Design of experiments for thermodynamic model discrimination applied to phase equilibria at high pressures

High Pressure Chemical Engineering Ph. Rudolf von Rohr and Ch. Trepp (Editors) 9 1996 Elsevier Science B.V. All rights reserved.

379

Design of Experiments for Thermodynamic Model Discrimination Applied to Phase Equilibria at High Pressures C. Dariva a, E. Cassel b and J.V. Oliveiraa aDepartment of Chemical Engineering, Federal University of Santa Catarina, 88010-970, Brazil bChemical Engineering Program, Federal University of Rio de Janeiro, 68592-970, Brazil ABSTRACT Traditionally, the screening of thermodynamic models in phase equilibria is carried out based on the full data set and, therefore, after having obtained all the experimental points. The aim of this work is to show that sequential model discrimination may be used as a guide for obtaining experimental data for systems at high pressures. We have compared two discrimination procedures in order to select the next experimental point, one of them considering only the predicted responses (Hunter and Reiner [ 1]) and the other taking also into account the error limit inherent to each model (Ferraris and Forzatti [2]). In order to test these methodologies, literature data were used along with current thermodynamic models. It is shown that the sequential discrimination leads to the same conclusions as obtained when using the full data set, but with much less effort. 1. INTRODUCTION The design of experiments associated with model discrimination is a technique aimed to optimize experimental data collection and thus causing a cost reduction. The necessity of model discrimination arises as we observe the large number of existing equations of state (EOS) and also due to the possibility of optimizing data collection process. It has been shown by Cassel and Oliveira [3] that this rationalization is effective in vapor-liquid equilibrium at high pressures, making the sequential model discrimination an important tool for the screening of EOS. However, in such a investigation, these authors have not considered the error limit to each model and, hence, the discrimination procedure has been accomplished based only on the predicted responses. In this work we use the least square method to estimate the parameters and the Hunter-Reiner[1] and Ferraris-Forzatti[2] discrimination strategies along with the Fisher F-Distribution Function to proceed the model discrimination. The Peng Robinson[4] EOS (PR-EOS) with five different mixing rules was employed to evaluate the potential of the discrimination methodologies presented here.

2. THEORY The vapor-liquid equilibrium can be described by:

380

~Lxi: ~Vyi

(1)

where ~v and ~[ are the fugacity coefficients for the component "i" in the vapor and liquid phases, respectively; which are calculated in this work by the generalized PR-EOS. The calculation of a and b parameters of this equation was performed using five mixing rules in order to verify which of them best describes the phase equilibria at high pressures. We used the following mixing rules[5]: van der Waals 1, Schwartzentruber-Renon, van der Waals 2, Panagiotopoulos-Reid and Strijeck-Vera, as shown in Table 1. The sequential discrimination procedure is divided in two steps: the first one is the parameter estimation and the second step is accomplished by determining the place in the experimental space where the next experiment should be run. These steps are presented bellow. Table 1 PR-EOS mixin$ rules Mixing Rule Model 1 van der Waals 1 Model 2 Schwartzentruber-Renon

Functional Form

Parameters

a = Z Y'. zizj(aiaj)V2( 1- kij)

kji

i j

b = ~ z ib i i

[

a = ~'.Y'.zizj(aiaj) 1/2 1-kij - l mijz j i i ~+ mjiz i j b = )-'.zib i i

Model 3 van der Waals 2 Model

4

Panagiotopoulos- Reid Model 5 Strijeck-Vera

a=

"1

kij

J

lji

n~jzi - mj~zj(zi + zj)|

Z Y'.zizj(aiaj)l/2(1 - kij) i j

(b i + bj) () . l - k j i . b = Z9 Z.z i z j 2 J a = ~] ~] zizj(aiaj)l/2(1 - kij +(kij - kji)zi)

i j

b = ~ zib i i a = )-] if'. z i z j ( a i a j ) l / 2 ( 1 - zikij - z j k j i ) i j b = E zibi i

n~j kij kj~ kij

kji kU

k,,

2.1 Parameter Estimation

The parameters of the mixing rules are estimated using the least square method. In the calculations "Y" (P) is the output variable and the errors of the experimental measurements are considered to be normally distributed. Next, the fit is performed by minimizing the following objective function with respect to the vector of parameters: Nexp (y~Xp _ y~al)2 S= Z 2 (2) i=l

((~y)i

The variance of the estimation procedure is defined as: S2 :

1 N exp exp [al ) 2 (N exp- Npar) i=lE (Yi - Y

(3)

381 where Nexp is the number of experimental points, Npar the number of parameters and exp and cal refer to experimental and calculated values. The absolute average deviation is defined as: 100 N~p pexp p~l A A D - l~-xp

~i:l l i Pi -

7x~;

(4)

'

2.2 Sequential Model Discrimination There are many alternatives for modeling high pressure phase equilibria. Therefore, it is required a discrimination procedure capable to screen these models satisfactorily. In this work we have compared two discrimination methods in order to select the next experimental point. The first one is the Hunter-Reiner methodology that considers only the predicted response of each model according to the following equations:

(x) -[yi

y j(x)]

(5)

and, N mod N rood

D(x)-

Z i

~dij(x)

(6)

j=i+l

where d~j(x) is the squared difference between the responses of models "i" and "j" and D(x) is the difference among the responses of all models. The next experimental point should maximize D(x) to one value greater than the experimental error, otherwise the divergence between the models can be explained in terms of experimental error variance. The Ferraris-Forzatti methodology takes also into account the error limit inherent to each model, as presented by Equations (7) to (13) of reference [2]. According to Ferraris- Forzatti, an adequate indicator of the divergence among the model responses relative to the error limit is: N mod N rood

T(x)-

Z Z (Yi-Yj)2 i=1 j=i+l (7)

Nmod

(N mod- 1)(N mod 0 2 -k- Z

(~ i 2 (X))

i=l

being the estimated variances of the responses given by: t~i 2 (X) -- gi T ( x ) ( G i T G i ) - l g i ( x ) o

2

(8)

where Nmod is the number of models, T(x)is the difference among all models, 0 2 is the experimental variance, t~i2(X) the estimated variance of the responses, gi is a vector containing the model derivatives with respect to the parameters, giT its transposition and Gi is a matrix (Nexp rows and Npar columns) containing the gi vector calculated at experiment i. The next experimental point should maximize T(x) to one value greater than 1, otherwise the divergence among the models can be explained in terms of experimental error variance plus

382 the variance of the expected responses. In both cases, alter we have the input variables that maximize D(x) and T(x), the experiment is run and the parameters are estimated. These two steps - estimation and discrimination - are repeated sequentially until the mixing rule that best fits the experimental data is chosen or until D(x)
To verify, among the employed mixing rules, the one or ones that best represent the equilibrium data, the CO2(1)-Diphenyl(2) at 343.2 K and CO2(1)-n-Propylbenzene(2) at 353.2 K systems were used. The data and their conditions are presented in the works of Zang et al. [6] and Maurer et al. [7], respectively. The pure properties of the components are given in Reid et al.[8]. The standard deviation of the measured output variable is: Cv=0.05MPa and ~p=0.07MPa, respectively. The sequential planning for the systems CO2-Diphenyl and CO2-n-Propylbenzene is shown in Tables 2 and 3, respectively. In Figure 1 is depicted the Hunter-Reiner and Ferraris-Forzatti deviations for the CO2-n-Propylbenzene system. It can be noted that the selected points for both methodologies are (essentially) the same. This can be explained due to the very small estimated variances of the responses and, hence the Ferraris-Forzatti methodology reduces to the Hunter-Reiner, as can be verified by Equations (5) to (7). This occurred because of, for the systems studied here, the parameters were estimated with small uncertainties. So, the additional effort of the Ferraris-Forzatti methodology is not justified for these systems (the same result was obtained for the CO2-Diphenyl). However, the same can not be true when considering multiple responses. Table 2 Experimental plannin8 for the CO2-Diphenyl system T(MPa 2) CO2 mole fraction D(MPa 2) Next Point 38.75 5 0.795 345.25 24.56 6 0.656 214.73 4.59E-5 7 0.619 2.65E-4 Table 3 Experimental planning for the CO2-n-Propylbenzene system

Models Discarded

1 and 5

383

Next Point 5 6 7 8 9

CO2 mole fraction 0.365 0.360 0.404 0.362 0.720

D(MPa 2) 197.65 204.67 16.85 14.79 2.12E-5

T(MPa 2) 27.32 37.52 3.68 3.81 1.19E-6

Models Discarded 1 1 1 and 5

Figure 1. Deviations of the discrimination strategies as a function of the C02 mole fraction. To check the reliability of the results it was also performed the parameter estimation using the full data set. It can be noticed from Table 4 that there is no loss of fitting quality for the system CO2-Diphenyl and the parameters are approximately the same when we compare the conventional and the discrimination procedures.

384 Furthermore, an aspect much more relevant to be noticed is that the use of the sequential discrimination led to the same conclusions as obtained when using the full data set, but with much less effort. This statement can be verified by comparing the confidence for each model at the end of the discrimination procedure with the ones resulting from full data set (the same result was obtained for the CO2-n-Propylbenzene). Table 4 Estimated parameters and confidence level at the end of the discrimination procedure and using the full data set for the CO2-Diphenyl system. Number of van der Schartzentrub van der Panagiotopou Strijeck-Vera points Waals 1 er-enon Waals 2 los- Reid k12 7.993E-2 k12 8.265E-2 k12 9.902E-2 k12 9.902E-2 k12 1.056E-1 112 5.234E-2 k21 8.348E-2 k21 8.348E-2 k21-7.89E-2 m12 -7.89E-3 AAD 9.617 AAD .558 AAD .559 AAD .559 AAD 12.486 s2 1045.61 s2 1.298 s21.365 s2 1.365 s2 987.45 Conf. % 99.36 87.98 88.28 88.28 99.25 Full Data k12 8.501E-2 k12 8.768E-2 k12 9.915E-2 k12 9.915E-2 k12 9.871E-2 Set 112 4.985E-3 k21 8.345E-2 k21 8.345E-2 k21-6.987E-2 m12 -7.23E-3 22 AAD 18.231 AAD 1.424 AAD 1.465 AAD 1.465 AAD 18.445 s2 1556.7 s2 2.640 s2 2.680 s2 2.680 s2 1235.465 Conf. % 99.998 90.123 90.265 90.265 99.997 4. CONCLUSIONS We have presented a comparison between two model discrimination procedures using literature data for phase equilibria at high pressures. Though the application of these methodologies furnished essentially the same results for the systems investigated here, they seem to be useful for experimental design in the field of phase equilibria. A more rigorous treatment should consider the case of multiple responses and also the experimental error in the independent variables; these topics are now under investigation. REFERENCES

1 2. 3 4. 5 6. 7. 8.

W.G. Hunter and A.M. Reiner, Technometrics, 7 (1967) 307. G.B. Ferraris and P. Forzatti, Chem. Eng. Sci., 38 (1983) 225. E. Cassel and J.V. Oliveira, The Journal of Supercritical Fluids, In press. D.Y. Peng and D.B. Robinson, Ind. Eng. Chem. Fundam., 15 (1976) 59. A. Anderko, Fluid Phase Equilibria, 61 (1990) 145. D.L. Zhang; W. Wang; W. Sheng and B.C.Y. Lu, Can. J. Chem. Eng., 69 (1991) 1352. A. Bamberger and G. Maurer, The Journal of Supercritical Fluids, 7 (1994) 115. R.C. Reid; J. M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, Mc GrawHill Book Company, 4th edition (1987).