Reliable Phase Stability Analysis for Asymmetric Models

Transcription

1 Relable Phase Stablty Analyss for Asymmetrc Models Gang Xu 1, Wllam D. Haynes 2* and Mark A. Stadtherr 2 1 Invensys/SmSc-Esscor, Rancho Parkway South, Sute 1, Lake Forest, CA 9263 USA 2 Department of Chemcal and Bomolecular Engneerng, Unversty of Notre Dame, 182 Ftzpatrck Hall, Notre Dame, IN USA (March 25) (revsed, June 25) Keywords: Phase stablty; Phase equlbrum; Interval analyss; Valdated computng; Equaton of state; NRTL. * Current address: NFS Inc., 125 Banner Hll Road, Erwn, TN 3765 USA Correspondng author Tel.: ; Fax: ; E-mal: markst@nd.edu

2 Abstract A determnstc technque for relable phase stablty analyss s descrbed for the case n whch asymmetrc modelng (dfferent models for vapor and lqud phases) s used. In comparson to the symmetrc modelng case, the use of multple thermodynamc models n the asymmetrc case adds an addtonal layer of complexty to the phase stablty problem. To deal wth ths addtonal complexty we formulate the phase stablty problem n terms of a new type of tangent plane dstance functon, whch uses a bnary varable to account for the presence of dfferent lqud and vapor phase models. To then solve the problem determnstcally, we use an approach based on nterval analyss, whch provdes a mathematcal and computatonal guarantee that the phase stablty problem s correctly solved, and that thus the global mnmum n the total Gbbs energy s found n the phase equlbrum problem. The new methodology s tested usng several examples, nvolvng as many as eght components, wth NRTL as the lqud phase model and a cubc equaton of state as the vapor phase model. In two cases, publshed phase equlbrum computatons were found to be ncorrect (not stable).

3 1 1. Introducton A key step n the computaton of multphase equlbrum s phase stablty analyss. A relable technque for phase stablty analyss wll assure both that the correct number of phases s found, and that the phase splt computed corresponds to a global mnmum n the total Gbbs energy. That s, phase stablty analyss serves as a global optmalty test n solvng the global optmzaton problem that determnes phase equlbrum at constant temperature and pressure. However, phase stablty analyss s tself a global optmzaton problem that can be very dffcult to solve relably. A standard approach to the problem s the use of tangent plane analyss [1], n partcular the method mplemented by Mchelsen [2]. However, a very large number of other methods have been proposed, nvolvng a varety of equaton solvng and optmzaton technques. Some methods use local optmzaton and/or equaton solvng methods, perhaps n connecton wth some multstart approach, or the use of homotopy-contnuaton. Stochastc global optmzaton methods (e.g., smulated annealng, genetc algorthms, etc.) have also been frequently proposed n ths context. However, none of these technques s actually guaranteed to produce the correct results. Thus, there has been sgnfcant nterest n the development of determnstc technques that guarantee the correct soluton of the phase stablty problem, as revewed brefly below. These efforts have been focused prmarly on the case of symmetrc models (same thermodynamc model used for all phases). Work on determnstc stablty analyss for the asymmetrc case (dfferent models used for dfferent phases) has been lmted to cases nvolvng ether an deal gas vapor phase or a pure sold phase. We wll consder here a determnstc method for the more general asymmetrc case, focusng on the common stuaton n modelng vapor-lqud equlbrum n whch nondealtes are represented n the vapor phase by an equaton of state and n the lqud phase by an excess Gbbs energy model. One approach for determnstc phase stablty analyss, as demonstrated by McDonald and Floudas [3,4,5,6] for symmetrc cases n whch varous excess Gbbs energy models were used, s the use of determnstc global optmzaton technques, such as GOP [7,8] and branch-and-bound [9]. McDonald and Floudas [3,4,5,6] also consdered the asymmetrc case n whch an excess Gbbs energy model was used for lqud phases and the vapor was an deal gas. A more general branch-and-bound strategy, the α-

4 2 BB method, was appled by Hardng and Floudas [1] to symmetrc cases n whch cubc equaton of state models were used. The α-bb approach reles on the use of convex underestmatng functons to obtan lower bounds on the objectve functon. However, whether or not these are vald bounds depends on the proper choce of a parameter (α). Methods exst [11], based on an nterval representaton of the Hessan matrx, that can be used to guarantee a proper value of α, and ths approach was appled by Hardng and Floudas [1]. An alternatve determnstc procedure for phase stablty analyss s the use of an nterval- Newton approach [12]. Ths has been demonstrated for symmetrc cases usng excess Gbbs energy models by Stadtherr et al. [13], McKnnon et al. [14], and Tesser et al. [15], and for symmetrc cases usng cubc equatons of state by Hua et al. [16,17,18]. Recently Xu et al. [19] also appled ths approach to the symmetrc case n whch a statstcal assocatng flud theory (SAFT) model s used. The nterval- Newton procedure provdes a mathematcal guarantee that the phase stablty problem s correctly solved. Moreover, snce t uses nterval arthmetc throughout, and thus bounds roundng error, the nterval- Newton method also provdes a rgorous computatonal guarantee of global optmalty [2]. The nterval-newton approach wll be appled here to the asymmetrc case n whch both lqud and vapor phases are nondeal. In comparson to the symmetrc model case, the use of multple thermodynamc models n the asymmetrc case adds an addtonal layer of complexty to the phase stablty problem. To deal wth ths addtonal complexty we wll formulate the phase stablty problem n terms of a new type of tangent plane dstance functon, whch uses a bnary varable to account for the presence of dfferent lqud- and vapor-phase models. To then solve the problem determnstcally, we wll use an nterval-newton approach. The new methodology s tested usng several examples wth NRTL as the lqud-phase model and a cubc equaton of state as the vapor-phase model. In two cases, publshed phase equlbrum computatons were found to be ncorrect (not stable). In should be noted that, as recently demonstrated by Burgos-Solórzano et al. [21], procedures for determnstc phase stablty analyss, such as descrbed here, can be used n connecton wth any algorthm or software package for computng phase equlbrum, to valdate the computed results and to provde correctve feedback f needed.

5 3 2. Problem Formulaton The bass for tangent plane analyss for phase stablty s the test formulated by Baker et al. [1]. Assume that the phase to be tested has a composton (mole fracton) vector x and that a constant temperature T and pressure P have been specfed. Then consder the molar Gbbs energy vs. composton (mole fracton) surface g(x) and a hyperplane tangent to g(x) at x= x. If ths tangent plane ever crosses (goes above) the Gbbs energy surface, then the system beng tested s not stable (.e., t s ether unstable or metastable). Ths condton s often stated n terms of the tangent plane dstance functon D(x) that gves the dstance of the Gbbs energy surface above the tangent plane. Ths s gven by D( x) = g( x) g s ( x x ), (1) T where g = g( x ) and s = g( x ) are the Gbbs energy functon and ts gradent evaluated at the feed composton x. If D(x) s negatve for any value of x, then the phase beng tested s not stable. To determne f D s ever negatve, ts mnmum s sought. If a statonary pont (local mnmum) of D s found for whch D <, then ths ndcates that the phase beng tested s not stable. Actually, to show that a phase t not stable, t s suffcent to fnd any pont x for whch D <. However, statonary ponts wth D < are commonly sought snce they are useful n provdng ntal composton estmates for a possble new equlbrum phase or phases. Proof that the phase beng tested s stable s obtaned f the global mnmum of D s zero (correspondng to the tangent pont at the feed composton x ). Obvously ths procedure may fal f the global mnmum of the tangent plane dstance functon s not found. For nstance, f the optmzaton algorthm used returns a global mnmum of zero, whle the true global mnmum s negatve, the concluson that the phase s stable wll be ncorrect. The foregong assumes that there s a sngle functon g(x) that represents all phases that may be present n the system (though g(x) may be multvalued f an equaton of state model s used and there are multple compressblty roots). Unlke ths symmetrc model case, n the asymmetrc model case there wll be dfferent g(x) functons for dfferent types of phases. We wll consder here only the stuaton n whch vapor and lqud phases are possble, and use g V (x) to represent the Gbbs energy of a vapor phase and g L (x) to represent the Gbbs energy of lqud phases. Xu et al. [22] and Scurto et al. [23] have

6 4 descrbed an approach for determnstc phase stablty analyss for the asymmetrc case of sold-flud equlbrum, n whch one model s used for flud phases, and another for pure sold phases. In tangent plane analyss for phase stablty, snce the goal n testng a phase s to detect alternate states that have a lower Gbbs energy, the Gbbs energy surface that must be used s gven by whchever of g V (x) and g L (x) s lowest. That s, n Eq. (1), g(x) = mn[g V (x), g L (x)], and evaluatons at x must be done on the lower of the Gbbs energy surfaces. Thus, D( x) = mn[ g ( x), g ( x)] g s ( x x ) = mn = V L T V T g x g s x x L T g x g s x x D ( ) ( ) ( ) ( ) D V L mn[ ( ), ( )]. x x (2) Note that the vapor and lqud tangent plane dstance functons, L L T D ( x) = g ( x) g s ( x x ) and V V T D ( x) = g ( x) g s ( x x ), respectvely, are both based on the same values of g and s, as determned from whchever Gbbs energy surface s lower at x. The mnmzaton problem that then must be solved s V L mn{mn[ D ( x), D ( x)]} x n s.t. 1 x =. = 1 (3) In order to avod the dffcultes assocated wth the nondfferentable objectve functon, t s convenent to reformulate the problem. We defne the pseudo tangent plane dstance functon V L D ( x) = θd ( x) + (1 θ) D ( x), where θ {,1} s a bnary varable whose value wll be determned as part of the optmzaton problem. Assumng that an equaton of state model f( Z, x ) = s used for the vapor phase, and treatng the compressblty Z as an ndependent varable, the mnmzaton problem that must be solved can now be expressed as mn{ θ V L D ( Z, x) + (1 θ ) D ( x) } x, Z, θ s.t. 1 x = n = 1 f( Z, x) = (4) θ {,1}.

7 5 Usng Lagrangan analyss, ths optmzaton problem can also be represented by the equvalent system of nonlnear equatons D D =, = 1,, n 1 x x n 1 x = = 1 n f( Z, x) = θ(1 θ) =. (5) Note that the last equaton s equvalent to θ {,1}. Ths s an (n+2) (n+2) system of nonlnear equatons that can be solved for the statonary ponts n the optmzaton problem. As noted above, t s common to solve the phase stablty problem by seekng such statonary ponts, snce they may be useful n provdng ntal composton estmates for a possble new equlbrum phase or phases. The ntroducton of the bnary varable θ s a key feature of ths problem formulaton, as t provdes the capablty to combne any two dfferent thermodynamc models that mght be used n an asymmetrc model of phase behavor. Whle θ appears as a contnuous varable n Eq. (5), t wll be treated explctly as a bnary varable when ths system s solved, as explaned below. The (reduced) Gbbs energy functons are expressed here relatve to the pure components as lquds at system temperature and pressure. For a lqud phase n L E g = x x + g = 1 ( x) ln ( x ), (6) where g E ( x ) s the excess Gbbs energy as gven by some approprate model. The model used here s NRTL, g E n x jτ jg n j j= 1 ( x ) = x, (7) n = 1 xg Aj where Gj = exp( α jτ j ) and τ j =. In the examples below, the parameters A j and RT k = 1 k k α j are taken from varous lterature sources. Though we wll use NRTL here, the computatonal methodology descrbed can be appled n connecton wth any model for the excess Gbbs energy (e.g., UNIQUAC, Wlson, etc.). For a vapor phase,

8 6 n n n sat V ˆ L sat sat P g ( Z, x) = xln x + xln φ( Z, x ) x v ( P P ) + lnφ + ln, (8) = 1 = 1 = 1 P where ˆ φ ( Z, x ) s the fugacty coeffcent of component n the mxture at system T and P, L v s the molar volume of pure as a lqud at system T (assumed ndependent of pressure and evaluated at saturaton), a vapor at sat sat P s the vapor pressure of pure at system T, and φ s the fugacty coeffcent of pure as P and system T. The mxture fugacty coeffcents ˆ φ ( Z, x ) are determned here from a sat cubc equaton of state model of the form 3 2 f( Z, x) = Z + b( x) Z + c( x) Z + d( x ) =. (9) In the examples below, ether the Peng-Robnson (PR) or Soave-Redlch-Kwong (SRK) equaton of state s used, wth standard van der Waals mxng rules. An equaton of state can also be used to determne sat φ ; however, snce sat P s lkely to be relatvely small, t s often reasonable to assume that φ sat = 1, and that s what s done here. Sources of data for the vapor-phase model parameters are dscussed n the examples below. 3. Problem Solvng Methodology For solvng Eq. (5), an (n+2) (n+2) nonlnear system, for the statonary ponts n the phase stablty problem, we use a method based on nterval mathematcs, n partcular an nterval-newton approach combned wth generalzed bsecton (IN/GB). Ths s a determnstc technque that provdes a mathematcal and computatonal guarantee that all the statonary ponts are found, and thus that the global mnmum n the pseudo tangent plane dstance functon D s found. For general background on nterval mathematcs, ncludng nterval arthmetc, computatons wth ntervals, and nterval-newton methods, there are several good sources [2,24,25]. Detals of the basc IN/GB algorthm employed here are gven by Schnepper and Stadtherr [12] and Hua et al. [18]. An mportant feature of ths approach s that, unlke standard methods for nonlnear equaton solvng and/or optmzaton that requre a pont ntalzaton, the IN/GB methodology requres only an ntal nterval, and ths nterval can be suffcently large to enclose all feasble results. Thus, n the case of phase stablty analyss, all composton varables (mole fractons) x can be ntalzed to the nterval

9 7 [, 1]. For the vapor-phase compressblty factor Z an ntal nterval of [.5, 1] s used. The lower lmt of.5 effectvely elmnates any lqud-lke compressblty roots (the equaton of state s used to represent the vapor phase only) and n most crcumstances s reasonable for the relatvely low pressures at whch asymmetrc models are typcally used. If the pressure was so hgh that the vapor-phase compressblty was less than.5, then a symmetrc model usng an equaton of state for both vapor and lqud phases would lkely be used. Of course, the ntal nterval for Z can easly be modfed as desred to ft other crcumstances, as dscussed further n Secton 4.4 below. Fnally, for the bnary varable θ, the ntal nterval s set to [, 1]. Intervals are searched for statonary ponts usng powerful root ncluson tests based on the nterval-newton method. Ths method can determne wth mathematcal certanty f an nterval contans no statonary pont or f t contans a unque statonary pont. In the algorthm used here, we frst apply nterval-newton usng the pvotng precondtoner descrbed by Gau and Stadtherr [26]. If necessary, ths s followed by a root ncluson test usng the standard nverse-pont precondtoner [12]. If, after both of the nterval-newton tests, t cannot be proven that that nterval contans a unque statonary pont or no statonary pont, then the nterval s bsected and the root ncluson tests appled eventually to each subnterval. For the bnary varable θ, a specal bsecton rule s used. If θ s chosen by the IN/GB algorthm as the varable to be bsected, then t s bsected nto the degenerate (thn) ntervals [, ] and [1, 1]. Note that thus θ can be bsected only once. In ths way, θ s treated explctly as a bnary (rather than contnuous) varable n solvng the equaton system. On completon, the IN/GB algorthm wll have determned narrow enclosures of all the statonary ponts of D, ncludng the local and global optma, and thus the global mnmum can be readly determned. Alternatvely, IN/GB can be appled n connecton wth a branch-and-bound scheme, whch wll lead drectly to the global mnmum wthout fndng any of the other statonary ponts. However, as noted above, f the tested phase s not stable, knowledge of the statonary ponts can be useful for ntalzng phase splt computatons.

10 8 4. Results and Dscusson Usng ths problem solvng methodology, together wth the concept of pseudo tangent plane dstance, as ntroduced above, we now consder several test problems. In each case, an asymmetrc model s used, wth NRTL for the lqud phase, and a cubc equaton of state (PR or SRK) for the vapor phase. The frst two example problems are very smple and serve to demonstrate the key concepts of the methodology. 4.1 Problem 1 Ths problem nvolves the bnary mxture of tetrahydrofuran (component 1) and 2,2,4- trmethylpentane (component 2). Du et al. [27] performed vapor-lqud equlbrum measurements for ths system at P = 11.3 kpa, and then modeled ther results usng NRTL and SRK. They determned NRTL parameters by ft to expermental data, obtanng A 12 = cal/mol and A 21 = cal/mol, wth α 12 = α 21 fxed at.2. For SRK, they used the pure component propertes (crtcal temperature, crtcal pressure and acentrc factor) T c1 = K, P c1 = 5188 kpa, ω 1 =.2264, T c2 = K, P c2 = 2568 kpa, and ω 2 =.331. The bnary nteracton parameter was taken to be k 12 =. For ths example, we wll focus on phase stablty at T = 35 K. At ths temperature, from the physcal property models used by Du et al., the vapor pressure values are kpa, and the saturated lqud molar volumes are L v 1 = cm 3 /mol and sat P 1 = kpa and L v 2 = cm 3 /mol. sat P 2 = For ths model, the Gbbs energy surface g for a phase of composton x 1 (and x 2 = 1 x 1 ) s shown n Fgure 1. Note that ths surface s determned from the vapor-phase model g V and lqud-phase model g L usng g = mn[g V, g L ]. For an overall (feed) composton of x,1 =.6, the pseudo tangent plane dstance (PTPD) functon D s shown n Fgure 2. Ths shows that there are two statonary ponts n the PTPD, one at x 1 =.6 and θ = 1 (the feed pont; vapor phase) and the other at x 1.33 and θ =. Because D s negatve at ths latter pont, ths system s clearly not stable as a sngle phase. Snce θ = at the pont where D s negatve, ths ndcates that the total Gbbs energy can be lowered by ntroducng a lqud phase. The nterval-newton methodology outlned above was appled to compute the statonary ponts for several feed compostons for ths system, wth the results as shown n Table 1. The values reported

11 9 here (and n the subsequent tables) for the statonary ponts are pont approxmatons of very narrow ntervals that have been proven to contan a unque soluton to Eq. (5). The feed composton of x,1 =.3 s shown to be stable as a lqud (θ = ) and x,1 =.9 s shown to be stable as a vapor (θ = 1), snce for these cases the global mnmum of the PTPD s zero (at the feed pont). The feed compostons of x,1 =.5, x,1 =.6, and x,1 =.65 are shown to be not stable snce they exhbt negatve values for the PTPD. These results are consstent wth the phase dagram computed by Du et al. [27] usng the model descrbed above. For each feed pont consdered, the computaton tme requred to compute all the statonary ponts was approxmately.1 s. These tmes, and all computaton tmes reported below, were determned usng an Intel Pentum 4 CPU (3.2 GHz). 4.2 Problem 2 For ths problem we consder the bnary mxture of dmethyl carbonate (component 1) and cyclohexane (component 2) at T = K, the phase behavor of whch was measured and modeled by Cocero et al. [28]. NRTL parameters were determned from expermental data as A 12 = cal/mol and A 21 = cal/mol, wth α 12 = α 21 =.47. Vapor-phase nondealtes were modeled usng PR, wth pure component propertes T c1 = 539. K, P c1 = 463 kpa, ω 1 =.462, T c2 = K, P c2 = 47 kpa, and ω 2 =.212, and bnary nteracton parameter k 12 =. At the system temperature of K, the vapor pressure values used by Cocero et al. are volumes were taken to be L v 1 = cm 3 /mol and sat P 1 = 7.19 kpa and L v 2 = cm 3 /mol. sat P 2 = kpa. Lqud molar The nterval-newton methodology was appled to compute the statonary ponts for several feed compostons for ths system at P = 16.5 kpa (close to a homogeneous azeotrope at about 16.8 kpa), wth the results as shown n Table 2. Feeds at x,1 =.125, x,1 =.325, and x,1 =.75 are shown to be stable as sngle-phase lqud, vapor, and lqud, respectvely, snce n each case the global mnmum of the PTPD s zero. For the feeds at x,1 =.275 and x,1 =.45, negatve values of the PTPD are observed, so these are not stable as a sngle phase. These results are all n agreement wth the work of Cocero et al. [28]. For each feed pont consdered, the computaton tme requred to compute all the statonary ponts was agan approxmately.1 s.

12 1 4.3 Problem 3 In ths example, the system studed s the bnary mxture of 2,3-dmethyl-2-butene (component 1) and methanol (component 2). Expermental vapor-lqud equlbrum measurements were made for ths system recently by Uus-Kyyny et al. [29] at atmospherc pressure. The expermental pressure vared slghtly, but for most measurements was P = 11.2 kpa, and ths s the value used here. The expermental results were modeled by Uus-Kyyny et al. usng SRK as the vapor-phase model, and usng NRTL, Wlson, and UNIQUAC as dfferent lqud-phase models. Parameters n each of the lqud-phase models were estmated by mnmzng the sum of the absolute values of the relatve errors between the measured actvty coeffcent and the actvty coeffcent calculated from the model. To then test the lqud-phase models, Uus-Kyyny et al. used them to perform bubble-pont calculatons at each of the lqud-phase compostons x 1 on the expermental vapor-lqud envelope. Comparng the computed results for the vapor-phase compostons y 1 and temperature T to the expermental values showed that the average errors were y 1 =.59 and T =.14 K for the Wlson equaton, y 1 =.124 and T =.41 K for NRTL, and y 1 =.194 and T =.56 K for UNIQUAC. So the Wlson equaton apparently provded the best ft, followed by NRTL, and then UNIQUAC. We wll use the NRTL model here, wth the parameters A 12 /R = K and A 21 /R = K (α 12 = α 21 =.4), as determned by Uus-Kyyny et al. For SRK, the pure component propertes used are T c1 = 524. K, P c1 = 316 kpa, ω 1 =.2333, T c2 = K, P c2 = 896 kpa, and ω 2 =.5656, and the bnary nteracton parameter used s k 12 =. Pure component vapor pressures come from the Antone equaton (wth sat P n MPa and T n K), = exp B sat P A T + C, (1) wth A 1 = 6.574, B 1 = 25.8, C 1 = 64.19, A 2 = , B 2 = and C 2 = The lqud molar volumes used are L v 1 = cm 3 /mol and are exactly as gven by Uus-Kyyny et al. L v 2 = 4.7 cm 3 /mol. All the model parameters used here As a test problem for the phase stablty analyss procedure descrbed above, we frst chose a feed composton on the expermental vapor-lqud envelope, namely x,1 =.6233 and T = K. Our

13 11 expectaton was that ths feed would ether be just slghtly outsde the phase envelope predcted by NRTL/SRK, n whch case the feed would test as stable, or t would be just slghtly nsde the phase envelope, n whch case the feed would test as unstable, wth one statonary pont near the expermental vapor-phase composton (y 1 =.4758) showng a negatve value of PTPD. However, what we actually computed for ths feed composton was qute dfferent, as shown n the frst row of results n Table 3. Ths feed tested as not stable, but n addton to a statonary pont wth negatve PTPD near the expermental vapor-phase composton, there were also two other statonary ponts wth negatve PTPD. Usng NRTL/SRK to do a bubble-pont calculaton for ths lqud-phase composton (x 1 =.6233) shows that there s a soluton to the bubble-pont problem at y 1 =.4684 and T = K. Ths s very near the expermental values of.4758 ( y 1 =.74) and K ( T =.377 K), and so presumably ths s the soluton obtaned by Uus-Kyyny et al. Now testng ths result for phase stablty, wth the results shown n the second row of data n Table 3, as well as n the PTPD plot n Fgure 3, shows that ths soluton to the bubble pont problem s n fact not stable. Whle ths may be a mathematcally correct soluton to the bubble pont problem, t s not thermodynamcally correct, and t s not a pont on the vapor-lqud envelope predcted by NRTL/SRK, as was beleved by Uus-Kyyny et al. In fact, a phase splt calculaton, followed by phase stablty analyss of the results (thrd row of results n Table 3), shows that for a feed of x,1 =.6233 at T = K, NRTL/SRK predcts lqud-lqud equlbrum, wth one lqud phase of composton x 1 =.2973 and another lqud phase wth composton x 1 = In Fgure 4, we show the entre phase dagram for ths system when calculated from Uus-Kyyny et al. s SRK/NRTL model, along wth the expermental phase equlbrum data. These calculatons were valdated by usng the methodology for phase stablty analyss descrbed above. Stablty analyss results are gven for some selected ponts n Table 3 (for each feed tested n ths table the computaton tme was less than.5 s). Clearly the phase dagram calculated from the model does not closely match the expermental data. Expermentally, a homogeneous azeotrope s observed; however, the model predcts a heterogeneous azeotrope (VLLE lne). The model predcts that there should be no lqud phases observed wth compostons n the range from roughly x 1 =.298 to x 1 =.858 (end ponts on the VLLE lne), but n fact there are many expermental lqud-phase ponts seen wthn ths range. These dscrepances

14 12 between model and experment are planly due to the fact that Uus-Kyyny et al. s model predcts a lqud-lqud splt, whle expermentally ths apparently does not occur. The predcton of a lqud-lqud splt s due solely to the lqud-phase model used, and so t s clear that the NRTL parameters gven by Uus-Kyyny et al. are not correct. Uus-Kyyny et al. [29] were msled nto thnkng ther NRTL parameters were reasonable values because the phase stablty procedure they used (f any) was apparently not relable, and thus they dd not recognze that ther model predcted a lqud-lqud splt. Ths does not necessarly mean that NRTL s a poor choce of model for the lqud, only that a poor choce of parameters has been made. 4.4 Problem 4 Ths example s based on the work of Kang [3], who dd measurements and modelng of vaporlqud equlbrum for the system dchlorodfluoromethane (CFC-12) (component 1) and hydrogen fluorde (component 2) at T = K. Measurements of equlbrum pressure and average lqud phase composton show that ths system exhbts a maxmum-pressure heterogeneous azeotrope, and a model s presented whch appears to provde a good predcton of the pressure and the lqud-lqud phase splt at the azeotrope. Kang s model uses NRTL for the lqud phase. For the vapor phase, the base model s Peng-Robnson, but there s an addtonal contrbuton to the equaton of state to account for the assocaton of HF molecules, as descrbed by Łenko and Anderko [31]. For the computatons done here, we wll use the same lqud-phase model (NRTL) as Kang, but for the vapor phase, we wll use the base PR model only, wthout the assocaton terms. Thus we would expect our calculatons to match lqudlqud phase splt results computed by Kang, but not to match Kang s results for vapor-phase compostons or equlbrum pressures. For the NRTL model, the parameters obtaned by Kang are A 12 = cal/mol and A 21 = cal/mol, wth α 12 = α 21 =.425. For PR, the pure component propertes used are T c1 = 385. K, P c1 = 4129 kpa, ω 1 =.179, T c2 = 461. K, P c2 = 648 kpa, and ω 2 =.372, and the bnary nteracton parameter used s k 12 =. At the system temperature of K, the pure component vapor pressure values used by Kang are sat P 1 = kpa and sat P 2 = 144. kpa. Lqud molar volumes were taken to

15 13 be L v 1 = cm 3 /mol and gven by Kang. L v 2 = 14.9 cm 3 /mol, from the modfed Rackett equaton wth parameters Kang computes a maxmum-pressure heterogeneous azeotrope (VLLE lne) at about 868 kpa, wth the lqud splttng nto one lqud phase wth composton x 1.6 and other lqud phase wth composton x 1.9 (values estmated from a plot). Snce the lqud-phase model (NRTL) s pressurendependent, the same lqud-lqud splt should be observed for pressures above the maxmum-pressure heterogeneous azeotrope, where there should be lqud-lqud equlbrum only (no vapor). As an ntal test pont for our phase stablty analyss procedure, we chose a pressure (95 kpa) somewhat above the heterogeneous azeotrope, and a composton (x,1 =.54) n the mddle of the presumed two-phase (lqudlqud) regon. Results of applyng phase stablty analyss for ths pont are shown n the frst row of data n Table 4. Ths shows that actually ths pont s a stable lqud (sngle phase), and not n a two-phase regon as Kang s computatons would ndcate. Further analyss shows that there s n fact a soluton to the lqud-lqud equlbrum condtons (equal actvty condtons) wth x 1 =.652 for the frst lqud phase and x 1 =.8993 for the second lqud phase. Ths s clearly the phase splt found by Kang for the heterogeneous azeotrope. Phase stablty analyss shows, however, that ths s not a stable lqud-lqud phase splt, as ndcated n Table 4 (second row) and n the PTPD plot n Fgure 5. In fact, when Kang s model s correctly solved, by use of the relable phase stablty analyss procedure gven here, there are two VLLE lnes (one a heterogeneous azeotrope) found. At the heterogeneous azeotrope (slghtly lower n pressure than Kang s ncorrect azeotrope) and pressures above t, the lqud splts nto one lqud phase wth x 1 =.5566 and another lqud phase wth x 1 =.913. At the other VLLE lne (slghtly lower n pressure than the heterogeneous azeotrope) and pressures above t, the lqud splts nto one lqud phase wth x 1 =.647 and another lqud phase wth x 1 =.536. Both these phase splts were valdated usng phase stablty analyss, as ndcated n Table 4 (thrd and fourth lnes) for tral ponts (95 kpa) somewhat above the VLLE pressures. Snce ths type of phase behavor s not observed expermentally, and the predcton of a lqud-lqud splt s due solely to the lqud-phase model used, t s apparent that the NRTL parameters determned by Kang are not approprate. In a stuaton smlar to that seen n the prevous example, apparently the phase stablty procedure used by Kang (f any) was not relable, leadng

16 14 to an ncorrect lqud-lqud phase splt result, and msleadng Kang nto thnkng that hs NRTL parameters were reasonable. Fgures 6 and 7 show a phase dagram computed for the pressure range near the three-phase lnes. These calculatons were valdated by dong phase stablty analyss usng the methodology descrbed above. Stablty analyss results are gven for some selected ponts n Table 4 (for each feed tested n ths table the computaton tme was less than.1 s). It should be re-emphaszed that the vapor-phase model used for these computatons s not the same vapor-phase model used by Kang, and thus pressure and vapor-phase composton results are not drectly comparable to Kang s results. In fact, snce Kang s lqud-phase model s poor, as determned above, and our vapor-phase model s poor snce t does not account for the assocaton of HF, the phase dagram n Fgures 6 and 7 s perhaps best regarded as that of a hypothetcal system descrbed by the NRTL/PR model gven above. Another ssue that should be dscussed n connecton wth ths problem s the ntal nterval chosen for the vapor-phase compressblty Z. As dscussed above, we generally specfy a lower bound of.5 for Z. For standard cubc equatons-of-state, such as PR and SRK, at the low pressures for whch NRTL or other actvty coeffcent models are lkely to be used, ths s a very safe assumpton. However, for a strongly assocatng flud, such as HF, standard cubc equatons of state are nadequate, and the vapor-phase compressblty could easly be below.5 even at condtons for whch Z s close to one for most other compounds [31]. Snce, n our hypothetcal model, PR s used wthout vapor-phase assocaton, we were stll able to use a lower bound of.5. However, f the Łenko and Anderko model [31], whch does account for vapor-phase assocaton, was used, we would have needed to use a smaller lower bound for Z. We antcpate that, n most such cases, the modeler wll be able to set the lower bound on Z based on physcal knowledge of the system beng modeled. If ths s not possble, then the lower bound on Z can smply be set to the lowest feasble value consstent wth the EOS beng used. If ths s done, however, statonary ponts from the EOS model (θ = 1) mght be found that have a lqud-phase, not vapor-phase, compressblty. Snce the EOS s ntended to model the vapor phase only, these statonary ponts would need to be screened out. Ths can be done by solvng the EOS for the vapor-phase

17 15 compressblty at the composton of the statonary pont, and then elmnatng that statonary pont f t does not correspond to the vapor phase. 4.5 Problem 5 Ths problem nvolves the ternary mxture of methyl acetate (component 1), methanol (component 2) and water (component 3) at P = 11.3 kpa, the phase behavor of whch was measured and modeled by Martn and Mato [32]. For the lqud phase, the model used s NRTL-m, a varaton of NRTL due to Mato et al. [33] n whch α j s not an ndependent parameter but s nstead determned from 1 α j =, (11) 2 + GG wth G j as defned above. NRTL-m parameters were determned by ft to expermental data to be A 12 /R = K, A 21 /R = K, A 13 /R = K, A 31 /R = K, A 23 /R = K and A 32 /R = K. For the vapor phase, the model used s PR, wth pure component propertes T c1 = K, P c1 = 4691 kpa, ω 1 =.327, T c2 = K, P c2 = 896 kpa, ω 2 =.569, T c3 = K, P c3 = kpa, ω 3 =.348, and all bnary nteracton parameters set to zero. The lqud molar volumes used are j j L v 1 = cm 3 /mol, L v 2 = cm 3 /mol, and v L 3 = 18.7 cm 3 /mol. Pure component vapor pressures come from the Antone equaton (wth sat P n kpa and T n K), wth A 1 = , B 1 = , C 1 = , A 2 = , B 2 = , C 2 = 49.55, A 3 = , B 3 = , and C 3 = , as gven by Martn and Mato (whose expresson for the Antone equaton contans a typographcal error). The nterval-newton methodology was appled to compute the statonary ponts for several feed compostons for ths system, wth the results as shown n Table 5. Each feed pont tested s one on the expermental vapor-lqud envelope, and thus we would expect t to be ether just outsde or just nsde the phase envelope predcted by the model. If the feed s just nsde the predcted phase envelope, t should test as not stable, wth a statonary pont havng a negatve PTPD near the composton of the phase on the other sde of the expermental te lne; or, f the feed s just outsde the predcted phase envelope, then t should test as stable. For each feed pont tested, ths expectaton s met. For example, an expermental te lne at T = K connects a lqud phase of composton (.472,.323,.25) and a vapor phase of composton (.663,.263,.74). Usng the lqud phase as the feed pont, as shown n the frst row of

18 16 data n Table 5, the stablty test ndcates that t s not stable, and there s a statonary pont wth a negatve PTPD at (.6543,.2719,.739) and θ = 1 (vapor), very near the expermental vapor phase composton. For each feed tested, the computaton tme was less than.2 s. 4.6 Problem 6 In ths problem, we consder the fve-component mxture of n-propanol (component 1), n-butanol (component 2), benzene (component 3), ethanol (component 4) and water (component 5). For the lqud phase, NRTL s used wth the parameters as gven by Tesser et al. [15]. For the vapor phase, PR s used wth the pure component propertes (T c, P c and ω) gven n Table 6. Snce we are prmarly nterested n ths system as a test of computatonal performance, and not n the accuracy of the model, all bnary nteracton parameters k j are assumed to be zero. The values of the bnary nteracton parameters do not affect the mathematcal form of the equaton system to be solved, only the coeffcent values, and thus by assumng all k j =, the problem s not necessarly made any more or less dffcult. Table 6 also provdes, for each component, the lqud molar volumes used, as well as constants for the Antone equaton (wth sat P n mmhg and T n K). We used the nterval-newton methodology to determne statonary ponts for several feed mxtures for ths system. Table 7 gves selected results, at dfferent T and P, for feeds wth composton (.148,.52,.52,.1,.18). The frst row of data, for T = K and P = kpa, corresponds to condtons used n one of the test problems gven by Tesser et al. [15], who correctly assumed that there would be no vapor phase at these condtons. The lqud-phase statonary pont results correspond drectly to those gven by Tesser et al. Computaton tme requrements are also gven n Table 7, and range from 7.3 to 29.9 s, dependng prmarly on the number of statonary ponts that exst. 4.7 Problem 7 Ths fnal example s ntended prmarly to nvestgate the computatonal performance of the nterval-newton methodology as problem sze (number of components) contnues to ncrease. Hypothetcal mxtures of up to eght components are used. The NRTL A j and α j parameters for these mxtures are gven n Table 8. For the vapor phase, PR s used wth the crtcal propertes, acentrc factor,

19 17 lqud molar volumes, and Antone equaton coeffcents (for sat P n mmhg and T n K) gven n Table 6 for components 1 5, and n Table 9 for components 6 8 (note that though the pure component data used for components 1 5 are the same as used n Problem 6, the NRTL parameters used for components 1 5 are not the same as n Problem 6). The sx component mxture studed nvolves components 1 6 and has a feed composton of (.148,.52,.4,.1,.18,.12). The seven component mxture comprses components 1 7 wth a feed composton of (.148,.52,.3,.1,.18,.12,.1). The eght component mxture conssts of components 1 8 and has a feed composton of (.148,.52,.3,.1,.18,.12,.6,.4). Table 1 gves results at selected temperatures and pressures for the three feeds and problem szes consdered. These results are representatve of the range of computaton tme requrements encountered n solvng these problems for several other temperatures and pressures. For each problem sze, computaton tmes range over roughly an order of magntude, wth hgher computaton tme not necessarly correspondng to a larger number of statonary ponts. These results also reflect the exponental complexty that may be assocated wth determnstc global optmzaton (n general, an NPhard problem). Whle mult-hour computaton tmes, such as seen n the next to last problem n Table 1 for the eght component problem, may seem large, at least n ths context, ths tme can be well spent f t means avodng the computaton of a phase equlbrum wth the wrong number of phases. 5. Concludng Remarks We have addressed here the problem of determnng phase stablty, n a relable and determnstc way, for the case n whch asymmetrc modelng (dfferent models for vapor and lqud phases) s used. To do ths, we ntroduced a new type of tangent plane dstance functon, whch uses a bnary varable to account for the presence of dfferent lqud and vapor phase models. To then solve the problem determnstcally, we used an nterval-newton approach, whch provdes a mathematcal and computatonal guarantee that the global mnmum, as well as all the statonary ponts, n the tangent plane dstance functon are found. The new methodology was successfully tested usng several examples, nvolvng as many as eght components, wth NRTL as the lqud phase model and a cubc equaton of

20 18 state as the vapor phase model. In two cases, publshed phase equlbrum computatons were found to be ncorrect (not stable). The guarantee that the phase stablty problem s correctly solved clearly comes at the expense of addtonal computaton tme, whch may become large as problem szes become large. Thus, ths procedure s not well suted for stuatons n whch t would be used repeatedly nsde some other teratve calculaton, such a process smulator, or a stand-alone code for phase equlbrum. Instead, the most approprate use for ths procedure s as a fnal valdaton step, to determne whether the process smulator, or stand-alone phase equlbrum code, has n fact reached a correct result. The use of determnstc phase stablty analyss n ths context has recently been demonstrated by Burgos-Solórzano et al. [21], who also show how the statonary pont results can be used to provde correctve feedback n the case that a phase equlbrum result s determned to be ncorrect. Because of the addtonal computatonal expense, a modeler may ultmately need to consder the trade-off between the computaton tme expense and the rsk of gettng the wrong answer to the phase equlbrum problem of nterest. As the example problems consdered demonstrate, ths rsk s not nsgnfcant. The addtonal computaton tme used to valdate a result may be especally well spent n msson crtcal stuatons, n whch errors n the number or composton of equlbrum phases present s not tolerable. Acknowledgement Ths work has been supported n part by the donors of The Petroleum Research Fund, admnstered by the ACS, under Grant AC9, by the Natonal Scence Foundaton Grant EEC CRCD, by the Envronmental Protecton Agency Grant R and by The State of Indana 21 st Century Fund under Grant #

21 19 Lst of Symbols A A j B C D D g G j k j n P R T v s x Z Antone equaton coeffcent NRTL parameter Antone equaton coeffcent Antone equaton coeffcent tangent plane dstance pseudo tangent plane dstance molar Gbbs energy NRTL parameter bnary nteracton parameter number of components pressure gas constant temperature molar volume gradent of g composton (mole fracton) vector compressblty Greek letters α j θ τ j φ ˆ φ ω NRTL parameter bnary varable used to defne pseudo tangent plane dstance NRTL parameter fugacty coeffcent mxture fugacty coeffcent acentrc factor

22 2 Subscrpt c ndcates crtcal property ndcates evaluaton at feed composton Superscrpts E L sat V ndcates an excess property ndcates quantty for the lqud phase ndcates evaluaton at saturaton ndcates quantty for the vapor phase

23 21 References [1] L. E. Baker, A. C. Perce, K. D. Luks, Soc. Petrol, Eng. J. 22 (1982) [2] M. L. Mchelsen, Flud Phase Equlb. 9 (1982), [3] C. M. McDonald, C. A. Floudas, AIChE J. 41 (1995) [4] C. M. McDonald, C. A. Floudas, Comput. Chem. Eng. 19 (1995) [5] C. M. McDonald, C. A. Floudas, Ind. Eng. Chem. Res. 34 (1995) [6] C. M. McDonald, C. A. Floudas, Comput. Chem. Eng. 21 (1997) [7] C. A. Floudas, V. Vsweswaran, Comput. Chem. Eng. 14 (199) [8] C. A. Floudas, V. Vsweswaran, J. Optm. Theory Appl. 78 (1993) [9] J. E. Falk, R. M. Soland, Manag. Sc. 15 (1969) [1] S. T. Hardng, C. A. Floudas, AIChE J. 46 (2) [11] C. S. Adjman, S. Dallwg, C. A. Floudas, A. Neumaer, Comput. Chem. Eng. 22 (1998) [12] C. A. Schnepper, M. A. Stadtherr, Comput. Chem. Eng. 2 (1996) [13] M. A. Stadtherr, C. A. Schnepper, J. F. Brennecke, AIChE Symp. Ser. 91(34) (1995) [14] K. I. M. McKnnon, C. G. Mllar, M. Mongeau, n State of the Art n Global Optmzaton: Computatonal Methods and Applcatons (C. A. Floudas, P. Pardalos, Eds.), Kluwer Academc Publshers, Dordrecht, The Netherlands (1996), pp [15] S. R. Tesser, J. F. Brennecke, M. A. Stadtherr, Chem. Eng. Sc. 55 (2) [16] Z. Hua, J. F. Brennecke, M. A. Stadtherr, Flud Phase Equlb. 116 (1996) [17] J. Z. Hua, J. F. Brennecke, M. A. Stadtherr, Comput. Chem. Eng. 2 (1996) S395-S4. [18] J. Z. Hua, J. F. Brennecke, M. A. Stadtherr, Ind. Eng. Chem. Res. 37 (1998) [19] G. Xu, J. F. Brennecke, M. A. Stadtherr, Ind. Eng. Chem. Res. 41 (22) [2] E. Hansen, G. W. Walster, Global Optmzaton Usng Interval Analyss, Marcel Dekker, New York, 24. [21] G. I. Burgos-Solórzano, J. F. Brennecke, M. A. Stadtherr, Flud Phase Equlb. 219 (24)

24 22 [22] G. Xu, A. M. Scurto, M. Caster, J. F. Brennecke, M. A. Stadtherr, Ind. Eng. Chem. Res. 39 (2) [23] A. M. Scurto, G. Xu, J. F. Brennecke, M. A. Stadtherr, Ind. Eng. Chem. Res. 42 (23) [24] R. B. Kearfott, Rgorous Global Search: Contnuous Problems, Kluwer Academc Publshers, Dordrecht, The Netherlands (1996). [25] A. Neumaer, Interval Methods for Systems of Equatons, Cambrdge Unversty Press, Cambrdge, UK (199). [26] C.-Y. Gau, M. A. Stadtherr, Comput. Chem. Eng. 26 (22) [27] T.-B. Du, M. Tang, Y.-P Chen, Flud Phase Equlb. 192 (21) [28] M. J. Cocero, F. Mato, I. García, J. C. Cobos, H. V. Kehaan, J. Chem. Eng. Data 34 (1989) [29] P. Uus-Kyyny, J.-P. Pokk, Y. Km, J. Attamaa, J. Chem. Eng. Data 49 (24) [3] Y. W. Kang, J. Chem. Eng. Data 43 (1998) [31] M. Łenko, A. Anderko, AIChE J. 39 (1993) [32] M. Martn, R. B. Mato, J. Chem. Eng. Data 4 (1995) [33] F. A. Mato, R. B. Mato, F. Mato, Ind. Eng. Chem. Res. 28 (1989)

25 23 Table 1. Results for selected feed compostons n Problem 1. Feed ( x,1, x,2 ) Statonary Ponts a (x 1, x 2 ); θ ; Z (.3,.7) (.3,.7); (.5696,.434); 1;.9641 (.5,.5) (.5,.5); (.7229,.2771); 1;.9684 (.6,.4) (.6,.4); 1;.965 (.3336,.6664); (.65,.35) (.65,.35); 1;.9664 (.3953,.647); (.9,.1) (.9,.1); 1;.9729 (.811,.189); PTPD ( D ~ ) Result Stable (L) Not Stable Not Stable Not Stable Stable (V) a A value of the compressblty Z s gven only for vapor-phase statonary ponts (θ = 1).

26 24 Table 2. Results for selected feed compostons n Problem 2. Feed ( x,1, x,2 ) Statonary Ponts a (x 1, x 2 ); θ ; Z (.125,.875) (.125,.875); (.268,..732); 1;.9917 (.275,.725) (.275,.725); 1;.9917 (.1351,.8649); (.325,.675) (.325,.675); 1;.9917 (.2789,.7211); (.45,.55) (.45,.55); (.3474,.6526); 1;.9917 (.75,.25) (.75,.25); (.3927,.673); 1;.9917 PTPD ( D ~ ) Result Stable (L) Not Stable Stable (V) Not Stable Stable (L) a A value of the compressblty Z s gven only for vapor-phase statonary ponts (θ = 1).

27 25 Table 3. Results for selected feed compostons and temperatures n Problem 3. Feed ( x,1, x,2 ) T (K) Statonary Ponts a (x 1, x 2 ); θ ; Z PTPD ( D ~ ) Result (.6233,.3767) (.6233,.3767); (.4678,.5322); 1; Not Stable (.2923,.777);.6359 (.8551,.1449);.484 (.6233,.3767) (.6233,.3767); (.4684,.5316); 1;.9692 Not Stable (.2914,.786);.6428 (.8559,.1441);.4878 (.85822,.14178) (.85822,.14178); (.2973,.7297); Stable (LL) (.4691,.539); 1; (.6125,.3875);.5537 (.47,.53) (.47,.53); 1;.9692 (.393,.697);.4763 Stable (V) (.5913,.487);.8723 (.8618,.1382);.1917 (.46,.54) (.46,.54); 1;.9695 (.2448,.7552);.2933 Not Stable (.763,.2397);.1694 (.7954,.246);.1692 (.48,.52) (.48,.52); 1;.9689 (.8845,.1155);.144 Not Stable a A value of the compressblty Z s gven only for vapor-phase statonary ponts (θ = 1).

28 26 Table 4. Results for selected feed compostons and pressures n Problem 4. Feed ( x,1, x,2 ) P (kpa) Statonary Ponts a (x 1, x 2 ); θ ; Z PTPD ( D ~ ) Result (.54,.46) (.54,.46); Stable (L) 95. (.8151,.1849); 1; (.2247,.7753);.64 (.649,.9351);.3998 (.7796,.224);.2569 (.8985,.115);.121 (.65255, ) (.652,.9348); Not Stable 95. (.8993,.17); (.8152,.1848); 1; (.2228,.7772);.5488 (.5446,.4554); ).8581 (.7762,.2238);.1485 ( , ) (.5566,.4434); Stable (LL) 95. (.913,.987); (.8156,.1844); 1; (.2181,.7819);.7156 (.659,.9341);.248 (.7672,.2328);.18 ( , ) (.647,.9353); Stable (LL) 95. (.536,.464); (.815,.185); 1; (.2264,.7736);.5776 (.7826,.2174);.2775 (.8978,.122);.155 (.816,.184) (.816,.184); 1;.827 Stable (V) 94. (.2133,.7867);.9177 (.667,.9333);.4454 (.57,.43);.1125 (.7569,.2431);.2413 (.933,.967);.2516 (.8,.2) (.8,.2); 1;.821 Not Stable 94. (.47,.953);.7318 a A value of the compressblty Z s gven only for vapor-phase statonary ponts (θ = 1).

29 27 Table 5. Results for selected feed compostons and temperatures n Problem 5. Feed ( x,1, x,2, x,3 ) T (K) Statonary Ponts a,b (x 1, x 2, x 3 ); θ ; Z PTPD ( D ~ ) Result (.472,.323,.25) (.472,.323,.25); Not Stable (.6543,.2719,.739); 1; (.11,.827,.72) (.11,.827,.72); Stable (L) (.261,.7144,.255); 1; (.788,.75,.137) (.788,.75,.137); 1;.9743 Stable (V) (.4547,.846,.466);.3545 (.152,.433,.415) (.152,.433,.415); Not Stable (.4983,.3946,.171); 1; (.762,.136,.12) (.762,.136,.12); Not Stable (.775,.1546,.74); 1; a Mole fractons may not sum precsely to one, due to roundng of computer output durng transcrpton to the table. b A value of the compressblty Z s gven only for vapor-phase statonary ponts (θ = 1).

30 28 Table 6. Pure component physcal property data used n Problem 6. Antone equaton coeffcents are for the case of P n mmhg and T n K. sat = 1 = 2 = 3 = 4 = 5 T c (K) P c (kpa) ω L v (cm 3 /mol) A B C

31 29 Table 7. Results for selected feeds n Problem 6. The feed composton n each case s (.148,.52,.52,.1,.18). Feed T (K) P (kpa) Statonary Ponts a,b (x 1, x 2, x 3, x 4, x 5 ); θ ; Z PTPD ( D ~ ) CPU tme c (s) Result (.148,.52,.52,.1,.18); (.343,.1,.681,.979,.1857); 1; Not Stable (.1625,.532,.276,.1284,.3854);.8219 (.1633,.564,.3969,.1161,.2674);.111 (.26,.6,.19,.383,.9332);.8658 (.796,.263,.784,.577,.524); (.148,.52,.52,.1,.18); 1;.9751 (.3353,.2881,.2273,.528,.964); Not Stable (.148,.52,.52,.1,.18); 1;.9764 (.3221,.2757,.2584,.529,.91); Stable (V) (.148,.52,.52,.1,.18); (.859,.67,.4516,.1297,.3261); 1; Not Stable (.1647,.548,.3722,.1169,.2914);.53 (.52,.3,.81,.542,.8845); (.148,.52,.52,.1,.18); 1;.8526 (1655,.149,.637,.436,.383); Not Stable a Mole fractons may not sum precsely to one, due to roundng of computer output durng transcrpton to the table. b A value of the compressblty Z s gven only for vapor-phase statonary ponts (θ = 1). c Intel Pentum 4 CPU (3.2 GHz)