1669
Ind. Eng. Chem. Res. 1991, 30, 1659-1666 dictions using the method of Englezos and Bishnoi (1988). It is seen that the predictions agree very well with the experimental data In Figure 4 the maximum and smalleat absolute deviations between the experimental and the calculaM hydrate formation preasurea are 7.07 and 0.76%, respectively. The average absolute deviation is 3.19%. In Figure 5 these quantities are 4.58, 0.22, and 2.13%, and in Figure 6 they are 2.78, 0.08, and 1.53%. 6. Conclusions
An experimental apparatus for the determination of the vapor-liquid-hydrate incipient equilibrium conditions was built. The apparatus was used to obtain equilibrium hydrate formation data for ethane in aqueous solutions of NaC1, KC1, CaC12, and KBr in the temperature and pressure ranges of 265.36-282.98 K and 0.488-2.188 m a . The experimental data were compared with the predictions by using the method of Englezos and Bishnoi (1988) and were found to be in excellent agreement.
Acknowledgment The financial support from the Izaak Walton Killam Memorial scholarship trust and the Natural Science and Engineering Research Council of Canada (NSERC and NSERC-Strategic) is greatly appreciated. Registry No. NaCl, 7647-14-5; KCl, 7447-40-7; CaC12, 10043-52-4; KBr, 7758-02-3; ethane hydrate, 14485-33-7.
Literature Cited Berecz, E.; Balla-Ache, M. Some technological Aspecta of the Role and Application of Gas Hydrates. In Studies in Inorganic Chemistry 4: Gas Hydrates; Elsevier: Amsterdam, 1983; pp 267-297. Bond, D. C.; Russell, N. R. Effect of Antifreeze Agenta on the Formation of Hydrogen Sulfide Hydrate. Pet. Technol. 1949, 192-198. Davidson, D. W. Clathrate Hydrates. In Water: A Comprehensive Treatise; Frank, F., Ed.; Plenum Press: New York, 1973; Vol. 2, Chapter 3, pp 115-133.
Deaton, W. M.; Frost, E. M., Jr. “Clathrate Hydratee and Their Relation to the Operations of Natural Gas Pipelines”. US. Department of Interior, 1946. Dholabhai, P. D.; Englezos, P.; Kalogerakis, N. E.; Bishnoi, P. R. Equilibrium Conditions for Methane Hydrate Formation in Aqueous Mixed Electrolyte Solutions. Can. J . Chem. Eng. 1991, in press. Englezos, P. Gas Hydrate Equilibria. Ph.D. Dissertation, The University of Calgary, Calgary, 1990. Englezos, P.; Bishnoi, P. R. Prediction of Gas Hydrate Formation Conditions in Aqueous Electrolyte Solutions. AZChE J . 1988,34, 1718-1721. Hammerschmidt, E. G. Preventing and Removing Hydrates in Natural Gas Pipe Lines. Gas 1939, 15, 30-34. Holder, G. D.; Hand, J. H. Multi-Phase Equilibria in Hydrates from Methane, Ethane, Propane and Water Mixtures. AZChE J . 1982, 28,440-447. Holder, G.D.; Zetta, S. P.; Pradhan, N. Phase Behavior in Systems Containing Clathrate Hydrates. Rev. Chem. Eng. 1988, 1-69, Knox, W. G.;Hess, M.; Jones, G. E.; Smith, H. B. The Hydrate Process. Chem. Eng. Prog. 1961,57,66-76. Kubota, H.; Shimizu, K.; Tanaka, Y.; Makita, T. Thermodynamic Properties of R13 (CClF8), R23 (CHF8), R152a (CzH4Fl),and Propane Hydrates for Desalination of Seawater. J . Cham. Eng. Jpn. 1984,17,423-429. Makogon, Y. F. Use of Gas Hydrates. In Hydrates of Natural Gas; Translation by Cieslewicz, W. J.; Penn Well Publishing Co.: Tulsa, OK, 1981; pp 213-224. Menten, P. D.; Parrish, W. R.; Sloan, E. D. Effect of Inhibitors KCl, CaCIz,and CHSOHon Cyclopropane Hydrate Formation Conditions. Znd. Eng. Chem. Process Des. Deu. 1981,20, 399-401. Paranjpe, S. G.;Patil, S. L.; Kamath, V. A,; Godpole, S. P. Hydrate Equilibria for Binary and Ternary Mixtures of Methane, Propane, Iaobutane, and n-Butane: Effect of Salinity. SPE Pup. 16871 1987,379-389; Presented at 62nd Annual Technical Conference of Society of Petroleum Engineers, Dallas, TX, Sept 27-30. Roo, J. L.; Peters, G. J.; Lichtenthaler, R. H.; Diepen, G. A. M. Occurrence of Methane Hydrate in Saturated and Unsaturated Solution of Sodium Chloride and Water in Dependence of Temperature and Pressure. AZChE J. 1983,29,651-657. Sloan, E. D., Jr. Molecular Structure and Similarities to Ice. In Clathrate Hydrates of Natural Gases; Marcel Dekker New York, NY, 1989, pp 24-66.
Receiued for review September 19, 1990 Accepted February 25,1991
The PVT Region of Linear IsoK Lines Eugene M. Holleran Chemistry Department, St. John’s University, Jamaica, New York 11439
Many common gases are shown to exhibit a fairly extensive PVT region in which T versus p is linear for constant values of K = (2- l ) / p . The linearity is shown to be accurate within the uncertainty of good experimental data. An equation of state explicitly incorporating this linearity is introduced, and its coefficients are evaluated for 13 data sets for 11gases from PVT data in the literature. The isoK linearity allows extrapolation from data a t moderate densities to the critical density. Extrapolation down to zero density on the assumption of continued linearity allows the evaluation of the virial coefficients B and C. In reverse, the known second and third virial coefficients for an intermolecular potential can be used to find its equation of state in the linear isoK region. This leads to the possibility of testing the suitability of an intermolecular potential for a given gas by the direct fitting of PVT data.
Introduction The unit compressibility line (UCL) is the locus of PVT points along which P = pRT. On the UC1 the compressibility factor, 2,defined as P/pRT, equals unity, and the quantity, K, defined as (2- l)/p, equals zero. It has been found (Morsy, 1963; Holleran, 1967; Powles, 1983; BenAmotz and Herschback, 1990; Holleran, 1990) that along the UCL, the temperature, T, is accurately linear in den0888-5885/91/ 2630-1659$02.50/0
sity, p, for many gases from low densities to beyond the critical density. Above the UCL, in the PVT region where 2 > 1 and K > 0, lines of T versus p at constant K (isoK lines) exhibit a positive curvature that rapidly becomes very large as K increases. Below the UCL, i.e. where 2 < 1 and K C 0, the expected negative curvature does not immediately appear. Instead, there is a fairly extensive PVT region in which the isoK lines seem to be linear. This Q 1991 American Chemical Society
1660 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 Table I. Argon Reduced Temperatures at the Indicated Reduced Densities and Reduced K Valueso P K = 0.0 K = -0.1 K = -0.2 K = -0.3 K = -0.4 K = -0.5 0.3 2.471 2.115 1.857 1.661 1.508 1.384 1.622 2.055 1.809 1.473 1.353 0.4 2.393 0.5 2.315 1.996 1.762 1.582 1.438 1.322 1.542 1.404 1.291 1.937 1.714 0.6 2.237 1.369 1.259 0.7 2.159 1.878 1.667 1.50 1.334 1.228 0.8 2.081 1.818 1.619 1.462 1.299 1.197 1.571 1,422 0.9 2.004 1.759 1.265 1.166 1 .o 1.927 1.700 1.523 1.382
K = -0.6 1.282 1.254 1.225 1.196 1.167 1.139 1.110 1.081
OThese temperatures were interpolated from polynomial leaat-squares fits of 1/Tversus K at constant density from data tabulated by Levelt (1960) I
phenomenon has never been thoroughly investigated, although the linearity of isoK lines has been used to help evaluate the coefficienta in an equation of state (Holleran and Hammes, 1975) and has been proposed as the basis for a method of evaluating the virial coefficients (Holleran, 1970). It is the purpose of the present investigation to find out how accurate this apparent linearity is, how extensive a PVT region it includes, how common it is, and whether it can be put to good use.
Preliminary Procedures and Results There are many seta of experimental PVT data for various gases available in the literature that are appropriate for this study (cf. Table 11). Usually the data are presented for constant values of temperature and often at constant densities. A preliminary procedure used in this work was to interpolate in these data to find temperatures and densities at selected constant values of K. Table I illustrates the results obtained by this method from the data for argon, as presented by Levelt (1960). This data set was selected for illustration for the following reasons. First, the data have a good accuracy, with 2 reported to four decimal places with only a small uncertainty in the last digit. Second, this data set is very extensive and turns out to include the entire PVT region of interest in this investigation. And finally, the data are tabulated at constant temperatures and densities reduced by the critical temperature and density, so that the region of our interest can be seen in this reduced context. The interpolation was carried out on the isochores, along which it was found to be convenient to represent 1 / T as a function of K because of the small curvature. On most isochores a third- or fourth-order polynomial provided a very good fit of the data points. Reduced temperatures calculated from these polynomials for the various reduced densities at the indicated constant values of K are shown in Table I. For ease of inspection, the temperatures are rounded to three decimal places. One can see from the incrementa that the linearity is nearly perfect in this data region. Densities below 0 . 3 were ~ ~ omitted because at low densities the experimental values of K become unreliable, since in the expression (2- l ) / p the numerator and denominator both approach zero and experimental errors are magnified. In this work we assume that the linearity that is observed ~~ down to zero for densities between pc and 0 . 3continues density. Intercept temperatures and slopes found on this assumption from Table I simply by inspection would appear to be accurate to the third decimal place. For example, the zero-density temperature intercept for K = -0.4 is clearly within 0.001 or 0.002 of 1.612. Figure 1shows the isoK lines in these reduced variables, with the gas-liquid two-phase region for reference. The region of ieoK linearity is bounded by zero density on the left, the critical density on the right, the K = 0 line at the
2.q
1
c__---
_ _ _ _ - -----_ --
, . y e
8
0
2
6
4
a
1
1 2
P Figure 1. T,p region of linear isoK lines in reduced variables. The isoK lines pass through the experimental pointa and are extrapolated to zero density. Reduced K values extend, by intervals of 0.1, from -0.7 for the lowest line to zero for the topmost. The dashed saturation line passes through the critical point (1,l).
top, and a temperature of 10 or 20% above critical at the bottom. The K = 0 line continues to be linear to about twice the critical density (Holleran, 1990), but as seen in the figure, the lines for K < 0 have leas negative slopes and must eventually curve downward to avoid intersection. This expected downward curvature finally becomes evident at about the critical density beyond which data were excluded. In spite of the rounding off, the numbers in Table I provide a good feel for the rather remarkable linearity of the isoK lines. When an additional decimal place is included in the temperaturea interpolated from the isochorea, then linear least-squares fits of T versus p for these isoK lines give excellent results. For example, at K = -0.4, T is found to be 1.6122 - 0.3477~with an R2of 0.999999 and a standard deviation of 0.O001. However, these fits are for isolated isoK lines. In the following section, interdependent fits will be found in the form of an overall equation of state. Before proceeding with this, it is interesting to observe that one consequence of the linearity of the isoK lines is that only two isochores are needed to establish the PVT behavior in this region. Thus any two of the rows in Table I, for example those at p = 0.4 and 0.8, can be used to fill in all the rest of the table almost perfectly by simple linear interpolation and extrapolation. Extending the extrapolation to zero density again gives, for example, the intercept temperature of 2 X 1.473 - 1.334 = 1.612 for K = 4.4. Two isotherms would not serve the same purpose, because they do not overlap much in K if the 7"s are usefully separated, as can be seen in Table I.
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1661
Equation of State From the observation by the above preliminary interpolation procedure that for argon the isoK lines over a considerable PVT region are linear, possibly within the experimental error, it was clear that a much better and simpler investigative procedure would be to fit all the data in this region simultaneously to a single equation, an equation of state. The following form was used aT + bp = 1 (1) in which a and b are functions of K only, thus incorporating in advance the assumed linearity of the T versus p isoK lines. Simple power series were used to express a and b: a = 00 + a1K + ~ 2 K 2 (la)
b = bo + b1K + b2K2 + b3K3 (1b) On the basis of the experience in fitting isochores, 1/T was selected as the dependent variable and fitted by multiple least-squares aa 1/T = a. + alK + a@ + b@/T + b,pK/T + b@P/T + bspK9/ T , thus yielding the seven coefficients for eqs 1. (The least-squares procedure applies the same mathematical treatment to y = co + clxl + c2x2 + ..., whether the xi are powers of a single x variable, as for a polynomial fit, or various combinations of several variables, as above. Of course, in our case, the minimization of the squares of the deviations is for the variable 1/T; fortunately, as seen below, these Coefficients also provide excellent fits of Z(p,T)that are close to the optimum, as was confirmed by the fact that small trial and error adjustments in the coefficients did not significantly improve the fits.) Results are shown in Table I1 for 13 data sets for 11 different gases. Unless limited by the extent of the data, these fits included densities up to about pc and temperatures from about l.lTc or 1.2Tcto a little above the UCL. Some values of 2 > 1 were included, but usually limited to about 1.05 at low densities and 1.02 at high densities. The temperature and density limits were tested by repeated fits. Densities below about 0 . 2 or ~ 0~ . 3 were ~ ~ excluded for the reason given in the previous section but were included in the later tests for goodness of fit. Except in two cases where the data covered only a small part of the region in question, eq l a had to be quadratic and eq l b had to be cubic. In no case were higher powers needed. The quality of the least-squares fits was excellent, with adjusted R2values ranging from 0.999 981 to 0.999 999, and a standard deviation in 1/T of about 2 or 3 parts in 1OOOO. Equation 1 was tested as an equation of state by using the coefficients in Table I1 to calculate the compressibility factor, 2,at the experimental values of T and p. 2 was found as 1 + pK, and K was found from the equation cJP + cJP+ clK + co = 0, in which co = aoT + bop - 1, c1 = alT + blp, c2 = a2T + b g , and c3 = b3p. The results for the various data sets are included in Table 11. For example, the 165 data points for argon in the designated region are reproduced by eq 1 with an average absolute deviation (AAD) of less than 1 in the fourth decimal place in 2 and a maximum deviation of 2, except at the fringe. Multiplying the AAD by roughly 1.5 would give the average absolute fractional deviation, since 2 in our range lies between about 0.4 and 1.0. Beyond the limits of our range the deviations quickly become very large. For example, at p = 1.2pC,errors in calculated 2 begin to appear in the ~ ~ reach the second third decimal place, and at p = 1 . 5 they decimal place. The data used for methane are those of Douslin et al. (1964). Observed 2 was found from the tabulated pres-
sures and densities by using T = t + 273.15 rather than the thermodynamic temperature scale used by the authors. Table I11 lists the differences between these observed 2 values and those calculated by eq 1 with the coefficients listed in Table 11. The densities extend from 0.75 to 11 mol/L (p, = 10.15 mol/L), and the temperatures range from 273 to 623 K at intervals of 25 K. An intermediate isotherm at 303.15 K was omitted for being slightly inconsistent with the others. The agreement for the 174 points in Table I11 is seen to be excellent, with the differences averaging about 1 unit in the fourth decimal place, or about 1.5 parts per 1oooO. The experimental error is reported as ranging from 0.03% at low T and p to 0.3% at high T and p. The data for fluorine are due to Pry& and Straty (1970). Most of the points in our region lie on 18 pseudoisochores at densities of about 1.5 to about 15.6 mol/L and at temperatures from 170 K (1.18TJ to the experimental maximum of 300 K. Equation 1 fits these 270 points with an AAD in 2 of 1.2 X lV. Pry& and Straty give an equation of state for their low-density data (up to 6 mol/L) in the form 2 = 1 + Bp + cp2 (2) with the T dependence of B and C given by empirical expressions using 11 coefficients. Their equation extends to lower temperatures than eq 1, and eq 1 extends to higher densities, but in the region of overlap (105 points ranging in T from 170 to 300 K and p from 1.5 to 6 mol/L) both show the same AAD of 1.2 X lo-' from observed 2. The two equations have similar difficulty with the pseudoisochore at p = 1.5 mol/L ( 0 . 1 ~ ~ Omitting ). this from the overall fit of eq 1, and also omitting the points at p = 15.6 mol/L, which is beyond pc = 15.1 mol/L, leaves 245 points that are fitted by eq 1 with an AAD of 0.9 X lo-'. This is much below the experimental error, estimated at 0.1 %, but is more consistent with the experimental precision, given as 0.02% . The data set for oxygen is the one used by Schmidt and Wagner (1985), as compiled from several sources. As such, the points can be expected to have a poorer precision than those of the individual sources. The set is limited in T to 300 K except for UCL points extrapolated to 406 K. Equation 1 fits the 330 points in our range ( T down to 170 K (1.lOT.J and p up to the critical density of 13.6 mol/dm3) with an AAD of 1.9 X lo-'. For the purpose of evaluating the virial coefficients, Wagner, Ewens, and Schmidt (1984) (WES) fitted the data up to 6.8 mol/dm3 ( 0 . 5 ~ to ~ ) the truncated virial equation, eq 2. They used an optimization procedure that selected from a bank of terms five terms for B and three for C to represent their T dependences from 68 to 406 K. In the region where this equation (WES) overlaps eq 1; that is, from 170 K up and 6.8 mol/dm3 down, the two equations give the same AAD of 1.9 X lo-'. Their equation is better for densities near 6.8 mol/dm3 but is poorer at fitting the UCL points. Otherwise, the two equations agree well, especially for groups of points for which the fit is relatively poor. For example, five points near 200 K, six near 233 K, and seven near 273 K deviate almost identically from both equations, by as much as 10 X lo-' in 2 and with an AAD of 6 X lo-'. The data set used for nitrogen was obtained in a different way. The points were calculated from the 33 constant, modified Benedict-Webb-Rubin (MBWR) equation of state, given by Younglove (1982), which had been fitted to multiple data sets. Ranging from 160 K (1.2TJ to 320 K, and from 1 to 11 mol/dm3 (p, = 11.2), 154 pointa found from the MBWR equation were fitted by eq 1 with an AAD of 1.5 X lo-' in 2. This is much better than could
1662 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991
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Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1663 have been expected from the 0.2% maximum deviation from the experimental data reported for the MBWR equation. For CO, the data of Goodwin (1985) were used. Equation l fits 77 points ranging in T from 160 K (1.2TC)to the experimental maximum of 300 K and in p on pseudoisochores from 0.5 to 10.8 mol/L (p, = 10.85 mol/L) with an AAD in 2 of 1.7 X lo-'. The eight points on the pseudoisochore at 5.8 mol/L are somewhat inconsistent with the rest of the data, having an AAD of 5.1 X lo4. The remaining 69 points have an AAD of 1.3 X lo4. For ethylene the data of Douslin and Harrison (1976) were used. Equation 1fits 90 points, on six isotherms from 298 K (1.15TJ to the experimental maximum of 448 K and at 15 densities from 0.8 to 7.5 mol/dm3 (p, = 7.635 mol/ dm3), with an AAD of 1.7 X lo4 in 2. For ethane 11 isotherms beginning at 348 K (1.147',), as reported by Douslin and Harrison (19731, were used. The highest temperature isotherm (623 K)was omitted because it gave deviations as high as 10 X lo-' in 2,which, though below the reported maximum experimental error of 30 X lo4, is much above the average deviation. The lowest density points, especially at 0.75 mol/dm3, also showed relatively large negative (calc - obs) deviations. However, in a simultaneous fit of the Douslin and Harrison data with those of Michels et al. (19541, the latter do not show a similar trend. The points at 0.75 mol/dm3 were therefore omitted. Finally, without four high-density points for which 2 > 1.02, the remaining 139 points, extending from 1.0 to 7.0 mol/dm3 (p, = 6.88 mol/dm3), have an AAD of 1.6 X lo-' in 2. The data of Thomas and Harrison (1982) were used for propane. Equation 1fits the 90 points (10 densities on 9 isotherms) ranging from T from 323 K (1.14TJ to the experimental maximum of 623 K and in p from 0.8 to 5.0 mol/dm3 (p, = 5.13 mol/dm3) with an AAD of 1.7 X lo4. Omitting the lowest isotherm leaves 80 points that have an AAD of 1.3 X lo-'. The data for higher hydrocarbons include only the lower portion of our region. But 36 points for isobutane (Beattie et al., 1950) have an AAD of 1.8 X lo4, and 23 points for neopentane (Beattie et al., 1952) have an AAD of 1.7 X 10-4. From all these examples we can conclude that for most common gases eq 1 is an accurate equation of state and that, within the accuracy of good experimental data and within this rather extensive PVT region, the isoK lines are linear. Wagner's Methane and Nitrogen Data The last two data sets to be considered are for methane and nitrogen, as recently reported by Wagner and coworkers (Kleinrahm et al., 1988; Duschek et al., 1988). They cover only limited ranges of T and p (from 273 to 323 K for both gases; from 0.2 to 3.6 mol/dma for N2and from 0.05 to 4.1 mol/dm3 for CH,), but these lie within our linear-isoK region, and they are included here because of their high quality. Although one can find by inspection pairs of points that are discrepant by about 1 X lo-' in 2, the overall accuracy and precision seem to be much better than that. For methane, Wagner et al. represent the data by a truncated virial equation (to be referred to as WAG) of the form of eq 2, again with the temperature exponents selected from a bank of terms by an optimization procedure. Their equation fits the 169 data points with an AAD of only 0.36 X lo-' in 2,and the equation is considered more reliable than the individual experimental values. For the fit of eq 1 to the same data, the cubic term in eq l b was found to be unnecessary because of the limited range
Table IV. Worst Points Deviations in 2 from WAG Eauations and Eauation 1. 104(Calc- Obs) T,"C P, m o W WAG eq 1 Methane 0 0.044 60 -1.0 -1.0 0 0.088 83 1.6 1.6 1.9 1.9 0 0.155 55 0 0.311 14 2.0 2.0 -0.9 0 0.623 93 -1.0 -1.2 10 0.15606 -1.3 -1.5 -1.3 10 0.311 51 1.2 20 0.155 63 1.1 2.0 2.2 20 0.312 89 1.5 40 0.313 54 1.3 0 0
10 20 20
Nitrogen 0.222 44 0.442 11 1.500 24 0.500 01 0.503 23
1.1 1.0 1.2 1.1 1.8
1.1 1.0 1.4 1.2 1.9
of the data. With the six coefficients listed in Table 11, eq 1 fits the data with an AAD of 0.37 X lo4, in close agreement with WAG. The two equations also agree in identifying the worst points. In Table IV are listed all the experimental points whose deviations in 2 from the WAG equation are 1 X lo-' or more. Their deviations from eq 1are also listed and are seen to be nearly the same. This agreement of two very different equations of state might allow contident pruning or adjusting of these worst points, which would in turn permit a refitting and refining of the equation of state. (This was not done however.) The density of methane at STP calculated from eq 1is 0.71745 kg/m3, in close agreement with the value of 0.717 46 + 0.0007 kg/m3 found from WAG. The AAD of the two equations of state from each other is 0.16 X lo4. This is much smaller than the experimental error and scatter, but, even so, the differences vary systematically with density. This is due to the difference in mathematical form and reflecta the fact that, on the isotherms, K versus p is linear for WAG but slightly curved for eq 1 and, on the isoK lines, T versus p is linear for eq 1 but slightly curved for WAG. The results for their nitrogen data are similar. Here the WAG equation includes a fourth term in the virial expression for 2. It fits the 127 data points with an AAD of 0.32 X lo4. Equation 1,with the six coefficients listed in Table 11, shows the same AAD of 0.32 X lo4. Again the two equations agree on the worst points, which are included in Table IV. And again they agree well with each other (AAD = 0.14 X lo4), with tiny differences that are systematic in density. They also agree exactly in their calculation of the STP density. Virial Coefficients As p r o p e d by Holleran (19701, the linearity of the isoK lines can be used to evaluate the second and third virial coefficients, B and C. The virial equation of state can be written as K = B + Cp + Dp2 +
..,
(3)
in which the virial coefficients, B, C, D,etc., are functions of temperature. At p = 0, K = B, and, according to eq la, 1/T = a.
+ alB + a2B2
(4)
Since the coefficients in eq 4 were found by fitting eq
1 to the experimental data, the evaluation of T(B)and hence B(T) from eq 4 is equivalent to a concerted ex-
trapolation of isoK lines from moderately high densities
1664 Ind. Eng. Chem. Res., Vol. 30, No. 7,1991 Table V. Oxygen Virial Coefficients from Equation 1 and Their Differences from the WE8 Equation B, a, C, Ac, ~ T,K cms/mol cma/mol ( ~ m ~ / m o l )(cma/mol)2 1901 103 -56.04 -0.19 190 1735 65 -49.97 -0.11 200 36 -0.07 1601 210 -44.66 13 4.05 1492 220 -39.98 -5 -0.02 1403 230 -35.81 -20 0.00 1330 240 -32.07 1268 -34 0.04 250 -28.69 1216 -45 -25.63 0.07 260 -55 0.09 1172 270 -22.84 1134 -64 -20.28 0.13 280 1101 -71 -17.93 0.15 290 -79 0.18 1071 300 -15.76 1021 -93 -11.88 0.21 320 0.21 980 -105 340 -8.53 -116 0.18 945 360 -5.59 914 -127 0.13 380 -2.99 887 -136 400 -0.68 0.05 862 -146 1.39 -0.06 420
to zero density on the assumption of their continued linearity. We note that 1/T is quadratic in B within experimental error in our T range (about 1.2 to 2.7 times T,, or about 0.45 to 1.0 times the Boyle temperature, TB).A cubic term is not needed in eq l a and therefore not in eq 4. From the form of eqs 1, it is clear, however, that for isochores at densities other than zero a quadratic expression for 1/T versus K is not sufficient. We also note that, because of the slope of the isoK lines,the lowest temperature at which B is evaluated (that is, the lowest zero-density intercept temperature) Will be somewhat (Perhaps 10% above the lowest temperature of the data set used, as can be seen in Figure 1. Equations 3 and 1 provide expressions for the slope of the isoK lines. Thus (dT/d~= ) ~-C/(dB/dT) = -b/a (5) With U / d T evaluated from eq 4, this allows the third virial coefficient C to be found at any T in our range as C = b/T(a, + 2 a 8 ) (6) with K = B in eq lb. If the data set being fitted includes a good portion of the UCL, then the following characteristic physical constants of the gas can be evaluated from the coefficients of eq 1. The Boyle temperature, at which B is zero, is given by l / a @ The Boyle volume, defined as T(dB/dT) at TB
is -ao/al. The extrapolated zero temperature intercept of the UCL, pB, is l/bo. The third virial coefficient a t TB, designated by CB, equals VB/pB, or -aobo/al. And the dimensionless constant k~ defined as PBVBis -ao/albo. The use of eqs 4 and 6 to determine the second and third virial coefficients will be illustrated for the case of oxygen. This data set was chosen because the results can be compared with those from the equation of state of WES, which was constructed expressly to evaluate the virial coefficients, and with which our equation of state agrees very well, as noted above. Table V shows B and C from eqs 4 and 6, together with the differences of these values from those of WES. The values of B agree within WES’s estimated uncertainty of 0.3 cm3/mol. But the differences in C considerably exceed WES’s estimated uncertainty of 20 ( ~ m ~ / m oand l ) ~are definitely systematic. This shows how tricky the evaluation of virial coefficients can be. Here we have two equations of state based on two different approaches, which in their region of overlap exhibit similar goodness of fit of the PVT data, as noted above. The main difference for the evaluation of virial coefficients is that the two eauations extraDolate differently to zero density. The resulk given by M S are based the assumption that i s o K ~ vemus lines, fitted at moderate densities to the equation K = B + cp,continue with the same slope to zero density. The results given here are based on the assumption that the i s o T ~ versus lines fitted at moderate densities to the equation a~ + bp 1, continue with the same slope to density. Neither assumption may be true, though both may be correctwithin uncertainty. However, it should be noted that the UC1 that is incorporated into the data set, namely, T = 406.3 - 8.33p, is notwell by the WES equation, in fad gives a nonlinear UCL. This is not important in the fit of 2,because the term p~ in 2 = 1+ p~ is near the UCL. But it is important in the extrapolation to zero density to find the virial coefficients. For example, if the UCL is linear,eq 5 requires that C at TB should be -dB/dT times the UCL slope, -8.33, which it is for eq 1but not for bo, as discussed in the next d o n , B and c fomd from eq 1represent that equation and contain the same information, namely, the equation of state for the entire linear-isoK region, whereas B and C found from eq 2, for example the WES equation, do not.
Equation of State for an Intermolecular Potential Earlier it was noted how all isochores in the linear-isd region could be interpolated and extrapolated from any
Table VI. Equation of State, Z(T*,p*),for the Exp-6,16 Intermolecular Potential Z ( T l , p * ) for given P* TI 0.1 0.15 0.2 0.25 0.35 0.45 0.05 0.3 0.4 3.0 1.0018 1.0053 2.9 LOO04 1.0025 1.0066 2.8 0.9988 0.9994 1.0020 1.0066 2.7 0.9971 0.9961 0.9970 0.9999 1.0051 2.6 0.9953 0.9925 0.9916 0.9927 0.9961 1.0018 1.0102 2.5 0.9933 0.9885 0.9857 0.9849 0.9863 0.9900 0.9963 1.0055 1.0180 2.4 0.9912 0.9842 0.9792 0.9763 0.9765 0.9771 0.9812 0.9881 0.9982 2.3 0.9888 0.9794 0.9721 0.9668 0.9637 0.9630 0.9647 0.9691 0.9766 2.2 0.9861 0.9742 0.9642 0.9664 0.9507 0.9474 0.9465 0.9483 0.9530 2.1 0.9832 0.9683 0.9555 0.9448 0.9363 0.9302 0.9264 0.9253 0.9270 2.0 0.9799 0.9618 0.9458 0.9319 0.9203 0.9110 0.9042 0.8999 0.8984 1.9 0.9762 0.9544 0.9348 0.9174 0.9023 0.8896 0.8793 0.8715 0.8665 1.8 0.9720 0.9461 0.9225 0,9011 0.8821 0.8654 0.8513 0.8397 0.8308 1.7 0.9672 0.9367 0.9084 0.8825 0.8590 0.8380 0.8195 0.8036 0.7904 1.6 0.9617 0.9258 0.8923 0.8612 0.8326 0.8066 0.7831 0.7623 0.7444 1.5 0.9553 0.9132 0.8735 0.8364 0.8020 0.7702 0.7411 0.7147 0.6912 1.4 0.9478 0.8983 0.8515 0.8073 0.7660 0.7274 0.6917 0.6588 0.6290 1.3 0.9389 0.8806 0.8251 0.7725 0.7229 0.6762 0.6326 0.5921 0.5547
0.5
0.55
0.6
0.65
1.0119 0.9874 0.9609 0 .9318 0.8999 0.8644 0.8248 0.7801 0.7292 0.6706 0.6021 0.5204
1.0023 0.9725 0.9401 0.9046 0.8654 0.8217 0.7727 0.7170 0.6530 0.5783 0.4895
0.9884 0.9523 0.9129 0.8698 0.8220 0.7685 0.7080 0.6386 0.5578 0.4618
1.0098 0.9692 0.9265 0.8781 0.8258 0.7677 0.7022 0.6273 0.6405 0.4376
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1666 two isochores. This can be thought of as the two-point form of the isoK lines. The extrapolation was extended to include the zero-density isochore and hence the virial coefficients. The reverse procedure is also possible; that is, all the isochores in this region, from zero density to the critical density, and hence the equation of state, can be found from the two virial coefficients, B and C. This is the slope-intercept form of the isoK lines, with the intercept temperature given by eq 4 and the slope by eq 5. One important consequence of this is that we now have a means of determining the equation of state in this region for any of the various intermolecular potentials for which the virial coefficients, B* and C*, have been computed as functions of temperature, P ,assuming that the potential is realistic enough to mimic the observed behavior of real gases. (m indicates reduction by the potential parameters e / k and a.) As an example, Table VI shows the equation of state for the exp-6,16 potential, for which values of B* and C* were reported by Sherwood and Prausnitz (1964a,b). The table lists values of 2 as a function of T* and p* in the linearisoK region. To outline this region, it was first necessary to estimate the values of the critical constants, Tc* and pc*. This potential has TB* = 2.935, pB* = 2.51, and k B = 2.13 (Holleran, 1969). A correlation of critical constants (Holleran et al., 1975) yields from these the approximate values: Tc* = 1.14 and pc* = 0.68 for this potential. Accordingly, Table VI lists values of 2 for P from 1.3 (1.14Tc*) to 3.0 and for p* up to 0.65. Holleran (1969) and Holleran and Hammes (1975) have shown how a three-parameter intermolecular potential can be selected for a given gas by its kB value, with its f/k and u evaluated from its T B and p B Now we can see how good such a potential is at actually reproducing PVT data in the linear isoK region. In this way, also, two different forms of potential with the same kB can be compared and perhaps differentiated by their goodness of fit of the data. And finally, data sets that lie in our region but do not include the UCL (and so do not provide reliable values of k ~ TB, , and PB) can still be used to select a potential and evaluate its parameters by a direct fitting of the experimental data. This will be undertaken shortly. Conclusions The goals set forth in the introduction, namely, to discover the quality, extent, and usefulness of the linearity of the isoK lines, have been achieved by this investigation. It has been shown that this linearity (a) is a property of many common gases, (b) covers a fairly wide range of pressure, temperature, and density, (c) is accurate within the experimental error of good data, and (d) has a number of uses, including the following. (1) It provides an accurate equation of state for this region, with the same form of all gases, explicitly incorporating the linearity. In principle, data on two isochores suffice to evaluate the coefficients of this equation, but of course more are better. (2) It allows reliable extrapolation by this equation from moderate densities up to the critical density. This is in contrast to the truncated virial equation fitted at the same moderate densities. (3) It also allows extrapolation from moderate to lower densities on the assumption of continued linearity. Extended to zero density, this provides values of the virial coefficients, which differ somewhat from those provided by a truncated virial equation fitted at the same moderate densities. (4) By the same assumption of isoK linearity, it provides the equation of state in this region for any intermolecular
potential for whith B * ( P ) and C * ( P )have been evaluated. This allows the testing of the suitability of a given potential for a given gas by the direct fitting of PVT data, Nomenclature a = coefficient of T in eq 1, K-' a, = coefficients of Kn in eq la, (mol/dm3)" K-' AAD = average absolute deviation b = coefficient of p in eq 1, dm3/mol b, = coefficients of K" in eq lb, (dm3/mol)'" B = second virial coefficient, d d / m o l B* = second virial coefficient for an intermolecular potential, reduced by u C = third virial coefficient, (dm3/mo1)2 C B = third virial coefficient at TB, (dm3/mol)2 C* = third virial coefficient for an intermolecular potential, reduced by u2 D = fourth virial coefficient, (dm3/mo1)3 K = (2- l)/p, dm3/mol k = Boltzmann constant, J K-' kB = PBVB,a dimensionless constant P = pressure, MPa R = gas constant, J K-' mol-' R2 = statistical correlation coefficient T = temperature, K TB = Boyle temperature, at which B = 0, K T, = critical temperature, K T* = temperature reduced by t/k UCL = unit compressibility line, along with 2 = 1 V = volume, dm3/mol VB = Boyle volume, T(dB/dT) at TB, dm3/mol 2 = compressibility factor, P pRT, dimensionless u = molecular volume, 27rN /3, dm3/mol Greek Symbols e = energy parameter of an intermolecular potential, J p = density, mol/dm3 pB = zero temperature intercept of the UCL, mol/dm3 pc = critical density, mol/dm3 p* = density with volume reduced by u u = molecular diameter parameter of an intermolecular potential, dm Registry No. Argon, 7440-37-1; methane, 74-82-8; fluorine, 7782-41-4; oxygen, 7782-44-7; nitrogen, 7727-37-9; carbon monoxide, 630-08-0; ethylene, 74-85-1; ethane, 74-84-0; propane, 7498-6; isobutane, 75-28-5; neopentane, 463-82-1.
d
Literature Cited Beattie, J. A.; Marple, S.; Edwards, D. G.T h e Compressibility of and an Equation of State for Gaseous hbutane. J. Chem. Phys. 1960, 18, 127-128.
Beattie, J. A,; Douslin, D. R.; Levine, S. W. The Compressibility of and an Equation of State for Gaseous Neopentane. J. Chem. Phys. 1962,20,1619-1623.
Ben-Amotz,D.; Herschbach, D. R. Correlation of %no (2 = 1) Line for SupercriticalFluida with Vapor-Liquid Rectilinear Diametem. Isr. J. Chem. 1990, 30, 59-68.
Douslin, D. R.; Harrison, R. H. Pressure, Volume, Temperature Relations of Ethane. J. Chem. Thermodyn. 1973, 6, 491-612. Douelin, D. R.; Harrison, R. H. Pressure, Volume, Temperatwe Relations of Ethylene. J. Chem. Thermodyn. 1076,8,301-830. Douslin, D. R.; Harrieon, R. H.; Moore, R. T.; McCullough,J. P. PVT Relations for Methane. J. Chem. Eng. Data 1964, 9, 368-363.
Duschek, W.; Kleinrahm, R.;Wagner, W.; Jaeechke, M.Measursment and Correlation of the (Pressure, Density, Temperature) Relation of Nitrogen in the Temperature Range from 273.16 K to 323.15 K at Pressures UD to 8 MPA. J.Chem. Thermodvn. 1988, 20, 1069-1077.
Goodwin. R. D. Carbon Monoxide ThermoDhveical Prowrtiw from 68 to lo00 K at Pressures to 100 MPa. j.k y s . Chem. Ref. Data 1986,14,849-932.
Holleran, E. M. Linear Relation of Temperature and Density at Unit Compressibility Factor. J. Chem. Phys. 1067,47, 6318-6924.
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1666
Holleran, E. M. A Dimensionless Constant Characteristic of Gases, Equations of State, and Intermolecular Potentials. J. Phys. Chem. 1969, 73, 167-173. Holleran, E. M. Accurate Virial Coefficients from PVT Data. J. Chem. Thermodyn. 1970,2, 779-786. Holleran, E. M. The Overall Unit CompressibilityLines for Real and Simulated Fluids. Znd. Chem. Eng. Res. 1990, 29, 632-636. Holleran, E. M.; Hammes, J. P. A Three-Parameter Equation of State for Gases. Cryogenics (Feb) 1975,95-102. Holleran, E. M.; Walker, R. E.; Ramos, C. M. A Correlation of Critical Points. Cryogenics (April) 1975, 210-216. Kleinrahm, R.; Duschek, W.; Wagner, W.; Jaeschke, M. Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Methane in the Temperature Range from 273.15 K to 323.15 K at Pressures up to 8 MPa. J.Chem. Thermodyn. 1988, 20,621-631. Levelt, J. M. H. The Reduced Equation of State, Internal Energy and Entropy of Argon and Xenon. Physica 1960,26, 361-377. Michels, A.; van Straaten, W.; Dawson, J. Isotherms and Thermodynamical Functions of Ethane at Temperatures Between 0 OC and 150 OC and Pressures up to 200 Atm. Physica 1954, 20, 17-23. Morsy, T. Ideal Curves. Dissertation Technische Hochschule Karlsruhe, 1963.
Powles, J. G. The Boyle Line. J. Phys. C Solid State Phys. 1988, 16,503-514. Prydz, R.; Straty, G. C. PVT Measurements, V i Coefficients, and Joule-Thomson Inversion Curve of Fluorine. J. Res. Natl. Bur. Stand. 1970, 74A, 747-760. Schmidt, R.; Wagner, W. A New Form of the Equation of State for Pure Substances and Ita Application to Oxygen. Fluid Phaee Equilib. 1985, 19, 175-200. Sherwood, A. E.; Prausnitz, J. M. Third Virial Coefficient for the Kihara, Exp-6, and Square-Well potentials. J. Chem. Phys. 1964a, 41,413-428. Sherwood, A. E.; Prausnitz, J. M. Intermolecular Potential Functions and the Second and Third Virial Coefficients. J. Chem. Phys. 1964b, 41, 429-437. Thomas, R. H. P.; Harrison, R. H. Pressure-VolumeTemperature Relations of Propane. J. Chem. Eng. Data 1982,27, 1-11. Wagner, W.; Ewers, J.; Schmidt, R. An Equation of State for Oxygen Vapour-Second and Third Virial Coefficients. Cryogenics (Jan) 1984, 175-200. Younglove, B. A. Thermophysical Properties of Fluids. J. Phys. Chem. Ref. Data 1982, 11 (Suppl 11, 1-347.
Receiued for reuiew October 15, 1990 Accepted February 4, 1991
RESEARCH NOTES Prediction of the McAllister Model Parameters from Pure Component Properties for Liquid Binary n -Alkane Systems A new method for predicting the McAllister viscosity model parameters from pure component properties for binary n-alkane liquid systems is reported. The resulta of this method are compared with experimental data.
Introduction Solution of many engineering problems requires the knowledge of the dependence of kinematic viscosities on composition. Moreover, Viscosities of liquid mixtures help in elucidating the fundamental behavior of liquid systems. However, a general and reliable theory for predicting the kinematic viscosities of liquid mixtures from pure component properties is not available yet. Consequently, information on the dependence of viscosity on composition continues to depend on costly and time-consuming experimental measurements. Several models (McAllister, 1960; Auslander, 1964; Heric, 1966; Wei and Rowley, 1984, 1985; etc.) for the prediction of the dependence of viscosities of liquid mixtures on composition have been reported in the literature. McAllister’s model is based on Eyring’s absolute rate theory assuming three-body or four-body interactions. For three-body interactions, the equation reported by McAllister is In b’ = XA3 In b’A + 3xA2xB In b’m+ 3xAxB2 In b’BA + xB3 In b’B - In [xA + x$MB/MA] + 3xA2xB In [(2 + MB/MA)/3] + 3xAxB2 In [(I + 2MB/MA)/3] + xB3 [MB/MAI (1) where XA and xg are the mole fractions of components A and B, respectively, MAand MB are their respective molecular weights, and U A , uB, and Y are the kinematic viscosities of the pure components and the liquid mixture,
respectively. The model given by (1) contains two adjustable parameters um and YBA. These adjustable parameters are determined by fitting experimental kinematic viscosity-composition data to (1). The McAllister four-body interaction model is In Y = x A 4 In VA + 4xA3xBIn b’m+ ~ X A ~ XhB b ~ ’m~ 4xAxB3 In b’BBBA + xB4 ln YB ln [xA + x$MB/MA] + 4xA3xB In [(3 + MB/MA)/4] + ~ ~ A In ~ [(I X + B MB/MA)/2] ~ + 4XAxB3 In [(I + 3MB/MA)/4] + xB4 In (k?B/MA) (2) This contains three adjustable parameters, v u , vmB, and VBBBA, which again are determined from kinematic viscosity-composition data. The major drawback of both of the McAllister models is the presence of the adjustable parameters. This is bec a w the determination of these parameters requires costly experimental data. Therefore, the development of a technique for predicting the values of the McAuister model parameters from pure component properties would be a significant improvement. The unsatisfactory state of the art with respect to the structure of liquids led one of the present authors to break liquid solutions into three classes, viz., regular solutions, n-alkane solutions, and associated solutions (Asfour, 1980). Such a classification led to success in tackling molecular diffusion problems in liquids, for example (Asfour, 1985; Dullien and Asfour, 1985; Asfour and Dullien, 1986). Such
0888-5885/91/2630-1666$02.50/0 0 1991 American Chemical Society
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