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Procedure for checking and fitting experimental liquid-liquid equilibrium data
FLUIDPHASI: EQUILIBRIA ELSEVIER
Fluid Phase Equilibria 129 (1997) 15-19
Procedure for checking and fitting experimental liquid-liquid equilibrium data V. Gomis *, F. Rulz, J.C. Asensi, M.D. Saquete Departamento de Ingenierla QMmica, Universidad de Alicante, Apartado 99. Alicante, Spain
Received 17 May 1996; accepted 11 September 1996
Abstract An objective method was developed to check and fit liquid-liquid equilibrium data obtained experimentally. The analytical concentrations are changed slightly within the interval given by the uncertainties of the determinations in order to satisfy the material balances. The method consists of a minimisation with constraints of a proposed objective function. © 1997 Elsevier Science B.V. Keywords: Experiments; Method; Liquid-liquid
1. Experimental determination of liquid-liquid equilibrium data A review of papers published in recent years regarding determinations of liquid-liquid equilibrium data shows that the method currently used to obtain a tieline consists of: 1. preparation by weighing of a global heterogeneous mixture; 2. analysis of the components of each of the phases into which the global mixtures splits. It is not actually necessary to analyse all the components in each of the phases. In an n-component system, if the composition of the global mixture is known Xl~, x2o .... x,G, it is only necessary to analyse n - 1 components in one of the phases and one component in the other phase to accurately determine the tieline. The rest of the compositions can be calculated using the following equations which represent the material balances (total (Eq. (I)) and for each component (Eq. (2))): ml + m 2 = m G
* Corresponding author. E-Mail: [email protected]. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S 0 3 7 8 - 3 81 2 ( 9 6 ) 0 3 1 8 5 - 8
(1)
16
V. Gomis et a l . / F l u i d Phase Equilibria 129 (1997) 15-19
m l X l l --I--m 2 x 1 2 = m G X l G
mlx21 + m2x22 = m° x2° mlXnl
"4- m 2 X n 2
(2)
= m G XnG
where m e, m 1, and m 2 represent the weights of the global mixture (known), of phase 1 (unknown) and of phase 2 (unknown) respectively, xio represents the mass fraction of the component in the global mixture (known), and xi~ and xi2 are the mass fractions of component i in phases 1 and 2 respectively (n - 1 compositions of phase 1 and one of phase 2 are known; one composition of phase 1 and n - 1 of phase 2 remain as unknowns). Obviously, the previous equations can be formulated in terms of the number of moles (instead of weight) and mole fraction (instead of mass fraction). These equations can be combined to obtain: xil = 1 and ~ xi2 = 1 i=l X i l -- XiG
(3)
i=l
= constant(i= 1,2 .... n)
(4)
XiG -- Xi2
Eqs. (3) and (4) allow all the unknown compositions to be calculated. The problem with this method is that a small error in the experimental determination of some of the components can produce very large errors when the non-analysed components are calculated with the previous equations. For example, considering the determination of a tieline of a ternary system 1 - 2 - 3 , a global mixture has been prepared with the following composition: x~c = 0.473 x2o = 0.060 x3G = 0.467 Phasel x2t = 0.100 x31 = 0.046 The compositions determined experimentally were: Using Phase2 x22 = 0.011 Eqs. (3) and (4), the following composition can be calculated:xll = 0.8540 xl2 = 0.0063 x32 = 0.9827 The experimental determinations are obviously not perfectly accurate and the values obtained experimentally can have a great influence on the calculated values. The relative accuracies of the weight or mole fraction measurements used to be in the range 0.5-2%. Assuming that the value obtained for x2~ should have been 0.101 instead of 0.10 (which represents an error of 1%), with this value of x2~ the calculated values of the unknown concentrations should be xll = 0.8530, x~2 = 0.0177 and x32 = 0.9730, which differ from those previously calculated by 0.1%, 180% and 1% respectively. It can be seen that the relative errors in the majority components are similar to those of the experimental determination. However, the relative error in the minority component is much greater. Consequently, it is more accurate to determine the compositions of all the components in each of the phases (or at least all the minority components). It is worth noting that a review of the papers published in this field in recent years shows that it is common practice to determine all the minority components in each of the phases into which the global heterogeneous mixture splits.
2. Checking experimental equilibrium data When more of the strictly necessary compositions are analysed, owing to the experimental uncertainties of all the determinations it is difficult for the results obtained to perfectly satisfy the
17
V. Gomis et al. / Fluid Phase Equilibria 129 (1997) 15-19
material balances (Eqs. (3) and (4)). In this case, two possible practices are either to publish the obtained results directly or to correct them taking into account the accuracy of the determinations. In current papers, it is not usual (as it was previously) to show the compositions of the global heterogeneous mixtures used to carry out the determinations. Therefore, it is not possible to check whether the experimental results satisfy Eqs. (3) and (4) corresponding to the material balances. In any case, if the published experimental data satisfy these equations, the method for correcting the direct experimental results is never indicated and supposedly depends on the criteria of the researcher. In this work, an objective method is proposed that fits the liquid-liquid equilibrium data obtained experimentally, taking into account the accuracy of the determinations. The method can also be used to check the validity of the experimental data, rejecting those that do not satisfy the material balances in the tolerance interval of experimental errors.
3. Proposed method to fit the experimental data The proposed method slightly modifies the experimental results within the interval given by the uncertainties of the determinations, in order that the final results satisfy the material balances. The method, known as "data reconciliation" or "validation", has been used for years in the process industries to handle plant measurements (Heyen et al., 1996; Rao and Narasimhan, 1996), but it has not been presented in the framework of laboratory analysis. The input data to apply the method are: xic - - the concentrations of all the components in the global mixture; these values are very accurate since they are usually obtained directly by weighing; xe~p and Xi2exp - - the concentrations of all the components in each of the phases; 6i~ and 6 n - - the absolute accuracy of the determinations of each component i in each phase. The problem is calculating new values of x/l and x n in the interval [r Xijexp -- ¢~ij, X e j p "~ 6ij] which verify the material balances (Eqs. (3) and (4)). It is, therefore, a constrained minimisation problem with the following objective function:
Xil-1
F=10
xi2-1
+10
i =1
i
+ i=l j=l
x i l - xi~
xJl-- xjC-
XiG - - X i 2
XjG - - X j2
(5)
where the factors 10 have been introduced to give more weight to the sum of mass fractions. Moreover, the solution is not unique and it is convenient to add to the objective function the penalty function: 1 E 10 j=l i = l
exp
(6)
xij
in order to obtain a solution which is closer to the experimental concentrations. Again, a factor of 1 / 1 0 has been introduced to decrease the weight of these differences. Obviously, if 6ij is too small, the intersection of the set of points verifying the mass balances and the interval [ x~j.xp - 6;j, xijexp + 3ij] may be empty: the mass balance equations will not be satisfied at the solution and the algorithm will not be able to fit the concentrations. In this case either 1. the experimental determinations have to be repeated since the presence of eventual errors made by the operator has led to inaccurate data, or
V. Gomis et al./Fluid Phase Equilibria 129 (1997) 15-19
18
2. the estimated 6ij have to be revised since the real uncertainties of the analytical method are greater than supposed. In the literature, there are several references describing data reconciliation algorithms to minimise objective functions of this type. For instance, the methods of Crowe et al. (1983) when the constraints are linear equations and Crowe (1986) for the non-linear case (and especially for bilinear equations, as in this work) are efficient algorithms for solving the minimisation problem. All of them involve matrix algebra and projection matrix techniques, which are very useful when the number of variables to fit is great. However, as the number of variables involved in Eqs. (5) and (6) is small, the subroutine ZXMWD from the well known and widely used IMSL Library (1983) is preferred. This subroutine transforms the objective function with xij belonging to the interval (Xijmin, Xijmax ) = (X7~"p6ij, xijexp + 6ij) in a new objective function depending on new variables t i implicitly defined as Xi j = Ximin j .~_ ( Xijmax -- x i j i n ) sm • 2 tij = X iexp j -- 6ij +
2 6ij sinXtij
(7)
where tij a r e now unconstrained. This objective function is minimised using a quasi-Newton method to obtain the tij values, and from these the x o that verify the material balances. For example, assuming that the composition of a global mixture for the determination of a tieline is x ~ - - 0 . 4 7 3 0 , x2c = 0.0600, x3c = 0.4670, the concentrations of each component in each phase determined experimentally are: xll = 0.8480 _+ 0.0080
xl2
X21 = 0 . 1 0 0 0
X22 = 0 . 0 1 1 0
"]- 0 . 0 0 1 0
x31 = 0.046 + 0.0005
= 0.0065 _+ 0.0002 ~ 0.0002
x32 = 0.9760 + 0.0080
Obviously, this solution is not accurate since the experimental values do not perfectly satisfy the material balances. The minimisation of the objective function (Eq. (5)) with the penalty function (Eq. (6)), and the constraints in the experimental concentrations given by the experimental error using the procedure described above, allow the following concentrations to be obtained: Xll
0.8540
x12 = 0.0065
x21 = 0.1000
X2z = 0.0110
x31 = 0.0460
x32 = 0.9825
=
These concentrations, which are close to the experimental ones, perfectly verify the material balances. The program also checks the accuracy of the experimental data: if errors greater than 6ij have been made in the experimentation, the computer program will not be able to fit the concentrations.
Acknowledgements The authors wish to thank the DGICYT (Spain) for the financial aid of Project PB93-0946
V. Gomis et al. / Fluid Phase Equilibria 129 (1997) 15-19
19
References Crowe, C.M., Campos, Y.A.G. and Hrymak, A., 1983. Reconciliation of process flow rates by matrix projection, Part I: Linear case. AIChE J, 29: 881-888. Crowe, C.M., 1986. Reconciliation of process flow rates by matrix projection, Part II: The nonlinear case. AIChE J., 32: 616- 623. Heyen, G., Marlchal, E. and Kalitventzeff, B., 1996. Sensitivity calculations and variance analysis in plant measurement reconciliation. Comput. Chem. Eng., 20: $539-$544. IMSL, 1983. The IMSL Library: a set of fortran subroutines for mathematics and statistics. IMSL Inc., Houston, TX. Rao, R.R. and Narasimhan, S., 1996. Comparison of techniques for data reconciliation of multicomponent processes. Ind. Eng. Chem. Res., 35: 1362-1368.
Fluid Phase Equilibria 129 (1997) 15-19
Procedure for checking and fitting experimental liquid-liquid equilibrium data V. Gomis *, F. Rulz, J.C. Asensi, M.D. Saquete Departamento de Ingenierla QMmica, Universidad de Alicante, Apartado 99. Alicante, Spain
Received 17 May 1996; accepted 11 September 1996
Abstract An objective method was developed to check and fit liquid-liquid equilibrium data obtained experimentally. The analytical concentrations are changed slightly within the interval given by the uncertainties of the determinations in order to satisfy the material balances. The method consists of a minimisation with constraints of a proposed objective function. © 1997 Elsevier Science B.V. Keywords: Experiments; Method; Liquid-liquid
1. Experimental determination of liquid-liquid equilibrium data A review of papers published in recent years regarding determinations of liquid-liquid equilibrium data shows that the method currently used to obtain a tieline consists of: 1. preparation by weighing of a global heterogeneous mixture; 2. analysis of the components of each of the phases into which the global mixtures splits. It is not actually necessary to analyse all the components in each of the phases. In an n-component system, if the composition of the global mixture is known Xl~, x2o .... x,G, it is only necessary to analyse n - 1 components in one of the phases and one component in the other phase to accurately determine the tieline. The rest of the compositions can be calculated using the following equations which represent the material balances (total (Eq. (I)) and for each component (Eq. (2))): ml + m 2 = m G
* Corresponding author. E-Mail: [email protected]. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S 0 3 7 8 - 3 81 2 ( 9 6 ) 0 3 1 8 5 - 8
(1)
16
V. Gomis et a l . / F l u i d Phase Equilibria 129 (1997) 15-19
m l X l l --I--m 2 x 1 2 = m G X l G
mlx21 + m2x22 = m° x2° mlXnl
"4- m 2 X n 2
(2)
= m G XnG
where m e, m 1, and m 2 represent the weights of the global mixture (known), of phase 1 (unknown) and of phase 2 (unknown) respectively, xio represents the mass fraction of the component in the global mixture (known), and xi~ and xi2 are the mass fractions of component i in phases 1 and 2 respectively (n - 1 compositions of phase 1 and one of phase 2 are known; one composition of phase 1 and n - 1 of phase 2 remain as unknowns). Obviously, the previous equations can be formulated in terms of the number of moles (instead of weight) and mole fraction (instead of mass fraction). These equations can be combined to obtain: xil = 1 and ~ xi2 = 1 i=l X i l -- XiG
(3)
i=l
= constant(i= 1,2 .... n)
(4)
XiG -- Xi2
Eqs. (3) and (4) allow all the unknown compositions to be calculated. The problem with this method is that a small error in the experimental determination of some of the components can produce very large errors when the non-analysed components are calculated with the previous equations. For example, considering the determination of a tieline of a ternary system 1 - 2 - 3 , a global mixture has been prepared with the following composition: x~c = 0.473 x2o = 0.060 x3G = 0.467 Phasel x2t = 0.100 x31 = 0.046 The compositions determined experimentally were: Using Phase2 x22 = 0.011 Eqs. (3) and (4), the following composition can be calculated:xll = 0.8540 xl2 = 0.0063 x32 = 0.9827 The experimental determinations are obviously not perfectly accurate and the values obtained experimentally can have a great influence on the calculated values. The relative accuracies of the weight or mole fraction measurements used to be in the range 0.5-2%. Assuming that the value obtained for x2~ should have been 0.101 instead of 0.10 (which represents an error of 1%), with this value of x2~ the calculated values of the unknown concentrations should be xll = 0.8530, x~2 = 0.0177 and x32 = 0.9730, which differ from those previously calculated by 0.1%, 180% and 1% respectively. It can be seen that the relative errors in the majority components are similar to those of the experimental determination. However, the relative error in the minority component is much greater. Consequently, it is more accurate to determine the compositions of all the components in each of the phases (or at least all the minority components). It is worth noting that a review of the papers published in this field in recent years shows that it is common practice to determine all the minority components in each of the phases into which the global heterogeneous mixture splits.
2. Checking experimental equilibrium data When more of the strictly necessary compositions are analysed, owing to the experimental uncertainties of all the determinations it is difficult for the results obtained to perfectly satisfy the
17
V. Gomis et al. / Fluid Phase Equilibria 129 (1997) 15-19
material balances (Eqs. (3) and (4)). In this case, two possible practices are either to publish the obtained results directly or to correct them taking into account the accuracy of the determinations. In current papers, it is not usual (as it was previously) to show the compositions of the global heterogeneous mixtures used to carry out the determinations. Therefore, it is not possible to check whether the experimental results satisfy Eqs. (3) and (4) corresponding to the material balances. In any case, if the published experimental data satisfy these equations, the method for correcting the direct experimental results is never indicated and supposedly depends on the criteria of the researcher. In this work, an objective method is proposed that fits the liquid-liquid equilibrium data obtained experimentally, taking into account the accuracy of the determinations. The method can also be used to check the validity of the experimental data, rejecting those that do not satisfy the material balances in the tolerance interval of experimental errors.
3. Proposed method to fit the experimental data The proposed method slightly modifies the experimental results within the interval given by the uncertainties of the determinations, in order that the final results satisfy the material balances. The method, known as "data reconciliation" or "validation", has been used for years in the process industries to handle plant measurements (Heyen et al., 1996; Rao and Narasimhan, 1996), but it has not been presented in the framework of laboratory analysis. The input data to apply the method are: xic - - the concentrations of all the components in the global mixture; these values are very accurate since they are usually obtained directly by weighing; xe~p and Xi2exp - - the concentrations of all the components in each of the phases; 6i~ and 6 n - - the absolute accuracy of the determinations of each component i in each phase. The problem is calculating new values of x/l and x n in the interval [r Xijexp -- ¢~ij, X e j p "~ 6ij] which verify the material balances (Eqs. (3) and (4)). It is, therefore, a constrained minimisation problem with the following objective function:
Xil-1
F=10
xi2-1
+10
i =1
i
+ i=l j=l
x i l - xi~
xJl-- xjC-
XiG - - X i 2
XjG - - X j2
(5)
where the factors 10 have been introduced to give more weight to the sum of mass fractions. Moreover, the solution is not unique and it is convenient to add to the objective function the penalty function: 1 E 10 j=l i = l
exp
(6)
xij
in order to obtain a solution which is closer to the experimental concentrations. Again, a factor of 1 / 1 0 has been introduced to decrease the weight of these differences. Obviously, if 6ij is too small, the intersection of the set of points verifying the mass balances and the interval [ x~j.xp - 6;j, xijexp + 3ij] may be empty: the mass balance equations will not be satisfied at the solution and the algorithm will not be able to fit the concentrations. In this case either 1. the experimental determinations have to be repeated since the presence of eventual errors made by the operator has led to inaccurate data, or
V. Gomis et al./Fluid Phase Equilibria 129 (1997) 15-19
18
2. the estimated 6ij have to be revised since the real uncertainties of the analytical method are greater than supposed. In the literature, there are several references describing data reconciliation algorithms to minimise objective functions of this type. For instance, the methods of Crowe et al. (1983) when the constraints are linear equations and Crowe (1986) for the non-linear case (and especially for bilinear equations, as in this work) are efficient algorithms for solving the minimisation problem. All of them involve matrix algebra and projection matrix techniques, which are very useful when the number of variables to fit is great. However, as the number of variables involved in Eqs. (5) and (6) is small, the subroutine ZXMWD from the well known and widely used IMSL Library (1983) is preferred. This subroutine transforms the objective function with xij belonging to the interval (Xijmin, Xijmax ) = (X7~"p6ij, xijexp + 6ij) in a new objective function depending on new variables t i implicitly defined as Xi j = Ximin j .~_ ( Xijmax -- x i j i n ) sm • 2 tij = X iexp j -- 6ij +
2 6ij sinXtij
(7)
where tij a r e now unconstrained. This objective function is minimised using a quasi-Newton method to obtain the tij values, and from these the x o that verify the material balances. For example, assuming that the composition of a global mixture for the determination of a tieline is x ~ - - 0 . 4 7 3 0 , x2c = 0.0600, x3c = 0.4670, the concentrations of each component in each phase determined experimentally are: xll = 0.8480 _+ 0.0080
xl2
X21 = 0 . 1 0 0 0
X22 = 0 . 0 1 1 0
"]- 0 . 0 0 1 0
x31 = 0.046 + 0.0005
= 0.0065 _+ 0.0002 ~ 0.0002
x32 = 0.9760 + 0.0080
Obviously, this solution is not accurate since the experimental values do not perfectly satisfy the material balances. The minimisation of the objective function (Eq. (5)) with the penalty function (Eq. (6)), and the constraints in the experimental concentrations given by the experimental error using the procedure described above, allow the following concentrations to be obtained: Xll
0.8540
x12 = 0.0065
x21 = 0.1000
X2z = 0.0110
x31 = 0.0460
x32 = 0.9825
=
These concentrations, which are close to the experimental ones, perfectly verify the material balances. The program also checks the accuracy of the experimental data: if errors greater than 6ij have been made in the experimentation, the computer program will not be able to fit the concentrations.
Acknowledgements The authors wish to thank the DGICYT (Spain) for the financial aid of Project PB93-0946
V. Gomis et al. / Fluid Phase Equilibria 129 (1997) 15-19
19
References Crowe, C.M., Campos, Y.A.G. and Hrymak, A., 1983. Reconciliation of process flow rates by matrix projection, Part I: Linear case. AIChE J, 29: 881-888. Crowe, C.M., 1986. Reconciliation of process flow rates by matrix projection, Part II: The nonlinear case. AIChE J., 32: 616- 623. Heyen, G., Marlchal, E. and Kalitventzeff, B., 1996. Sensitivity calculations and variance analysis in plant measurement reconciliation. Comput. Chem. Eng., 20: $539-$544. IMSL, 1983. The IMSL Library: a set of fortran subroutines for mathematics and statistics. IMSL Inc., Houston, TX. Rao, R.R. and Narasimhan, S., 1996. Comparison of techniques for data reconciliation of multicomponent processes. Ind. Eng. Chem. Res., 35: 1362-1368.