TESTING THE CONSISTENCY OF VAPOR-LIQUID EQUILIBRIUM DATA LUH C . TAO
A un$ed method tests local and over-all consiste n y of linear paths in binaiy and multicomponent systems.
It analyses efccts of vapor non-
ideality and integral heats and volumes of mixing when an apparent inconsistency occurs nterest in vapor-liquid equilibrium data stems from
I their wide application in process industries and their
implications in physicochemical studies. Reliability of experimental data is often tested for internal consistency because experimental errors such as a nonequilibrium steady state in a recirculation type still, and incorrect sampling are involved. The very existence of a consistency test implies that the number of independent variables in a set of data is more than that required to define a system and that compatibility between the measured variables is a necessary though not sufficient condition to assure the assumed equilibrium state. Commonly used methods, which are based on the GibbsDuhem equation in its integral form over a whole composition range, further limit the test result to an over-all and not necessarily local consistency. I n addition, apparent inconsistencies caused by nonideality of vapor, integral heat or volume of mixing, and random errors are usually not analyzed. Therefore, the results obtained in many test methods are inconclusive. The objective of this report is to unify many available concepts into a test method which is applicable to binary as well as multicomponent systems for their local and over-all consistency, and investigate possible causes of apparent deviations. Whether or not the data are consistent within the bounds of noise or random experimental error, and the conditions under which the data are consistent can be determined. Theory
A true equilibrium of any component i distributed between two phases requires an equality of fugacities of i in both phases. This results in Equation 1 which is usually used to compute the liquid phase activity coefficient. From activity coefficients, the amount of excess free energy of liquid compared with the ideal solution may be defined in a dimensionless form as Q in 36
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Equation 2. The use of dimensionless forms has an obvious advantage of independence of units. Differcntiation of this dimensionless free energy according to Equation 5 results in a generalized Gibbs-Duhem equation (72). For a linear path in any multicomponent system, the derivative of dimensionless free energy is Equation 3 ( 7 7 ) . Linear paths are used to avoid complications of dimensionality and yet to be able to cover any multidimensional space by mapping many linear paths. I n addition, this makes it possible to plot graphically Q and a1 us. X I , a composition coordinate representing the path. The subscripts a and b signify any two points on
Since Q and a1 can be calculated directly from experimental data according to Equations 2 and 3 and also indirectly from Equations 4 and 5, an integral method with Equations 2 and 4, and a differential method with Equations 3 and 5 are available to test an over-all linear path as well as local points. For the special case of binary systems, the integral method with x l a = 0 and x l b = 1 simplifies to the Redlich-Kister method (9). This general idea of cross checking for a binary was used by Black (2) in developing a correlation. Since all experimental results have noise, any method tests only consistency within the noise bound. For convenience, it is the usual practice to neglect ea, A (for isobaric data), and B (for isothermal data). This practice in fact assumes that the true effects of these AUTHOR L u h C. T a o is Associate Professor of Chemical Engineering at the University of Nebraska, Lincoln, N e b . Assistance of Dauid E . Farlow and comments by D r . Cline Black of Shell Deuelojvnent Co. are gratefully acknowledged.
bounds are thus *E(?,), *E(Q), and *E(al) which represent the maximum errors of those variables in parentheses. Computations by these equations are straightforward,and they aid experimenters in improving data precision by minimizing selectively the dominating measurement errors. The use of this noise bound is found convenient for the test method.
Tail Method
factors are small and within the probable noise bound. If apparent inconsistency is observed outside the noise bound, examination of the neglected tern is warranted to determine if an inclusion of these terms would move the indirectly computed values into the area of noise bound. If the values do not then fall within the noise bounds, a definite conclusion of inconsistency of data is firmly established. Therefore, some simple computations of noise bound for experimental data are useful and should be included in any consistency test. Utilizing the principles of the least squares method, the most probable smooth curves on plots of Q and 011 us. x1 can be obtained by minimizing the sum of squares of deviations (ssd) of data points from a proposed curve. From ssd, statistical variance of the data may be estimated by Equation 6 in assuming all data points as samples from the same population. This assumption 1 is made to avoid complex statistical treatment. Finally, the noise bound may simply be set as *Cs around the smooth curve where C is related to the normal ditribution curve and C = 1, 2, 3 for, respectively, the probability limits including 2/3, 19/20, 997/1000 of data points.
9 = ssdjm
(6)
Also, it is possible to estimate the bound by a nonstatistical approach. Since maximum measurement e m r s E(T),E(P), and E(x) are often known, the propagation of these errors to the computed thermodynamic quantities can be calculated from Equations 7, 8, and 9 a8 obtained by taking total derivatives of Equations l, 2, and 3 not including e, A, and B . The noise
The operation sequence is illustrated by a flowchart (Figure 1) and it leads definitely to one of three conclusions: consistency, inconsistency, conditional consistency. Smce data on H or V are not available for many systems, a conditional consistency is placed as one of the conclusions. The condition is that the true H or V be within the bounds of those calculated by Equation 10.
H=-
- 011.
* E(al)]RT*/(dT/dxl)for isobars (1W
V = [wo - ala
E(at)]RT/(dP/dxl)for isotherms (10W
In many cases it is possible to estimate the magnitude of H or V by direct comparison with known systems of similar molecular structures, by extrapolating values of homolog compounds or by estimating intermolecular forces. Thus, further judgment on data consistency may be made but assurance of the results requires experimentally determined H or V values. For systems far below the critical temperatures of its components, V is usually negligible and B(xl) vanishes in EquziGon 3 for an isothermal system. In general, for all relatively low pressure systems, the effect of H is a more dominating cause of apparent inconsistency than the nonideality of vapor. As the p m u r e of a system increases, the nonideality of vapor becomes more significant and contributions of et to apparent inconsistency should be investigated. Estimation of e, is usually made from an equation of state for vapor mixtures. The most popular ones are the equations of Benedict-Webb-Rubin (1) for hydrocarbons VOL 56
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