1122
Ind. Eng. Chem. Res. 1998, 37, 1122-1129
Effect of Data Type on Thermodynamic Model Parameter Estimation: A Monte Carlo Approach Victor R. Vasquez and Wallace B. Whiting* Chemical & Metallurgical Engineering Department, University of NevadasReno, Reno, Nevada 89557-0136
Parameters for the UNIQUAC (universal quasi-chemical) model were regressed from VLE and LLE data for the systems: diisopropyl ether + acetic acid + water; 1,1,2-trichloroethane + acetone + water; and chloroform + acetone + water. The results show a very significant effect of the data type used in the regressions (binary versus ternary) on the uncertainty of the predicted performance of sample liquid-liquid extraction units. Introduction In thermodynamics, activity coefficient models are very common. Most of them are used for vapor-liquid and liquid-liquid equilibrium prediction, allowing designers and engineers to simulate and design industrial chemical processes. One of the major concerns in this kind of model is how to regress appropriate parameters. For example, for the prediction of liquid-liquid equilibrium in ternary systems (in particular systems of type I for which only one binary is partially miscible whereas the other ones are completely miscible), the equilibrium prediction is not a very easy task (Anderson and Prausnitz, 1978). Generally, the use of binary vaporliquid data for the completely miscible components and mutual solubility data for the partially miscible components is recommended for the regression of the binary interaction parameters. (Higashiuchi et al., 1987; Prausnitz et al., 1987; Sørensen et al., 1979). It is often possible to find experimental data sets for liquid-liquid tertiary systems (Sørensen, 1980; Wisniak, 1981), allowing the regression of binary interaction parameters for models such as UNIQUAC (universal quasi-chemical) and NRTL (nonrandom two-liquid) from ternary data for liquid-liquid applications. This approach is useful because the parameters regressed from binary data are highly correlated for the UNIQUAC and NRTL models (Anderson and Prausnitz, 1978). However, it is known that the binary interaction parameters regressed in this way should not be used for vaporliquid applications (Abrams and Prausnitz, 1975). On the other hand, binary parameters regressed from vapor-liquid data are often used for the prediction of liquid-liquid equilibrium. Sometimes, combinations of binary and tertiary data from both vapor-liquid and liquid-liquid equilibria give good results in the binary parameter regression (Anderson and Prausnitz, 1978; Sørensen et al., 1979). Another factor that must be taken into account is the mathematical procedure used for the parameter regressions. In this work, the maximum likelihood approach was used. This method is widely used for thermodynamic model regressions (Nova´k et al., 1987; Sørensen et al., 1979). In general, there are always variations in the final prediction of the equilibrium of chemical systems because of the regression technique used, the data available, and of course the approach and model selected.
So, for a specific chemical system, the designer or process engineer is faced with a menu of routes for the prediction of the thermodynamic properties of the system. The final decision always relies on the experience and knowledge of the designer, but there is still a question remaining: How much does the variation “normally accepted” in the prediction of the thermodynamic equilibrium affect the final results of process design and simulation? In this work, we studied the effect of using two approaches to regress the binary interaction parameters of the UNIQUAC model. The first one was using binary data including vapor-liquid and liquid-liquid equilibria, and the second one was using only liquid-liquid ternary data. The parameters obtained were used to simulate a liquid-liquid extractor and to study its performance for extraction operations using three systems: chloroform + acetone + water, diisopropyl ether + acetic acid + water, and 1,1,2-trichloethane + acetone + water. One hundred simulations were performed for each set of parameters for each system. The Monte Carlo approach was used to do the simulations, followed by the analysis of the cumulative frequency distribution curves for the output variable; in this case, the percentage extracted of the component of interest. The results show that the approach followed to regress the parameters is very important, because the uncertainty of the predicted unit performance varies greatly with the regression approach used. More details about the results are discussed in the following sections. The UNIQUAC Model The UNIQUAC equation (Abrams and Prausnitz, 1975) is widely used for predicting vapor-liquid and liquid-liquid equilibrium. Some authors (Sørensen et al., 1979) consider this equation to be better than the NRTL model for liquid-liquid equilibrium; however, some limitations are reported by Nova´k et al. (1987), particularly for highly nonideal systems. According to Anderson and Prausnitz (1978), the effect of parameter uncertainty is not large for the UNIQUAC model for typical applications; however, this model has the disadvantage of not having unique parameters, and the parameters regressed from liquidliquid systems sometimes cannot be used for vaporliquid prediction.
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Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1123
The equation for the UNIQUAC model is:
ln γi )
Φi
θi z + qi ln - q′i ln t′i - q′i xi 2 Φi
∑j θ′jτij/t′j +
li + q′i -
Φi xi
∑j xjlj
(1)
where
θi ) qixi/qT;
qT )
∑k qkxk
θ′i ) q′ixi/q′T;
q′T )
∑k q′kxk
τij ) aij + bij/T + cij ln T + dijT Φi ) rixi/rT;
rT )
∑k rkxk
z li ) (ri - qi) + 1 - ri 2 t′i )
∑k θ′kτki
z ) 10 aij * aji bij * bji cij ) cji dij ) dji For this work, the binary parameters are bij and bji, with aij, aji, cij, and dij set equal to zero. The maximum likelihood regression method was used. The objective function was the sum of squares of the differences between the measured data and the model prediction (for all variables), weighted by the following estimated standard deviations: 0.01 °C, 0.1% for the compositions of the first two components, and zero for the third component. The regressions were done using the ASPEN PLUS (trademark of Aspen Technology Inc., Cambridge, MS) simulator. An estimate of the standard error for each parameter set was evaluated. The Standard Error of Estimate (SEE) was calculated as the square root of the objective function divided by the number of tie lines. In this work, there are two kind of parameters: (a) those regressed from liquid-liquid tertiary systems, where all parameters are correlated; and (b) those regressed from vapor-liquid and liquid-liquid data but using binary systems, where only two parameters are correlated per binary system. Uncertainty of Model Parameters As already mentioned, the UNIQUAC model is often considered to have a small uncertainty in its parameters. Despite that, the main purpose of this work is to study how the uncertainty present in those parameters affects the predicted performance in a specific unit operation.
For this study, the uncertainty for the input variables (bij and bji binary interaction parameters) was taken directly from the regression technique. Each BIP (Binary Interaction Parameter) has its variance, and the variance-covariance matrix correlates all of them. Another uncertainty source is the experimental data set itself, and there are several data sets for a given system available in the literature. Basically, two references were used for that purpose: Sørensen and Arlt (1980) and Stephenson (1992). The vapor-liquid binary data selected were those that passed the thermodynamic consistency tests based on the area and point methods performed by the ASPEN PLUS Simulator. For the liquid-liquid systems, the data used were those with the least sum of squares in the regression of the UNIQUAC equation, for a given system. The Monte Carlo Approach In uncertainty studies, one of the major concerns is how to do the sampling of the input parameters to get reliable estimators of the moments and other properties of the output variables. Two major factors are involved in this process: (a) the number of samples and (b) the sampling technique on the input variable distributions. The number of samples basically depends on both the cost associated with running the simulations and what the results will be used for. In this case, the cost of simulations is not very important for the case of small process simulations. For the second item, the main purpose is to study the cumulative distribution of the output variable, so it is necessary to compute the number of samples based on a specific confidence level desired. Details for computing the number of samples in this way can be found in Morgan and Henrion (1990) and Reed and Whiting (1993). Approximately 100 simulations gives a 90% confidence region of 0.85-0.95 on the 0.90 fractile; this is the sample size used in this work. The sampling technique used is stratified sampling, where the sample space for the input variables is divided into intervals of equal probability and one sample is taken at random from within each interval. This method is a variation of the traditional crude Monte Carlo simulation (Morgan and Henrion, 1990), which does the sampling at random within the whole distribution of the variables. An advantage of stratified sampling is that it produces a faster and better approximation of the output probability distribution compared with the traditional Monte Carlo. The most popular method of stratified sampling is the Latin Hypercube Sampling Technique (Iman and Shortencarier, 1984). This approach for sampling has been widely used in recent years (Morgan and Henrion, 1990; Reed and Whiting, 1993; Whiting et al., 1993; Vasquez and Whiting, 1997); however, it has some disadvantages, specially for cases where all the parameters are correlated. Usually, several unfeasible situations are found in a set of 100 simulations for a specific process or unit operation. This means that some of the samples do not have physical or practical sense at all and other ones are probably useless from an engineering point of view. The UNIQUAC model has highly correlated parameters, and this correlation must be considered for sampling purposes. Iman and Connover (1982) describe a method to generate samples from n variables with a specified rank-order correlation. Generally, using this
1124 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998
Figure 1. LLE for chloroform + acetone + water ternary system at 25 °C. The predicted value is by the UNIQUAC model with parameters regressed from ternary data. Experimental data are from Bancroft and Hubard (1942).
method, the target correlation matrix for the input variables is close enough to the correlation matrix of both the samples and their ranks. The Iman and Conover method was used in this work. Chloroform (1) + Acetone (2) + Water (3) This is a well-known system and several references report experimental data for both liquid-liquid and vapor-liquid equilibria (Sørensen and Arlt, 1980; Gmehling et al., 1993; Ohe, 1989). For the ternary system, Bancroft and Huband (1942) originally reported the data, but also these data can be found in Sørensen and Arlt (1980). The data for the binary system acetone + chloroform are reported by Gmehling et al. (1993), and the two experimental vapor-liquid data sets used in this work were originally reported by Kogan and Deizenrot (1975) and Apelblat et al. (1980). Several sources were chosen with the purpose of studying the effect of changing the experimental data source for one of the binary systems and looking at the effect of having different variances of the parameters regressed. The experimental vapor-liquid data for the system acetone + water were obtained from Eduljee (1958), also reported by Gmehling and Onken (1977), and the data for the liquid-liquid chloroform + water system are from Stephenson (1992). All the experimental data just mentioned above were used to regress the UNIQUAC binary interaction parameters that were used for predicting the equilibrium of the system in a process operation. In Figure 1, the ternary system and the equilibrium predicted by the UNIQUAC model are shown. The predicted values are from using the parameters regressed from the ternary data (all parameters correlated). Figure 2 shows the same experimental data as Figure 1, but the predicted values were computed using the UNIQUAC model with the binary interaction parameters regressed from both liquid-liquid and vapor-liquid binary data. The two curves shown in Figure 2 present the difference obtained from using different data sources for the binary system acetone + chloroform keeping the other two binary interaction
Figure 2. LLE for chloroform + acetone + water ternary system at 25 °C. The predicted value is by the UNIQUAC model with parameters regressed from binary data. See section on Chloroform (1) + Acetone (2) + Water (3) and Table 1 for details of the data sources used. Table 1. Binary Parameters bij and bji Regressed for the UNIQUAC Model for the System Chloroform (1) + Acetone (2) + Water (3) i 1 1 2
j 2 3 3
bij (K)
std. dev. (K)
bji (K)
Ternary LLE Set SEE ) 378.30 241.54 267.32 -94.22 -1064.71 941.83 -332.77 -129.72 118.15 15.78
std. dev. (K) 352.29 37.92 34.91
Binary Sets SEE (Set 1) ) 488.52 1 1 1 2
2 2 3 3
-15.99a -254.28b -813.32 -330.45
SEE (Set 2) ) 436.38 45.40 100.76a 540.81 272.70b 275.31 -342.50 22.76 35.82
38.70 249.51 118.502 11.45
a Binary interaction parameter used in Figure 2 for the “BINARY 02” curve; parameters regressed using data from Apelblat et al. (1980). b Binary interaction parameter used in Figure 2 for the “BINARY 01” curve; parameters regressed using data from Kogan and Deizenrot (1975).
parameter sets constant. Later, it will be shown how this difference affects the performance for a specific unit operation. The parameters regressed for each set of experimental data used are shown in Table 1. Also, the values of SEE are shown. The smallest SEE is for the ternary system, meaning that the ternary system is the best from the statistical point of view. The other two SEEs are very similar to each other. Their correlation coefficients are given in Appendix A. Illustrative Case. To illustrate the effect of using different sets of binary interaction parameters in a specific chemical process, we selected an example of a liquid-liquid extraction operation reported by Smith (1963). In this example, water is used to separate a chloroform + acetone mixture in a simple countercurrent extraction column with two equilibrium stages. The feed contains equal amounts of chloroform and acetone on a weight basis. The column operates at 25 °C and 1 atm. A solvent:feed mass ratio of 1.565:1 is used. The output variable for this case is the percentage of acetone extracted. The operation was simulated in the ChemCAD (trademark of Chemstations Inc., Houston, TX)
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1125
Figure 3. Uncertainty of percentage of acetone extracted in the liquid-liquid extractor, using different sets of parameters for the UNIQUAC model.
Figure 4. LLE for diisopropyl ether + acetic acid + water system at 23.5 °C. The predicted value is by the UNIQUAC model. See section Diisopropyl Ether (1) + Acetic Acid (2) + Water (3) for details of the data sources used.
process simulator. One hundred runs were made for each set of parameters, using the same conditions in the extractor. Nine simulations resulted in unfeasible situations for the parameters regressed from the ternary data and one unfeasible situation for the other sets. Figure 3 shows the cumulative frequency curve for the performance of the extraction operation simulated. We clearly can see the effect of the two parameter sets regressed for the binary system acetone + chloroform. The distribution frequency curves BINARY 01 and BINARY 02 are quite different, particularly their broadness. This behavior is only due to the difference between the variances of the parameters for the acetone + chloroform system. The cumulative frequency curve for the parameters regressed from ternary data (ternary in Figure 3) is broader than for BINARY 02; that difference is significant for the process performance, and it might increase as the complexity of the process increases. According to Abrams and Prausnitz (1975), fitting of binary parameters from ternary data using connodal line and tie line data is supposed to produce excellent results, or, in other words, this approach has to be better than predicting liquid-liquid equilibria of ternary systems using parameters regressed from binary data. However, analyzing the results presented in Figure 3, we can see that it is not easy to conclude which set of parameters is the best. The importance of having small variances in the parameters regressed is well addressed with this example, and these variances depend mainly on the quality of the experimental data and the thermodynamic model chosen. Also, it is important to separate the uncertainty in the model and uncertainty in the experimental data, to draw conclusions about the precision and exactness of the thermodynamic model; however, this is not an easy task.
Table 2. Binary Parameters bij and bji Regressed for the UNIQUAC Model for the System Diisopropyl Ether (1) + Acetic Acid (2) + Water (3)
Diisopropyl Ether (1) + Acetic Acid (2) + Water (3) Experimental data for this system were taken from Sørensen and Arlt (1980) for the ternary liquid-liquid system, originally reported by Othmer et al. (1941).
i
j
bij (K)
std. dev. (K)
bji (K)
1 1 2
2 3 3
Ternary Set SEE ) 78.95 171.14 14.98 9.15 -608.81 0.74 -126.44 158.30 0.76 107.40
1 1 2
2 3 3
-152.34 -694.67 170.33
std. dev. (K) 8.00 0.19 24.85
Binary Set SEE ) 700.31 95.06 50.48 98.06 -124.65 46.90 -101.86
73.86 31.90 68.53
Stephenson (1992) reports the data for the binary liquid-liquid system diisopropyl ether + water. Brusset et al. (1968) report experimental vapor-liquid equilibrium data for the system acetic acid + water, also found in Gmehling and Onken (1977), and Gmehling et al. (1979) reports the vapor-liquid equilibrium for diisopropyl ether + acetic acid system, originally reported by Molochnikov et al. (1970). The data cited were used for the regression of the binary interaction parameters for the UNIQUAC model for this specific system. The approach followed is the same as described for the previous system (chloroform + acetone + water), with the exception that only one set of parameters was regressed for each binary system. All the binary parameters were regressed with the purpose of predicting the liquid-liquid equilibrium for the system shown in Figure 4. In this case, the values predicted by UNIQUAC using parameters regressed from ternary and binary data are presented. We can see that the difference between the two binodal curves is very small. But, how important could this difference be in the performance of a specific process? This issue will be discussed later in this section using an illustrative case. The parameters regressed for each set of data are shown in Table 2 and the correlation matrixes are shown in Appendix B. It seems that the differences in the values for the parameters between the ternary set and the binary set are not very large, and similar behavior is observed for their standard deviations. However, the values of SEE show an important differ-
1126 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 Table 3. Binary Parameters bij and bij Regressed for the UNIQUAC Model for the System 1,1,2-Trichloroethane (1) + Acetone (2) + Water (3) i 1 1 2
j 2 3 3
bij (K)
std. dev. (K)
bji (K)
Ternary Set SEE ) 102.88 257.37 29.80 -495.60 -1170.19 247.36 -282.15 -86.75 22.67 -80.35
std. dev. (K) 82.33 6.97 27.35
Binary Sets SEE (Set 1) ) 132.87 1 1 2 2
2 3 3 3
-12.31 -674.84 -297.07a -330.45b
SEE (Set 2) ) 362.76 31.28 46.44 444.31 -312.14 147.73 54.78a 22.76 35.82b
32.29 205.95 46.81 11.45
a Binary interaction parameter used in Figure 6 for the “BINARY 01” curve; parameters regressed using data from Kojima et al. (1968). b Binary interaction parameter used in Figure 6 for the “BINARY 02” curve; parameters regressed using data from Eduljee (1958).
Figure 5. Uncertainty of percentage of acetic acid extracted in the liquid-liquid extractor, using different set of parameters for the UNIQUAC model.
ence. From the statistical point of view, the ternary set parameters are superior. Illustrative Case. In this case, the example selected is from Treybal (1981); it consists of 8000 kg/h of an acetic acid-water solution, containing 30% acid, which is to be countercurrently extracted with diisopropyl ether to reduce the acid concentration in the solventfree raffinate product. The column has eight equilibrium stages, and the solvent feed is 12 500 kg/h. The column operates at 23.5 °C. The output variable is the percentage of acetic acid extracted at steady-state conditions in the column or extractor. The extraction was simulated using the ChemCAD simulator. One hundred simulations were performed for each set of parameters, using the same conditions in the column. None of the simulations resulted in an unfeasible situation for the two parameter sets studied. The simulation results for the parameter sets are shown in Figure 5. The cumulative frequency distribution is broader for the parameters regressed from binary data than for those regressed from ternary data, but basically the difference in their broadness is not very important compared with the difference in the mean for the distributions. At first sight, in particular looking at Figure 4, it seems that the difference in the two equilibrium curves would not result in a big difference in the predicted performance of a specific process like the one used in this example. However, the results show the reverse conclusion. In this case, the prediction of the plait point for this kind of ternary system (type I) could create that remarkable difference. There is a difference of ∼12 in the mole percentage of water at the predicted plait point in the two curves presented in Figure 4. Once again, the effect of regressing thermodynamic parameters either using binary or ternary data is very significant for the prediction of liquid-liquid equilibrium in ternary systems. We saw with the chloroform + acetone + water system that the effect of the parameter variance on the performance of the extraction simulated was significant; but, in this case, the variances are quite small compared with the ones in the chloroform + acetone + water system. Thus, for this particular case, the differences
obtained in the distribution curves are mainly due to the approach followed for the parameter regression. 1,1,2-Trichloroethane (1) + Acetone (2) + Water (3) Sørensen and Arlt (1980) report the experimental data for the ternary liquid-liquid system, originally reported by Treybal et al. (1946). The data for the binary liquid-liquid system 1,1,2-trichloroethane + water were taken from Stephenson (1992). Gmehling and Onken (1977) report experimental vapor-liquid equilibrium data for the system acetone + water; the data from Eduljee (1958) and Kojima et al. (1968) were used. One set of binary interaction parameters for the UNIQUAC model was regressed using the ternary data and another one was obtained using the binary data with the variant that the parameter-pair for the system 1,1,2-trichloroethane + acetone was regressed using the ternary data, keeping the other two parameter pairs constant with the values regressed from the binary systems. This procedure was necessary because of the lack of experimental vapor-liquid data in the literature for the 1,1,2-trichloroethane + acetone equilibrium. Two sets of binary interaction parameters were regressed for the binary system acetone + water, and one of them (set 2) is the same used previously for the study of the system chloroform + acetone + water. The parameters regressed are shown in Table 3. Looking at SEE values, the smallest is for the ternary set, followed by the value of binary set 1; however, the difference between the ternary set and binary set 1 is not very large. The correlation matrixes for these regressions are shown in Appendix C. Figure 6 shows the experimental tie lines for the ternary system and the predicted compositions using the UNIQUAC model for the three parameter sets regressed. Looking at Figure 6, it seems that there is a better fit of the binary set 1 than the ternary set. A deeper analysis of the equilibrium prediction by these two sets shows that the binary set 1 does a poorer prediction than the ternary set in the following cases: (1) composition of TCE on the left phase; and (2) composition of TCE, acetone, and water on the right phase. The criterion used for the previous analysis is based on the average absolute deviation in the estimation of all the experimental data compositions. It is
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1127
Figure 6. LLE for 1,1,2-trichloroethane + acetone + water system at 25 °C. The predicted value is by the UNIQUAC model. See section on 1,1,2-Trichloroethane (1) + Acetone (2) + Water (3) and Table 3 for details of the data sources used.
important to mention that the differences of the average absolute deviation between these two parameter sets are not very large, a result that is in agreement with the small difference in their respective SEE values. Additionally, there is a difference of ∼5 percentage points (based on water composition) in the estimation of the plait point between the ternary set and the binary set 1. With an illustrative example, we will show how the differences in the three sets affect the predicted performance of a liquid-liquid extraction operation. Illustrative Case. Schweitzer (1979) reports the example selected. For this case the feed is 45 kg/h (50% acetone-50% water on a weight basis), and the solvent is 13.5 kg/h of 1,1,2-trichloroethane. The column is a countercurrent extractor with five equilibrium stages operated at 25 °C. The output variable for this case is the percentage of acetone extracted. The extraction was simulated using the ChemCAD simulator. One hundred simulations were performed for each set of the parameters reported in Table 3. Fourteen of the 100 simulations resulted in unfeasible situations for the ternary set, 16 for the binary set 1, and 12 for the binary set 2; therefore, the cumulative frequency curves do not start at the value of zero on the ordinate axis. The reason for these unfeasible situations is mainly that the composition of the feed is close to the equilibrium curve. For certain parameter sets, the initial global composition is calculated to be outside of the equilibrium curve; therefore, there is only one phase in the extraction column and no separation occurs. The cumulative frequency distributions for the results are shown in Figure 7. The shape of the output distribution is broader for the binary set 1 than for the other two. For the case of the ternary set and the binary set 2, the cumulative frequency curve of the first one is a little broader than the second one. From Figure 7, the significant role of the variance of the parameter on the broadness of the curves can be seen. The change of the broadness is only due to the change of the variance in the parameters b23 and b32, in particular for b23. Also, there is a significant difference between both the mean and median for the predicted percentage acetone extracted, indicating in this case that the experimental
Figure 7. Uncertainty of percentage of acetone extracted in the liquid-liquid extractor, using different set of parameters for the UNIQUAC model.
data type and source are very significant for the parameters regression. We can see in this case that having a good fit (apparent from looking at Figure 6 for the binary set 1) in the composition prediction does not necessarily mean that we obtain the “true values” for the binary parameters. An appropriate control of the variances of the parameters is necessary to reduce the uncertainty. Such control can only be done by separating the uncertainty analysis for the model and for the experimental data to see which uncertainty can be reduced or decreased. The experimental data source itself is another important origin of uncertainty. A more detail discussion about this topic can be found in Vasquez and Whiting (1997). Conclusions The approach followed to regress the binary interaction parameters for the UNIQUAC model is very important if the uncertainty in process simulation and design are considered as valuable criteria for decision making. The complexity of correlating liquid-liquid equilibria for ternary systems with reliable results is still high, as shown in this work. Regressing binary interaction parameters from binary data (liquid-liquid and/or vapor-liquid) is more risky than using ternary data for ternary liquid-liquid applications. This conclusion is based on an analysis of the uncertainty effect on the predicted performance of specific processes. Basically, the cumulative frequency distribution for the simulations based on UNIQUAC with parameters regressed from binary systems tend to be broader (greater uncertainty). The results also show that regression of UNIQUAC (or other model) parameters using binary or ternary data can lead to predicted ternary phase equilibria that are significantly in conflict with ternary phase equilibria data, which is another important source of uncertainty in process simulation calculations. That is, the illustrative cases described in this paper point out not only the deviations resulting from different parameter sets but also the deviations resulting from the inadequacy of the model. When ternary data are available, they of course
1128 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 Table 4. Correlation Matrix for Ternary Set {Chloroform (1) + Acetone (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.966678 -0.683223 0.629709 0.894373 -0.288668
-0.966678 1 0.569817 -0.472073 -0.857803 0.380218
-0.683223 0.569817 1 -0.819872 -0.822454 0.472799
0.629709 -0.472073 -0.819872 1 0.593751 -0.059529
0.894373 -0.857803 -0.822454 0.593751 1 -0.628827
-0.288668 0.380218 0.472799 -0.059529 -0.628827 1
Table 5. Correlation Matrix for Binary 01 Set {Chloroform (1) + Acetone (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.999869 0.000000 0.000000 0.000000 0.000000
-0.999869 1 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 1 0.970452 0.000000 0.000000
0.000000 0.000000 0.970452 1 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 1 -0.975409
0.000000 0.000000 0.000000 0.000000 -0.975409 1
Table 6. Correlation Matrix for Binary 02 Set {Chloroform (1) + Acetone (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.996232 0.000000 0.000000 0.000000 0.000000
-0.996232 1 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 1 0.970452 0.000000 0.000000
0.000000 0.000000 0.970452 1 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 1 -0.975409
0.000000 0.000000 0.000000 0.000000 -0.975409 1
Table 7. Correlation Matrix for Ternary Set {Diisopropyl Ether (1) + Acetic Acid (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 0.957947 0.790750 -0.622814 -0.776765 0.996002
0.957947 1 0.732245 -0.797958 -0.691843 0.979069
0.790750 0.732245 1 -0.550230 -0.838947 0.788379
-0.622814 -0.797958 -0.550230 1 0.259277 -0.725883
-0.776765 -0.691843 -0.838947 0.259277 1 -0.769585
0.996002 0.979069 0.788379 -0.725883 -0.769585 1
Table 8. Correlation Matrix for Binary Set {Diisopropyl Ether (1) + Acetic Acid (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.995861 0.000000 0.000000 0.000000 0.000000
-0.995861 1 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 1 -0.328013 0.000000 0.000000
0.000000 0.000000 -0.328013 1 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 1 -0.998715
0.000000 0.000000 0.000000 0.000000 -0.998715 1
Table 9. Correlation Matrix for Ternary Set {1,1,2-Trichloroethane (1) + Acetone (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.248919 -0.383428 0.563370 -0.017538 0.580678
-0.248919 1 0.191183 -0.366476 0.108484 0.413762
-0.383428 0.191183 1 -0.236748 -0.579942 0.180832
0.563370 -0.366476 -0.236748 1 -0.489250 0.463732
-0.017538 0.108484 -0.579942 -0.489250 1 -0.518154
0.580678 0.413762 0.180832 0.463732 -0.518154 1
Table 10. Correlation Matrix for Binary Set 1 {1,1,2-Trichloroethane (1) + Acetone (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.952950 0.000000 0.000000 0.000000 0.000000
-0.952950 1 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 1 0.991120 0.000000 0.000000
0.000000 0.000000 0.991120 1 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 1 -0.975409
0.000000 0.000000 0.000000 0.000000 -0.975409 1
should be used directly in the simulations. However, even the experimental uncertainties of these data can lead to significant and unexpected uncertainties in predicted process performance, which can be revealed through Monte Carlo uncertainty analysis. As a tool, the approach followed in this work allows a broader understanding of the effect of uncertainties on the thermodynamic models used for simulation and
design. Specifically, design engineers can develop uncertainty plots for predicted performance, based on the information readily available from the parameter regression. These plots can then be used for (1) risk avoidance via rational over-design, (2) quantification of the value of additional data, and (3) gaining a healthy respect for the uncertainty of the calculated results. An important conclusion is that calculation of merely the
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1129 Table 11. Correlation Matrix for Binary Set 2 {1,1,2-Trichloroethane (1) + Acetone (2) + Water (3)} b12 b21 b13 b31 b23 b32
b12
b21
b13
b31
b23
b32
1 -0.952950 0.000000 0.000000 0.000000 0.000000
-0.952950 1 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 1 0.991120 0.000000 0.000000
0.000000 0.000000 0.991120 1 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 1 -0.944089
0.000000 0.000000 0.000000 0.000000 -0.944089 1
median or most likely result is not good enough to give the design engineer a sense of either the precision or the accuracy of the calculation. Acknowledgment This work was supported, in part, by National Science Foundation grant CTS-96-96192. Appendix A Each element of the correlation matrix is defined by:
Fi,j )
cov(i,j) σiσj
(2)
where Fij is the correlation coefficient for the elements i and j in the correlation matrix, cov(i,j) is the covariance between elements i and j, and σi is the standard deviation for the element i and σj for element j (see Tables 4-6). Appendix B Correlation matrixes for the regressions in the system diisopropyl ether (1) + acetic acid (2) + water (3) (see Tables 7 and 8). Appendix C Correlation matrix for the regression in the system 1,1,2-trichloroethane (1) + acetone (2) + water (3) (see Tables 9-11). Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 1. Anderson, T. F.; Prausnitz, J. M. Application of the UNIQUAC Equation to Calculation of Multicomponent Phase Equilibria. 2. Liquid-Liquid Equilibria. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 4. Apelblat, A.; Tamir A.; Wagner, M. Fluid Phase Equilib. 1980, 4, 229. Brancroft, W. D.; Hubard, S. D. J. Am. Chem. Soc. 1942, 64, 347. Brusset, H.; Kaiser, L.; Hocquel, J. Chim. Ind., Genie Chim. 1968, 99, 207. Eduljee, H. E., Rad. Ind. Eng. Chem, Chem. Eng. Data Ser. 1958, 3, 44. Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection. Aqueous-Organic Systems; DECHEMA: Frankfurt/Main, Germany, 1977; Vol. I, Part 1. Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection. Aldehydes, Ketones and Ethers; DECHEMA: Frankfurt/Main, Germany, 1979; Vol. I, Parts 3 and 4.
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Received for review June 18, 1997 Revised manuscript received December 2, 1997 Accepted December 3, 1997 IE970444C
