Subscripts = evaluated at the wall
ze
literature Cited
(1) Bogur. D. C., Ph.D. ‘thesis, Univ. of Delaware, Newark, Del., 1 wx
(2) Brodnyan, J. G.: Gaskins, F. H., Philippoff, LV., Trans. SOG. Rheol. 1. 109 (1957). (3) BrodniTan, J. G.; Gaskins, F. H., Philippoff, W., Lendrat, E. G., Ibbid., 2, 285 (1958). (4) Dodge, D. W., Ind. Eng. Chem. 51, 839 (1959). (5) Dodge. D. \V., Ph.D. thesis, Univ. of Delaware, Newark, Del.,
1957. (6) Dodge, D. W., Metzner, A. B., A.I.Ch.E. J . 5, 189 (1959). (7) Fabula, A. G., unclassified Naval Ordnance Rept., 1961. (8) Gfanville, P. S., U. S. Navy Dept., David Taylor Model Basin Rept., 1961. (9) Hinze, J. O., “Turbulence,” McGraw-Hill, New York, 1959. (10) Houghton, LV. T.: M.Ch.E. thesis, Univ. of Delaware, Newark, Drl., 1961. (11) Hurd, R. E., B.Ch.E. thesis: Univ. of Delaware, Newark, Del., 1962. (12) Knudsen, J. G., K,itz, D. L., “Fluid Dynamics and Heat Transfer,” McGraw-Hill, New York, 1958. (13) Kotaka, T., Kurata, M., Tamura, M., J . Appl. Phys. 30, 1703 (1959) ; Bull. Chenz. SOC. Japan 32, 471 (1959). (14) Laufer, J., Natl. Advisory Comm. Aeronautics Rept. 1174, 1954. (15) Lummus, J . L., Fox:. J. E., Jr., Anderson, D. B., Oil Gas J . 59, 87 (Drc. 11, 1961).
(36) Metzner, A. B., in “Handbook of Fluid Dynamics,” V. L. Streeter, ed., McGraw-Hill, New York, 1961. (17) Metzner, .4.B., Reed, J. C., A.I.CI1.E. J . 1, 434 (1955). (18) Miller, B., T r a m . ASME 71, 357-67 (1949). (19) Millikan, C. B.: “Proc. Fifth Intern. Congr. Appl. Mech..“ 386-92, [Viley, Piew York, 1939. (20) Nikuradse, J., Forsch. Gebiete Ingenieurw. .\us. B., Bd. 3, 1-36 (1932). (21) Ousterhout, R . S., Hall, C. D., SOC.Petrol. Engrs. Meeting. Denver. Colo.. October 1960. (22) Pai,’S. I.? “Viscous Flow Theory,” Vol. 11, Van Nostrand. New York, 1957; J . Franklin Inst. 256, No. 4, 337 (1953): J . Appl. ,%fech. 20, 109 (1953). (23) Randall, B. V., Pan .4merican Petroleum Corp., ~. Tulsa, Okla.. private communication, 1960. (24) Reichardt, H., Z. angeuj. Math. und Mech. 31, No. 7, 208 (1951). (25) Sailor; R. .4., Univ. of Delaware. Ne\rark, Del., unpublished work, 1960. (26) Schlichting, H., “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960. (27) Shaver, R. G.?Merrill, E. W., A.Z.Ch.E. J . 5, 181-8 (1959). (28) Thomas, D. G., in “Progr. in Intern. Research on Thermodynamic and Transport Properties,“ p. 669, Am. SOC.Mech. Engrs. and Academic Press, New York, 1962. (29) Wilkinson, W. L., “Non-Newtonian Fluids,” pp. 59, 62, Prrgamon Press, New York, 1960. ’
RECEIVED for review August 2, 1962 ACCEPTED February 27, 1963 Work supported by Ethyl Corp. and National Science Foundation fellowships.
ESTIMATES OF SATURATED FLUID DENSITIES AND CRITICAL CONSTANTS STANLEY
H. F I S H T I N E
20 Sutton St., Mattapan 26, Mass.
The existing literature has no accurate generalized method for determining the critical properties of both organic and inorganic substances.
The method discussed and evaluated in this paper utilizes and relates
four existing empirical correlations to obtain the desired result. The method i s quite accurate for critical temperatures and critical volumes, though only fair for critical pressures. The procedure is particularly accurate for rionpolar substances, but is not recommended for nitriles, metallic elements, or for compounds which associate in the vapor state. For 32 substances studied, the average errors in T,, V,, and P, are 1.6, 2.2, and 7.376, respectively. N e w equations for calculating densities of saturated liquids and vapors are also presented. Good accuracy is obtained for both vapor and liquid phases over a wide temperature range extending all the way to the critical point.
D
of saturated liquids and saturated vapors above atmospheric pressure are often required in chtmical piocess design. Except for the refrigerants, low molecular ueight hydrocarbons, and some heating mediums-such as steam and the Dowtherms-high pressure fluid densities are not commonly found in the literatuie. \I hen the critical temperature, critical pressure, critical volume, and a low pressure liquid density are known, vapor and liquid densities may be accurately determined a t any saturation point from the several generalized charts and tables of Lydersen. Greenkorn and Hougen (74, 2 3 ) . T h e principal disadvantage of the corresponding state approach used by Lydersen and coworkers is the necessity of having large-scale charts available for calculations in the vicinity of the critical point. Based on the earlier work of Lydersen ( I d ) , Hanson (72), and Watson ( d o ) , TA (27) recently developed a n empirical EYSITIES
relationship Irhich enabled him to condense the generalized properties of liquids into a single chart. Although Lu’s chart alloivs one to estimate the densities of both saturated and pressurized liquids over a wide temperature range, the method rapidly loses accuracy near the critical point. To overcome these difficulties. four new equations have been derived to alloLt accurate Lalculations of saturated fluid densities over the entire range of tv o-phase equilibrium. Densities in the Moderate Pressure Range
Density equations for the range below a reduced temperature of 0.85 may be derived from the Haggenmacher equation of state ( 7 7 ) .
VOL. 2
NO. 2 M A Y 1 9 6 3
149
and from the Goldhammer correlation (8).
When the reference state for Equation 2 is selected a t or below the normal boiling point, p g is very small compared with p L and may be neglected. Thus Equation 2 may be rewritten,
pressures above the normal boiling point may be estimated with surprisingly good accuracy by drawing a straight line between the critical point and the atmospheric boiling point on a log P us. 1/T plot. For more accurate estimates of vapor pressure above to use the reduced state correlation proposed by Riedel (30). Densities in the High Pressure Range
(3)
The prime is added to the density and temperature in the denominator to denote a low pressure reference state for which the liquid density is known. An earlier work by the author (3) revealed that the exponent of the Goldhammer equation is not the same for all substances, but actually varies from 0.237 to 0.343 for 26 liquids studied. Recommended values of the exponent are given in Table I. Equation 3 may now be written in the more general form,
Densities of saturated liquids and vapors in the range between T , = 0.85 and the critical point may be determined by using a two-step approach: First calculate p g and p L a t a reduced temperature of 0.85, using Equations 6 and 7. Then use the law of rectilinear diameters to calculate p o and p L at the desired saturation temperature. The law of rectilinear diameters states that the sum of the densities of a pure liquid and its saturated vapor is a linear function of temperature. Mathematically this can be expressed as follows:
D (4)
Equation 4 is very accurate up to a reduced temperature equal to 0.85. Above this temperature the exponent n increases in value and no longer remains constant for a given substance. Rewriting the Haggenmacher equation in terms of densities instead of specific volumes, we obtain
pu
+
where ( p o
PL =
+
[(PO
+
4 G Av PL'
[ +
PL'G 2 (1
4
+ 11
Recommended Exponents for the Goldhammer Equation Compound Group n
Alcohols and water Hydrocarbons and ethers All other organic compounds All inorganic compounds except water
150
I&EC FUNDAMENTALS
(8)
sum of fluid densities calculated a t Ti= 0.85 The desired expressions for p Q and p t may be derived from an algebraic solution of Equations 5 and 8. p L ) ~ r= 0.85 =
I/po
-
+
I/PL PL =
AU
=
D
(5) (8)
Thus. for saturated vapors in the temperature range T , = 0.85to T , = 1.0.
+ 2
1 - ! [D?+ 4 1"' 2
Av
+
Now what are the limitations of Equations 6 and 7? First of all, they should not be used to calculate fluid densities above T , = 0.85 because of the changing value of n above this temperature. Second, Equations 6 and 7 may not be used to calculate the fluid densities of substances having variable vapor molecular weights (the carboxylic acids, sulfur, phosphorus, nitrogen dioxide, etc.). Third, values of the critical pressure and critical temperature must be fairly accurate for density calculations close to T , = 0.85. Equations 6 and 7 are rarely limited in use to substances for which experimental values of T , and P, exist. The critical constants of a large number of organic compounds may be estimated rapidly and with good accuracy by means of the Lydersen group contribution method (6, 22, 35). The calculation method described later in this paper may be used to determine the critical constants for a great many inorganic and organic compounds. Stull (26, 36) lists vapor pressure data above 1 atm. for approximately 100 substances. Where no data exist, vapor Table I.
pc
(AU)'
(9)
and for saturated liquids in the same temperature range. D l l I 2 P L = 2- - Av [D'
and for saturated liquids. PL =
(T - T) T , -I-
~ J 0.15
2
pB
Equations 4 and 5 are then solved algebraically to yield the desired expressions for p B and p L . For saturated vapors,
-2P
PL)T? = 0.85
0.25 0.29 0.31 0.333
+ (&]
For density calculations close to the critical point, the input necessarily be very accurate. data-Tc. Pc,p c and P-must For calculations above T , = 0.90, reliable experimental values of the critical temperature and the critical pressure must be available. Estimated values of T , and P, are definitely not recommended. However. in most cases estimated values of p c can be used without decreasing accuracy significantly. Equations 6, 7, 9, and 10 have been tested for six compounds a t 24 data points ranging from T , = 0.75 to the critical point. The method of this paper has also been compared with that of Lydersen, Greenkorn. and Hougen. The results are shown in Table 11. For the substances studied Lydersen's method is more accurate for the vapor phase and less accurate for the liquid phase. Generalized Method for Calculating Critical Constants of Organic and Inorganic Substances
During the last 20 years, a large number of excellent correlations have appeared in the technical literature for estimating the critical constants of organic compounds and the elements. T,, P,, and V, are often used in many engineering correlations for estimating other properties such as liquid and vapor density, gas viscosity, thermal conductivity, surface tension, vapor pressure. and the latent heat of vaporization a t high temperatures and pressures. The best over-all calculation systems for organic compounds appear to be the group contribution methods of Lydersen (22) and Riedel (28-30). A third group contribution method, proposed by Eduljee, has been highly recommended by Gambill (5. 6). However, all three methods may
Table 11.
Comparison between Calculated and Experimental Saturated Fluid Densities
Reference State PL‘:
1b.l ft. 42.2
-20
Dichlorodifluoromethane (Freon 1 2 ) Ethane
92.6
-20
33.0
-100
Methanol
49.4
68
Sulfur dioxide
92.4
0
Compound
Ammonia
Water
CU.
62.4
t’,
’ F.
60
Calculated by Method of Equations 6, 7, 9, and 10
Literature
t, F. 80 120 80 130 -50 0 40 60 80 230 320 400 420 440 120 200 250 280 300 420 530 600 650 690
Tr 0.740 0.795 0.780 0.852
l?) ft.
CU.
0.511 0.955 2.36 4.81 0 . 7 4 5 0.738 0.837 1.74 0.910 3.26 0.946 4.62 0.982 7.09 0.747 0.336 0.845 1.25 0.932 3.52 0.954 4.71 0.975 6.63 0.749 0.134 0.851 4.06 0.916 7.75 0.955 11.8 0.981 1 6 . 7 0.755 0,665 0.850 1.96 0.911 3.75 0.954 6.20 0.987 1 0 . 5
Average error
be very inaccurate for high molecular \\eight compounds or for compounds containing more than one highly polar substituent group. Thodos (37) has developed a remarkably accurate procedure for predicting the critical temperature and critical pressure of satiirated and unsaturated aliphatic hydrocarbons. Gates and Thodos (7) have proposed a very accurate equation for predicting the critical temperatures of the elements. Their equation is based on 17 experimental values of T , for the nonmetals, and the!; suggest that the correlation is applicable to the metals as well. However. since there is no reliable experimental data to back up this supposition, extrapolation of the Gates and Thodos equation for the metallic elements may be a very risky procedure. The author knows OF no satisfactory method of estimating the critical properties of inorganic compounds. A calculation method ivill be presented which is generalized in scope-that is, it is applicable to organic compounds, inorganic compounds, and the elements. Development of t h e Method. We have just seen how saturated fluid densities may be accurately calculated when the critical constants are known. We know that many relationships between other physical properties and the critical constants already exist. Logically, then, the reverse may be possible-that T,, P,, and V, may be determined accurately from easily measured physical properties. The calculation method described below involves the simultaneous solution of four equations. The method is not a t all complicated, the actual computations involve successive approximations-no more than three trials are requiredand the calculations generally take no longer than 1 hour to complete. The only daia required are the latent heat of vaporization a t the normal boiling point (&) and a liquid density ( p L ’ ) a t or below the normal boiling point. The first equation considered is the Lydersen equation for 2, (ZZ), the compressibility factor a t the critical point.
l:.? cu,ft. 37.5 35.3 81.3 74.5 30.8 28.0 25.1 22.9 19.7 43.8 39.5 33.8 31.4 28.6 81.0 71.6 63.5 56.6 50.0 51.5 47.2 42.4 37.4 30.5
lbqy ft. 0,503 0,943 2.49 5.11 0,740 1.82 3.47 4.86 7.61 0.324 1.17 3.38 4.49 6.50 0,135 4.15 7.93 12.1 17.2 0.661 1.96 3.83 6.02 11.1
CU.
Error,
%
-1.5 -1.2 4-5.5 4-6.2 +0.3 +4.4 4-6.4 +5.2 4-7.3 -3.6 -6.2 -4.0 -4.7 -2.0 +0.7 f2.2 +2.3 f2.5 4-3.0 -0.6 0.0 +2.1 -2.9 4-5.7 3.3
PL,
1b.l ~ ~ . f 36.2 34.0 81.8 75.2 31.0 28.4 25.9 22.8 19.3 43.5 39.4 32.8 30.4 28.0 80.0 70.3 62.6 55.9 49.2 51.5 47.2 41.6 36.7 29.5
Lydersen, et al. PL,
1b.l ft. 37.4 0.495 -3.1 36.4 0.923 - 3 . 4 2.43 f 3 . 0 80.8 f 0 . 8 74.0 +0.9 4.85 +0.6 0,730 - 1 . 1 30.3 f1.4 1.72 -1.1 27.4 f3.2 3.26 0 . 0 24.0 -0.4 4.41 -4.6 22.4 -2.0 6.95 -2.0 19.8 -0.7 0.339 + 0 . 9 4 2 . 6 -0.3 1.26 f0.8 37.8 -3.0 3.42 -2.8 33.0 -3.3 4.57 -3.0 30.9 -2.1 6.51 -1.8 28.0 -1.2 0.133 -0.7 81.1 -1.8 4.03 -0.7 72.5 -1.4 7.48 - 3 . 5 64.2 -1.2 11.4 -3.4 57.7 -1.6 16.4 -1.8 51.2 0.0 0.668 f 0 . 4 49.2 0.0 1 . 9 7 + 0 . 5 43.7 -1.9 3.75 0 . 0 39.9 -1.9 5.97 -3.7 35.5 -3.3 10.7 f1.9 29.1 1.7 1.8 l?) Error, ft. 90
Error, t .% -3.5 -3.7 +0.6
CU.
Error,
CU.
2, = P,VJRT, = 1/(3.43 f 0.0067 Lob2)
%,
-0.3 -2.6 -0.6 -1.1 -1.6 -2.1 -4.4 -2.2 +0.5 -2.7 -4.3 -2.4 -1.6 -2.1 +O.l 1-1.3 4-1.1
4-1.9 +2.4 -2.5 -7.4 -5.9 -5.1 -4.0 2.5
(11 )
The units of LLbhere are kcal. per gram mole. Lydersen developed this equation specifically for organic compounds, but it is also quite accurate for a great many inorganic compounds as well. Except for the nitriles and the carboxylic acids, the average deviation of Equation 11 for 129 organic compounds is reported by Lydersen to be 3.1% and the ‘.9jY, reliability” 9.0%. The second leg of the calculation method is the Benson equation ( I ) , which relates the reduced liquid density a t the normal boiling point to the critical pressure. PTb = P L b / p c =
1.981 f 0.422 log P c
(12)
The units of P, are atmospheres absolute. Equation 12 may be written in terms of molar volumes (V). V , = V~b(l.981
0.422 log P,)
(13)
Equation 13 holds within 4 ~ 3 % for associated as well as nonpolar liquids. I n most cases chemical or engineering handbooks do not contain liquid densities (or specific volumes) a t the atmospheric boiling point. However, VLbmay be calculated very accurately by means of the modified Goldhammer equation (Equation 4). Neglecting p g and writing the equation in terms of molar volumes for consistency, we obtain
where VL’ = M / p L ‘ . The final link in the calculation method is the Klein equation (78), which relates the molal latent heat of vaporization a t the normal boiling point to the critical temperature and the critical pressure.
Here again the units of P, are atmospheres. VOL. 2
NO. 2
The units of
MAY 1963
151
L z b are calories per gram mole when T , and T, are in
K., and B.t.u./lb. mole when the temperatures are in O R . Sama (33) found that the Klein equation gives more accurate results when multiplied by a correction factor (ITIcL) equal to 1.04. Recent studies by the author indicate that K,, is a function of the normal boiling point. Sama's correction factor should be used for substances boiling above 200" K . Below this temperature K K L= 1.02. Equation 15 may be written in a more useful form, and with the K,c factor introduced,
At low pressure the Haggenmacher square-root term is very nearly identical to the compressibi1it)- factor of the saturated vapor. This value is equal to 0.95 z 0.02 for most substances a t the normal boiling point. Equations 11, 13, 14: and 16 may now be used to calculate T,, P,, V,, and the critical compressibility factor 2,. The method is outlined in the following example : Calculate the critical constants of pyridine.
heat values obtained during this period often exceed 10yG, and in some cases run as high as 307,. The "International Critical Tables" note when and how the determinations were made, but unfortunately the modern handbooks do not give this information. Latent heat data for the hydrocarbons tabulated by Lange and Perry are with few exceptions taken from A.P.I. Research Report S o . 44 (32),and these values are of the highest order of precision. Both Lange and Perry also list heats of vaporization for about 180 inorganic compounds and the elements. Only 10 values are experimentally determined; the other values were calculated from vapor pressure data by Kelley ( 1 6 ) in 1935. The calculated latent heats are mostly quite reliable, but several values are in error due to poor vapor pressure data. In his original study, Kelley comprehensively analyzed the reliability of each value reported, but because of space limitations this discussion could not be incorporated along u i t h the tables of latent heats given by Perry and Lange. Where latent heat data are not available, L , , may be calculated ivith less than 3970 error for nonpolar substances by means of the Kistiakowsky equation (77).
D.4~4 PL'
= 0.982 gram/cc. a t 20' C. (293' K.)
tb
2
115.4' C.
Tb
=
388.6' K .
iM
= 79.1 L L .= ~ 107.4 cal.//gram or 8520 cal./gram mole
SOLUTION Solve for Z , from Equation 11. 2, = 0.255 Trial 1. Select some reasonable value for P, and solve for T , from Equation 16. .4ssume the Haggenmacher factor = 0.95 in this first approximation. Assume P, = 70.0 K K , = 1.04 T , = 627' K. from Equation 16 Solve for V L ~from . Equation 14. Use the value of T , just estimated. n = 0.31 from Table I VL' = 1.29 cu. ft./lb. mole V L= ~ 1.44 cu. ft./lb. mole from Equation 14 Solve for V , from Equation 13. V , = 4.15 cu. ft./lb. mole Solve for P, from Equation 11. P, = Z,RT,/V, = 50.5 atm. Trial 2. I--sing the values of P , and T , estimated in Trial 1, calculate the Haggenmacher factor. Then proceed to determine new values for T,, V L ~V,, , and P,. You will find the change in P, from the value obtained in Trial 1 is very small in this case, so there is no need to make a third approximation. The results of the computation are given in Table 111.
COMMENTS The results for pyridine are typical. Calculated values of T , and V , are normallv auite accurate. (The error of T , in this case though is rather' high.) Since the critical pressure occurs as a log function in Equation 16, values determined for P, are very sensitive to errors in T,. Therefore, errors in P, will usually be much larger than the corresponding errors for the critical temperature. As we shall see in Table 111, deviations of P, from the experimental values are almost directly proportional to errors in the Lydersen value of 2,.
Common Sources and Reliability of Input Data. Lange (20) and Perry (26) both give liquid densities for a very large number of compounds. With rare exceptions these data are very accurate. The same is true of boiling point data. Lange and Perry have also tabulated heats of vaporization for a large number of substances. Hoivever, latent heats given in the common handbooks are inaccurate in many cases. Data for the nonhydrocarbons measured after 1910 are reasonably accurate (normally less than 5% error in L J , but experimental determinations made prior to that date are mostly unreliable due to the crude techniques used. Errors in latent 152
l&EC FUNDAMENTALS
L,b
=
Tb(8.75 -k 4.576 log Tb)
(17)
The units of Lcb in Equation 17 are cal./gram mole and T, is in O K. The Kistiakoivsky equation, however, cannot be used for substances lvhich undergo partial vapor phase association-such as Na? K: and probably R b and Cs as well. For polar organic compounds which do not associate by hydrogen bonding L t b may be calculated, again with less than 3% error, by a modified Kistiakowsky equation. L,b = K~iTb(8.75f 4.576 log Tb)
(18)
The correction factor ( K K a )may be taken equal to 1.02 for weakly polar compounds such as aliphatic ethers, sulfides, mercaptans: and mixed halides. Lrb may of course be calculated from vapor pressure data. ,4n equation highly recommended for this purpose is one proposed by Haggenmacher (70) :
where
B =
(t2
-k C ) ( t l tz - t l
+ C) log (PZlPl)
B and C are the constants of the Antoine vapor pressure equation. I t is beyond the scope of this paper to discuss methods for calculating or estimating these constants, but the interested reader may consult references (4) and (39) for detailed information on the subject. Calculation of the critical constants using the method previously described. along \I;ith Equations 19 and 20, would necessarily be very time consuming. Several trial and error calculations would be required. In any case, the latent heat value used in the calculation method, whether experimental or calculated, must be accurate t o within 3Y0 for reliable estimates of T,, P,,and V,. Accuracy of the Calculation Method. Table I11 compares calculated and experimental critical constants for 12 inorganic and 20 organic substances. Calculations were made wherever possible using only the most reliable values of Lob. In four cases, latent heats were calculated by means of the KistiakoLvsky equation. The method was given a rigorous test by including several highly associated compounds-namely, ammonia, water, methylamine, nitromethane. and five alcohols. The method described is very accurate in predicting the critical temperature and the critical volume. For the 32 substances studied, the average deviation for T , is 1.6% and
4 3
r
VOL. 2
NO. 2
M A Y 1963
153
the 95% reliability (the maximum deviation expected in 95% of the cases) is 4.470. The average deviation for V, is 2.2% and the 95y0 reliability 5.6%. The results for P, are just fair, the average deviation being 7.3% and the 95% reliability 20.4%. In some cases though, the experimental values appear to be more questionable than the calculated ones. A few examples are cited. BROMINE.Note that the experimental 2, factor for bromine is 0.307, as opposed to a value of 0.268 calculated from the Lydersen equation. The 2, factor determined from the measured critical constants is abnormally high. The unusually high value of Z, may be the result of a positive error in the experimental value of P, and/or a negative error in V,. Errors in critical temperature measurements are generally not very large. It is the opinion of the author that the experimental value of P , is seriously in error. This conclusion is supported by the fact that the calculated and experimental Z , factors for chlorine are in close agreement. CARBOSDISULFIDE.The experimental 2, factor for CS2 of 0.292 also appears to be abnormally high. The apparent P , estimated by the error in the calculated P, is -10.370. Lydersen group contribution method is 73.4 atm.. or an apparent error of -5.9%. I t is the author’s conclusion that the measured value of P, is a t least 6% too high. ISOPROPYL ALCOHOL. Only a single measurement of P , for isopropyl alcohol has ever been recorded, and this by Nadejdine (24) in 1883. Analysis of critical pressure data by the author for other compounds (from reference 79) indicates that Nadejdine’s values are invariably high. The experimental Z , value of 0.280 is unusually high for the alcohols (see Table 111), and this indicates that the accuracy of the measured critical pressure for isopropyl alcohol is very questionable. BUTYL ALCOHOL.Only one experimental value of P, exists for n-butyl alcohol (73). Unfortunately, the critical volume for this compound has never been measured. so Z, cannot be calculated from the experimental data. The critical pressure calculated by Lydersen (22) appears to be -10.5% in error. But in the opinion of the author, the measured value of P , for n-butyl alcohol is a t least 10% too high. Deviations of T , and P, for methanol and nitromethane are both exceptionally large. However, two values of the critical pressure have been measured for methanol. and the agreement is good. Only a single determination has been made for nitromethane ( 9 ) , but it is a recent one. Although the experimental 2,factor for nitromethane appears very low, first members of homologous series often behave abnormally in many respects. Therefore. the measured critical constants of these compounds must be assumed correct in the light of present knowledge. Excluding bromine, carbon disulfide, isopropyl alcohol, and n-butyl alcohol, the mean deviation for P, is reduced to 6.1y0, If nitromethane is also excluded, the average deviation for P, is further reduced to 5.3%. A summary of the statistical data is given in Table I V . This calculation method is particularly accurate for nonpolar or slightly polar substances. (A slightly polar compound is defined by the author as one having a dipole moment less than 1.0 but greater than zero.) Br?, CS?, CC14, C12, G e C L R n , Xe, cyanogen, isobutane, 2.2,4-trimethylpentane. cyclohexane, and benzene are all nonpolar. Phosphine, trimethylamine, and furfuran are slightly polar. For these 15 substances, the average deviation for T , is 0.9%. for P, 4.2%, and for V, 2.8%. Limitations of the Calculation Method. Three groups of substances must be omitted from the calculation procedure described in this paper. NITRILES. T h e Lydersen equation for 2, apparently predicts high values of the critical compressibility factor for 154
I&EC FUNDAMENTALS
H C N and the nitriles. 2, factors for these compounds calculated from the experimental data are abnormally low. 2, = 0.198 for H C N and 0.183 for acetonitrile. One possibility for this abnormal behavior is that H C N and the nitriles may associate chemically in the vapor state to a considerable degree even a t the critical point. This would lead to calculation of low values for the molar volume, using the formula V , = M / p c (where p c is the measured critical densitv and M is the simple formula weight). Thus 2,calculated from the experimental data would also be too low. CARBOXYLIC ACIDS. These acids are known to associate appreciably in the vapor state, and even at the critical point the fraction of dimers in the vapor may be substantial. None of the correlations used would be expected to hold for substance of this type. Several inorganic compounds and elements also exhibit partial and varying degrees of association in the vapor state. Some of these have been noted earlier. METALS.Although the Klein equation should be accurate for all substances except those noted above, there is at the present time no information to show that the Benson equation or the Lydersen equation apply for metallic elements. Therefore, no suggestion is made that the method described in this paper should be used to estimate the critical constants of the metals. A test of this calculation method for the metals must wait until the critical properties of mercury have been substantiated by new experimental data. The reader mav be interested in a comparison of experimental and calculated values for mercur?. Table V shows that the problem is far from resolved.
Table IV. Summary of Statistical Data .Vumber of Remarks Substances Tc Pc VC
Average Deviation, % 1, 6 7.3 2 . 2 A11 substances in Table I11 1 .6 6.1 1 .9 All substances excluding Brz, CS2, isopropyl alcohol and n-butyl alcohol 27 for T , and P, 1 , 4 5.3 1 .7 A11 substances except 21 for V, the four named above and also CHaN02 15 for T, and P, 0 . 9 4 2 2 . 8 All nonpolar and 11 for V, slightly polar substances ( p < 1.0) 2.9 2 . 6 All nonpolar and 13 for T , and P, 0 . 9 slightly polar sub9 for V, stances except Brz and CS?
32 for 25 for 28 for 22 for
T , and P, V, T , and P, V,
32 for 25 for 28 for 22 for
T, and Pc 4 . 4 V T , and Pr 4.2 V,
27 for T , and P, 21 for V,
Table V.
957, Reliability, 7c 20.4 5 . 6 All substances in Table I11 18.2 5.2 All substances except Brz, CS?.isopropyl alcohol and n-butyl alcohol 3.5 14.6 5 . 0 All substances except the four named and also CHzNOz
The Critical Constants of Mercury
Source
Calculated bv Gambill ( 6 ) Calculated b i Gates and Thodos (7) Calculated by the author Experimental (2)
VO,
Tc> K. 1173
Atm.
1135 1555 1733
549 1588
Pc,
179.9 134
Cu. Ft./ Lb. Mole 0.715
3.24 0.792
...
Nomenclature
B, C
D G KIcl KKi L, M
= constants of the Antoine vapor pressure equation = sum of fluid densities a t temperature T = Goldhammer factor =
correction factor for the Klein equation
= correction factor for the Kistiakowsky equation
molal latent heat of vaporization molecular weig!nt exponent of the Goldhammer equation n P vapor pressure R gas constant t = temperature T = absolute temperature u = specific volume Av = difference in specific volumes of saturated liquid and saturated vapor V = molal volume VL’ = molal volume of the saturated liquid a t some temperature T’ at or below the normal boiling point p = density p ~ ’ = liquid density corresponding to VL’ p = dipole moment = = = = =
(8) Goldhammer, D. A., Z.physik. Chem. 71, 577 (1910). (9) Griffin, D. N., J . Am. Chem. Sod. 71, 1432 (1949). (10) Haggenmacher, J. E., Znd. Eng. Chem. 40, 436 (1948). (11) Haggenmacher, J. E., J . Am. Chem. SOC. 68, 1633 (1946). (12) Hanson, E. S., Znd. Eng. Chem. 41, 96 (1949). Neukirch, E., Z. physbk. Chem. 104, 433 (1923). (13) Herz, W., (14) Hougen, 0. A,, Watson, K. M., Ragatz, R. A., “Chemical Process Principles-Part 11,” 2nd ed., pp. 574-83, Wiley, New York, 1959. (15) Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley, New York, 1956. (16) Kelley, K. K., U. S. Bur. MinesBull. 383,1935. (17) Kistiakowsky,W., Z.physik. Chem. 107,65 (1923). (18) Klein, V. .4., Chem. Eng. Progr. 45, 675 (1949). (19) Kobe, K. A., Lynn, R. E., Chem. Rev. 52, 117 (1953). (20) Lange, N. A.: ed., “Handbook of Chemistry,” 8th ed., Handbook Publ., Sandusky, Ohio, 1952. (21) Lu, B. C., Chem. Eng. 66,137 (May 4, 1959). (22) Lydersen, A. L., University of Wisconsin Eng. Expt. Sta. Rept. No. 3, 1955. (23) Lydersen, A. L., Greenkorn, R. A., Hougen, A . O., University of TYisconsin EnF. EXD.Sta. ReDt. No. 4. 1955. (24) Nadejdine, A,: Be;bl. Ann. Piysik. 7, 698 (1883). (25) Osborne, D. W., J . Am. Chem. Soc. 64, 169 (1942). (26) Perrv, J. H., ed., “Chemical Engineers’ Handbook,” 3rd ed., McGraw-Hill, New York, 1950. (27) Pitzer, K. S., Givinn, W.D., J . Am. Chem. Soc. 63,3313 (1941). (28) Riedel, L., Z . Elektrochem. 53, 222 (1949). (29) Riedel, L.. Chem.-Zngr.-Tech. 24, 353 (1952). (30) Zbid.: 26, 83, 259 (1954). (31) Rohm & Haas Co., Philadelphia, “The Methylamines.” (32) Rossini, F. D., Pitzer, K. S.,Arnett, R. L., Braun, R. M., Pimentel, G. C., “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, Pa., 1953. (33) Sama. D. A , , M.S. thesis in chem. eng., Mass. Inst. Technol., Cambridge, Mass., 1955. (34) Scott, D. LV., et al., J . Am. Chem. Soc. 74, 4666 (1952). (35) Sherwood, T. K., Reid, R. C., “The Properties of Gases and Liouids.” DD.8. 18. McGraw-Hill. New York. 1958. (36) Stuli, D. R.iind. Eng.’Chem. 39,157 (1947). (37) Thodos, G., A.Z.CI1.E. J . 1, 165, 168 (1955). (38) Zbid., 2, 508 (1956). (39) Thomson, G. W.. Chem. Reu. 38. 1 (1946). (40) \Vatson, K. M.. Znd. Eng. Chem.‘35,‘398 (1943). ~
Subscripts b = normal boiling point c = critical point g = saturated vapor L = saturated liquid = reduced state-refers to the ratio of the given property r to the same property a t the critical point 1 = lo\rer temperature level 2 = higher temperature level T , = 0.85 = reduced temperature equal to 0.85 literature Cited
Colloid Chem. 52, 1060 (1948). (1) Benson. S. \V.. J . P h y . (2) Birch. F.. Phjs. Rez’.41, 641 (1932). (3) Fishtinr. S. H., Chen. Eng. 69, 154 (Sept. 3>1962).
(4) (5j (6) (7)
Gambill. \ V . R.. Zbid.. 64. 262 (Dec. 1957). Zbad.. 66, 181 (June 15, f959).‘ Zbad.. 66, 157 (Julv 13, 1959). Gates. D. S.. Thodos. G., A.Z.Ch.E. J . 6, 50 (March 1960).
RECEIVED for review May 21, 1962 ACCEPTEDDecember 28, 1962
COM M U N I C A T I ON
S E M I MICRODETER M I NATION OF VAPOR- LlQUI D EQUI LI B R I U M A semimicronlethod for the determination of isothermal vapor-liquid equilibrium in multicomponent systems is described. Gas chromatography was used to analyze the phases. Only about 2 ml. of sample is needed to determine one experimental point. The proposed method is advantageous, especially for systematic choice of extraction agents for extractive distillation.
THE experimental
determination of vapor-liquid equilibrium is usually performed in circulating stills ( 3 ) which in the normal arrangement require about 30 to 200 ml. of liquid mixture to yield one experimental point. T h e proposed method is based on the sampling of very small volumes of the vapor phase which is analyzed by gas chromatography and makes possible the determination of equilibrium data on a considerably smaller amount of the liquid mixture (about 2 ml. for one value). ‘The quantity of substance in the sample is proportional to the peak area; therefore, it is
possible to calculate the relative volatility from chromatograms directly by comparing the ratios of peak areas of components in the liquid and vapor phases. I t is not even necessary to reach the equilibrium partial pressures because it is sufficient to measure the concentration ratio of components in order to determine the relative volatility. T h e small amount of the vapor phase sampled makes it possible to obtain a considerably greater number of analytical data while leaving the concentration in the liquid phase unchanged. Using the still described and a system with relative volatility 01 = 4, ten VOL. 2
NO. 2
MAY 1963
155
