0 = -0.26
-t 11.19
=
10.93 minutes
Tad
it’hen the recommended approximation (exp. ( x ) / x . 1x1 > 1 5 ) \$as used to obtain the exponential integrals of 18.86 and 19.38, an answer of 10.92 minutes was obtained. Since the approximation is in greatest error for low values of the argument and since the values in question were near the lower limit, the second term in a series (5) (of which the recommended approximation is the first term) was also evaluated. Including the second term, changed the values of the exponential integrals by about 57:, but only changed the final results from 10.92 to 10.93 minutes.
-
Nomenclature
+
rR
aA bB a: h = = a,, b,, = aA! aR A,, A , .4* =
+ .rS
stoichiometric coefficients initial amounts of A and B activities of A and B frequency factors in Arrhenius equation, having the same units as the corresponding values of k . C A ~CR,, ? = initial concentrations of A and B = average heat capacity (B.t.u./lb./” F.) CP E = energy of activation in the Arrhenius equation, k = Ae-E/RT
S:
Ei(x) =
exp.(t)dt, t = first order exponential integral a
exp.(t)dt, t Z = second order exponential integral
E ~ Y=) m
F
feed rate (mass/time) heat of reaction (B.t.u./mol. of A converted) = reaction velocity constants when the rate equation is written in terms of concentrations, activities, and partial pressures, respectively n.4 = n.4” - .YA = mole:; of A unconverted per unit mass of feed n.4”. nB.,, no = initial moles of A and B and total moles of feed per mass, of feed n, = n, 6u.d = total moles of reacting system per unit mass of feed P = pressure in (atm.) r = reaction rate in (moles/volume/time) R = gas constant = initial temperature in O R . T,, T = To ( - J H ) ( C A ~- C A ) / P C = ~ To + , ( - A H ) ( ) i t o - n.4)/Cp - absolute temperature in (” R.) = =
(-AH?; k,. k >A-
+
+
TodBi,
Vb, VI?
Tv, 11’0 XA
Y
2,Z ,
= To
+ +
+
(-AH)Ca,/pCp = T o XnA, in Tables I and 11, respectively = To (-AH)aCB,/pbCp = To aXnB,/b in Tables I and 11, respectively = volume of system per mass of charge and volume of reactor, respectively = ( E / R )(1/.T- 1 / T a d ~ o )(, E / R )(1/To - l / T o d ~ J , respectively = moles of A converted per mass of reactor feed = =
+
E/RT ( E / R ) ( l / T- 1 / T d and ( E / R ) ( l / T o- l / T o d , respectively
Greek 6
- r+s-a-b -
8
=
x
= (-AH)/Cp = (r s
a
Residence time in flow reactor or reaction time in batch reactor
+ +
) - sum of the stoichiometric coefficients of the reaction products = mass and molar densities in (mass/volume) and p , pm (moles/volume), respectively For some svmbols suggested units are given, but any consistent set of units may be used. V
literature Cited
(1) Billingsley. D. S.,,McLaughlin, LV. S.. Jr., LVelch, N. E., Holland, C. D., Znd. Eng. Chem. 50, 741 (1958). (2) . , Bilous,. O.,. ;\mundson, N. R., A.Z.Ch.E. Journal 2, 117 (1956). (3) Deindorrfer, F. H., Humphrey, A. E.. Appl. Microbiol. 7, 256 (1959). (4) Gratch, S.. Birmingham, Mich., March 1957, personal communication, 32475 Bingham Rd., Birmingham, Mich., March 1957. (5) Jahnke, E., Emde. F.: Losch, F., “Tables ofHigher Functions,” p. 23, McGraw-Hill, New York, 1960. (6) Handbook of Chemistry and Physics, pp. 289-90, C. D. Hodgman, cd., Chemical Rubber Publishing Co., Cleveland, O., 1961. (7) Hougen, 0. -4.; \Vatson, K. M., “Chemical Process Principles,” pp, 816-40, Vol. 111, Kinetics and Catalysis, \Viley, New York, 1947. (8) Parts, A. G., Australian J . Chem. 11, 251 (1958). (9) Smith, J. M., “Chemicd Engineering Kinetics,” pp, 100, 128, McGraw-Hill, Kew York, 1956. RECEIVED for review Sovember 6, 1961 ACCEPTEDMarch 7, 1962
G E N E R A L EQUATIONS FOR TESTING CONSISTENCY OF MULTICOMPONENT VAPOR-LIQUID EQUILIBRIUM DATA LU H C
.
T A 0 , Department of Chemical Engineering, University of Nebraska, Lincoln, Xeb.
General equations are presented for testing the internal consistency along piecewise continuous composition paths in a mullticomponent system b y methods similar to those already used for binary systems. Convenient numerical methods are illustrated b y application to a ternary and a quaternary system. HE DEGREE O F COMPLEXITY in a multicomponent system Tincreases with the total number of components. If the composition of a mixture is represented as a point in a space with coordinates representing concentrations of each component. treatment of data may be simplified to that of a binary system by considering clnly linear paths in a multidimensional space. This geometrical concept is used to obtain general
equations for testing internal consistency of multicomponent, vapor-liquid equilibrium data. Several restricted integration methods have been proposed for testing multicomponent data by using the Gibbs-Duhem equation at constant temperature and pressure; usually they imply a test only for the whole path under consideration. Krishnamurty and R a o (3) suggested a method for linear paths VOL.
1
NO. 2 M A Y 1 9 6 2
119
in a ternary system kvith one component a t a constant concentration. Prausnitz and Snider’s equation ( 4 ) may be used for tests along linear paths connecting to pure components. Alignment of d a t a points along these paths is difficult to arrange in experimental work. T h e following general equations are applicable to all linear paths. to integration and differentiation methods and to cases where the heat or volume change of mixing may be significant. Illustrations are given in which these equations are used for testing data along successive short composition intervals in a ternary and a quaternary system.
If a quantity a1 is defined by Equation 8, Equation 9 may be obtained by combining Equations 7 and 5.
L‘pon integration, Equation 8 may also be written as Equation l @ .
1Q
Theory
T h e movement of a point along any linear path in a space Lvith composition coordinates may be construed as the change of composition of a mixture by adding successive amounts of one solution to another, each with a definite composition. T h e two solutions, one located a t the intersection of the linear path and the surface of x ? = 0 a n d the other a t the intersection of path with the surface of x l = @ , are defined as initial solutions “1” and ”2>”respectively. T h e initial solution ‘‘1” has no component 2 and the initial solution “2” contains no component 1. T h e constant mole ratios of any component k to the components 1 a n d 2 in their respective initial 2-free a n d 1-free solutions are noted as u f l and uh?. By definition 6 2 1 = 6 1 2 = 0:and uil = u : , ~= 1.0. T h e change of x k in a mixture and the change of the sum of all mole fractions d u e to changes of x1 and x 2 along a path for which the u’s are constant may be represented by the material balance Equations 1 and 2. dxk = Z akidxi
where
2:
Ukl = U I I
= (x:
Z
Thus, dxp
=
-
Uk2
UkidXi
f
(1)
uk:,dX?
+ Z ~ n z d ~=?0 + + . . . + unl
(2)
ua1
+ + x;
+
,
x:)/xP
= 1/x;
(3)
= 1/x;
(4)
Q(X1) - Q ( x ; ) =
c
= e1 dxl
(10)
From these equations. the dimensionless thermodynamic quantities, Q and a l . can be calculated directly from the experimental data and each can also be derived indirectly from the other by differentiation or integration. Therefore, Equations 6 : 9: 10 a n d 6>8: 9 suggest integration methods and differentiation methods, respectively. similar to those frequently used for binary systems to test consistency. Similar methods can now be applied to multicomponent dala by comparingvalues of a1 and Q calculated directly and indirectly from the same data. Also, a combination of Equations 9 and 10 can be easily simplified into the equations (3-5) by imposing special restrictions. I n Equation 9> the assignment of any coniponent as component 1 is arbitrary. Since aldxl= d Q = a,dxi, the same value of AQ would result for any linear segment regardless of which component is selected as the first one. For isothermal systems far below critical pressures of components, the d7‘ d?r~ term is zero a n d the dP,’dxl is negligible due to small Vvalues for liquids. For isobaric systems. the dP,’d.rl term is zero but the dT dxl cannot al\vays be neglected, especially for precise work. T h e log term for each component consists of the 1 For a particular linear path log y g and the d . ~ ~ ” d xparts. segment, the contribution of a component to a1 is large if the net composition change along the linear segment a n d the average coefficient along the segment are both large.
(x ,“jxp)dx,, and dxi; = [ut1
-
ut2
(x;/x;)Idxl
(ja)
xy and 2; have the physical significance of being
.XI and x:, in initial solutions .’1” and “2.”respectively. T h e mole ratios u k 1and ug2 can be calculated from the following material balance equations by using any t\vo points “a” and “6” on the linear p a t h :
UklXla
f
Uk2x2a
=
UklXlb
f
Uk2X2h
= XkL
Geometrical Interpretation
The relationship between Q and aIcan be interpreted as in F i p r e 1 for a ternary system. T h e Q-values of pure com-
Xka
e
If these u k l and uh.2as functions of xl, xp, x g a t points u and b are substituted into Equations 3, 4, and 5,: the following equation is obtained (5)
This simple relationship is indeed the defined linearity of the path. Treatment of vapor-liquid equilibrium data usually uses the concept of excess free-energy of a solution as defined by Equation 6 and the corresponding differential relationship, Equation 7, as derived by Van Ness (8) for multicomponent systems. Q = GE/2.3RT
= Z xi:
log
-/k
Q 1)
I c
, ’ I
1
(6)
Figure 1. Geometrical interpretation of Q and a1 in a ternary system 120
I&EC FUNDAMENTALS
components 1 and 2 of a polyhedron. Geometrically, such linear paths in a quaternary system are shown in Figure 2. Liquid compositions along c2 may be synthesized by mixing various amounts of initial solutions c and d. Both solutions have the same xi and x i . For a high order system, initial solutions 1 and 2 would have the same x3, x4> x5,. . . x i . 0
0
0
Applications
T h e local test of comparing the derived and the experimental values at various points is definitely more desirable than the over-all test suggested in most methods because local consistency a t all points along a line implies over-all consistency. However, unless a proper equation can be fitted for the Q curve, the derived cyl can only be determined graphically. T h e precision of graphical differentiation is usually poor, especially when some scatter of Q data exist. Since ~ y ~ - u s . -curves x~ for most binary systems are nearly straight, a local test will be made for a n xl-increment of finite length instead of at a point using numerical methods. A consistent set of data thus implies local consistency in each segment of all linear paths in the multidimensional composition space of a system. T h e simplest numerical method for integration and differentiation involves the trapezoid rule. For an increment ;;b in Figure 1, the derived values may be computed by the following equations :
cy1
Figure 2. Linear paths in a quaternary system for simplified test calculations
ponents are zero by definition. .4 linear path segment across the triangular base of a ternary composition space is represented by c2. T h e Q-profile along this linear path is the intersection curve of the Q-surface and the vertical plane containing 2. T h e relationship between Q and cy1 for any multicomponent system is shown in Figure 1. Consistency of a set of data implies, therefore, that a t each point on the path a1values derived directlv and indirectly should agree and
sp
cyldxl = Q d
- Qc.
a ] , e l p and Qexp are computed directly from data by using Equations 9 and 6, respectively. T h e permissible size of Axl-e.g., ab depends on the curvature of cy1 curve. -4 large deviation from linearity requires small A x , . If many data points along a linear path are available, a plot of c y l , e x p us. XI would immediately indicate the appropriate size of hxl to be used. For designing experiments, it is advisable ta provide data points evenly spaced at a distance of 0.05 to 0.10 mole fraction intervals along a few linear paths so that precise evaluation of consistency by this method can be made. Illustrations are given in two examples to show the workability of the proposed equations and methods. Scarcity of experimental data on complete isothermal systems and heat of mixing data for isobaric systems limits extensive and rigorous test of most of the presently available multicomponent data.
T h e former is usually called the local
test and the later is the over-all test. A test program for any region or the entire multicomponent system may consist of choosing paths systematically to cover a space under consideration. Since the coefficients of log yc in Equation 9 vanish according to Equation 5 for components c 3 , with constant x, along a linear path, a saving in computation time can be achieted by choosing paths with more components as c's. Equation 9 then has the simplest form of ai = - [ ( H 2 . 3 R T 2 ) ( , d T ' d x i ) ] [ ( V 2 . 3 R T ) ( d P ,d x J ] log(y1 ' 7 2 ) for those patiis parallel to the edge connecting pure i(
+
+
Table I.
Calculated Results of Example 1
Methanol(11-Benzene(2)-Carbon Tetrachloride(3) at 34.68' C.
A.
Point 1 2 3 4 5
x1
XP
0.207!j 0 2110 0.3781 0,554:3 0.759')
0.1900
0.3879 0,3122 0.2078 0.1076
P , mm. Hg 291.11 302.13 307,23 308.13 298.80
Y la 3,472 3,503 2,061 1.484 1.167
B. Path
K*b
Kab
DATA
ffl,
7 2"
Y3 O
1.542 1.477 1.848 2,536 3.949
1.135 1.159 1.428 1.848 2,962
Qexp
0,1810 0,2063 0.2499 0,2424 0.1776
RESLJLTS a18
E l , *xp
El, oal
A Q ~ I
AQW
a+b 56.542 -57.542 8.010 0.0253 0.0253 6.434 7,222 7.228 2 3 -0.4531 - 0,5469 0.433 0.109 0.0453 0,0436 0.271 0,261 3 4 -0.5925 - 0,4075 0.093 -0,177 -0.042 0,0074 0,0075 -0,042 4 5 -0.4874 - 0,5126 -0,162 -0.465 0.0646 0,0648 -0,314 -0,315 3 1 0.7163 - 1.7163 0.240 0,581 0.415 0.404 -0,0708 -0,0689 Calculated from yi =: yiP/xip; wherep; = 198.83,pi = 146.26, a n d p i = 153.55 mm. Hg ( 2 ) . 011 = log y t K Plog y~ K Blog ?a. Per cent over-alldiscrepa.ncy = 100 Z ( A Q c a l - AQeXp)/Z(AQc.l) = (100) (0.0006)/(0.2133) = 0.28%.
+
VOL.
+
1 NO. 2 M A Y 1 9 6 2
121
P--
-
I n the following examples, the distances between successive experimental data points are rather large. However, deviations owing to the use of trapezoid rule are apparently small. Example 1. T h e vapor-liquid equilibrium data for a ternary system at 34.68’ C. by Scatchard and Tichnor ( 6 ) are used in this illustration. Since the effects of vapor phase nonideality and increase of liquid volume due to mixing are small, liquid-phase activity coefficients are calculated directly as y z = y I P / x , p i and a1 is calculated by using Equation 9 M ithout H a n d V terms. Five experimental points are shown in Figure 3. Data and computation results are summarized in Table I. A sample calculation for the path segment 1+2 is given in the following steps: Coefficients K,,in CY^: 1 (0.1900 - 0.3879) (0.2075 - 0.2110) = 56.542 (0.6025 - 0.4011)/(0.2075 - 0.2110) = -57.542 log 71 +56.542 log 7 2 -57.542 log y3 Step 2. @ l a and by substitution: @lo = log 3.472 56.542 log 1.542 - 57.542 log 1.135 = 8.010 sib = log 3.503 56.542 10s 1.477 - 57.542 log 1.159 = 6.434 Step 3. 51. and E I . ex,,: 81.elp = (8.010 6.434) 2 = 7.222 = (0.2063 - 0.1810)/(0.2110 - 0.2075) = 7.228 51, (by numerical differentiation) Step 4. hQcaland AQexp AQcal = (7.222) (0.2110 - 0.2075) = 0.0253 (integration by trapezoidal rule) (0.2063 - 0.1810) = 0.0253 AQexp
Step 1.
ITl
= k-2 = KS = 011 =
I .o
1.0
MeOH(I)
(2)C,H,
Figure 3.
+
Linear paths tested in Example 1
+
+
T h e tabulated results clearly indicate consistency of these experimental data, considering the rather large linear distances between successive points. The over-all discrepancy is only 0.28% and summing AQWJ along the closed path 123 results (Summing AQerp along a net error of closure of only -0.0003. any closed path obviously gives zero always.) Example 2. Quaternary system data are rather scarce and the data (7) were chosen because they were obtained from an improved equilibrium still. Existence of a n H-effect and nonideality of the vapor are anticipated for this quaternary system but the extent of these effects has not been studied. Because of the absence of quantitative data on H a n d suitable equations of state of vapor mixtures, only a n approximate
k6H6(2)
e85
Figure 4. Linear paths tested in Example 2 projected on = 0.1 5 plane
Xd
Table II. Calculation Results of Example 2 Methyl Cyclopentane( 1 )-Benzene(l)-Ethanol(3)-n-Hexane(4)
A. Point
t,
1 2 3 4 5
O
c.
61.7 60.8 61.3 63.2 65.4
XI
x2
2 3
0,656 0.510 0.284 0.184
0.114 0.180 0.298 0.516 0,659
0.068 0.154 0,268 0.153 0,085
0.110
1 2 2 3 3 4 4 5 a 011
122
= log YI
f Kz log
Kaa
K3’
K2a
-0.452 -0.522 -2.180 -1.932
0,041 0,026 0,030 0.013
-0.590 -0.504 1.150 0.919 YZ
+ K3 log + K4 log y4.
ILEC FUNDAMENTALS
7 3
DATA Yl
Y2
YJ
Y4
QeXP
1.06 1.16 1 .37 1.31 1.30
1.33 1.34 1.39 1.18
8.08 4.20 2,50 3.66 5.07
1.06 1.20 1.43 1.36 1.35
0,0965 0.1641 0.2114 0.1646 0.1162
7.4
0.162 0.156 0.150 0.148 0.146 B.
Path a+b
‘Y1a
-0,565 -0,314 0,287 0,498
at 760 mm. Hg
1.09
RESULTS ‘Ylb
-0,357 -0,134 0.612 0.692
% over all discrepancy
El,
enp
-0,461 -0,224 0.450 0.595
011, ea1
-0,463 -0,209 0.468 0.654
Qea I
0,0673 -0,0507 -0,0450 -0,0440
= (100) (0.0093)/(0.2070) = 4.5%.
QW
0,0676 0.0473 -0,0468 -0,0484
test is made by assuming negligible effects of H and nonideality of vapor. T h e tabulated calculation results in Table I1 show a n over-all discrepancy of 4.570 and small but consistently positive deviations of ( E l , e x p - E l , tal) for all local segments. H AT TheH-effect may b e e s t i m a t e d a s ~ : , , , , - E l , , , ~ = ~2.3 R T 2A X ~ from Equation 9 by using 300 cal./g.-mole as the approximate heat of mixing from ihe ethanol-benzene system ( 7 ) . T h e estimated H-effect and those from Table I1 are compared as follows:
Nomenclature GE = excess free energy
H = integral heat of mixing
P = total pressure p" = vapor pressure of component i
Q = dimensionless function defined
in Equation 6
R = gas constant
v=
integral volume increase of mixing
T = absolute temperature mole fraction in liquid phase
x
=
Y
= mole fraction in vapor phase
a = dimensionless function defined in Equation 8
y = liquid phase activity coefficient Path Segment
2-3 3-4 4-5
E l , exp
-
El,
0 015 0 018 0 059
,.a1
Estimated Contribution H-Effect 0 001 0 011 0 017
T h e agreement in sign indicates that H-effect probably does exist; the remaining differences may be caused by the use of large A x , values in this q'stem, by vapor-phase nonideality or by actual H-value being larger than 300 cal./g.-mole, in addition to the possible real inconsistency. t\n investigation of suitability of A.xl size requires additional experimental points alone; these paths. If available, equations of state of the vapor mixture could be used to correct the apparent y's in Table I1 into actual y's and H-data could be substituted into Equation 9 for precise testing. Therefore, with additional experimental data, the proposed equations and methods will enable one to conclude whether small deviations such as these shown in Table I1 are caused by actual inconsistency of measurements.
u = mole ratio of components in a solution
Superscript 0 = initial solutions Subscripts k = identity of component k in a system Literature Cited (1) Brown, I., Foch, W., Smith, F., Australian J . Chem. 9, 364 (1956) . (2) Jordon, T. R., "Vapor Pressures of Organic Compounds," Interscience, New Y o r k , 1954. (3) Krishnamurty, V. V. G., Rao, C. V., J . Sci. Znd. Research (India) 14B, 188 (1955). (4) Prausnitz, J. M., Snider, G. D., A.Z.CI2.E. Journal 5 , 75 (1959). (5) Redlich, O., Kister, A. T., IND.ENG.CHEM.40, 345 (1948). (6) Scatchard, G., Tichnor, L. B., J . Am. Chem. Soc., 74, 3724 (1952). (7) Sinor, J. E., Weber, 3. H., J . Chem. Eng. Data 5 , 243 (1960). (8) Van Ness, H. C., Chem. Eng. Sci. 10, 225 (1959).
RECEIVEDfor review March 13, 1961 ACCEPTEDSeptember 25, 1961
A N EQUATION OF STATE INVOLVING T H E CRITICAL RATIO LEO F. EPSTEIN
General Electric Co., Vallecitos Atomic Laboratory, Pleasanton, C a y .
Modern statistical mechanics makes it possible to compute the second and third virial coefficients, 6 and C, in the equation of state of a gas in terms of fundamental intermolecular interactions. By using theoretical methods to obtain these parameters and applying the concept of corresponding states with the values of the critical constants P,, Vc, and T,, an equation is obtained which gives good agreement with experimental data and in which each of the five constants used is directly related to the physical properties of the gas.
\VAALS (26) in 1873 first formulated the equation of state of a gas in terms of the attractive and repulsive components of the forces acting between molecules, there have been many attempts to extend and generalize these concepts Partington end Shilling ( 2 4 ) listed 56 equations of state in 1924 I t is not improbable that the number of modifications proposed since then is a t least as great. Some of the resulting empirical formulations have become quite elaborate. T h e equation of state x e d for water vapor by Keenan and Keyes ( 1 3 ) . for example, contains at least nine arbitrary constants. T h e recognition of tl-e continuity of the liquid and gaseous phases and the law of corresponding states, which may be deduced from this behavi3r, has been combined frequently with the van der It'aals model to yield equations which attempted to
S
INCE v k h DER
represent the P-V-T relations of gases over a n extensive range of the system parameters in terms of the critical constants, P,, V,, T,-for example, the article by Lydersen, Greenkorn, and Hougen (16). I n this study, charts and tables are presented based on the van der Waals model but using L, as a n empirical property of each gas rather than the value 2, = 8/3 predicted by this simple theory. For most real nonpolar gases z, is about 3.68 ( 3 ) and for polar substances tends to be somewhat higher. For example, the following values have been given (22): Compound
Diethyl ether Ethyl alcohol Water
Formula ( CzHs)2
*C
0
CzHbOH H2 0 VOL. 1
NO.
3.814 4.115 4.458 2 M A Y 1962
123
