C. O. MILLER
ANDR. C. BARLEY1
Case S ch ool o f A pplied S cien ce, Cleveland 6, Ohio
Empirically it has been found that, for the n-paraffin hy
drocarbons from propane through n-octane, the vapor fugacity / „ is a single function o f total pressure at condi
tions o f constant liquid fugacity. It was known previously that, for a given substance, liquid fu g a c ity /, is a single function o f temperature. By employing these two rela
tions, values derived from the experimental vapor-liquid equilibria data o f Katz and Hachmuth (4) have been cor
related. The correlation is presented in the form o f an alignment chart with scales for temperature, pressure, and
equilibrium constants (K — y/ x), and points representing various light hydrocarbons. This chart includes n-paraf
fin hydrocarbons Ci to Cn and several lower olefins over a temperature range of —10° to +700° F. and a pressure range o f 1 to 500 pounds per square inch absolute. It is recommended for use with mixtures o f adjacent hydro
carbons— e.g., Ci to Ci as in distillation, and for lighter components absorbed in heavier. Other incidental use*
are in estimation o f vapor pressures and extrapolation of vapor-liquid equilibria data.
V
APOR-liquid equilibria data are used in designing equipment for distillation, condensation, or absorption opera
tions. The correlation presented here should be useful when it is desired to extend the range of meager vapor-liquid equilibria data or to approximate unknown vapor-liquid equi
libria data from such other properties as may be known— e.g., va
por pressure data.
Vapor-liquid equilibria data are usually expressed in the form of equilibrium constants. These constants may be determined experimentally as follows: A liquid hydrocarbon mixture is brought to equilibrium with its vapor at a definite temperature and pressure. The equilibrium constant, K, of a given substance is then found by analyzing vapor and liquid and dividing the mole fraction in the vapor, y, by the mole fraction in the liquid, x.
Among the first to measure equilibrium constants (K — y/x) were Souders, Selheimer, and Brown (14). More recently Katz and Hachmuth (4) measured equilibrium constants for mixtures of natural gas and mid-continent crude. They pre
pared charts of equilibrium constants for methane through
“ heptanes and heavier” on which the temperature range is —30° to + 270° F.
and the pressure range is 5 to 3000 pounds per square inch absolute. Brown and White (7), working with a naphtha and distillate furnace oil, determined equilib
rium constants through the relatively large temperature range 0° to 1000° F.
(approximate) and pressure range 10 to 1000 pounds per square inch absolute (approximate). Standing and Katz (15) made similar measurements on mixtures of natural gas and crude oil in which pressures up to 8200 pounds per square inch were used. The thermodynamic properties of systems such as propane- n-pentane were evaluated by Sage, Lacey, and colleagues (11). As a result of their work, the vapor-liquid equilibria relations of many relatively simple systems have been fixed.
Vink ci al. (17) and Kirkbride and Bertetti (6) studied the effect of solvent
on equilibrium “ constants” . The correlation presented here de
pends on data obtained with mid-continent crude as the solveni or base.
C A LC U LA TIO N O F E Q U IL IB R IU M C O N STA N TS The preceding paragraphs refer to equilibrium constants ex
perimentally determined by analysis of liquid and vapor in equilibrium. These constants may also be determined ap
proximately by calculation from the fugacity. The fugacity concept is described by Lewis and Randall (7). The develoi>- ment from the equilibrium constant expressed in terms of mole fractions to that in terms of fugacities will be briefly reviewed since fugacity is used in the correlation to be presented here.
Raoult’s law may be combined with Dalton’s law in the form.
p, = P ya = P°xa (1)
where p„ — partial pressure of component a over a mixture in vapor-liquid equilibrium
P = total pressure
PI - vapor pressure of pure component a at tempera
ture of mixture
1 Present address, E. B. Badger A Sons
Com pany, Boston 14, Mass. Figure 1. Fugacities o f Hydrocarbons Plotted on a Reduced Basis
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November, 1944 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
Figure 2. Fugacities o f Liquid Hydrocarbon*
Since Pya
in regions of relatively low pressure when the Raoult and Dalton laws apply. At higher pressures the simplicity of Equation 2 vapor pressure at temperature of system
fugacity of a pure component in vapor at temperature and total pressure of system
A liquid fugacity fz is sometimes called a
“ corrected” vapor pressure; and a vapor fugacity is sometimes called a “ corrected” total pressure although, as Weber (18) indicates, it is prob
ably better to state that fugacity approaches pressure as pressure approaches zero.
Fugacities of hydrocarbons to be used as in
dicated above can be obtained by graphical in
tegration of equations involving compressibility data. These data have been determined ex
perimentally from the P -V -T relations of the hydrocarbons. Lewis and Kay (8) correlated fugacity data for light hydrocarbons as shown in Figure 1. From this figure vapor fugacity /„
of Equation 3 may be obtained. From data in
cluded in Figure 1, Kay (5) prepared a plot (Figure 2) similar to a Cox chart (2) from which liquid fugacity/* of Equation 3 may be obtained.
(It is interesting to note that the liquid fugaci
ties could be arranged on an Othmer type of plot (9), using a substance such as n-pentane for reference. The slope of the resulting straight line for a given hydrocarbon would then be equal to the ratio of the heats of evaporation of that hydrocarbon and n-pentane.] For illustration, the equilibrium constant for n-pentane at 5 at
mospheres absolute pressure and 200° F. may be calculated by the use of Figures 1 and 2. The critical temperature T c of n-pentane is 470.3° K., the critical pressure P c is 32.8 atmospheres:
IOOO gathered the vapor-liquid equilibria data for the light hydro
carbons on a single chart. Gilliland and Scheeline (S) recently presented a correlation of minimum values of equilibrium con
stants for the less volatile components occurring in varioui- Fugacity and Total Pressure
7*
K ' - = / f . h fy
(7)
F igure 4. Liquid F u gacity vs. E q uilibriu m C on stan t at Pressures
relation made by Othmer (10), calculated equilibrium constants are correlated by vapor-pressure data of a reference substance.
It will be recalled that liquid fugacity / , is sometimes likened to a corrected vapor pressure and that vapor fugacity f u is some
times likened to a corrected total pressure. From Figure 2 it is evident that temperature fixes the liquid fugacity of a given hydro
carbon. It wa8 found empirically that total pressure fixes, with satisfactory approximation, the vapor fugacity for a given hydro
carbon for pressures up to about 500 pounds per square inch absolute at conditions of constant liquid fugacity. Thus, if temperature and total pressure are known, the former fixes the numerator in K - /*//»> and the latter fixes the denominator.
The manner in which it was found that/ „ = <i>(P) follows:
Some preliminary calculations of equilibrium constants for n-butane and n-pentane were made from fugacities by choosing a total pressure and then choosing temperatures such that the liquid fugacity of n-butane was equal to that of n-pentane.
Under such conditions the equilibrium constants of the two hydro
carbons were always equal. To avoid tedious calculations, graphical methods were used.
A total pressure was chosen. Then temperatures were read from Kay’s plot (5) such that the liquid fugacities of the various hydrocarbons were equal. By using plots derived from the Taylor and Parker calculations (16), a single value (with slight variations) of the equilibrium constant was found at given liquid fugacities for all the hydrocarbons from propane to n- octane. This procedure was repeated to obtain values for the calculated curves of Figure 3. Similarly, K ay’s plot (6) and the plots of Katz and Hachmuth (4) were used to obtain values for the Katz curves of Figure 3. Since the Katz data are for mixtures o f normal and isoparaffins, it was deemed justifiable to extend the calculated curves parallel to the Katz curves in the higher pressure regions. The result was the set of curves used in the correlation.
Following the determination of a set of curves based on fugacities in the lower pressure regions and dependent on the repräsentative experimental data of Katz and Hachmuth (4) in the higher pressure regions, a cross plot was made which resulted in the straight isobars (lines of constant pressure) of Figure 4.
The ratio of liquid fugacity to equilibrium constant at points along a given isobar was found to give a single value of vapor fugacity:
At various total pressures, corresponding vapor fugacities were determined and plotted on Figure 5. Thus it was shown by graphical methods that f v = 4>(P) for the light paraffin hydrocarbons propane through n-octane at conditions of con
stant liquid fugacity. The development of this relation made the present correlation possible.
Since the relation fy = <j>(P) is a unique property of the fugacity data when a derivation is made by the step3 indicated, its use is intended only in the limited sense implied in the derivation.
An alignment chart is presented in Figure 6 with temperature and pressure on one scale, equi
librium constants on a second, and liquid fugaci
ties on a reference scale (fx divisions not shown).
Table I compares equilibrium constants for n- pentano from the Katz and Hachmuth data (4), the calculated data (16), and the present correla
tion. The alignment chart values agree with the calculated data at pressures up to 80 pounds per square inch absolute with the exception of methane, ethylene, and ethane. The points for n-Ct through n-Cu are based solely on calculated data.
USES
The alignment chart condenses the equilibrium constant data from the correlation curves of Figures 3 and 4. The chart has several incidental usee, a few of which follow. Estimation of vapor pressures is possible by aligning temperature and the point for a substance, and determining at what pressure the equilibrium constant is equal to unity. Estimation of boiling points is pos
sible by aligning pressure with an equilibrium constant equal to Various
/
o .to 0 . 4 0 1.0 4 . 0 10 2 0
VAPO R F U G A C IT Y , , ATM OSPHERES
Figure 5. Equilibrium Constant vs. Vapor Fugacity at Conditions of Constant Liquid Fhigacity
unity and running an index through the point for a substance to the corresponding temperature. Meager vapor-liquid equi
libria data may be extended by plotting the few data available and determining their point locus. This procedure has been used successfully on such systems as hydrogen chloride and n- butane. The point locus for such a system does not fall on the straight line connecting the n-paraffin hydroearbons, illustrating the fact that the equilibrium constant for a given substance at a given temperature and the total pressure varies from one solvent
1021
Figure 6. Alignment Chart Presenting Correlation of Equilibrium Constants
to another. This variation is not great, fortunately, from a sol
vent such as a gas oil to one such as a mid-continent crude in the case of light hydrocarbon solutes.
It is to be hoped that further experimentation on vapor- liquid equilibria of multicomponent hydrocarbo» systems will,
perhaps with the aid of a correlating method such as the one here presented, allow more uni
versal correlation for engineering purposes of the relations governing these systems. Corre
lation is particularly desirable at higher pres
sures— 500 to 10,000 pounds per square inch absolute— where for a given temperature and pressure the nature of the system involved mixture in vapor-liquid equilibrium Subscripts