Error analysis in Barker s method: extension to ternary systems

Transcription

1 Fluid hase quilibria rror analysis in Barer s method: extension to ternary systems I.M.A. Fonseca, L.Q. Lobo ) Departamento de ngenharia Quımica, UniÕersidade de Coimbra, 3000 Coimbra, ortugal Received 1 December 1994; accepted 7 November 1998 Abstract An extension of Barer s method to ternary systems is briefly outlined. xpressions for the standard deviations of the excess molar Gibbs free energy and the equilibrium pressure as functions of composition are obtained. These expressions are applied to accurate experimental data for the cryogenic Ž liquid. mixture CH 3F q NOq Xe recently measured in our laboratory. q 1999 lsevier Science B.V. All rights reserved. Keywords: VL low-pressure; Ternary systems; rrors; Barer s method 1. Introduction In the experimental determination of vapour liquid equilibrium Ž VL. using isothermal static cells the measured properties are the equilibrium pressure exp of the mixture and the amount of substance ni of each component introduced into the cell. Reduction of these data to obtain x y phase diagrams is usually made by using Barer s method wx 1. This method has been widely used for binary and ternary systems but in most cases the statistical analysis of the errors associated with it has not been reported. Goral wx described a procedure to carry out such analysis for binary systems, and wx Kolasinsa et al. 3 applied it assuming the Wilson model for the excess molar Gibbs free energy G. In this paper we have extended this ind of study to ternary mixtures, and applied it to VL data for the system CH FŽ 1. qn OqXe Ž 3. and its constituent binaries, at K wx 3 4, recently obtained in our laboratory by a modified version of the accurate low-temperature technique introduced and developed by Davies et al. wx 5 and Fonseca and Lobo wx 6. Since the equilibrium pressure is a nonlinear function of the adjustable parameters in the assumed G model expression, Barer s method embodies a nonlinear regression technique. Therefore, the ) Corresponding author. Tel.: q ; fax: q r99r$ - see front matter q 1999 lsevier Science B.V. All rights reserved. II: S

2 06 ( ) I.M.A. Fonseca, L.Q. LoborFluid hase quilibria uncertainties associated with such parameters are to be obtained from the variance covariance matrix wx 7. Using the propagation law of errors the standard deviations surfaces for and G as functions of composition for ternary systems can be calculated. In this paper we report on such calculations.. Barer s method: outline of application to ternary mixtures In essence Barer s method optimizes the parameters in the assumed expression for the liquid composition dependence of G through an iterative procedure which minimizes the sum of the squares of the pressure residuals R, R s y. Ž 1. exp The calculated equilibrium pressure is obtained from Ý i i i i i s x g ) b, where xi is the mole fraction of component i in the liquid mixture, gi is its activity coefficient, and b is a factor which Ž mainly. i taes into account the vapour-phase non-ideality, usually through a two-term virial equation of state, ½½ ÝÝ ) b sexp Ž V )yb. y y y y d yd :RT. Ž 3. i i ii i i ji j j In these expressions y represents the mole fraction of i in the vapour, ) and V ) i i i are the vapour pressure and the liquid molar volume of pure component i, respectively, Bii is the second virial coefficient of i, and dijsbijybiiyb jj. Initial estimates for the values of the mole fractions in the vapour yi are found by assuming Raoult s law, and in the next iterations by taing y s xy ) r b. Ž 4. i i i i i xpressions for the activity coefficients are derived from the rigorous thermodynamic relationship L L ln gis n G r n i T,,n, 5 j where n L sý n L and the superscript L i i refers to the liquid mixture. For ternary systems we have Ž chosen to represent the reduced quantity g sg rrt. in the form Ý ij 13 all binaries g s g qg, Ž 6. where the binary terms g, and the ternary one g the mole fractions in the liquid, and ij 13 ½ 5 are assumed to be Redlich Kister functions of ij i j ij ijž i j. ijž i j. g sx x A qb x yx qc x yx, 7 g sx x x c yc x yc x

3 ( ) I.M.A. Fonseca, L.Q. LoborFluid hase quilibria In these expressions A, B, C, c, c and c are the adjustable parameters mentioned above. From ij ij ij 0 1 qs. 5 and 6 the activity coefficient of component 1 becomes ln g1sx Ž A1qB1qC1. x1ž 1yx1. q Ž A1yB1qC1. xž 1y x1. q4c1 x1 xž 3 x1y. qx3 Ž A13qB13qC13. x1ž 1yx1. q Ž A13yB13qC13. x13 = Ž 1y x1. q4c13 x1 x3ž 3 x1y. qx x3 1C3 x x3yž A3yB3qC3. x3 yž A3qB3qC3. x qx x3 c0ž 1y x1. yc1 x1y Ž 1y3 x1.ž c1 x1qc x.. Similar expressions for ln g and ln g are obtained from q. Ž 9. 3 by circular change of subscripts. The activity coefficients are functions of the liquid composition and of the parameters present in the G model. Values for the parameters A ij, Bij and Cij are arrived at from binary data alone while those for the parameters c 0, c1 and c are found by regression of the experimental data for the ternary mixtures, i.e., by solving the matrix equation where L d c s T, Ž 10. ž / ž /ž / ž /ž c c c c c / Ý Ý Ý ž /ž / ž / ž /ž / ž c /ž c / ž c /ž c / ž c / 0 1 ÝR pž c 0 / 1 ÝR pž c 1 / ÝR pž c / L s Ý Ý Ý, c c c c c Ý Ý Ý dc 0 d c s dc, and T s. Ž 11. dc All the summations in these matrices are taen over all the M ternary experimental points. The elements of matrix wlx are obtained directly from q. Ž.. The elements d c Ž s 0,1,. represent increments in the c parameters, being obtained from q. Ž 10. by inversion: y1 d c s L T. Ž 1. Ž 9.

4 08 ( ) I.M.A. Fonseca, L.Q. LoborFluid hase quilibria Table 1 Values of the parameters in qs. Ž.Ž. 6 8 with their respective standard deviations wx 4 A s0.148" B sy0.086" C s0.0645" A s1.7470" B sy0.1467" C s0.1957" A3s1.16" B3s0.0715" C3s0.0506"0.01 c0sy0.6711" c1sy0.4088" csy0.5585" Ž The iterations start by assuming c sc sc s0 i.e., g s , and the process is then recycled with new values for the element matrices wlx y1 and wtx by taing corrected values of c given by cscž last iteration. qdc, Ž 13. until a situation is reached for which the d c become smaller than a pre-defined value Ž e.g., y6 dc -10 in this wor.. 3. rror analysis The calculation of matrix wlx y1 wd x, immediately yields wx 7 the so called variance covariance matrix D s L s, Ž 14. y1 where M Ý 1 s s R rž Mym., Ž 15. represents the variance of the equilibrium pressure, and m is the number of adjustable parameters in q. Ž 8.. The diagonal elements D in the symmetrical matrix wdx ii are the variances of the parameters. Both Dii and the covariances Dij Ž i.e., the non-diagonal elements. are affected by the experimental uncertainty incorporated in s as shown by q. Ž 14. p. Three independent variance covariance matrices for the binary systems are obtained exactly in the same way. They are also independent from that for the ternary system. Indeed each set of binary data Ž and the ternary one. concerns a different mixture embodying different experimental uncertainties and originating different values of s Ž 15.. p Table Values of the covariances of the parameters in qs. Ž. 6 Ž. 8 covž A, B. s1.58=10 covž A, C. sy6.953=10 covž B, C. sy3.605=10 covž A, B. s3.7=10 covž A, C. sy.449=10 covž B, C. sy1.559=10 covž A, B. sy.46=10 covž A, C. sy5.519=10 covž B, C. sy8.334=10 covž c, c. sy7.401=10 covž c, c. s8.109=10 covž c, c. sy4.179=10 y5 y5 y5 1 1 y6 1 1 y6 1 1 y y y y4 3 3 y3 3 3 y3 3 3 y

5 ( ) I.M.A. Fonseca, L.Q. LoborFluid hase quilibria The uncertainty associated with any other variable which is itself a function of the optimized parameters mentioned above, namely the G function, is readily obtained as follows. Let f be a function of parameters a, b, and c, fsfž a,b,c., Ž 16. whose uncertainties may assume any values between ys and qs Ž sa, b, c.. Then, through the propagation law of errors, the variance s is given as wx 7 f s sf s qf s qf s f a a b b c c q FF a bcovž a,b. q FFcov a c Ž a,c. q FFcov b c Ž b,c. Ž 17. where the usual convention has been used to indicate the partial derivatives, e.g., F sž Fra. a b,c. xpressions similar to q. Ž 17. omitting the covariance terms have been used. However, these terms can often mae a significant contribution to the estimated value of s f, and therefore should not be neglected wx 8. Applying q. 17 to g defined by q. 6, an expression for the standard deviation surface of G as a function of the composition of the liquid mixture is obtained: ½ ½ 4 G 1 A 1 1 B 1 1 C s s RT x x s q x yx s q x yx s q x yx cov A, B 3 Ž 1. Ž 1 1. Ž 1. Ž 1 1. q x y x cov A,C q x y x cov B,C q... similar terms for the binaries Ž 1,3. and Ž, qž x x x. = s qs x 1 3 c0 c1 1 1r qsc xycov c 0,c1 x1ycov c 0,c xqcov c 1,c x1x. 18 The total number of parameters whose uncertainties contribute to s G for each binary mixture, and three for the ternary term. 55 is, therefore, twelve: three Fig. 1. erspective view of the standard deviation surface s calculated from q. Ž 18. G, with parameters listed in Tables 1 and. s is represented as a function of the mole fractions of the three components. G

6 10 ( ) I.M.A. Fonseca, L.Q. LoborFluid hase quilibria In a similar way the standard deviation surface of the equilibrium pressure s liquid composition is obtained by applying q. Ž 17. to as defined in q. Ž.: ½ ž / A1 ž / B1 ž / C1 ž /ž / 1 1 A1 B1 C1 A1 B1 s s s q s q s q covž A, B. ž /ž / ž /ž / q covž A 1,C1. q covž B 1,C1. A C B C q... similar terms for the binaries Ž 1,3. and Ž, as a function of the ž / c c c Ž ž / 1 ž / c c c ž c /ž c / ž c / r c c1 c q s q s q s q cov c,c q = ž / ž /ž / covž c,c. q covž c,c.. Ž Results and conclusions The values of the parameters in q. Ž 6. obtained by Barer s method for CH FŽ 1. qn 0qXe Ž 3. 3, at K, using experimental data recently obtained in our laboratory wx 4 are summarized with their respective standard deviations in Table 1. The calculated covariances are listed in Table. The standard deviations surfaces s G and s are represented as functions of the liquid mole fractions in Figs. 1 and, respectively. Both perspective views in these figures clearly show that the standard deviations in the parameters for the binary mixtures affect the calculated values of s G and s for Fig.. erspective view of the standard deviation surface s calculated from q. Ž 19., with parameters listed in Tables 1 and. s is represented as a function of the mole fractions of the three components.

7 ( ) I.M.A. Fonseca, L.Q. LoborFluid hase quilibria the ternary system. The higher values of these deviations near the N O qxež 3. binary region are a consequence of the comparatively much higher values of the parameters standard deviations for this binary mixture. The value of s for the equimolar ternary mixture is about 6 J mol y1 G, half of which arises from s G 13. References wx 1 J.A. Barer, Aust. J. Chem. 6 Ž wx M. Goral, Z. hys. Chemie, Leipzig 58 Ž wx 3 G. Kolasinsa,. Oracz, Z. hys. Chemie, Leipzig 60 Ž wx 4 I.M.A. Fonseca, L.Q. Lobo, Fluid hase quilibria 113 Ž wx 5 R.H. Davies, A.G. Duncan, G. Saville, L.A.K. Staveley, Trans. Faraday Soc. 63 Ž wx 6 I.M.A. Fonseca, L.Q. Lobo, Fluid hase quilibria 47 Ž wx 7 G... Box, W.. Hunter, J.S. Hunter, Statistics for xperimenters, Wiley, New Yor, wx 8 W.. Wentworth, J. Chem. duc. 4 Ž