Empirical Correction for Compressibilitv Factor and Activitv Coefficient Curves J
RALPH A. MORGEN AND J. HOWARD CHILDS' University of Florida, Gainesvilla, Flo.
0
Preliminary calculations N E of the handiest tools In determining compressibility factor s and activity coefshowed that a correction to in chemical engineering ficient f/p, correction factors for the law of corresponding T Rmade the largest contribucalculations is the use of an states are indicated. Ethylene and nitrogen are used as tion to bringing the individual approximation method when the reference, and the curves of 5 us. PR and f/p us. PRare values onto the general curve. empirical data are lacking. given for these gases. A chart (Figure 3) gives a positive A correction to P R was found Many of these approximation correction factor for those gases (TR > 1) for which the remethods (8, 4, 6, 14, 16, 16, to be second in importance, ciprocal of the compressibility factor at the critical point and 18) depend on the law of cora correction to V Rwas, in (1/5,J is 3.35 or less; the correction factor may be neglected responding states which, in most cases, of minor imporfor those vapors (TR < 1) for which the reciprocal of the essence, says that gases and tance. The only exception is compressibility factor at l/s, is 3.35 or less. A chart vapors the same relative disin the vicinity of the actual (Figure 4) gives a negative correction factor for those gases tances from their critical critical region, T R = 1 and ( T R > 1) for which the reciprocal of the compressibility states behave alike. PR = 1, where the correction factor at l/sc is 4.0 or greater( Figure 4 gives a positive These methods have led to factor for V R becomes a p correction factor for those vapors ( T R < 1) for which the the use of reduced temperapreciable. reciprocal of the compressibility factor at l/ao is 4.0 or It was next observed that ture, pressure, or volume greater. For those compounds whose l/sc values are bethose substances whose critirelations plotted against some tween 3.35 and 4.0, the curve for s andf/p for ethylene and factor whose value is to be cal constantswere muchlower nitrogen should be used with the knowledge that values determined. T h e reduced than the substances for drift in the direction indicated by the correction factors. which the general curves v a l u e i s d e f i n e d as t h e were drawn, had deviations actual value divided by the critical value: Le.. reduced above the reduced temperature isotherms when T R was greater than 1 and below the isotemperature, b ~is,actual T divided by critical T i n absolute units. I n using the general reduced state curves for determining z, the therms when TRwas less than 1. On the other hand, those substances whose critical constants were greater than the reference compressibility factor in the equation, PV = zRT, i t has been substances deviated in exactly the opposite direction. noted (8,4, 6 ) that there is a drift in the error, depending on which gases are used to make the curves, This is also true in This suggested that if a curve were plotted using the material with the lowest critical constants, helium, all the deviations would determining fjp, the activity coefficient (16). The purpose of be in the same direction. However, there were insufficient this paper is to analyee this drift and to find a general factor to reliable data to make this curve, and the correction factor would correct for it. Newton (16) recognized that hydrogen, helium, be too cumbersome for substances now satisfactorily handled and neon needed a correction factor and suggested that an emwith the present curves. pirical constant of +8 be added to the critical temperature and If the law of corresponding states were absolute, then at the pressure of each of these gases. This is shown to be true for only critical point, c = PoVc/RTewould be the same for all suba limited range. Early in this analysis it was recognized that both z and f / p restances. By plotting l/z, against To for various substances, quired three correction factors dependent on T , P, and v, respecthere is a steady drift from l/zc for helium = 3.28 and hydrogen = tively. For z the reason is obvious from its definition, z = 3.27 with T , a t 5.26' and 33.3' K.,respectively, to l/zofor water = 4.30 and ammonia = 4.12 with To at 647.3" and 405.56' K., PV/RT. Activity coefficient f/p is determined by evaluating Equation 1 from Lewis and Randall (18) by graphical interespectively. It was decided, therefore, to make a curve for a gration : substance with an intermediate value of l/ze. Ethylene (9, &I), with a l/zc value = 3.58,was chosen; and in the range where RT In J = ap at constant T ( 1) data could not be obtained for ethylene, values (10) for nitrogen, P l/zo = 3.43,were used. I n the overlapping range the two check where u RT/P - V as closely as the data justify. The curves of Figures 1 and 2 were obtained. Within the limits of accuracy of these correction factors, the These curves agree closely with those published by Dodge (69, above equation is within the modification of Tunnell's Equation 1 Weber ( I O ) , Hougen and Watson ( Y ) , and Newton (16). How(17). If reduced values for P are plotted against f/p at constant ever, values for the individual hydrocarbons on the family from reduced T , generalized activity coefficient curves are obtained. methane (CHJ, l/zc = 3.46, to decane (CloHn), l/z, = 3.91, The error and drift of values in these curves were found to be of shift progressively from this curve with a regular deviation. the same order of magnitude and direction as those for the z The error is usually to indicate too high a value of z or f/p when curves. T Ris 1 or more, and too low a value when TRis less than 1. For 1 Preaent addrase, National Advisory Committee for Aeronautics LaboraMost purposes the error is not large enough to warrant a calcutoria, Cleveland. Ohio.
-r
667
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 37, No. 7
July, 1945
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
669
Iated c o r r e c t i o n f a c t o r except t o have a knowledge of the direction of the error. At its maximum, the deviation usually does not exceed 2 or 3%. With water, ammonia, and other substances for which l / z c > 4.0, the error at several places in the curves becomes appreciable. Two correction factors were therefore made, one for substances for which 1/20 < 3.35, and the other for substances for which l/zc > 4.0. Since the equations of these factors are complicated, the corrections were plotted and appear as Figures 3 and 4. CORRECTION FACTORS
TEMPERATURE TI WHBN l / Z c < 3.35. Figure 3 was o b tained by taking a s e r i e s of P V T values for hydrogen (8) and adding a correction factor, TI, of such size t h a t a pseudo v a l u e of t h e reduced temperature, Ti, would fit the - p , curve for ethylene and nitrogen. When plotted on logarit,hmic paper, these values (Table I) give a P proximately a straight line whose slope is 2.03 and intercepts the line for pseudo redyed temperature T, 1 at T, = 1.05. The equation for this line is, therefore,
Vol. 37, No. 7
INDUSTRIAL A N D ENGINEERING CHEMISTRY
670 TABLEI.
CORRECTION FACTORS FOR FIGURE 2 (BASEDON HYDROGEN DATA)
T:
TI
P
T;
1.2 1.4 2.0
$1.5 $2.5 +3.5
10.8
3.0 4.0 6.0
+1.3 f1.8
TI +10 +17 +28
Pf
+ + 58 ,. :0
f/p
f14.0
~
TABLE 11. CORRECTION FACTORS FOR FIGURE 4 (BASEDON WATERAND AMMONIA DATA) -------WaterTR 1.0 1.2 1.4 0.88
-Ammonia-
T/ - 7 - 13 -21 +12
p/ 3.5 - 6.5 -10.5 f 6.0
-
TR
Tf
1.2 1.4
- 12
- 20
+lo
0.9
Pf - 6 10 $ 5
-
Arbitrary values for helium and neon were chosen, and they check the use of this curve very well. It was therefore assumed that this correction factor could be used for any gases for which l/zc = 3.35 or less. I n the vapor region, T R < 1, the factor becomes negative. However, the data are so meager and the temperatures and pressures so low that it was not deemed advisable to apply any correction in that range. PRESSURE Pf WHENl/zo < 3.35. The pressure correction factor, Pf, to obtain the pseudo reduced pressure, PR, was taken in each case as one half the value of TI. No explanation is offered for these values except that their use makes the values for hydrogen and the other gases with low critical constants fit the general curves. Newton's correction factor (16) of $8 becomes the same as this method for the region T R = 2.5 to 3.0, and is too large for lower values of T R or too small a t higher values of TR. Arbitrary substitutions will quickly verify these facts. TI AND PI WHENl/z, > 4.0. For those substances for which the correction factor is negative when T R iB greater than 1 and positive when T Ris less than 1, Figure 4 was obtained. The same method was used as that for gases with low critical constants, except that PVT values for both water ( l a ) and ammonia (3) were utilized (Table 11). The negative values plotted in the positive direction give a V-shaped curve made up of two straightline segments. The numerical value of the correction factor is larger than that for Figure 3, but because of the higher T , and P, of these gases, the percentage correction is considerably less. The vapor correction curve is given in this case because these substances are often used in this range. I t is believed that this curve can be used for all substances when l/z, is greater than 4.0.
The authors were unable to obtain a aingle correction factor to make all gases fit the same reduced condition curves. A general shift in the values for different substances is shown as the critical temperature and the critical pressure increase. The correction factor for gases for which l/z, < 3.35 has the widest use when applied to hydrogen. Hydrogen enters into so many organic reactions that a simple method of determining ita deviation is most useful. This factor is more accurate than that proposed by Newton (16). The correction factor for gases for which l/zc > 4.0 is interesting in showing a trend. Usually when the results justify making the correction factor for substances like ammonia and water, the actual value can be obtained just as readily from published PVT tables. In that case fugacity can be obtained from the relation,
RT In f / j o = AH - T A 8 where fo can be assumed equal to p a t the lowest pressure in the table. However for other materials for which l/z, is greater than 4.0, the correction is helpful, and this curve gives a useful approximation when empirical data are lacking. NOMENCLATURE
f
I nPf = & L p a W '
= 0.1495
OF COMPRESSIBILITY DATA TABLE 111. COMPARISON CORRECTIONS
Gas
T,
K. 273.1 47.7
z
Mor en and Chid8 Cor., Fig. 1 , 2 , 4 . 2 1.065b 0.67
Newton Cor. and Fig. 1, z 1.09 0.44
b Tfand
1.11* 0.9Oh 0.893
Pfare obtained from Figure 3 and the relation T:
--
--
When Ti = 30, T ; = 5.25and A -0.93 4.30 and A = 4-0.83 When TI = 20, Ti Tf = 12.6 By interpolation, Tf 25 and Pf 273.1 100 T* - 33.2 + 25 = 4 . 6 1 and E'; 1 2 . 8 f 12.5
- 3 273.1 m -
A
3'9
and from Figure 1, I 1.065. Data oi Bartlett (1). d Using the method of Kay ( 1 1 ) for mixtures, Pseudo T o = (33.2 8) (0.5) 126.1 (0;5) = 83.05' K. ) atm. Pseudo P o = (_'2A8.+8 ) (0.5) 33.5 ( 0 . ~ =27.15 Pseudo T R = 83 o5 = 3.29, pseudoPR = 7.37 27 15 and from Figure 1, z = 1.14. e Find Tf and P/ from Figure 3 for H2 6s above = 25 and 12.5, respectively: 126.1 (0.5) = 92.15' K. Pseudo T o = 33 2 25) (0.5) Pseudo P o = 1 1 2 i 12.5) (0.5) 33.5 (0.5) = 29.4 atm. Pseudo TR = 2.96; pseudo P R = = 6.8
+ ' c
++
++
=
iF4
-
andfromFigure 1, a = 1.11.
306.5
- 33.2 + T,
Empirical
1.OP Hs Hz 0.6Qa 50 mole % HxandNn 200 273.1 1.104, 1.14d NHs 100 648.1 0.90/ 0.860 0.90g Ha0 27.2 5.9.4 0.8% From P V T data in International Critical Tables (8).
:E-6
2. Using Figure 2 and correcting with Figure 3,
f Using the data of Beattie and Lawrence (S).
A
when T = 30, T i = 5.25, A = 0.40 when = 20, Tg = 4.30, A = -1.61 By interpolation TI = 28 and P, =
Ii/
P, Atm. 100 36.8
++
j / p = 1.160
I !
= fugacity
= pseudo reduced temperature, ' K. Tf = correction factor for the critical temperature to get T:
T:
a
1. Em irically by plotting cy against P for hydrogen at 33.4' C. &06.a0 k.),graphically integrating, and dividmg by RT, In f/p is found:
1.15
DISCUSSION
EXAMPLES
To verify the curves and illustrate the use of the correction factors, five examples are given in Table 111. The user is cautioned that the units of TI are ' K. and of P, are atmospheres. One illustration of the use of Figure 2 to get corrected activity coefficients f/p using Figure 3 for adjustment, will be given. At 33.4' C. and 240 atmospheres what is the activity coefficient of hydrogen?
=
3. As a further check the actual value of f/p for nitrogen waa calculated at its real T R = 5.0 and P R = 8.95-i.e., 357.3" C. (630.5' K.) and 300 atmospheres. Then j / p was found to be 1.148.
No correotion factor and Figure 1. A Correction factor Figure 4 (note thk is a negative value) and Figure 1. i Data of Keenan (1.2) and F. G.Keyes. i Correction factor Figure 4 (note this is a positive value) and Figure 1.
0
= 14
July, 1945
Pz PI
z z4 a
-
5
INDUSTRIAL AND ENGINEERING CHEMISTRY
pseudo reduced pressure, atmospheres
= correction factor for critical ressure to get P i compressibility factor, PV&T 5 com ressibilit factor at the critical state P,Vo/RT,
( R ~ P-)v (ieviation from perfect gas law) LITERATURE CITED
(1) Bartlett, E. P.,J . Am. Chem. SOC.,49,687,1966(1927). (2) Beattie, J. A., and Lawrence, C. K., Ibid., 62,6(1930). (3) Brown, 0. O., Souders, M., and Smith, R. L., IND.ENGCRWM., 24,613 (1932). (4) Cope, J. Q., Lewis, W. K., and Weber, H. C., Ibid., 23, 887 (1931). (6) Dodge, B. F.,Ibid., 24,1353(1932). (6) Dodge, B. F., “Chemical Engineering Thermodynamics”, pp. 161,238,New York, McGraw-HillBook Co., 1944. (7) Hougen, 0. A., and Watson, K., “Industrial Chemical Calculations”, 2nd ed., p. 398,427, New York, John Wiley & Sons, 1936.
671
(8) International Critical Tables, Vol. 111, p. 4, New York, McGrsw-Hill Book Co.,1929. (9) Zbid., Vol. 111,p. 14. (10) Ibid.,Vol.III.p. 18. (11) Kay, W.B., IND.ENG.CHIPM., 28,1014 (1936). (12) Keenan, I. H.,and Keyes, F. G., “Thermodynamic Properties of Steam”, New York, John Wiley & Sone, 1936. (13) Lewis, G. N., and Randall, M., “Thermodynamics”, p. 194, New York, McGraw-Hill Book Co., 1923. (14) Lewis, W. K., IND.ENG.CHB~M., 28,269(1936) (15) Newton, R. H., Ibid., 27,302(1936). (16) O t h e r , D.F.,Ibid., 32,841 (1940). (17) Tunnell, G., J . Phys. Chem., 35,2886 (1931). 23, 360 (1931); 35, 398 (18) Watson, K. M., IND.ENG.CHB~M., (1943). (19) Weber, H. C., “Thermodynamics for Chemical Engineers”, p. 198,New York, John Wiley & Sons, 1939. (20) York, R., and White, E. F., Trans. Am. Inst. C h m . Enur8., 40, 227 (1944).
CONVERSION OF AROMATICS Alkyl Group Transfer in the Presence of Silica-Alumina Catalysts R. C. HANSFORD, C. G. MYERS, AND A. N. SACHANEN Socony-Vacuum Oil Company, Inc., Paulsboro, N. J. The conversion of alkylaromatic hydrocarbons in the presence of a typical petroleum cracking catalyst under conditions similar to those encountered in commercial catalytic cracking and reforming operations has been studied. Xylene is converted to toluene and trimethylbenzenes by disproportionation of methyl groups. Trimethylbenzenes are converted to toluene, xylene, and polymethylbenzenes, and methylnaphthalene is converted to naphthalene and polymethylnaphthalenes by the same mechanism. Methylbenzene is converted to ethylbenzene and ethylene; no triethylbenzenes are detected in the products. In the presence of benzene, alkyl groups from alkylbenzenes are transferred- to this hydrocarbon with the formation of higher yields of monoalkylbenzene than in the disproportionation reaction. Methyl groups are not, however, trans€erred from methylnaphthalene to benzene.
T
HE reactions of alkylaromatic hydrooarbone in the presence of various catalysts, particularly those of the Friedel-Crafts type, have been the subject of many investigations during the past sixty years. The primary interest in some of these investigations has been the conversion of polyalkylbenzenes to monoalkylbenzenes, g.a exemplsed in the conversion of xylene to toluene. The first commercial interest in this type of conversion was aroused during World War I, when the demands for larger quantities of toluene than could be obtained directly from coal carbonization forced attention to the problem of converting polymethylbenzenes to toluene. Many yeam prior, however, considerable work had been reported on the reactions of xylene, pseudocumene, mesitylene, and other polyalkylbenzenes in the presence of aluminum chloride. Thua, Anschlitz (1) reported that nt-xylene can be converted to toluene in yields as high as 25%, with the simultaneous formation of trimethylbenzenes, by refluxing with about 30% aluminum chloride for 2-3 hours. Pseudocumene wag reported to yield about 20% each of xylene and toluene. The reaction of polyalkylbenzenes to form monoalkylbenzenes (or alkyl
benzenes having a smaller number of corresponding alkyl groups) with the simultaneous formation of more highly alkyrated benzenes is an excellent example of the type of reaction commonly termed “disproportionation”. Jacobsen (8) reported fair yields of toluene from xylene and of toluene and xylene from pseudocumene, through the displacement of one or more methyl groups as methyl chloride, on passing anhydrous hydrogen chloride into the refluxing polyalkylbenzene in the presence of aluminum chloride, This type of reaction is not one of disproportionation in the true sense. Later efforts toward possible commercial conversion of polymethylbenzenes to toluene were directed mainly to the thermal cracking reaction at very high temperatures and to the transfer of methyl groups to benzene in the presence of aluminum chloride. The results reported on the latter reaction were generally negative. Boedtker and Halse (8) found no toluene in the reaction of xylene with benzene in the presence of aluminum chloride at the reflux temperature after 6 hours. Fischer and Niggemann (4) reported that the results of their‘investigation on this reaction were inconclusive. However, these investigators did report yields of toluene up to 12% from xylene by decomposition (undoubtedly disproportionation) at the reflux temperature in the presence of 34% aluminum chloride. A high-temperature noncatalytic cracking proces for the conversion of coal-tar solvent naphtha to 13-14% toluene and 8% benzene was described by Egloff (3). A comprehensive review of the literature on the action of aluminum halides on alkylbenzenes, including alkylation and dealkylation reactions and intermolecular and intramolecular rearrangements, was compiled by Nightingale (7). More recently Pitzer and Scott (8) published a notable contribution on the thermodynamics and structure of benzene, toluene, and xylene. They reported equilibrium measurements on the reaction of two moles of toluene to give a mole of benzene and a mole of xylenes at 60’ C., the catalyst being aluminum bromide promoted with anhydrous hydrogen bromide. Calculated equilibrium values for this reaction at 298.16’ to 1500’ K.are also reported in their paper.
