544
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
M c , = molal heat capacity at constant pressure Mc, = molal heat capacity at constant volume N = number of moles of gas
absolute pressure heat withdrawn from surroundings and added to system = line compression ratio, P ~ / P Igas ; law constant in Equation 16 R’ = cylinder compression ratio S = entropy, B. t. u./(lb.)(: R) or B. t. u./(lb. mole)(” R.) T = absolute temperature, K. or O R. V = gas volume VO = volumetric rate of gas flow corrected t o Po pressure, Equation 9 W = work done by system on surroundings S. T. = ‘suction temperature c = heat capacity per unit mass k = exponent in P V m = number of stages n = exponent in PVn = residual entropy function, defined by Equation 18 u = value in final state - value in initial state A isentropic work factor, defined by Equation 17 + == Compressibility factor, Equation 16
P
Q R
= =
Subscripts c = critical propert o = measuring conJtion p = constant pressure ‘ R = reduced property S = constant entropy v = constant volume 1 = initial state, suction condition 2 = final state, discharge condition
Vol. 34, No. 5
Literature Cited (1) Brown, G. G., Proc. Natural Gas Assoc. Am., 19, 60 (May, lQ40). ~. ,~~
Brown, G. G., Souders, M., and Smith, R. L., IND. ENG.CHEM., 24, 513 (1932). Cope, J. Q.,Lewis, W. K., and Weber, H. C., Ibid., 23, 887 (1931). Dddge, B.F., Ibid., 24, 1353 (1932). Edmister, W. C., Ibid.,30, 352 (1938). Ibid., 32,373 (1940). Gill, T. T., “Air and Gas Compression”, p. 52, New York, John Wiley & Sons, 1941. Hougen, 0. A., and Watson, K. M., “Industrial Chemical Calculations”, p. 398, Kew York, John Wiley & Sons, 1936. Kats, D. L., and Brown, G. G., IND. Exo. CHEM.. . 25.. 1378 (1933) Kay, W. B., Ibid.,28, 1014 (1936). Refer, P. J., and Stuart, M. C., “Principles of Engineering Theymodynamics”, pp. 67, 472, New York, John Wiley & Sons, 19.70.
Laverty, F. W., Oil Gas J.,38, 32 (Nov. 2, 1939).
Perry, J. H., Chemical Engineers’ Handbook, 2nd ed., p. 1398, New York, McGraw-Hill Book Co., 1941. Sage, E.H., Budenholeer, R. A., and Lacey, W. K.,1x0. ENQ. CHEM.,32, 1262 (1940). Sage, B. H., and Lacey, W. N., IND.ENU.CHEM.,31, 1497 (1939). Sage, B: H., and Lacey W. N., Oil Gus J., 38, No. 27, 189 195 (1939). Weber. H. C., “Thermodynamics for Chemical Engineers”, New York, John Wiley & Sons, 1939. York, R., and Cheverton, E., to be published. York, R., and Weber, H. C., IND. ENG.CREM.,32,388 (1949).
Generalized Thermodynamic Properties of Gases at High Pressures Case School of Applied Science, Cleveland, Ohio
THE
ever increasing tendency toward high pressures in gas reactions has greatly stimulated interest in and the need for accurate and extensive information on the compressibilities and thermodynamic properties of gases at high pressures. For many calculations of both theoretical and applied nature, not only are P-V-T data required, but also such quantities as activity coefficients, heats and entropies of compression, heat capacities at high pressures, and Joule-Thomson coefficients. In order to obtain this needed information for gases, the following methods are generally available: (a) direct experimental study of compressibilities and other thermodynamic properties; (b) experimental measurement of compressibilities, and evaluation from these of the various other properties by graphical methods based on thermodynamic relations; (c) fitting of equations of state to P-V-T data and calculations of the thermodynamic properties by analytic counterparts of the graphical methods; ( d ) estimation of compreseibilities and thermodynamic properties from generalized graphical correlations based on the theorem of corresponding states; and ( e ) estimation of these quantities from analytical correlations based upon the same theorem. The purpose of the present paper is to discuss briefly the ad-
vantages and limitations of the first four of these methods, and to consider in greater detail the possibility of generalized analytic correlation of the thermodynamic properties of pure gases at high pressures. Experimental Estimation of Compressibilities and Thermodynamic Properties
Despite the voluminous literature on P-V-T relations of gases (26), the compressibilities of most gases are still known over only comparatively narrow pressure and temperature ranges. This fact, and the unreliability of some of these data, constitute a serious handicap to the chemical engineer engaged in high-pressure work. Furthermore, as a result of difficult technique and expense of apparatus required, P-V-T data at the higher pressures are forthcoming very slowly. Consequently, it is not at all unusual to find that, for many reactions conducted a t high pressures, compressibility data are either very incomplete or are lacking for one or more constituents of a reacting mixture. Direct experimental determinations of heats of compression, Joule-Thomson coefficients, and heat capacities at high pres-
May, 1942
INDUSTRIAL A N D ENGINEERING CHEMISTRY
sures are even more difficult to carry out, especially with accuracy. The result is that experimentally measured values of these quantities are very meager in the literalure. I n fact, outside of several notable exceptions, such data are practically nonexistent. Graphical Evaluation of Thermodynamic Properties
The various thermodynamic properties of gases are usually evaluated from compressibility data by graphical methods based on thermodynamic relations. If, following Newton $3) and others (16, 18), the ratio of fugacity to pressure, f / P , is defined as the activity coefficient, y, then
545
lead directly to the thermodynamic properties explicit in P and T, and allow also the direct estimation of the gas density. Again, the theoretical equations proposed so far do not cover the extended ranges of pressure and temperature frequently required. For these reasons recourse must be had to purely empirical equations of state which generally permit wider correlation than do some of the theoretical ones. Of these, possibly the most convenient form is the one suggested by Kammerlingh Onnes: PV
=
RT
+ a,P + a*PZ + asp* ...
(6)
As will be shown later, such an equation can be fitted to experimental data covering a wide range of temperatures and pressures.
where the value of the integral may be obtained graphically from P-V-T data by a plot of ( R T / P )--V against P. From the y values thus obtained, the heats and entropies of compression follow from the graphical differentiations called for by the relations:
Finally, knowing AH a t various temperatures and pressures, the heat capacity change on compression at any pressure and temperature can be evaluated graphically from (4)
and the Joule-Thomson coefficient, p, from (5)
AH, AS, and ACp are generally referred to the values of these quantities for a gas at one atmosphere pressure. These methods, in various forms, have been employed frequently. They are particularly well illustrated in the graphical evaluations of f , AS, p, and AC, carried out by Deming on nitrogen, carbon monoxide, and hydroand Shupe (6,7,9) gen from the extensive compressibility data of Bartlett and co-workers (1). Although Deming and Shupe showed that precise information may be obtained by these methods, the processes of plotting, cross plotting, and graphical integration and differentiation required are tedious and laborious. Furthermore, the data necessary for such operations must be complete and of a high order of accuracy. Use of Equations of State
Whenever possible it is highly desirable to represent compressibility data by equations of state. Such equations condense the experimental results into compact form and permit analytic evaluation of the thermodynamic properties without recourse to graphical manipulations. Another advantage lies in the fact that the thermodynamic properties are expressed by analytic functions which facilitate the calculation of these properties under any desired conditions within the range of the equations. Theoretical or semitheoretical equations of state deduced from kinetic considerations lead to expressions which give the pressure as a function of volume and temperature. Such equations are not convenient for engineering purposes for they yield the thermodynamic properties in terms of V and T rather than P and T. As Dodge (1.2)pointed out, it is preferable to have V as a function of P and T,since such equations
Various general methods for the estimation of thermodynamic properties of gases at high pressures are classified and discussed. Particular emphasis is given to the theorem of corresponding states, and deductions are made from it. A critical rbsumd of graphical generalized correlations of compressibilities and other thermodynamic properties of gases is included. The authors' work on the generalization of the BeattieBridgeman equation of state i s reviewed and extended, and equations are deduced to show which properties may b e generalized in terms of P, and T, and their dependence on these variables. A new empirical equation of state of the form PV = RT 4- alP arP2 aaPc aaP4 i s presented, and generalized equations for various thermodynamic propertier are deduced from it. With this equation it is possible to calculate, from T, and Pc of a gas only, the compressibilities of gases up to 1000 atmospheres pressure over a temperature interval of at least 1.5-71 the generalized activity coefficients deduced from this equation are in substantial accord with Newton's generalized plots over the same temperature and pressure interval. Indications are also given that the generalized equations presented will reover the interval covered b y the produce AS, AH, and equation with good or reasonable accuracy, and that AC, values may b e estimated from the equation at T, = 9.0 or above.
+
+
+
All empirical equations involve the evaluation of constants from compressibility data for each gas separately. These constants can be made to reproduce the data for the temperature and pressure range for which they were calculated, but they will not, as a rule, permit successful extrapolation, especially to higher pressures and lower temperatures. However, the authors showed (21) how it is possible, by assuming the theorem of corresponding states, to proceed from the constants of the Beattie-Bridgeman equation of state for a gas like nitrogen to those of any other gas; it is thereby possible to set up with reasonable accuracy the equation of state of a gas without any greater knowledge than the critical temperature and critical pressure of the gas. How the same treatment can be applied to Equation 6 will be discussed later. Once the equation of state is available, the thermodynamic properties can readily be obtained as analytic functions of pressure and temperature by the application of Equations 1 through 5. An instance of such a treatment, applied to nitrogen, has been reported (22).
.
546
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 34, No, 5
Generalized Graphical Correlations
Since compressibility data are lacking in so many cases, and since empirical equations of state are valid only for the substances and pressure and temperature ranges for which they were determined, it is highly desirable to have means of estimating the compressj bility and thermodynamic properties of a gas without being directly dependent upon the compressibility data of the gas. It has been found that compressibilities can most readily be correlated by the theorem of corresponding states. This theorem postulates, in essence, that all gases should obey a universal equation of state when P , V , and T are expressed as ratios of the actual values to the critical constants of the particular gas. Thus, defining V r = V/V,, T , = T / T c , P, = P/P,, the theorem states that for all gases V , = 4 (P,, T r ) ,where the function, 4, is the same in all cases. This theorem,. originally put forward by van der Five-Stage Waals, is readily derivable from any equation of state in three arbitrary constants. I t is, no doubt, based upon many assumptions and is only an approximation, but it has proved extremely useful in the correlation and generalization of compressibilities and thermodynamic properties of gases. From this theorem it may readily be deduced (16) that the compressibility factor, 2 = P V / R T , which is in general a complex function of P and T , should be the same for all gases at the same values of Tr and Pr. The argument is briefly as follows. By definition,
At P, = 0 the perfect gas law is obeyed, 2 = 1, and therefore:
Since, according to the theorem, the reduced equation of state must be the same function for all gases, the quantity RT,/ PcVc must be a constant identical for all gases, and consequently Z should be the same function of T , and P, for all gases obeying the theorem. Thus if 2 for a gas is plotted against P, at various values of T,, the plots should be equally applicable to any other gas; therefore it should be possible to predict from such curves the P-V-T relations of any gas obeying the theorem from a knowledge of its critical pressure and temperature only. This type of correlation has been used extensively by W. K, Lewis and others (3,4,6,16,17,18,19) for estimating the compressibilities of hydrocarbons. Lewis (16) believes that such a correlation is particularly applicable to hydrocarbons because of their constitutional similarity and because the ratio Te/P.V, is constant within experimental error for hydrocarbons containing more than three carbon atoms. He gives a plot, constructed from data on pentane and isopentane a t T , values of about 1.0 and below and from data on ethylene a t higher values of T,, showing 2 as a function of Pr and T , and covering the range T , = 0.75 to 1.7 and P, up to 8-9. Comparison of experimental 2 values for methane, ethylene, and isopentane with the plot showed good agreement in most cases, the greatest deviation amounting to 8 per cent. A similar graph was constructed by Dodge ( l a ) from data on methane, ethylene, ammonia, hydrogen, nitrogen, and carbon dioxide. This graph is not confined to hydrocarbons and extends up to T,= 17 and P, = 95. When tested a t one tem-
Norwalk Compressor Which
C a n Develors Pressures
30,000 Pounds per Square Inch
UD to
’
perature only with data on ethylene, the agreement was found to be very good the maximum deviation being 5.5 per cent. If 2 is a generalized function of P, and T,, it follows that certain other thermodynamic properties should also be subject to generalization in terms of these variables. Thus Watson and Smith (SO) showed that y, AHIT, AC,, and pC, (PJT,) should be generalized functions of T , and P,. They constructed a compressibility chart based on a large number of gases and covering the range T , = 0.6 to 15 and P , up to 30, and then proceeded to develop from it by graphical methods plots for the quantities mentioned as functions of T, and P,. Although the plots are given, no comparison is presented with experimental data. An extensive graphical correlation of activity coefficients in terms of the reduced variables was made by Newton (IS). Newton evaluated from compressibility data the activity coefficients of a large number of gases of various types, plotted them against P, a t constant T,, and drew average curves through the points. His published charts give y as a function of P, a t even values of T , over the interval T , = 0.7 to 35 and P, up to 100. Extended comparisons between the y values read from the curves and those determined graphically from compressibilities show a concordance remarkably good in most cases. I n general the deviations are of the order of 2 per cent, although in several instances these rise to as much as 10 per cent. One point that should be mentioned in this connection is Newton’s observation that for hydrogen, neon, and helium the reduced temperatures and pressures must be defined as T,’ = T / ( T , 8) and Pr’ = P / ( P , 8) in order to bring these gases into correlation with the other gases. Once these definitions are made, the gases mentioned fall in very well with the generalized curves. Newton and Dodge (24) called attention to the fact that Newton’s generalized y values may be utilized for the estimation of other generalized properties, and proceeded to apply them to the calculation of Joule-Thomson coefficients. The estimated coefficients when compared with observed values in several instances showed deviations ranging from 5 to 25 per cent. The pressures a t which tests were made did not exceed 200 atmospheres. Other graphical generalized correlations given in the literature involve principally hydrocarbons. The various authors
+
+
May, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
(16-19, 26, 28) who give 2 plots for hydrocarbons usually accompany them with y plots covering the same T,and P r range. Using the compressibility factors for the higher hydrocarbons evaluated by Brown, Souders, and Smith (4), Watson and Nelson (29) formulated an empirical equation for AH/T in terms of P, and T,, and presented a plot covering the interval T r = 1.0 to 1.5 and P, up to 6. A single comparison for a naphtha fraction a t a pressure around 65 to 70 atmospheres gave an agreement of the order of one per cent. A somewhat different approach to a generalized correlation of thermodynamic properties of hydrocarbon vapors was given recently by Edmister (13). He calculated first for a number of hydrocarbons the reduced volume residuals, a r , defined by the relation,
where
ac =
547
of state because this equation was shown to reproduce well compressibility data over a fairly wide range of temperatures and pressures, and because the equation, a t least in approximate form, can be obtained explicit in volume, as shown by Beattie (2). The volume form of this equation is
where 8, yB, and 6 are virials given by:
yB =
RTBob
+ Aoa - RBQC ~2
critical volume residual
and then placed the smoothed arvalues for the various gases on a generalized plot covering a range of T,values from 0.8 to 2.5 and P, values up to 6. He next calculated the maximum deviations in the vapor volumes estimated from his a,us. P, charts and compared them with the maximum deviations from the 2 plots of Brown (4) and Lewis (16). He showed thus that the residual plots yielded deviations from 3.5 to 9.8 per cent, while the 2 plots gave deviations from 2.8 to 14.5 per cent. Edmister also determined graphically from his archarts other generalized thermodynamic properties proportional to y, AC,, AS, and AH, and demonstrated that his estimated values of AH agreed with those of Sage, Webster, and Lacey (27) within about 20 calories. I n summarizing the attempts at graphical generalization of the thermodynamic properties of gases, it may be said that, while extensive tests have not been made in the case of most properties, the indications are that the thermodynamic properties of hydrocarbons may be estimated from generalized plots with satisfactory accuracy up to pressures of 200-300 atmospheres. I n the case of other gases, Newton showed that the activity coefficients may be obtained with good accuracy in most instances over a wide range of T, and Prvalues. Finally, although extensive comparisons with generalized correlations have not been carried out for compressibilities and properties other than y for nonhydrocarbon gases, Newton's success with the generalized activity coefficients would seem to indicate that other properties as well may be estimated with reasonable accuracy by a generalized approach. Generalized Analytical Correlations
Newton's ability to correlate the activity coefficients of gases in terms of Tr and P, over wide ranges of temperature and pressure suggested to the authors (BO, 91) that it should be possible to deduce the nature of the generalized functions for y, compressibilities, and other thermodynamic properties from a suitable equation of state explicit in volume. If this could be accomplished, such a treatment offered a number of inviting prospects. First, there was the possibility that a generalized equation of state could be developed which would permit the calculation of P-V-T data of gases from critical data alone. Secondly, analytical relations could be deduced for the thermodynamic properties which would allow the estimation of these properties for many gases with a minimum of information. Finally, such a treatment, based on a single gas as a reference, would offer a more critical test of the principle of corresponding state than the present practice of averaging graphically the properties of a number of gases. After considering various known equations, it was decided to try out the idea first with the Beattie-Bridgeman equation
Ao,BO,a, b, and c are constants characteristic of each gas and must be evaluated from the compressibility data of the gas. To obtain a generalized analytic equation for y from Equation 10, we proceed as follows: Inserting Equations 10A, 10B, and 1OC into 10, and then substituting this value of V into Equation 1, we obtain on integration for In y:
To convert P and T t o the reduced variables P, and Tr, we can make the substitutions P = P,P,and T = TcTr. Then Equation 11 becomes:
The coefficients of the reduced variables in Equation 12 are in terms of the Beattie-Bridgeman and critical constants of a specific gas. These coefficients may be evaluated from the constants of any gas obeying the law of corresponding states, and if the law is valid, these coefficients should be the same then for any other gas. The authors (20) chose to evaluate these coefficients by using the Beattie-Bridgeman and critical constants for nitrogen as given by Deming and Shupe (8). Nitrogen ww selected because, according to Newton's correlation ($8) it obeys the law of corresponding states fairly well, and because the Beattie-Bridgeman equation for this gas as set up by Deming and Shupe is valid over a Tr interval of 1.6 - 7 and P, up to 7 at the lower temperatures and P, up to 30 a t the higher temperatures. With these constants Equation 12 reduces to: 0.06477
loglo Y =
(T
0.1706 - 004334
-T:
+
.
0 002715
T ) '(
+
T:
Equation 13 should be the generalized expression for y and, if correct, should be the equation for the curves given by Newton. This equation was checked (20) against Newton's plots over a T, range from 0.7 to 10 and over a P, range up to 15. The agreement was found to be good except at the higher
548
INDUSTRIAL AND ENGINEERING CHEMISTRY
pressures near the critical temperature. At T , = 1.3 and above, the deviation was no greater than 4 per cent up to Pr = 13. It has already been pointed out that, according to the theorem of corresponding states not only y but also 2,AH/T, AC,, and pC, (P,/T,) should be functions of P, and T , only. It should be possible, therefore, t o derive from Equation 13 expressions for these quantities. We may illustrate the derivation of 2 by utilizing the following thermodynamic relations (15): F-F"=RTlnf=RTln~P
where F
Fo
= =
Vol. 34, No. 5
generalized and the relations which connect them to the generalized activity coefficients. Introducing T = T,T, into Equation 2 we have for AHJT,: AH =
- RT:
T,
b In (&)
Pr
From Equation 3 it may be shown that (AS be a function of T , and P, only:
+ R In P) should
(14)
Further, since 21 as:
molar free energy at pressure P molar free energy at unit fugacity
3 AX [m,]
Pr
=
T,
ACp follows from Equation
Assuming P oto be independent of pressure (i. e., y = 1when P = l), and differentiating Equation 14 with respect to P at constant T , we find:
Finally, on solving for fiC, from Equation 5 ,
From Equations 15 and 16 it follows that,
Evaluating
[e],
from Equation 20, and inserting into
Equation 23, we obtain for pC,Pc/Tc: or in terms of the reduced variables,
On evaluating from Equation 13 and inserting in Equation 18, the generalized equation for 2 becomes: Z =1
0.04334 + 2.303 (T0.06477 - 0.1706 ~ , -a T ) p 4.606
o.oo2715
r
0.005165 o.ooo3i3i - -T9- r ) P P
( 7 +
+
-
6.909 (0:00002669) T," p:
(19)
Knowing T , and P , for a gas, it is possible from Equation 19 either to calculate the compressibility factors of the gas or to write an equation of state for the gas giving V as a function of P and T. The authors did not test Equation 19 directly. Instead, a method was developed ( d l ) from Equation 12 whereby i t became possible t o estimate the Beattie-Bridgeman constants of any gas from those of nitrogen with a knowledge of P, and T o of the gas only. This method was tested, in conjuction with the Beattie-Bridgeman equation of state, on a large number of gases and vapors, including ammonia, nitric oxide, hydrogen, oxygen, carbon dioxide, methane, ethane, ethylene, propane, methyl chloride, ethanol, and methanol, by comparing the compressibilities calculated by means of these constants with those observed and those evaluated from the van der Waals equation. The results indicated that, with the exception of the critical region, these constants along with the Beattie-Bridgeman equation of state reproduced the compressibility data. of the gases listed much better than the van der Waals equation; and, further, that this method of arriving a t the Beattie-Bridgeman constants could be employed to calculate compressibilities of various gases up to 100-400 atmospheres. The pressure up t o which the calculations could be pushed depended on the value of T,. Generalized equations for the remaining thermodynamic properties can be arrived a t in a manner similar t o the one given for 2. These derivations will not be given in detail beyond indicating the nature of the functions which may be
Thus, from Equation 13 or from any other relation giving y as a function of T , and P,, it is readily possible through Equations 18, 20, 21, 22, and 24 to obtain the indicated thermodynamic properties as generalized functions of P, and T , only. The relations were set up in terms of the y values rather than in other mays because it is for this property more than for any other that the principle of corresponding states has been established to be quite generally valid. No further attempt was made t o extend the calculations beyond the point mentioned for a reason which will become apparent. However, the results obtained in the calculation of generalized activity coefficients and compressibilities of gases indicate that generalized equations derived from more extended equations of state are capable of reproducing compressibilities and other thermodynamic properties of gases much more accurately than the simpler equations of state involving only two constants. Furthermore, by following the procedure outlined above, it is readily apparent that any equation explicit in V may be generalized in the same manner as the modified Beattie-Bridgeman equation to give the thermodynamic properties of gases as functions of Pr and Tr only. In view of these conclusions and in view of the fact that these results were obtained with the approximate form of the Beattie-Bridgeman equation which holds a t best only to P, values of 6-7 at the lower temperatures, the authors felt that the accuracy of the correlations and the pressure range covered could be increased by a more extended equation of state for nitrogen. An attempt was made, therefore, t o obtain for this gas an equation in V = $(P, T ) which would be applicable up to 1000 atmospheres over the temperature range -70" to 600" C. The equation chosen was of the Kammerlingh Onnes type: PV
=
RT
+ aiP + CUP +
asp3
+ e4P4
(25)
where al,a2,etc., are virial coefficients which depend on the temperature only. The virials were evaluated a t a number of temperatures by least squares from the smoothed nitrogen data of Bartlett and co-workers as tabulated by Deming and
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1942
Shupe (9). The simplest functions relating the virials to T were then sought, and it was found that the following equations gave the most satisfactory agreement:
The constants ax, a2 . . . ., bl, bz . .. ., etc., were also evaluated by the method of least squares and are listed in Table I.
---
TABLE I. CONSTANTS OF EQUATION 25 FOR NITROGEN -6.660 x 10-8 3.920 X 102 -2.089 x 107 2.703 X 10-8 dz = -0.2829 da = 1.314 X 10'
8 1 = 3.835 X 10-2 az = -10.07. 88 -2.449 x lob ba 6.594 ba = -1.217 X 106 bs = 8.217 X l o 9
c1 ca c8 di
--
549
(me)
(es)
Letting the constant coefficients alPe = AI, bi P a . -, . etc., Equation 27 may be written more simply as:
"
=
BI,
I
Equation 27 should be a generalized equation for y. The coefficients of this equation were evaluated from the constants of Table I and the critical constants of nitrogen, and some of the y values calculated were compared with those read from Newton's curves. These comparisons (Table 11) must still be considered fragmentary, but they do indicate what may be expected. At Tr = 1.8 and above, the two sets of values agree to about 3 per cent. At T. = 1.6, the lower end of the correlation, and a t T, = 1.5 the agreement is only within about 8 per cent. I n general, however, the results of Equation 28 agree much more satisfactorily with Newton's curves over a wider P r range than do those calculated from Equation 13. A generalized equation for 2 may be obtained from Equation 28 by performing the operation indicated in Equation 18. The result is:
The application of Equation 25 to the calculation of the molar volumes and the thermodynamic properties of nitrogen was published elsewhere (22). Suffice it to say here that this equation reproduces the experimental volumes for nitrogen over the entire temperature interval and up to 1000 atmospheres pressure with a maximum deviation of less than 3 per cent, an accuracy for the whole range considerably higher than that attained by Deming and Shupe with the exact form of the Beattie-Bridgeman equation. Thus, the reduced interval covered by Equation 25 is Tr = 1.6 to 7 and P,up to about 30.
By applying to Equation 29 a treatment similar to the one used before with Equation 19, it should be possible to deduce from it equations of state €or various gases from a knowledge of Toand P, of these gases alone. Such equat,ions may be expected to be valid over the same T, and P, interval as Eauation 25. To test this point. eauations of state were formulated for &carbon*monoxide, methane, oxygen, and hydrogen from Equation OF CALCULATED ACTIVITY COEFFICIENT.5 WITH TABLE 11. COMPARISON 29 and the critical constants of these gases only. THOSIE OBTAINEDPROM NEWTON'S CURVES Although the details of the procedure cannot be -Tr = 1.5-Tr 1.6-Tv = 1.8-Tr 2.0-Tr 5.0pr yoslod. ycalod. yn yoalod. yn yoalod. yU yoalod. YU discussed here, the results thus obtained may 2 0.82 0.86 0.87 0.90 0.92 0.93 0.96 0.96 .. . be indicated. Table I11 gives the volumes of t h i 4 0.73 0.76 0.79 0.82 0.88 0.88 0.94 0.93 6 0.68 0.72 0.75 0.79 0.86 0.86 0.93 0.92 . ... gases mentioned calculated a t 0" C. up to 1000 8 0.66 0.71 0.74 0.78 0.86 0.87 0.94 0.93 10 0,67 0.73 0.75 0.81 0.87 0.89 0.96 0.96 i.*i6 i.'i7 atmospheres by means of these equations of state, 12 0.70 0.76 0.78 0.83 0.90 0.93 1.00 1.01 1.20 1.21 and the percentage deviation of these calculated 15 0.78 0.82 0.84 0.90 0.97 1.00 1.07 1.10 1.26 1.28 20 0.97 1.01 1.02 1.09 1.15 1.19 1.25 1.29 1.37 1.40 volumes from the observed (10,11, 14). The 25 ... agreement in all cases except methane must be 30 ... ... ... ... ... ... ... ... ... ... .,. ::: ::: i:: i:: considered very good. I n case of methane the maximum deviation, about 7 per cent, must still be considered satisfactory when it is remembered This equation can now be applied to the calculation of y and that 0" C. for methane corresoonds to a reduced temoerature the other generalized properties. Substituting for V in of 1.43, a temperature consfderably lower than t h i lowest Equation 1 from Equation 25 and performing the indicated temperature (T,= 1.6) for which Equation 25 was set up. integration, we find: The results quoted are typical of those obtained at other temperatures, both below and above 0" C. If anything, the results improve as the temperature is raised, the maximum I n 7 -&[alp+ ( T ) P z + (3)Ps+ (26) deviation for methane being only 3.3 per cent at 1000 atmosInserting the virials from Equations 25A to 25D and setting pheres and 100" C. P = Pep,and T = TcTr, Equation 26 becomes: The superiority of the relations deduced from Equation 29 over such equations of state as the Berthelot, van der Waals, 1n 7 pr or Dieterici is borne out by Table IV. THe observed compressibility factors are given for nitrogen, hydrogen, and bePa 1 oxygen a t 0" C., and for comparison the compressibility fao+ (2*) + -k tors as calculated by our equations and the equations of state mentioned. The agreement for nitrogen is expected to be close, but the excellent concordance of oxygen and hydrogen supports the contention that extended equations of state (27) along with the principle of corresponding states, may be em6
P
... ..... ...
(:)PI
[(go) + (%)++ (g) &] +
(s) &Ip:
550
Vol. 34, No. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
r
interval with a maximum deviation of 103 calories per mole, although in most cases the differences were of the order of 10 t o 30 calories. The Joule-Thomson coefficients, except for pressures above 200 atmospheres a t -70" c., were P. Atm. Vdod. tion Vcalod. tion vdod. tion Voaiod. tion satisfactory throughout. Finally, the heat 0.9104 25 0.8842 A O . 0 ... ... ,.. *O.O ,., o,;ig5 ~ 0 : capacities ~ for nitrogen came out satisfactorily ... ... 30 0.59.83 2 0 1 1 at all pressures above 0' C. At 0' C. and be36.2 O.Yi23 d0:O 0.39.56 50 0.4i82 *0:1 low, the agreement was good only a t the lower ... '612 ... 54.74 0.39.01 .0:1 75 0.29io .0:2 ... ... 0.3f30 db:O o,yi44 .il2 pressures. Comparisons were made with the 80 100 0.21'84 '0:3 0.2%~ '012 0.1827 4.0 values obtained graphically by Deming and 150 0.1477 0.2 ... . * . O.is'85 ... LO:O ... o.'ifoe '1317 Shupe (6,7,9). 160 200 O.ii'40 -012 O.ibi9 '6:5 O.'f%9 :6:2 0.0897 5.1 From the results obtained with nitrogen, from 0.0672 1 . 5 0 . 3 0.0898 0.0714 0 . 6 1 . 0 0.0829 300 400 0.0694 -0.8 0.0577 - 2 . 1 0.0714 -0.4 0.0598 2.2 the ability of Equation 28 to predict generalized 5.0 0.0507 0.0003 -0.4 0.0506 -2.3 0.1 0.0623 500 o.0579 1.3 o.0466 -1.7 o,0530 -0.5 0.0547 7.1 activity coefficients, and from the success of 600 700 o.'d523 .i:2 0.0440 o,0422 -0.7 o.2 o.;i40 *6:1 ,,.;iO5 '+:o Equation 29inyielding compressibilitiesof gases 800 0.0408 1.1 up to 1000 atmospheres pressure, we may expect 900 o.Oi81 '219 0.0395 1.5 o.O&a 'i:i o.bk'55 'i:s 1000 that the generalized equations here deduced will reproduce A S values with good accuracy and AH values with fair accuracy over the entire reduced pressure and temperature range covered by Equation 25. Good concordance in p may also be ployed effectively for the generalized correlation of compressiexpected over most of the range. However, the values of AC, bility data. Through the relations given in Equations 20,21,22, and 24 as predicted by Equation 32 may be expected to be accurate we may readily derive from Equation 28 the generalized exa t the high pressures only a t T , = 2.0 or above. The authors propose to test these expectations, in so far as possible, pressions for the other thermodynamic properties. Thus we in the near future. find that: I n the meantime, the indications are that the analytical AH 2A 4A 3B1 5B 7B equations for the generalized thermodynamic properties given -= in this paper may be employed to good advantage as follows: $ 4) T , pr 4- ( F f p: T, For T,values of 1.5 and above and for P, up t o 20-30, Equa50 70 tions 25 to 33 may be used for the estimation of the properties + + p: + + + (30) given. For T, values of 1.2 to 1.5 and P, up to 10-12, Equation 13, based on the modified Beattie-Bridgemsn 3A 4B 6B equation and the relations deduced from it may be used. (ds4- In )' 7$ pr + p: 4- For values of T,a t and near the critical temperature, the 7$ latter relations can be applied only t o much lower values of P, . A more specific definition of the range of applicability c ithese equations or the accuracy t o be expected will have to be postponed until further calculations in progress in this laboratory are more advanced. AC, = - R [ ( $ + p, + + 20B --F;-" + 42B3 -)p; + TABLE 111. COMPARISON OF CALCULATED AND OBSERVED VOLVMES OF SEVERAL GASESAT 0" C. Ht,Tr 6.85. CHI, Tr a 1.43 0 2 , Tr 1.77 co. TI= 2 . 0 5 "lo. 4 &via% deviadeviadevia-
-
-
w
$ $) (% -$
[
(s $$ @)
I:')$.
(%
)
[($
$)
q)
($$+
20Cz
($$ T , T: 6D 200 p: -I- (4 T, +7 T , '+ F) P:]
e)
+ 42C
(32)
Nomenclature P = pressure, atmospheres
T
+ (%+ ! . ! +! E $ $) i p, +
2Az + 4A -3)
=
absolute temperature,
V = volume, liters
' C.
R = gas constant, liter-atm./o C./mole HI F , S , C, = heat content, free energy, entropy, and heat capacity per mole of a gas at pressure P T: T, P: (TI F* T 3 ) P:] (33) H O , F O , S O , C; = same quantities for a gas at 1atm. pressure A H , AF, AS, AC, =,change in heat content, free energy, entropy, and heat capacity of a mole of gas on compression from_] atm. t o any pressure P These equations have not yet been tested on gases other than j = fugacity of a gas the reference gas, nitrogen. I n the case of nitrogen Equation y = activity coefficient of a gas = f / P = ComPressibilitYfaCtorofagas = PV/RT 25, from which these relations are derived, gave AS values with satisfactory accuracy over the entire pressure and tem~ , , : ' ~ temperature ' ~of gas ~ perature range. AH values were reproduced over the entire P., v,, T. = critical pressure, volume, and temperature of a gas p C pP Oz =
-R [(A~ + 15cz 21c
(3+
+ +) +
120
+
200
+
280
Fr,
TABLE Iv.
COMPARISON O F OBSERVED
P,Atm. 100 200 300 400 500 600 800 1000
Obsvd. 1.00 1.04 1.14 1.26 1.39 1.52 1.80 2.06
VALUES WITH
THOSECALCULATED B Y VARIOUS EQUATIONS
N2
? . .
Calcd. 0.99 1.05 1.13 1.25 1.39 1.54 1.52 2.06
Berthelot 0.98 0.99 1.03 1.10 1.18 1..26 1.46 1.66
0 2
Di$t,e- Van der rici Waals 0.93 0.91 0.98 1.10 1.26 1.43
1.78
2.14
0.95 0.99 1.13 1.30 1.48 1.67 2.05 2.34
Obsvd.
Calod.
0.93 0.91 0.96 1.08 1.15 1.27 1.50
0.93 0.92 0.96 1.03 1.13 1.25 1.51 1.76
1.74
Berthelot
DiFteria
Van der Waals
Obsvd.
0.93 0.87 0.85
0.86 0.77 0.78 0.87 1.00 1.13 1.42 1.70
0.89 0.86 0.96 1.10 1.24 1.40 1.72 2.01
1.06 1.13 1.20 1.28 1.35 1.43 1.58 1.72
0.88 0.93 1.00 1.16 1.34
O F STATE AT 0' H, Berthe- DieteCalcd. lot rici
1.06 1.13 1.20 1.27 1.35 1.42 1.57 1.75
1.03 1.13 1.19 1.25 1.32 1.39 1.62 1.64
1.05 1.20 1.31 1.42 1.54 1.65 1.89 2.12
c.
-
Van der Waals 1.04 1.17 1.27 1.39 1.50 1.62 1.86 2.08
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1942
8, rB,6 = virial coefficients in Beattie-Bridgeman equation of
state Ao, Bo,a , b, c state
=
constants of Beattie-Bridgeman equation of
a1 an, as, a4 = virial coefficients in Equation 25 al, an . . . , bl, bz . . . , etc., = constants in Equations 25A to 2 5 0 AI, An, . , B1,B,, . . ., etc. = constants in Equation 28 as defined by 27 Literature Cited
..
(1) Bartlett, Cupples, and Tremearne, J . Am. Chem. SOC.,50, 1275 (1928). (2) Beattie, Proc. Natl. Acad. Sci., 16, 14 (1930). 26, 825 (1934) (3) Brown, Lewis, and Weber, IND.ENQ.CHEIM., (4) Brown, Souders, and Smith, Ibid., 24, 513 (1932). (5) Cope, Lewis, and Weber, I W . , 23, 887 (1931). (6) Deming and Deming, Phys. Rev., 45, 111 (1934). (7) Ibid., 48,448 (1935). (8) Deming and Shupe, J. Am. Chem. SOC.,52,1382 (1930). (9) Deming and Shupe, Phys. Rev., 37, 638 (1930). (10) Ibid., 38, 2245 (1931). (11) Ibid., 40,848 (1932). I
551
(12) Dodge, IND.ENQ.CHEM.,24, 1353 (1932). (13) Edmister, Ibid., 30,352 (1938). (14) International Critioal Tables, Vol. 111, pp. 3-17, New York, McGraw-Hill Book Co., 1928. (15) Lewis, G. N., and Randall, Merle, “Thermodynamics”, New York, MoGraw-Hill Book Co., 1923. (16) Lewis, W. K., IND.ENQ.CHEM.,28, 257 (1936). (17) Lewis, W. K., and Kay, W. C., Oil Gas J., 32, No. 45,40 (1934). 25,725 (1933). (18) Lewis, W. K., and Luke, C. D., IND.ENQ.CHEM., (19) Lewis, W. K., and Luke, C. D., Trans. Am. Soc. Mech. Engrs. 54, 65 (1932). Maron and Turnbull, IND.ENQ.CHEM., 33, 246 (1941). Zbid., 33,408 (1941). Maron and Turnbull, J . Am. Chem. SOC.,64,44 (1942). Newton, IND.ENQ.CHEM., 27, 302 (1935). Newton and Dodge. Ibid.. 27.577 (1935). Piokering, Bur. Standards, C&. 279 (1925). Sage, Schaafsma, and Laoey, IND.ENQ.C H ~ M26, . , 1218 (1934) Sage, Webster, and Lacey, Ibid., 29. 658 (1937). Selheimer, Souders, Smith, and Brown, Ibid., 24, 515 (1932). Watson and Nelson, Zbid., 25,880 (1933). Watson and Smith, Natl. Petroleum News, July 1. 1936.
Degrees of Freedom in Multicomponent Absorption and Rectification Columns c. R. q
u
e. c. R&
Massachusetts Institute of Technology, Cambridge, Mass.
A general analysis of multicomponent interphase contacting systems i s presented to illustrate the restrictions and limitations inherent in any method of rectifier, absorber, or extractor process design. This type of fundamental analysis should be of value to the design engineer in formulating specifications which are consistent with the limitations imposed b y the general laws governing the system in question. The major difficulties encountered in these calculations arise from the practical necessity of “fixing” more variables than are independent in order to expedite the process design as a whole.
WITHIN
the last ten years many articles on multicomponent absorption and rectification have appeared, each advocating some special system of design (1, 8, 3). The general problem is complex, and many of these contributions have proposed the use of numerous simplifying engineering approximations to arrive at a practical solution. I n most cases the work has been handicapped by the absence of a rigorously correct analysis of the degrees of freedom for the general multicomponent case of countercurrent multistage interphase contact. The purpose of this article is to develop such a theory for the case of equilibrium contacting equipinent, the results being applicable to any case involving countercurrent physical interaction (mass transfer) between two streams.
Before design can proceed on any chemical engineering system, it is necessary to conduct an analysis to determine how many of the design variables may be arbitrarily h e d before the system as a whole becomes physically fixedthat is, before the remaining design variables can be computed from the equations controlling the operation of the system. For example, in a simple rectifying column operating on a given binary mixture a t atmospheric pressure, the composition and condition of the feed, the composition of the distillate and bottoms, and the reflux ratio above the feed plate are fixed ; then the usual simplifying assumptions made on an ordinary McCabe and Thiele diagram make i t possible to calculate the optimum’ number of theoretical plates required to effect the separation. The simplifying assumptions (7,8) in this case lead to so few and simple equations that there is no doubt regarding the number of variables which must be fixed to fix the system. Introduction of a third component results in considerable complication, even though the usual simplifying assumptions be retained. Experience shows, for instance, that having specified feed composition and condition, column pressure, and reflux ratio, it is no longer possible to specify the complete composition of distillate and bottoms; the concentration of one component in the overhead and in the bottoms is sufficient to fix the optimum number of plates absolutely. Specification of one more terminal concentration will fix the number of plates a t some value not necessarily the optimum, and specification of a fourth terminal concentration would in general be inconsistent with the first three concentrations 1 “Optimum” is not here used in the sense of an eoonomic optimum resulting from oomplete eoonomic balance but in the sense of the minimum number of plates resulting ahen feed is introduced on the proper plate.
