Ind. Eng. Chem. Res. 1993,32,3143-3145
3143
Empirical Representation of the Sudden Change in the PVT Behavior of Compressed Fluids Eugene M. Holleran Chemistry Department, St. John’s University, Jamaica, New York 11439
The recently reported fairly sudden change in the PVT behavior of fluids near twice the critical density is given an empirical analytical representation by requiring the three reducing parameters in the reduced equation of state to undergo a continuous but rapid change from their constant lower-density values to their constant higher-density values over a relatively short density range. The resulting equation of state then agrees with the published MBWR (modified Benedict-WebbRubin) equations for argon, oxygen, and ethane with an average absolute difference in density of less than 0.01 % The striking changeover in PVT behavior which this new formulation represents explicitly is therefore implicit in the MBWR equations for these fluids.
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Table I. Constants and Coefficients
Introduction
ethane 474 120-600 5-70 7-21 761.79 25.257 0.072 333 790.5 24.63 0.069 418 2.8 14.86 14.719 14.84 1.421 484 0 -0.694 280 0.272 796 5.868 818 -8.841 139 2.129 165 -6.507 504 -0.708 951 0 -0.714 793 0 8.453 246 2.020 042 0.597 100 6.867 970 -8.424 471 -1.203 931 -0.160 565
In a recent paper (Holleran, 1992) it was shown that the PVT behavior of dense fluids undergoes a fairly rapid change from that of an extensive range below a dividing density (the L range) to that of an extensive range above it (the H range). The changeover occurswithin a relatively narrow boundary region near twice the critical density (the B range). The representation of this behavior given in that paper made use of two different sets of three reducing constants in a single reduced equation of state, one set for the L region and another for the H region. This then described the PVT behavior within experimental error in those two regions, but not in the boundary region. The purpose of the present paper is to provide an accurate representation that includes the boundary region. This is done by giving the three reducing parameters an empirical form according to which they change rapidly but continuously from their L to their H values as the density increases through the narrow range of change. The Equation of State The equation of state used in this work is
P = pRT(l+ p K ) (1) with K given in reduced form as K = K/Ko, and ~ ( 6 , sgiven ) by
+
K = (6 6 - l)/W(6,6) (2) where w(6,6) is a function of the reduced temperature, 6 = T/To, and reduced density, 6 = p/po. The quantity w is taken to be the s u m of terms of the form u(n,m)Onbrn. The coefficients, u(n,m), found for argon, ethane, and oxygen are listed in Table I. The three reducing parameters (TO, PO,KO)have constant values in the extensive L range of density, and different constant values in the extensive H range. In the B range they are made to change rapidly from their L to their H values as p increases. Thus, TOis given by
To = [To(H) + F,T&)I/[l+
FTI
(3)
where exp[C(CT - ~ 1 1 (4) in which C is a constant that determines the width of the B range. The other two reducing parameters, PO and KO, are represented by equations similar to eqs 3 and 4, with the same value for C, but with the constant CT replaced
F,
argon 542 105-400 10-100 13-37.5 406.36 47.134 0.042 772 413.93 46.304 0.041 933 2 23.64 23.435 23.64 1.367 503 -0.231 364 -0.211 395 0.075 256 -0.778 068 2.348 408 -1.535 477 -1.861 585 1.647 299 1.245 371 -0.367 118 0 -3.429 794 -2.306 684 0.587 523 0 3.187 842 1.326 058 -0.306 321
oxygen 697 100-400 15-120 13-40 406.79 48.597 0.040 442 419.8 47.5 0.039 264 1.45 27.8 27.502 27.55 0.568 969 0.393 952 0 0.037 079 -1.944 604 6.713 280 -5.877 848 3.984 006 -2.879 031 0.362 633 -0.163 991 -14.530 059 7.638 081 -1.254 196 0.258 948 11.953 951 -5.763 554 1.147 907 -0.151 336
by C, or CK,and FTby F,or FK. The values of the constants for eqs 3 and 4 are included in Table I. The resulting rapid variation in the reducing parameters is illustrated in Figure 1for po versus p for ethane. Accuracy
As in the earlier work, published MBWR (modified Benedict-Webb-Rubin) equations of state were used to provide data sets for ethane (Youngblood, 19821, argon, and oxygen (Youngblood and Ely, 1987). Points were taken at the intersections of isotherms and isobars; this gave a convenient spacing in density. Densities were allowed to range from about pc to about 3pc to give a rough balance on both sides of B. Pressures were taken at intervals of 5 MPa up to the maximum covered by the MBWR equations, and temperatures were taken at 5,10, or 15 K intervals up to the MBWR limits. Some poorly
o a g a - 59312632-3143~04.0010 ~~ 0 1993 American Chemical Society
3144 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 25 3
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Density (mollL) Figure 1. Variation with density of the reducing parameter po for ethane, as given by the analog of eq 4 for use in the reduced equation of state. The other two reducing parameters, TOand KO,undergo similar sudden changes.
fitting points at the lowest pressures and densities were eliminated from the fringes of these data sets because, as explained in Holleran (1992),the purpose here was not to replace the MBWR equation of state, but to represent accuratelythe PVT surface of these fluids over an extensive enough range to demonstrate its strange behavior. For ethane, 474 points were used, for temperatures from 120 to 600 K, pressures from 5 to 70 MPa, and densities from 7 to 21 mol/dm3 (pc = 6.9 mol/dm3). For argon, 542 points were used, ranging in temperature from 105 to 400 K, in pressure from 10 to 100 MPa, and in density from 13 to 37.5 mol/dm3 (p, = 13.4 mol/dm3). For oxygen, 697 points were used for temperatures from 100 to 400 K, pressures from 15 to 120 MPa, and densities from 13 to 40 mol/dm3 (p, = 13.6 mol/dm3). The oxygen points were derived from the MBWR equation of Younglove (1982) because of its reportedly more extensive coverage of T , p, and P than the equation of Schmidt and Wagner (1985). Figure 2 shows the ethane set in terms of K versus p with the density ranges L, B, and H indicated. The constants and coefficients for eqs 1-4 are listed in Table I. They were evaluated for the three fluids as described below. Then the accuracy of these equations in representing the data was confirmed as follows. For each of the 1317 T,P points, a value of p was found by iteration and compared with the MBWR value. The results were excellent. For all three ranges (L, B, H), together and separately, the average absolute deviation (AAD)in p was less than 0.01% for all three fluids. The maximum deviations were 0.03 or 0.04% in L and H, and 0.02 or 0.03% in B.
Evaluating the Constants The values of the constants and coefficients for eqs 2-4 were determined using for each fluid the data sets mentioned above, in conjunction with a set of points calculated along the unit compressibilityline (UCL),which is the locus of points at which the compressibility factor, 2,defined as PIpRT, equals unity, and K, defined as (2 - l)/p, equals zero. The UCL points are referred to below as the K = 0 set and the large data sets as the all-K sets. The following five-step procedure was used to evaluate the constants and coefficients.
6
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re
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p (moliL)
Figure 2. All-K data set for ethane. The approximate width of the B range is indicated. The 70-MPa isobar is at the top; the 40-MPa isobar is roughly tangent to the UCL ( K = 0). The upper left edge is the 600 K isotherm, and the 130K isotherm forms most of the right edge. The critical density is 6.9 mol/L.
1. This first step determines the constant values of TO and po in H and L, and also estimates the width of the B range. Either data set can be used. With the all-K set, T is fitted by multiple least squares to
T = co + clp + KW (5) in which W is the sum of terms of the form c(n,rn)T"p" with the same powers as for w in eq 2. The separate L and H fits are extended not to a common dividing density as in Holleran (1992) but to the edge of a B range which is left as a gap. TOis given by CO, and PO by -CO/CI. The values of the constants depend slightly on the range used, and so differ somewhat from those in earlier work. This, and an increase in the number of terms in W, yielded a slight improvement in the fit of eq 5. Essentially the same constants can be found using the K = 0 data and minimizing the average absolute value of (0 + 6 - 1)in the L and H regions. The B range is taken to be that in which lo4@+ 6 - 1)dips below -3, reaching a minimum in the middle of B if the L constants are used below this midpoint and the H constants above it. 2. The values of C, CT,and C, are found using the K = 0 set. First a trial value of C is selected, and the other two constants are adjusted by trial and error to minimize the average absolute value of (0 + 6 - 1) in the B range. For large values of the constant C (say 20 or 30 dm3/mol), the changeover from the L to the H values of TOand po is nearly discontinuous, and the 0 + 6 - 1values in B are large. As the selected value of C is made smaller, the changeover is more gradual, and the fit in B improves. The value of C was lowered until the fit in B was as good as it was in L and H. For all three fluids this yielded for 104(8+ 6 - 1)an average absolute value of 1 in all three ranges and a maximum of 2 in B and 3 in L and H. If C is made too small, the values of the reducing parameters in the L and H regions begin to be affected, and the fits there worsen. CTand C, are necessarily near the midpoint of B, but C, always has to be slightly (about 1% ) smaller than CT. 3. The ratio Ko(H)/Ko(L) is found using the H and L ranges of the all-Kset simultaneously. For each of several trial values of the ratio, a least-squares fit of eq 6 was carried out:
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3146
e + 6 - 1 = KRW
(6)
Here W is the sum of terms of the form b(n,m)Onbmwith the same powers as in eqs 2 and 5, and KR is taken as K in L and K times the KOratio in H. The best value of the Koratioyielded u(e + 6) < 1X 10-4,R2= 0.999 997 or better, and a temporary set of b,, coefficients. 4. The constant CK is found using the B range of the all-K set. For each T,P point in B, the value of p given by the new equation of state with the temporary b(n,m) was found by iteration using a trial value of CK. Then CK was varied until the average absolute deviation (AAD) in density was minimized. 5. The final values of the Coefficients, b(n,rn),were found using all the points in the all-K set in a least-squares fit of eq 6 with KRnow varying according to the analog of eq 3. This gave the same excellent R2and standard deviation as in step 3. Ko(L) was then set equal to l/[b(1,0) + b(2,O) + b(-1,0) + b(-2,0)1, and Ko(H) was set toKo(L) times the KOratio. Finally, the coefficients for eq 2 were found as u ( R , ~= ) b(n,m)Ko(L).
Discussion The goal of this work was achieved by means of eqs 3 and 4 and their analogs, which provide a rapid variation with density for the three reducing parameters, TO,PO, and KO,as illustrated in Figure 1,which clearly shows the sharpness of the change. This then allows the single reduced equation of state, eq 2, to follow the changeover in the PVT behavior in the B range with as great an accuracy there as it has in the L and H ranges (AAD = 0.01 % ). Because eq 2 so exactly reproduces the MBWR equation, the agreement of the two equations with experiment in these regions is essentially identical. They both incorporate the same strange behavior, but it is explicit and thus much more obvious in this new formulation. It is also significant that the MBWR equation does not require any particular form for the UCL. This indicates that the striking pattern of behavior of its UCL, and its PVT surface in general, is a property of these fluids themselves and not a quirk or anomaly of the MBWR equation. In fact, the tendency of any such multiconstant equation of state when fitted to the experimental data would be not to exaggerate but to slightly minimize by smoothing any sudden change. Therefore the change must in reality be at least as abrupt as it appears. It should also be noted that the density range, B, over which the change in behavior occurs appears to be independent of the temperature. The three fluids discussed here provide the best illustrations of this change. Other fluids, such as methane, show a more gradual changeover between the two linear segments of their MBWR UCL's (Holleran, 1990).
Nomenclature coefficients of On6" in the expression for w(e,6) in eq 2 AAD = average absolute deviation b(n,m) = coefficients of in the expression for W(O,6)in eq 6 B = the relatively narrow range of density in which the reducing constants, TO,PO, and KO,change from their lower density (L) values to their higher density (H)values according to eqs 3 and 4 and their analogs co = empirical constant in eq 5 , equal to TO,K c1 = empirical constant in eq 5, equal to -To/po, K dm3/mol u ( R , ~= )
c(n,m) = coefficients of Ppmin the expression for W(T,p)in
eq 5, K1-n (dm3/mol)'-m C = empirical constant in eq 4 and its analogs, dm3/mol CK,CT,C, = empirical constants in eq 4 and its analogs, mol/ dm3 FK,FT,Fp = exponentialfunctions defined by eq 4 and used in eq 3 to describe the sudden change in KO,TO,and PO H = designation for the density range above B, roughly from 2Pc to 3Pc K = function of T and p defined by eq 1 and equal to zero on the UCL, dm3/mol KO= reducing parameter for K, constant in the H and L regions, dm3/mol Ko(H),Ko(L) = constant values of KOin H and L, dm3/mol KR Ko(H)/Ko(L) L = designation for the density range below B, roughly from Pc to 2Pc m = exponents of p or 6 in the expressions for W or w in eqs 2, 5 , and 6 R = exponents of T o r 0 in the expressions for W or w in eqs 2, 5 , and 6 P = pressure, MPa R = gas constant, J K-1 mol-' R2 = statistical correlation coefficient T = temperature, K TO= reducing parameter for T, K To(H),To(L) = constant values of TOin H and L; zero-density intercepts of the linear segments of the T vs p UCL, K UCL = unit compressibility line, along which K = 0 V = volume, dms/mol w = in eq 2, empirical polynomial with terms of the form u(n,m)enP
W = in eq 6, empirical polynomial with terms of the form b(n,m)e%m, dimensionless, and in eq 5 with terms of the form c(n,m)T"pm, K.mol/dm3 Greek Symbols
6 = reduced density, 6 = p / p o K
= reduced K, K = K/Ko
0 = reduced temperature, 0 = T/To p = density, mol/dm3 PO = reducing parameter po(H), po(L) = constant
for p , mol/dm3 values of PO in H and L; zero-T intercepts of the linear segments of the T vs p UCL, mol/ dm3 p c = critical density, mol/dm3 a = standard deviation
Literature Cited Holleran, E. M. The OverallUnit CompressibilityLines for Real and Simulated Fluids. Znd. Eng. Chem. Res. 1990,29,632-636. Holleran, E. M. An Abrupt Change in the PVT Behavior of Fluids at Densities near Twice the Critical Density. J. Phys. Chern. 1992, 96,8568-8571. Schmidt, R.; Wagner, W. A. A New Form of the Equation of State for Pure Substances and Its Application to Oxygen. Fluid Phase Equilib. 1985,19, 175-200. Younglove,B. A. ThermophysicalProperties of Fluids.J. Phys. Chem. Ref. Data 1982, 11 (Suppl. l), 1-347. Younglove, B. A,; Ely, J. F. Thermophysical Properties of Fluids. J. Phys. Chem. Ref. Data 1987,16, 577-798. Received for review May 28, 1993. Abstract published in Aduance ACS Abstracts, November 1, 1993. @
