I
LUH C. TAO Department of Chemical Engineering, University of Nebraska, Lincoln, Neb.
How fo Compute
Binary Vapor-Liquid Equilibrium Compositions
. .. from
€xperirnental P-x or f-x Dafa This method contributes to the reduction of time consumed in
these
operations without
THE
.
usual study of vapor-liquid eqoilibrium, which is important for the design and study of the distillation operation, consists of obtaining corresponding phase compositions directly from various laboratory equilibrium stills. I n general, a static-type apparatus usually has sampling difficulties of vapor phase and a dynamic-type ensures only steady state, not necessarily the equilibrium state assumed. Subsequently, most of the thermodynamic test methods detect only the necessary condition of internal consistency but not the sufficient condition of attainment of actual equilibrium state. Since precise measurement of t-x or P-x data may be achieved in simple reflux-type apparatus, it is interesting to investigate computation methods which may produce precise phase equilibrium data from experimental t-x or P-x data. The numerical method developed here is based on rigorous material balance and thermodynamic considerations and requires vapor-phase activity coefficients and AHor Av data as used in consistency tests. This method, like many other numerical methods, can be programmed in a high-speed computer, and the results will be useful in assuring attainment of phase equilibrium in an apparatus, extending data accurately from one single x-y point, and bypassing or minimizing operations of equilibrium still for cases where difficulties of construction and operation of a suitable still, chemical unstability of the system, or other factors are encountered.
Theory The vapor phase activity coefficients and integral heat or volume of solutions may be obtained by P-V-T and calorimetric measurements. In this method, these quantities are assumed to be known values. Since the equilibrium state is defined by equal fugacities of a component in liquid phase and that in the vapor phase, then YiXifii"
= f e L = f i V = y,@
(1)
a
sacrifice
For a binary system, components 1 and 2 compose the whole system. Therefore, the material balance of vapor phase is : y1 + y z = 1.0
in
accuracy
n = 1 up to n, and subsequent transformation yields the following form:
(2)
By combining Equations 1 and 2, the following equation contains only two unknowns-Le., y1 and yz:
By combining Equations 7 and 3, basic Equation 8 of this method containing
, is derived.
only one unknown, The exact thermodynamic equation of excess free energy of solutions is defined in Equation 4, and a rigorously derived form for a solution with no chemical reactions is also developed by Ibl and Dodge (4) as Equation 5 : AGE/RT =
where
r
=
XI
In
- _fl_H RTZ
sure data, and
y1
dxl Av
+ RT
temperature data.
~
f x z In
YZ
(4)
for constant presdP for constant dxi
Since the curvature
of d(GE'RT) us. x1 is usually small, intedxi
gration by trapezoid rule is suitable and more precise computation may be carried out conveniently by using small Ax1 in a digital computer. Substitution of an assumed constant increment of Ax1 = E , x ; = (n-l)e, and x; = ne into Equations 4 and 5 gives the following after rearrangement:
With the aid of ro = 0 and yz,O = 1.0, the summation of all Equations (6) with
The y1,O in Equation 9 can be obtained by equations derived by Gautreaux and Coates (2) using P-x or t-x data. The equations are shown in exact forms as 12 and 13 and approximate forms for 1~w pressures as 14 and 15 :
From one of these equations, y1,o is obtained by substituting the experimental VOL. 53, NO. 4
0
APRIL 1961
307
(g)p (E) or
at
=
x1
0 and then
in Equation 8 is solved by iterations starting at n = 1. By substituting the solution into Equation 3, y z and are obtained. From these activity coefficients, the vapor compositions are computed by Equation 1. In other words: this method is based on solution of two unknowns y1 and yz described by two simultaneous equations, 3 and 8. Since Equation 8 is a first-order ordinary differential equation, another equationone of 12 to 15-must be used to provide y,,o, an integration constant. Method
The method is more effectively presented by the flow-chart below to sholv iterations of this numerical method.
Discussion
Agreement between computed and experimental values in these two examples indicates both feasibility and accuracy of this proposed method. In order to ensure computed results, either 4 and AH or Av data must be available or an assumption of negligible effect must be confirmed. The confirmation may be achieved by comparison of a computed y value with an experimental y a t the same x point, where these factors
impose a maximum deviation, or by actual measurement and computation of relative deviations caused by these factors. l f y l , cannot ~ be determined accurate11 by Equations 12 to 14 owing to a fictitious value o f f o to be assumed or to the difficulty of obtaining the terminal slope, one experimental x-y value may be used to guide the selection of a correct y1,0 value. For such computations, the use of a high-speed computer \ \ o d d be essential.
Example- Ethyl Alcohol(1)-Water( 2) System The t - x data of ethyl alcohol-water system at 760 mm. Hg has been studied (3), and activity coefficients fit well in Van Laar equation ( 7 ) . = 6.92 was obtained b y Coates (2)from Equation 14. Computation results are tabulated below. A comparison of computed and experimental values of y in the figure below shows good agreement.
I n t e r p o l a t e and store T(or PI,
$ (or
o,,
), 0,,
A H ( o r A v ) a t 0. E . 2 E
,
.,
1.0 ~
4
Compute and store , 7 , (one of Eq 12 t o 15)
.b
I1
C o m p u t e Cn ( E q . 3 ) An (Eq. 10)
Bn (Eq. 11)
I C o m p u l e and s t o r e 7 , ( E q 3 ) ’ I -r
[Compute and s t o r e
?‘,I
n
x1
0 1 2 3 4 5 6 7 8 9 10
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
11 12 13 14 15 16 17 18 19 20 a
t,
oc.
Rn
A 1L
... 0.0832
100 90.8 86.8 84.5 83.2 82.4 81.8 81.3 80.8 80.4 80.0 79.7 79.35 79.0 78.76 78.56 78.37 78.22 78.16 78.23 78.32
...
0.7106 0.6189 0.5636 0.5277 0.4960 0.4647 0.4295 0.3928 0.3549 0.3165 0.2791 0.2408 0.2036 0.1690 0.1358 1.1044 0.0754 0.0484 0.0232 0
0.1539 0.2248 0.2990 0.3729 0.4528 0.5261 0.5946 0.6601 0.7218 0.7775 0.8248 0.8655 0.9015 0.9335 0.9575 0.9790 0.9990 1.0123 1.0000
C,
... 1.0495 1.1243 1.1865 1.2416 1.2860 1.3202 1.3421 1.3544 1.3597 1.3555 1.3430 1.3226 1.2969 1.2668 I . 2330 1.1945 1.1564 1.1171 1.0702 1.0248
YdY2 6.92 3.89 3.06 2.48 2.02 1.69 1.39 1.20 1 .oa 0.94 0.83 0.737 0.676 0.635 0.584 0.530 0.523 0.501 0.423 0.420 0.375
Reference ( 2 ) .
A flow-chart of the proposed numerical method
Particular attention is called to transforming C,-l to C, in Equation 9 by
(%-
introducing 1)e step prior to reset n.
0
EXPERIMENTAL Y,
x
COMPUTED
Y,
obtained in a
Illustrations
Owing to scarcity of AH, A v , and C$ data, this method is tested only for systems where these factors are negligible. Since there are a number of commonly used correlations-e.g., Van Laar, Margule, Scatchard, and others-two systems of known correlations, one for Van Laar and the other for Margule, Hill be used to test the method and to confirm generality of this method. e is assumed to be 0.05 to facilitate calculations in a desk calculator. (See Examples.)
308
INDUSTRIAL AND ENGINEERING CHEMISTRY
,
1 I I .2 * 4xI or y,
,
T
.E
Ethyl alcohol (I)-water (2) system
Y1
YZ
I41
6.92n 4.04 3.16 2.62 2.216 1.930 1.676 1.518 1.414 1.310 1.228 1.162 1.120 1.093 1.063 1.036 1.092 1.024 1.003 1.002 1.000
1 .oo 1.015 1.032 1.058 1.097 1.142 1.206 1.265 1.309 1.394 1.480 1.576 1.657 1.722 1.821 1.955 1.974 2.044 2.371 2.386 2.665
0.00 0.323 0.432 0.496 0.534 0.560 0.575 0.595 0.621 0.636 0.654 0.673 0.698 0.729 0.757 0.784 0.827 0.867 0.897 0.948 1.000
VA PO R-L IQ U I D E Q U ILlBRlUM The Van Laar, Margule, and Scatchard equations are definitely simple in form and utilize only two constants for determining y and y in the whole composition range. However, they cannot be used for accurate computation because each involves specific assumptions. The applicability of any of these equations is thus limited to correlation of known activity coefficients rather than to accurate computation of equilibrium
compositions. Similarly, the RedlichKister equation (5) is a generalized correlation in polynomial form in which those constants must be determined by experimental x-y data. Apparently, the shapes of P-x or t-x determine the suitability of a particular form of a correlation equation. Therefore, this method appears to offer a logical and nonrestricted approach for obtaining phase equilibrium
Example-Chloroform(1)-Ethyl Altohol(2) System The P - x data of chloroform-ethyl alcohol system a t 30" C. is obtained from Scatchard (6). The activity coefficients apparently follow the Margule equation rather than the Van Laar ( 7 ) . yl,o = 1.480 is obtained b y using the slope calculated from experimental points a t x1 = 0 and x2 = 0.0062. Computation resdts are in the tabulation below which also lists correlated values (6) for comparison. Values of y and y from computation and experiments are compared in the figure.
n 0
XI
p, Mm. Hg 102.78 121.58 143.23 166.80 190.19 210.00 228.88 244.10 257.17 268.00 276.98 284.10 290.20 295.10 298.53 302.10 304.22 306.30 306.84 304.89 295.11
An
... 0.1226
Bn
Cll
YdY2
... 1.0098
... 0.8110
1.480 1 1.653 0.2134 0.6688 1.0356 1.754 2 0.10 3 0.15 0.2827 0.5597 1.0652 1.845 0.3409 0.4749 1.0984 1.885 4 0.20 5 0.25 0.3983 0.4161 1.1338 1.827 0.4520 0.3673 1.1685 1.776 6 0.30 7 0.35 0.5088 0.3291 1.2026 1.678 0.5665 0.2959 1.2342 1.566 8 0.40 0.6254 0.2662 1.2622 1.443 9 0.45 10 0.50 0.6849 0.2385 1.2856 1.314 11 0.55 0.7457 0.2125 1.3052 1.174 12 0.60 0.8028 0.1863 1.3157 1.047 13 0.65 0.8572 0.1607 1.3187 0.920 14 0.70 0.9087 0.1356 1.3132 0.787 15 0.75 0.9506 0.1104 1.2976 0.675 16 0.80 0.9874 0.0859 1.2724 0,552 17 0.85 1.0115 0.0621 1.2351 0.453 18 0.90 1.0275 0.0398 1.1871 0.340 19 0.95 1.0344 0.0189 1.1249 0.220 20 1.00 1.0429 0.0000 1.0429 0.180 Interpolated values in ref. (6). 6 Calculated from P at 0.00 0.05
Q
,
YI
Yl
7 2
Yla
...
1. 480b 1.000 0.0000 1.646 0.996 0,2006 0.2000 1.742 0.993 0.3589 0.3588 1.821 0.987 0.4833 1.853 0.983 0.5750 0.5754 1.810 0.991 0.6359 1.774 0.999 0.6862 0.6844 1.707 1.017 0.7221 1.633 1.043 0.7496 0.7446 1.558 1.080 0.7720 1.484 1.129 0.7906 0.7858 1.409 1.200 0.8050 1.341 1.281 0.8182 0.8136 1.278 1.389 0.8307 1.214 1.543 0.8401 0.8378 1.164 1.725 0.8527 1.113 2.017 0.8637 0.8618 1.075 2.374 0.8804 1.037 3.051 0.8976 0.8976 1.004 4.564 0.9232 0.9304 e.. 1.000 5.556 1 * 0000 x1 = 0 and 0.0062 of ref. (6)
... ...
...
... ...
... ... ...
4.7
e
4 -
0
300-
3 -
x
250-
EXPERIMENTAL 'r2
EXPERIMENTAL
>,
COMPUTED
?,
data for the whole composition range without using conventional equilibrium stills. Also, computation may yield more accurate x-y data from t-x or P-x data which are obtainable from simple equipment, with less time-consuming operation and more precise measurements provided other data of 4, AH, or Av are available. Such a method may also find uses where difficulties occur in operating or constructing a proper equilibrium still, in chemical unstability of the system. or in analysis of samples. Since the method is rigorous and error of numerical computation can be diminished by using a small increment in a high speed computer, the computed values ofy would be accurate and precise. Therefore, they may be used to check not only thermodynamic consistency of data but also the degree of attainment of equilibrium state itself in the apparatus. If the attainment oi equilibrium can be assumed in an apparatus, Equations 8 and 9 may also be used to produce 9, AH, or AU values, which are relatively scarce in the literature. Nomenclature (Since all equations are combined in dimensionless groups, no units are given) AGE = excess free energy of a liquid solution AH = molal integral heat of solution n = number of increments bo = vapor pressure of a pure com- ponent P = total vapor pressure of the system R = gas constants t = temperature Av = molal integral volumetric change of solution = mol fraction of a component in x liquid phase = mol fraction of a component in y vapor phase 6 = increment of x 1 6 = gas phase activity coefficient y = liquid phase activity coefficient r = defined by equations following Equation 5
0
x
Subscripts
1, 2
'
= designation of component 1 and
2 ne, ka = composition of
XI
where n or k
are integers 9
literature Cited (1) Carlson, H. C., Colburn, A. P., IND. ENG.CHEM.34, 581 (1942). (2) Gautreaux, M. F., Coates, J., A.1.Ch.E. Journal 1, 496 (1955). (3) Jones, C. A., Schoenborn, E. M., Colburn, A. P., IND.ENG. CHEM.35, 666
?-
200-
2 -
X
+.
E
d
*e*.
0 .X
z
c
t
150-
x
I
*x
l i
ex
*
o EXPERIMENTAL Y,
(I 947)
x I[
* x
100-
x
*
COMPUTED Y,
X
&
I
90, 0
,
6
'
,
Yo%--
I . , . *
Ib 0.9
.
_. 10
* .
x
*X
e.*
*
6
.
A
*
x - "~
'
I
(4f-Ibf,'N. V., Dodge, B. F., Chem. Eng. Sci. 2, 120 (1953). (5) Redlich, O., Kister, A. T., IND.END. CHEM.40, 345 (1948). (6) Scatchard, G., Raymond, C. L., Gilman, H. H., J. Am. Chem. Soc. 60, 1278 (1938).
RECEIVED for review November 7, 1960 ACCEPTEDJanuary 25, I961 VOL. 53, NO. 4
APRIL 1961
309
