An Improved Group Contribution Volume Translated Peng-Robinson Equation of State

Transcription

1 An Iproved Group Contrbuton Volue Translated Peng-Robnson Equaton of State By Alan Foster Unversty of KwaZulu-Natal, Howard College Capus For the degree Master of Scee (Checal Engneerng) Progress Report May 2009 The fnaal assstae of the Natonal Research Foundaton (NRF) towards ths research s hereby acknowledged. Opnons expressed and colusons arrved at, are those of the author and are not necessarly to be attrbuted to the NRF.

2 ABSTRACT The descrpton of thero-physcal propertes of pure coponents and xtures, especally the equlbru between two or ore phases s of great portae for the desgn and sulaton of checal processes and any other applcatons. The Volue Translated Peng-Robnson (VTPR) odel provdes an accurate eans by whch to calculate these propertes for pure and xed subcrtcal, as well as supercrtcal coponents, wth relable estatons for syetrc and asyetrc systes. However, VTPR requres group nteracton paraeters and currently the VTPR group nteracton paraeter atrx s sall, as regresson of these paraeters usng experental data requres great care n order to obey all known boundary condtons. In contrast though, the odfed UNIFAC group nteracton paraeter atrx, whch has been contnuously extended and proved se 983, contans consderably ore nforaton and would be of great use n the VTPR ethod. The odfed UNIFAC paraeters however are unavalable for use n the VTPR ethod due to ther teperature dependee, whch leads to orrect teperature extrapolatons when used together wth the VTPR xng rule. The new group contrbuton equaton of state VGTPR ntroduces an excess Gbbs energy translaton futon nto the xng rule, whch allows the cobnaton of the volue translated Peng-Robnson EOS wth the odfed UNIFAC group contrbuton ethod, and allows the use of the large aount of nforaton assocated wth odfed UNIFAC. In the VGTPR ethod, the equalty of g E s obtaned by teratve adjustent of the xture a- paraeter of the EOS and s guaranteed at any teperature. A fxed reduced densty of the pure coponents and the xture s used for extenson to supercrtcal condtons. The pressure dfferee between ths state and the saturated state changes g E only slghtly due to the low pressure dependee of g E n the lqud state. Ths odel therefore allows the cobnaton of the volue translated Peng-Robnson EOS wth the odfed UNIFAC ethod (wth teperature dependent group nteracton paraeters). The VGTPR ethod s farly new and untl now lttle has been done n ters of testng the odel thoroughly. The work perfored wll provde a re-dervaton of the g E translated xng rule n an attept to splfy the current algorth and perfor coprehensve tests on the VGTPR ethod, evaluatng ts perforae utlzng the large aount of experental data stored n the Dortund Data Bank (DDB). Ths report covers a coprehensve revew of a nuber of exstng g E xng rules, lookng at the theoretcal coepts related to ther dervatons along wth the advantages and dsadvantages assocated wth each and follows the developent of the feld over the last 30 years. The dea behnd the VGTPR odel s ntroduced and dscussed before layng out a work plan for future research requred.

3 PREFACE The work presented n ths progress report was perfored at the Unversty of KwaZulu-Natal, Durban fro January 2009 to May The work was supervsed by Prof. D. Rajugernath and Prof. Dr. J. Rarey. Ths progress report s presented as a partal requreent for the degree of MSc n Checal Engneerng. All the work presented n ths report s orgnal unless otherwse stated and has not (n whole or part) been subtted prevously to any tertary nsttute as part of a degree. Alan Foster ( )

4 TABLE OF CONTENTS ABSTRACT... PREFACE... TABLE OF CONTENTS... LIST OF FIGURES...v NOMENCLATURE... v. INTRODUCTION MIXING RULES A REVIEW Coposton-Dependent Cobnng Rules Mxng Rules Usng Excess Gbbs Energy (g E )/ Actvty Coeffcent Models Vdal Huron-Vdal Mollerup Modfed Huron-Vdal Wong-Sandler PSRK LCVM VTPR THE VGTPR MODEL FUTURE WORK REFERENCES... 47

5 LIST OF FIGURES FIGURE Varaton of q or h-futon wth respect to α for the SRK EOS also showng the lnear ft n the range 0 < α < 3: (---) q or h-futon, ( ) lnear ft... 7 FIGURE 2 Average absolute % error (AAE) n predcted bubble pont pressure for ethane/n-alkane systes as a futon of the λ value (taken for Boukouvalas et al. (994)) FIGURE 3 Predcton of the bubble pont pressure for the syste ethane / nc 28 at 373 K (taken fro Boukouvalas et al. (994)) FIGURE 4 Change n the quotents r alkane /r ethane and b alkane /b ethane n dependee of the degree of asyetry of the syste: (o) paraeter b (PR EOS); (*) relatve van der Waals volue paraeter r (taken fro Ahlers and Gehlng (2002a)) Fgure 5 Current status of the odfed UNIFAC paraeters avalable for use n the VTPR odel (taken fro DDBSP - Gehlng et al. (2009)) Fgure 6 Current status of the UNIFAC paraeters avalable for use n the PSRK odel (taken fro DDBSP - Gehlng et al. (2009)) Fgure 7 Current status of the odfed UNIFAC (Dortund) paraeters avalable for drect use n the new VGTPR odel (taken fro DDBSP Gehlng et al. (2009)) v

6 NOMENCLATURE Letters a - Equaton of state attracton paraeter or Helholtz energy b - Equaton of state co-volue B - Second vral coeffcent c - Equaton of state volue translaton ter C - Thrd vral coeffcent D - Fourth vral coeffcent or splfyng futon for ethod of Wong and Sandler (992) defned by Equaton (2.59) f - Futon defned n dervaton by Mollerup (986) equvalent to u (reduced lqud volue) f c - Futon defned n dervaton by Mollerup (986) F - Surface area to ole fracton rato used n UNIFAC g - Molar Gbbs free energy G - Expresson used n ethod of Twu et al. (99) defned by Equaton (2.4) h - Futon used n dervaton by Mchelsen (990) defned by Equaton (2.34) or olar enthalpy H - Expresson used n ethod of Twu et al. (99) defned by Equaton (2.3) k - Bnary-nteracton paraeter l - Bnary-nteracton paraeter - Bnary-nteracton paraeter n - Nuber of oles - Total nuber of coponents P - Pressure v

7 q - Futon used n dervaton by Mchelsen (990) defned by Equaton (2.33) or when accopaned by a subscrpt, the relatve van der Waals surface area Q - splfyng futon for ethod of Wong and Sandler (992) defned by Equaton (2.58) r - Relatve van der Waals volue R - Unversal gas constant T - Teperature u - Reduced lqud phase volue - defned by Equaton (2.35) v - Molar volue V - Volue to ole fracton rato used n UNIFAC z - Mole fracton Z - Copressblty factor Greek Sybols α - Shortcut notaton for a/brt β - Bnary-nteracton paraeter γ - Actvty coeffcent φ - Fugacty coeffcent λ - Relatve dstrbuton constant for use n the LCVM xng rule Λ - Equaton of state specfc constant used n g E xng rules or constant of approxatng futon used n dervaton of MHV and MHV2 xng rules Ψ - Constant of approxatng futon used n dervaton of MHV and MHV2 xng rules Γ - Constant of approxatng futon used n dervaton of MHV2 xng rule Subscrpts - Infnte pressure referee 0 - Zero pressure referee v

8 γ - Property calculated fro an actvty coeffcent/g E odel c - Crtcal property calc - Result fro a calculaton cob - Cobnatoral part EOS - Property calculated fro an equaton of state exp - Result fro experental work f - Coponent referee letter - Coponent referee letter j - Coponent referee letter k - Coponent referee letter l - Coponent referee letter - Property of the xture (as a whole) MHV - Calculated by MHV xng rule res - Resdual part trans - Translaton ter V - Calculated by ethod of Vdal (978) Superscrpts E - Excess property ^ - Property of a partcular coponent n a xture v

9 . INTRODUCTION The descrpton of thero-physcal propertes of pure coponents and xtures, especally the equlbru between two or ore phases s of great portae for the desgn and sulaton of checal processes and any other applcatons. Hstorcally, two dfferent approaches have been used to estate pure coponent propertes and phase equlbra: the equaton of state (EOS) approach and the excess Gbbs energy (g E ) odel approach. Cubc equatons of state are the ost popular and wdely used type of EOS n ndustry due to ther relatve splcty and accuracy n predctng pure coponent and xture therodynac propertes. A great nuber of g E odels exst, however recent developents have seen the ntroducton of fully predctve group-contrbuton odels such as ASOG, UNIFAC and odfed UNIFAC whch do not requre bnary-nteracton paraeters found fro bnary experental data and are therefore very attractve to engneers nvolved n process desgn and sulaton. Obvously both the equaton of state ethod and the actvty coeffcent ethod have advantages over the other and outperfor the other n specfc areas. g E odels are sple and robust whle equatons of state allow calculatons for supercrtcal systes. A desre to lnk the two ethods and ake use of the advantages of both, has drven over 30 years of extensve research aed partcularly at utlzng the g E odel to descrbe xture paraeters of cubc equatons of state. The an advantage of cubc equatons of state s not n ther ablty to represent pure coponents but rather ther value when representng flud xtures. Sengers et al. (2000) beleve that representaton of propertes of flud xtures s the an, f not the only, purpose of cubc and generalzed van der Waals equatons of state and that when a pure flud coponent s of nterest, equatons of ths type are not the preferred ethod for representaton. In any case, t s wdely regarded that there s lttle roo for proveent n the use of cubc equatons of state for pure coponents as ths feld has had a vast aount of te and effort nvested nto t by past researchers. A nuber of ethods have been developed for predctons of varous types of pure coponents and therefore cubc equatons of state ay be used for any polar and non-polar or assocatng copounds. Ths has created an opnon that very lttle progress can be ade wth regards to pure coponent representaton by equatons of state; however ths s by no eans the case when t coes to EOS use n ultcoponent systes.

10 In order to extend the pure coponent EOS odels to xtures one can use the van der Waals one-flud xng rule. The one-flud theory of xtures s based on the assupton that the EOS for a xture s the sae as that for a hypothetcal pure flud that has EOS paraeters whch depend on the coposton of the xture. The relatonshp used to establsh xture paraeters ust descrbe the coentraton dependee of theses paraeters and s coonly known as a xng rule. Fndng the CEOS xture paraeters s of utost portae when representng a flud xture n order to calculate accurate results. Ths portae s hghlghted by Sengers et al. (2000) whch coludes that the establshent of the xture paraeters s ore portant than the actual PVT relatonshp eboded wthn a partcular EOS. A nuber of dfferent xng rules have been developed, each one ang at provng predctons of flud xtures usng cubc equatons of state. Generally each new xng rule proves on prevous rules n certan areas, however they fal n others. Soe xng rules have also been developed to prove predctons of specfc types of xtures but stll fal when used to represent other types. Most portantly though, wth regards to the work presented n ths report, a nuber of xng rules have been publshed whch use a g E odel to descrbe the coposton dependee of the EOS paraeters; therefore provdng the desred lnk of the two ethods entoned above. The g E takes nto account lqud phase non-dealty of the syste under nvestgaton and ntroduces t nto the EOS odel. Ths results n ore relable representaton of systes contanng polar or assocatng copounds when usng equatons of state. Furtherore, t has been shown that should predctve g E odels be used, fully predctve EOS odels ay be developed. 2

11 2. MIXING RULES A REVIEW At present one unversal xng rule s not avalable and research s contnung n order to establsh a flexble and uoplcated rule that could predct propertes of ost ultcoponent systes wth a reasonable level of accuracy. The followng secton nvestgates a nuber of ethods for extendng cubc equatons of state to xtures through the calculaton of xture a, b and c (n the case of volue-translated equatons of state) coeffcents. The revew wll start wth the earlest and ost sple xng rule (the quadratc xng rule) before nvestgatng soe eprcal odfcatons that were ntroduced to prove flexblty of the xng rules. The fnal secton provdes a coprehensve revew of the g E xng rules whch use the g E odels to defne the coposton dependee of the xture paraeters. Due to the fact that the cubc Peng-Robnson EOS s used n the current ethod, only rules descrbng xtures n assocaton wth cubc equatons of state wll be dscussed below. Mxng rules used n other types of equatons of state are not revewed here. 2. Quadratc Mxng Rules The classcal quadratc xng rule s by far the ost popular due anly to ts splcty and the relatvely hgh level of accuracy acheved for xtures contanng nonpolar or only slghtly polar copounds. The orgn of ths xng rule les n basc statstcal therodynacs whch s used to descrbe the coposton dependee of the vral coeffcents. The coposton dependee of the vral coeffcents for a xture contanng coponents s gven by: B C D j j = j= j k jk = j= k = j k l jkl = j= k = l= etc. = = = z z B z z z C z z z z D (2.) where z f s the ole fracton of coponent f and B j, C jk, D jkl etc. are sets of vral coeffcents dependent solely on teperature. Obvously B ff, C fff, D ffff etc. are just the pure coponent vral 3

12 coeffcents of f, whle the reanng coeffcents are known as nteracton vral coeffcents. The subscrpt ndcates a xture property. The van der Waals equaton of state for a flud xture can be expanded nto a power seres around zero densty to gve an expresson closely reseblng the vral equaton: b a Z = + = v RTv 2 3 a 2 3 = + b + b + b +... RT v v v (2.2) where Z s the copressblty factor, a and b are the substae specfc equaton of state energy (attracton) and co-volue paraeters, the subscrpt ndcates a xture property, R s the unversal gas constant, T s the teperature and v s the olar volue. Coparson of the densty vral equaton and Equaton (2.2) reveals the followng relatons between the vral coeffcents and the EOS paraeters: B C D etc. a = b RT = b 2 = b 3 (2.3) Fro Equaton (2.) one can see that the second vral coeffcent B has a quadratc dependee on coposton. The EOS coeffcents a and b are lnked to B by Equaton (2.3) and so n order to antan consstey, the coposton dependee of a and b ay be at ost a quadratc futon. Conversely, the thrd vral coeffcent has a cubc coposton dependee whch poses the strcter constrant that the EOS paraeter b should be only a lnear futon of coposton. Slar results ay be found when consderng other equatons of state; however the splest futon for a (fro the van der Waals equaton) s gven by: = z z jaj (2.4) = j= a here z f ay be the ole fracton of pure coponent f n the vapour or lqud phase as cubc equatons of state are capable or representng both phases. a and a jj are the pure coponent a 4

13 paraeters calculated usng crtcal propertes and the equaton assocated wth the EOS beng used, whle a j (=a j ) s the cross ter and s related to a and a jj by the geoetrc ean rule (a cobnng rule) whch contans one adjustable bnary-nteracton paraeter k j (=k j ) and no coposton dependee: 0.5 ( ) ( ) a = a a k (2.5) j jj j When =j, the bnary-nteracton paraeter equals zero and Equaton (2.5) produces the pure coponent a paraeter. The b paraeter s represented ost coonly by the lnear xng rule due to ts splcty: = zb (2.6) = b The equaton above s reasonable as the paraeter b only represents the closest packng volue possble for a partcular type of olecule n a xture and does not requre a coplcated futon for extenson fro pure coponents to a xture. The b paraeter ay also be calculated by an equaton slar to (2.4) and usng the cobnng rule: bj = ( b + bjj )( lj ) (2.7) 2 l j (=l j ) s a bnary-nteracton paraeter that ust be ftted to experental data. Ths added coplexty s only requred for xtures contanng coponents that are hghly asyetrc wth respect to sze and s often not utlzed. As stated above, the quadratc xng rule s, for ost cases, sutable for representaton of phase equlbra n ultcoponent systes whch contan ether non-polar or slghtly polar coponents, however ths xng rule begns to fal severely when appled to systes that contan strongly polar and assocatng copounds. In addton t wll always requre bnarynteracton paraeters specfc to the syste beng nvestgated whch requres experental results and a fttng procedure, reovng the possblty of a fully predctve odel. 5

14 2.2 Coposton-Dependent Cobnng Rules In order to account for the shortcongs of the quadratc xng rule a nuber of eprcal odfcatons have been ade. These odfcatons a at reasng flexblty of the xng rule to correlate phase behavor of xtures contanng strongly polar or assocatng copounds (non-deal solutons), whch s a trat greatly lackng n the orgnal quadratc xng rule. These odfcatons are pleented to prove the calculaton of the cross ters (a j ) and n effect ntroduce soe coposton dependey nto the orgnally coposton-ndependent cobnng rule (Equaton (2.5)) of the classcal quadratc xng rule. The xng rule for a (Equaton (2.4)) reans the sae for ost ethods and only odfcatons to the cross ter calculaton are ade. Chao and Robnson (986) and Stryjek and Vera (986) pleented cobnng rules that requre two bnary-nteracton paraeters k j and k j, as opposed to the sngle bnary-nteracton paraeter (k j =k j ) found n the orgnal quadratc xng rules. The proposed cobnng rules were respectvely: ( ) 0.5 ( ) ( ) a = a a k + k k z (2.8) j jj j j j a 0.5 = ( a a ) j jj k k j j ( zkj + z jk j ) (2.9) The cobnng rule of Chao and Robnson (986) was later extended by Schwartzentruber et al. (987) to lude a thrd paraeter ultately reasng the flexblty and accuracy of the quadratc xng rule but at the sae te reasng ts coplexty. The proposed cobnng rule was gven as: where k j =k j, l j =-l j, j =- j and k =l =0. ( ) 0.5 a = a a k l j jj j j z z j j j z + z j j j (2.0) Whle the coposton-dependent cobnng rules do provde a sple ethod for extenson of equatons of state to xtures contanng non-polar or assocatng copounds, and do so wth a 6

15 satsfactory level of accuracy, they have been found to contan a serous defect. Mchelson and Kstenacher (990) nvestgated coposton-dependent cobnng rules and found that they provde dfferent results when a syste s consdered to be coposed of ts actual coponents to when t s consdered to be coposed of subcoponents of the actual coponents. Sengers et al. (2000) explans the proble as follows: If a bnary xture wth coposton (x, x 2 ) s treated as a ternary syste wth coposton (x, x 2, x 3 ), where the ternary xture s fored by dvdng coponent 2 nto two pseudocoponents wth dentcal propertes, a dfferent value for the paraeter a wll result. Therefore, the calculated propertes wll depend on the nuber of pseudocoponents, whch s n contrast to experental evdee. Many researchers attepted to overcoe the Mchelsen-Kstenacher proble whle antanng the obvous advantages of the coposton-dependent cobnng rules. Mathas et al. (99) produced an equaton that overcae ths proble by addng a new coposton-dependent ter to the classcal quadratc xng rule and not just alterng the cobnng rule as was done by prevous authors assocated wth the flawed coposton-dependent cobnng rules. The new xng rule proposed was: a = z z a a k + z z a a l = j= = j= 0.5 /2 /3 /3 ( ) ( ) j jj j j ( jj ) j (2.) l j ay or ay not equal l j dependng on the level of accuracy and coplexty requred. Twu et al. (99) also proposed a revsed xng rule that overcae the Mchelsen- Kstenacher proble and contaned an extra ter to that of the classcal quadratc one: 3 ( ) ( ) /3 /6 H G a a z j j jj j /2 ( ) ( j= j jj j ) = j= = Gj z j j= a = z z a a k + z 3 (2.2) where: H j k k j j = (2.3) T Gj = exp( βjhj ) (2.4) 7

16 Ths xng rule can ether be used as a 2-paraeter odel by fttng both k j and k j to bnary experental data or t can be used as a ore accurate 4-paraeter odel whereby β j and β j are also ftted. 2.3 Mxng Rules Usng Excess Gbbs Energy (g E )/ Actvty Coeffcent Models The classcal quadratc xng rule can at ost only be appled to xtures of slght soluton non-dealty. In order to prove ths, alteratons to ths rule was necessary whch requred a hgher level of coplexty along wth the fttng of bnary experental data to fnd bnarynteracton paraeters and led to the developent of the coposton-dependent cobnng rules. In soe cases as any as 4 bnary-nteracton paraeters need to be ftted to experental data, whch s obvously not deal. Hghly non-deal solutons (contanng non-polar and assocatng copounds) have been descrbed usng g E (or actvty coeffcent) odels wth great success. Therefore uch effort has been dedcated to cobnng these odels wth equatons of state n order to extend EOS applcablty to non-deal solutons, and consequently utlze the attractve features of both classes of odels. The g E of a soluton can be calculated n two ways: wth the use of an approprate g E odel (ost coon) and wth the use of an EOS va fundaental therodynac equatons. Therefore the EOS xng rule coposton dependee for the lqud phase can be reflected by a desrable g E odel f the dfferent expressons are atched as follows: E E g g EOS γ = (2.5) The subscrpt γ ndcates calculaton by a g E odel (actvty coeffcent γ odel) whle the subscrpt EOS ndcates calculaton by eans of an EOS. If one assues that the lnear xng rule for b (Equaton (2.6)) apples, the a paraeter ay then be found by assung that Equaton (2.5) holds true. 8

17 A nuber of odels exst whch can be used to calculate E g γ such as NRTL, UNIQUAC, UNIFAC and any ore. E g EOS s calculated usng a specfc EOS and the followng equaton: g RT z ϕ E EOS = lnϕ ln = (2.6) where φ and φ are the xture and pure coponent fugacty coeffcents respectvely and are calculated fro the followng equaton whch s specfc to pure coponents: ln Pv ln Pv RT ϕ = + P dv RT RT RT v v (2.7) Equaton (2.7) ay be used to calculate φ as the xture s assued a pure flud wth pure coponent EOS paraeters equal to a and b. It s portant to note that ths expresson ay not be used to calculate fugacty coeffcents of a partcular coponent n the xture. The P ter s represented by a pressure explct EOS and s therefore a futon of v. The ore recent advaeent n the feld of g E odels, through ASOG, UNIFAC and odfed UNIFAC, has seen the developent of accurate odels based on the group-contrbuton coept. These odels do not requre bnary experental data, they only requre group-group nteracton paraeters, and as a result are fully predctve. If a predctve g E odel s used n Equaton (2.5) to develop the g E -xng rule, a predctve group-contrbuton EOS (GCEOS) results. If a predctve ethod s used to deterne the coposton dependee of the EOS paraeters, no bnary experental data s requred and the EOS becoes fully predctve too. Ths s obvously a ajor advantage that was not evdent when the nvestgaton nto g E xng rules frst began due to the aturty of the feld of predctve g E odels, however at present t contnues to drve further research nto ore accurate ethods. 9

18 Cobnng cubc equatons of state wth group-contrbuton g E odels to obtan a xng rule extends the applcablty of cubc equatons of state to the predcton of VLE n three ajor areas:. Polar systes at low pressure, where the GCEOS n essee atches the perforae of the g E odel wth the γ-φ approach. 2. Polar systes at hgh pressure, where conventonal xng rules have been found to fal. 3. Systes that contan supercrtcal coponents, wheren gas olecules are consdered as new groups. The followng sectons wll cover ajor developents ade n the feld of g E xng rules fro the frst dea proposed by Vdal (978) to the developent of the ost recent VTPR odel (developed n early 2000) Vdal Vdal (978) was the frst to use a g E odel to establsh a xng rule for an EOS n order to allow odelng of hghly non-deal systes. The Redlch-Kwong EOS was used and as a result applcaton of Equaton (2.6) yelded the followng expresson: ( b ) E P( v b ) P v geos = RT ln z ln + Pv zpv +... RT = RT = a v + b a v + b... ln + z ln b v = b v (2.8) where v s the xture olar volue, a and b the pure coponent EOS paraeters of coponent and v s the pure coponent olar volue of coponent. Equaton (2.8) stll contans a nuber of unknowns (naely b, v and all v ) that would restrct ts use n fndng a useful expresson for a. In order to elnate these ters fro Equaton (2.8) and arrve at an explct expresson for a an nfnte-pressure lt was appled. Due to the fact that the xture EOS paraeters are ndependent of pressure, calculatons at dfferent pressures should not affect the calculated paraeters assung the paraeters calculated away fro the syste pressure (n ths case at nfnte pressure) are calculated correctly. The nfnte pressure lt allows one to ake the assupton that the flud s copressed to such an extent that the olar volue of the 0

19 flud would be equal to the closest packng volue of the olecules (.e. the olecules are copressed so as to be n contact wth each other, and the only nterolecular space exsts between contactng olecules). The b (co-volue) EOS paraeter represents the closest packng volue of the olecules and therefore the followng assupton could be ade, whch effectvely reoves the unknown flud olar volues (v and v ) fro Equaton (2.8): v = b (2.9) v = b (2.20) It was also noted that n order for the g E to be fnte (.e. not be nfnte) as pressure approaches nfnty, the excess volue v E ust be assued to be zero. If v E s not zero then g E wll approach nfnty as pressure approaches nfnty. Ths can be seen by nvestgatng the followng expresson (whch s developed fro the fundaental property relatons): E E E g = a + Pv (2.2) where a E s the excess Helholtz energy. The above deducton ples that the lnear van der Waals xng rule ust be used for the b paraeter. Substtuton of the lnear b xng rule along wth Equaton (2.9) and (2.20) nto Equaton (2.8) followed by rearrangeent produces the followng expresson whch apples only at nfnte pressure: E a g a = b z (2.22) = b ln 2 where E g s the g E at nfnte pressure and ay be calculated usng a g E odel (assung Equalty (2.5) holds). In usng an nfnte pressure lt t was assued that g E s ndependent of pressure. The recoended odel for g E calculaton was NRTL and the result fro usng ths odel was used drectly n Equaton (2.22) even though the odel paraeters were ftted usng data obtaned fro low to oderate pressure systes. By akng ths assupton one could use exstng odel paraeters to calculate the E g ter wthout havng to reft odel paraeters to hgh

20 pressure data. Although t was known that g E does have a dependey on pressure t was orgnally decded that the dependey was so slght that the tedous odel paraeter fttng process could be avoded. Obvously ths assupton s orrect and wll lead to erroneous results, however the xng rule can be used to provde satsfactory results for hghly non-deal systes f the g E odel paraeters are reftted usng hgh pressure data so that a ore correct value for E g s obtaned. Havng sad ths though, Sengers et al. (2000) ponts out a nuber of theoretcal and coputatonal dffcultes assocated wth the Vdal xng rule. These lude naccurate representatons for non-polar hydrocarbon xtures, falure of the second vral coeffcent boundary condton at the low-densty lt (quadratc coposton dependee see Equaton (2.)) and, as already dscussed, the need to reft g E odel paraeters to account for elevated pressure condtons. Sengers et al. (2000) also states that the g E odel paraeters are strongly dependent on teperature so that, whle t s good for correlatons, t has lted extrapolatve or predctve capablty Huron-Vdal The work of Vdal was later extended slghtly by Huron and Vdal (979), who dd not alter or extend the coepts proposed by Vdal but dentfed that Equaton (2.22) can be generalzed to other equatons of state. The general for proposed by Huron and Vdal was: E a g a = b z (2.23) = b Λ where Λ s a nuercal constant that depends on the partcular EOS that s used. Huron and Vdal (979) derved g E xng rule expressons for a usng the van der Waals, Soave-Redlch- Kwong and Peng-Robnson equatons of state usng the sae procedure as Vdal (978) (whch nvestgated the Redlch-Kwong EOS). 2

21 The followng expressons for Λ were found: van der Waals: Λ = Redlch-Kwong: Λ = ln 2 Soave-Redlch-Kwong: Λ = ln 2 Peng-Robnson: Λ = ln Mollerup It soon becae evdent, followng the work of Vdal and Huron and Vdal, that the g E calculated fro an EOS and fro a g E odel needed to be lnked at low pressure rather than at nfnte pressure so that the large aount of exstng (low pressure) actvty coeffcent odel paraeters could be utlzed, therefore reovng the need to easure data and reft odel paraeters at elevated pressures. Mollerup (986) was the frst to ove the feld n ths drecton by atchng the g E fro the EOS and an approprate odel at zero pressure. In dervng the xng rule Mollerup anaged to avod the assuptons ade by Vdal that g E calculated fro an EOS at nfnte pressure equals that calculated by a g E odel (usng low pressure paraeters) and that the co-volue paraeter b equals the volue v at nfnte pressure. Mollerup stll assued that v E s zero, however ade the assupton that Equaton (2.5) apples at a pressure of a few atospheres or less whch s far ore reasonable as the g E odel paraeters are ost coonly establshed usng low to oderate pressure data. Mollerup (986) derved the followng expresson usng the van der Waals EOS: E g EOS b a b a = ln + z ln + z +... RT v RTv = v = RTv E v Pv... + z ln + = v RT (2.24) If one then assues that saturated lqud volues are ndependent of pressure (a reasonable assupton by all accounts), the pressure ter n the van der Waals EOS ay be neglected (P = 0) and the equaton can be solved for b/v to gve: 3

22 b 4RTb = + v 2 a /2 (2.25) whch ay be appled to both the xture and pure coponents. Mollerup (986) states that at low pressures (less than a few atospheres) Equaton (2.25) s accurate to wthn percent. The followng ters were defned n the dervaton: f b v = (2.26) f b = (2.27) v f c v b = v b (2.28) whch allowed the soluton of Equaton (2.24) wth respect to (a /b ) to be found (also applyng the assupton that v E = 0): E a a f g0 RT fcb = z + z ln b = b f f f = b (2.29) E where g 0 s g E at the zero referee pressure. f (for pure coponents and xtures) s only a weak futon of teperature and n the case of xtures s also dependent on coposton, however Mollerup (986) states that f ay be regarded as a constant for practcal applcatons. Followng ths, Mollerup dentfed that f b/v for the pure coponents and for the xture can be assued equal, then f = f and f c =, and found that at the noral bolng pont f s n the regon of 0.8 for lquds when usng the van der Waals EOS. All of ths appled to Equaton (2.29) results n the followng explct expresson for a : 4

23 a E b a = b z gγ RT z ln = b 0.8 = b (2.30) The assupton that f = f and the assupton that both are equal to soe constant found fro experental data was a huge step forward and has been used n the developent of any g E xng rules se Modfed Huron-Vdal Mchelsen (990) extended the dea proposed by Mollerup and atched the g E at a referee pressure of zero usng the Soave-Redlch-Kwong EOS. In dong so the followng expresson was developed whch n contrast to the Huron-Vdal xng rule s not explct: g RT b ln ( ) ( α ) E 0 + z = q α zh = b = (2.3) where α s a shortcut notaton used to cobne varables n the followng way: a α = b RT a α = b RT (2.32) q and h are futons of α and α respectvely and are gven by: q u + = u,0,0 ( α ) ln ( u ) α ln h,0 u + = u,0,0 ( α ) ln ( u ) α ln,0 (2.33) (2.34) 5

24 u,0 and u,0 are the reduced lqud phase volues of the xture and pure coponents at zero pressure respectvely: u u,0,0 v = b P= 0 v = b P= 0 (2.35) Equaton (2.33) and Equaton (2.34) dsplay the dependee of q and h on the paraeters α and α. u,0 and u,0 n these futons can be expressed as futons of α by frst convertng the Soave-Redlch-Kwong EOS nto the followng general for: Pb α = RT u u( u + ) (2.36) At the referee pressure of zero Equaton (2.36) reduces to: 0 = α u u ( u + ) (2.37) Solvng Equaton (2.37) for u 0 and takng the sallest (lqud) root, produces the followng expresson for u 0 as a futon of α: 2 ( ( 6 ) ) /2 u0 = α α α + (2.38) 2 vald for α > Equaton (2.38) s used to represent both u,0 and u,0. Mchelsen (990) states that g E odel paraeters are based anly on bnary xtures at or near atospherc pressure and under these condtons α are far reoved fro the ltng value, wth typcal values (at the noral bolng pont) rangng fro 0 to 3. As a result of ths, Mchelsen (990) nvestgated the behavor of the q and h-futon wthn ths range, and notced that they vary alost lnearly wth respect to α. Ths can be seen n Fgure below. 6

25 -4 a ) (a h o r ) (a q FIGURE Varaton of q or h-futon wth respect to α for the SRK EOS also showng the lnear ft n the range 0 < α < 3: (---) q or h-futon, ( ) lnear ft The dea of Mchelsen (990) was then to replace the q and h-futon gven by Equatons (2.33) and (2.34) wth lnear approxatons: q( α ) q + qα (2.39) 0 ( α ) 0 h h + hα (2.40) By substtutng these straght lne approxaton nto Equaton (2.3), the rght-hand sde of the equaton s approxated as: q( α ) z h( α ) q + q α h h z α (2.4) 0 0 = = By coparng Equaton (2.39) and Equaton (2.33) one can see: q q ( u ) = ln 0,0 u +,0 = ln u,0 (2.42) 7

26 slarly, for Equaton(2.40) and Equaton (2.34): h h 0,0 ( u ) = ln u +,0 = ln u,0 (2.43) and followng the assupton of Mollerup (986) that the reduced lqud phase volue (tered f n the work of Mollerup) s constant and the sae for the pure coponents as for the xture,.e.: v v,0 = u,0 = = b b P= 0 P= 0 u (2.44) the followng coluson ay be drawn: q q = h = Ψ 0 0 = h = Λ (2.45) where Ψ and Λ are constants. Ths then results n a uch spler verson of Equaton (2.4): q( α ) z h( α ) Λ α z α = = (2.46) Equaton (2.3) can therefore be rearranged nto the followng explct for for calculaton of a : a E b a = b z + gγ + RT z ln = b = b Λ (2.47) Λ s found by fttng a straght lne to a plot of the q-futon (or h-futon) between α-values of 0 and 3 and establshng the slope. Λ depends only on the EOS used as ths deternes the type of q-futon (or h-futon) obtaned n achevng an expresson of for slar to Equaton (2.3). The q and h-futon of Equaton (2.33) and (2.34) are specfc to the Soave- Redlch-Kwong EOS and yelds a Λ value of Mchelsen (990) dd a slar analyss to that of the Soave-Redlch-Kwong EOS usng the Peng-Robnson EOS and the van der Waals 8

27 EOS and found Λ values of and respectvely. Equaton (2.47) becae known as the Modfed Huron-Vdal Frst-Order xng rule (MHV) due to ts slarty to the Huron-Vdal xng rule and the fact that a lnear (frst order) approxaton s used to represent the q and h- futon. A Modfed Huron-Vdal Second-Order xng rule (MHV2) was frst proposed by Mchelsen (990) and later pleented by Dahl and Mchelsen (990), wheren a second order (quadratc) approxaton s used to represent the q and h-futon. Ths akes sense as these futons are not perfectly lnear (see FIGURE above) and ay be better approxated by a quadratc expresson, therefore producng better results. The quadratc approxatons are: q( α ) q + q α + q α (2.48) ( α ) α α h h + h + h (2.49) 0 2 Oe agan one fnds that the constants n the pure coponent and xture approxatons are dentcal: q q q = h = Ψ 0 0 = h = Λ = h = Γ 2 2 (2.50) where Γ s a constant. Use of ths approxaton n equaton (2.3) does provde proved results over the MHV ethod, however there s added coplexty as the resultant expresson s not explct and as stated by Mchelsen (990) the neatness assocated wth a sple explct xng rule s lost. The MHV2 equaton s: E 2 2 gγ b Λ α zα + Γ α zα = + z ln = = RT = b (2.5) The unversal Λ and Γ paraeters can be found by fttng a second-order polynoal to the EOSspecfc approxaton futon. 9

28 The recoended values are: Soave-Redlch-Kwong EOS: Λ = Γ = Peng-Robnson EOS: Λ = Γ = Both MHV and MHV2 have been found not to satsfy the second-vral coeffcent boundary condton. Havng sad ths though, both odels provde very reasonable correlatons and predctons of data obtaned fro experent for systes that are hghly non-deal Wong-Sandler Wong and Sandler (992) proposed a new ethod to lnk the g E odel results wth EOS coputatons n order to obtan a xng rule for the EOS a and b paraeters. Attepts to atch g E at zero pressure were abandoned and the fact that excess Helholtz energy a E s vrtually ndependent of pressure was nvestgated, resultng n a E calculated fro an EOS beng used to develop the xng rule. There are two ajor advantages of usng a E nstead of g E. The frst s that the assupton that v E = 0 s no longer requred as when usng g E and the second, as stated already, s that a E s not as strongly dependent on pressure as g E. The bass of the work done by Wong and Sandler s suarzed by the followng expresson: E (, =, ) = (, =, ) E = a ( T, P = low, z ) E = g ( T, P = low, z ) a T P z a T P z E EOS (2.52) The followng arguent s used n order to arrve at Equaton (2.52): At suffcently low pressures the Pv E ter of Equaton (2.2) s very sall. Ths ples that g E s equvalent to a E at low pressure. a E s essentally ndependent of pressure (or densty) and as a result a E at low pressure s equvalent to a E at nfnte pressure. Therefore the a E of a syste calculated at nfnte pressure usng an EOS ay be equated to the g E of the syste calculated usng a g E odel, whch s essentally a low pressure calculaton (due to the orgnal fttng of odel paraeters usng low-pressure data). The equalty between the g E at low pressure and the a E at nfnte pressure s used to establsh the coposton dependee of the xture EOS paraeters. Wong and Sandler had to also use the 20

29 coposton dependee of the second vral coeffcent to relate the pure coponent a and b paraeters to the equvalent xture paraeters. Equaton (2.) and (2.3) ay be lnked to gve an expresson representng the second vral coeffcent coposton dependee: b a zz jbj RT = j= = (2.53) The cross ter B j s calculated by: aj a a jj B = b = b b k RT 2 + RT RT ( ) j j jj j (2.54) where k j s a bnary paraeter whch s ost coonly regressed usng low-pressure experental data. Calculaton of a E fro a van der Waals type EOS at nfnte pressure results n the followng expresson: a a E = Λ z b = b a (2.55) where Λ s a constant dependent on the EOS used. For the Soave-Redlch-Kwong and Peng- Robnson EOS Λ s equal to and respectvely. Usng Equaton (2.52) Wong and Sandler were able to convert Equaton (2.55) nto the followng for: g E a γ a = b z = b Λ (2.56) The b paraeter s not calculated by the sple lnear xng rule n the Wong-Sandler ethod, nstead t was ensured that the second vral coeffcent coposton condton s satsfed. Ths was acheved by substtutng the expresson for a (Equaton (2.56)) nto the equaton representng the second vral coeffcent coposton dependee (Equaton (2.53)) and rearrangng to get an explct futon for b : 2

30 b x x B j j = j= = gγ ( T, low P, z ) a + z RT = b RT E (2.57) If Q and D are defned as follows: Q = zz jbj (2.58) = j= a g D = z + RT = b E γ ( T, low P, z ) Λ (2.59) then the Wong-Sandler xng rule ay expressed as: a D = RTQ D (2.60) b Q = (2.6) D PSRK Holderbau and Gehlng (99) developed a group-contrbuton EOS that cobned the SRK EOS and the UNIFAC ethod. The ethod s known as Predctve Soave-Redlch-Kwong (PSRK) due to ts predctve abltes (as there was no ntroducton of new paraeters whch would requre a fttng procedure, only exstng UNIFAC group-nteracton paraeters and pure coponent paraeters are requred). The PSRK odel can be used for predctons of VLE over a teperature and pressure range uch wder than that possble wth UNIFAC, and ay also be easly extended for use n supercrtcal systes, whch s not possble wth the use of a g E odel. Holderbau and Gehlng (99) reveals that the PSRK odel uses the sple MHV xng rule (Equaton (2.47)) dfferng only n the value of Λ whch s changed fro to , however not uch nsght s provded as to how ths value was deterned. 22

31 The PSRK xng rule s: a E b a = b z gγ + RT z ln = b = b (2.62) The new Λ value though was found to provde uch proved results at elevated pressure whch led Holderbau and Gehlng (99) to colude that the PSRK equaton s especally suted for condtons, where use of a γ-φ-approach s dffcult (.e. when the real behavor of the vapour phase s unknown and not neglgble) or nadequate (.e. when supercrtcal coponents are present). Followng the nnovatve work of Wong and Sandler (992), Fscher and Gehlng (996) provded an alternatve dervaton of the PSRK odel whch was based on a E as opposed to the fugacty coeffcents. Ths dervaton provdes ore nsght nto the value of Λ. In dervng the PSRK equaton Fscher and Gehlng (996) akes two an assuptons:. The excess volue v E s zero (neglgble), whch s an assupton ade durng the dervaton of any g E xng rules. 2. The reduced lqud phase volue u s assued constant,.e.: = = v = v = (2.63) u u u b b The second assupton was valdated by calculatng lqud olar volues of a large nuber of pure coponents at noral pressure and bolng teperature and subsequently dvdng ths value by b (whch s calculated usng pure coponent data). The values of u were found to vary only slghtly fro 0.9 for hghly polar coponents (ethanol and water) to.2 for non-polar coponents (ethane, propane, butane etc.), wth an average value of.. If one consders that at nfnte pressure u =, a value of. at atospherc pressure s not unreasonable due to the fact that lquds ay be copressed only slghtly. As a result of u beng estated at atospherc pressure the referee pressure of the PSRK odel s n the regon of atosphere (and not zero as s the case n the MHV xng rules). 23

32 Through ths alternatve dervaton t s seen that the value of Λ n Equaton (2.47) s equvalent to a futon of u as follows: u Λ = ln u + (2.64) Substtuton of u =. results n the value of Λ equal to whch s the value prescrbed earler by Holderbau and Gehlng (99). Fro Equaton (2.64) one can also see that the constant u value assupton nherent n the MHV odel requres a value of.235 for u at zero pressure, whch s also not unreasonable as the olar volue of the lqud phase would rease slghtly under reduced pressure condtons. So, the MHV odel uses a zero pressure referee, whle the PSRK odel assues a referee state at atospherc pressure. g E odel paraeters are ost coonly ftted usng low pressure VLE data (not zero pressure data), therefore by usng a referee pressure n the regon of atosphere ore accurate results wll be produced. In developng the PSRK odel, other than alterng the xng rule for a, Holderbau and Gehlng (99) also decded to replace the orgnal teperature dependee of the pure coponent a paraeter (α futon) gven by Soave (972) wth that gven by Mathas and Copean (983). Ths odfcaton was ade n order to extend the applcablty of the PSRK odel to polar xtures, as the orgnal teperature dependee gven by Soave fals to provde suffcently accurate vapour pressure data for polar substaes. The Mathas-Copean expresson s found to provde uch proved representaton of pure coponent vapour pressures, whch obvously proves the relablty of predctons for polar xtures. The only downsde of ths s that the Mathas-Copean expresson requres three adjustable paraeters whch ust be ftted to pure coponent vapour pressure data (whch ay not be readly avalable), whle the Soave expresson only requres the pure coponent acentrc factors and crtcal teperatures. As ponted out by Ahlers and Gehlng (2002a), the Mathas-Copean α futon also fals at elevated teperatures. There are a nuber of advantages assocated wth the PSRK odel such as the ablty to provde accurate predctons of VLE over a large pressure range (.e. use UNIFAC paraeters ftted at low pressure for predctons at hgh teperature and pressure) and the fact that paraeters assocated wth any g E odel do not have to be altered but ay be used drectly n the odel. For exaple, should the NRTL odel be used nstead of UNIFAC, the exstng nteracton 24

33 paraeters ay be used wthout odfcaton. Holderbau and Gehlng (99) also states that the an advantage of equatons of state n coparson to g E odels (γ-φ approach) s ther ablty to represent phase equlbra of systes that contan supercrtcal coponents and because of ths the UNIFAC nteracton paraeter table has been extended to lude gases (eg. CO 2, CH 4, N 2 etc.) for use n the PSRK odel. Ths extenson has been contnuous and had vast aounts of research nvested nto t by Holderbau and Gehlng (99), Fscher and Gehlng (996), Gehlng et al. (997), Horstann et al. (2000) and Horstann et al. (2005) to nae a few, and as a result has seen the addton of well over 30 new groups to the orgnal UNIFAC groups. Ths extenson has obvously greatly reased the range of applcablty of the PSRK odel and ths, along wth the fact that t predcts relable results, has ade t a very portant tool to checal engneers. Havng sad ths though, there are also a nuber of ltatons related to the PSRK odel. Fscher and Gehlng (996) dentfes two ajor shortcongs of the ethod. The frst s n the ablty of PSRK to descrbe water-alkane systes (a proble nherent n the UNIFAC ethod) and second s the predcton of too hgh bubble pont pressures n systes that contan coponents that dffer greatly n sze. Ahlers and Gehlng (2002a) dentfes 4 ajor probles wth the PSRK odel and n dong so strengthens the arguent for the developent of an proved odel (see VTPR below): ) Predcted lqud denstes devate fro experental values n a slar way to the basc EOS (the SRK EOS). 2) The Mathas-Copean α futon provdes unreasonable results at hgher reduced teperatures. 3) Predctons of VLE for asyetrc systes are often unsatsfactory. 4) Predctons of excess enthalpes (h E ) and nfnte dluton actvty coeffcents (γ ) are poor LCVM Boukouvalas et al. (994) proposed an nterestng odel that ade use of both the orgnal g E xng rule proposed by Vdal (978) and that proposed by Mchelsen (990) (MHV). The α- ter produced by both odels s lnked va a lnear futon, and for ths reason the odel of Boukouvalas et al. (994) s called the lnear cobnaton of Vdal and Mchelsen (LCVM) xng rule. The LCVM odel akes use of a odfed and translated Peng-Robnson EOS and 25

34 the orgnal UNIFAC g E odel, however any EOS and g E odel ay be used n ths xng rule. The developent of ths odel was drven by the falure of the PSRK odel when predctng VLE of xtures that contan coponents that dffer greatly n sze (hghly asyetrc systes). As a result the LCVM odel has been found to provde satsfactory results for systes of dsslar coponent sze and systes that contan non-polar and polar coponents at low and hgh pressure. The forulaton of the LCVM xng rule was based on two fundaental observatons: ) Both the Vdal and MHV xng rules ay be used at any pressure, rrespectve of the referee pressure used n ther developent (nfnte pressure for Vdal, zero pressure for MHV). 2) The atheatcal expresson representng both odels are very slar (see Equatons (2.23) and (2.47)), the only dfferees beng the nuercal value represented by Λ and the presee of Σz (b /b ), a Florry-Huggns-type ter n the MHV xng rule. The reason for cobnng the two ethods s due to the fact that n hghly asyetrc systes the Vdal odel has been found to under-predct bubble pont pressures whle the MHV odel has been found to over-predct bubble pont pressures whch ay be seen clearly n FIGURE 3 below. If the two ethods were cobned n such a way that the over-predcton of the MHV odel was copensated for by the under-predcton of the Vdal odel, accurate results could be obtaned overall. It was therefore proposed to have a lnear cobnaton of α calculated fro the Vdal odel (sybolzed by α V ) and α calculated fro the MHV odel (sybolzed by α MHV ), n order to calculate the true α. The proposed cobnaton was: α = λα + ( λ) α (2.65) V MHV where λ s a constant that deternes the relatve contrbutons to α by α V and α MHV. When λ=0 α s sply α MHV and when λ= α s sply α V. The LCVM xng rule ay also be represented as follows by substtutng the α-for of the Vdal and MHV xng rules nto Equaton (2.65): α g α (2.66) E λ λ γ λ b = + + z ln + z ΛV Λ MHV RT Λ MHV = b = 26

35 where Λ V and Λ MHV are the Λ values of the Vdal and MHV xng rules respectvely. Λ V reans uhanged fro the orgnally proposed value of for the Peng-Robnson EOS. On the other hand though, Boukouvalas et al. decded to ft the lnear approxaton of the q- futon found n the dervaton of the MHV xng rule for the Peng-Robnson EOS over a wder nterval of α, reasng the nterval fro (0, 3) to (6, 20). Ths causes a change n the slope of the straght lne approxaton (and hee the Λ MHV paraeter) fro to In order to deterne the value of λ (.e. establsh to what degree α MHV and α V contrbute to α ), results for the bubble pont pressure of any ethane/n-alkane systes were calculated usng values of λ rangng fro 0 to. Systes contanng ethane (sall) and large alkanes were nvestgated as the a of the LCVM ethod was to overcoe the proble assocated wth representaton of systes contanng coponents that dffer greatly n sze. The average absolute error for each λ was then establshed by coparson wth experental data. The result of ths analyss by Boukouvalas et al. ay be seen n FIGURE 2 below r e r e s u 60 P le b 50 u B 40 ) n (% 30 E A C 2 /nc 5 (377 o K) ? FIGURE 2 Average absolute % error (AAE) n predcted bubble pont pressure for ethane/n-alkane systes as a futon of the λ value (taken for Boukouvalas et al. (994)) Fro the results t s obvous that n systes contanng coponents of slar sze, such as C 2 /nc 5, MHV (λ=0) provdes the best results but the results provded by Vdal (λ=) are stll 27

36 farly good wth an average absolute error (AAE) well below 0%. However, as the sze dfferee between coponents begns to rease t becoes evdent that there exsts a value of λ for whch the AAE can be nzed as the Vdal and MHV odels begn to fal severely. For exaple f one nvestgates just the C 2 /nc 28 curve of FIGURE 2, the Vdal odel produces results wth an AAE of around 70% and the MHV odel yelds results wth an AAE n excess of 80%, however the correct cobnaton of the two odels allows predctons wth an AAE n the regon of 0%. Boukouvalas et al. (994) suggests an optu value (or reasonable coprose) for λ of 0.36 when usng UNIFAC whch was found by nvestgatng dfferent bnary systes (ludng systes of slar and greatly dfferent coponent sze) at hgh and low pressure, usng varyng values of λ and focusng specfcally on acceptable predcton of bubble pont pressures and vapour phase copostons. The result of usng λ = 0.36 for the ost asyetrcal syste tested, the ethane/nc 28 syste, ay be seen n FIGURE 3. In coparson to the Vdal and MHV odels, the results of the LCVM odel are seen to be uch ore accurate whch confrs the relablty of ths odel for predctons of asyetrc systes. A slar nvestgaton revealed that f odfed UNIFAC s used, the value of λ falls n the range fro 0.65 to a r ) (b 50 r e r e s u 40 P le b u 30 B 20 0 Experental MHV LCVM Vdal x ethane FIGURE 3 Predcton of the bubble pont pressure for the syste ethane / nc 28 at 373 K (taken fro Boukouvalas et al. (994)) 28

37 Voutsas et al. (996) provdes an extensve coparson of the 4 odels: MHV2, PSRK, Wong- Sandler and LCVM, nvestgatng the predcton of VLE n asyetrc systes by these odels. It was coluded that LCVM was the only odel of the four to provde satsfactory results. The sae paper also states that the LCVM odel has been successfully used to predct VLE n a range of systes of varyng coplexty VTPR It has long been known that cubc equatons of state lack the requred accuracy when t coes to representaton of saturated lqud denstes. Ahlers and Gehlng (200) dentfed ths and realzed that by startng fro an proved CEOS (one that was better equpped to calculate saturated lqud denstes) an proved group contrbuton EOS could be developed. It was wth ths n nd that Ahlers and Gehlng began developent of a group contrbuton EOS that would prove the probles assocated wth the already hghly regarded and successful PSRK odel. Over the next 3 years, Ahlers et al. (Ahlers and Gehlng (2002a), Ahlers and Gehlng (2002b), Wang et al. (2003), Ahlers et al. (2004)) developed ths coept further n a 5-part seres and cae up wth the already successful Volue Translated Peng-Robnson (VTPR) group contrbuton EOS capable of copletely replacng PSRK. As stated already, Ahlers and Gehlng (200) realzed that proveents could be ade to exstng g E xng rules by sply usng a better EOS. The coept of applyng a volue translaton to an exstng EOS, proposed by Peneloux and Freze (982), was therefore utlzed n conjuton wth the Peng-Robnson EOS as follows: P RT a = v + c b ( v + c )( v + c + b ) + b ( v + c b ) (2.67) where c s the translaton paraeter and effectvely shfts or translates each v ter n the EOS. The volue translaton has no effect on VLE calculatons and sply provdes sgnfcant proveents n the descrpton of saturated lqud denstes. 29

38 The pure coponent c paraeter can be deterned by calculatng the dfferee n experental and calculated denstes at a reduced teperature T r = 0.7: c = v v (2.68) exp calc where the subscrpts exp and calc represent olar volues v obtaned fro experent and calculated fro the EOS respectvely. Ahlers and Gehlng (200) found that, should no experental data for lqud denstes be avalable, c ay also be calculated drectly fro crtcal data as follows: RTc c = (.5448Z c ) (2.69) P c Equaton (2.69) was found by a fttng procedure (explaned by Ahlers and Gehlng (200)) whch nvolved nvestgatng lqud denstes of 44 pure coponents of dfferent fales (alkanes, aroatcs, ketones, alcohols and refrgerants). The results for predctons of lqud denstes by the VTPR EOS were copared to that of the Peng-Robnson and Soave-Redlch- Kwong EOS and t was found to be by far the ost accurate odel. Over a teperature range 0.3 < T r < the VTPR EOS (usng Equaton (2.69)) was found to have the lowest devaton n lqud densty fro experental results wth a value of 4.%. The Soave-Redlch-Kwong and Peng- Robnson equatons of state devated fro experental results by 3.3% and 6.9% respectvely. Near the crtcal teperature the change n v wth change n T (.e. the slope dv/dt) s extree and as a result the volue translaton coept fals, therefore VTPR ay not be used for the entre teperature range. Ahlers and Gehlng (2002a) says that t s not recoended to use the VTPR EOS to calculate pure coponent and xture lqud denstes at reduced teperatures greater than 0.8. Ahlers and Gehlng (200) also dentfed ths proble and as a result nvestgated the use of a teperature dependent translaton ter c(t) as they began explorng the use of volue translaton for an proved group contrbuton EOS. Through ths nvestgaton an alternate ethod to the VTPR EOS was developed whch used the teperature dependent volue translaton and was called the T-VTPR EOS. The T-VTPR ethod was found to provde accurate predctons of lqud denstes rght up to the crtcal pont (T r = ) however representatons of VLE at hgh pressure proved to be hghly unreasonable. As a result t was decded to base future developents on the VTPR EOS and not the T-VTPR EOS. 30

39 Obvously the proved descrpton of pure coponent lqud denstes also leads to better representaton of xtures, therefore elnatng one of the ajor probles found when usng the PSRK odel. In tryng to provde an proved g E xng rule and replace PSRK, Ahlers and Gehlng (2002a) dentfed 3 other areas n whch the PSRK shows weaknesses:. The Mathas-Copean α futon, used n the calculaton of pure coponent a paraeters, s therodynacally orrect at hgher reduced teperatures 2. The predcton of asyetrc systes often delvers unsatsfactory results. 3. Predctons of excess enthalpes h E and nfnte dluton actvty coeffcents γ are poor. In order to ensure that the VTPR odel was an proveent over the PSRK odel, a nuber of alteratons were ade to deas used n PSRK. Frstly, the Mathas-Copean α futon (Mathas and Copean (983)) was replaced wth the Twu α futon (Twu et al. (99), Twu et al. (995)) for the calculaton of pure coponent EOS a values. The Twu α futon provdes uch ore relable reproducton of pure coponent vapour pressures and has been shown to operate reasonably at elevated teperatures (.e. shows reasonable teperature extrapolatons). At very hgh teperatures the α futon should approach zero and t s under these condtons that the Mathas-Copean α futon fals. In order to prove predctons of hghly asyetrc systes (weakness 2 above), Ahlers et al. dentfed that developent of the VTPR odel requred the ntroducton of proved xng rules for calculaton of the a and b paraeters. Iproveents to the xng rule for a revolved around the fact that n the PSRK odel there are two paraeters whch represent a slar property but have dfferent values. These paraeters are the co-volue of the pure coponents b (used n the EOS) and the relatve van der Waals volue r (used n the UNIFAC odel). As the degree of asyetry reases these values are found to becoe reasngly dfferent (see FIGURE 4 below). 3

40 35 30 a n e /b eth b a n e r /r eth Nuber of Carbon Atos FIGURE 4 Change n the quotents r alkane /r ethane and b alkane /b ethane n dependee of the degree of asyetry of the syste: (o) paraeter b (PR EOS); (*) relatve van der Waals volue paraeter r (taken fro Ahlers and Gehlng (2002a)) Ratos relatve to ethane are used n FIGURE 4, as the an focus of ths study was behavor at dfferent levels of asyetry. Based on proveents ade to the PSRK odel by L et al. (998), whch ntroduced an eprcal correcton for ore relable predctons of asyetrc systes, t was decded that the relatve van der Waals volue r should be replaced by the covolue b n the proved xng rule odel. Usng ths alteraton, Ahlers and Gehlng (2002a) proved that the PSRK xng rule ay be greatly splfed and proved for asyetrc syste predctons. The followng arguent was presented to do so:. g E calculated usng the UNIFAC odel consst of two parts, the cobnatoral and the resdual part: E E E gγ = gcob + gres (2.70) The cobnatoral part s calculated as follows usng pure coponent relatve van der Waals volues r and surface areas q : E gcob = RT z lnv + 5 zq ln F = = (2.7) 32

41 where V and F are the volue to ole fracton rato and the surface area to ole fracton rato respectvely, calculated by: F = V = j= j= q r z q j z r j j j (2.72). The Florry-Huggns ter Σz ln(b /b ) n the PSRK xng rule (Equaton (2.62)) can be rearranged nto the followng for: b = ' z ln z lnv = b = (2.73) where: V ' = j= b z b j j (2.74). Now applyng the fact that r s replaced wth b, one can see that the altered for of the Florry-Huggns ter (Equaton (2.73)) s the negatve equvalent of the frst suaton ter n the cobnatoral part of the UNIFAC odel (Equaton (2.7)) and therefore the two ters can be caeled n the PSRK xng rule. In addton, the second suaton n the cobnatoral part provdes only a sall contrbuton (relatve to the frst suaton ter) and s therefore regarded as neglgble. As a result of the above explanaton, the g E ter found n the PSRK xng rule need only be represented by the resdual part of the UNIFAC odel (whch also eans that the relatve van der Waals volue paraeter r s no longer requred n the xng rule) as the Florry-Huggns ter dsappears. 33

42 The VTPR xng rule for a s therefore a splfed verson of PSRK gven by: E a g res a = b z (2.75) = b g E res s calculated n the sae way as n the UNIFAC odel. The constant Λ found n MHV (Equaton (2.47)) s calculated fro a futon specfc to the EOS used and depends on the referee state of the xng rule (whch affects the reduced lqud phase volue u). For the Peng-Robnson EOS Λ s calculated fro: ( 2 ) ( ) u + Λ = ln 2 2 u (2.76) The referee state of the VTPR ethod s atospherc pressure (sae as PSRK) and under these condtons u was deterned for 75 copounds n a slar way to that descrbed for PSRK above, but usng the Peng-Robnson EOS nstead of the Soave-Redlch-Kwong EOS. u was assued constant (as n PSRK Equaton (2.63)) and an average value of u = was calculated. Usng ths value of u the Λ paraeter n VTPR s To further prove the perforae of the VTPR odel wth respect to asyetrc systes, Ahlers and Gehlng (2002a) appled the work of Chen et al. (2002), whch dentfed that n asyetrc systes the xng rule for b has a larger nfluee than prevously antcpated and as a result proposed a quadratc xng rule for the b paraeter: = zz jbj (2.77) = j= b whereby the followng cobnng rule apples: b j 3/4 3/4 b + b j = 2 4/3 (2.78) The above xng rule replaces the lnear verson used n the PSRK odel and has been found to sgnfcantly prove predctons of asyetrc systes. 34

43 The volue translaton paraeter c s extended for use n xtures by a sple lnear xng rule: = zc (2.79) = c No ndcaton s gven n the lterature coverng the developent of the VTPR odel as to why ths rule was selected, however t s very sple and results produced wth the odel ndcate that the rule works adequately. The last area of weakness found n the PSRK odel that Ahlers et al. aed to elnate n the new VTPR odel was the predcton of excess enthalpes h E and nfnte dluton actvty coeffcents γ. To do ths they proposed to use teperature-dependent nteracton paraeters whch would be sultaneously ftted to VLE, h E and γ data. Ahlers and Gehlng (2002b) gve detaled nforaton as to how ths fttng procedure was perfored and provdes nsght nto the objectve futon used. The reason for usng teperature-dependent nteracton paraeters s to ensure relable predctons for gas-alkane systes, whch cover a large pressure and teperature range. As a result, the odfed UNIFAC odel s used n VTPR however the group nteracton paraeters used are specfc to VTPR (.e. exstng odfed UNIFAC paraeters are not used). 35

44 3. THE VGTPR MODEL To date the PSRK odel has becoe by far the ost successful g E xng rule developed and s found n ost checal engneerng sulaton software. Its success ay be attrbuted anly to ts relablty, avalablty of paraeters and range of applcablty. However, as ponted out by Ahlers and Gehlng (2002a) and dscussed n secton above, the PSRK odel does exhbt soe serous defects. In order to overcoe these, a nuber of odfcatons were ade to PSRK, resultng n the developent of the VTPR odel. The odfcatons ade n developent of the VTPR odel provded an proved xng rule over PSRK. The VTPR odel allowed better: predctons of saturated lqud volues (through a volue translated EOS), representaton of asyetrc systes (by alterng the xng rules), pure coponent vapour pressure representaton (wth the use of the Twu α futon) and calculatons of h E and γ (by refttng odfed UNIFAC (Dortund) paraeters to a larger experental data base). Although the VTPR xng rule provdes a ajor step forward n the feld of xture representaton n equatons of state, t does contan one ajor dsadvantage whch has resulted n PSRK reanng as the xng rule of choce. VTPR uses the odfed UNIFAC (Dortund) ethod to calculate g E, however t s unable to utlze the large aount of exstng odfed UNIFAC (Dortund) group-group nteracton paraeters. Cubc equatons of state contan ther own teperature dependee whch n cobnaton wth the teperature dependee of the odfed UNIFAC (Dortund) paraeters leads to erroneous results. To overcoe ths proble the odfed UNIFAC (Dortund) group-group nteracton paraeters have to be re-regressesed specfcally for use n the VTPR odel. In order to prove calculatons of a nuber of propertes when usng the VTPR odel, these paraeters are regressed usng an objectve futon that takes nto account the dfferees n experental and calculated values of the followng: - VLE of noral and hgh bolng coponents - Gas solubltes - Infnte dluton actvty coeffcents - Excess enthalpes - Excess heat capactes 36

45 - Lqud-lqud equlbra - Sold-lqud equlbra Lookng at the lst above, t s understandable that ths regresson s very coplcated and requres great care to obey all specfc boundary condtons, whch results n a very te consung exercse. As a result the VTPR odel offers only a very lted nuber of groupgroup nteracton paraeters, whch severely lts ts avalablty for use n calculatons. Although new paraeters are beng contnuously added, t has taken approxately 8 years for the VTPR paraeter atrx to reach ts current state dsplayed n Fgure 5 below. The PSRK odel on the other hand has a very large paraeter atrx (whch ay be seen n Fgure 6) due to contnuous extenson over the past 8 years and as a result t wll take any ore years untl the VTPR odel reaches the aturty level of PSRK. Untl ths happens VTPR wll contnue to fall n the shadow of PSRK despte the obvous advantages t has over the PSRK odel. 37

46 FIGURE 5 Current status of the odfed UNIFAC paraeters avalable for use n the VTPR odel (taken fro DDBSP - Gehlng et al. (2009)) 38

47 FIGURE 6 Current status of the UNIFAC paraeters avalable for use n the PSRK odel (taken fro DDBSP - Gehlng et al. (2009)) 39

48 FIGURE 7 Current status of the odfed UNIFAC (Dortund) paraeters avalable for drect use n the new VGTPR odel (taken fro DDBSP Gehlng et al. (2009)) The dea nvestgated n ths work then nvolves keepng the VTPR odel, whle however utlzng the vast aount of exstng (and contnuously advang) odfed UNIFAC (Dortund) odel paraeters nstead of undertakng the tedous and dffcult task of reregressng the for use n VTPR. Ths wll ensure that the new odel wll not only nhert the advantages of VTPR but wll also rease ts range of applcablty up to and even beyond that of PSRK. In other words the ltng factor assocated wth the VTPR odel wll be elnated. 40