg/m3 = bare gas-hydrocarbon interaction parameter
Superscripts
IG = ideal gas
- = reduced variable, dimensionless
* = pure component molecular property
Subscripts
1 = water 2 = hydrocarbon - = molar property r = reference condition
Appendix Calculation of the Solubility of Water in n-Octane at T = 250 OC and P = 19.6 MPa. The fugacity of water is calculated from eq 2 with G = H - TS. From the steam tables (Grigull, 1990) G = 34.78 MJ/kmol. The ideal gas
values are calculated in terms of a heat capacity expression cubic in temperature (Sandler, 1989):
SIG= JT, cp(nd T - R ln(
E)
(All
giving G I G = 42.01 MJ/kmol and f = 3.71 MPa. The reference conditions are the triple point of water in all calculations. From eqs 5 and 6,62 = 0.66 and p2 = 534.23 kg/m3, and from eq 3 k ~ - l 5.42 x lo4 kgwabr/(mwtane3Pa).The concentration as calculated from eq 1 is then C1 = 20.11 kgWabr/moctane3, giving a solubility of (CdPd = 0.038 kgwater/kg-e*
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* Abstract published in Aduance ACS Abstracts, March 15, 1994.
