August 1953 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 1815

for water in 2-methyl-1-propanol and 8.29 weight % 2-methyl-l- propanol in water, which are considered satisfactory checks.

The mutual solubilities of the 1-pentanol-water system were not determined in the investigation. However, Ginnings and Baum (6) and also Butler ( 2 ) have presented data on the mutual solubilities of the 1-pentanol-water system a t 25' C.

A tendency to periodicity can be noticed in the distribution data for some of the solutes such as caproic acid and potassium acid phthalate in the 2-methyl-1-propanol-water system. When plotted on log-log coordinates the data for all the systems, except that of methylamine, lie on straight lines with a slope very close to unity, which is necessary for systems with constant distribution coefficient.

Collander ( 4 ) has published one or two values for the distri- bution coefficient between 2-methyl-1-propanol and water for a number of substances at about 20' C. He states that these might be in error by as much as 10 to 20%. The present data, for 25' C., agree with his within his stated error. The agreement

between the present results and those of Archibald ( 1 ) for the 1-pentanol-water system is not particularly close.

ACKNOWLEDGMENT

The author is indebted to the Standard Oil Co. (Ind.) and the Kimberly-Clark Corp. for fellowship support, given during the period of the investigation.

LITERATURE CITED

(1) Archibald, R C., J . Am. Chem. Soc., 54, 3178 (1932). (2) Butler, J. A. V., Thomson, D. W., Maclennan, W. H., J . Chem.

( 3 ) Colburn, A. P., and Welsh, D. G . , Trans. Am. Inst . Ckem. Engrs.,

(4) Collander, Runar, Acta Chem. Scand., 4, 1085 (1950). (5) Ginnings, P. M., and Baum, R. , J. Am. C h a . Soc., 59, 1111

(6) Landolt-Bornstein Tabellen, including I and I11 Erganzungs-

RECEIVED for review December 8, 1952. A c c E a E n April 17, 1953.

SOC., 1933, p. 674.

38, 179 (1942).

(1937).

band, 5 Auflage, Berlin, Julius Springer, 1923.

Composition of Vapors from Boiling Binarv Svstems

J J

NEW METHODS OF REPRESENTING AND PREDICTING EQUILIBRIUM DATA

DONALD F. OTHMER, LOUIS G. RICCIARDI', AND MAHESH S. THAKAR2 Polytechnic Znstitute of Brooklyn, Brooklyn 2, N . Y .

UMEROUS attempts have been made to present thermody- N namically the vapor-liquid equilibrium relationships of volatile liquids, using as a basis the Gibbs-Duhem equation. However, this equation applies strictly to constant temperature systems, while engineering operations-e.g., distillation columns- are a t constant pressure.

In the following, simple thermodynamic derivations are pre- sented to represent constant pressure vapor-liquid equilibrium data. Close interrelationship of boiling points and equilibrium vapor compositions with their latent heats of vaporization is also indirated.

A method is shown for representing vapor-liquid equilibrium data as straight lines on a logarithmic paper. These straight-line plots are shown t o be useful in conjunction with the thermody- namic equations for the correlation and prediction of the equi- librium data.

PREVIOUS METHODS

In 1885 the Gibbs-Duhem differential equation was derived shoM ing the relation between the composition in moles and the partial pressures of the individual constituents of a multicompo- nent system. Margules (64) in the same year integrated this equa- tion as an infinite series. I n 1900, Zawidski (62) found a method of determining the coefficients of the equation by the slopes at the ends of the total pressure and composition relation-Le., P ws. x a t constant T.

Dolezalek (10) and van Laar (19, 20) modified the Margules ex- pressions using molecular association and van der Waals' equa- tion, while Bose ( I ) , Marshall (25), and Krichewskil and Kazar- novskii (18) offered methods for solving the Duhem equation

1 Present address, Colgate-Palmolive-Peet Co., Jersey City, N. J. 2 Present address, The Indian Aluminum Co , Ltd, 31 Chowranghee Road,

Calcutta, India.

directly without the Margules' formula. These methods, how- ever, are very laborious and have had little practical application.

In 1914 Rosanoff et al. (39) proposed a semiempirical formula for calculating these equilibrium relations, which Levy (26) showed tto be inaccurate.

In 1924 Lewis and Murphree (23) derived a thermodynamic equation for constant temperature systems and developed a trial and error method for stepwise integrating the Duhem equation. Leslie and Carr (21) proved in 1925 that the Duhring rule is ap- plicable to some solutions, and using this rule calculated the equilibrium relation from Duhring lines. Othmer (29) showed a simplification with improved accuracy using a refrence substance logarithmic plot. Othmer and Gilmont (SO) showed that vapor composition activities, equilibrium constants, and relative vola- tilities may be plotted a t constant x (or constant y) as straight lines on a logarithmic sheet against a temperature scale derived from a reference substance. They (SO) also showed that these properties could be plotted directly in many cases on logarithmic paper against total pressures to give straight lines; and this is a simple and effective correlation when data are available a t dif- ferent pressures. In a later paper ( S I ) , they gave improved methods for expressing such P, T, x, and y data based on the critical constants, and with associates (14) they used activity co- efficients to derive a general equation.

Levy in 1941 (85') improved Zawidski's method in determining the slopes of the total pressure curve. Carlson and Colburn ( 5 ) in 1942 correlated and predicted vapor-liquid equilibrium values for binaries using a modified form of the van Laar equation.

Clark ( 6 ) proposed an empirical method for predicting and presenting x, y relationships for any system by using the equation of two hyperbolas. These two equations do not, of course, present a complete expression of the equilibrium relations, as they give no

1816 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 Vol. 45, No. 8

information concerning the variation of temperature or prc~surc with composition.

In 1948 Redlich and Kister (38) proposed several thcrmody- namic equations which were successfully used in correlating vapor-liquid equilibrium for both binary and multicomponent systems.

In 1949 Hirata ( 1 5 ) offered an empirical method of integrating the Duhem equation stepwise, which is very similar to the one proposed by Clark (6). Hirata, however, showed that for many binaries the relation between the values X and Y , where X =

z/(1 - x j and Y = y / ( l - y), may be represented by three straight lines when plotted on logarithmic paper. ThiF is a plot for a liquid and a gas phase comparable to that of Othiner and Tobias (33) for two liquid phases, wherein it was shown that this same function of phase compositions gave straight lines on logarithmic paper.

Prahl in 1951 (87 ) by means of three constants which can be derived from a straight-line plot developed an algebraic relation betn-een z and T / .

DERIVATIOK OF EQUATIONS RELATING COXIPO TEMPERATURE A T CONSTANT PRESSURE

For binary systems the general form of the Gibbs-Duhem cqua- tion in terms of moles is ( I S ) :

idp pi f 72dp2 = -sd2' + Vdp (1)

This equation is thermodynamically sound, and relates p with 7, T2, and P.

For a binary mixture with liquid and vapor phases in equilib- rium a t coristant pressure, and considering the liquid phase and its composition 5, there results:

xidpi + xzdpz = -sLd2' (2)

This equation rigorously relat,es p as a function of 2' and x a t a

For a system in thermal equilibrium (7' = aonst,atit). as well as given pressure, P.

a t constant pressure, the following holds:

pf p f ; =

For any binary vapor-liquid equilibrium system at, constant pressure, the temperature changes as the liquid composition, x, is changed. But a t any given liquid composition, x, at a fixed pressure, the temperat.ures of the liquid phase and the vapor phase are the same and the Eystem is in complete thermal equilibrium. Thus, even though the equilibrium temperature changes as com- position changes from point to point until the complete equilib- rium curve is described, the conditions as set down by Equation 2a are fulfilled and therefore applicable for each individual point or equilibrium value.

In an equilibrium system held at constant pressure, there is only one degree of freedom. Any change in x will be automatically followed by changes in w and T . If, on the other hand, one of the variables is held constant-for example, x--2' and /I become auto- matically fixed. In such a case ( d p / d T ) p represents a complete derivative of p. Since it has been shown t,hat for each point in equilibrium pf = p.2" and pb = p:, it is permissible to write the following:

(":) =(G) dT P dT P

Therefore, Equation 2 becomes:

Equation 2b is rig~rgus and represents the variation of ~ with T and z at, constant P

As by definition /I? = p; + R2' log y l x 1 ~ P ~ and i.8 = p; + RT log ayl, by proper substitution and manipulation, the follow- ing fundamental relationship is obtained:

Since the vapor prcssure of pure components, PO, is only it func- tion of temperature up to moderately high pressures, and IS in- dependent of composition. x, using the Clausius-Clapeyron cqua- tion :

(2d) dT x - RT2

and the fatst that:

where

and by properly substituting in Equation 2c. there is obt:tinc.tl 1 Iir following equation:

The last bracket in Equation 2g.

is equal to zero, sinre the change in activity a t constant !/' aiitl 1' is zero (conventional Gibbs-Duhem equation). The remhining two brackets on the right-hand side of Equation 2g reprcserit thv actual latent heat of vaporization of the solution per molc.. Thrrc- fore, Equation 2g reduces to:

Equation 2h is completely rigorous, thermodynamically, and represents the relation between x, y, and T within the limitations of Dalton's partial pressure law and the Clausius-Clapeyrori equa- tion.

Equation 2h can be rearranged in a more usable forin, as follon,s :

A similar relationship was derived by Redlich and Kiiter ( 3 8 ) which was obtained by converting the Gibbs-Duhem equation a t constant T and P to an equation representing x, y, T at constant pressure. I t involved, however, an assumption whereby the variation of activity coefficients with temperature a t constant composition was assumed to be negligible. No such assuniption is necessary in the derivation proposed in this paper. L, in Equation 3 represents the actual latent heat of vaporization (in- cluding the heat of solution) per mole of solution

Similarly, starting with the equilibrium vapor phase, thrrc miiv be developed:

August 1953 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 1817

For ternary systems a t constant pressure, similar relations may be derived:

or

and

or

All the above equations are thermodynamically sound and, in fact, may be derived by other and more involved methods ( S I ) . The only assumptions are Dalton's law of partial pressures and the Clausius-Clapeyron equation.

PREDICTING VAPOR-LIQUID EQUILIBRIUM VALUES FROM TEMPERATURE-COMPOSITION DATA

The solution of any of the above differential equations makes possible the prediction of vapor composition data from vapor pressure or boiling point data. The differential equations, how- ever, do not yield a direct mathematical solution.

Any evaluation of these equations may use vapor-pressure data or boiling point-composition data to avoid the experimental de- termination of values of vapor compositions. A very simple method of obtaining adequate vapor pressure data is to employ a family of straight-line reference logarithmic plots (89) which re- quire only a few data points to delineate P, T, x data for the en- tire system. Furthermore, from the slopes of the family of vapor pressure lines on the plot (29, Figure 16), the molal latent heats of the solutions of different compositions may be obtained im- mediately.

Using this method (89) of evaluating vapor pressures and latent heats, the above differential equations may be solved by evalua- tion of the right side through substitution of the data, and then proceeding by one of two methods.

METHOD I. STEPWISE INTEGRATION. From the boiling point composition ( T , x ) diagram, and starting a t x = 0, y = 0, i t is possible by using Equation 3 and taking small increments of A x and AT' to obtain the entire y , x curve (see Table I). The pre-

TABLE I. STEPWISE INTEGRATION OF EQUATION 3 FOR ALCOHOL- WATER SYSTEM (4)

s - Y

X AT 0.0 t o 0.05 9 . 0 0 . 0 5 to 0.10 4 . 5 0 .10 to 0 .15 2 . 2 0 . 1 5 t o 0 . 2 0 1 . 3 0 .20 to 0 . 2 5 0 . 9 0 .25 to 0 . 3 0 0 . 7 0 .30 to 0 . 3 5 0 . 5 0 .35 to 0 . 4 0 . 5 0 . 4 t o 0 . 4 5 0 . 3 5 0 .45 to 0 . 5 0 . 4 0 . 5 t o 0 . 5 5 0 . 3 5 0 .55 to 0 . 6 0 .35 0 . 6 t 0 0 . 6 5 0 .25 0 .65 to 0 . 7 0 . 4 0 . 7 t o 0 . 7 5 0 25 0.75 t o 0 . 8 0 . 1 5 0 . 8 to 0 .85 0 . 1 0 .85 to 0.9

?/(I - Y) *

0 . 9 t o 0 . 9 5 0 ' 2

I, RT'

LZ 9590 9580 9570 9560 9550 9540 9530 9520 9510 9500 9490 9480 9470 9460 0450 9440 9430

n i i o

(Equation 3)

T 364.0 359.5 357.4 356.1 355.2 354 5 354.0 353.5 353.15 352.75 352.4 352.05 351.80 351.4 351.15 351 .0 350.9

350.9 . . . .

Y 0.285 0.415 0.479 0.519 0.547 0.572 0.592 0.617 0.632 0 .653 0.672 0 .703 0.730 0.771 0.800 0.817 0.868 0.884 0 .930

100

90 - I I I I I I I I

0 0 IO 20 30 40 50 60 70 00 90 100

M O L % E T H A N O L IN L I Q U I D

Figure 1. Plot of Mole Per Cent Ethanol in Vapor us. Mole Per Cent Ethanol in Liquid

cision will depend, of course, on how small the increments are chosen. Such integration was carried out for the system ethanol- water ( 4 ) by taking increments of A x = 5%; the results are shown in Figure 1.

METHOD 11. ALGEBRAIC RELATIONS. Beginning with the findings of Hirata (15), which were based on an empirical state- ment of the relationships of X and Y , a solution to Equation 3 was found.

Hirata plotted the vapor-liquid equilibrium data on logarithmic paper for many binaries, including those forming azeotropes. In- stead of plotting the conventional x , y mole fractions, he plotted mole ratios, Y -=. y/(l - y ) us. X = x / ( 1 - x), and found that each binary can be represented by three straight lines (see Figure 2). When the system is ideal, the data by this method can be represented by a single straight line with a slope of 45". The data for two ideal systems using this method of plotting are also shown in Figure 2.

With nonideal systems the slope of the straight line a t high values of x was almost always unity; the slope a t low values of x was found to be 1 in most cases or slightly less than 1. To be con- sistent with basic molecular theory, so that the relative volatility approaches finite values, a slope of unity was taken also for low values of x . Thus, the slope from experimental data is very close to the theoretically correct one.

(7) (8)

(9)

Y = OX, low values of x

Y = alX, high values of x Y = a,Xm, intermediate values of x

Straight lines are obtained in the lower and upper portions of the curve because Henry's and Raoult's laws apply in the dilute regions. ,As in the binary systems described above, it was found that ternary systems, such as acetone-chloroform-methyl iso- butyl ketone ( l 7 ) , can also be represented as straight lines on logarithmic paper (see Figure 3).

In Equations 7, 8 and 9, the slope of the middle line, m, is characteristic of each system; and it has now been found that when it is plotted on logarithmic paper versus (a~ /ao - 0.03) a straight line is obtained (Figure 4) having the following equa- tion:

[ C Y I / ~ O - 0.031 = r n Z . 6 7 (10)

The data plotted in Figure 4 we tabulqted in Table 11.

1818 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 Vol. 45, No. 8

0 ACETONE-WATER

l o k - -

Y 1.0 - - - - - - - - - .I 0 -- I D E A L SYSTEMS - - - - - 9 BENZENE-TOLUENE

o BENZENE -ETHYLENE DICHLORIDE - - - -

.o I u .IO 1.0 IO 100 .001

X Figure 2. Plot of Y us. X on Logarithmic Paper for Ideal and Nonideal

Binary Systems

It can be seen that a wide variety of systems have been plotted and Equation 10 is extremely general in application.

As indicated above, the equilibrium relationship can be repre- sented by Y us. X , where Y and X are connected with x and y by the following equations:

Using these relations, Equation 3 is changed as follows:

1 + X X - Y dY f ( T ) = x ~ 1+Y Y x d x

wheref( 5") represents a symbol, arbitrarily selected, and which is

set equal Since -- = m -, differentiating Equa-

tions 7 , 8, and 9, and substituting in Equation 11, the following equations are obtained:

dY Y d X X

for small and large values of X .

m(1 + X ) 1 + y

x - Y x 7 = fm ( T )

for intermediate values of X .

Hence, it follows, substituting Y = ~ O X , Y = alX, and Y = a,X" in the corresponding above equations, that the following relationshipe are obtained :

where xo is the arithmetic mean from x = 0 and a small increment of xo

where x1 is the arithmetic mean from x = 1 and a small increment of XI

where xm is the arithmetic mean from some intermediate value of xm and a small increment of xm

There a0 and a, are the intercepts of Y vs, X lines on logarithmic paper a t z -+ 0 and x --c 1, respectively, and fa( T ) and f i (T ) are the values calculated from the 7' - z diagram as z -+ 0 and z -+ 1, respectively.

METHOD O F PREDICTING BINARY SYSTEMS

By following the steps given below and-knowing Equations 14, 15, and 16 and the T , x diagram, the whole equilib-

rium curve can be defined:

1. fa(T) and f , (T ) are determined from the T , z diagram a t x 0 and z + 1, respectively. At x -+ 1, L,, is the latent heat of vaporization of pure component 1, and TI 1s the absolute boiling

1. 2. 3. 4. 5. 6. 7. 8.

9.

10. 11. 12. 13 14. 15. 16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

33. 34. 35. 36 37. 38.

TABLE 11. (3 - 010

System Water-acetic acida Acetone-water Acetone-methanol Acetone ethanol Ethanol-benzene Benzene-ethylene dichloride Carbon tetrachloride-benzene Carbon tetrachloride-ethyl ace-

Carbon disulfide-carbon tetrachlo-

Carbon disulfide-acetone Acetone-chloroform Ethanol-water Ethyl acetate-ethanol Water-ethylene glycol Isopropyl alcohol-isopropyl ether iMethanol-water Water-hydrazine n-Butanol-acetaldehyde dibutyl

n-Hexane-benzene n-Heptane-toluene n-Butanol-n-butyl acetate Methanol-water b Acetone-methanol Ethanol-water Ethyl ether-acetone Carbon disulfide-benzene Benzene-isopropyl alcohol Chloroform-ethanol Benzene-cyclohexane Methanol-water Carbon tetrachloride-benzene Carbon tetrachloride-ethyl ace-

tate

ride

acetal

Benzene-acetic acid

0.03 us. m 1 [" ao - 0 . 0 3 1 m

0.789 0.935 0.0164 0.19 0.417 0.71 0.497 0.75 0.0254 0.22 0.97 1 .0 0 .88 0.935

0.62 0 .83

0.81 0.935 0.0395 0.34 2.76 1 .35 0.092 0.42 0.27 0.59 2.60 1.39 0.103 0.47 0.37 0.68 3.62 1.57

0.20 0.56 0.47 0 .77 0 .7 0.87 0.337 0.70 0.52 0 .72 0.327 0 .63 0.095 0.37 0.346 0.75 0 .54 0.80 0.04 0 .23 0.094 0.44 0.606 0.85 0 .33 0.62 0.88 0.935

0.69 0.86 0.535 0.80 0.22 0.56 0,033 0.29 4.07 1 .45 0.163 0.54 0.202 0 .58

a Systems 1 through 21 determined at constant pressure. b Systems 22 through 38 determined at constant temperature.

August 1953 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 1819

h

10

TERNARY SYSTEM I- -- 1.0

X

.IO

ACETONE CHLOROFORM -

METHYL ISOBUTYL KETONE (MIK) - I I l l l l l l l I I l l l l l l l I I 1 1 1 1 1 L

- 01 JO to 10 100

Figure 3. Plot of Y vs. X on Logarithmic Paper for Ternary System Acetone-Chloroform-Methyl Isobutyl

Ketone

Y

m Figure 4. Plot of (2 - 0.03) us. m on

Logarithmic Paper

point of this same pure substance. At x + 0, L,, is the latent heat of vaporization of pure component 2 , and Tz is the absolute boiling point of this pure substance, 2 2 .

010 and a1 are determined from Equations 14 and 15, which are the intercepts of the Y = ax curve on logarithmic paper a t low and high values of x, respectively.

From the ratio (CYI/LYO - 0.03), m is determined for the middle curve.

At an intermediate value of X , the value o f f m ( T ) is deter- mined from the T - x diagram. The values of fm(T) and m are substituted in Equation 16, and am is calculated.

There are plotted on logarithmic paper two lines, each having a slope of 1, and intercepts equal to a0 and ~ ( 1 , respectively. The two lines represent the vapor-liquid equilibrium a t extremely low values of x1 as XI - 0 and high values of xi as x1 ---c 1. The equations of the two straight lines are Y = OCOX and Y = a l X .

2.

3.

4.

5.

The middle straight line is obtained by knowing the slope, m, and the intercept, an. The three straight lines then complete the X , Y curves, from which the conventional x, y graph can readily be obtained, by reversing the calculating process.

Sometimes x, y data obtained from the part of the curve where a sharp bend occurs in the logarithmic X , Y plots will require smoothing of the final x, y curve to eliminate similar sharp bends From this operation very little accuracy, if any, is lost.

For purpose of clarification, an illustrative example of the sys- tem ethanol-water ( 4 ) is worked out in Tables I11 and IV.

Two illustrations, including a nonideaI but nonazeotropic sys- tem (acetone-water) (4), and a minimum boiling, azeotropic sys-

TABLE 111. ETHANOL-WATER (4) f ( T ) = (E) (I

z T m RT2 0.0 100 0.019 95.5 dx = 0.019 (-4.5) = d T 1 .0 -8.42 10.1 xo = 0.0095 TaV. = 370.8

1 .0 78 .3 0.8943 78.15 dx = 0.1057 (0.15) = dT 1 . 0 0.054 0.947 xi = 0.9471 TaV. = 351.2

0.3965 80.7 0.3273 81 .5 d z = 0.0692 zm = 0.3619 Tsv. = 354.1 X m = 0.508

0.440 2.04 (-0.8) = dt 0 . 4

Q’ - 0.3 = 0.094m = 0.4 010

TABLE IV. COMPARISON OF EXPERIMENTAL AND EMPIRICAL VALUE OF y FOR ALCOHOL-WATER SYSTEM (4)

X X Y ( E m p i r i d ) y (Empirical) y (Exptl.) 0.05 0 .10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95

0.54 0.78 1.15 1.45 1.75 2.04 2.45 2.95 3.80 8 .60 18 .5

0.35 0.44 0.52 0.58 0 .63 0.67 0.71 0.75 0.80 0.895 0.945

0.32 0.44 0.50 0.58 0.62 0 .67 0.70 0.75 0.81 0.895 0.940

0 10 20 30 40 50 60 70 60 90 100 M O L % M O R E V O L A T I L E C O M P O N E N T I N L I Q U I D

Figure 5. Plot of Mole Per Cent More Volatile Com- ponent in Vapor US. Mole Per Cent More Volatile

Component in Liquid

tem (alcohol-water) ( 4 ) are given in Figure 5 , where the calcu- lated x, y data are compared with experimental data.

CHECKlNG AND CORRELATlNG VAPOR-LIQUID EQUlLlBRlUM DATA

The aforementioned thermodynamic equations may be re- garded as a fundamental tool not only for predicting equilibrium data, but also for correlating and checking experimental vapor- liquid equilibrium data at constant pressure.

Thus, by substituting in the left side of Equation 3 the proper values of x and y, and in the right side the corresponding boiling point and latent heat of vaporization, identical values should be

1820 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 Vol. 45, No. 8

as shown in Figure 7 , and many other systems which have been correlated but not shown here. m;m ;iK compositions It is possible, without also, to reference correlate this to the function boiling of temperatures vapor and liquid by

division of Equation 4 by Equation 3 to eliminate temperature. - 'e There follows:

(17)

5

I

I Y(1- Y) ; Y 2 U - Y) (d log rd ; Y(1 - Y ) 2

x ( l - 2) y - z dx Y - X

0 1 2 3 4 5 0 1 2 3 4 5 (d log rz) - L, Q R H S R H S dx L, dx

Figure 6. Plot of Left-Hand US. Right-Hand Side of Equation

Left-Hand Side Right-Hand Side L, d T

RTa dz - -.

X

120

100

80

- CALCULATED o ETHANOL- WATER (EXPTL.)

110 METHANOL- WATER (EXP'TL.) I t 0 120

9

70 70 dX

40 :EL*<#$ 30

20 10 0 0 .I .2 .3 A 5 .6 7 B .9 1.0

Equation 17 does not lend itself readily to checking or extra- polating experimental data. It is, however, demonstrated in Figure 8, for the same systems, and has been shown equally well for many other systems.

In Figure 9 a plot is made of the ternary system acetone-chloro- form-methyl isobutyl ketone (1'7) using Equation 5.

By dividing Equation 6 by 5, the temperature function may be eliminated.

CONCLUSION

The above equations thermodynamically represent the rela- tionships between y, x , and T for both the binary and ternary vapor-liquid equilibrium systems under constant pressure con- ditions. They can be used for correlating and checking available data. The methods indicated to solve this equation are useful in predicting ?J, x data from T , x data (or vice versa) for binary sys- tems.

The results will depend on the accuracy of the T , x data avail- able. Similar methods of solution for ternary systems can be ob- tained by further investigation, and if developed, will be ex- tremely useful for a ternary system, as only the boiling point and comDosition data would be needed to define the entire vanor-

X liquid equilibrium data. The very involved experimental analyses otherwise required would therefore be avoided.

The above equations can be applied for predicting and correlat- ing 2, y data for vacuum and moderately high pressure systems, within good accuracy, since Dalton's law of partial pressures and the Clausius-Clapeyron equation, which were the only assump- tions made in deriving the thermodynamic equations, are appli- cable for vacuum and for moderately high pressures.

dT Plot of --- dx us. x Using Equation

[;- LIA] & = L, dT

Figure 7.

(1 - y) d~ RT2 dz

obtained. Such a check ensures thermodynamic consistency of

This result may be shown to be true for all systems, which indi- the experimental data. NOMENCLATURE

L dT RTZ dx

cates that there is only one true value of the left-hand side of the f ( T ) = ~ - equation for any given value of the right-hand side.

"The interrelation or equality Figure 6, where Equation 3 is water ( 4 ) and methanol- water (SS)]. A straight line with a slope of 45' through- out the entire range of liquid compositions indi- cates the validity of this e q u a t i o n b y p r o v i n g i n every case the equality which the equation de- mands.

Another method of check- ing the x , y, T data available for a system is to plot ( d T / d x ) calculated from Equation 3 against d T / d x o b t a i n e d experimentally. Excellent checks were ob- tained for the same two systems of ethanol-water ( 4 ) and methanol-water (361,

of x, g> and T function is tested in plotted for two systems [ethanol- J ~ ( T ) = k2 a t x + 0 f i ( ~ ) = k% gat x - 1 - I O

METHANOL- WATER 1 1 ETHANOL- WATER

B I N A R Y S Y S T E M S

1 P I I

.IO 0. I 1.0 IO 0. I 1.0 IO

R H S R H S

Logarithmic Plot for Binary Systems of Two Sides of Equation Figure 8. Left-Hand Side Right-Hand Side

August 1953 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 1821

I .o I I I11111~ I I ACETONE CHLOROFORM

Figure 9. Loga- r i t h m i c P l o t f o r T e r n a r y S y s t e m Acetone-Chloro- form-Methy l Iso- butyl Ketone of Two Sides of Equation

I) I I1111111 I I I l l I) I .IO 1.0

R, H S Left-Hand Side ’ Right-Hand Side zidyi ~ 2 d ~ 2 zrdya - LE d T

y2dz1 yadzi RTZ dzi yldzl 4- - + - - - -

Lz dT RTZ dx fm(T) = -- - a t intermediate value of x

L = actual molal latent heat of vaporization including heat

= partial molal latent heat of vaporization of each com- - ponent

L, = Llx + Lz (1 - x) = molal latent heat of vaporization of For ternary:

of solution

solution based on liquid composition. - -

L, = zixi - Lzxz - Lax3 .-

L, = Lly + zz (1 - y ) = molal latent heat of solution based on

L, = Liyi + Lzyz + Lava

water composition. For ternary:

= heat of solution For solutions not far from ideal, the latent heat of vaporization

of the solution can be estimated from the latent heats of pure com- ponents. Thus:

L, = L1x - Lz(1 - 2)

L , = h y - LA1 - Y )

m = slope of line Y = amXm for intermediate values of x when plotted on logarithmic paper

= moles of a component = pressure

pi, p 2 = Dartial messures of comDonents 1 and 2 Po R

T Ti TZ TW. V

Y

X

Y

S SL

X

a0

a1

a m

Y lr lrl0

n-

= ;apor pressure of pure component = gas constant = extensive entropy = entropy per mole of liquid = absolute temperature = absolute temperature of component 1 = absolute temperature of component 2 = average temperature (absolute) = volume = mole fraction of component in liquid = mole fraction of component in vapor

X = - 1 - x

Y 1--Y

= constant in equation Y = a0 X for low values of x. It will be the intercept when Y is spotted US. X on a logarithmic paper.

= constant in equation Y = 011 X for high values of x. It will be the intercept when Y is plotted vs. X on a

= -

logarithmic paper. = constant on equation Y = amXm for intermediate values

of 5. = activity coefficient = chemical potential of a component = chemical potential at standard condition of tempera-

ture and pressure = total pressure

Subscripts 1, 2, 3, etc., signify components 1, 2, 3, etc. Superscripts L and G stand for liquid and vapor phases, respec- tively. i = i th component of x and/or y

LITERATURE CITED

(1) Bose, E., Physik. Z., 8, 353 (1908). (2) Bromiley, E. C., and Quiggle, D., IND. ENG. CHEM., 25, 1136

(3) Brunjes, A. S., and Furnas, C. C., Ihid., 27, 396 (193’5). (4) Carey, J. S., and Lewis, W. K., Ihid., 24,882 (1932). (5) Carlson, H. C., and Colburn, A. P., Ihid., 34, 581 (1942). (6) Clark, A. M., Trans. Faraday SOC., 41,718 (1945). (7) Conner, A. Z. , Elving, P. J., and Steingiser, S., IND. ENG.

(8) Cornell, L. W., and Montanna, R. E., Ihid., 25, 1331 (1933). (9) Dobson, H. J. E., J . Chem. Soc., 127,2866 (1925).

(1933).

CHEM., 40, 497 (1948).

(10) Dolezalek, F., 2. physik. Chem., 64, 727 (1908). (11) Ferguson, J. B., and Funnell, W. S., J . Phys. Chem., 33, 1

(12) Furnas, C. C., and Leighton, W. B., IND. ENG. CHEM., 29, 709

(13) Gibbs, J. W., “Collected Works,” Vol. I, p. 88, Equation 96,

(14) Gilmont, R., Weinman, E. A. , Miller, E., Hashmall, F., and

(15) Hirata, M., Chem. Eng. (Japan) , 13, 138 (1949). (16) Ibid., 14, 65 (1950). (17) Karr, A. E., Scheibel, E. G., Bowes, W. M., and Othmer, D F.,

(18) Krichewskii, I., and Kazarnovskii, I., 2. unorg. Chem., 218, 49

(19) Laar, J. J. van, 2. physik. Chem., 72, 723 (1910). (20) Ihid., 83, 599 (1913). (21) Leslie, E. H., and Carr, A. R., IND. ENG. CHEM., 17, 810 (1925). (22) Levy, R. M., Ihid., 33, 928 (1941). (23) Lewis, W. K., and Murphree, E. V., J . Am. Chem. Soc., 46, 1

(24) Maraules. M.. Sitzher. Akad. Wyiss. Wien. Math. naturw. Kl..

(1929).

(1937).

New York, Longmans, Green, and Co., 1928.

Othmer, D. F., IND. ENG. CHEM., 42, 120 (1950).

IND. ENG. CHEM., 43, 961 (1951).

(1934).

(1924).

11; 104; 1243 (1895). (25) Marshall, J. M., J . Chem. SOC. (London) , 89, 1350 (1906) (26) Miller, H. C., and Bliss, H., IND. ENG. CHEM., 32, 123 (1940). (27) Morton, D. S., J . Phys. Chem., 33, 384 (1929). (28) Olsen, A. L., and Washburn, E. R., Ibid., 41,457 (1937). (29) Othmer, D. F., IND. ENG. CHEM., 32, 841 (1940). (30) Othmer, D. F., and Gilmont, R., Ihid., 36, 858 (1944). (31) Ihid., 40, 2118 (1948). (32) Othmer, D. F., and Morley, F. R., I6id., 38, 751 (1946). (33) Othmer, D. F., and Tobias, P. E., Ihid., 34, 690 (1942). (34) Paul, M. A., “Principles of Chemical Thermodynamics,” p. 335,

(35) Perry, J. H., “Chemical Engineer’s Handbook,” 2nd ed., New

(36) Ibid., 3rd ed., p. 574. (37) Prahl, W. H., IND. ENG. CHEM., 48, 1767 (1951). (38) Redlich, O., and Kister, A. T., Ihid., 40, 341, 345 (1948). (39) Rosrtnoff, M. A., Bacon, C. W., and Schulze, J. F. W., J . Am.

(40) Rosanoff, M. A., and Easley, E. W., Ihid., 31,979 (1909). (41) Sameshima, J., Ihid., 40, 1483, 1503 (1918). (42) Scatchard, G., and Raymond, C. L., Ihid., 60, 1275 (1938). (43) Scatohard, G., Wood, S. E.. and Mochel, J. M., J . Phys. Chem.,

(44) Schutz, P. W., J. Am. Chem. Soc., 61, 2691 (1939). (45) Smith, E. R., and Matheson, H., J . Research Natl. Bur. Stand-

(46) Tongberg, C. O., and Johnston, F., IND. ENG. CHEM., 25, 733

(47) Trimble, H. M., and Potta, W., Ihid., 27,66 (1935). (48) Uchida, S., et al., private communication. (49) Uchida, S., and Ksto, H., J . SOC. Chem. Ind . Japan, 37, 525

(50) Uchida, S., and Ogawa, Chem. Eng. (Japan) , 11, 2 (1947). (51) Wrewsky, 8. 2. physik. Chem., 81, 1 (1912). (52) Zawidski, J., Ihid., 35, 129 (1900).

New York, McGraw-Hill Book Co., 1951.

York, McGraw-Hill Book Co., 1941.

Chem. SOC., 36, 1993 (1914).

43, 119 (1939).

urds, 20, 641 (1938).

(1933).

(1934).

RECEIVED for review February 21, 1952. ACCEPTDD April 20, 1953. Previous papers in this series have appeared in IND. ENQ. CHEM., 20. 743 (1928); 35, 614 (1943); 36, 1061 (1944); 37, 299, 895 (1945); 38, 751 (1946); 39,1175 (1947); 40,168 (1948); 41,572 (1949); 43, 707, 711, 961, 1607 (1951); IND. ENQ. CEDIU., ANAL. ED., 4,232 (1932); Anal. Chcm., 20, 763 (1948).

C O R R E S P O N D E N C E

Lifting and Blowoff of Flames from Short Cylindrical Burner Ports

SIR: Reference is made to Channing W. %%on’s paper, “Lift- ing and Blowoff of Flames from Short Cylindrical Burner Ports,” IND. ENG. CHEM., 44, 2937 (1952)l. Although the experimental data apparently correlatevery well, there are several inconsistencies in the theoretical treatment.

Equation 1 is valid for either laminar or turbulent flow, but the resistance or frictional coefficient, X, in Equation 1 is twice the value of X used in Equation 2. Since Equation 2 is used in the development of the final correlation, Equation 2 should be written

in order t o be consistent with the X defined in the equatiuns for Cases A, B, and C on page 2941.

Equation 1 is applicable only to “established” flow since it is derived on the basis that the velocity near the wall is directly proportional to the pressure gradient along the tube. This condition is not satisfied along the entrance length of any tube. Presumably, the ve!ocity distribution and pressure losses have not been quantitatively determined for a square-edged entrance. Consequently, the distance from a square-edged entrance for a flow t o become established cannot he predicted. Qualitatively, it is known that the pressure losses for a square-edged entrance are much greater than the losses for a rounded entrance and that the velocity distribution is more nonuniform for the square- edged entrance. In other words, the distance from the entrance for flow to become established is a t least as great for a square- edged entrance as for a rounded entrance.

Bassinesq [Compt. rend., 113, 9, 49 (189l)l has computed the entrance distance for a rounded entrance to be

X = 0.065 D Re

where D is the tube diameter and Re is the Reynolds number. This expression can still be found in modern treatments (Perry, J. H.. ed., “Chemical Engineer’s Handbook,” 3rd ed., p. 388, New York. McGraw-Hill Book Co., 1950).

The application of the equation to Wilson’s most favorable experimental condition-i.?, for Re = 400 and l / r = 10- shows that the burner ports are only one fifth of the necessary length for flow to become established. For all other experi- mental conditions, this fraction of the necessary length is even smaller.

In summary. it appears that the right side of Kilson’s Equation 2 should be divided by 2 so that X would have consistent definition throughout the paper. Furthermore, the experimental evalua- tion of X by Equation 4 even when multiplied by 2, cannot be substituted into Equation 1 since the flow at the exit of the burner ports is not established for any of the experimental conditions.

Another minor point: In tho title of Figure 3 the word “parts”

PHILIP E. BocQuwr should read “ports.”

UBIVERSITY OF MICHIGAX DEPARTMENT O F P H Y S I O L O G Y

A N N A R B O R , M I C H . . . . . . SIR: I appreciate Xr . Bocquet’s calling attention to an un-

Equation 4 will accountable omission of a 2 in Equation 2. then be, accordingly

It was made clear in an earlier paper [Wilson, C. W. and Hawkins, K. J., IND. ENG. CHEM., 43, 2129 (1951)] and indicated briefly in the present retercnce, that the full streamline or turbu- lent velocity profile, respectively, is not developed in these short burner ports, whether they have a rounded or a square- edge entrance. They are similar to the entrance section of a tubc in which there is a transition from a uniform velocity to the estab- lished flow profile. The length of the ports n’as in all cases much shorter than the transition length.

A theoretical treatment by Schiller [Gebiete Inqenieurw., 248, 5-36 (1922)l permitted him to estimate theoretically the pressure drop coefficient, 1, in the entrance region of a tube having a rounded entrance. This treatment considered the momentum change accompanying the change in velocity profile, the pressui e loss, and the frictional forces. He found agreement between experimental measurements of pressure loss and his theory, and found that for a given length of the entrance section the value of the coefficient, A, is a function of the Reynolds number.

In the absence of a similar theoretical treatment, which would be very difficult for a square-edged entrance, it was not considered unreasonable to attempt the application of values of X derived from the experimentally determined pressure 1 0 ~ 8 , corrected for the kinetic energy change, to the velocity profile Equation 1. The correlation of the blowoff data by means of this procedure seems satisfactory, within the range of variation of the experi- mental conditions encompassed. However, it should not be concluded that these results can be used to establish the validity of any hypothesis regarding the character of flow in channels of this form. Although there is uncertainty concerning the length and effect of the vena contracta, the limited number of pressure loss measurements which mere made suggest a relationship be- tween X and Re similar to that deduced by Schiller for rounded entrances. I t is regretted that these points were not made more clear in the paper R E s r A R C H DEPARTMEUT C H A N N I N C I\‘. WILSON COXSOLIDATED GAB ELECTRIC LIGHT A X D POWER Co. OF BALTIMORE B A L T I M O R B 3, RfD.

1822