VLE Thermodynamic Consistency: Herington Test background

Test 1: Herington Test (Area Test). Integration of the equation

over composition x1 at constant T or p gives
,

where and .

The integration term containing ε can be neglected for isothermal systems, because the absolute value of ε is typically less than 3×10-5, as reported by Kurihara et al.19 However, ε for isobaric systems can be large19 (as high as 4×10-2) and cannot be neglected. For the evaluation of the integration term containing ε for isobaric systems, variation of the excess enthalpy HE with temperature and composition is required. Evaluation and integration of HE causes additional difficulties, due to the degree of availability and reliability of HE data sets. Herington20 provided an empirical estimate of the integration term containing ε for isobaric systems by use of the total boiling range of the mixture. Wisniak21 slightly modified the criteria provide by Herington. Empirical criteria require two values for the test:
,
where A is the area above the zero line on the plot of ln(γ1/γ2)against x1, and B is the area below the line. According to Wisniak,21 the test criteria are met for isothermal data sets with D < 5, while for isobaric data sets, the condition for passing is (D-J) < 10. Kojima and coworkers19,22 presented slightly different criteria for the test; for |A*| < 3 , the test is passed, and otherwise, it is not. These criteria are especially useful when the given mixture is nearly ideal. For such mixtures, values of |A*| are very small, but D (or D-J) can remain high. In this research, we combined the two criteria in our software implementation.

In TDE, if |A*| < 3, the test is passed. Otherwise, for an isothermal data set, if D<5 , the test is passed, and for an isobaric data set, if (D-J) < 10 the test is passed.

For integration of the experimental data, a polynomial equation with order between 2 and 6 is selected automatically based on the correlation coefficient of a fit to the data.
,
where d is the order of the polynomial, and ai are the coefficients of the polynomial.

The Herington Test indicates compliance with the Gibbs-Duhem equation over the whole composition range. It has the advantage of simple implementation, and a single plot of ln(γ1/γ2) against x1 shows the overall quality of a VLE data set. The quality factor for the Herington Test , Ftest1, is calculated with the values of (D or D-J) obtained in the test.

For isothermal data sets; Ftest1 = 5/D, with the limits 5 <D<50
and for isobaric data sets; Ftest1 = 10 / (D - J), with the limits 10 <(D-J)< 100.

Herington Test Results: Output screen details