Ind. Eng. Chem. Res. 1990, 29, 125-128 k 2 = second-order reaction rate constant, dm3 mol-’ min-’ K = equilibrium constant Registry No. EA, 112-86-7; EM, 112-84-5; EN, 73170-89-5;
125
Ekpenyong, K. I.; Okonkwo, R. 0. Determination of Acrylonitrile/ Methylmethacrylate Copolymer Composition by Infrared Spectroscopy. J . Chem. Educ. 1983,60, 429-430. Gunstone, F. D. Oils and fats; production, consumption, availability and chemical reactions. Chem. Znd. 1987,43-44. Pasto, D. J.; Johnson, C. R. Organic Structure Determination; Prentice-Hall: Englewood Cliffs, NJ, 1969; pp 121, 127, 130. Reid, E. E.; Marven, L. W.; John, W. Esterification Reactions. In Unit Operations in Organic Synthesis, 5th ed.; Groggins, P. H., Ed.; McGraw-Hill: New York, 1958; pp 694-749.
NH3, 7664-41-7.
Literature Cited Buswell, K. M.; Link, W. E. The Quantitative Determination of Small Amounts of Nitrile in fatty Acids. J. Am. Oil Chem. SOC. 1964, 41, 717-719. Cavalli, P.; Cavani, F.; Manenti, I.; Trifiro, F.; El-Sawi, M. Kinetic and Mechanistic Analysis of Toluene Ammodixation to Benzonitrile on Vanadium-Titanium Oxides. Znd. Eng. Chem. Res. 1987,26, 804-810.
Received for review January 9, 1989 Revised manuscript received September 1, 1989 Accepted October 20, 1989
COMMUNICATIONS Vapor Pressures of Pure Substances A generalized vapor pressure equation which is applicable to all substances has been developed. In addition to the critical pressure, temperature, and density (or Z,) it was necessary to introduce two other critical properties: S,, the slope of the critical density isometric for the gas phase; and K , a parameter that relates the apparent slope of the vapor pressure curve a t the critical point to the ratio S,/Z,. The equation contains two terms: the first applies to the noble gases and the second to the quantum gases, whereas all other substances contain both terms. For nonpolar and slightly polar substances, the five critical properties can be accurately calculated from vapor pressure and liquid density data in the vicinity of the normal boiling point. For any substance, the accuracy of the equation is comparable to the best available equation developed specifically for the substance. The object of this investigation was to develop an accurate generalized vapor pressure equation that uses as parameters the pressure-volume-temperature (PVT) properties at the critical point. At the critical point, thermodynamic considerations dictate that the slope of the vapor pressure curve (dP/dT)Tcis equal to the slope of the gas-phase critical density isometric (dP)dT)d,T,: (w/d ) T~,= (dp/ T )d , ~ , (1)
By use of the data for argon, hydrogen, nitrogen, propane, heptane, ammonia, and water, the following equation was developed:
Using the slope of the gas isometrics in dimensionless (reduced) form ( S ) ,we have s = (dP/dT)d/Rd (2)
For hydrogen and helium, the first bracketed term must be set equal to zero. Since S , for the noble gases is close to 1.782, the second bracketed term has a value close to zero for these gases. K is the ratio of the “apparent” slope of the vapor pressure curve at the critical point to the true slope (SC/Zc).The determination of K for nitrogen is shown in Figure 1. Note that in the critical region (T, > 0.99) the slope of the vapor pressure curve increases to a value that is 6% higher than the apparent slope. The term 0.94 in eq 5 is the value of K for the noble gases and the nonpolar diatomic gases. The apparent slope at the critical point (K(S,/ZJ)will be used instead of the true slope so the determination of K for a substance is required in order to relate the vapor pressure to the PVT properties S , and 2,. Also, how the molecular structure affects the value of K became an important part of this investigation. The integral form of eq 5 is as follows:
Since Rd = P / T Z where 2 is the compressibility factor, then (dP/dT),j(T/P) = S / Z
(3)
or (dpr/dTr)d,(Tr/pr) = S / z
(3’ )
where d,, P,, and T, are the reduced density, pressure, and temperature: d, = d / d c P, = P/P, TI = T/T,
At the critical point, (@r/dTr)Tc = (dpr/dTr)dc = Sc/zc
L
J
(4)
Thus, critical point criteria dictate that a generalized vapor pressure equation must include both S, and 2, as parameters. 0888-5885/90/2629-0125$02.50/0
3[
2
- 1](T;’
0 1990 American Chemical Society
-1
+ In T,)
(6)
126 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 1 .o
.90
Table I. Comparison of S o Obtained from PVT Data and I from Vapor Pressure Data
I .go
, .95
J 1 .o
substance hydrogen argon nitrogen oxygen methane ethane propane n-butane isobutane n-pentane neopentane heptane ethylene propylene isobutylene water
P V T data 1.565 1.797 1.833 1.812 1.794 1.871 1.917 1.966 1.964 1.990 1.966 2.052 1.863 1.900 1.937 1.907
vapor pressure 1.518 1.803 1.834 1.821 1.799 1.872 1.915 1.950 1.946 1.989 1.959 2.046 1.867 1.924 1.951 1.907
diff, % -3.00
+0.33 +0.05 +0.50 +0.28
+0.05 -0.10 -0.82 -0.92 -0.05 -0.36 4.29 +0.21 +1.26 +0.72
0
Reduced Temperature
Figure 1. Determination of the parameter K for nitrogen. The vapor pressure slope function ((dP,/dT,)(T,/P,)) divided by the slope of the critical isometric (S,/Z,) plotted versus the reduced temperature.
The integral form of the equation for hydrogen and helium is as follows: In O/PJ -
value of the ratio PI/Tr2 Once the vapor pressure equation for the two substances has been determined, then corresponding states having the same value of PIIT, are easily determined. It was found that for most pairs of substances the ratio of their values for 2, was inversely proportional to the square of the ratio of their reduced liquid densities at corresponding states:
K(S,/Zc)
This equation can be rearranged to the following form:
Empirical Determination of the Vapor Pressure Equation It is assumed that T,, P,, and the slope of the vapor pressure curve at the normal boiling point are known. The differential form (eq 5) and the integral form (eq 6) of the equation are solved simultaneously to determine the values for the terms K(S,/Z,) and S J K . If 2, is known, then K and S, can be calculated from the values for these terms. Alternatively, if a value for K can be assumed, then S, and 2, can be calculated. Note that the vapor pressure equation has been determined without requiring values for S,, Z, or K , only P,, T,, the normal boiling temperature, and the slope of the vapor pressure curve a t the normal boiling point are required. As a test of the validity of eq 5, values for S, calculated from the vapor pressure equation and from PVT data2are compared in Table I. Both methods have an estimated accuracy of about f0.5%, so the difference between the two values is expected to be about f1.070. Only hydrogen is appreciably outside this error estimate. Using the above method to determine the terms K (S,/Z,) and S,/K will result in an equation that has a perfect fit to the experimental data for a considerable range around the boiling point and at the critical point. Between these two points, the error will seldom exceed 1.0% with an average error a t most a few tenths of 1%. Below the normal boiling point, the error remains less than 1% down to pressures of about 0.1 atm and will only be a few percent at pressures as low as 0.001 atm.
Note that the liquid density of substance 2 (d2)in eq 9 is not a reduced density. This relationship is applicable over a wide range of values for the ratio PJT, provided the reduced density of the liquid is greater than 2.0. It has an accuracy of about &0.5%. A comparison of calculated and experimental values for 2, is included in the supplementary material. The exceptions to this relationship are the organic acids, alcohols, and amines, substances that are both highly polar and far from being spherical molecules. The method is quite accurate for the organic esters, which are much less polar than the acids and alcohols.
Correlation for Predicting 2, Ideally the ratio of the values of 2, for two substances should correlate with the ratio of their liquid densities at low temperatures when this ratio is obtained a t "corresponding states". Previous work on corresponding states in the gas phase suggested that two substances would be in corresponding states when they have the same
Discussion of the Vapor Pressure for 110 Substances In the supplementary material, the parameters in the vapor pressure equation for 110 substances are tabulated. The following is a summary of the values of K for different classes of substances: monatomic and diatomic gases (except H2 and He), 0.93-0.95; straight chain hydrocarbons, 0.93-0.95; branched hydrocarbons, 0.95-0.98; halogenated methanes, 0.93-0.96; halogenated ethanes, 0.94-0.96; aliphatic esters, 0.91-0.95; cyclic hydrocarbons and halogenated derivatives, 0.94-0.95; polar substances, 0.87-0.95; water, 0.925; ammonia, 0.911. The values for K are quite sensitive to the values used for P,, d P / d T at the normal boiling point, d,, and especially T,. A 1% increase in the values used for these parameters will result in the following approximate changes in K: + L O , -1.7, +0.5, and -6.7, respectively. For those substances for which an accurate value of the critical density has been obtained by the law of rectilinear diameters, the estimated accuracy of the values for K is f1.3%. If K is independent of the molecular structure, the value for K should be in the range from 0.93 to 0.96. Aside from hydrogen, helium, and neon, the values for K of all of the nonpolar and slightly polar substances are in this range
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 127 except for some of the branched aliphatic hydrocarbons. Accurate critical density data are available for only a few branched hydrocarbons, and for these, the value of K is in the expected range. If the value of K is higher than 0.96, then the vapor pressure data are suspect. Most likely the value for the critical pressure is too high. For polar compounds, the effect of molecular structure on K is rather confusing. About all that can be said is that the values for K tend to be lower than for the nonpolar compounds. The range of the values for K is from 0.87 to 0.95.
Comparison with Other Generalized Vapor Pressure Equations Reid et a1.8 recommend three generalized equations for calculating vapor pressure: the Gomez and Thodos equation, the Lee and Kesler equation, and a two reference fluid variation of Lee and Kesler's method. Gomez and Thodos use the normal boiling point in addition to P, and T, to determine the constants in their equation. There are three separate equations, one for nonpolar compounds, a second for polar compounds that do not hydrogen bond, and the third for those compounds that do hydrogen bond. In addition, separate correction terms are required for hydrogen, helium, and neon. Lee and Kesler use the Pitzer w (acentric) factor as a third parameter in addition to P, and T,. w is calculated from the vapor pressure at T , = 0.7. This method is only accurate for nonpolar compounds. An accurate equation for calculating w can be obtained by substituting eq 6 into the defining equation for w : w =
K(S,/Z,)[0.1696
+ 0.0937((0.94SC/1.782K)- l)]- 1
Note that w is a hybrid term involving both S, and 2, (and K). Equation 6 requires, in addition to the normal boiling point, the vapor pressure at another temperature in order to determine the two parameters K(S,/Z,) and S,/K. Preferably, however, the slope of the vapor pressure curve at the normal boiling point is used with eq 5 instead of the second boiling point since its value is usually reported in the literature. It is apparent that a three-parameter equation such as Lee and Kesler's can only apply to nonpolar substances or to one of several classes of substances, as in Gomez and Thodos. A four-parameter equation such as the one developed in this work applies to all classes of substances, including the quantum gases hydrogen and helium. The particular four-parameter equation developed in this work is of special interest since it uses only critical point data as parameters. Table I1 taken from Reid et a1.8 is a comparison of the vapor pressure of acetone calculated by using eq 6 (eq 6' for T I < 0.58) with values calculated by using the above three methods. Also included are the results calculated by using the Wagner equation. This equation is not a generalized equation. It contains four constants that are determined for each substance by using a constrained optimization method. Hence, extensive experimental data are required. Since acetone is a polar substance, the Gomez-Thodos equation should give superior results to the two methods that use w as a third parameter. Such is the case particularly for temperatures below the boiling point (Tt = 0.648). Equation 6 values are superior to the GomezThodos values and are nearly identical with the Wagner equation values. Results of comparable accuracy were obtained for water when compared with the equation of Keenen et al.5 and for hydrogen, nitrogen, propane, and
Table 11. Comparison of Equations for Calculating the Vapor Pressure of Acetone (Example of Reid et aL8): Error, Percent method Gomez Lee two vapor and and ref pressure, Wagner eq 6 Thodos Kesler fluids bar Tr 0.510 0.538 0.571 0.631 0.691 0.768 0.879 0.926 1.00
0.04267 0.09497 0.21525 0.74449 2.01571 5.655 17.682 26.628 47.00
0.2 -0.1 0.2 0.1 0 0.2 0 -0.1 0
0.2 -0.1 0.1 0 0.2 0.4 0.1 0.1 0
1.1 0.2 0.1 0.1 0.3 0.8 0.6 0.3 0
-7.9 -6.2 -3.7 -0.2 1.9 2.2 0.7 0.3 0
-5.5 23.9 -2.0 0 0.8 1.3 0.6 -0.2 0
monochlorodifluoromethane with the tables prepared by the Center for Applied Thermodynamic Studies, University of Idaho.' In conclusion, eq 6 will give very accurate results provided accurate normal boiling point and critical point vapor pressure data are available. The use of normal boiling point data to determine the third parameter in the Gomez-Thodos equation suggests that correlations must exist between S,, Z, and the normal boiling point (Trb)for each class of compounds. For nonpolar and slightly polar organic compounds, the following correlations were developed: (S,/Tr,,)/P,0.058 = 2.825 f 0.025 (10) S,(Z,/K) = 0.795 f 0.015 1 - Z,/K The critical pressure in eq 10 is in MPa. Equation 10 can be used to determine S,. Then with K = 0.94, eq 11 can be used to determine 2,. It is preferred, however, to use eq 9 to calculate 2, from liquid density data. In addition, these correlations can be used to accurately estimate the critical temperature and pressure for nonpolar and slightly polar substances. The method to be presented converges because the parameter s,/ TIbin eq 10 is only slightly dependent on the critical pressure. First, the left-hand side of eq 5 is rearranged as follows:
(%) P i ; ;
Trb)
where dP/dT is the known slope of the vapor pressure curve at the normal boiling point (Tb). By use of estimated values for T , and P,, S, is calculated from eq 10. Since only 2, in eq 5 is not known, its value can now be calculated. By use of the eq 6, PI is calculated a t the normal boiling point, and the critical pressure is calculated by using P, = Pb/Pr. We now have a preliminary set of critical constants and the corresponding vapor pressure equation. Since the liquid density at a temperature TI is known, the value of PI a t this temperature is calculated. The corresponding liquid density for the reference substance (propane) is obtained at the same value of the ratio Pr/Tr.The following equation can be used to calculate this density for values of Pr/Trless than 0.01: d,(mopane) = .- 4'02736 + 6.18318 (In (Pr/Tr))0~'125
P r / T , < 0.01
The value of 2, is now calculated by using eq 9 and compared with the value obtained from eq 5. By use of the calculated value for P, and a new assumed value for T,, the above procedure is repeated until the values for 2,
Ind. Eng. Chem. Res. 1990,29, 128-133
128
agree. As an example of the use of this method, the critical constants were calculated for 2,3-dimethylhexane: tabulated values 0.9594 566.2 2.696 2.0642 0.2700
K
proposed method 0.94 566.8 2.658 2.0525 0.2675
difference, 70 -1.66 +0.12 -1.41 -0.57 -0.94
conThe vapor pressure equation using these stants gives results that are essentially identical with the original equation UP to a temperature Of o.8* The maximum error occurs at the critical point and is 2.3%.
Modified Equation for Very Low Pressures not be used For the highest accuracy, eq and for reduced temperatures below 0.55. The following Correction terms were developed based on the data for propane, which has a very low triple point ( T , = 0.231): correction term for eq 5 +(9/8)T:(1
-
T1)4(6(1 - TI) - 15/8)
T, < 0.55
correction term for eq 6
TI < 0.55
A simpler form of eq 5 and 6 has been developed for reduced temperatures of less than 0.58:
-
Supplementary Material Available: Critical and normal boiling point data for 110 substances (3 pages). Ordering information is given on any current masthead page. Literature Cited
+(9/8)(1 - T1)5((l- T,) - 3/81
3[
pressure equation so that it applies to all substances. Four of these five parameters (Pc,T,, S, and 2,) are required to accurately correlate the-PVT properties for all substances. For nonpolar substances, only three of the four are required. P, and T, are always used. A number of suggestionsfor the third parameter include 2, by Meissner and Seferian; S, by Bloomer and Peck? and w by Pitzer et ala7 While correlations using 2, and w are only accurate for a class of compounds such as the nonpolar compounds, the correlation using st are believed to be accurate for all substances but only for reduced densities less than 1.0. The development of an accurate corresponding state PVT correlation for polar substances must include both S , and 2, as parameters. Such a correlation may not be possible because of the paucity of experimental data for those polar substances that are far from being spherical molecules.
z L - 1](1
T,)
T I < 0.58 ( 5 ’ )
1](T;I- 1 + In T,)
T I < 0.58 (6’)
-
J[
Pressure-Volume-Temperature and Thermodynamic Properties The most useful method of predicting the PVT and related thermodynamic properties of gases and liquids is the law of corresponding states based on the critical properties. This investigation has shown that five critical properties are required to accurately generalize the vapor
American Society of Heating, Refrigeration and Air Conditioning Engineers. Fundamentals Handbook. 1985. Bloomer, 0. T.; Peck, R. E. Modified Law of Corresponding States for Gases. AIChE J. 1960, 6, 240. Curtis, C. F.; Boyd, C. A.; Palmer, H. B. Thermodynamics of the Critical Point. J . Chem. Phys. 1951, 19, 801. Dreisbach, R. R. Physical Properties of Chemical Compounds-11. Adv. Chem. Ser. 1959, No.-22. Keenan. J. H.: Kevs. F. G.: Hill. P. G.: Moore. J. G. Steam Tables: John’Wiley and Sons: New York, 1978. Meissner, H. P.; Seferian, R. PVT Relations of Gases. Chem. Eng. Prog. 1951, 47, 579. Pitzer, K. S.; Lippmann, D. Z.; Curl, R. F.; Huggins, C. M.; Peterson, D. E. The Volumetric and Thermodynamics of Fluids, I1 Compressibility Factor, Vapor Pressure and Entropy of Vaporization. J . Am. Chem. SOC.1954, 77, 3433. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. Timmermans, J. Physico-Chemical Constants of Pure Organic Compounds; Elsevier Publishing: New York, 1950.
Oscar T.Bloomer 83 Coronado Drive Rochester, New York 14617 Received for review April 12, 1989 Revised manuscript received September 26, 1989 Accepted October 9, 1989
Gas-Liquid Mass Transfer in a Three-phase Fluidized Bed Containing Low Density Par tides Gas-liquid mass-transfer behavior in a three-phase fluidized bed containing particles with properties comparable t o those of bioparticles used in biological processes was examined. The volumetric gas-liquid mass-transfer coefficients, kLa, decrease with increasing solid concentration and with increasing terminal velocity of particles. An increase in the liquid velocity significantly increases hLa, but only slightly increases the gas holdup, thus suggesting a significant liquid velocity effect on k L . Three-phase fluidized beds containing low-density particles are widely used in biological processes such as wastewater treatment and fermentation (Fan, 1989). Fundamental transport phenomena of these systems, however, have not been extensively studied. The hydrodynamic characteristics of three-phase fluidized beds of
low-density particles have been reported by the authors (Tang and Fan, 1989). The three-phase fluidized bed of low-density particles exhibits a significant axial variation in the individual phase holdups, in contrast to axially constant phase holdups in a fluidized bed of heavy particles.
0888-5885/90/2629-0128$02.50/0 0 1990 American Chemical Society
