Ind. Eng. Chem. Res. 2005, 44, 639-644
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Semiempirical Model of the Vapor-Liquid Phase Behavior of the Hydrogen Chloride-Water System B. Todd Brandes* Albemarle Corporation, 451 Florida Street, Baton Rouge, Louisiana 70801-1765
A simple, accurate, and widely applicable semiempirical vapor-liquid equilibrium model is developed for the HCl-water system. The model is based on the modified Raoult’s law, adjusted to include a function known as the “pseudo-vapor pressure.” The resulting set of equations relate total pressure, liquid bulk (nonionized) composition, vapor composition, and temperature. The parameters were fit to a wide range of data available in the literature (-10 to 200 °C, 0.0026866.1 atm, and 2-54.9 wt % HCl) that includes the supercritical region. The final equations are demonstrated to allow straightforward calculation of azeotropes and dew points. Introduction The vapor-liquid phase behavior of the hydrogen chloride (HCl)-water system is very complex relative to that of simple organic solvent systems or even as compared to that of many other aqueous-inorganic systems. There is an extremely high affinity between HCl and water, and HCl is a very strong acid and, as such, is essentially fully ionized in water with Ka ≈ 107.1 Because HCl is a very commonly used material in the chemical process industry, considerable research has been devoted to experimentally characterizing its phase behavior with water, and the results are published widely (e.g., see Zeisberg,2 Schmidt,3 Kao,4 Rosenberg,5 Austin et al.6). Because so many modern process design techniques are employed using computer methods, such as the design and scale-up of distillation and absorption columns, flash units, and other equilibrium-controlled unit operations, a widely applicable and accurate mathematical representation of the system is useful. Numerous previous researchers over several decades have undertaken independent fundamental approaches to the modeling of aqueous electrolyte systems, dissociating gases in solvents, and the HCl-water system specifically, and these models have provided various ranges of applicability and accuracy. Some pioneering work7-10 provided a basis for considerably more modeling research that has employed physical principles to develop appropriate mathematical forms to fit experimental data. Fundamentally, it is the nonionized form of HCl in solution that exerts a partial pressure over aqueous solutions. Because this quantity is such a small fraction of the total H+ and Cl- in solution, it is clear that the relationship between ionization and volatilization is so intimate, complex, and highly nonideal that Henry’s law does not have the appropriate functional form to model HCl-water VLE. Additional complexities of HCl include the formation of various hydrates with water depending on concentration and temperature (Rosenberg5), transition to the supercritical state above 51.6 °C, and formation of minimum-boiling azeotropes. The modeling efforts presented in the literature generally fall into two categories: those based on equations of state and those based on a combination of traditional * Tel.: (803) 539-5136. Fax:
[email protected].
(803) 536-0981. E-mail:
excess Gibbs free energy and Pitzer/Debye-Hu¨ckel theory. Many take into account specified, fairly detailed microscopic aqueous electrolyte phenomena with equations of varying degrees of mathematical complexity to achieve a designed model form. Several have been applied directly to the HCl-water system and found to provide good fits of the data. Following is a brief but representative review of selected prominent aqueous electrolyte modeling research. Equation of State Based Models. Only a small portion of the HCl-water modeling research has been devoted to equations of state. The approach has appeal in its rigorous treatment of supercritical fluids, broadness in ability to simultaneously model other physical properties, and relatively simple functional backbone. However, in practice, these equations have generally required fairly complex correlations for the parameters and mixing rules. Nonetheless, some work has been successful for some regions of data. The Patel-Teja cubic equation of state11 is an extension of the van der Waals equation specifically developed for VLE and density correlation for pure fluids. The model requires the critical temperature and pressure, plus two additional component-specific parameters. A novel attribute of the model is its applicability to polar and nonpolar fluids, making it an advancement over the Soave-Redlich-Kwong and Peng-Robinson equations of state for this feature. Additionally, the model offers significant advancement in the development of the correlations of the native parameters to critical constants. Further advancement of the mixing rules for polar fluids was achieved by Panagiotopoulos and Reid.12 Delano13 specifically applied the model to HClwater and clearly showed a fit of the general trend, but the model was fit to a small set of data and is a little low in prediction of the bubble-point total pressure (predicts about 60% of Kao’s4 measured pressures). The Peng-Robinson-Stryjek-Vera (PRSV) equation of state was fit to HCl-water VLE data14 with four component-specific adjustable parameters. The model offers a good fit for HCl bulk mole fractions below 0.16 but experiences large inaccuracies for more concentrated mixtures. The PSRK group contribution equation of state15-18 combines the Soave-Redlich-Kwong19 equation of state, pure-component parameter prediction with group contribution theory via UNIFAC20 and further enhance-
10.1021/ie049218a CCC: $30.25 © 2005 American Chemical Society Published on Web 01/08/2005
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ment by Mathias-Copeman density-dependent local composition theory to calculate mixture parameters.21 Input to the model includes critical data, the accentric factor, and group contribution interaction parameters. Horstmann18 indicates that HCl-water system regression is currently in progress. Traditional Excess Gibbs Free Energy and Pitzer/Debye-Hu 1 ckel Based Models. The use of traditional excess Gibbs-free-energy-derived expressions have appeal in modeling HCl-water VLE for their likeness to standard activity coefficient equations typically studied in chemical engineering. The equations are generally familiar and have been well-studied and successfully applied to many mixture systems throughout the literature. However, the treatment of ionization and supercriticality offers specific challenges to this approach. Fortunately, several novel theories have allowed applicability of traditional activity coefficients to the HCl-water system with some mathematical creativity and complexity. The largest portion of VLE modeling research for HCl-water-type systems has incorporated Debye-Hu¨ckel theory of ionic solutions22 with excess Gibbs free energy models, following that pioneered by Pitzer.23 The approach has generally been able to circumvent many of the thermodynamic pitfalls described above, but again is often mathematically detailed. Engels and Bosen’s24 work combined a dissociation model and a local composition equation for characterizing electrolyte system VLE. The model allows for short- and long-range electrostatic interactions and empirically incorporates hypothetically incomplete dissociation of HCl where, generally, only water surrounds the H+ and Cl- ions. The development included an empirical expression for chemical equilibrium, a modified form of Raoult’s law,25 and activity coefficients given by Wilson’s equation.26 The model gives good agreement above 5 mol % HCl, but the fit is limited to low pressure (up to about 1 bar) and up to about 100 °C. Also, because the vapor pressure of the components is included, HCl is extrapolated into the critical region. Chen and Evans27 analogously studied the electrolyte NRTL28 model, including incorporating provisions for the degree of ionization of the solute. Their work was applied to the HCl-water system up to 42 wt % HCl and up to about 1 atm. Liu and Gre´n29 adapted and extended Engels and Bosen’s24 work. They retained Wilson’s equation26 for short-range interactions, added an additional model for long-range interaction, and assumed complete dissociation of the electrolyte species. The final model was fit to HCl-water data below 1 atm and 110 °C. Brandani et al.30 adapted Pitzer’s theories with the reference state of HCl given by Henry’s law, thereby avoiding the ambiguity of calculating VLE in the critical region for HCl. They assumed complete dissociation of HCl and designed thermodynamic consistency into their model all the way to infinite dilution, claiming distinction from Engels and Bosen’s work24 for this feature. The resulting model was fit to data and found to represent well bubble and dew points up to 100 °C and up to about 1.5 bar. Sander et al.31 combined Debye-Hu¨ckel theory with a modified form of UNIQUAC10 for salt systems. Macedo et al.32 and Kikic et al.33 provided extensions of Sander’s work. Li et al.34 developed a different model with a modified Raoult’s law backbone and a complex expres-
sion for the activity coefficient. The model incorporated ion charge interaction, Debye-Hu¨ckel theory, UNIQUAC, and multiple electrostatic interaction terms based on osmotic pressure. Their model is specifically applicable to mixed solvent systems but is not rigorous for supercritical systems. Yan et al.35 extended Li’s work by substituting UNIFAC. Wu and Lee36 incorporated full ionization, osmotic cell pressure accountability, and the mean spherical approximation (from statistical mechanics) for the effect of salt concentration. Wang et al.37 grouped electrolyte systems into three classes: fully associated (similar to nonelectrolyte systems), completely dissociated (which is the most common for electrolyte systems), and speciation-based (simultaneous solution of chemical and phase equilibria). They developed a new model for this third class, analogous to the work by Engels and Bosen24 and Chen and Evans.27 Their model is based on a three-contribution expression for the excess Gibbs free energy coupled with an expression for chemical equilibrium and includes provisions for the infinite-dilution limit. The model was applied to the HCl-water system and provided a good fit of literature data. Other researchers did similar work before Wang et al.38,39 Overall, many significantly different approaches in the literature have been devoted to modeling aqueous electrolyte and HCl-water VLE with varying extents of applicability and accuracy. Most of the work, though, is mathematically complex, not readily supported by commercially available process simulators, and accurate in only selected regions. In this study, a single and much simplified model was sought for a wide range of conditions and for the HCl-water system only. Because the system is so complex, involving so many thermodynamic phenomena described above, a distinctly novel, semiempirical approach was adopted here. The following sections detail the development of the model and provide examples of its practical use. Model Development The classical equation used very widely to model a mildly nonideal liquid solution in equilibrium with an ideal gas mixture of the same components is the modified Raoult’s law25
yiP ) xiγiPsat i
(1)
where yi and xi are the mole fractions of component i in the vapor and liquid phases, respectively; P is the system total pressure; γi is the activity coefficient of component i in the liquid phase; and Psat is the vapor i pressure of component i at the system temperature. This model applies well to weakly associating, nonionizing species at low to moderate pressures and far from the critical region. Although it is not fundamentally applicable to the HCl-water system, its functional form gives rise to a semiempirical model that does fit the data very well. Vapor pressure is often correlated with temperature, T, by an expression of the following form as derived from the Clausius-Clapeyron equation25
ln(Psat i ) ) g +
h T
(2)
where g and h are unique for a given component. Likewise, the activity coefficient is typically related to
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 641
composition by an equation originating from the excess Gibbs free energy. One of the simplest of such forms is the two-suffix Margules equation,10 which is secondorder in xi
k ln(γi) ) (1 - xi)2 T
(3)
Upon inspection of eqs 2 and 3, it follows that f1(T) f2(xi,T) e γiPsat i ) e
(4)
with f1 given by the right-hand side of eq 2 and f2 given by the right-hand side of eq 3. Because f1 already includes 1/T, the form of eq 3 is therefore simplified to exclude T. Substituting, eq 4 then transforms to
(
γiPsat i ) exp g +
h + k - 2kxi + kxi2 T
)
Figure 1. Variation of Z with liquid mole fraction of HCl.
(5)
Adding back one more term from eq 3 to include the xi-T interaction and taking each coefficient in eq 5 as independent, the final form results
(
γiPsat i ) exp Fi +
)
Gi xi + Hixi + Iixi2 + Ji T T
(6)
Denoting HCl with the subscript H and water with the subscript W, eqs 1 and 6 can be conveniently expressed in the forms
yWP ) xWe
R(T,xW)
Z(T,xH)
yHP ) xHe
(7) (8)
with xW and xH denoting bulk compositions expressed as if the HCl did not ionize. The author has described eR and eZ to be the “pseudo-vapor pressures” of water and HCl, respectively, with R and Z given by
R ) FW + Z ) FH +
GW xW + HWxW + IWxW2 + JW T T
(9)
GH xH + HHxH + IHxH2 + JH T T
(10)
Parameter Determination A wide range of vapor-liquid equilibrium data were obtained for the HCl-water system from the references cited above. The data span temperatures from -10 to 200 °C, pressures from 0.00268 to 66.1 atm, and compositions from 2 to 54.9 wt % HCl and were transformed to R and Z using eqs 7 and 8. Figures 1 and 2 show the nearly linear variation of Z with the liquid mole fraction of HCl and R with the liquid mole fraction of water, respectively. Equations 9 and 10 were fit to the data, and the resulting parameters obtained are given in Table 1 (for x in bulk mole fraction neglecting ionization, P in mmHg, and T in K). Least-squares regression was employed to fit the parameters using Minitab40 statistical software, which minimized the sum of squares of the residuals as the objective function.41 The overall fit provided R2 values of 99.3% and 99.6% for Z and R, respectively. Each of the 10 parameters were statistically significant with p values of 0 (less than 0.05 means the parameter is statistically significant). Overall, given
Figure 2. Variation of R with liquid mole fraction of Water. Table 1. Parameters for Equations 9 and 10 for HCl-Water F G H I J
HCl (Z)
water (R)
25.6515 -9019.56 1.08161 -46.4394 17506.4
14.4486 -8482.2 18.4698 -12.8854 3495.15
the total system pressure and liquid composition, the model predicts the temperature with an average error of 2.6 °C and a 95% confidence interval of (0.49 °C, and it predicts the vapor mole fraction HCl with an average error of 0.015 and a 95% confidence interval of (0.0041. Results and Application The model obtained as described above is very useful for calculating HCl-water phase behavior over a wide range that is amenable for use with engineering calculations. With its relatively simple form compared to many models available in the literature, physical properties can be determined with straightforward numerical methods and implemented in a wide variety of engineering software. To illustrate the fit, Figures 3 and 4 were generated with a simple spreadsheet and compare the model predictions with a selection of the data from the literature. Produced in a similar manner, Figure 5a shows the phase diagram of the HCl-water system as calculated with the model showing very smooth predictions at different conditions of temperature and composition. As a comparison, Figure 5b provides the curves at 0 and 70 °C calculated using the model along with literature data. The model matches essentially indistinguishably the results provided by Wang et al.,37 which appears to be one of most accurate (and most complex) of the models discussed earlier. Application to Calculation of Azeotropes. A very useful application of the resulting VLE model for the
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Figure 3. Model prediction of HCl partial pressure (squares from Zeisberg,2 circle from Austin et al.,6 diamonds from Schmidt3).
Figure 4. Model prediction of water partial pressure (squares from Zeisberg,2 circle from Austin et al.6).
HCl-water system as described above is the calculation of the azeotrope given the system pressure. The calculated results include liquid composition and the temperature. Equations 7-10 can be easily transformed by setting
yi ) xi
(11)
according to the definition of a homogeneous azeotrope,23 as well as
yH + yW ) 1
(12)
xH + xW ) 1
(13)
Figure 5. Calculated phase diagram for the HCl-water system: (a) for various temperatures, (b) for 0 and 70 °C with literature data (circles from Austin et al.,6 diamonds from Kao4).
to give the following result
ln(P) ) R(T,1-xH) ) Z(T,xH)
(14-15)
where R and Z are given by eqs 9 and 10. To illustrate, Figure 6 shows the temperature of the HCl-water azeotrope as a function of total pressure, and Figure 7 shows the liquid composition as a function of total pressure. The squares on each figure are independent literature data.5 Application to Calculation of Dew Points. Another interesting application of the VLE model for the HCl-water system is the determination of dew points. Often, anhydrous HCl is processed and stored in carbon steel.6 If small amounts of water, in the part-per-million range, contaminate anhydrous HCl, a dew is formed that is on the order of 50 wt % HCl and is highly corrosive to carbon steel.42 This dew point is highly dependent on pressure and temperature and can be easily determined using the model obtained above. Figure 8 shows the calculated water concentration in
Figure 6. Calculated azeotropic temperature as a function of the total pressure for the HCl-water system.
the vapor at the dew point for various temperatures and pressures according to the parameters obtained for eqs 9 and 10 above. Conclusions A novel, simple, and accurate semiempirical vaporliquid equilibrium model was developed and fit to a wide range of literature data for the HCl-water system. Despite the many thermodynamic complexities of the system, the successful model was based on the modified Raoult’s law with only the simple functional forms for
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 643 yi ) mole fraction of component i in the vapor phase γi ) activity coefficient of component i in the liquid phase
Literature Cited
Figure 7. Calculated azeotropic liquid composition as a function of the total pressure for the HCl-water system.
Figure 8. Dew point water concentration in HCl-water vapor (squares from Zeisberg,2 triangle from Schmidt,3 circles from Austin et al.,6 diamonds from Kao4).
the vapor pressure and two-suffix Margules activity coefficient being transformed to a single function known as the “pseudo-vapor pressure.” The resulting model, which relates total pressure and liquid bulk (nonionized) composition to vapor composition and temperature, can determine the temperature within a 95% confidence interval of (0.49 °C and the vapor mole fraction HCl within a 95% confidence interval of (0.0041. The results can be readily employed for many computer-based engineering applications, such as the design and scaleup of distillation and absorption equipment. As examples, the model was illustrated by calculating the HCl-water azeotrope as a wide function of total pressure, as well as dew points of HCl-water gas mixtures. Because of its semiempirical nature, the form of the model derived is not intended for direct extension to multicomponent or other systems with the parameters obtained. An extension to these cases is possible, but a large amount of regression data is needed or the model would require a fundamental adaptation to unify its applicability to multiple systems. Because the HCl-water system is so common and important in industry, though, this model has proven to be a useful result for its simplicity, accuracy, and breadth. Notation eR ) “pseudo-vapor pressure” of water eZ ) “pseudo-vapor pressure” of HCl Fi-Ji ) constants for component i f1 ) vapor pressure functional form f2 ) activity coefficient functional form k ) constant P ) system total pressure Psat ) vapor pressure of component i at the system i temperature T ) absolute temperature xi ) mole fraction of component i in the liquid phase
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Received for review August 26, 2004 Revised manuscript received November 8, 2004 Accepted November 18, 2004 IE049218A
