Guggenheim, E. A., “Mixtures,” Chap. 4, p. 31, Oxford University Press, London, 1952. Heric, E. L., Posey, C. D., J. CHEM.ENG.DATA9, 35 (1964). Hildebrand, J. H., Scott, R. L., “The Solubility of Nonelectrolytes,” 3rd ed., Chap. 8, p. 149, Chap. 1 2 , p. 201, Reinhold, New York, 1950. Lewis, G. N., Randall, M., “Thermodynamics,” 2nd ed. revised by K. S.Pitzer, L. Brewer, Chap. 18, p. 228, Chap. 21, pp. 283-4, McGraw-Hill, New York, 1961. Linard, J., Bzill. SOC.Chim. Belges 34, 363 (1925). McLaughlin, E., Ubbelohde, A. R., Trans. Faraday SOC.53, 628 (1957). Natl. Bur. Std. ( U . S.) “Selected Values of Properties of Hydrocarbons,” Circ. C-461, 49 (November 1947). Rowlinson, J. S., “Liquids and Liquid Mixtures,” Chap. 4, pp. 147, 152, Butterworths, London, 1959.
(12) Satterfield, R. G., Haulard, M.,J. CHEM.ENG.DATA 10, 396 (1965). (13) Schroeder, I., 2. Phys. Chem. 11, 457 (1893). (14) Topping, J., “Errors of Observation and Their Treatment,” Chap. 3, p. 88, Reinhold, New York, 1957. (15) Ward, H. L., J . Phys. Chem. 38, 761 (1934). (16) Weimer, R. F., Prausnitz, J. PI., Ibid., 42, 3643 (1965). (17) Weissberger, A., Proskauer, E. S., Riddick, J. A., Troops, E. E., “Organic Solvents,” 2nd ed., p. 194, Interscience, New York, 1955.
RECEIVED for review September 13, 1968. Accepted September 27, 1969. Presented, in part, a t the Southeastern Regional Meeting, ACS, Atlanta, Ga., November 1967.
Gas-liquid Equilibrium of the Oxygen-Carbon Dioxide System AAGE FREDENSLUND’ and G. A. SATHER Department of Chemical Engineering, University of Wisconsin, Madison, Wis.
53706
Gas-liquid equilibrium compositions for the 0,-CO, system were determined over the -SOo to + l o o C. temperature range and from 130 atm. down to the saturation pressure of CO,. Activity coefiicients are calculated, and a thermodynamic consistency test i s applied to each isotherm.
T H E GAS-LIQUID equilibrium of t h e oxygen-carbon dioxide system h a s been studied along a few isotherms (10, 12, 1 6 ) , b u t until now no isobaric d a t a have been reported. I n t h i s work, g a s a n d liquid equilibrium phase compositions were determined f o r t h e 02-C0, system at t h e -50”, -40”, -3O”, -20”, - l o ” , 0 ” , a n d + l o ” C. isotherms at pressures f r o m 130 a t m . down t o t h e s a t u r a t i o n p r e s s u r e of CO,, in 10-atm. increments. Thus, both isothermal a n d isobaric phase diagrams m a y be constructed. Activity coefficients based on t h e unsymm e t r i c normalization a r e calculated f o r each isotherm. T h e activity coefficients f o r C 0 2 are r e f e r r e d t o t h e p u r e component at t h e t e m p e r a t u r e of t h e system, whereas t h e activity coefficients f o r O2 a r e r e f e r r e d to t h e infinitely dilute solution. T h e activity coefficients a r e f u r t h e r m o r e r e f e r r e d to a constant pressure via t h e Poynt i n g correction factor, making possible t h e application of t h e isothermal-isobaric f o r m of t h e Gibbs-Duhem equation. T h i s equation, i n t h e f o r m of a n area t e s t f o r thermodynamic consistency, is applied separately t o each isotherm.
EX PER IMENTAL
T h e a p p a r a t u s used h e r e i s of t h e vapor recirculation t y p e ; t h e gaseous phase is circulated t h r o u g h a s t a t i o n a r y liquid phase enclosed i n a chamber. The flow d i a g r a m of t h e s y s t e m is shown in F i g u r e 1, where t h e m a i n vapor circulation p a t h i s indicated by a heavy line. T h e vapor phase, a f t e r leaving t h e pump ( a L a p p model CPS-2 diaphragm compressor), e n t e r s t h e equi1 Present address, Instituttet f o r Kemiteknik, Danmarks Tekniske Hfijskole, Lyngby, Denmark
librium cell via cooling coils, both cell and coil being immersed in a constant-temperature bath. T h e vapor phase bubbles t h r o u g h t h e liquid phase in t h e cell a n d finally goes back t o t h e pump. The design of t h e cell, which is provided w i t h windows t o p e r m i t visual observation of t h e two phases, is given by Fredenslund ( 7 ) . Before obtaining t h e gas a n d liquid phase samples, t h e lines leading to bypass-valves B P 1 a n d B P 2 ( F i g u r e 1) a n d the liquid sample line leading from t h e bottom of t h e cell t h r o u g h valve L S 2 a r e flushed. T h e g a s circulation t h r o u g h t h e liquid i s resumed, a n d t h e g a s a n d liquid sampling spaces a r e evacuated. A t t h e t i m e of gas sampling, BP1 is opened. W i t h V S 1 a n d VS2 closed, t h e gas located i n t h e space between VS1, VS2, VS3, a n d VS4 is sealed off, while gas circulation continues via BP1. To t a k e t h e liquid sample, B P 2 is opened, MS2 closed, a n d L S 2 opened t o t r a n s f e r about 0.5 cc. of liquid sample into t h e space between valves LS2, LS3, and LS4, which are all located inside the cryostat. The g a s a n d liquid samples may now be expanded into t h e i r respective sample chambers. T h e liquid sample i s pumped t h r o u g h t h e p a t h LS3-LS4-chamber t o e n s u r e t h a t t h e g a s composition i n t h e chamber i s t h e s a m e as t h a t i n t h e lines immersed i n t h e cryostat. F r o m t h e sample chambers, t h e samples are s e n t to a gas chromatograph equipped w i t h a 6-foot silica gel column f o r analysis. Refrigeration of t h e constant-temperature b a t h i s accomplished by f o r c i n g a regulated amount of liquid nitrogen t h r o u g h cooling coils located i n t h e b a t h fluid. A Bayley t e m p e r a t u r e controller f u r n i s h e s t h e final fine control by a d d i n g a small amount of h e a t v i a s i x resistance coils located in t h e bath. It proved possible t o control the t e m p e r a t u r e t o w i t h i n *0.005” C. f o r proJournal of Chemical and Engineering Data, Vol. 15, N o . 1, 1970
17
VENT
Jk
/
VI
A
C I
Figure 1 . System flow diagram M a i n system line Sampling or pressure measurement line Vacuum line PPressure gage numberRD Rupture disk G.C. Gas chromatograph
-----
Valves
MSVPVS-
LSVACBPVENT-
FEED-
longed periods. The absolute temperature i s measured to within i0.02" C., using a Hewlett-Packard Model Dy-2801A quartz thermometer. Using three Bourdon pressure gages ( P l , P2, and P3, F i g u r e 1 ) of different maximum readings, t h e pressure was measured to within 0.55% throughout the range of this investigation. Checks were made for t h e over-all reproducibility of the phase composition data by completing four r u n s at -30" C., 100 atm., and -20" C., 40 atm., respectively. The determined compositions were within 0.004 mole fraction f o r t h e vapor a t -30" C., 100 atm., and within 0.002 mole fraction f o r t h e liquid samples and t h e vapor sample at -20" C., 40 atm. E n t r a i n m e n t was minimized by using a low vapor circulation r a t e and by keeping the liquid level in the cell low. Reproducibility tests w i t h t h e vapor circulation r a t e increased by a factor of f o u r did not change the vapor composition significantly. RESULTS The experimental data f o r the 0,-CO, system a r e shown i n Table I and F i g u r e s 2 and 3. The data a t 0" C. a r e in good agreement with those of the previous investigators (10, 12, IS), a s may be seen from F i g u r e 4. Agreement is closest with the d a t a of Muirbrook and Prausnitz (12). I n spite of precautions taken, perhaps entrainment did occur a t 0" C., 110 atm., t h u s accounting f o r t h e 18
Journal of Chemical and Engineering Data, Vol. 15, No. 1, 1970
Main system valve numberValve for pressure gage numberVapor sampling valve numberLiquid sampling valve numberVacuum system valve numberBypass valve numberVenting valve numberValve to feed-
relatively large deviation from t h e previous data a t t h a t point. The smooth data f r o m t h e isothermal and isobaric phase diagrams, Figures 2 and 3, a r e presented in Table 11.
THERMODYNAMIC ANALYSIS The condition f o r phase equilibrium i s t h a t the fugacity of each component, i, in the gaseous phase i s equal to i t s fugacity in t h e liquid phase
The fugacity coefficient, &, i s now defined by
+i may be calculated f r o m a n equation of state. T h e activity coefficient, yi, i s defined by
where f i 0 is some ( a r b i t r a r y ) reference fugacity. Prausnitz e t al. ( 2 , 1 3 , 1 4 ) have developed a n internally consistent technique for the calculation of liquid phase activity coefficients a t high pressures. The activity coefficients of light, supercritical components a r e referred to the infinitely dilute solutions, whereas t h e
Table I. Equilibrium Compositions for the OaCOz System
Mole Fraction Oxygen in Liquid and Vapor Phases Temperature, Pressure, Atm. 10.0 f0.03 20.0 f0.2 30.0 f0.3 40.0 fO.2 50.0 *0.2 60.0 f0.2
1 V
1 V
1 V
1 V
1 V
1 V
70.0
1
f0.2 80.0 Zk0.3 90.0 f0.3
V V
100.0
1
~k0.3 110.0 f0.4 120.0 f0.4 130.0 f0.5
V
1 1 V
1 V
1 V
1 V
- 50.0 *0.02
-40.0 3z0.02
0.008 0.312 0.025 0.617 0.049 0.712 0.071 0.761 0.096 0.789 0.117 0.809 0.150 0.814 0.189 0.818 0.220 0.820 0.267 0.819 0.293 0.797 0.334 0.780 0.393 0.762
0.118 0.021 0.419 0.041 0.589 0.064 0.651 0.087 0.695 0,120 0.723 0.141 0.741 0.167 0.757 0.205 0.766 0.248 0.767 0.274 0.760 0.328 0.730 0.380 0.702
-30.0 f0.02
C.
-20.0
-10.0 fO.01
3zo.01
$10.0 &0.01
0.0
10.01
... 0.010
0.275 0.031 0.463 0.052 0.551 0.080 0.609 0.104 0.644 0.140 0.671 0.162 0.693 0.191 0.689 0.236 0.694 0.266 0.682 0.306 0.671
0.006 0.061 0.030 0.302 0.042 0.400 0.062 0.500 0.092 0.540 0.125 0.560 0.141 0.574
0.009
0.096 0.035 0.263 0.056 0.346 0.079 0.417 0.110 0.463 0.134 0.493 0.162 0.496 0.195 0.500 0.243 0.498 0.299 0.446
0.170
0.601 0.220 0.599 0.255 0.601 0.300 0.570
0.010 0.099 0.042 0.214 0.062 0.298 0.090 0.336 0.126 0.372 0.147 0.398 0.185 0.401 0.247 0.362
0.028 0.115 0.040 0.161 0.071 0.218 0.087 0.263 0.115 0.294 0.187 0.263
Figure 2. Isotherms for the Oz-C02 system Figure 3. Isobars for the 0 2 - C 0 2 system
activity coefficients of heavy, condensable components a r e referred t o t h e pure components. I n t h e following, component 2 i s taken to be a supercritical component (oxygen) and component 1 a condensable component (carbon dioxide). Making use of Equation 3, t h e reference fugacity of component 2 becomes
+
A
X 60atm. 80 otm.
ZOatm. 40otm.
[Izld In + x2d In fz fi
~O~otm.
120otm.
= jp v1
dP
1,
(5)
+
Since vi = zlvl x2G2 ( t h e b a r indicates a partial molar q u a n t i t y ) , i t follows t h a t
H a i s the Henry's law constant a t the temperature of t h e system. The isothermal Gibbs-Duhem equation f o r binary mixtures i s
Now one defines a n adjusted fugacity,
fl(Po),
by
Journal of Chemical and Engineering Data, Vol. 15, No. 1, 1970
19
gacity. Then Equation 8 becomes, at constant tempera-
c
ture,
For component 2, '2y2p exp{
Figure 4. The
X
0" C. isotherm
A
This work Muirbrook
0
Zenner and Dana Kaminishi
-
[d In fl(P")
=
d In fl - RT " dP]
Jd'
&i dP}]
(11) T
T h e reference pressure, P o , is taken to be the vapor pressure of C 0 2 a t the temperature of t h e system. H2(po) is Henry's law constant evaluated a t the temperature of the system and the saturation pressure of COP. The equation of state used f o r the calculation of t h e fugacity coefficients and partial molar volumes is a modified version of the Redlich-Kwong equation suggested by Chueh and P r a u s n i t z ( 4 ) . The equation of state is
MOLE FRACTION OXYGEN
0
-
T
a n d similarly for component 2. A t pressure Pa,f i ( p " ) = fl, so integrating Equation 7 f r o m pressure Po to any pressure, P , one obtains upon rearrangement
p=-
RT v -b
a
- T ' / ~ (Vv
+ b)
(12)
For b i n a r y mixtures, t h e constants a and b a r e given by QaiR2T,iz'6 a =
yiylaij; i
ai< =
j
PCi
a t,l . -
+ RZT 2 ..P6
,
CZl
Pcij
S&.RT,< b = .X yibi;
T h e pressure-adjusted activity coefficient, y i ( p o ) ,is defined by
where
f l p,,re(Pn) is
the pure-component reference fu-
z
bi =
Pci
TCiand PCi denote t h e pure-component critical temperatures and pressures, respectively ; T,,,and PCijdenote the mixture pseudocritical temperature and pressure. Oat and Obi a r e parameters which a r e curve-fitted
Table II. Smoothed Compositions for the Oz-COz System
Mole Fraction Oxygen in Liquid and Vapor Phases Temperature, C.
Pressure, Atm. 10
- 50
-40
1
0.006
V
0.300
0.001 0.100
1
0.0265 0.617 0.0485 0.718 0.071 0.761 0.0955 0.789 0.122 0.806 0.1515 0.815 0.1835 0.820 0.2183 0.821 0.255 0.817 0.294 0.805 0.338 0.785 0.393 0.758
20
V
30
V
40
V
50
V
1 1 1 1
60
V
70
V
80
V
1 1 1
90
V
100
V
110
V
120
V
1 1 1 1
130 20
V
0.0205 0.425 0.0415 0.585 0.0635 0.652 0.087 0.695 0.112 0.723 0.1405 0.744 0.170
0.757 0.2035 0.766 0.2385 0.767 0.276 0.758 0.320 0.738 0.380 0.698
-30
0.0105 0.285 0.031 0.463 0.053 0.550 0.0765 0.609 0.102 0.646 0.129 0.675 0.159 0.691 0.191 0.696 0.226 0.695 0.2635 0.685 0.306 0.665
Journal of Chemical and Engineering Data, Vol. 15, No. 1, 1970
-20
0.001 0.050 0.021 0.302 0.0425 0.422 0.066 0.495 0.092 0.537 0.119 0.566 0.1465 0.586 0.1765 0.598 0.209 0.600 0.247 0.595 0.300 0.570
-10
0
0.009 0.097 0.033 0.263 0.056 0.359 0.080 0.418 0.106 0.463 0.133 0.493 0.162 0.504 0.195 0.505 0.243 0.493 0.299 0.446
0.0135 0.099 0.038 0.214 0.062 0.293 0.089 0.341 0.118 0.377 0.147 0.398 0.184 0.401 0.247 0.380
+10
0.015
0.087 0.039 0.170
0.065 0.225 0.092 0.263 0.119 0.294 0.187 0.263
volume of t h e liquid mixture. Fortunately, d a t a a r e available on the molar volumes of the saturated liquid mixture a t 0" C. a s a function of pressure ( 1 2 ) . These d a t a were fitted to the modified Redlich-Kwong equation of state, again using a nonlinear least squares procedure, and leaving and abi f o r o2 and CO, a s paramet e r s to be determined. The saturated liquid mixture molar volumes were then determined a t all the temperat u r e s and pressures of interest from t h e modified Redlich-Kwong equation of state, with the new nai and abi incorporated in the coefficients of Equation 12. Once t h e molar volume i s known, t h e partial molar volumes may be determined using t h e equation of s t a t e , Equation 12, as discussed by Chueh and P r a u s n i t z ( 3 ) :
f r o m s a t u r a t e d vapor or liquid P-V-T d a t a f o r each of the p u r e components. Note t h a t t h e liquid phase paramet e r s a r e not necessarily equal to those of the g a s phase. F o r liquid phase calculations, xi is substituted f o r yi in t h e above combining rules. The suggested mixing rules f o r Pcij and T r i ja r e
where
vcij =
and z,,,
=
1 8
(vci1'3
+
v,,1/3)3
0.291 - 0.04
(w,
+ w,)
and 0, a r e the acentric factors f o r components i and j. K~~ is a measure of t h e deviation of Tell f r o m t h e geometric mean f o r t h e given system. I t i s t h e only parameter entering into t h e equation of state, Equation 12, which depends on b i n a r y interactions. I t m a y be obtained f r o m independent sources such as the second cross-virial coefficient, BL3,and is relatively independent of temperature, pressure, and composition ( 5 ) . On the basis of B,, d a t a f r o m references ( 5 , 8 , 1 1 ) , i t was found t h a t K,] f o r the 02-C0, system i s 0.03. T h e calculation of the fugacity coefficients i s performed a s described by Chueh and P r a u s n i t z ( 4 ) : O,
ln$k
=
In
V
+ 3-
2
1.
yiaik
bk
abk
( P C F i = exp
+
Lr
dP)
j$,
(17)
In 2 ,
v+b ~ (In 2 2 )-
R
v+b
and nbcozf o r t h i s calculation a r e given i n ( 3 ) , ~ % O Z and Ruoz and abozwere obtained f r o m t h e pure-component, saturated-liquid P-V-T d a t a given in references (1, 1 5 ) . I n the calculation of the Poynting correction factors,
b
m)- In
Pv
(15)
The volume, V , of the saturated vapor m i x t u r e was calculated from Equation 1 2 a f t e r and f&lcoz were obtained from ( 4 ) , and Q a O 2 and a b O z were determined by a nonlinear least squares fit of Equation 12 t o pure, s a t u r a t e d vapor P-V-T d a t a obtained f r o m ( 1 5 ) . The resulting fugacity coefficients a r e given in Table 111. Only calculated results f o r t h e 0" C. isotherm a r e presented here. T h e complete Table I11 m a y be obtained f r o m the American Society f o r Information Science. I n the calculation of partial molar volumes in t h e liquid phase, the first problem is to obtain v, the molar
it i s assumed t h a t Gii s a function of t e m p e r a t u r e and composition, b u t not pressure. T h e partial molar volumes and Poynting correction factors a r e shown in Table 111. The reference fugacity f o r C 0 2 was calculated at each temperature by the generalized methods discussed by Hougen et al. (9). Comparison with the d a t a available at O " , l o " , and 20" C. (6) was satisfactory. T h e reference fugacity f o r O2 is Henry's law cons t a n t , evaluated a t t h e temperature of the system and t h e saturation pressure of COB. The details of the graphical determination of H,(p") in accordance with Equation 4 a r e given in ( 7 ) . All of the information needed to calculate the activity coefficients according to Equations 10 and 11 is now
Table Ill. Vapor and Liquid Phase Calculations
T = 0" C. 0
Fugacity Coefficients Pressure, P 40.0 50.0 60.0 70.0
80.0 90.0 100.0 110.0
$02
1.1010 1.0923 1.0871 1.0933 1.1016 1.1205 1.1641 1.2600
Liquid Partial Molar Volumes, Cc./Gram-Mole
9Cod
voz
0.7404 0.6827 0.6301 0.5791 0.5314 0.4845 0.4353 0.3815
87.96 91.30 94.74 99.58 105.83 112.82 125.86 171.24
Poynting Correction Factors
VCO?
PCFO~
47.25 46.76 46.12 45.19 43.83 41.98 38.36 23.71
1.0222 1.0656 1.1143 1.1713 1.2402 1.3229 1.4453 1.7817
PCF C
O ~
1.0119 1.0331 1.0541 1.0744 1.0933 1.1097 1.1188 1.0832
Activity Coefficients r
10
*
2
1.0094 0.9221 0.8838 0.7997 0.7253 0.6594 0.5608 0.3824
T c02
0.9997 1.0096 1.0110 1.0208 1.0272 1.0370 1.0682 1.1929
At 0" C., fco2
= 26.74 atm., H ,JPo) = 313 atm., % inconsistency = -2.1 q. Calculated results a t other temperatures are available from ASIS. Per cent inconsistency at other temperatures were: - 50" C., +9.5'%; -40" C., f14.57,; -30" C., +11.97,; -20" C., -8.57,; -10" C., -2.9%; +lo" C., -15.47,.
a
Journal of Chemical and Engineering Data, Vol. 15,
No. 1, 1970
21
v = molar volume, cc. per gram-mole x = liquid phase mole fraction y = vapor phase mole fraction x = compressibility factor Greek letters
y = activity coefficient K
= deviation from geometric mean
4 = fugacity coefficient 0 = parameter in Redlich-Kwong equation of state 0
= acentric factor
Subscripts YOLE FRACTIW OXYQEN,
\
Figure 5. Activity coefficients at the 0” C. a n d -50” C. isotherms
A
available, a n d t h e results are given in Table 111. F i g u r e 5 shows the activity coefficients at -50” a n d 0” C. as a function of t h e composition. T h e d a t a a n d over-all calculation may be tested f o r internal consistency, since t h e isothermal-isobaric Gibbs-Duhem equation applies to ~ ~ ( ~ ‘ T1h .e thermodynamic consistency test ( 2 ) becomes
where x ~ , i s~ the , ~maximum ~ value of xg2 for which d a t a were taken. L e t
and =
100
AREA1 - AREA2
T h e trapezoidal rule is used i n t h e integration t o obtain
AREA,. T h e calculated per cent inconsistencies a r e given in Table I11 f o r each isotherm. Considering t h e difficulty i n obtaining accurate d a t a f o r high-pressure gas-liquid equilibria i n t h e critical region a n d t h e uncertainties i n the calculations, these results are quite satisfactory.
NOMENCLATURE a = constant in Redlich-Kwong equation of state b = constant in Redlich-Kwong equation of state 6 = second virial coefficient f = fugacity, atm. H = Henry’s law constant, atm. P = pressure, atm. PCF = Poynting correction factor R = gas constant T = temp., C.
22
Superscripts
= reference property partial molar property 1 = liquid v = vapor ( P o )= evaluated a t P o ’” - unsymmetrical convention used .,. O
o‘c. -50°C.
c/c inconsistency
i, j, k = components of a mixture c = critical property
Journal of Chemicol and Engineering Data, Vol. 15, No. 1, 1970
-
LITERATURE CITED Barron, R. F., “Cryogenic Systems,” p. 635, McGrawHill, New York, 1966. Chueh, P. L., Muirbrook, N. K., Prausnitz, J. M., A.Z.Ch.E. J . 11,1097 (1965). Chueh, P. L., Prausnitz, J. M., Ibid., 13, 1099 (1967). Chueh, P. L., Prausnitz, J. M., Znd. Eng. Chem. Fundamentals 6, 492 (1967). Ibid., 60, No. 3, 34 (1968). Deming, W. E., Deming, L. S., Phys. Rev. 56, 108 (1939). Fredenslund, A., Ph.D. thesis, University of Wisconsin, Madison, Wis., 1968. Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” pp. 202, 1114, Wiley, New York, 1954. Hougen, 0. A,, Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” P a r t 11, pp. 569-605, Wiley, New York, 1959. Kaminishi, G., Toriumi, T., Kogyo Kagaku Zasshi 69, No. 2,175 (1966). Magasanik, D., Ellington, R. T., paper presented at A.1.Ch.E. Annual Meeting, Houston, Tex., December 1963. Muirbrook, N. K., Prausnitz, J. M., A.I.Ch.E. J., 11, 1092 (1965). O’Connell, J. P., Prausnitz, J. M., Ind. Eng. Chem. Fundamentals 3, 347 (1964). Prausnitz, J. M., Chem. Ens. Sci. 18, 613 (1963). Stewart, R. B., Ph.D. thesis, University of Iowa, Iowa City, Iowa, 1966. Zenner, G. H., Dana, L. I., Chem. Ens. Progr. Symp. Ser. 59, No. 44, 36 (1963).
RECEIVEDfor review September 23, 1968. Accepted October 27, 1969. This work was supported by fellowships and grants from the Union Carbide Corp., the Amoco Chemicals Corp., the National Science Foundation through the University of Wisconsin Research Committee, and the University of Wisconsin Engineering Experiment Station. E’or complete Table 111, order NAPS Document NAPS-00687 from ASIS National Auxiliary Publications Service, c/o Information Sciences, Inc., 22 West 34th St., New York, N. Y. 10001; remit $1.00 f o r microfiche or $3.00 for photocopies.
