Relable omputato of Mxture rtcal Pots Beto A. Strad, Joa. Breecke, James P. Koh ad Mark A. Stadtherr epartmet of hemcal Egeerg Uversty of Notre ame Notre ame, IN 46556 USA ecember 999 (revsed, July 000) Topcal Headg: Thermodyamcs Keywords: rtcal Pots, Hgh-Pressure Phase Equlbrum, Iterval Aalyss, omputatoal Methods Author to whom all correspodece should be addressed. ax: (9) 63-8366; E-mal: markst@d.edu
Abstract The determato of crtcal pots of mxtures s mportat for both practcal ad theoretcal reasos the modelg of phase behavor, especally at hgh pressure. We preset here the frst completely relable method for locatg all the crtcal pots of a gve mxture. The method also verfes the oexstece of a crtcal pot f a mxture of a gve composto does ot have oe. The methodology used s based o terval aalyss, partcular a terval Newto/geeralzed bsecto algorthm that provdes a mathematcal ad computatoal guaratee that all mxture crtcal pots are located. The procedure s talzato-depedet, ad thus requres o a pror kowledge of the umber of mxture crtcal pots or ther approxmate locatos. The techque s llustrated usg several example problems volvg cubc equato of state models; however, the techque s geeral purpose ad ca be appled coecto wth other thermodyamc models.
Itroducto As revewed more detal by Sadus (994), the determato of crtcal pots of mxtures s mportat for a varety of both practcal ad theoretcal reasos, ad thus has bee wdely studed (e.g., eters ad Scheder, 976; Boberg ad Whte, 96; Hurle et al., 977a,b; Nagaraa et al., 99a,b; Stockfleth ad ohr, 998; Spear et al., 97; Eck et al., 985; Greves ad Thodos, 96; Muoz ad hmowtz, 993; Tea ad Kropholler, 975; Tea ad Rowlso, 973), especally the cotext of hydrocarbo producto ad processg (e.g., Hedema, 975, 983; Baker ad Luks, 980; Luks et al., 987; Boshkov ad Yelash, 967; Rochocz et al., 997). or example, kowledge of vapor-lqud crtcal pots s useful determg whether retrograde codesato or evaporato s possble. Kowledge of crtcal states ca also be used schemes for classfyg the overall phase behavor of mxtures, as exemplfed by the well-kow scheme of va Koyeburg ad Scott (980). Measuremet of crtcal pheomea s ofte expesve, thus crtcal pots are frequetly calculated from models, typcally equato-of-state (EOS) models. A mxture of a gve composto may have oe, more tha oe, or o crtcal pots. learly t s desrable that methods used to compute crtcal pots be capable of relably fdg all crtcal pots, or verfyg wth certaty that there are oe the doma of terest. Ths task s made partcularly dffcult by the complex olear form of the crtcalty codtos, especally sce there s geerally o kowledge a pror cocerg the umber of crtcal pots to be foud, or deed f there are ay. Perhaps the most relable ad wdely used techques curretly used for computato of crtcal pots are the methods of Hcks ad Youg (977) ad Hedema ad Khall (980). The approach used by Hcks ad Youg (977), whch s based o searchg for sg chages the crtcalty codtos, ca be very relable, s capable of fdg multple crtcal pots, ad
requres o talzato, oly the specfcato of tervals of temperature ad volume over whch to search. However, as oted by Hcks ad Youg (977), t may mss solutos to the crtcalty codtos ad thus provdes o guaratee that all crtcal pots wll be foud. The method of Hedema ad Khall (980), whch employs two ested sgle-varable terato loops usg a local equato solver (Newto-Raphso), s very relable for locatg a sgle hghtemperature vapor-lqud crtcal pot, requrg essetally o talzato (sce ay suffcetly hgh guesses of temperature ad volume wll suffce, ad such guesses ca easly be geerated automatcally). However, f there are other crtcal pots to be foud, ths must be doe by choosg as the tal guess a volume ear the desred root (Hedema ad Khall, 980). Thus, some a pror kowledge of the umber ad approxmate locato of the mxture crtcal pots s eeded, wthout whch there s o guaratee that all crtcal pots wll be foud. or example, f t s kow that there are exactly three crtcal pots, the oe ca try as may dfferet tal guesses as eeded to fd all three, stoppg whe the thrd has bee foud. However, f t s ot kow advace how may crtcal pots to look for, the oe would ot be able to tell whe to stop tryg dfferet tal guesses, as the ext oe tred could coverge to a crtcal pot that had ot bee foud before. To estmate the umber ad locato of crtcal pots, Hedema ad Khall (980) ad others suggest usg a approach based o lookg for sg chages the crtcalty codtos; however, as oted by Hcks ad Youg (977), ths type of approach may fal to detfy all crtcal pots. A excellet mplemetato of the method of Hedema ad Khall (for fdg a sgle hgh-temperature vapor-lqud crtcal pot oly) s avalable the route RITPT that s a part of the IV-SEP smulato package (Hytoft ad Ga, 996; Mchelse, 984; Mchelse ad Hedema, 98). We descrbe here the frst completely relable approach for locatg all crtcal pots of mxtures, ad for verfyg whe oe exst. The techque s mathematcally ad
computatoally guarateed to fd (or, more precsely, to eclose wth a very arrow terval) ay ad all solutos of the crtcalty codtos. Lke the method of Hcks ad Youg (977), ad ulke the method of Hedema ad Khall (980), the ew method requres o talzato ad o a pror kowledge of the umber of crtcal pots to be foud, requrg oly the specfcato of tervals of temperature ad volume over whch to search. Ulke the method of Hcks ad Youg (977), however, the ew method descrbed here provdes a guaratee that o solutos to the crtcalty codtos wll be mssed. We wll demostrate the techque here usg a geeralzed cubc EOS model. However, the methodology s geeral purpose ad ca be appled coecto wth other models of phase behavor. Problem ormulato The crtcalty codtos ca be stated terms of the Gbbs free eergy G or the Helmholtz free eergy A. However, for a pressure-explct EOS, a formulato based o A, wth temperature ad volume as depedet varables, s geerally preferred. Based o a Taylor seres expaso of A, ad cosderg the secod ad thrd order terms, Hedema ad Khall (980) formulated the crtcalty codtos for a mxture of compoets as k I Eq. (), the matrx Q has elemets Q 0 () A 0. () k k Q A A T, V 3
where ad dcate compoet mole umbers, ad (,,..., ) T represets a ozero perturbato the compoet mole umbers, ad Eq. (), A k 3 A. k T, V To assure that t s ozero, the perturbato vector s ormalzed T 0. (3) The dervatves A ad A k are evaluated at the gve mxture composto 0 (,0,,0,...,,0 ) T. Thus, Eqs. ()-(3) represet a system of equatos the varables T, V, ad, the soluto of whch gves the crtcal temperature T c, crtcal volume V c ad a correspodg. The crtcal pressure P c ca the be computed drectly from the pressureexplct EOS. The Hedema-Khall formulato of the crtcalty codtos, as summarzed above, s etrely equvalet to the ofte see determat-based formulatos. It has bee wdely used due to ts clear theoretcal foudato ad because t s relatvely smple to mplemet computatoally, ad s the formulato we wll use here for the computato of mxture crtcal pots. It should be oted that the crtcalty codtos solved, whether ther determat form or the Hedema-Khall form, may yeld pots that are stable, ustable or metastable. Thus, computed solutos of the crtcalty codtos should be checked for phase stablty. A completely relable method for phase stablty aalyss usg cubc EOS models, also based o terval aalyss, has bee preseted ad demostrated by Hua et al. (996,998). The EOS model used here s a geeralzed cubc EOS RT P v b a ( v b)( v b ) 4
5 ad stadard va der Waals mxg rules wth a bary teracto parameter k are used (see Appedx for otatoal detals). Of course, a cubc EOS s aalytc ad caot gve the theoretcally correct behavor at the crtcal pot. The rego mmedately aroud a crtcal pot ca be accurately descrbed by scalg laws that clude crtcal expoets (e.g., Levelt Segers, 99). Nevertheless, EOS models are useful provdg a geeral descrpto of hgh pressure phase behavor, ad crtcal pots computed from EOS models are useful detfyg approprate operatg codtos a varety of process applcatos. or the case of the geeralzed cubc EOS gve above, Eqs. () ad () ca be expressed usg the expressos of Mchelse ad Hedema (98). Eq. () becomes A ( ) ( ) 0 6 5 3 a a b a N y RT α α (4) ad Eq. () becomes k k k A 0 ) 3 ) )( ( (3 ) ( ) ( 3 6 4 3 6 3 3 3 a b a N y RT α (5) The otato used these equatos s explaed detal the Appedx. We ote that Eq. (5) s a corrected form of the equato gve by Mchelse ad Hedema (98), whch cotas
a error the last term. Methodology We apply here terval mathematcs, partcular a terval Newto/geeralzed bsecto (IN/GB) techque, to fd, or, more precsely, to fd very arrow eclosures of, all solutos of a olear equato system, or to demostrate that there are oe. Recet moographs whch troduce terval computatos clude those of Neumaer (990), Hase (99) ad Kearfott (996). The algorthm used here has bee descrbed by Hua et al. (998), ad gve more detal by Schepper ad Stadtherr (996). Properly mplemeted, ths techque provdes the power to fd, wth mathematcal ad computatoal certaty, eclosures of all solutos of a system of olear equatos or to determe wth certaty that there are oe, provded that tal upper ad lower bouds are avalable for all varables (Neumaer, 990; Hase, 99, Kearfott, 996). Ths s made possble through the use of the powerful exstece ad uqueess test provded by the terval Newto method. Our curret mplemetato of the IN/GB method for the crtcal pot problem s based o approprately modfed routes from the ORTRAN-77 packages INTBIS (Kearfott ad Novoa, 990) ad INTLIB (Kearfott et al., 994). The key deas of the methodology used are summarzed very brefly here. osder the soluto of a olear equato system f(x) 0, where x X (0) (terval quattes are dcated upper case, pot quattes lower case). The soluto algorthm s appled to a sequece of tervals, begg wth the tal terval vector X (0) specfed by the user. Ths tal terval ca be chose to be suffcetly large to eclose all physcally feasble behavor. or a terval X (k) the sequece, the frst step the soluto algorthm s the 6
fucto rage test. Here a terval exteso (X (k) ) of the fucto f(x) s calculated. A terval exteso provdes upper ad lower bouds o the rage of values that a fucto may have a gve terval. It s ofte computed by substtutg the gve terval to the fucto ad the evaluatg the fucto usg terval arthmetc; ths s the approach used here, wth terval arthmetc mplemeted usg the INTLIB package (Kearfott et al., 994). If there s ay compoet of the terval exteso (X (k) ) that does ot cota zero, the we may dscard the curret terval X (k), sce the rage of the fucto does ot clude zero aywhere ths terval, ad thus o soluto of f(x) 0 exsts ths terval. Otherwse, f zero s cotaed (X (k) ), the processg of X (k) cotues. The ext step s to apply the terval Newto test to the curret terval (X (k) ). Ths ( k) ( k) ( k) ( k) requres solvg the system of lear terval equatos ( X )( N x ) f ( x ) for a ew terval, the mage N (k). Here (X (k) ) s a terval exteso of the Jacoba of f(x) over X (k), ad x (k) s a pot X (k), usually take to be the mdpot. omparso of the curret terval ad the mage provdes a powerful exstece ad uqueess test (Neumaer, 990; Kearfott, 996). If N (k) ad X (k) have a ull tersecto, ths s mathematcal proof that there s o soluto of f(x) 0 X (k). If N (k) s a proper subset of X (k), the ths s mathematcal proof that there s a uque soluto of f(x) 0 X (k). If ether of these two codtos s true, the o coclusos ca be made about the umber of solutos the curret terval. However, t s kow (Kearfott, 996) that ay solutos that do exst must le the tersecto of N (k) ad X(k). If ths tersecto s suffcetly smaller tha the curret terval, oe ca proceed by reapplyg the terval Newto test to the tersecto. Otherwse, the tersecto s bsected, ad the resultg two tervals added to the sequece of tervals to be tested. These are the 7
basc deas of a terval Newto/geeralzed bsecto (IN/GB) method. Whe mache computatos wth terval arthmetc operatos are doe, as the procedures outled above, the edpots of a terval are computed wth a drected outward roudg. That s, the lower edpot s rouded dow to the ext mache-represetable umber ad the upper edpot s rouded up to the ext mache-represetable umber. I ths way, through the use of terval, as opposed to floatg pot, arthmetc ay potetal roudg error problems are elmated. Overall, the IN/GB method descrbed above provdes a procedure that s mathematcally ad computatoally guarateed to eclose all solutos to the olear equato system or to determe wth certaty that there are oe. It should be emphaszed that the eclosure, exstece, ad uqueess propertes dscussed above, whch are the bass of the IN/GB method, ca be derved wthout makg ay strog assumptos about the fucto f(x) for whch roots (zeros) are sought. The fucto must have a fte umber of roots over the search terval of terest; however, o specal propertes such as covexty or mootocty are requred. It s assumed that f(x) s cotuous; however, t eed ot be cotuously dfferetable. Istead, as show by Neumaer (990), f(x) eed oly be Lpschtz cotuous over the terval of terest; thus, fuctos wth slope dscotutes ca be hadled. I order to apply the method, t must be possble to determe a terval exteso of the Jacoba matrx (or of the Lpschtz matrx f f(x) s ot cotuously dfferetable). I geeral, ths requres havg a aalytc expresso for f(x); thus, the terval approach s ot sutable f f(x) s some kd of black box fucto. Oe dffculty wth the terval Newto approach s that f a soluto occurs at a sgular pot (.e., where the Jacoba of f(x) s sgular), the t s ot possble to obta the result that detfes a uque soluto. or such a case, the evetual result from the IN/GB algorthm wll be a very arrow terval for whch all that ca be cocluded s that t may cota oe or more solutos. I other words, the algorthm wll ot mss the soluto (so 8
the guaratee to eclose all solutos remas), but rather, wll eclose t wth a arrow terval whch ca the be examed usg a alteratve methodology (e.g., Kearfott et al., 000). Results ad scusso To solve the crtcalty codtos, Eqs. (3)-(5), for a geeralzed cubc EOS, the IN/GB method, as outled above, was used. Use of scaled volume ad temperature varables was foud to be both coveet ad effectve. The volume was scaled usg the va der Waals volume parameter b the EOS, ad the tal terval used was v/b [., 4.0]. The temperature was scaled by a factor of 00 K, ad the tal terval used was T/00 [0.55, 4.0]. These tal tervals were cosdered reasoable for the rage of example systems cosdered (Hedema ad Khall, 980). or the mole umber perturbato varables, the tal tervals used were [0, ] ad [-, ]. Restrctg to be postve elmates a duplcate set of roots whch dffer oly that the mole umber perturbatos are opposte sg (ths occurs sce both ad satsfy the crtcalty codtos). Because s ormalzed to a legth of oe by Eq. (3), the rage of the mole umber perturbato varables ca be safely restrcted to values of magtude less tha oe. or each of the example problems cosdered below, all of the solutos to the crtcalty codtos were foud for mxtures of specfed composto. The pure compoet propertes (crtcal temperatures, crtcal pressures ad acetrc factors) requred the EOS models were take from Red et al. (987). I the tables of results, we preset the computed mxture crtcal states (T c, v c, P c ) ad the correspodg mole umber perturbato varables. The crtcal pressures are computed from the EOS after the crtcalty codtos have bee solved for the crtcal volumes ad temperatures. It should be oted that whle pot approxmatos of T c, v c, 9
ad are gve the tables here, we have actually determed verfed eclosures of each root. Each such eclosure s a extremely arrow terval kow to cota a uque root, based o the terval Newto uqueess test descrbed above. or example, the frst crtcal temperature reported Table s gve as T c 90.8 K, but s actually computed as T c /00 [0.95435690884, 0.95435690885] K, ad ths terval s kow to eclose a uque crtcal temperature root. Whle ths level of precso s certaly ot eeded ths applcato, because the EOS parameters are kow oly to much lower precso, t s dcatve of the potetal power of ths problem-solvg tool. The degree of precso desred the fal results has lttle mpact o the overall computatoal effort. Several of the crtcal pots foud occur at egatve pressures. A codesed phase at egatve pressure s metastable ad ca be thought of as occurrg whe the materal s teso (.e., stretched). Though metastable, these states ca be observed expermetally, ad, fact occur ature (e.g., the flow of sap plats). rtcal pots at egatve pressure correspod to the demxg of a lqud teso to two other lqud phases teso. Good dscussos of the thermodyamcs of egatve pressures have bee provded recetly by Imre et al. (998) ad ebeedett (996). All of the crtcal pots at postve pressures were checked for phase stablty usg the method of Hua et al. (998) ad are stable uless dcated otherwse. Also preseted the tables s the PU tme requred to solve each problem, gve secods o a Su Ultra 30 workstato. The computato tmes see here are much hgher tha what s requred by local methods such as the method of Hedema ad Khall (980) ad Mchelse ad Hedema (98). However, such local methods do ot relably fd all crtcal pots. Thus, there s a choce betwee faster methods that are ot fully relable, or ths completely relable but slower method that s guarateed to gve the correct aswer, fdg all crtcal pots. 0
Example : Methae ad Hydroge Sulfde I ths frst example problem we cosder mxtures of methae () ad hydroge sulfde (). The Soave-Redlch-Kwog (SRK) EOS model was used wth a bary teracto parameter of k 0.08. Ths example problem, as well as Examples ad 3, are also used by Hedema ad Khall (980). However, drect comparso of computed results s ot meagful sce Hedema ad Khall do ot provde the model parameters used ther computatos. I our experece, the locato of low temperature (lqud-lqud) crtcal pots may be extremely sestve to small chages model parameters such as the k used. The crtcal states computed usg the terval approach are show Table. Note that at some compostos there s oe crtcal pot, at some there are two crtcal pots, at some there are three crtcal pots, ad at some there are o crtcal pots. I order to verfy that we correctly mplemeted the Hedema-Khall crtcalty codtos we used the route RITPT, a part of the IV-SEP smulato package (Hytoft ad Ga, 996; Mchelse, 984). RITPT uses the same problem formulato as used here, but the soluto method s the local method of Hedema ad Khall (980). Ths method s based o a automatc talzato that s sutable for fdg hgh-temperature crtcal pots. I order to use RITPT to fd other crtcal pots, we modfed the code to dsable the automatc talzato ad to allow the user to provde the talzato. Usg ths modfed code, wth the same model parameters as used the IN/GB code, ad wth talzatos that were carefully selected (guded by the results obtaed wth the terval approach), we were able to fd the same crtcal pots gve Table, as well as those gve as results other examples that follow. Of course, sce RITPT s a local, talzatodepedet solver, whe t s used, eve ths modfed form, there are o guaratees that all crtcal pots wll be foud. Whe the terval approach descrbed here s used, such a guaratee s provded.
or the mxtures that have o crtcal pot, the terval method provdes mathematcal proof that ths s the case. Ths s mportat, sce wth local solvers, such as RITPT, much effort could be wasted by tryg a very large umber of dfferet talzatos searchg for a soluto that does ot exst, ad oe could ever be sure that there was ot really a soluto. Example : arbo oxde ad -Octae I ths example we cosder mxtures of carbo doxde () ad -octae (), a classc O /alkae system. The SRK EOS model was used wth a bary teracto parameter of k 0. The crtcal states determed, ad computato tmes requred, usg the IN/GB method ad the Hedema-Khall crtcalty codtos are gve Table for several dfferet mxture compostos. The behavor see s dfferet from that of the prevous example (methae/hydroge sulfde) that the computed hgh temperature (vapor-lqud) crtcal locus s cotuous betwee the crtcal pots of carbo doxde ad -octae. or ths example, a determat-based form of the crtcalty codtos (Red ad Beegle, 977) was also used, addto to the Hedema-Khall formulato. The IN/GB approach was used to solve the determat-based crtcalty codtos, yeldg the same crtcal states as the Hedema-Khall formulato. However, usg the Hedema-Khall formulato was sgfcatly more effcet computatoally, eve though t volves four ( ) depedet varables, versus oly two whe the determat-based formulato s used. The s dcatve of the fact that, whle coutertutve, reducg the dmesoalty of olear equato solvg problems may fact make them more, ot less, dffcult to solve. Example 3: arbo oxde ad -Hexadecae The fal bary system cosdered volves mxtures of carbo doxde () ad - hexadecae (). The SRK EOS model was used wth a bary teracto parameter of k 0.
The crtcal states computed usg the IN/GB method are gve Table 3 for several dfferet mxture compostos. The computato tmes reported here ad the other examples are actually qute good for a geeral-purpose approach offerg a verfed soluto. To see ths we also solved the problems ths example usg the commercal package Numerca (ILOG), whch also offers a verfed soluto. Ths code (va Heteryck et al., 997) combes deas from terval aalyss, such as used here, wth techques from costrat satsfacto programmg (SP). O the frst mxture Table 3 ( 0.97), Numerca requred 3947 s to fd ust the frst crtcal pot (T c 386K), ad to allow ths crtcal pot to be foud wth ths amout of tme, t was ecessary to start wth a arrow tal terval of wdth oly 0 K ad 0 cm 3 cotag the crtcal pot. The approach used here requred oly about 85 s to fd all three crtcal pots whe startg wth the wde tal terval dcated at the begg of ths secto. Example 4: Terary Mxtures (SRK) Terary mxtures are ofte used as models to represet more complex mxtures the petroleum dustry. The frst two terary systems cosdered here, methae()/troge()/ hexae(3) ad methae()/o ()/hexae(3), volve compouds that serve to model mxtures foud the cryogec processg of lquefed atural gas (Merrll, 983). The thrd system cosdered s the same as the secod except that hexae s replaced by hydroge sulfde. The SRK EOS was used to model these mxtures usg the bary teracto parameters show Table 4. The computed crtcal states for mxtures wth varous compostos are gve Table 5. or two cases, the crtcal pots were also computed usg the Peg-Robso (PR) EOS. It s terestg to ote that for the thrd system (volvg H S), whle there are relatvely small 3
dffereces betwee the two models predctg the hgher of the two crtcal pots, there s a very large dfferece the predcto of the lower temperature crtcal pot. Not oly s the locato of low temperature crtcal pots sometmes very sestve to the form of the model, but, as oted above, t may also be very sestve to the model parameters used. Ths s a dcato that models should be valdated carefully before use for specfc applcatos. We observe here, as well as prevous examples, that computg the crtcal pots of mxtures for whch oe or more crtcal pots has egatve pressure ofte requres much more computg tme compared to other cases. If fdg crtcal states wth egatve pressure s ot of terest, the t would be easy to modfy the algorthm to lmt the search by elmatg regos over whch the pressure s egatve. Ths could be doe as a part of the fucto rage test by computg the terval exteso P(X (k) ) from the EOS, ad the dscardg X (k) f the upper boud of the terval P(X (k) ) s egatve. Example 5: Terary Mxtures (PR) Ths secod set of terary systems cossts of problems cosdered prevously by Peg ad Robso (977). These are represetatve of model systems studed hydrocarbo processg. The PR EOS was used to model these mxtures usg the bary teracto parameters show Table 6. The systems used are methae()/o ()/H S(3), ethae()/butae()/-heptae(3), ethae()/-petae()/-heptae(3), ad methae()/propae()/ troge(3). The crtcal pots computed usg the terval approach are show Table 7. These results are excellet agreemet wth those computed by Peg ad Robso (977), though the problems solved here we have lkely used slghtly dfferet k parameters. I addto, we foud a crtcal pot for the frst terary mxture that had ot bee reported before, though Peg 4
ad Robso (977) may ot have bee lookg for crtcal pots wth P < 0. Ths demostrates the ablty of the terval approach to fd all solutos of the crtcalty codtos. ocludg Remarks We have descrbed here the frst completely relable method locatg all crtcal pots of mxtures, ad for verfyg the oexstece of crtcal pots f oe are preset. As see several example problems, the method requres o pot talzato ad o a pror kowledge of the umber of crtcal pots, ad ca fd both hgh ad low temperature crtcal pots. The techque s based o terval aalyss, partcular a terval Newto/geeralzed bsecto algorthm, whch provdes a mathematcal ad computatoal guaratee that all crtcal pots are located. Ths guaratee comes at the expese of a sgfcat PU requremet. Thus, there s a choce betwee fast local methods that are ot completely relable, or ths method that s guarateed to gve the complete ad correct aswer, fdg all crtcal pots of a mxture. The modeler must make a decso cocerg how mportat t s to get the correct aswer. Recet expermets mprovg the IN/GB algorthm (Gau et al., 999) have show sgfcat promse reducg the computato tme requremets of the method o a varety of modelg problems, ad we beleve these techques wll also be applcable to the computatoal of crtcal pots. I the work preseted here, the mxtures were modeled usg cubc equatos of state, wth stadard mxg rules. However, the problem solvg techque used s geeral purpose ad ca be appled coecto wth other thermodyamc models. I addto to the soluto of crtcal pot problems, the methodology used here ca also be appled to a wde varety of other problems the modelg of phase behavor (e.g., Stadtherr et al., 995; Hua et al., 998; 5
Maer et al., 998; Tesser et al., 000; Gau ad Stadtherr, 999, Xu et al., 000), ad the soluto of process modelg problems (Schepper ad Stadtherr, 996). Ackowledgmets Ths work has bee supported part by the Evrometal Protecto Agecy Grats R86-734-0-0 ad R84-73-0-0, the Natoal Scece oudato Grats TS-95835 ad EE97-00537-R, ad the doors of The Petroleum Research ud, admstered by the AS, uder Grat 304-A9. Oe of the authors (BAS) gratefully ackowledges fellowshp support by the Hele Kellogg Isttute for Iteratoal Studes ad the oca ola ompay. 6
Refereces Baker, L. E. ad K.. Luks, rtcal Pots ad Saturato Pressure alculatos for Multcompoet Systems, Soc. Pet. Eg. J., 0, 5 (980). Boberg, T.. ad R. R. Whte, Predcto of rtcal Mxtures, Id. Eg. hem. udam.,, 40 (96). Boshkov, L. Z. ad L. V. Yelash, losed-loops of Lqud-Lqud Immscblty Bary Mxtures Predcted from the Redlch-Kwog Equato of State, lud Phase Equlb., 4, 05 (967). ebeedett, P. G., Metastable Lquds: ocepts ad Prcples, Prceto Uversty Press, Prceto, New Jersey (996). eters, U. ad G. M. Scheder, lud Mxtures at Hgh Pressures. omputer alculatos of the Phase Equlbra ad the rtcal Pheomea lud Bary Mxtures from the Redlch-Kwog Equato of State, Ber. Buseges. Physk. hem., 80, 36 (976). Eck, R., G.. Holder ad B. I. Mors, rtcal ad Three Phase Behavor the arbo oxde/trdecae System, lud Phase Equlb.,, 09 (985). Gau, -Y., R. W. Maer ad M. A. Stadtherr, New Iterval Methodologes for Relable Process Modelg, preseted at AIhE Aual Meetg, allas, TX, Oct. 3 Nov. 5, 999. Gau,.-Y. ad M. A. Stadtherr, Nolear Parameter Estmato Usg Iterval Aalyss, AIhE Symp. Ser., 94(30), 445 (999). Greves, R. B. ad G. Thodos, The rtcal Temperature of Terary Hydrocarbo Systems, Id. Eg. hem. udam.,, 45 (96). Hase E., Global Optmzato usg Iterval Aalyss, Marcel-ekker, New York (99). Hedema, R. A., The rtera for Thermodyamc Stablty, AIhE J.,, 84 (975). 7
Hedema, R. A ad A. M. Khall, The alculato of rtcal Pots, AIhE J., 5, 769 (980). Hedema, R. A., omputato of Hgh Pressure Phase Equlbra, lud Phase Equlb., 4, 55 (983). Hcks,. P. ad. L. Youg, Theoretcal Predcto of Phase Behavor at Hgh Temperatures ad Pressures for No-Polar Mxtures: I. omputer Soluto Techques ad Stablty Tests, J. hem. Soc. araday II, 73, 597 (977). Hua, Z., J.. Breecke ad M. A. Stadtherr, Relable Predcto of Phase Stablty Usg a Iterval-Newto Method, lud Phase Equlb., 6, 5 (996). Hua, J. Z., J.. Breecke ad M. A. Stadtherr, Ehaced Iterval Aalyss for Phase Stablty: ubc Equato of State Models, Id. Eg. hem. Res., 37, 59 (998). Hurle, R. L.,. Joes ad. L. Youg, Theoretcal Predcto of Phase Behavor at Hgh Temperatures ad Pressures for No-polar Mxtures, J. hem. Soc. araday II, 73, 63 (977a). Hurle, R. L., L. Toczylk ad. L. Youg, Theoretcal Predcto of Phase Behavor at Hgh Temperatures ad Pressures for No-polar Mxtures, J. hem. Soc. araday II, 73, 68 (977b). Hytoft, G. ad R. Ga, IV-SEP Program Package, amarks Tekske Uverstet, Lygby, emark (996). Imre, A., K Martas ad L. P. N. Rebelo, Thermodyamcs of Negatve Pressures Lquds, J. No-Equlb. Thermody., 3, 35 (998). Kearfott, R. B. ad M. Novoa, Algorthm 68: INTBIS. A Portable Newto/Bsecto Package, AM Tras. Math. Software 6, 5 (990). 8
Kearfott, R. B., M. awade, K.-S. u ad.-y. Hu, Algorthm 737: {INTLIB}, A Portable {ORTRAN 77} Iterval Stadard ucto Lbrary,'' AM Tras. Math. Software, 0, 447 (994). Kearfott, R. B., Rgorous Global Search: otuous Problem, Kluwer Academc Publshers, ordrecht, The Netherlads (996). Kearfott, R. B., J. a, ad A. Neumaer, Exstece Verfcato for Sgular Zeros of Nolear Systems, SIAM J. Numer. Aal., press (000). Levelt Segers, J. M. H., Thermodyamcs of Solutos Near the Solvet s rtcal Pot, Supercrtcal lud Techology: Revews Moder Theory ad Applcatos (T. J Bruo ad J.. Ely, eds.), R Press, Boca Rato, L (99). Luks, K.., E. A. Turek ad L. E. Baker, alculato of Mmum Mscblty Pressure, SPE Res. Eg. J., November, 50 (987). Maer, R. W., J.. Breecke ad M. A. Stadtherr, Relable omputato of Homogeeous Azeotropes, AIhE J., 44, 745 (998). Merrll, R.., Lqud-lqud-vapor Pheomea ryogec Lquefed Natural Gas Systems, Ph.. Thess, Uversty of Notre ame, Notre ame, Idaa (983). Mchelse, M. L. ad R. A. Hedema, alculato of rtcal Pots from ubc Two- ostat Equatos of State, AIhE J., 7, 5 (98). Mchelse, M. L., alculato of rtcal Pots ad Phase Boudares the rtcal Rego, lud Phase Equlb., 6, 57 (984). Muoz,. ad E. H. hmowtz, rtcal Pheomea Mxtures. I. Thermodyamc Theory for the Bary rtcal Azeotrope, J. hem. Phys., 99, 5438 (993). 9
Nagaraa, N. R., A. S. ullck ad A. Grewak, New Strategy for Phase Equlbrum ad rtcal Pot alculatos by Thermodyamc Eergy Aalyss. Part I. Stablty Aalyss ad lash, lud Phase Equlb., 6, 9 (99a). Nagaraa, N. R., A. S. ullck ad A. Grewak, New Strategy for Phase Equlbrum ad rtcal Pot alculatos by Thermodyamc Eergy Aalyss. Part II. rtcal Pot alculatos, lud Phase Equlb., 6, (99b). Neumaer, A., Iterval Methods for Systems of Equatos, ambrdge Uversty Press, ambrdge, UK (990). Peg,.-Y. ad. B. Robso, A Rgorous Method for Predctg the rtcal Propertes of Multcompoet Systems from a Equato of State, AIhE J., 3, 37 (977). Red, R.., J. M. Praustz ad B. E. Polg, The Propertes of Gases ad Lquds, 4th Ed., McGraw-Hll, New York (987). Red, R.. ad B. L. Beegle, rtcal Pot rtera Legedre Trasform Notato, AIhE J., 3, 76 (977). Rochocz, G. L., M. aster ad S. I. Sadler, rtcal Pot alculatos for Sem-otuous Mxtures, lud Phase Equlb., 39, 37 (997). Sadus, R. J., alculatg rtcal Trastos of lud Mxtures: Theory vs. Expermet, AIhE J., 40, 376 (994). Schepper,. A. ad M. A. Stadtherr, Robust Process Smulato usg Iterval Methods, omput. hem. Eg., 0, 87 (996). Spear, R. R., R. L. Robso ad K.-. hao, rtcal States of Terary Mxtures ad Equatos of State, Id. Eg. hem. udam., 0, 588 (97). Stadtherr, M. A.,. A. Schepper ad J.. Breecke, Robust Phase Stablty Aalyss Usg Iterval Methods, AIhE Symp. Ser., 9(304), 356 (995). 0
Stockfleth, R. ad R. ohr, A Algorthm for alculatg rtcal Pot Multcompoet Mxtures whch a Be Easly Implemeted Exstg Programs to alculate Phase Equlbra, lud Phase Equlb., 45, 43 (998). Tea, A. S. ad H. W. Kropholler, rtcal states of Mxtures whch Azeotropc Behavor Perssts the rtcal Rego, hem. Eg. Sc., 30, 435 (975). Tea, A. S. ad J. S. Rowlso, The Predcto of the Thermodyamc Propertes of luds ad lud mxtures-iv. rtcal ad Azeotropc States, hem. Eg. Sc., 8, 59 (973). Tesser, S. R., J.. Breecke ad M. A. Stadtherr, Relable Phase Stablty Aalyss for Excess Gbbs Eergy Models, hem. Eg. Sc., 55, 785 (000).. va Heteryck, P., L. Mchel ad Y. evlle, Y., Numerca: A Modelg Laguage for Global Optmzato, MIT Press, ambrdge, Mass. (997). va Koyeburg, P. H. ad R. L. Scott, rtcal Les ad Phase Equlbra Bary va der Waals Mxtures, Phlos. Tras. Roy. Soc. Lodo A, 98, 495 (980). Xu, G., A. M. Scurto, M. aster, J.. Breecke, M. A. Stadtherr, Relable omputato of Hgh Pressure Sold-lud Equlbrum, Id. Eg. hem. Res., 39, 64 (000).
APPENIX Ths Appedx explas detal the otato used Eqs. (4) ad (5) the text, whch are derved from a geeralzed cubc EOS model. Eqs. (4) ad (5) are repeated here for coveece. A ( ) ( ) a a b a N y RT α α 6 5 3 (4) k k k A ) 3 ) )( ( (3 ) ( ) ( 3 6 4 3 6 3 3 3 a b a N y RT α (5) Here : total umber of compoets. A : secod order dervatve of the Helmholtz free eergy wth respect to the umber of moles of speces ad. : chage total umber of moles of speces. Ths s used the cotext of a Taylor seres expaso of the Helmholtz free eergy terms of composto. R : deal gas costat. T : absolute temperature.
: : N : y : a : total umber of moles. umber of moles of speces. Aalogous meag for other subdces. parameter used the equatos for the computato of crtcal pots, t s defed as, N mole fracto of compoet. Aalogous meag for other subdces. average eergy parameter of the equato of state for the mxture of compoets. Ths s computed usg stadard va der Waals mxg rules: a a a ( a a ) 0.5 ( k ) a : eergy of teracto parameter betwee speces ad. a : eergy parameter of speces. The meag s the same for other subdces. It s computed usg the followg correlato: a ( RTc ) P c η c T T c 0.5 η 0.4748 for the Soave-Redlch-Kwog (SRK) Equato of State. η 0.4574 for the Peg-Robso (PR) Equato of State. w : acetrc factor of speces. T c : crtcal temperature of speces. P c : crtcal pressure of speces. c c 0.48.574w 0.76w 0.37464.546w 0.699w for SRK EOS for PR EOS k : bary teracto parameter betwee speces ad. 3
α k : parameter used the equatos for the computato of crtcal pots, t s defed as, α k y a a k α : parameter used the equatos for the computato of crtcal pots, t s defed as, α α a : parameter used the equatos for the computato of crtcal pots, t s defed as, a a a b : average va der Waals molar volume the Equato of State. Ths s computed usg stadard va der Waals mxg rules, b b b : va der Waals molar volume of speces. Ths s computed as follows, b 0.08664RT / P for SRK EOS c c b 0.07780RT / P for PR EOS c c : parameter used the equatos for the computato of crtcal pots, t s defed as, b b : parameter used the equatos for the computato of crtcal pots, t s defed as, 4
5 : 6 auxlary fuctos used the computato of crtcal pots, they are defed as, K K K 3 K K 3 3 4 K K 5 l K K 6 l K K K K K : dmesoless volume, t s defed as, b v b V K : parameters of the equato of state. Ther values are calculated wth the followg formulas, 4 o o o w u u 4 o o o w u u where, 0, o o w u for the SRK EOS, o o w u for the PR EOS δ : Kroecker delta operator, whch s defed as, 0 δ δ δ
Table. omputed crtcal pots for mxtures of methae () ad hydroge sulfde (). eed omposto Volume (cm 3 /gmol) rtcal rtcal Temperature (K) rtcal Pressure (bar) Total tme (s) H 4 H S 0.998 0.00 0.9999 0.004 4.6 90.8 46.3 0.5 0.97 0.03 0.9977 0.068 07.70 96.74 50.4.4 0.9475 0.055 0.996 0.9 0.8 0.37 53.9 3.5 0.94 0.06 0.9884 0.57 00.30 0.86 55.* 4.06 0.93 0.07 0.9830 0.837 97.75 04.78 56.7* 5.64 0.685 0.7858 44.7 4.77-83.6 0.86 0.4 0.865 0.506 77.95 3.7 65.5* 0.9 0.6003 0.7998 56.59 8.3 -.7 0.85 0.5 0.889 0.5739 74.03.99 64.5* 5.4 0.639 0.785 59.4 90.98.5* 0.84 0.6 NP 39.0 0.75 0.5 NP 4.5 0.53 0.47 NP 9.6 0.5 0.48 0.07-0.9795 59.6 70.0 46. 35.0 0.874-0.9578 54.94 60.7 49.0 0.5 0.49 0.35-0.9908 63.37 79.5 45.0 7.8 0.3896-0.90 50.3 49.0 60. 0.49 0.5 0.684-0.739 34.89 08.5 800. 6.4 0.59-0.854 44.6 3.67 30.4 0.0846-0.9964 67.59 88.86 44.0 0.36 0.64 0.068-0.9999 83. 33.07 3.6 8.87 0.9 0.77 0.0435 0.9999 95.58 345.50 7. 4.87 0.4 0.76 0.04 0.9999 94.58 343.86 8.4 5.4 0.09 0.9 0.097 0.9999 07.9 363.58 00.0.95 NP: o crtcal pot was foud. * rtcal pot s ot stable. 6
Table. omputed crtcal pots for mxtures of carbo doxde () ad -octae (). eed omposto Volume (cm 3 /gmol) rtcal rtcal Temperature (K) rtcal Pressure (bar) Total tme (s) O 8 H 8 0.90 0.0 0.7377 0.675 8.03 384.70 37.0 38.9 0.9895-0.447 55.50 8.43-006.3 0.88 0. 0.6093 0.799 36.55 398.50 40.5 35.5 0.990-0.403 58.75 6.7-00.3 0.878 0. 0.5964 0.807 37.43 399.83 40.8 36. 0.990-0.399 59.07 5.90-0. 0.875 0.5 0.577 0.867 38.76 40.8 4. 34.9 0.990-0.394 59.53 5.48-04.9 0.87 0.9 0.557 0.834 40.55 404.43 4.4 34.6 0.9903-0.388 60.4 4.90-08.9 0.80 0.0 0.97 0.9756 74.8 444.6 36.8 7.6 0.993-0.35 70.7.90-8. 0.60 0.40 0.04 0.9999 83.70 50.0 94.7 4.5 0.40 0.60 0.06-0.9999 398.67 540.74 6.7 7.46 0.0 0.80 0.0689-0.9999 55.50 557.93 39.9.9 0.0 0.90 0.0344-0.9999 574.3 563.9 3.7.53 7
Table 3. omputed crtcal pots for mxtures of carbo doxde () ad -hexadecae (). rtcal eed omposto Volume (cm 3 /gmol) rtcal Temperature (K) rtcal Pressure (bar) Total tme (s) O 6 H 34 0.97 0.03 0.6058 0.7956 00.95 368.00 70.7 84.8 0.998-0.0606 58.0 73.3-593.6 0.9958-0.09 50.97 88.30-0.5 0.94 0.06 0.6773-0.7357 7.7 440.54 30.3 70.4 0.9985-0.055 69.93 59.95-677.8 0.93 0.07 0.79-0.690 37.6 463. 37.7 66.6 0.9985-0.0544 73.50 55.07-70. 0.58 0.4 0.358-0.9673 606.39 686.9 74.0.4 0.50 0.50 0.890-0.980 78.5 696.59 58.3 7.5 8
Table 4. Bary teracto parameters (k ) used Example 4. Bary teracto parameters () (k ) O N H S H 4 6 H 4 0.0989 0.0933 0.78 O 0.0 0.0974 0.099 0.00 N 0.0 0.078 0.496 H S 0.0989 0.0974 H 4 0.0933 0.099 6 H 4 0.78 0.00 0.078 0.496 0.0 0.08 () 0.0 0.08 () 0.0 0.0 0.0374 0.04 0.0374 0.04 0.0 () () Parameters for the SRK EOS are pla face type, ad for the PR EOS bold face type. Except as oted, these values are from the Aspe Plus (verso 9.3) database. Value used by Hua et al. (998). 9
Table 5. omputed crtcal pots for terary mxtures Example 4. rtcal eed omposto Volume 3 (cm 3 /gmol) rtcal Temperature (K) rtcal Pressure (bar) Total tme (s) H4 N 6 H 4 0.46 0.07 0.47 0.37 0.0485-0.993 8.83 46.64 7.0 0 0.5 0.08 0.40 0.55 0.0655-0.986 96.63 448.58 4.5 6 0.68 0.0 0.30 0.6 0.07-0.9648 53.8 45.3 76. 4 H4 O 6 H 4 0.70 0.0 0.0 0.370-0.045-0.979 3.53 36.39 06. 936 0.74 0.0 0.5 0.3360-0.037-0.949 3.40 39.35 96. 39 0.78 0.5 0.07 0.9-0.707-0.3748 49.4 4.70-4.0 346 0.768-0.533-0.6494 60.68 08.9 77.7 0.78 () 0.5 0.07 0.906-0.756-0.3740 43.88 43.58-55.9 04 0.6594-0.90-0.697 54.7 0. 80.0 0.79 0.05 0.6 0.546-0.068-0.085 93.0 33.67 6.0 78 H4 O H S 0.50 0.0 0.40 0.389-0.090-0.967 50. 35.8 08.3 609 0.75-0.3-0.990 68.00 74.0 3.4 0.50 () 0.0 0.40 0.47 0.384 0.9645 7.35 85.58 3.8 44 () omputed usg the PR EOS. 0.654-0.0606-0.7540 3.38 44.36 8.9 30
Table 6. Bary teracto parameters (k ) used Example 5. Bary teracto parameters () (k ) O N H S H 4 H 4 3 H 8 4 H 0 5 H 7 H 6 O 0.0 0.0974 0.099 N 0.0 0.03 0.085 H S 0.0974 0.0 0.0 H 4 0.099 0.03 0.0 0.0 0.040 H 4 0.0 0.0096 0.0078 0.0067 3 H 8 0.085 0.040 0.0 4 H 0 0.0096 0.0 0.0033 5 H 0.0078 0.0 0.0074 7 H 6 0.0067 0.0033 0.0074 0.0 () Values from Aspe Plus (verso 9.3) database. 3
Table 7. omputed crtcal pots for terary mxtures Example 5. eed omposto 3 Volume rtcal (cm 3 /gmol) rtcal Temperature (K) rtcal Pressure (bar) Total tme (s) H4 O H S 0.07 0.66 0.34 0.038 0.800 0.575 98.6 3.5 83.3 55 0.0730 0.44-0.9035 35.35 3. -44.7 H 6 4 H 0 7 H 6 0.49 0.373 0.98 0.96 0.68 0.6686 37.95 439.54 63. 68 0.76 0.7 0.03 0.765 0.436 0.4738 74.03 388.64 74.9 74 0.54 0.4 0.074 0.445 0.855 0.7407 05.9 403.54 6.5 54 H 6 5 H 7 H 6 0.80 0.064 0.35 0.7737 0.99 0.605 70.0 394.7 8.9 665 0.6 0.7 0.7 0.504 0.7359 0.459 4.5 44.77 70.4 78 0.65 0.96 0.089 0.5098 0.796 0.3368.7 49.54 69.39 709 H4 3 H 8 N 0.45 0.54 0.043 0.64 0.999-0.0 39.36 37.47 87.0 9.9 0.360 0.545 0.095 0.069 0.9939-0.076 37. 38.5 9.5 45.9 0.453 0.5005 0.0465 0.49 0.9887-0.04 3.06 3.7 9.8 37. 0.45 0.5030 0.0855 0.38 0.9908-0.066 30.04 3.76 96. 5. 3