Correlation of Vapor-Liquid
Equilibria Data for Hydrocarbons C. 0. MILLER AND R. C. BARLEY’ Case School of Applied Science, Cleveland 6, Ohio Empirically i t has been found that, for the n-paraffin hydrocarbons from propane through n-octane, the vapor fugacity f v is a single function of total pressure a t conditions of constant liquid fugacity. I t was known previously that, for a given substance, liquid fugacity f. is a single function of temperature. By employing these two relations, values derived from the experimental vapor-liquid equilibria data of Kata and Hachmuth ( 4 ) have been correlated. The correlation is presented in the form of an alignment chart with scales for temperature, pressure, and
equilibrium constants (K = y / x ) , and points representing various light hydrocarbons. This chart includes n-paraffin hydrocarbons CI to Clr and several lower olefins over a temperature range of -10’ to +700° F. and a pressure range of 1 to 500 pounds per square inch absolute. I t ib recommended for use with mixtures of adjacent hydrocarbons-e.g., CI to Ce as in distillation, and for lighter components absorbed in heavier. Other incidental use8 are in estimation of vapor pressures and extrapolation nf vapor-liquid equilibria data.
APOR-liquid equilibria data are used in designing equipment for distillation, condensation, or absorption opsrations. The correlation presented here should be useful when i t is desired to extend the range of meager vapor-liquid equilibria data or to approximate unknown vapor-liquid equilibria data from such other properties as may be known-e.g., vapor pressure data. Vapor-liquid equilibria data &re 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 a t a definite temperature and pressure. The equilibrium constant, IC, of a given substance i s then found by analyzing vapor and liquid and dividing the mole fraction in the vapor, y, by t,he mole fraction in the liquid, z. Among the first to measure equilibrium constants ( K = y/z) were Souders, Selheimer, and Brown (14). More recently Kate and Hachmuth (4) measured equilibrium constants for mixtures of natural gas and mid-continent crude. They prepared charts of eauilibrium constants for methane throiigh “heptanes and heavier” on which the bemperature range is -30” to 270” F. and the pressure range is 5 to 3000 pounds per square inch absolute. Brown and White (I), working with a naphtha and distillate furnace oil, determined equilibrium ‘constants through the relatively large temperature range 0’ to 1000° F. (approximate) and pressure range 10 to LOO0 pounds per square inch absolute (approximate). Standing and Katz (16) 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 aa propanen-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. Vi& et al. (17) and Kirkbride and Bertetti (6) studied the effect of solvent
on equilibrium “constants”. The correlation presented here depends on data obtained with mid-continent crude as the solveni or base. CALCULATION OF EQUILIBRIUM CONSTANTS
The preceding paragraphs refer to equilibrium constants experimentally determined by analysis of liquid and vapor in equilibrium. These constants may also be determined agproximately by calculation from the fugacity. The fugacit! concept is described by Lewis and Randall (7). The develop ment from the equilibrium constant expressed in terms of mole fractions to that in terms of fugacities will be briefly Jeviewed since fugacity is used in the correlation to be presented here. Raoult’s law may be combined with Dalton’s law in the forilr.
Pa Pya p.0~ (1) where p , = partial pressure of component a over a mixture in vapor-liquid equilibrium P = total pressure P.”= vapor pressure of pure component a a t tempera ture of mixture
+
1 Preaent address, E. B. Badger & Bone Compsny, Boston 14, Mass.
PR
Figure 1.
Fugacities of Hydrocarbons Plotted on a Reduced Basis
1018
INDUSTRIAL A N D ENGINEERING CHEMISTRY
November, 1944
1019
T
rl.
=
366*4 T R = 470.3 - = 0.778
5 P,
e
PR
9
5.0 32.8 ~
p
0.1526
From Figure 1, fu/P = 0.88; 5 = 4.40 atm. From Figure 2, a t 200* F (93.3’ C.) fz = 4.27 atm.
/,, = 0.88 X
K =& fu
Figure 2.
Since P y ,
= Pzx0
then y,/x., = P.“/P
or
K,
4.40
P
(ti)
Equilibrium constantv for methane through octane, calculated according to the above procedure by Taylor f, , ATMOSPHERES and Parker (16), are tabulated by Sherwood ( 1 X ) . Fugacities of Liquid Hydrocarbone Before proceeding with thr p r e s e n ; correlation which makes use of the fugacity, four previous correlations will be mentioned.
= P.“/P
(2)
in regions of relatively low pressure when the Raoult and Dalton laws apply, At higher pressures the simplicity of Equation 2 may be kept by using the modification:
K
4.270.97
I
=fdfu
(3)
where fa = fugacity of a pure liquid component under its own vapor pressure a t temperature of system ju= fugacity of a pure component in vapor a t temperature and total pressure of system A liquid fugacity .f is sometimes called a “corrected” vapor pressure; and a vapor fugacity is sometimes called a “corrected” total pressure ulthough, as Weber (18) indicates, it is probably better to state that fugacity approaches pressure as pressure approaches zero. Fugacities of hydrocarbons to be used as indicated above can be obtained by graphical integration of equations involving compressibility data. These data have been determined experimentally 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 fu of Equation 3 may be obtained. From data included in Figure 1, Kay (5) prepared a plot (Figure 2) similar to a Cox chart (a) from which liquid fugacity f . of Equation 3 may be obtained. [It is interesting to note that the liquid fugacities could be arranged on an Othmer type of plot @), 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 a t 5 atmospheres absolute pressure and 200’ F. may be calculated by the use of Figures 1 and 2. The critical temperature T,of n-pentane is 470.3” K., the critical pressure P, is 32.8 atrnnspheres:
CORRELATION
By the use Of *pentane a reference substance, S h b h (18) gathered the vapor-liquid equilibria data for the light hydrocarbons on a single chart. Gilliland and Scheeline (3)recently presented a correlation of minimum values of equilibrium CODstants for the less volatile components occurring in varioum binary systems. Brown and White ( 1 ) made a plot similar t o Gilliland’s Figure 10 (a) which includes data for additional binary mixt,ures and for complpx mixtures, In another cor-
K *
Figure 3.
f,/f,
Equilibrium Constant Data Plotted in Terms of Liquid Fugacity and Total Pressure
1020
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 36, No. 11
At various total pressures, corresponding vapor fugacities were determined and plotted on Figure 5. Thus it was shown by graphical methods that j v = +(P)for the light paraffin hydrocarbons propane through n-octane at conditions of constant liquid fugacity. The development of this relation made the present correiatian possible. Since the relation jv = + ( P )ie a unique property of the fugacity data when a derivation is made by the steps indicated, its w e is intended only in the limited sense implied in the derivation. An alignment chart is presented in Figure B with temperature and pressure on one scale, equilibrium constants on a second, and liquid fugacities on a reference scale Cr, divisions not shown). Table I compares equilibrium constants for n1.0 4.0 IO 40 I00 4 00 pentane from the Kat2 and Hachmuth data (0, K the calculated data (16), and the present correlaFigure 4. Liquid Fugacity V S . Equilibrium Constant at Various tion. The alignment chart values agree with the Pressures calculated data at pressures up to 80 pounds per sauare inch absolute with the excePtion of methane, ethylene, and ethane. The points for n-Co through relation made by Othmer (1 O), calculated equilibrium constants n-Clr are baaed solely 011 calculated data. are correlated by vapor-pressure data of a reference substance. I t will be recalled that liquid fugacity ft is sometimw likened USES to a corrected vapor pressure and that vapor fugaoityf, is sometimes likened to a corrected total pressure. From Figure 2 it is The alignment chart condenses t,he equilibrium constant data evident that temperature fixes the liquid fugacity of a given hydrofrom the correlation curves of Figures 3 and 4. The chart has carbon. It waa found empirically that total pressure fixes, with several incidental uses, a few of which follow. Estimation of satisfactory approximation, the vapor fugacity for a given hydrovapor pressures is possible by aligning temperature and the point carbon for pressures up to about 500 pounds per square inch for a substance, and determining a t what pressure the equilibrium absolute at conditions of constant liquid fugacity. Thus, if constant is equal to unity. Estimation of boiling points is postemperature and total pressure are known, the former fixes sible by aligning pressure with an equilibrium constant equal t~ the numerator in K = j z / j uand , the latter fixes the denominator T h e manner in which it was found thatf, = +(P)follows: Same 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 fugaaity of n-butane was equal to that of n-pentane. Under such conditions the equilibrium constants of the two hydrocarbons were always equal. To avoid tedious calculations, graphical methods were used. A total pressure was chosen. Then temperatures were read from Kay's plot (6) such that the liquid fugacities of the various hydrocarbons were equal. By using plots derived from the Taylor and Parker calculations (16),a single valuk (with slight variations) of the equilibrium constant was found at given liquid fugacities for all the hydrocarbons from propane to noctane. This procedure was repeated to obtain values for the calculated curves of Figure 3. Similarly, Kay's plot (6) and the plots of Kata and Hachmuth (4) were used to obtain values for t h e Katz curves of Figure 3. Since the Katz data are for mixtures of normal and isoparaffins, i t was deemed justifiable to extend the calculated curves parallel to the Kata curves in the higher pressure regions. The result was the set of curves used in the aorrelation. Following the determination of a set -of curves based on fugacities in the lower pressure regions and dependent on the representative experimental data of Katz and Hachmuth (4) i n the higher pressure regions, a cross plot w a ~made which resulted in the straight isobars (lines of constant preswre) of Vigure 4. The ratio of liquid fugacity to equilibrium constant a t points along a given isobar was found to give a single value of vapor fugacity :
400 vi
v)
4
-
(D
