Consistency testing

Comprehensive data collection for more than 6000 binary and multicomponent mixtures at moderate pressures. Data correlation and consistency tests are given for each data set. 

The sum of the squared differences between calculated and measures pressures is minimized as a function of model parameters. This method, often called Barker s method (Barker, 1953), ignores information contained in vapor-phase mole fraction measurements such information is normally only used for consistency tests, as discussed by Van Ness et al. (1973). Nevertheless, when high-quality experimental data are available. Barker s method often gives excellent results (Abbott and Van Ness, 1975). 

Literature dealing with adsorbent—adsorbate interactions in Uquid phase is largely confined to patents (11—43). Although theoretical consistency tests exist for such data (44), the search for an adsorbent of suitable selectivity remains an art. 

Penetration (ASTMD5). This is a commonly used consistency test. It involves the deterrnination of the extent to which a standard needle penetrates a propedy prepared sample of asphalt under definitely specified conditions of temperature, load, and time. The distance that the needle penetrates in units of mm/10 measured from 0 to 300, is the penetration value. Soft asphalts have high penetration values. 

Impression plasters are formulated to produce a thin, fluid slurry when mixed with the proper amount of water. A satisfactory impression plaster should have a setting time of 4 1.5 min fineness, ie, 98% should pass a number 100 sieve (ca 0.15 mm), and 90% pass a number 200 sieve (ca 0.07 mm) setting expansion at 2 h should be <0.15% the compressive strength at one hour should be 5.9 2 MPa (855.5 290 psi) and testing consistency as determined by the diameter of the slump in the consistence test should be 90 3 mm. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

The well-known Gibbs-Duhem equation (2,3,18) is a special mathematical redundance test which is expressed in terms of the chemical potential (3,18). The general Duhem test procedure can be appHed to any set of partial molar quantities. It is also possible to perform an overall consistency test over a composition range with the integrated form of the Duhem equation (2). 

Many additional consistency tests can be derived from phase equiUbrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubiUty, and solubiUty of water in chemicals are related to solution activity coefficients and other properties through fundamental equiUbrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equiUbrium models may permit a missing property value to be calculated from those values that are known (2). 

Thermodynamic consistency tests for binary vapor-liquid equilibria at low pressures have been described by many authors a good discussion is given in the monograph by Van Ness (VI). Extension of these methods to isothermal high-pressure equilibria presents two difficulties first, it is necessary to have experimental data for the density of the liquid mixture along the saturation line, and second, since the ideal gas law is not valid, it is necessary to calculate vapor-phase fugacity coefficients either from volumetric data for... 

A consistency test described by Chueh and Muirbrook (C4) extends to isothermal high-pressure data the integral (area) test given by Redlich and Kister (Rl) and Herington (H2) for isothermal low-pressure data. [A similar extension has been given by Thompson and Edmister (T2)]. For a binary system at constant temperature, the Gibbs-Duhem equation is written... 

The three areas are found by graphical integration. The thermodynamic consistency test consists of comparing the sum of the three areas [left-hand side of Eq. (81)] with the right-hand side of Eq. (81). The three areas depend upon equilibrium data for the composition range x2 = 0 to x2 = x2. However, the right-hand side of Eq. (81) depends only on equilibrium data at the upper limit x2 = x2. The comparison indicated by Eq. (81) should be made for several values of x2 up to and including the critical composition. 

To illustrate this thermodynamic consistency test, Figs. 15, 16, and 17 show plots of the appropriate functions needed to calculate Areas I, II, and III, respectively, for the nitrogen-carbon dioxide system at 0°C the data are taken from Muirbrook (M5). Fugacity coffiecients were calculated with the modified Redlich-Kwong equation (R4). 

Thermodynamic Consistency Test for Carbon Dioxide (I)-Nitrogen (2) at 0°C... 

The thermodynamic consistency test for binary systems described above can be extended to ternary (and higher) systems with techniques similar to those described by Herington (H3). The necessary calculations become quite tedious, and unless extensive multicomponent data are available, they are usually not worthwhile. 

Sutton, T.L., and J.F. MacGregor, "The Analysis and Design of Binary Vapour-Liquid Equilibrium Experiments Part I Parameter Estimation and Consistency Tests", Can. J. Chem. Eng., 55, 602-608 (1977). 

The availability of data on KAW, KoW and KqA raises the possibility of a consistency test. At first sight it appears that KqA should equal K0W/KAW, and indeed this is often approximately correct. The difficulty is that in the case of KAW, the water phase is pure water, and for KqA the octanol phase is pure dry octanol. For KoW, the water phase inevitably contains dissolved octanol, and the octanol phase contains dissolved water and is thus not dry. Beyer et al. (2002) and Cole and Mackay (2000) have discussed this issue. 

This illustrates the general philosophy of taxometrics. For something to be deemed a taxon, it needs to clear several hurdles, which arguably makes taxometrics the most rigorous analytic approach to the study of taxonomy. The last two conditions (e and f) are external consistency tests and apply to all taxometric studies. On the other hand, a, b, c, and d are internal consistency tests. Some of them are specific to MAXCOV—other procedures have their own unique internal consistency tests—but a and b can be performed with any of the CCK methods. We will now consider the first three internal consistency tests (a, b, and c) in detail and postpone discussion of the distribution of taxon membership. 

Clearly, the GFI is a useful index. How should one interpret a low GFI The only way to do so is by considering the results of the other consistency tests. If the nose count test failed, or cleared the threshold by a close margin,... 

MAMBAC offers two standard internal consistency tests, the nose count test and the base rate variability test. MAMBAC allows for two analyses per pair of indicators, which means that two indicators produce two plots, three indicators produce six plots, four indicators produce 12 plots, five indicators produce 20 plots, etc. Waller s MAMBAC does not do this automatically, and the investigator has to run each analysis independently. Ruscio s software, on the other hand, is fully automatic. 

Nose count and base rate variability consistency tests are possible with MAXSLOPE, although it is not yet clear how these tests behave when the underlying distributions are of the difficult kind. Luckily, MAXSLOPE puts less emphasis on internal consistency testing and stresses external consistency testing instead. MAXSLOPE is different from other taxometric algorithms and thus can provide a strong test of external consistency for other procedures. 

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