INDUSTRIAL AND ENGINEERING CHEMISTRY
408
Design of Deodorizing Vessel The optimum size and shape of the deodorizing vessel have always been a matter of some discussion and conjecture. From the preceding results it would seem possible that these are, within wide limits, comparatively unimportant. It is obvious that increasing the depth of the vessel will tend to raise the vaporization efficiency, since factor t in Equation 10 is Drouortional to the oil deDth. As mentioned above. however, there is some evidence that good efficiency is attained even under adverse conditions of pressure. If this is the case, similarly good efficiency should be obtained with shallow bodies of oil. The use of shallow vessels will permit a comparatively greater rate of steaming, since the maximum rate is more or less a function of the surface area of the oil body. For the same reason deodorization may be conducted more rapidly in small vessels than in large ones. Distribution of the steam in the form of fine bubbles is indicated, since factor A in Equation 10 is thereby increased. I n view of the high vaporization efficiencies obtained with
Vol. 33, No. 3
ordinary steam distribution, however, the value of elaborate distributing or diffusing devices would appear to be questionable. Acknowledgment The author is indebted to R. H. McKinney and R. F. Krage for preparation of the figures in this article, and to A. R. Diehl and Tales Newby for assistance in conducting the plant tests and experiments. Literature Cited Brash, J. Sac. Chem. I n d . , 45, 73T (1936). I b i d . , 45, 331T (1926). Dean and Chapin, Oil & S o a p , 15, 200 (1938). Haller and Lassieur, Compt. rend., 151, 697 (1911). Haller and Lassieur, J. SOC.Chem. I n d . , 28, 719 (1909). Lee and King, Oil & S o a p , 14, 263 (1937). Lewis and Whitman, IND.ENG.CHEM.,16, 1215 (1924). Salway, J . Chem. Soc., 111, 407 (1907). Singer, Seifensieder-Ztg.,65, 487, 507 (1938). ENG.CHEX.,26, 98 (1934). Souders and Brown, IND. Thurman, Ibid., 15, 395 (1923). Whiton. Cotton Oil Press, 7. No. 10, 32 (1924).
Calculating Beattie-Bridgeman Constants from Critical Data SAMUEL H. MARON AND DAVID TURNBULL Case School of Applied Science, Cleveland, Ohio NE of the most accurate and useful equations of state for gases is that of Beattie and Bridgeman (2) :
0
where p,
yB, and 6
are defined by:
where A , B , C, D, E , F , and G are given in terms of the Beattie-Bridgeman constants and the critical constants P, and T,by: A =- Bop, 2.303RT, -AJ'o 2.303 (ET,) =- -cP, 2.303RTC4 -BobPo' = 2 X 2.303(RT0)Z AoaPC2 = 2 X 2.303(RTJ8 -BocPO2 = 2 X 2.303R*TC5 BobcPO3 = 3 X 2.303R3TC6
B =
c Ao, Bo,a, b, and c are constants independent of temperature and pressure, which must be evaluated empirically for any given gas. The purpose of this paper is to show how, on the basis of certain assumptions, these five constants may be evaluated for any gas from its critical constants and the Beattie-Bridgeman and critical constants of some reference gas, and also to test the calculated constants for each gas with known compressibility data. Derivations of Equations
In the preceding paper ( I O ) the authors derived a n expression relating log,, y , where y is the activity coefficient of a gas, to the Beattie-Bridgeman constants and the reduced pressure and temperature, P, and T,. The derived expression was as follows:
D E
F G
Newton (11) found that when values of y obtained graphically were plotted against P, a t constant T,, a n average curve could be drawn through the points from which the activity coefficients of most of the gases studied deviated by no more than 2 per cent. This finding of Newton's may be generalized to state that a t the same values of T,and P, all gases, with possibly a few exceptions, have the same activity coefficient. Applying this condition to Equation 2, it must follow that
March, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
once the constants A through G are evaluated for a reference gas, the same set of constants should hold as well for any other gas, as was demonstrated in the preceding paper (IO). If this is so, then Equation 2A applied t o two different gases may be written as, A=A’ BQP, _ _ _ . = _ _ _Bo’Pe’
2.303RT,
2.303RTC
(3)
from which follows the relation for BOof one gas in terms of Bo’ of another gas as: Bo
=
BQ‘
(g)(g)
(4-4)
Extending the same process to the other constants in Equation 2, we obtain for the remaining Beattie-Bridgeman constants the relations: 2 P ‘ A , = Ao’ 4 P’
(g)(g)
c = e’
(2)(5)
a = a’ b
=
($)(g)
b’($)(g)
The Beattie-Bridgeman constants calculated by this procedure are, in general, different from the constants determined empirically; thus coefficients of Equation 1 other than those selected from the empirical data may reproduce the compressibility behavior of gases. It should be pointed out, however, that although these new constants may reproduce the compressibility data of various gases, the accuracy of reproduction cannot be expected to be so high as with constants evaluated empirically. The accuracy will be conditioned both by the assumptions made in the derivation of Equation 2 and by the validity of Newton’s generalization.
409
From a general equation for log,, y as a function of T,and P, for any gas, expressions are derived relating the BeattieBridgeman constants of any gas to the Beattie-Bridgeman constants of a reference gas in terms of critical pressure and critical temperature only. The Beattie-Bridgeman constants thus calculated for various gases are used to evaluate the compressibilities of these gases, and the calculated results are compared with observed data and with the van der Waals equation. Compressibilities calculated by the method described agree satisfactorily with observed data over a wide temperature and pressure range, and are, in general, superior to the results of the van der Waals equation at all temperatures except in the interval T , = 1.00-1.25.
these constants were used t o calculate a series of pressures for comparison with observed data and with the van der Waals equation. The results of these calculations are summarized in Tables I1 and 111, where the percentage deviation from the observed pressure is given both for the method of this paper and for the van der Waals equation. The sources of P-V-T data on which the comparisons are based are listed in column 2 of Table I.
Test of Calculated Constants
Discussion The constants Ao, Bo, a, h, and c were calculated from Table I1 embraoes calculations on methyl chloride, methanol, and ethanol a t temperatures of T, = 0.8 - 1.00. The Equations 4A to 4E for a number of gases by selecting nitrogen as the reference gas. The Beattie-Bridgeman conresults show that at T,less than 1.00 the deviations between stants for nitrogen used were those due to Deming and calculated and observed pressures are no greater than 5 per Shupe (6); the critical data were taken from International cent up to pressures 0.8-0.9 of saturation, the agreement imCritical Tables (8): Following Newton’s practice, the conproving a t lower pressures. I n nearly all cases the pressures calculated in this temperature range by the method outlined stants for hydrogen were calculated on the basis that P,” = P, 8, and T,“ = T, 8, a procedure Newton found necesare in substantially better agreement with observed pressures than are those calculated from the van der Waals equation. sary for gases having low critical pressures and temperatures. However, a t values of T,very close to unity the deviation The constants thus calculated for twelve different gases are summarized in Table I. is less than 5 per cent only up to pressures 0.7-0.8 of the These constants were then employed to evaluate 0, yB, and critical; here, also, the van der Waals equation gives 6 a t various temperatures; in conjunction with Equation 1, in most cases better agreement with observation than does the Beattie-Bridgeman eauation with calculated constants. For values of T, above 1.00, the result shown CONSTANTS FOR V A R i o U s TABLE I. CALCULATED BEATTIE-BRIDGEMAN in Table 111may be divided into three groups. GA0ES First, for the temperature range T, = 1.00-1.25, Source of Gas P-TI-T Data Ao Bo a b C pressures are reproduced within 5 per cent NP (6) (1.254) (0.04603) (0.01868) (-0,02588) (6.16 X 10‘) up to 60-75 atmospheres, an accuracy somewhat NHa 3.904 0.04454 0.01805 -0.02500 1.99 X 10‘ c0* 3.355 0.06104 0.02069 - 0.02864 9.6 x 106 less than that given by the van der Waals equa0.2170 0.02431 0.00985 -0,01365 1.15 X 103 H 2 tion. Secondly, for T , = 1.25-1.50, the calcuC2H6 5.049 0.07658 0.03104 -0.04298 1.46 X 10: 0.03801 0.01541 -0,02134 9.35 X 10 1.268 CH4 0 2 2,099 o.05095 o,02065 -o,02s60 2,36 lated constants reproduce data within 5 per cent CZH4 4.180 0.06806 0.02758 -0.03820 1.03 x I O B up t o pressures of 100 atmospheres or better, a n 0.1079 0.04372 -0.06053 3.65 X 106 CsHs (4) 8.624 NO 1.304 0.0337’2 0.01366 -0.ois92 1.29 x 106 accuracy on the whole better than that given by 0.03138 -0.04348 3.74 X 106 6.965 0.07744 CHnCl C ~ H ~ O H (12) 11.18 0,1002 0.04061 - 0,05624 9,22 lor the van der Waals equation. Finally, a t T , above CHsOH (19) 8.857 0.07987 0,03237 -0.04483 7.22 X 106 1.50, the deviation between observed and calculated pressures is no greater than 5 per cent up a Reference gas. to 200400 atmospheres, and is considerably
+
+
L
81
13$1
410
Vol. 33, No. 3
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Summary
e OF CALCULATED AND OBSERVED PRESTABLE 11. COMPARISON SURES FOR Tr BELOW 1.00 Pobavd.,
---
Atm.
--yo
Deviation-
Pobavd.,
Atm.
Poalod. P v s n der Wasla
CHaC1T 358.1' K.; TI 3 0.86 16.19 f 3 . 3 7.6 22.24 f 3 . 4 f10.7 T = 388.1' K.; Tr 0.93 18.26 +2.6 5.4 22.99 +2.5 6.4 33.53 1 4 . 4 +13.1 40.66 -2.2 +10.4 4 z H s O H T = 413.1' K.; Tr = 0.80 4.53 -0.22 f0.44 6.25 f 1 . 3 f3.0 7.14 + 2 . 1 f4.6 T = 453.1' K.; TI = 0.88 7.11 -1.4 *o.o 11.83 -0.25 f2.2 17.35 +2.4 f6.6 T P 483.1' K.; TI 0.94 6 . 4 1 -2.4 1.7 14.73 -2.3 0.27 26.01 -0.19 4.4 34.91 +1.8 d-11.2
Deviation-
7 %
Posled. P v a n der Waals
CaHaOH-
7 -
T = 519.1' K.; TI = 1.01
+
6.94 16.29 26.88 39.38
++
-2.7 -3.6 -3.7 -4.5
-0.43 -1.4 -1.3 -0.20
p C H s O H T = 433.1' K.: Tr = 0.84 7.89 +0.63 +2.0 10.53 13.16 f+2.0 3.7 + +4.0 6.8 = 473.1' K.; Tr = 0.92 5.26 -0.95 0.38 15.79 -0.57 f 1.3 26.32 f 1 . 5 5.3 36.84 +4.8 +12.6 T = 513' K.; TI = 1.00 15.79 - 3 . 4 2.5 26.32 -3.3 1.5 9.4 47.37 -2.4 60.52 - 3 . 9 -10.1
T
-
+ --
Although the constants as found from Equation 4 do not reproduce the experimental data so closely as empirically fitted constants, they do reproduce the pressures of all the gases investigated over a wide temperature and pressure range with a n accuracy better than that given by the van der Waals equation a t all temperatures below and above T, = 1.00, except for the interval T,= 1.00-1.25. The method outlined here is not suggested for gases for which precise empirically evaluated constants are known. It should be valuable, rather, for gases for which extensive compressibility data are lacking or for which Beattie-Bridgeman constants are not available. I n such cases use of Equation 4 yields BeattieBridgeman constants from critical data alone which permit the calculation of compressibilities of all types of gases with reasonable accuracy within the temperature and pressure intervals established above.
Nomenclature A,,, Bo = Beattie-Bridgeman constants of a gas for P in atmospheres and V in liters/mole + a', b', c', Ao', BO' = Beattie-Bridgeman constants for reference gas A , B , C, D,E , F, G = coefficients of Equation 2, independent of T,and P , TABLE 111. , COMPARISON OF CALCULATED AND OBSERVED PRESSUEES FOR T, P = pressure, atmospheres ABOVE 1.00 P, = critical pressure poi,,vd., -% DenationPobsvd., -% DeviationPobpvd., -% DeviationP, = reduced pressure Atm. Pcslod. P v . d. W. Atm. Peslod. P v . d. W. Atm. Podd. P v . d . W. R = molar gas constant, liter-atmos7 N I . L -1 -He---. pheres/' K. T = 273.1' K.: Tr 1.77 T = 373.1' K.; Tr = 1.22 T 473.1' K.; Tr 1.17 T = temperature,OK. 41.89 -0.41 f0.05 36.20 -0.22 -1.3 27.28 14.43 4-0.07 -0.11 $0.14 -0.22 T, = critical temperature 54.74 fO.20 -2.0 92.57 -1.6 -0.61 T, = reduced temperature 100.00 -0.30 -0.30 49.03 -1.1 -1.6 Tr 1.47 T 598.1' K.; V = volume, liters/mole 200.00 1-0.20 -6.0 58.29 -1.9 -2.6 300.00 -0.13 -1.1 -0.29 -0.58 31.22 p , Y B ,6 = virial coefficients of the BeattieT 473.1° K.; TI = 1.55 400.00 -2.8 +6.3 -0.85 -1.3 55.24 Bridgeman Equation 1, functions of 36.44 +0.03 -1.3 -1.3 -2.3 89.69 T = 372.6' K.; Tr =.a 2.41 69.38 -0.39 -3.4 T only for a given gas 130.4 -1.8 -3.5 100.00 -1.0 -3.6 130.2 -1.2 -7.3 y = activity coefficient of a gas = N -O200.00 -0.4 -5.3 160.8 -2.0 -8.1 fugacity/pressure 300.00 +0.07 -5.5 T 194.2' K,; T, = 1.08
-
-
-
-
30.00 50.00 70.00
-
-
-0.83
+1.1
+3.0
233.1' K.; Tr 1.30 30.00 -0.83 -0.50 50.00 -1.1 -0.66 70.00 -1.2 -0.71 100.00 -2.3 -1.3 120.00 -3.5 -2.1
T
-
T
282.1' K.; Tr +0.23 30.00 +0.64 70.00 +o.oo 100.00 4-0.58 140.00 +1.1 160.00
-
= 1.58
-0.10 -0.36 -0.75 -2.1 -1.9
203.18O K.; Tr = 4.92 0.54 50.00 f0.06 1.3 +O.lO 100.00 3.3 150.00 -15.6 300.00 6.62 T 273.18' _ . K.: TI +0.92 50.00 f0.16 +1.4 100.00 +0.30 -0.40 200.00 +0.80 -2.5 300.00 +0.67
T
i8:% -
873.18' K.: 3-0.16 50.00 CO.50 100.00 +0.65 200.00 +1.2 300.00
T -~
+-
-
TP =
9.04 +0.36 4-0.80 4-1.0 t1.5
c,
5
-
p
-2.9 -3.2 -6.4
a, b,
400.00
-0.18
200.00 300.00
400.00
-
-
-4.7
T = 472.6' K.; Tr +0.20 +0.67 fl.2
3.06 -1.6 -3.7 -6.6
-CO*---323.1' K.: Tr 50.00 0.38 3.2 75.00 100.00 -18.9
T
T
-
+ -
373.1' K.; 50.00 -1.3 75.00 -2.5 100.00 -4.6 125.00 -7.7 T 471.1' K.: -1.2 75.00 -1.7 100.00 -2.3 125.00 200.00 -3.8
-
-
P C Z H 4 T 333.1' K.; 50.00 1.8 100.00 -19.0
T
-
-
TI
= 1.06
+1.7 +0.13 -2.4
-
1.23
-1.0 -2.3 -4.2 -6.5
TF = 1.55 -2.8 -4.0 -5.4 -8.6
TI =
1.18 -0.8 -3.8
373.1' K.; Tr 1.32 50.00 -1.6 -2.4 100.00 -3.7 -5.5
f = 471.6' K.: T v 50.00 100.00 200.00 300.00
-0.72
-1.1 -1.1
-2.6
1.67 -2.4 -4.7 -6.5 +0.33
-CHI-
T = 273.1' K.; T I = 1.46 32.30 -0.22 49.44 -0.24 76.88 -0.42 92.7 -0.76 108.0 -0.83 T 373.1' K.; 46.47 -0.15 73.44 -0.03 120.9 +0.33 150.9 +0.60 181.8 11.2
-
-1.0 -1.7 -3.2 -4.1 -4.6 TI 1.96 -1.7 -2.6 -4.1 -4.7 -4.9
T = 473.1' K.; TI 60.49 97.05 164.0 2 n n ~ __I._ 254.3
f0.08 -0.01 f0.43
1.0.58
+iIi-
-
2.48 -1.8 -2.7 -4.0 -4.5
-4.6
Literature Cited (1) Bartlett, Cupples, and Tremearne, J. Am. Chem. Soc., 50, 1275 (1928). (2) Beattie and Bridgeman, PTOC.Am. Acad. Arts Sci., 63, 229 (1928). (3) Beattie, Hadlock, and Poffenberger, J . Chem. Phys., 3, 93 (1935). (4) Beattie, Kay, and Kaminsky, J . Am. Chem. Soc.. 59, 1589 (1937). (5) Beattie and Lawrence, Ibid., 52, 6 (1930). (6) Deming and Shupe, Ibid., 52, 1382 /, nc)n\ (lJOU,.
(7) International Critical Tables, Vol. 111, p C s H s pp. 3-17, New York, McGraw-Hill T 548.1'K. T I 1.48 Book Co.. 1928. 41.23 -1.1 -1.9 (8) Ibid., Vol. 111, p. 248. -2.4 -4.6 77.08 (9) Keyes and Burks, J . Am. Chem. SOC., -7.1 110.5 -3.4 49, 1403 (1927). (10) Maron and Turnbull, IND. ENG. CHEM.,33, 246 (1941). (11) Newton, Ibid., 27, 302 (1935). (12) Ramsay and Young, Trans. R o y Soe. (London), A177, 123 (1886). (13) Ibid., A178, 314 (1887).
-
less than this figure in most instances. For these higher temperatures Equation 1 in conjunction with Equation 4 gives, in all cases investigated, substantially better agreement with observed pressures than does the van der Waals equation of state.
PREBENTED before the Division of Industiial and Engineering Chemistry at the 100th Meeting of the American Chemical Society, Detroit, Mich.
Correction-Specific Heats of Organic Vapors The following correction applies t o Table I of our paper, published in INDUSTRIAL AND ENGINEERING CHEMISTRY, 30, 1029 (1938): In column 2 replace the fifth figure, 0.299, with 0.229. PAUL FUGASSI AND C. E. RUDY,JR.
