Computing Homogeneous Azeotropes æ Why í Identify limitations in separation operations í Construction of residue curve maps for design and synthesis o

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1 A New Approach for Reliable Computation of Homogeneous Azeotropes in Multicomponent Mixtures Robert W. Maier, Joan F. Brennecke and Mark A. Stadtherr æ Department of Chemical Engineering University of Notre Dame, Notre Dame, IN Paper Number 75` AIChE Annual Meeting, Los Angeles, CA, November 1997 æ Author to whom all correspondence should be addressed; Phone: ; Fax: è219è ; è219è

2 Computing Homogeneous Azeotropes æ Why í Identify limitations in separation operations í Construction of residue curve maps for design and synthesis of separation operations í Evaluation of thermodynamic models æ How í Solve systemèsè of nonlinear equations derived from equifugacity condition í These equation systemèsè often have multiple andèor trivial roots, or may have no solutions 1

3 Formulation : Simultaneous Approach x i,ln P, ln P sat i 1, X i2c, ln æ iæ = 0; i 2 C x i = 0 æ C is the set of all N components æ Ideal vapor phase æ P sat i and æ i are functions of T æ All k-ary azeotropes èk ç N è are solutions, as are all of the pure components ètrivial rootsè æ Need solution method guaranteed to ænd all solutions 2

4 Formulation : Sequential Approach æ If x i 6= 0 ln P, ln P sat i X, ln æ i = 0; i 2 C nz 1, i2c nz x i = 0 æ C nz is a set of k nonzero components æ All k-ary azeotropes èk ç N è for the chosen C nz are solutions; there may be no solutions æ Solve èunorderedè sequence of problems : For k = 2! N: For all combinations of k nonzero components, solve for all k-ary azeotropes æ Need solution method guaranteed to ænd all solutions of all problems, and to determine with certainty when there are no solutions 3

5 Formulation : Other Issues æ T dependence of æ i í Treat explicitly using T-dependent parameters in æ i model í Guess a reference temperature T ref and treat T-dependent parameters as constants evaluated at T ref è No guarantee all azeotropes will be found, even if equations solved correctly æ Solutions of equifugacity equations may not be stable phases èliquid may splitè í Need to check stability of liquid phase at azeotropic composition and temperature í Interval analysis also provides guaranteed method to determine stability èstadtherr et al., 1994è 4

6 Some Current Solution Methods æ Various local methods Fast, but initialization dependent and hard to ænd all roots æ Fidkowski et al. è1993è use a homotopycontinuation method í Simultaneous approach with explicit T- dependence of æ i í Improved reliability but no guarantee that all roots are found æ Harding et al. è1997è use a branch and bound method í Simultaneous and sequential approaches, but T ref approach for T-dependence of æ i í Reformulation as a global optimization problem using convex underestimating functions í Mathematical guarantee that all roots are found 5

7 Interval Approach æ Interval NewtonèGeneralized Bisection èinègbè í Given a system of equations to solve, an initial interval èbounds on all variablesè, and a solution tolerance í INèGB can ænd with mathematical and computational certainty either all the solutions or that no solutions exist. èe.g., Kearfott 1987,1996; Neumaier 1990è æ A general purpose approach : requires no simplifying assumptions or problem reformulations æ Details of algorithm given by Schnepper and Stadtherr è1996è æ Implementation based on modiæcations of routines from INTBIS and INTLIB packages èkearfott and coworkersè 6

8 Example Problems æ Solved using both simultaneous and sequential approaches èwith same results for azeotropic composition and temperatureè æ Solved for case of T-dependent æ i -model parameters and for the case of constant æ i -model parameters æ Problem 1 í Ethanol, Methyl Ethyl Ketone, Water í 1 atm, æ C, Wilson Equation æ Problem 2 í Acetone, Chloroform, Methanol í 16.8 atm, æ C, NRTL Equation æ Problem 3 í Acetone, Methyl Acetate, Methanol í 1 atm, æ C, Wilson Equation æ Times reported for Sun Ultra 1è140 7

9 Results - Problem 1 Azeotropes æ i parameters E MEK W T è æ Cè Constant èt ref =73.7 æ Cè T-dependent CPU Times èsecè Constant T dependent Sequential Simultaneous E = Ethanol; MEK = Methyl Ethyl Ketone W = Water 8

10 Results - Problem 2 Azeotropes æ i parameters A C M T è æ Cè Constant èt ref =152.4 æ Cè T-dependent CPU Times èsecè Constant T dependent Sequential Simultaneous A = Acetone; C = Chloroform; M = Methanol 9

11 Results - Problem 3 Azeotropes æ i parameters A MA M T è æ Cè Constant æ T dependent CPU Times èsecè Constant T dependent Sequential Simultaneous A = Acetoneè1è; MA = Methyl Acetateè2è M=Methanolè3è æ æ 12 = 0.480, æ 21 = 1.550, æ 13 = æ 31 = 0.566, æ 23 = 0.544, æ 32 =

12 Concluding Remarks æ Have solved many other problems using Wilson, NRTL and UNIQUAC activity coeæcient models with up to N=5 æ Interval analysis provides an eæcient, general purpose approach for solving azeotrope problems, providing a mathematical and computational guarantee of reliability æ Interval analysis provides powerful problem solving techniques with many other applications in the modeling of phase behavior and in other process modeling problems 11

13 Acknowledgments æ PRF: AC9 æ NSF: DMI , CTS æ EPA: R æ DOE: DE-FG07-96ER14691 æ Sun Microsystems, Inc. 12