INDUSTRIAL AND ENGINEERING CHEMISTRY
1738
HORIZONTAL SECTIOK.Assume average p in this section = 1500 lb./sq. ft.; average pA = 0.054 lb./cu. ft. If only air were flowing, head loss due to friction would be:
Vol. 40, No. 9
volving suspensions of fine particles in large pipe, as no experimental data are available for this range. NOMENCLATURE
dI/a(w
- p)
gda = d I / a
v
X 62.4 X 1.28 X 0.054 X 32.2 (0.158/122 0.018 X 0.000672
E
848
A
D f
= empirical constants = ipe diameter, feet
== Flocal riction factor acceleration due to gravity
A = 26,000 (from Figure 7)
g g.
k = 0.93 (from Figure 8)
mass pounds-feet 32’17 force pounds-(seconds) L,k , , k , = empirical constants R = ratio of average axial particle velocity to the average fluid velocity L = length, feet m,n = mass ratio of flow of fluid and solid, respectively, pounds per second p = pressure, pounds per square foot r = weight ratio of solid t o fluid flowing ua, up = average velocity of fluid and solid, respectively, feet per second ur = velocity of fluid relative to that of particle (u,- u,l v = specific volume, cubic feet per pound W = energv, feet per pound w = solid density, pounds per cubic foot X = vertical distance above a given datum, feet a: = relative pressure drop, dimensionless p = viscosity of fluid, pounds per foot second p = fluid density, pounds per cubic foot
a:
-1
a:
-
=
A
(6 x g)k
2);(
1 = 26,000
(.3- )z 0 108
o!
(0l4 79.8
98,000
= 4.75
Wj.lirt = 4.75 X 1160 X 0.185 = 1020 ft. lb./sec. Substituting in 17a gives: 10,300 X
$: t:ii
+ 5.08 x
1 0 8 [pi -
&]
= -1020
By trial and error: p2 = 1309 lb./sq. ft. The ressure at the blower intake is then 1309 pounds per square root of 9.1 pounds per square inch absolute and the pressure drop through the line is 5.6 pounds per square inch.
= conversion factor in Newton’s second law =
BIBLIOGRAPHY
CONCLUSIONS
The data of this investigation together with published data on the pneumatic conveying of wheat indicate that the pressure drop due to friction in flowing suspensions can best be represented by the folloying type of equation:
where A and k are functions of the dimensionless group 4 1 : 3 ( ~ p)pg@
IJ
The data are insufficient t o evaluate the effect of particle shape but indicate that this factor is not of primary importance. The above equation should not be used for design work in cases in-
(1) Badger, W. L., and McCabe, W. L., ”Elements of Chemical En-
gineering,” New Yorlc, McGran~-HillBook Co., 1936. (2) Cramp, W., J.SOC.Chem.lnd.(London), 44.207-B (1925). (3) Cramp, W., and Priestly, J. F., Engineer, 137, 34 (1924). (4) Cramp, W., and Priestly, J. F., J. SOC.Brts., 69,253 (1924). (5) Gasterstadt, H., 8.D.I. Forschungsarbeiten, No. 265 (1924). (6) Hudson, W. G., “Conveyors and Related Equipment,” New York, John Wiley & Sons, 1944.
(7) Perry, J.,H., “Chemical Engineer’a Handbook,” New York, McGraw-Hill Book Co., 1934. (5) Segler, W.. “Untersuchungen an Kornergeblason und Grundlagen fur ihren Berechnung,” Mannheim, Wiebold Co., 1934. (9) Van Driest, E. R., J . A p p l i e d Mechanics, 12, A 3 4 (1946). (10) Wood, S . A , , and Bailey, A . , Proc. Inst. M e c h . Eng. (London),
142, 149 (1939).
RECEIVED J u n e 11, 1947.
Fugacities in Gas Mixtures JOSEPH JOFFE Nezcark College of Engineering, ivewark, N . J .
F
UGBCITIES of components of gas mixtures are of importance in the study of chemical equilibria in gaseous systems at high prbssures (.5). In view of the difficulties inherent in the determination of fugacities of individual coinponents in mixtures, the Lewis-Randall fugacity rule, which is based on the law of additive volumes: has come into general use. Accordi:ig to this rille the fugacity of the ifk component in the mixture, j ~is, given by j &= fpxi (1) wheref: is the fugacity of the pure ith component a t the temperature and total pressure of the mixture, and zi is the mole fraction of the ith component in the mixture. FUGACITIES OF CORIPOSENTS OF GAS MIXTURES
By employing the Levi-is-Randall fugacity rule in conjunction with generalized fugacity charts, n’ewton and Dodge have shown
that the effect of pressure on the equilibrium constant of a gaseous reaction may be predicted with the aid of critical data alone ( 1 7 ) . While the method of Kewton and Dodge gives good results for the ammonia equilibrium which they studied, its validitv rests on tho applicability of the Lev, is-Randall rule, which is known to fail in some gaseous systems at high pressures (18). In this paper a method is presented for the evaluation of fugacities of components with the aid of generalized charts, which is similar in principle to the mrthod developed by Gamson and Watson (3)in connection with high pressure vapor-liquid equilibria, but which is believed to be applicable over a different range of conditions. This method does not involve the assumption of the Leais-Randall fugacity rule and should prove useful in thc study of chemical equilibria at high pressures. It is assumed, 111 accordance with the suggestion macle by Kay (11), that the pseudocritical temperature, To, and the pseudocritical pressur c, P,, of the mixture are given by the relations:
1139
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1948
I n addition, we have the relations ( 1 ) :
A
-=
method is presented for the calculation of fugacities of' individual components in a gas mixture. The method is similar to that of Gamson and Watson, but is applicable over a different range of conditions. Further, an exact thermodynamic relation for the fugacity of a gaseous mixture has been derived. By applying tho Lewis-Randall fugacity rule, an approximate relation has been oljtained which can be used to compute the fugacity of the mixture from fugacities of the pure components at moderate pressures.
3Tr
. T,(H"
- H)/RT2
(10)
in which H i s the molal enthalpy of the mixture a t the given temperature and pressure, H o is the molal enthalpy of the mixture at a low pressure a t which the gas is ideal, and z is the compressibility factor ( z = P V J R T ) for the mixture. Substituting Equations 8, 9, 10, and 11 into Equation 6, there follows : -dln = fm
a&
(T,
- T i ) ( H o- H ) / R T T , + (P, - Pi) (z
- l)/Pc
(12)
Substituting this result into Equation 6 and in view of Equations 2 and 3,
+ ~ z T z+ . . . P , = XIP1 + X2PZ + ' . . Tc = XiTi
(2)
(3)
..
in which xi, 2 2 , . are the mole fractions of the Components, T I , T 2 ,. , . are the critical temperatures of the components, and P1, Pz,. . are the critical pressures of the components. Kay's relations have been tested and have been found t o be sufficiently accurate for many applications (19). In a mixture of n components the partial molal volume of the first component is given by (1):
.
log
(71,'~i)
= logf,
+ (Tc - T i ) ( H " - H)/2.303RTTc + (Pc
- P ~ ) ( -z
1)/2.303P, (13)
Similar equations may be written for the fugacities of the other components. With the aid of Kay's relations for the pseudocritical temperature and pressure of the mixture, the fugacity of the mixture, fm, is obtained from a generalized fugacity chart (do), (H' --H) for the mixture is obtained from a generalized enthalpy-pressure chart (8, do), and P is read from a generalized compressibility factor chart (bo). Substituting these values into Equation 13, the fugacity of the component in the mixture is calculated. In connection with high pressure vapor-liquid equilibria Gamson and Watson have proposed the equation (S),
where V is the molal volume of the mixture. On' integrating both sides of Equation 4 with respect t o pressure, there follows from the definition of fugacity the relation:
wherefl is the fugacity of the first component in the mixture and the fugacity of the mixture. The derivatives of the fugacity of the mixture with respect to the composition variables may be evaluated by considering the fugacity a function of the reduced temperature, Tp, and reduced pressure, P,, of the mixture. A t any given temperature and pressure the reduced temperature and reduced pressure are functions of the mole fractions by virtue of Kay's Equations 2 and 3. By employing the usual techniques of partial differentiation we have:
fin is
a ~ , --+ax,
bln fm -=-.
bln fm
ax,
dT,
dlnf,
,
ap,
bP, axa
(6)
Remembering that there are TZ - 1independent composition variables, Equation 2 may be written:
To = XITI
+ ~ 2 T z+
'
t
+ (1 -
XI
- ~2
-
'
*
- ~n-1)Tn
(7)
I t follows that a r r =
axd
b (T/T.j axi
= -(T/TZ)(Ti
- Tn)
(8)
Similarly, -apr =
3XS
-(P/P:)(P,
- P,j
(9)
which is analogous to Equation 13. A special generalized chart (3) must be used for the evaluation of the integral appearing in Equation 14. Equation 13 has been tested against Gibson and Sosnick's data on fugacities of argon and ethylene in four argon-ethylene mixtures at 25' C. (4) and the data of Merz and Whittaker on fugacities of hydrogen and nitrogen in four hydrogen-nitrogen mixtures a t 0" C. (14). Only critical data and generalized charts (20)together with Equation 13 were used to calculate fugacities of the components in these mixtures. Deviations of calculated values from observed fugacities are presented in Tables I and 11. There are shown also for comparison the deviations of the fugacity values calculated by the method of Newton and Dodge with the aid of a generalized fugacity chart and Equation 1. Inspection of Tables I and I1 shows that the present method yields results superior t o those obtained by the method of Newton and Dodge, particularly in the case of argon-ethylene mixtures. This is in harmony with the observations of Gibson and Sosniclr ( 4 ) and Mer2 and Whittaker ( 1 4 ) ,that the Lewis-Randall fugacity rule yields relatively large errors for argon-ethylene mixtures and holds well for hydrogen-nitrogen mixtures. The method of Gamson and Watson could not be tested for the hydrogen-nitrogen mixtures because the reduced temperatures of these mixtures fall outside the range of Gamson's and Watson's generalized chart which must be used with Equation 14. This method, however, was tested for the argon-ethylene mixtures with the results shown in Table I. It is seen that in the case of argon-ethylene mixtures Equation 13 yields more satisfactory results than Equation 14.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1740
~
~~~~
TABLEI. DEVIATIONS OF- CALCULATED FUGACITIES IN ARGOSETHYLEXE MIXTURES AT 25 C. O
,
(Four mixtures of mole fractions 0.2, 0.4, 0.6,and 0.8, respectively) Av. % Deviation, Argon AP. % Deviation, Ethylene Pressure, Equation Equation Equation Equation Equation Equation Atm. 1 13 14 1 13 14 10 1.08 0.71 0 33 0.62 2.76 1 26 1.55 20 1.18 0.95 1.58 4.07 0,73 2.38 30 1.05 1.46 4.03 2.20 1.68 40 3.23 1.52 1.90 2.81 3.84 3 58 50 4.44 1.80 5.16 2.25 5,81 3.43 GO 5.73 0.54 8.64 4.40 3.30 7.04 80 9.02 1.36 12.59 3.52 3.84 8.20 100 12.43 1.77 16.32 4.91 10.3 1.51 128 15.12 5 82 3.05 12 0 18.07 1.99 Average for all points 6.11 1.42 2.78 7.28 3.28 5.77
Vol. 40, No. 9
FUGACITllES OF GAS IIIIXlUKES
Fugacities of gas mixtures find important application in determining the isothermal effect of pressure on enthalpies and entropies and in computing the work of isothermal compression of such mixtures ( 7 ) . A relation between the fugacity of a gas mixture and the fugacities of the Components may be derived in the following manner: the fugacity of the mixture is given by the expression (6)
while that of the i l h component ( i = 1,2,3, .
. . .n) is given by ( 6 )
TABLE 11. DEVIATIOXS O F CALCULATED FUGACITIES I N HYDROGEN-NITROGEN MIXTURES AT 0 C. O
(Four mixtures of mole fractions 0.2,0.4,0.6, and 0.8, respectively) Pressure, Av. % Deviation, Hydrogen Av. Yo Deviation, Nitrogen htm. Equation 1 Equation 13 Equation 1 Equation 13 50 3.26 1 .77 0.63 1.05 100 4.84 1.02 1.90 0.93 200 6.27 0.67 5.72 1.44 300 7.50 1 24 6.84 3.85 400 7.62 1.08 4.36 7.13 600 8.G9 1 .79 4.67 7.73 Average for all points 6.35 1.26 4.48 3.23
The proposed method has been tested also for the ammonia equilibrium studied by Newton and Dodge ( 1 7 ) . The euperimental data are those of Larson and Dodgc ( I S ) and of Larson (14). Since the mixtures used by Larson and Dodge contained a small percentage of argon, these were treated as mixtures of four components in the present study. Equation 13 and generalized charts (6,9, 16) were used to calculate the fugacities of the coinponents. Observed percentages of ammonia in the equilibrium mixtures were mcd in all calculations except a t 500" C. aqd 1000 atmospheres. Larson's extrapolated smoothed value for the percentage of ammonia was used in the absence of experimental data for this point. The thermodynamic equllibrium constant, K,, was calculated by means of the relation,
Table 111 shon-s the results of these calculations, the average error of the calculated values of IC, a t each pressure, and for coinparison, the average errors for the results of NeTvton and Dodge. It is seen that the present method yields reasonably good results for the ammonia equilibrium over the entire pressure range from 100 to 1000 atmospheres. The method oi Kevvton and Dodge, on the other hand, yields excessively large errors a t 1000 atmospheres, pi-obably because of the failure of the Lemis-Randall fugacity rule a t this pressui'e. The method of Garnson and Watson could not be tested for the ammonia equilibrium because the reduced temperatures of the mixtures investigated by Larson and Dodge fall outside the range of Clamson's and Watson's generalized chart.
TABLE111. THERMODYXAMIC EQUILIBRIUM CONSTAXTFOR AMXONIASYKTHESIS Temp., Accepted Value c. (KDa t 0 A h . j 450 0.00663 475 0.00499 500 0.00382 Av. % deviation of Kf Av. % deviation of Kr S e w t o n a n d Dodge (17)
100 a t m . 0,00632 0,00474 0,00366 4.6 3.3
Calculated Values of Kf a t 300 a t m . GOO atm. 1000 atm. 0.00629 O.OO63G O.OOGQ3 0.00406 0.00519 0,00475 0,00371 0,00397 0,00349 6.2 2.5 4.2
1.1
9.4
41.0
where V , is the molal volume of the mixture, r i is the mole fraction of the ithcomponent,, and p c is its partial molal volume. On substituting the well known thermodynamic relation (1)
into Equation 16, and in view of Equation 17, the following relation between the fugacities of the individual components and that of the mixture is obtained ( 3 ) : lnjqn= zlin(Sl/zl)
+ z21n(S2/x2j +
. . + xJn(?Z/zn)
(10)
Because of the difficulties inherent in the evaluation of fugacities of individual components, the assumption has been niadc often, especially when dealing with gaseous equilibria, that the mixtures are ideal solutions and that one may apply the LewisRandall fugacity rule, Equation 1. On substituting Equation 1 into Equation 19, there results the approximate relation:
Equation 20, like the Lewis-Randall rule, depends on additivity of component volumes and should fail at higher pressures. Fugacities of gas mixtures often are evaluated v-ith the aid of generalized fugacity charts, on which the ratio of fugacity to preesure is plotted against reduced pressure for a series of reduced temperatures ( 7 ) . This method requires that a pseudocritical temperature and a pseudocritical pressure be assigned l o the mixture. Following the suggestion of Kay ( 1 1 ) ,pseudocritical values are computed from the absolute critical temperatures and pressures of the components in accordance with the relations given in Equations 2 and 3. I t was shown by the writer in a previous paper (10) that in calculating compressibilities of binary gas mixtures somewhat better results have been obtained when pseudocritical values have been calculated by means of the relations: Tc/P1L2= Z1Ti/P11'
+ x2Ti/P1/Z'
(2L4)
Equations 21A and 21B and similar equations for multicomponent mixtures (10) should be valid also for fugacity calculations. I n the present investigation the accuracy of fugacity calculations based on pseudocritical values obtained with the aid of Equations 21A and 21B is compared with that of fugacity calculations based on Equations 2 and 3. I n addition, the range of validity of Equation 20 has been examined. Experimental data were provided by Gibson and Sosnick's results on fugacities of
September 1948
INDUSTRIAL AND ENGINEERING CHEMISTRY
argon and ethylene in four argon-ethylene mixtures a t 25’ C. (4) and by the results of Mer2 and Whittaker on the fugacities of hydrogen and nitrogen in four hydrogen-nitrogen mixtures at 0’ C. (14). The fugacities of these mixtures were calculated from those of the components with the aid of the exact thermodynamic relation, Equation 19. Fugacities so computed were considered to represent experimental values. Fugacities of these mixtures then were calculated from those of the pure components by means of Equation 20. The fugacities were computed also from generalized fugacity charts (go), first with the aid of Equations 2 and 3 and again with the aid of Equations 21A and 21R. I n the case of hydrogen pseudocritical constants recommended by Newton (16) were used in place of experimental critical constants. The per cent deviations of calculated fugacities from experimental values were determined. The results are shown in Tables IV and V.
1741
proposed method is not subject to the limitations of the LewisRandall fugacity rule, itJis believed to be of more general applicability than the method of Newton and Dodge. An exact thermodynamic relation between the fugacity of a gaseous mixture and the fugacities of the components in the mixture has been derived. By applying the Lewis-Randall fugacity rule, an approximate relation has been obtained which can be used to compute the fugacity of the mixture from fugacities of the pure components at moderate pressures. Comparison with experimental data on argon-ethylene and hydrogen-nitrogen mixtures indicates that at pressures in excess of 30 or 50 atmospheres generalized fugacity charts yield better results for fugacities of gaseous mixtures than the relation based on the Lewis-Randall fugacity rule. When fugacities of gaseous mixtures are evaluated with the aid of generalized fugacity charts, it is recommended that the pseudocritical constants, T , and P,, be calculated from the relations: T,/P:/‘ = Z.,.siT,/P:“ (224
O F CALCUL’4TED FUGACITIES O F ARGONTABLE Iv. DEVIATIONS ETHYLENE MIXTURES AT 25’ C.
(Four mixtures of mole fractions 0.2, 0 . 4 , 0.6, and 0.8, respectively) Av. % Deviation Pressure, Equations Equations Atm. Equation 20 2 and 3 21A and B 10 0.22 20 0.75 1.53 30 2.27 40 3.29 50 4.48 60 7.71 80 100 9.95 125 11.18
TABLE V. DEVIATIONS OF CALCULATED FUGACITIES OF HYDROGEN-NITROGEN MIXTURES AT 0 O C. (Four mixtures of mole fractions 0 . 2 , 0.4, 0.6, and 0.8, Av. % Deviation Pressure, Equations Atm. Equation 20 2 and 3 50 0.45 0.66 100 i.19 0.88 200 1 .Q8 0.66 2.72 1.22 300 400 2.18 3.36 3.70 600 3.81 800 3.90 1000 3.96 ~~
respeotively)
E$ytiIlg 0.64 0.70 0.66 0.93 0.89 0.47
..
..
previously given by the author in his paper on compressibilities of gas mixtures (IO). For binary mixtures Equations 2 2 8 and 22B reduce to Equations 21A and 21B. These equations have been found to give better results for argon-ethylene and hydrogennitrogen mixtures than the procedure for calculating pseudocritical constants suggested by Kay. Kay’s relations were used in the first part of this paper because of the very simple way in which they relate the pseudocritical constants to the composition of the mixture. LITERATURE CITED
Dodge, B. F., “Chemical Engineering Thermodynamics,” pp. 98, 105, 106, 121, 242, 240, New York, McGraw-Hill Book Co., 1944. I b i d . , pp. 161, 162, 239. Gamson, B. W., and Watson, K. M., Natl. Petroleum News, Tech. Set., 36, R623 (1944). Gibson, G . E., and Sosnick, B., J . Am. Chem. Soc., 49, 2172 (1927).
Gillespie, L. J., Chem. Rev., 18, 359 (1936). Glasstone, S., “Thermodynamics for Chemists,” Chapter XII, New York, D. Van Nostrand Co., 1947. Holoomb, D. E., and Brown, G. G., IND.ENG.CHEM.,34,590 (1942).
’
Inspection of results shows that Equation 20 is accurate at low pressures, but yields progressively larger errors a t higher pressures. Both methods of computing pseudocritical constants, in cbnjunction with the generalized fugacity chart, yield good results for argon-ethylene mixtures. However, for these mixtures the maximum error of any fugacity value computed with the aid of Equations 2 and 3 was found to be 2.85%, while Equations 21A and 21B gave a maximum error of 2.26%. With hydrogennitrogen mixtures Equations 2 and 3 yielded a maximum error of 5.95% whereas Equations 21A and 21B gave a maximum error of 1.57%. The results of this study, therefore, indicate that while Equations 2 and 3 are sufficiently reliable in many cases and are useful because of their mathematical simplicity as shown in the first part of this paper, Equations 21A and 21B and similar equations for multi-component mixtures ( I O ) , may be used to advantage in the calculation of fugacities of gaseous mixtures from generalized fugacity charts. SUMMARY
A method has been presented for the calculation of fugacities of individual components in a gas mixture with the aid of generalized charts and Kay’s relations for the pseudocritical constants of a gas mixture, similar to the method of Gamson and Watson, but applicable over a different range of conditions. Since the
Hougen, 0. A., and Watson, K. M., “Thermodynamics,” pp. 494, 495, New York, John Wiley & Sons, Inc., 1947. Ibid., pp, 494, 495, 622. Joffe, J., IND. ENC.CHEM.,39, 837 (1947). Kay, 15‘. B., Ibid., 28, 1014 (1938). Larson, A. T., J . Am. Chem. Soc., 46, 367 (1924). Larson, A. T., and Dodge, R. L., Ibid., 45, 2918 (1923). Merz, A. R., and Whittaker, C. W., I b i d . , 50, 1522 (1928). Morgen, R. A , , and Childs, J. H., IND.E m . CHEM.,37, 667 (1946).
Newton, R. H., Ibid., 27, 302 (1935). Newton, R. H., and Dodge, B. F., Ibid., 27, 577 (1935). Randall, M., and Sosnick, B., J . Am. Chem. SOC.,50, 987 (1928). Su, G. J., Huang, P. H., and Chang, Y . M.,Ibid., 68, 1403 (1946).
1
Weber, H. C., “Thermodynamics for Chemicsl Engineers,” pp. 198, 199, 219, 108, 109, New York, John W l e y & Sons, 1930. RECEIVED
July 7, 1947.
