~ ! ~:, , - *" I II ELSEVIER Fluid Phase Equilibria 110 (1995) 89 - 113 IIRM Thermodynamic consistency of data banks D. Mikl6s, S. Kem6ny**, G. Alm~sy, K. Kollfir-Hunek* Department of Chemical Engineering, *Department of Mathematics Technical University of Budapest, H-1521 Budapest, Hungary Abstract Data sets originated from different sources are typically used simultaneously either in data banks or in data bases of ftowsheeting programs. If these data sets concern different kinds of data, they ought to be in accordance with the laws of thermodynamics. This is not assured, however, and in many cases it is not even checked. Methods used for data compilation rely on approximate models, which may bias the residuals. Two model-flee methods are proposed, where the residuals of the thermodynamic differential equations are investigated. The first method is a g2 test based on a quadratic form of thefresiduals, involving variances of the elementary measurements, ff they are available. The second method is a rather sensitive trend/shift analysis of residuals, which does not require the knowledge of error variances. The examples are the Clausins-Clapeyron equation, a binary vapor-liquid equilibrium data set and a simultaneous treatment of PVT and caloric data. Keywords: theory, methods of calculation, data treatment, thermodynamic consistency, pure compounds, mixtures. INTRODUCTION Quality of thermodynamic data can be interpreted in many different ways. It may be examined within a data set, when outliers are detected and the variance is assessed. More information is revealed by cross-checking data sets. These data sets may refer to the same quantity but measured in different laboratories. If there is only one independent variable, as in the case of pure component vapor pressure data, the data sets may be plotted together. If there are several independent variables, experimental data are often projected to a common base. E.g., second virial coefficients are calculated from pure component PVT data at lower pressures and plotted against temperature; or the Bl2 cross second virial coefficients are plotted for mixture data (Starling et al., 1986). ** To whom all correspondence should be addressed, e-mail: [email protected]0378-3812195/$09.50 O 1995 - Elsevier Science B.V. All rights reserved SSDI 0378-3812(95)02745-9
Transcript
~ ! ~:,
, - * " I
II E L S E V I E R Fluid Phase Equilibria 110 (1995) 89 - 113
IIRM
Thermodynamic consistency of data banks
D. Mikl6s, S. Kem6ny**, G. Alm~sy, K. Kollfir-Hunek*
Department of Chemical Engineering, *Department of Mathematics Technical University of Budapest, H-1521 Budapest, Hungary
Abstract
Data sets originated from different sources are typically used simultaneously either in data banks or in data bases of ftowsheeting programs. If these data sets concern different kinds of data, they ought to be in accordance with the laws of thermodynamics. This is not assured, however, and in many cases it is not even checked. Methods used for data compilation rely on approximate models, which may bias the residuals. Two model-flee methods are proposed, where the residuals of the thermodynamic differential equations are investigated. The first method is a g2 test based on a quadratic form of thefresiduals, involving variances of the elementary measurements, ff they are available. The second method is a rather sensitive trend/shift analysis of residuals, which does not require the knowledge of error variances.
The examples are the Clausins-Clapeyron equation, a binary vapor-liquid equilibrium data set and a simultaneous treatment of PVT and caloric data.
Keywords: theory, methods of calculation, data treatment, thermodynamic consistency, pure compounds, mixtures.
INTRODUCTION
Quality of thermodynamic data can be interpreted in many different ways. It may be examined within a data set, when outliers are detected and the variance is assessed. More information is revealed by cross-checking data sets. These data sets may refer to the same quantity but measured in different laboratories. If there is only one independent variable, as in the case of pure component vapor pressure data, the data sets may be plotted together. If there are several independent variables, experimental data are often projected to a common base. E.g., second virial coefficients are calculated from pure component PVT data at lower pressures and plotted against temperature; or the Bl2 cross second virial coefficients are plotted for mixture data (Starling et al., 1986).
** To whom all correspondence should be addressed, e-mail: [email protected]
0378-3812195/$09.50 O 1995 - Elsevier Science B.V. All rights reserved S S D I 0 3 7 8 - 3 8 1 2 ( 9 5 ) 0 2 7 4 5 - 9
90 D. Miklbs et al., / Fluid Phase Equilibria llO (1995) 89 - 113
An interesting practice is the investigation of properties along homologous series of compounds, where the properties are usually "smooth" functions of the carbon number or the normal boiling point (Chase, 1984). Thermodynamic consistency means that the data undergo the rules of thermodynamics. In most cases different kinds of measured (or calculated) quantities are related through exact mathematical expressions, which are ordinary or partial differential equations, but in some cases a requirement is expressed for a single property. Due to the existence of random errors the differential equations are almost never exactly satisfied. The question is whether the data are subject to systematic errors as well. According to Bard (1974) there are three principal ways to compare experimental data with differential equations: a.) solve the differential equation analytically or numerically and compare the resulting function with the experimental data; b.) integrate the data numerically or graphically and compare them with the integrated form of the differential equation; c.) differentiate the data and compare them with the original form of the differential equation. All three approaches have been used for thermodynamic consistency test. Consistency test methods of thermodynamic data are practically developed for two
cases. • consistency of binary or multicomponent vapor-liquid equilibrium data by the
Gibbs-Duhem equation • consistency of binary vapor-liquid equilibrium and heat of mixing data by the
Gibbs-Helm_holtz equation. The first one is an example of checking a single group of propelties, the second one is a relation between two different thermodynamic quantities.
A REVIEW OF MODEL-FREE TEST METHODS
Those methods are considered as model-free that do not use other constraints than the (differential) equations expressing the laws of thermodynamics. Tests for vapor-liquid equilibrium data are based on the Gibbs-Duhem equation. Its general form is (Van Ness and Abbott, 1982)
Ex, d~,-(d~-p) dp_(d~T ) dr=0 i T,_x P,x
(1)
where M is the molar amount of any extensive thermodynamic property, M is the partial molar quantity related to that property, P is the total pressure and T is the temperature; x stands for mole fractions. Let M be the reduced excess Gibbs free energy
G E
- -=Exihr~ (2) RT
D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 91
V E H a
~ x i d l l l Y i i - RT d P + ~-- i -d T =0 (3)
where y is the activity coefficient, /_/z is the molar heat of mixing, V e is the molar volume change on mixing.
Differential tests
The appropriate form of the differential equation for a binary vapor-liquid equilibrium problem is
d l n y I d l n y 2 V a d P H a d T x l - + x 2 - ~- - - - - =0 (4)
d x 1 d x 1 RT d x I RT 2 dx~
Under isothermal conditions the fourth term is zero, and the third term is usually negligible, thus the above formula reduces to the well-known Duhem-Margules equation. The test procedures based on this equation are called slope tests; their crucial part is the determination of derivatives (either numerically or graphically) from experimental data. Van Ness and Mrazek (1959) proposed a suitable transformation in order to make the curve smoother. An equivalent form is used by Kojima, Moon and Ochi (1990):
d ( - G E / VU H E ( R T ) In y' d P d T
d x 1 Y2 RT d x I RT 2 d x 1 - 0 ( 5 )
They propose an arbitrary acceptance criterion: 1~16 , l < 5, where 6 i is the deviation
of the computed left hand side from zero at the i-th measurement point.
Integral tests
Both local and integral tests are based on the integration of the Gibbs-Duhern equation. Two kinds of integral tests are used: the local or two-point tests and the area tests.
Local tests
The local tests, which are applied to multieomponent data as well, perform the inte- gration between two adjacent measurement points. A typical application is proposed by McDermott (McDermott, 1964, McDermott and Ellis, 1965), where the trapezoidal rule is used to integrate Eq. (3) between points a and b:
b M E b y x/, , + x j , b ( l n y / , b _ l n y j , a ) + i _ ~ 2 d T _ i V e d p = f ( a , b ) = O (6) / 2 ~ R T a RT
92 D. Miklbs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
Those a, b pairs giving high f(a,b) values are suspect. In fortunate cases both points appear at least in one additional pair so that the erroneous measurement point can be localized. Again for isothermal data the two last terms are usually neglected. McDermott (1964) proposed an acceptance criterion, based on an error propagation calculation using assumed error variances in x, T,y and P measurements. Papers by Ulrichson and Stevenson (1972), Samuels (1972), Samuels et al. (1972) are worth mentiomng as well. Dohnal and Fenclov~i (1985b) assume that thef(a,b) values follow the Gauss distribution, therefore their weighted sum of squares follows a X 2 distribution. They also emphasize the inherent correlation between the f(a, b) values, which is neglected earlier by Samuels et al. (1972). If the computed value of Z 2 exceeds the upper critical value taken from the appropriate table, there is either a systematic error present in the data set or the variances of the elementary (x, T,y,P) measurements have been underestimated. If the test statistic is below the lower critical Z 2 value, the variances are possibly overestimated. Thus the method heavily relies upon the proper assi~rnent of measurement precision.
A r e a t e s t s
The integral or, as it is also called, the area tests integrate Eq. (3) for a binary mixture from one pure component to the other one, i.e., fromxl=0 to xi=l. The result is
1 In Y l T,, V ~ p,~ H E J" dx 1 - J" dV+ j'--X~2 d T = 0 (7) o Y2 T,, RT p~RT
where p0 is the pure component vapor pressure, Tb is the boiling point. The method has been proposed by Redlich and Kister (1948) and Herington (1947) for mixtures under isothermal conditions, where only the first two terms were considered. Again Samuels et al. (1972) introduced acceptance limits, based on error propagation calculations. They also remark that with the integration taken for the whole concentration range one loses the possibility for checking randomness of local integrals. Van Ness et al. (1973) showed that the method is not sensitive at all to the measured total pressure data, while the deviation from zero is much influenced by the activity coefficients at the very dilute regions, where the experimental values are hardly accessible using the same methods as in the middle of the concentration range. Van Ness and Mrazek (1959) found that the area test always accepts data sets which were qualified as consistent by the local test, but it does not hold conversely. Herington (1951) semi-empirically extended the method to isobaric data without requiring explicit information on heat of mixing. Kojima et al. (1990) considered experimental HE data and proposed an arbitrary value as acceptance criterion for the deviation from zero.
D. Miklts et al., / Fluid Phase Equilibria I10 (1995) 89 - 113 93
Res idua l me thods
Here the coexistence differential equation, obtained from the Gibbs Duhem equation, is solved numerically to obtain ag(xl) function which allows calculation of activity
P = xlP~ exp lg +
L
+ x?P° exp[g - x~ L
X 2 + +
R T d x I R T 2 d x x
( d x g ) __Z E __dP q H g a T
R T d x I R T 2 d x I
(8)
where ~ is the vapor phase correction, g is the reduced excess Gibbs free energy
function g = G u / RT. According to the original proposal (Van Ness et al., 1973) the procedure involves discretization at equidistant x~ values, where P is considered as unmeasured and computed in the isothermal case. Thus a P(xl) fit is performed. This fit risks introducing systematic error, therefore the method has been developed further to skip that step (Koll~-Hunek et al., 1986). Van Ness et al. (1973) proposed the visual analysis of the residuals as plotted against x v In the absence of systematic errors they scatter around zero in a random manner. Note that nowadays computation methods are preferred to visual analysis. Christensen and Fredenslund (1975) proposed an acceptance criterion based on assuming uncertainty ofx and y measurements:
IAyI___ + O.Ol (9)
This method and criterion together with the area test are used throughout the DECHEMA volumes (Gmehling et al. 1977) to assess thermodynamic consistency of data. It is a common feature of test procedures discussed above that HE and V ~ contribute as corrections only, so that their approximate values are usually sufficient. For checking the coherence of vapor-liquid equilibrium and heat of mixing data, the Gibbs-Helmholtz equation is used (Palmer, 1972, Janaszewski et al., 1982, Olson, 1983):
94 D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
(10)
First G E values are computed from vapor-liquid equilibrium data through activity coefficients. They are then plotted against 1/T, and the derivative obtained is compared with the experimental/-ff value. Obviously the concentration should be the same for the two properties, but since vapor-liquid equilibrium data and ~ are measured at different laboratories, usually interpolation is required.
APPLICATION OF MODELS IN CONSISTENCY TEST PROCEDURES
In this paper model-free methods are proposed. This approach is not always accepted in the literature, in many cases models are involved in the test procedures (Anderson et al, 1978). The Ay residuals are easily obtained as deviations from fit of models. We have shown (Kemdny et al., 1982, Kolldr-Hunek et al., 1986) that the residuals obtained this way reflect the inadequacy of the model, in addition to the possible systematic errors in data. Thus the conclusion on consistency of data based on fitting models is dubious, as it is not clear that the systematic error observed is due to the model or data. This statement holds not only for molecular thermodynamic models (Wilson, UNIQUAC, etc.) but also for polynomials (e.g. to the Redlich-KJster equation). Polynomials may produce false waves leading to non-random residuals, the tendency for waving is increasing with the degree of the polynomial. Dolmal and Fenclovd (1985a) use statistical tests for checking randomness of residuals, but the procedure may be biased due to the model. An analogous approach is widely used with other thermodynamic data as well. The usual way (Starling, 1973) is termed as "mnltiproperty analysis" (Cox et al., 1971). A common model (a flexible equation of state) is fitted to different properties simultaneously and the residuals are examined by plotting. The objective function of data reduction for the example of density and enthalpy data is:
W p - + = (11)
where w h and wp are the weights, which may differ for each measurement point j or i, their value is based on statistical or heuristic considerations; p(O) and h(O) contain the model, and they are obviously related through the model; p and h are the experimental data themselves. This route is followed for pure component data in the IUPAC Tables (Angus et al., 1974, Angus, 1983, volumes published from 1971) and by Sytchev et al. (1977).
D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 95
Wagner and the researchers of the IUPAC Thermodynamic Tables Project Centre (Wagner, 1974,1977, de Reuck and Armstrong, 1979) developed a stepwise regression method using a flexible equation of state model composed of many terms. The main risk is here again that the error of fit may bias the residuals. This may occur especially when the model is differentiated in order to obtain certain thermodynamic properties. It is worth remarking that the data recalculated from a model automatically fulfil consistency criteria, therefore it is meaningless to check them for this. If data on different properties are smoothed separately, they may be subject to checks for consistency.
PROPOSED METHODS FOR CHECKING CONSISTENCY
A/m
The aim of this work was to develop methods to check thermodynamic data, contained in a data bank. More precisely, it is to be checked whether the data obey strict thermodynamic relations. This is important, since the user of a data bank may treat data sets, coming from different sources together, he or she may transform data on a certain thermodynamic property to another one, in order to supply missing data, using the offer of service programs. During this activity, coherence of different data sets is usually not checked, and typically the user is not supplied with methods for checking it. E.g., it does not meet difficulties if missing heat of mixing data for a certain temperature are calculated from phase equilibrium data and these calculated data are used together with experimental heat of mixing data measured at other temperatures. This kind of error may have quite serious consequence during flowsheeting calculations. The idea came from the analogy with data reconciliation for networks in chemical technology (Almfisy and Sztan6, 1975). The thermodynamic relations are considered as a system of mathematical equations, like balance equations in chemical engineering. The data reconciliation procedures offer methods both for smoothing data and for detecting gross errors, mostly the latter is relevant here. The purpose of methods presented here is to test if data sets contain systematic errors, or the systematic error is si~ificant in some temperature or pressure range. Two methods are presented and applied to two cases: the case of a single differential equation and that of two equations. The example for the single equation case is the Clausius-Clapeyron equation. The other example shows a simultaneous treatment of caloric and PVT data. Example of binary vapor-liquid equilibrium data is also discussed. Both methods proposed are model-free, that is they deal directly with the differential equations of thermodynamics, without the involvement of any other model. They belong to the family of differential test procedures.
96 D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
The two test problems
The test of vapor pressure and enthalpy of vaporization data by the Clausius- Clapeyron equation
The Clausius-Clapeyron equation for the relation of vapor pressure and enthalpy of vaporization of a pure fluid is as follows:
de) -FT ? e' (12)
where, for simplicity, H is used here the enthalpy of vaporization instead of the IUPAC A v e , P is the pure component vapor pressure.
The test of PVT and caloric data
The two differential equations for PVT and caloric data are as follows:
,W') =/~Cp, V,. (13)
(14)
where g = is the Joule - Thomson coefficient, H
Cp = is the isobaric heat capacity. P
Method l
Let us assume that the set of thermodynamic relaions is linear:
W X =0 (15)
where X stands for thermodynamic properties, derivatives or their functions, it is a column vector ofnx elements and W is a matrix of coefficients, with nf x n x elements. The X variables are not available in their error-fTee form, they are subject to experimental (direct or propagated) errors. If _x denotes the values measured or calculated from elementary measurements, then
D. Miklbs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 97
x = X + 6 (16)
It is assumed that the n x vector of errors e__ is a.) statistically independent of X b.) follows a multivariable Gauss distribution c.) its expected value is zero d.) its V covariance matrix is known. With the x variables, containing errors _.e, Eq.(15) is not fulfilled, but a deviation f is observed:
f =__Wx (17)
where f is a nf column vector. The following expression will be defined as "error measure" q2:
(18)
q2 is mvariant to scaling of Eq. (15), that is to changing the unitsx variables. If assumptions a.) - d.) hold, q2 follows a Z 2 distribution, with nf degrees of freedom. Thus, assumptions above, involving the absence of systematic errors, may be checked b y a z 2 t e s t .
Vapor pressure and enthalpy of vaporization data by Clausius-Clapeyron equation
For its derivation several assumptions have been made (molar volume of the liquid phase is negligible as compared to that of the vapor phase, the vapor obeys perfect gas law). Thus Eq. (12) can be used only at low pressures, far from the critical state. As it is used here for illustration purposes only, this limitation is not serious. The equation would be valid in error-free case to all i-th measurement points. As the experimental data are subject to error, the equations are not exactly fulfilled. The i-th element of the f error vector is calculated from the deviation as follows:
/ f ' = (19)
The elements of the x vector and the W matrix in the first, i-th and last points are given as
98 D. Mik l f s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
X =
H(T~ )
)
H(T.)
W =
"1 -1 0 0 ... 0 0 ... 0 0 : ; • . . , : • . , • .
• o
0 0 0 0 ... 1 -1 ... 0 0 : ; : . , . : : . . • :
• °
0 0 0 0 ... 0 0 ... 1 -1
(20)
where x is a 2n column vector, _.W is a (2n x 2n) matrix, with n the number of experi- mental points. The derivatives are to be determined by appropriate numerical methods. The Forsythe (1957) orthogonal polynomials were used here. A key property of the orthogonal polynomials is that their coefficients are statistically independent of each other. This is an advantage when the covariance matrix is computed. Covariance matrix _V contains the variances and the covariances between the elements
of vector X, that is between the (d P / dT) j and H(T k )Pk / RTk 2 variables, where
bothj and k go from 1 to n. The evaluation of the actual covariance matrix requires an error model. The following assumptions were made: a.) experimental errors in P, T and H are small b.) they follow a multivariate Gauss distribution c.) errors of P, T and H at different measurement points are mutually independent. When setting an error model, it is important to know which data are measured directly at points i and which of them are interpolated or available only in smoothed form. In the simplest case (and this will be dealt with here) all Pi, Ti and H i are direct experimental data, and Pi and H i are measured at the same T i temperature points. In more realistic cases at least one of them is interpolated. This affects the calculation of the covariance malxix profoundly. In the case investigated, experimental errors in Pi, Ti and Hi are independent of each
other at the i-th measurement point. The error in the derivative (d P / d T) ~ depends on
that committed not only in Pi and T i but at all P and T measurement points, since they all have been used for the fit.
D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 99
Simulation study
We intended to examine whether presence of systematic errors can be detected through the value ofq 2 error measure defined by Eq. (18). As the extent of presence of systematic errors is not known in real data sets, only simulated data were used. The steps of the calculations were as follow: 1. A P against T data set was selected from the Steam Table (Eisner et al., 1986), an orthogonal polynomial was fitted to them in logarithmic form, and then enthalpy of vaporization was calculated at each point using the d In P / d In T derivatives. These together were taken as error-free P, T and H values.
2. Independent Gaussian errors were generated with oJp, o~r and o-2~/ variances, respectively, and added to the error-flee P, T and H values. 3. Orthogonal polynomial of third degree was fitted to P against T data, leading to the d P / d T derivatives. 4. Elements of the V covariance matrix were calculated, together with the deviation vector f and finally the q2 error measure was obtained. The P and T pairs were taken at equidistant temperature values between 273 K and 323 K (611 Pa to 12335 Pa), in five ranges, 10 points each. The data in different ranges are treated separately, that is 5 data sets were analyzed. This made it possible to investigate the role of pressure and temperature in detecting systematic errors. The results are summarized in Table 1. If there are random errors only, the q2 error measure must follow a Z 2 distribution, with 10 degrees of freedom for each range. The lower and upper critical values for 5% significance level are 3.25 and 20.5. First the effect of improper assumption of error variance was investigated, on the example of that of the temperature measurement. If the uncertainty of the temperature
is relatively large (O~T = 0.01 K 2) the results are strongly biased by neglecting the error in temperature, and q2 is obtained in the critical region, without existence of any systematic error, however. This is shown by runs 1 and 2 in Table 1. If neglected uncertainties in temperature are lower (comparison is not shown here), the effect is undetectable. The method is obviously rather sensitive to the assumption on errors in P and H as well. A proportional systematic error in the pressure cannot be detected by this method (run 7), because both sides of Eq. (12) change identically by this kind of error, the deviation f will be unchanged. A constant absolute error of pressure is well detected, the error measure increases si~ificanfly (run 9). Obviously the effect is different in the ranges of lower and higher pressures. The systematic error in the enthalpy (AH) is also well detected (runs 11, 12, 13). It is well seen that the actual value of the q~ error measure depends heavily on the assumed variance value for elementary measurements. If these variances are assigned improperly, the conclusion on consistency is falsified. For real data sets these variances are hard to assess, especially if the experiments were performed in another laboratory, or if the data have been taken from the literature. The problem is analogous
100 D. Miklts et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
to that mentioned in connection with the method of Dohnal and Fenclovfi (1985b). That is, if the error variances are overestimated, the systematic errors are not discovered, even if they exist, while underestimated variances detect systematic errors, even if they do not occur.
Table I Results of the simulation study: Clausius-Clapeyron equation, Method I, 50 data points T (K), P (Pa), n (kJ/kg)
The need of a very detailed error model of data (which is usually hard to implement) accompanied with the very complicated calculation of the V covariance matrix is another difficulty of the application. Attempts were made to perform analogous calculations for the other problem involving two differential equations, but it proved extremely demanding of computer time and memory size.
Method H
Fig. 1 shows the elements of the _f deviation vector for all the 50 data points plotted against temperature. It is seen that the systematic error of pressure is reflected in a shift, whereas that of the enthalpy of vaporization causes a monotone increase in the deviations (trend). The trend and shift of the residuals is more characteristic of the existence of systematic errors, than the size of deviations. The latter determines the q2 error measure of the first method. The q2 considers equally the scattered large deviations and those having a trend or shift. Statistical test procedures proposed earlier for the residual method (Koll~r-Hunek et al., 1986) are successfully applied here as well. To check the trend in residuals the Abbe test (Linnik, 1962) or test of mean-square successive differences (Sachs, 1982) is used, while for the shift the well-known Student's test is uti|zed.
D. Mikl f s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
a.
mndom
-20 ~ 4 0
-6O
4 0 270 280 290 300 310 320 330
b
P +lOOPa P +300 Pa
20. 20.
t .2o r'2° ~ p ~ 0 - ~ O - - ' w - f -40 -40
- 6 0 ~ -60
-80 -80 270 200 310 330 270 290 310 330
101
C.
H +SO kJIkg H +200 kJIkg
2Oo ~ T • • 20 ~ ' r e "=t - "-" '":
::t 270 290 310 330 270 290 310 330
Fig. 1 Results of the simulation study, Clausius-Clapcyron equation, 50 data points a. random errors only b. random errors + AP systematic error c. random errors + AH systematic error The abscissa is T (K) everywhere,
Let the expected value of ~ be constant but unknown, denoted by a. hypothesis and the test statistic for the trend are as follow:
The null
102 D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
Ho: E(~:) = a
1 n-1 _ )2 - E ( 4 , + , +:,
R_n li=~
n -li=~
(21)
where ~ = ,=t .., and ~ is applied to the f deviations. n
If there is a trend m deviations, the subsequent values are closer than they would be in a random case, and thus the numerator is smaller. The null hypothesis is accepted if the actual value of the R test statistic exceeds the critical value, which is taken from a table for the n value and for the given significance level. If the data are consistent in the thermodynamic sense, deviations must not show either shift or any trend, i.e., a=O above. The null hypothesis and the test statistic for the shift a r e ;
t+o: E(+)=0
t = ( 2 2 )
Here the null hypothesis is accepted ift is smaller than the critical value. Both tests assume Gaussian distribution and constant variance of ~. Experience shows that the procedures are robust, i.e., they are not very sensitive to these assumptions. It is also true that the _f deviations are correlated, but it was found of minor importance.
Clausius-Clapeyron equation Here the tests were used forJi values, i=l,2,..,n obtained from Eq. (19). The simulation calculations performed for the previous Z 2 test on the q2 error measure have been utilized, as they simultaneously produce the_f vector of deviations. In the course of this test, the extent of trend and/or shill is compared with the standard
deviation of the residuals. If (d P / d T), is computed from the orthogonal polynomial
fitted to P against T data, values of the derivative, which is one of the x variables, do
D. MikMs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 103
not scatter at all. The uncertainty is expressed as the variance of the polynomial coefficients, meaning that by repeating experiments different polynomials would be obtained. In order to reflect scattering of data in the derivatives as well, the derivatives were computed from interpolating quadratic parabolas fitted to three adjacent error burdened points. This route was not followed if the scattering was very large, as e.g., for rtms 1 and 2 in Table 1. The number of evaluatedfdeviations was 9 in the first and last range (as the derivative could be evaluated only for interior points), and 10 in the remaining three intervals. Critical values for the Abbe's and Student's test are excerpted as Table 2 for different significance levels (Sachs, 1982). The test results are shown in Table 3 for the same runs as in Table 1.
Table 2 Critical values for the Abbe's and Student's test
1o4 D. Mikl6s et al.. / Fluid Phase Equilibria 110 (1995) 89 - 113
Run 7 demonstrates that a proportional additive error in P cannot be detected (R values in the Abbe's test are large, t values in the Student's test are small), the reason is the structural invariance of Eq. (12) to this kind of error. The constant absolute error of P is detected if it is large enough (300Pa, in run 10): at lower pressures t is significant (larger than 2.306 or 2.262, respectively), at higher pressures R becomes significant (it falls below 1.0244 or 1.0623). From runs 11 and 12, where the enthalpy of va- porization is subject to additive systematic error, it is seen that the Abbe's test becomes significant at higher pressures (in the fourth range for AH=5%, in the third range for AH=50 kJ/kg), but the random errors suppress the effect of systematic errors
at lower pressures. If o'2p is small (not shown here), the effects are more pronounced. At higher level of systematic error in the enthalpy of vaporization (200 kJ/kg) the Abbe's test detects the trend already in the first range. It is well seen from Fig. 1 that the temperature span of the ranges is narrow, thus a trend is hardly detectable if it is compared to the scattering o f f values. Calculations with another basic data set, closer to the outcome of a typical laboratory experiment, are shown in Table 4. Here 15 data points were simulated, so that the
number of derivatives (d P / d T)~ and the number of elements of the vector f was 13.
It is seen from Table 4 that either the Abbe's or Student's test or both detect the systematic errors added, and the visual observation of results plotted in Fig. 2 leads to similar conclusions. The reason of better performance of tests in this case is the larger span of temperature intervals.
Table 4 Results of the simulation study: Clausius-Clapeyron equation, Method II, 15 data points
oer =0.0001K2; o2e=100Pa2; o2n=100kJ z/kg 2
systematic error no AP=100 Pa AP=300 Pa AH=5 % AH=50 kJ/k 8 AH=200 kJ/kg
1.265 0.991 1.100 0.444 0.801 0.158
-0.240 -2.136 -5.788 -1.317 -1.050 -1.238
Simultaneous treatment of PVT and caloric data
Data for ethylene taken from tables by Sytchev et al. (1981) were used as "error-free" data at intersects of 14 isotherms (from 285 K to 400K) and 14 isobars (from 0.1 MPa to 8 MPa), that is altogether 196 data points. Both random errors of variance
o3~, o{~, and O'2v and selected systematic errors were added to the H, cp and V
values. Temperature and pressure were kept as error-free.
D. Miklfs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 ,05
b.
mndom
10.
-10 f-20
-30 40 -S0
270 290 310 I
330
P +100Pa P +300 Pa
1: T 1° I0 N • • -10~ ~ • N ~ e e e e e • l o
t -2o t -2o e e e e e e e e -30 ~ -30 -40 -40
-so I I I -so I I 270 290 310 330 270 290 310
C.
H +SDkJIkg H +200 kJIkg
l0 T I : T
-10 ~ • ~ I P % 0 0 0 • -10 ee •eeee f r -2o __•ae e -20
-30 -30 -40 .40 • • -so I I I -5o I I
270 290 310 330 270 290 310
I 330
I 330
Fig. 2 Results of the simulation study, Clausius-Clapeyron equation, 15 data points a. ramdom errors only b. random errors + AP systematic error c. random errors + All systematic error The abscissa is T (K) everywhere.
The two differential equat ions give two deviat ion vectors:
( c7¢ ) p~Cp F,. = - - + + - - (23)
(24)
106 D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
a.
random random
1.00E-OS T
12o:1 , ii w, I I ~ ~ ,
2 8 0 3 0 0 - - 3 2 0 3 4 0 3 6 0 3 1 t O 400
5.00E-05 T
3.00 E-05 -~
f2 1.00E-05 t
28O
A A •
• ' l I I I
300 320 340 360 380
4
V +0.05 m31 kg V +0.05 m31 kg '°'°'T/. 1.50E-04. 3.00E-05 -~-
I I I I I I 2 8 0 3 0 0 3 2 0 3 4 0 3 6 0 3 8 0 4 0 0
5.00E-05
3.00E-05
f2 1.00E-05
-1.00E-05 -
-3.00E-OS 280
cp +20%
• O 0 0 0 O O 00
I I I I I 300 320 340 360 380
d
5.00E-05 -
4.00F-00.
3.00E-06.
2.00E-05,
1.00E-05,
e.00E*O0
muL +4 E-6 led Pa
IIDoooo••OoO
I I I I I I 280 300 320 340 360 380 400
5.00E-05
3.00 E-0S t
f2 1.00E-05
-1.00 E-O5 l
-3.00E-05 , 280
mu +4 E-6 K/Pa
A , , , •
0 ' I I I I 300 320 340 360 380
D. Miklbs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 107
Fig. 3 Results of the simulation study, two differential equations, 2.5 MPa isobars (scales are not homogeneous!) a. random errors only b. random errors + AV systematic error c. random errors + A% systematic error d. random errors + AI.t systematic error The abscissa is T (K) everywhere.
To obtain f2i the second derivative of V with respect to T is required, and it was calculated numerically from the first derivatives, in the same way as it is described above. Thus from 14 data points along an isobar 12 f~ values could be calculated, while only 10 f 2 values were computed for the same isobar. Part of the results for the fl deviations are shown in Table 5, while those for the f2 deviations are seen in Table 6, both are plotted in Fig. 3. The basic data set seems to be slightly inconsistent (R values are small at higher pressures), but this is negligible if compared with those containing added systematic error. It is well seen that R values forfl residuals reflect the systematic errors introduced, the t values are generally less sensitive. This shows that the systematic errors investigated cause mainly trend inf,. The effect for f2 residuals are less stressed.
Table 5 Results of the simulation study: Two differential equations, Method II a. R values for the fl deviations along isobars
Eq. (5) is totally analogous to the Clausius-Clapeyron equation, thus it may be treated similarly. The steps of the calculation were as follow: 1. For each measurement point the ?1 and 3'2 activity coefficients were calculated from x, T, y, P data.
2. The derivatives of G ~ / RT with respect to Xl were computed numerically, using three-point interpolation. It is useful to follow the suggestion by Van Ness and Mrazek
(1959) and to obtain directly the derivatives of G E / RTx]x 2 and then calculate that of
G ~ / R T . 3. /_/z and ~ data are required for isobaric or isothermal data, respectively. In isothermal cases the term d H / d T may be neglected, in isobaric cases the derivatives should be computed as well. 4. The f vector, defined as deviation from Eq. (5), was calculated finally and plotted against x 1, then Abbe's R and Student's t test statistics were evaluated. The data set used is a simulated set for pyridine-tetrachloroethene mixtures. The number of data points is 19. The number of elements in the _f vector is 17. The results are compared in Table 7 and Fig. 4 with those obtained by the residual method (Kollfir-Hunek et al., 1986), where the number of residuals was 19. Both methods are able to detect systematic errors, except when additive errors were present in both the pressure and the temperature. The application outlined here is a
D. Miklbs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 109
refinement of the method proposed by Kojima et al. (1990), since not only the average absolute deviations are examined, but their possible trend and shift is also considered. This gives an increased sensitivity to systematic errors.
random random
0.8
0.4 0.2
l
.0. "0"4 t 41.6
0
0.04.
0.02 I
• _ l •-,* ~ • ~ dy 0 - w
• • -O',"WO o-eecm i - , - - e , , , u - -0.02 f -
/ I 1 .0.O4 I I I
0.5 1 0 0.5 1
b.
P +SmmHg P +5mmHg
• 0o, T
0.4_ ~ 0.02 ~ • • • • 0
f 0"20 dy 0 ~::~ m - 0 0 m O o O 0 -0.02
-0.e I I I -0,04 1 I I 0 O.S 1 0 O.S 1
C.
T +IK T+IK
0.8 - 0.04
o.4 0.o2 T Oo O.
f dy 0 t 0.~ ~..4 %.s O0~ Ool .0.2~ • • W • g I w - -- I -0.4 -0.02 -0.6
0 O.S 1 -0.04
Fig. 4 Results of the simulation study, binary vapor-liquid equilibrium. Comparison of results obtained by the proposed method (f) and the residual method (dy) a+ random errors only b. random errors + AP systematic error e. random errors + AT systematic error The abscissa is x (liquid mole fraction) everywhere.
110 D. Mikl6s et al., / Fluid Phase Equilibria I10 (1995) 8 9 - 113
Table 7 Results for the vapor-liquid equilibrium example Comparison of Method II with the residual method
Method II residual method systematic R t R t e r r o r
AP=-5 mmH 8 AP---5 mmH 8 AT=I K 0.294 0.028 0.709 -2.635 AT=-I K 0.138 0.229 0.484 0.647 5 mmHg + 1 K 0.842 0.029 2.158 -1.777
CONCLUSION
Two model-free methods are proposed for checking mutual thermodynamic consistency of experimental data sets. Both of them are differential procedures and do not require involvement of any model. The first method is a ~2 test based on a quadratic form of the f residuals from the thermodynamic differential equations. Its use requires explicit knowledge of error variances of the elementary measurements. If these are available, the method is able to detect systematic errors. The second method is a trend/shill analysis of residuals, which does not require the knowledge of error variances. As it does not involve computation of complicate covariance matrices of variables, the method is easy to implement.
LIST OF SYMBOLS
B12 cross second virial coefficient Cp isobaric heat capacity E expected value f residual or deviation from the differential equation, defined by Eq. (17) G E excess molar Gibbs energy
g = G ~" / R T h H H E
P
p0 q2
R r~
enthalpy simplified notation for enthaipy of vaporization excess molar enthalpy pressure (pure component vapor pressure in the Clausius-Clapeyron equation) pure component vapor pressure error measure defined by Eq. (18) gas constant; Abbe's test statistic, defined by Eq. (21) its critical value at (z significance level
D. Mikl6s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 111
T temperature t Student's test statistic, defined by Eq. (22) t~2 its critical value at (two-sided) ~ significance level V covariance matrix V molar or specific volume V E molar volume change on mixing W matrix of coefficients, defined by Eq. (15) x liquid phase mole fraction y vapor phase mole fraction Ay deviation of the vapor phase mole fraction in the residual method.
Greek letters
y activity coefficient p density
Joule-Thomson coefficient 6 2 v a r i a n c e
A systematic error
REFERENCES
Almfisy, G. and Sztan6, T., 1975. Checking and correction of measurements on the basis of linear system model. Problems of Control and Information Theory, 4: 57-69.
Anderson, T.F., Abrams, D.S., Grens, E.A., 1978. Evaluation of parameters for nonlinear thermodynamic models. AIChE Journal, 24: 20-29.
Angus, S., Armstrong, B., de Reuck, K.M., Featherstone, W., Gibson, M.R., 1974. International Thermodynamic Tables of the Fluid State, Ethylene. 1972, Butterworths, London
Angus, S., 1983. Guide for the preparation of thermodynamic tables and correlations of the fluid state, CODATA Bulletin No.51., Pergamon Press
Chase, J.D., 1984. The qualification of pure component physical property data. Chem. Eng. Progr. April, p. 63-67.
Christiansen, L.J. and Fredenslund, Aa., 1975. Thermodynamic consistency using orthogonal collocation for computation of equilibrium vapor composition at high pressures. AIChE J. 21: 49-59.
Cox, K.W., Bono, J.L., Kwok, Y.C., Starling, K.E, 1971. Multiproperty analysis. Modified BWR equation for methane from PVT and enthalpy dataInd. Eng. Chem. Fundam. 10: 245-250.
Dohnai, V. and Fenclova, D., 1985a. Verification of the accuracy of complete binary vapour-liquid equilibrium data. Fluid Phase Equilibria. 19: 1-12.
Dohnal, V. and Fenelova, D., 1985b. A new procedure for consistency testing of binary vapour-liquid equilibrium data. Fluid Phase Equilibria. 21:211-235.
112 D. Miklfs et al., / Fluid Phase Equilibria 110 (1995) 89 - 113
Eisner, N., Fischer, S., Klinger, J., 1986. Thermophysikalische Stoffeigenschaften yon Wasser, VEB Deutscher Verlag for Grtmdstoffindustrie, Leipzig
Forsythe, G. E., 1957. Generation and use of orthogonal polynomials for data- fitting with a digital computer. J. Soc. Ind. Appl. Math. 5: 74-88.
Gmehling, J., Onken, U., Arlt, W., 1977. Vapor-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt
Herington, EF.G., 1947. A thermodynamic test for internal consistency of experimental data on volatility ratio. Nature, 160:610-611.
Herington, E.F.G., 1951. Tests for the conistency of experimental isobaric vapour- liquid equilibrium data. J. Inst. Petrol., 37: 457-470.
Janaszewski, B., Oracz, P., Goral, M., Warycha, S., 1982. Vapour-liquid equilibria. I. An apparatus for isothermal total vapour pressure measurements: Binary mixtures of ethanol and t-butanol with n-hexane, n-heptane and n-octane at 313.15 K. Fluid Phase Equilibria 9: 295-310.
Kem6ny S., Skjold-Jorgensen, S., Manczinger J., T6th K., 1982. Reduction of thermodynamic data by means of the multiresponse maximum likelihood principle. AIChE J. 28: 20-30.
Kojima, K., Moon, H.M., Ochi, K., 1990. Thermodynamic consistency test of VLE data. Methanol - water, benzene - cyclohexane, ethyl-methyl ketone - water. Fluid Phase Equilibria 58: 269.
Koll/tr-Hunek K., Kem6ny S., H6berger K., Angyal P., Thury ]~., 1986. Thermodynamic consistency test for binary VLE data. Fluid Phase Equilibria 27: 405-425.
Lirmik, Yu.V., 1962. Method of least squares and principles of mathematical statistical data treatment (in Russian), Moscow
McDermott, C., 1964. PhD Thesis, Chem. Eng. Dept., University of Birmingham McDermott, C. and Ellis, S.R.M., 1965. A multicomponent consistency test.
equilibrium data from solution vapor-pressure measurements. Ind. Eng. Chem. Fundam. 4: 455-459.
Olson, J.D., 1983. Thermodynamic consistency testing of PTx-data via the Gibbs- Helmholtz equation. Fluid Phase Equilibria 14: 383-392.
Palmer, D.A. and Smith, B.D., 1972. Thermodynamic excess property measurements for acetonitrile - benzene - n-heptane system at 45 °C. J. Chem. Eng. Data 17: 71-76.
Redlich, O. and Kister, A.T., 1948. Algebraic representation of the thermodynamic properties and the classification of solutions. Ind. Eng. Chem. 40: 345-348.
de Reuck, K.M. and Armstrong, B., 1979. A method of correlation using a search procedure, based on a step-wise least-squares technique, and its application to an equation of state for propylene. Cryogenics, September, p. 505-512.
Sachs, L., 1982. Applied statistics. A handbook of techniques. Springer-Verlag, New York
D. Mik l f s et al., / Fluid Phase Equilibria 110 (1995) 89 - 113 113
Samuels, M.R., 1972. Interpreting thermodynamic consistency: How bad is "bad"? Ind. Eng. Chem. Fundam. 11:422-424
Samuels, M.R., Ulrichson, D.L., Stevenson, F.D., 1972. Interpretation of overall area tests for thermodynamic consistency: The effect of random error. AIChE Journal 18: 1004-1009.
Starling, K.E., Savidge, J.L., Kumar, K.H., 1986. PVT data base evaluation methodology for highly accurate equation of state development. Fluid Phase Equilibria 27: 203-219.
Ulrichson, D.L. and Stevenson, F.D., 1972. Effect of experimental errors on thermodynamic consistency and on representation of vapor-liquid equilibrium data. Ind. Eng. Chem. Fundam. 11: 287-293.
Van Ness, H.C. and Abbott, M.M., 1982. Classical Thermodynamics of nonelectrolyte solutions, McGraw Hill
Van Ness, H.C. and Mrazek, R.V., 1959. Treatment of thermodynamic data for homogeneous binary systems. AIChE J. 5: 209-212.
Van Ness, H.C., Byer, S.M., Gibbs, R.E., 1973. Vapor-liquid equilibrium. I. Appraisal of data reduction methods. AIChE Journal 19: 238-244.
Wagner, W., 1974. Ein neue Korrelationsmethode ffir thermodynamisehe Daten angewendet auf die Dampfdruckkurve von Argon, Stickstoff und Wasser. Habilitationschrifi, Fortsch.-Ber. VDI-Z, Reihe 3, Nr. 39
Wagner, W., 1977. A new correlation method for thermodynamic data applied to the vapour-pressure curve of argon, nitrogen and water. PC/T 15, IUPAC Thermodynamic Tables Project Centre, London
Acknowledgement This work has been supported by the Hungarian National Science Foundation (OTKA)
under grant numbers 178/91, 718/91 and T-007312/93.
