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Vapor and liquid equilibria in porous media
Fluid Phase Equilibria 166 Ž1999. 79–90 www.elsevier.nlrlocaterfluid
Vapor and liquid equilibria in porous media Youn-Ok Shin, Jana Simandl
)
Department of Chemical Engineering, McGill UniÕersity, Montreal, Quebec, Canada H3A 2B2 Received 10 May 1999; accepted 1 September 1999
Abstract The alteration of the vapor–liquid equilibrium ŽVLE. of volatile organic mixtures by placing porous media at the liquid–vapor interface was studied. Kelvin, assuming ideal behavior of fluids, first introduced the vapor pressure of liquid over a meniscus as a function of its surface tension and the radius of the curvature. A thermodynamic model ŽSS mod model. predicting the VLE of non-ideal organic mixtures in porous media was developed as a function of pore sizes. The model was used to predict the VLE of two aqueous alcohol solutions, ethanol–water and propanol–water, and two binary solutions, methanol–isopropanol and ethanol–n-octane. Experiments were conducted using sintered metal and fritted glass plates as porous media, and the results were compared with the model predictions. Using the actual diameter of the porous media, the model prediction showed good agreement with the experimental results. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Vapor–liquid equilibria; Vapor pressure; Curved liquid–vapor interface; SS mod model; PRSV equation of state
1. Introduction The alteration of the vapor–liquid equilibria Ž VLE. of volatile organic compounds is of interest since it allows the separation of organic mixtures that are difficult to distil. When a porous medium is placed at the liquid–vapor interface, the liquid surface forms a meniscus due to its tendency to minimize surface energy w1x. The capillary pressure exists at the interface and results in a pressure difference between the liquid and the vapor. Subsequently, the VLE in porous media differs from that established over a flat liquid–vapor interface. The vapor pressures in porous media have been studied extensively since Kelvin who first proposed that the vapor pressure over a meniscus is a function of the liquid surface tension and the radius of the curvature w2x. The Kelvin equation was developed assuming that the vapor and liquid phases behave ideally and that the curvature at the liquid interface is a fraction of a sphere. However, )
Corresponding author. Tel.: q1-514-398-4494; fax: q1-514-398-6678; e-mail: [email protected]
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 2 8 9 - 7
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
80
the experimental results available in literature show that the vapor pressures measured experimentally are orders of magnitude smaller than the values predicted by the Kelvin equation w3x. Yeh et al. w4x modified the Kelvin equation to include the liquid surface tension in porous media by estimating the dispersion and polar interactions at the solid–liquid interface. Shapiro and Stenby w5x introduced a new form of Kelvin equation that includes the non-ideality of the fluids, which cannot be ignored for oil–gas-reservoirs at high pressures. The study of the VLE in porous media was further extended in this work. A mathematical model, the SS mod model, which predicts the VLE in porous media as a function of pore sizes, was developed based on the pressure equations suggested by Shapiro and Stenby w5x. The model was used to predict the VLE of two aqueous alcohol mixtures, ethanol–water and propanol–water, and the VLE of two binary systems, methanol–isopropanol and ethanol–n-octane. The model predictions were compared with the experimental results, which were obtained by using a Genesis headspace autosampler and a Varian gas chromatograph ŽGC. combined with the glass vials. The glass vials were first designed by Wong w6x and contained sintered metal plates with a pore size of 120 mm or fritted glass plates with a pore size of 50 mm as porous media. The experimental results of the VLE of ethanol–n-octane solution measured without porous media showed excellent agreement with literature values w7x, indicating that the experimental techniques used in this study provided precise and reproducible data. The experimental measurements of VLE in porous media were compared with the SS mod model predictions and showed good agreement.
2. Background theories and the SS mod model The thermodynamics of VLE in porous media were first introduced by Kelvin assuming ideal behavior of fluids w2x. Since then, many attempts have been made either to modify or to develop a new vapor pressure equation better suited for non-ideal mixtures. Yeh et al. w4x modified the Kelvin equation to include the dispersion and the polar interfacial forces at the solid–liquid interface. However, the determination of the interfacial forces in such small pores remains a challenge. Furthermore, the non-ideality of solutions was still omitted in both Kelvin and Yeh et al. equations. Shapiro and Stenby w5x developed two new equations estimating the pressure exerted at the curved liquid–vapor interface: one for a non-ideal single component and the other for non–ideal multicomponent mixtures. These equations were developed for hydrocarbon mixtures in oil–gas-condensate reservoirs. The condition of equilibrium for the two phases can be written in terms of the chemical potential, m , at a given pressure, P:
m v Ž Pv . s m LŽ PL .
Ž1.
m v Ž Pd . s m LŽ Pd .
Ž2.
and
where subscript v denotes vapor phase, and L denotes liquid phase. Pd is defined as the dew pressure without porous media. After taking the difference of these two equations, the chemical potential of the vapor and the liquid can be written as:
m v Ž Pv . y m v Ž Pd . s m LŽ PL . y m LŽ Pd .
Ž3.
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
81
The chemical potential of a compound can be written in terms of HVdP at constant temperature. Thus, Eq. Ž3. can be transformed to: Pv
HP
Vv d P s
d
PL
VL d P
HP
Ž4.
d
By replacing Pcurv , the pressure over curved liquid–vapor interface, defined as: Pcurv s P L y Pv
Ž5.
and by assuming incompressibility of liquid, Eq. Ž4. can be written as: Pv
HP
Vv d P s VLŽ P L y Pd . s V L Pcurv q V LŽ Pv y Pd .
Ž6.
d
From Eq. Ž6., the SS mod model Ž modified version of Shapiro and Stenby equation. was developed. Since the molar volume of non-ideal solution can be easily calculated by the Peng–Robinson–Stryjek–Vera ŽPRSV. cubic equation of state w8x combined with the Sandoval–Wilczek–Vera–Vera ŽSWVV. mixing rule w9x, the chemical potential in the vapor phase can easily be estimated by: PV
HP
Vv d P s
d
PV
HP
d Ž PVv . y
d
VV
HV
PdVv s Ž PVv . p V y Ž PVv . pd y
d
VV
HV
PdVv
Ž7.
d
Subsequently, Pcurv for both single and multiple component solutions can be expressed as: Pcurv s
1
Vv
VL
Ž PVv . P y Ž PVv . P y H PdVv y VL Ž Pv y Pd . v
d
Ž8.
Vd
Then, the pore radius, r, creating a given capillary pressure can be calculated from: rsy
2 s cos u Pcurv
Ž9.
where s is the liquid surface tension, and u is the contact angle at the liquid–solid interface. Eq. Ž 8. combined with Eq. Ž9. is called the SS mod model and was used to predict the VLE in porous media in this work. The computation of the SS mod model was carried out by first evaluating Pcurv using Eq. Ž8., where HPdVv and molar volume of vapor were calculated by using the PRSV equation of state combined with the SWVV mixing rule. The dew point pressure was determined by Antoine’s equation. There are two possible approaches to estimate the molar liquid volumes: a revised parameter in the PRSV equation w10x or published liquid densities. The model curves presented in this paper are based on liquid molar volumes derived from density data in Perry’s Handbook w11x. Then, having Pcurv , the pore radius r was calculated from Eq. Ž9.. The liquid surface tension was evaluated by using empirical equations of Winterfield et al. and Tamura et al., as cited in Perry’s Handbook w11x. Finally, the capillary pressure exerted in the porous media was calculated from: Pact Pcurv
r s ract
Ž 10.
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Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
The contact angle is independent of the pore diameter. Since the overall calculation is a comparison between capillary and bulk situations, the contact angle is eliminated within the algorithm.
3. Experimental procedures The experiments determining the VLE were conducted using glass vials containing porous media, a Genesis headspace autosampler and a Varian 3400 GC. A sample of the vapor phase formed in the glass vial was withdrawn by the autosampler and sent to the GC for analysis. 3.1. Vials and porous media The glass vials were made by fusing the tops of two Pyrex glass vials to create openings at the top and the bottom. The vials were originally designed by Wong w6x. To fit into the autosampler, the vials were 70 mm high with a 21.5 mm inner diameter Ž Fig. 1. . Butyl rubber stoppers were used as septa and were secured by aluminum caps. Sintered metal and fritted glass plates were used as porous media. As shown in Table 1, their actual pore sizes measured by SEM were found to be one order of magnitude larger than stated nominal pore diameter provided by the manufacturer. Although the values given by manufactures may be adequate to predict what size particles will be trapped in tortuous pores, the nominal values are inappropriate for predicting the surface of liquid filling the pores. The upper limit of the pore size range was used in the SS mod model: 120 mm for sintered metal and 50 mm for fritted glass plates. This conforms the finding that the presence of larger pores negates the effect of smaller pores w6x. Another important aspect in the VLE experiments was to ensure that the liquid–vapor surface was formed within the plate. The liquid was pipetted directly onto the plate until a thin film of liquid was observed on its surface. The maximum liquid volume that could be held in each plate was found to be 0.2 ml. However, a liquid volume of 0.4 ml was required in each glass vial to keep the liquid concentration steady upon heating. Thus, each vial was prepared to contain two porous plates. Modified vials without porous media but containing the same volume of liquid were used as controls.
Fig. 1. Schematic Žnot drawn to scale. of glass vials: Ža. pyrex glass vial, unmodified, Žb. modified, Žc. with porous medium.
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
83
Table 1 Summary of porous plate specifications Plate type
Nominal pore diameter
Supplier
Actual pore diameter measured by Scanning Electron Microscope ŽSEM.
Pore diameter used in SS mod model
Sintered metal Fritted glass
40 mm 4–8 mm
Pall Canada Ace Glass
84"40 mm 35"16 mm
120 mm 50 mm
3.2. Headspace autosampler Genesis headspace autosampler consisting of a carrousel, a heated platen, a control panel and a septum needle adapter connected to a sample loop, was used for the sampling. The biggest advantage of using the autosampler is the precision of the sampling and injecting steps. A maximum of twelve vials can be heated simultaneously in the platen at constant temperature. Once the VLE was reached, each vial was raised onto the needle, and the vapor sample was withdrawn. Static pressure established in the vial upon heating forced the samples into the sample loop. The static vial pressure of 2 to 3 psig was recommended for reproducibility of data and safety w12x. The sample in the loop was then sent to the GC for analysis by Helium carrier gas. A summary of the autosampler parameters is shown in Table 2. The equilibrium time was determined experimentally by using the vials containing 60 mol% of methanol–isopropanol heated in the platen for 2–12 h, and plotting the vapor mole fraction of methanol as a function of heating time Ž Fig. 2. . The mole fraction of methanol had a standard deviation greater than 2% when heated for less than 750 min. Similar experiments were performed with ethanol–n-octane and ethanol–water mixtures, and the minimum equilibrium times of 900 and 350 min were obtained, respectively. Erring on the safe side, the maximum programmable time of 990 min of equilibrium time was used in all experiments, since no sample degradation occurs at these temperatures. The standard deviation at 990 min was 1%. In addition, all vials were kept at the required temperature in an oven for 24 h prior to being placed in the platen to ensure ample diffusion of vapor samples through the pores.
Table 2 Summary of autosampler parameters Model Carrier gas Carrier gas flow rate Platen temperature
Equilibrium time Sample loop volume Line and valve temperature Loop fill time
Genesis Headspace Autosampler with 50-position carrousel Ultra High Purity Helium ŽMatheson. 15 cm3rmin 558C Žfor methanol–isopropanol. 758C Žfor ethanol– n-octane. 608C Žfor ethanol–water. 990 min 5 ml 1758C 0.03 min
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
84
Fig. 2. Minimum equilibrium time for 60 mol% methanol–isopropanol solution with 95% confidence intervals.
3.3. GC The vapor sample sent from the autosampler was analyzed by a Varian 3400 GC and the Analog-to-Digital Converter Ž ADC. board controlled by a computer using the software Star Workstation. A summary of GC parameters used in the experiments is shown in Table 3. The peak areas obtained from the GC were converted into the number of moles of the sample via calibration by direct injection. These calibration factors were confirmed on a regular basis both with manual injections and with internal standards.
Table 3 Summary of GC parameters Model Column Column temperature Ramping temperature Detector Detector temperature Injector temperature Make-up gas Make-up gas flow rate Air flow rate Hydrogen gas flow rate
Varian 3400 GC DB-624 glass capillary column 658C Žfor ethanol–water. 958C Žfor binary alcohol mixtures. 1508C Flame Ionization Detector ŽFID. 2508C 1808C Ultra High Purity Helium 15 cm3rmin 300 cm3rmin 30 cm3rmin
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
85
Fig. 3. VLE of ethanol– n-octane without porous media Ž758C. with 95% confidence intervals.
3.4. Validation and error minimization The experimental apparatus and the techniques were verified by comparing the results with literature values. The VLE of ethanol–n-octane measured without porous media were plotted in Fig. 3. The 95% confidence intervals, produced from triplicate samples, were too small to be visible in this figure. These results indicate that the experimental procedures used in this study give precise and reproducible data. It was realized from the start that the differences between the vapor pressure with and without capillaries would range from 0 to 8%. In order to ensure precise results, efforts were made to minimize the impact of extraneous variables. All vials with porous media were handled simultaneously with unmodified vials containing the sample solution. The two were subjected to the same temperature and test conditions: any premature evaporation, gas dissolution, temperature fluctuations would thus be reflected in both. To avoid hysteresis, the order in which the compositions were tested was random. Three samples and controls were analyzed for each data point reported. The 95% confidence intervals are indicated in all figures. Due to their narrowness, they are not always visible.
4. Results and discussion The SS mod model was used to predict the vapor and liquid equilibria ŽVLE. in porous media for two aqueous alcohol mixtures, ethanol–water and propanol–water and two binary mixtures, methanol–isopropanol and ethanol–n-octane. These model predictions were compared with the experimental results conducted with sintered metal and fritted glass plates as porous media.
86
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
4.1. VLE in porous media as function of pore sizes As per the theory presented in Section 2, the VLE is affected by the pore sizes of porous media. Fig. 4 shows the model-predicted VLE of both aqueous and non-aqueous systems as a function of pore sizes. Note that each curve is specific for a given mixture at one concentration and temperature. For both aqueous and non-aqueous mixtures, the effect of curved liquid surface on the VLE decreases dramatically as the pore size increases. For pores greater than 90 mm, the difference in the vapor concentrations with and without porous media was predicted to be negligible. The curvature effect on the VLE increases rapidly as the pore size decreases. For ethanol–water and propanol–water mixtures, a pore size of 50 mm engenders a 4%–7% increase in the concentration of alcohol in the vapor mole phase. At this pore size, the percent increase for the methanol–isopropanol and ethanol–n-octane is less than 1%. At 5 mm, for aqueous mixtures, the alcohol concentrations are predicted to increase by 50%–60%. However, the liquid surface modification effect in the non-aqueous systems at 5 mm is only 5%–10%. 4.2. Comparison of model predictions with experimental results As shown in Fig. 4, no change in the vapor concentration was predicted when the porous were larger than 90 mm. Hence, when the 120 mm sintered metal plates were used, no change in the VLE was predicted. The experimental results confirmed this. For all four systems, the predicted and experimental values for 120 mm pores merely coincided with standard published VLE data. Using a pore size of 50 mm for the fritted glass plates, the model predicted an increase of 4%–6% in the ethanol mole fraction in the vapor phase Ž Fig. 5. . These predictions are in good agreement with the experimental results. The comparison for propanol–water with fritted glass plates is shown in
Fig. 4. Percent increase in vapor mole fraction as function of pore sizes predicted by the SS mod model.
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
87
Fig. 5. VLE of ethanol–water at 608C in fritted glass plate ŽVLE plane from Ref. w13x..
Fig. 6. The experimental results showed good agreement with the predicted increase of 7% in the propanol mole fraction in the vapor phase. From Figs. 5 and 6, one may conclude that the SS mod model predictions agree well with the experimental results. In comparison, both the Kelvin and Yeh equations predicted no change in the VLE of given solutions at this pore size due to their limitations in describing the behavior of real fluids. In order for the Kelvin and Yeh et al. equations to predict the same results as the SS mod model, the pore sizes would need to be orders of magnitude smaller than the
Fig. 6. VLE of propanol–water at 608C in fritted glass plate ŽVLE plane from Ref. w14x..
88
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
Fig. 7. VLE of methanol–isopropanol at 558C in fritted glass plate ŽVLE plane from Ref. w15x..
actual pore sizes. The model predictions for methanol–isopropanol and ethanol–n-octane are compared with experimental results in Figs. 7 and 8. Using the estimated pore size of 50 mm for fritted glass plates, the model predicted less than 1% change in the vapor phase compositions in both systems. The experimental results also confirmed that the pores were too big to have any effect on the equilibrium.
Fig. 8. VLE of ethanol– n-octane at 758C in fritted glass plate.
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
89
5. Conclusions The development of the SS mod model provided a tool for estimating the VLE in porous media as a function of pore sizes. The model was validated by a series of experiments conducted by using sintered metal and fritted glass plates as porous media. The actual pore diameter in chosen porous media was estimated, and the model prediction showed excellent agreement with the experimental results. The liquid surface modification effect was not visible in the non-aqueous solutions due to the pores that were too big to provide sufficient curvature at the liquid–vapor interface. The main limitation on the experimental procedure was the unavailability of porous media with very fine pores. The nominal sizes claimed by manufacturers do not correspond to the actual pore diameter. Porous media with finer and more uniform pores should be found to provide more pronounced changes in the vapor phase concentrations. List of symbols P pressure r radius V molar volume Greek letters u contact angle m chemical potential s surface tension Subscripts act actual curv curved liquid surface d dew point L liquid v vapor
Acknowledgements The GC expertise of Danielle Beland from Varian is truly appreciated. The support from Dr. J.H. ´ Vera for the model is greatfully acknowledged. Our research group and Dr. M.E. Weber provided helpful suggestions. The financial support of the Natural Sciences and Engineering Research Council ŽNSERC. and the Brace Research Institute are also appreciated.
References w1x P.W. Atkins, Physical Chemistry, Freeman, San Francisco, 1982. w2x R. Defay, I. Prigogine, A. Bellemans, D.H. Everett, Surface Tension and Adsorption, Wiley, New York, 1966. w3x G.C. Yeh, B.V. Yeh, B.J. Ratigan, S.J. Correnti, M.S. Yeh, D.W. Pitakowski, W. Fleming, D.B. Ritz, J.A. Lariviere, Desalination 81 Ž1991. 129–160.
90
w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
G.C. Yeh, B.V. Yeh, S.T. Schmidt, M.S. Yeh, A.M. McCarthy, W.J. Celenza, Desalination 81 Ž1991. 161–187. A.A. Shapiro, E.H. Stenby, Fluid Phase Equilibria 134 Ž1997. 87–101. N.S.J. Wong, Master’s Thesis, McGill University, Montreal, Canada, 1997. L. Boublikova, B.C.Y. Lu, J. Appl. Chem. 19 Ž1969. 89 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981. R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 Ž1986. 323–333. G. Sandoval, G. Wilczek-Vera, J.H. Vera, Fluid Phase Equilibria 52 Ž1989. 119–126. P. Proust, E. Meyer, J.H. Vera, Can. J. Chem. Eng. 71 Ž1993. 292–298. R.H. Perry ŽEd.., Perry’s Chemical Engineers’ Handbook, 7th edn., McGraw-Hill Book, 1997. Varian, Genesis Headspace Autosampler, Operator’s Manual, Varian Associates, 1991. V.V. Udovenko, L.G. Fatkulina, Zh. Fiz. Khim. 26 Ž1952. 1438 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981. D.C. Freshwatter, K.A. Pike, J. Chem. Eng. Data, 12 Ž1967. 179 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981. E. Shrfiber, E. Schuettau, D. Rant, H. Schuberth, Z. Phys. Che1 ŽLeipzig., 247 Ž1976. 23 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981.
Vapor and liquid equilibria in porous media Youn-Ok Shin, Jana Simandl
)
Department of Chemical Engineering, McGill UniÕersity, Montreal, Quebec, Canada H3A 2B2 Received 10 May 1999; accepted 1 September 1999
Abstract The alteration of the vapor–liquid equilibrium ŽVLE. of volatile organic mixtures by placing porous media at the liquid–vapor interface was studied. Kelvin, assuming ideal behavior of fluids, first introduced the vapor pressure of liquid over a meniscus as a function of its surface tension and the radius of the curvature. A thermodynamic model ŽSS mod model. predicting the VLE of non-ideal organic mixtures in porous media was developed as a function of pore sizes. The model was used to predict the VLE of two aqueous alcohol solutions, ethanol–water and propanol–water, and two binary solutions, methanol–isopropanol and ethanol–n-octane. Experiments were conducted using sintered metal and fritted glass plates as porous media, and the results were compared with the model predictions. Using the actual diameter of the porous media, the model prediction showed good agreement with the experimental results. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Vapor–liquid equilibria; Vapor pressure; Curved liquid–vapor interface; SS mod model; PRSV equation of state
1. Introduction The alteration of the vapor–liquid equilibria Ž VLE. of volatile organic compounds is of interest since it allows the separation of organic mixtures that are difficult to distil. When a porous medium is placed at the liquid–vapor interface, the liquid surface forms a meniscus due to its tendency to minimize surface energy w1x. The capillary pressure exists at the interface and results in a pressure difference between the liquid and the vapor. Subsequently, the VLE in porous media differs from that established over a flat liquid–vapor interface. The vapor pressures in porous media have been studied extensively since Kelvin who first proposed that the vapor pressure over a meniscus is a function of the liquid surface tension and the radius of the curvature w2x. The Kelvin equation was developed assuming that the vapor and liquid phases behave ideally and that the curvature at the liquid interface is a fraction of a sphere. However, )
Corresponding author. Tel.: q1-514-398-4494; fax: q1-514-398-6678; e-mail: [email protected]
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 2 8 9 - 7
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
80
the experimental results available in literature show that the vapor pressures measured experimentally are orders of magnitude smaller than the values predicted by the Kelvin equation w3x. Yeh et al. w4x modified the Kelvin equation to include the liquid surface tension in porous media by estimating the dispersion and polar interactions at the solid–liquid interface. Shapiro and Stenby w5x introduced a new form of Kelvin equation that includes the non-ideality of the fluids, which cannot be ignored for oil–gas-reservoirs at high pressures. The study of the VLE in porous media was further extended in this work. A mathematical model, the SS mod model, which predicts the VLE in porous media as a function of pore sizes, was developed based on the pressure equations suggested by Shapiro and Stenby w5x. The model was used to predict the VLE of two aqueous alcohol mixtures, ethanol–water and propanol–water, and the VLE of two binary systems, methanol–isopropanol and ethanol–n-octane. The model predictions were compared with the experimental results, which were obtained by using a Genesis headspace autosampler and a Varian gas chromatograph ŽGC. combined with the glass vials. The glass vials were first designed by Wong w6x and contained sintered metal plates with a pore size of 120 mm or fritted glass plates with a pore size of 50 mm as porous media. The experimental results of the VLE of ethanol–n-octane solution measured without porous media showed excellent agreement with literature values w7x, indicating that the experimental techniques used in this study provided precise and reproducible data. The experimental measurements of VLE in porous media were compared with the SS mod model predictions and showed good agreement.
2. Background theories and the SS mod model The thermodynamics of VLE in porous media were first introduced by Kelvin assuming ideal behavior of fluids w2x. Since then, many attempts have been made either to modify or to develop a new vapor pressure equation better suited for non-ideal mixtures. Yeh et al. w4x modified the Kelvin equation to include the dispersion and the polar interfacial forces at the solid–liquid interface. However, the determination of the interfacial forces in such small pores remains a challenge. Furthermore, the non-ideality of solutions was still omitted in both Kelvin and Yeh et al. equations. Shapiro and Stenby w5x developed two new equations estimating the pressure exerted at the curved liquid–vapor interface: one for a non-ideal single component and the other for non–ideal multicomponent mixtures. These equations were developed for hydrocarbon mixtures in oil–gas-condensate reservoirs. The condition of equilibrium for the two phases can be written in terms of the chemical potential, m , at a given pressure, P:
m v Ž Pv . s m LŽ PL .
Ž1.
m v Ž Pd . s m LŽ Pd .
Ž2.
and
where subscript v denotes vapor phase, and L denotes liquid phase. Pd is defined as the dew pressure without porous media. After taking the difference of these two equations, the chemical potential of the vapor and the liquid can be written as:
m v Ž Pv . y m v Ž Pd . s m LŽ PL . y m LŽ Pd .
Ž3.
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
81
The chemical potential of a compound can be written in terms of HVdP at constant temperature. Thus, Eq. Ž3. can be transformed to: Pv
HP
Vv d P s
d
PL
VL d P
HP
Ž4.
d
By replacing Pcurv , the pressure over curved liquid–vapor interface, defined as: Pcurv s P L y Pv
Ž5.
and by assuming incompressibility of liquid, Eq. Ž4. can be written as: Pv
HP
Vv d P s VLŽ P L y Pd . s V L Pcurv q V LŽ Pv y Pd .
Ž6.
d
From Eq. Ž6., the SS mod model Ž modified version of Shapiro and Stenby equation. was developed. Since the molar volume of non-ideal solution can be easily calculated by the Peng–Robinson–Stryjek–Vera ŽPRSV. cubic equation of state w8x combined with the Sandoval–Wilczek–Vera–Vera ŽSWVV. mixing rule w9x, the chemical potential in the vapor phase can easily be estimated by: PV
HP
Vv d P s
d
PV
HP
d Ž PVv . y
d
VV
HV
PdVv s Ž PVv . p V y Ž PVv . pd y
d
VV
HV
PdVv
Ž7.
d
Subsequently, Pcurv for both single and multiple component solutions can be expressed as: Pcurv s
1
Vv
VL
Ž PVv . P y Ž PVv . P y H PdVv y VL Ž Pv y Pd . v
d
Ž8.
Vd
Then, the pore radius, r, creating a given capillary pressure can be calculated from: rsy
2 s cos u Pcurv
Ž9.
where s is the liquid surface tension, and u is the contact angle at the liquid–solid interface. Eq. Ž 8. combined with Eq. Ž9. is called the SS mod model and was used to predict the VLE in porous media in this work. The computation of the SS mod model was carried out by first evaluating Pcurv using Eq. Ž8., where HPdVv and molar volume of vapor were calculated by using the PRSV equation of state combined with the SWVV mixing rule. The dew point pressure was determined by Antoine’s equation. There are two possible approaches to estimate the molar liquid volumes: a revised parameter in the PRSV equation w10x or published liquid densities. The model curves presented in this paper are based on liquid molar volumes derived from density data in Perry’s Handbook w11x. Then, having Pcurv , the pore radius r was calculated from Eq. Ž9.. The liquid surface tension was evaluated by using empirical equations of Winterfield et al. and Tamura et al., as cited in Perry’s Handbook w11x. Finally, the capillary pressure exerted in the porous media was calculated from: Pact Pcurv
r s ract
Ž 10.
82
Y.-O. Shin, J. Simandlr Fluid Phase Equilibria 166 (1999) 79–90
The contact angle is independent of the pore diameter. Since the overall calculation is a comparison between capillary and bulk situations, the contact angle is eliminated within the algorithm.
3. Experimental procedures The experiments determining the VLE were conducted using glass vials containing porous media, a Genesis headspace autosampler and a Varian 3400 GC. A sample of the vapor phase formed in the glass vial was withdrawn by the autosampler and sent to the GC for analysis. 3.1. Vials and porous media The glass vials were made by fusing the tops of two Pyrex glass vials to create openings at the top and the bottom. The vials were originally designed by Wong w6x. To fit into the autosampler, the vials were 70 mm high with a 21.5 mm inner diameter Ž Fig. 1. . Butyl rubber stoppers were used as septa and were secured by aluminum caps. Sintered metal and fritted glass plates were used as porous media. As shown in Table 1, their actual pore sizes measured by SEM were found to be one order of magnitude larger than stated nominal pore diameter provided by the manufacturer. Although the values given by manufactures may be adequate to predict what size particles will be trapped in tortuous pores, the nominal values are inappropriate for predicting the surface of liquid filling the pores. The upper limit of the pore size range was used in the SS mod model: 120 mm for sintered metal and 50 mm for fritted glass plates. This conforms the finding that the presence of larger pores negates the effect of smaller pores w6x. Another important aspect in the VLE experiments was to ensure that the liquid–vapor surface was formed within the plate. The liquid was pipetted directly onto the plate until a thin film of liquid was observed on its surface. The maximum liquid volume that could be held in each plate was found to be 0.2 ml. However, a liquid volume of 0.4 ml was required in each glass vial to keep the liquid concentration steady upon heating. Thus, each vial was prepared to contain two porous plates. Modified vials without porous media but containing the same volume of liquid were used as controls.
Fig. 1. Schematic Žnot drawn to scale. of glass vials: Ža. pyrex glass vial, unmodified, Žb. modified, Žc. with porous medium.
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Table 1 Summary of porous plate specifications Plate type
Nominal pore diameter
Supplier
Actual pore diameter measured by Scanning Electron Microscope ŽSEM.
Pore diameter used in SS mod model
Sintered metal Fritted glass
40 mm 4–8 mm
Pall Canada Ace Glass
84"40 mm 35"16 mm
120 mm 50 mm
3.2. Headspace autosampler Genesis headspace autosampler consisting of a carrousel, a heated platen, a control panel and a septum needle adapter connected to a sample loop, was used for the sampling. The biggest advantage of using the autosampler is the precision of the sampling and injecting steps. A maximum of twelve vials can be heated simultaneously in the platen at constant temperature. Once the VLE was reached, each vial was raised onto the needle, and the vapor sample was withdrawn. Static pressure established in the vial upon heating forced the samples into the sample loop. The static vial pressure of 2 to 3 psig was recommended for reproducibility of data and safety w12x. The sample in the loop was then sent to the GC for analysis by Helium carrier gas. A summary of the autosampler parameters is shown in Table 2. The equilibrium time was determined experimentally by using the vials containing 60 mol% of methanol–isopropanol heated in the platen for 2–12 h, and plotting the vapor mole fraction of methanol as a function of heating time Ž Fig. 2. . The mole fraction of methanol had a standard deviation greater than 2% when heated for less than 750 min. Similar experiments were performed with ethanol–n-octane and ethanol–water mixtures, and the minimum equilibrium times of 900 and 350 min were obtained, respectively. Erring on the safe side, the maximum programmable time of 990 min of equilibrium time was used in all experiments, since no sample degradation occurs at these temperatures. The standard deviation at 990 min was 1%. In addition, all vials were kept at the required temperature in an oven for 24 h prior to being placed in the platen to ensure ample diffusion of vapor samples through the pores.
Table 2 Summary of autosampler parameters Model Carrier gas Carrier gas flow rate Platen temperature
Equilibrium time Sample loop volume Line and valve temperature Loop fill time
Genesis Headspace Autosampler with 50-position carrousel Ultra High Purity Helium ŽMatheson. 15 cm3rmin 558C Žfor methanol–isopropanol. 758C Žfor ethanol– n-octane. 608C Žfor ethanol–water. 990 min 5 ml 1758C 0.03 min
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Fig. 2. Minimum equilibrium time for 60 mol% methanol–isopropanol solution with 95% confidence intervals.
3.3. GC The vapor sample sent from the autosampler was analyzed by a Varian 3400 GC and the Analog-to-Digital Converter Ž ADC. board controlled by a computer using the software Star Workstation. A summary of GC parameters used in the experiments is shown in Table 3. The peak areas obtained from the GC were converted into the number of moles of the sample via calibration by direct injection. These calibration factors were confirmed on a regular basis both with manual injections and with internal standards.
Table 3 Summary of GC parameters Model Column Column temperature Ramping temperature Detector Detector temperature Injector temperature Make-up gas Make-up gas flow rate Air flow rate Hydrogen gas flow rate
Varian 3400 GC DB-624 glass capillary column 658C Žfor ethanol–water. 958C Žfor binary alcohol mixtures. 1508C Flame Ionization Detector ŽFID. 2508C 1808C Ultra High Purity Helium 15 cm3rmin 300 cm3rmin 30 cm3rmin
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Fig. 3. VLE of ethanol– n-octane without porous media Ž758C. with 95% confidence intervals.
3.4. Validation and error minimization The experimental apparatus and the techniques were verified by comparing the results with literature values. The VLE of ethanol–n-octane measured without porous media were plotted in Fig. 3. The 95% confidence intervals, produced from triplicate samples, were too small to be visible in this figure. These results indicate that the experimental procedures used in this study give precise and reproducible data. It was realized from the start that the differences between the vapor pressure with and without capillaries would range from 0 to 8%. In order to ensure precise results, efforts were made to minimize the impact of extraneous variables. All vials with porous media were handled simultaneously with unmodified vials containing the sample solution. The two were subjected to the same temperature and test conditions: any premature evaporation, gas dissolution, temperature fluctuations would thus be reflected in both. To avoid hysteresis, the order in which the compositions were tested was random. Three samples and controls were analyzed for each data point reported. The 95% confidence intervals are indicated in all figures. Due to their narrowness, they are not always visible.
4. Results and discussion The SS mod model was used to predict the vapor and liquid equilibria ŽVLE. in porous media for two aqueous alcohol mixtures, ethanol–water and propanol–water and two binary mixtures, methanol–isopropanol and ethanol–n-octane. These model predictions were compared with the experimental results conducted with sintered metal and fritted glass plates as porous media.
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4.1. VLE in porous media as function of pore sizes As per the theory presented in Section 2, the VLE is affected by the pore sizes of porous media. Fig. 4 shows the model-predicted VLE of both aqueous and non-aqueous systems as a function of pore sizes. Note that each curve is specific for a given mixture at one concentration and temperature. For both aqueous and non-aqueous mixtures, the effect of curved liquid surface on the VLE decreases dramatically as the pore size increases. For pores greater than 90 mm, the difference in the vapor concentrations with and without porous media was predicted to be negligible. The curvature effect on the VLE increases rapidly as the pore size decreases. For ethanol–water and propanol–water mixtures, a pore size of 50 mm engenders a 4%–7% increase in the concentration of alcohol in the vapor mole phase. At this pore size, the percent increase for the methanol–isopropanol and ethanol–n-octane is less than 1%. At 5 mm, for aqueous mixtures, the alcohol concentrations are predicted to increase by 50%–60%. However, the liquid surface modification effect in the non-aqueous systems at 5 mm is only 5%–10%. 4.2. Comparison of model predictions with experimental results As shown in Fig. 4, no change in the vapor concentration was predicted when the porous were larger than 90 mm. Hence, when the 120 mm sintered metal plates were used, no change in the VLE was predicted. The experimental results confirmed this. For all four systems, the predicted and experimental values for 120 mm pores merely coincided with standard published VLE data. Using a pore size of 50 mm for the fritted glass plates, the model predicted an increase of 4%–6% in the ethanol mole fraction in the vapor phase Ž Fig. 5. . These predictions are in good agreement with the experimental results. The comparison for propanol–water with fritted glass plates is shown in
Fig. 4. Percent increase in vapor mole fraction as function of pore sizes predicted by the SS mod model.
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Fig. 5. VLE of ethanol–water at 608C in fritted glass plate ŽVLE plane from Ref. w13x..
Fig. 6. The experimental results showed good agreement with the predicted increase of 7% in the propanol mole fraction in the vapor phase. From Figs. 5 and 6, one may conclude that the SS mod model predictions agree well with the experimental results. In comparison, both the Kelvin and Yeh equations predicted no change in the VLE of given solutions at this pore size due to their limitations in describing the behavior of real fluids. In order for the Kelvin and Yeh et al. equations to predict the same results as the SS mod model, the pore sizes would need to be orders of magnitude smaller than the
Fig. 6. VLE of propanol–water at 608C in fritted glass plate ŽVLE plane from Ref. w14x..
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Fig. 7. VLE of methanol–isopropanol at 558C in fritted glass plate ŽVLE plane from Ref. w15x..
actual pore sizes. The model predictions for methanol–isopropanol and ethanol–n-octane are compared with experimental results in Figs. 7 and 8. Using the estimated pore size of 50 mm for fritted glass plates, the model predicted less than 1% change in the vapor phase compositions in both systems. The experimental results also confirmed that the pores were too big to have any effect on the equilibrium.
Fig. 8. VLE of ethanol– n-octane at 758C in fritted glass plate.
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5. Conclusions The development of the SS mod model provided a tool for estimating the VLE in porous media as a function of pore sizes. The model was validated by a series of experiments conducted by using sintered metal and fritted glass plates as porous media. The actual pore diameter in chosen porous media was estimated, and the model prediction showed excellent agreement with the experimental results. The liquid surface modification effect was not visible in the non-aqueous solutions due to the pores that were too big to provide sufficient curvature at the liquid–vapor interface. The main limitation on the experimental procedure was the unavailability of porous media with very fine pores. The nominal sizes claimed by manufacturers do not correspond to the actual pore diameter. Porous media with finer and more uniform pores should be found to provide more pronounced changes in the vapor phase concentrations. List of symbols P pressure r radius V molar volume Greek letters u contact angle m chemical potential s surface tension Subscripts act actual curv curved liquid surface d dew point L liquid v vapor
Acknowledgements The GC expertise of Danielle Beland from Varian is truly appreciated. The support from Dr. J.H. ´ Vera for the model is greatfully acknowledged. Our research group and Dr. M.E. Weber provided helpful suggestions. The financial support of the Natural Sciences and Engineering Research Council ŽNSERC. and the Brace Research Institute are also appreciated.
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G.C. Yeh, B.V. Yeh, S.T. Schmidt, M.S. Yeh, A.M. McCarthy, W.J. Celenza, Desalination 81 Ž1991. 161–187. A.A. Shapiro, E.H. Stenby, Fluid Phase Equilibria 134 Ž1997. 87–101. N.S.J. Wong, Master’s Thesis, McGill University, Montreal, Canada, 1997. L. Boublikova, B.C.Y. Lu, J. Appl. Chem. 19 Ž1969. 89 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981. R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 Ž1986. 323–333. G. Sandoval, G. Wilczek-Vera, J.H. Vera, Fluid Phase Equilibria 52 Ž1989. 119–126. P. Proust, E. Meyer, J.H. Vera, Can. J. Chem. Eng. 71 Ž1993. 292–298. R.H. Perry ŽEd.., Perry’s Chemical Engineers’ Handbook, 7th edn., McGraw-Hill Book, 1997. Varian, Genesis Headspace Autosampler, Operator’s Manual, Varian Associates, 1991. V.V. Udovenko, L.G. Fatkulina, Zh. Fiz. Khim. 26 Ž1952. 1438 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981. D.C. Freshwatter, K.A. Pike, J. Chem. Eng. Data, 12 Ž1967. 179 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981. E. Shrfiber, E. Schuettau, D. Rant, H. Schuberth, Z. Phys. Che1 ŽLeipzig., 247 Ž1976. 23 as cited by Gmehling and U. Onken, Vapor–Liquid Equilibrium Data Collection, Dechema, Deutsche Gesellschaft fur Chemisches Apparatewesen, 1981.