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2 14 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? Blanca E. García-Flores 1, Damler. Justo-García 2, Roumana P. Stateva 3 and Fernando García-Sánchez 1 1 Laboratory of Thermodynamcs, Research Program n Molecular Engneerng, Mexcan Petroleum Insttute, Mexco, D.F., 2 Department of Chemcal and Petroleum Engneerng, ESIQIE, atonal Polytechnc Insttute, Mexco, D.F., 3 Insttute of Chemcal Engneerng, Bulgaran Academy of Scences, Sofa, 1,2 Méxco, 3 Bulgara 1. Introducton In the past decades, natural gas processng has ncreasngly encompassed low temperatures wth the vew to recover ethane and heavy components from natural gas. Lquefed natural gas (LG) s a useful method for storng the gas n a small space at peak-shavng facltes of natural gas dstrbuton companes. The ncreased densty of LG facltes allows transportaton of natural gas va large ocean vessels from gas felds stuated far from ther potental markets. Also, n the past, natural gas has been an mportant source of chemcal feed stocks lke ethane, propane, butane, etc. In order to ncrease the recovery of these stocks, the gas processes have been shftng to lower temperatures where the recoveres are mproved. In all of the above processes and applcatons of natural gas, knowledge of the LG systems phase behavor and thermodynamc propertes s requred for the successful desgn and operaton of LG plants. The queston of how to predct and model phase equlbra behavor for natural gas systems s far from new and lqud-vapor equlbra problems have been successfully solved for many years now. At present, nterest has shfted to systems contanng not only speces n the smple paraffnc homologous seres, but also water, carbon doxde, hydrogen sulfde, hydrogen, ntrogen to menton a few. ot unrelated to ths growng varety of speces n cryogenc process streams s the occurrence of multphase phenomena. The stuaton of nterest n natural gas processng s often a methane-rch stream for whch there s the queston whether a second solvent can alter dramatcally the pattern of the phase behavor of the customarly lqud-vapor mxture and cause problems.

3 360 Advances n atural Gas Technology The actve nterest n the use of ntrogen gas to pressurze ol reservors to enhance recovery has resulted n natural gas process streams, rch n ntrogen, whch are lkely to dsplay complex phase behavor. Investgators have studed expermentally ternary prototype LG systems, contanng ntrogen as a second solvent, and a lot of excellent data have been publshed. However, n order to help further understandng the possble occurrence of multphase equlbra n LG process systems, t s necessary to acqure knowledge of ther phase behavor and of the varety of crtcal end pont boundares through an ablty to predct, model and calculate them. There are several aspects of separaton/refnement of natural and synthetc gases where multphase equlbra play a role. The occurrence of lqud-lqud-vapor (LLV) behavor durng the recovery of natural gas by low temperature dstllaton, especally from ntrogenrch gas mxtures, s just one such example. LLV behavor can also occur n the processng of gases contanng hgh quanttes of carbon monoxde that result from coal gasfcaton, as well as n hgh pressure absorpton processes for the removal of ether desrable or undesrable components from natural and synthetc gas mxtures. In these latter processes, methanol can be used as the absorber to separate out feed stocks (ethane, ethylene, propane, propylene, etc.), harmful components such as hydrogen sulfde, carbon dsulfde, and carbonyl sulfde, or smply unwanted speces such as carbon doxde. Obvously, the formaton of a second lqud phase can upset the expected performance of these processes. In the cryogenc processng of natural gas mxtures, speces such as CO 2 and the heaver hydrocarbons can form solds and foul the gas processng equpment. For example, the formaton of a sold phase can coat heat exchangers and foul expanson devces, leadng to process shutdown and/or costly repars. Knowledge of the precptaton condtons for gas streams s essental n mnmzng downtme for cleanup and repars. However, the appearance of a sold phase n a process s not always a lablty. Off-gas (prmary ntrogen) from power plants wth lght water reactors or from fuel processng plants can contan radoactve sotopes of krypton and xenon, whch may be removed by sold precptaton. The phenomenologcal aspects of LLV/sold-lqud-vapor (SLV) behavor are also nterestng because there s a need for a better understandng of the physcal nature of the thermodynamc phase space, especally the non-analytctes of crtcal pont regon. The success of the desgn and operaton of separaton processes n the ol and gas ndustry at low temperatures s crtcally dependent on accurate descrptons of the thermodynamc propertes and phase behavor of the concerned multcomponent hydrocarbon mxtures wth norganc gases. Consequently, t s mportant to apply approprate models wthn a thermodynamc modelng framework to predct, descrbe and valdate the complex phase behavor of LG mxtures. In ths case, equatons of state (EoS) are usually the prmary choce. The am n ths chapter s to examne and analyze the challenges and dffcultes encountered when modelng the complex phase behavor of LG systems, and to compare the capabltes of two numercal technques advocated for phase behavor predctons and calculatons of complex multcomponent systems. The chapter s organzed as follows. In Secton 2 the thermodynamcs and topography of the phase behavor of LG systems are presented. Secton 3 descrbes two computatonal technques, advocated by us, to predct and model complex phase behavor of nondeal mxtures, and ther applcaton to LG systems. Secton 4 s focused to the descrpton of two numercal methods to drectly calculate crtcal end ponts (K- and L-ponts). In Secton 5 two representatves of the

4 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 361 equatons of state (EoSs) type thermodynamc models, namely SRK EoS (Soave, 1972) and PC-SAFT EoS (Gross and Sadowsk, 2001) are used to represent the equlbrum flud phases. Results and dscusson of the multphase behavor modelng for selected ternary systems studed are presented n Secton 6. Fnally, conclusons derved from the work are gven n Secton Thermodynamcs and topography of the phase behavor of LG systems Though only a lmted number of mmscble bnary systems (methane n-hexane, methane n-heptane, to name the most promnent ones) are relevant to natural gas processng, LLV behavor can and does occur under certan condtons n ternary and hgher LG systems even when none of the consttuent bnares themselves exhbt such behavor. It s also known that the addton of ntrogen to mscble LG systems can nduce mmscblty and ths necessarly affects the process desgn for these systems. The qualtatve classfcaton of natural gas systems and the topography of the multphase equlbrum behavor of the systems n the thermodynamc phase space and the nature of the phase boundares wll be addressed n ths secton. Kohn and Luks (1976, 1977, 1978), who carred out extensve expermental studes on the solublty of hydrocarbons n LG and GL (natural gas lquds) mxtures, qualtatvely classfy natural gas systems (or any system) as one of four types (Kohn and Luks, 1976). These types of bnary systems show, for nstance, that a frst type system, methane-n-octane (Kohn and Bradsh, 1964), has a sold-lqud-vapor (S-L-V) locus whch starts at the trple pont of the solute and termnates at a Q-pont (S 1 -S 2 -L-V) near the trple pont of the solvent wth an S-V gap n the locus. The S-L-V branches termnate wth a K-pont where the lqud becomes crtcal wth vapor n the presence of a sold. On the contrary, methane-n-heptane (Kohn, 1961), a second type system, has a Q-pont (S-L 1 -L 2 -V) n the central porton of the SLV locus, at whch pont one loses the L 1 phase and gans the L 2 phase (solvent lean phase). These systems also have a L 1 -L 2 -V locus runnng from the Q-pont to a K-pont where the lqud L 2 becomes crtcal wth vapor n the presence of L 1. The other two types of systems have no dscontnutes between the trple ponts of the solvents; however, one of these types, methane-n-hexane (Shm and Kohn, 1962), a typcal thrd type system, has a L 1 -L 2 -V locus whch s termnated by crtcal ponts, ponts where the two lqud phases, or a lqud and vapor, become crtcally dentcal. A typcal system of the fourth type s ethane-n-octane (Kohn et al., 1976). The topographcal evoluton of multphase equlbra behavor s dscussed n detal by Luks and Kohn (1978). Ternary ( 3 ) or more complex systems exhbt smlar types of phenomena, but the loc and exact ponts of a bnary become 1 and 2 dmensonal spaces n the more complex systems. It s known that systems rch n methane can exhbt L 1 -L 2 -V behavor of the second and thrd type varety (e.g., methane-n-heptane and methane-n-hexane, respectvely). However, nvestgatons nto ternary S-L-V and more complex systems revealed that L 1 -L 2 -V behavor can occur n systems whose bnary pars exhbt no mmscblty (Green et al., 1976; Orozco et al., 1977). Hottovy et al. (1981, 1982) observed that systems of the frst type (methane-noctane) could be modfed to behave lke a second type system by addng a second solvent (e.g., methane-ethane-n-octane, methane-propane-n-octane, methane-n-butane-n-octane, and

5 362 Advances n atural Gas Technology methane-carbon doxde-n-octane). Merrll et al. (1983) reported the phase behavor of the ternary systems methane-n-pentane-n-octane and methane-n-hexane-n-octane, whch also exhbt L 1 -L 2 -V mmscblty. Addtonally, these authors studed the systems methane-nhexane-carbon doxde, n whch the carbon doxde s added to the par methane-n-hexane (thrd type) to nduce L 1 -L 2 -V mmscblty. In fve of these seven ternary systems, methane-n-octane-(ethane, propane, n-butane, n- pentane, and carbon doxde), the mmscblty regon s bounded by loc of type K, type Q and LCST (lower crtcal soluton temperature) ponts, whereas for the system methane-nhexane-n-octane the regon of L 1 -L 2 -V mmscblty s bounded, apart from the K, Q, and LCST ponts, by the L 1 -L 2 -V locus of the methane-n-hexane bnary system. In the case of the system methane-n-hexane-carbon doxde, the mmscblty regon s bounded by a locus of K and LCST ponts and by the L 1 -L 2 -V locus of the methane-n-hexane bnary system. It should be ponted out that the onset and evoluton of LLV behavor n mxtures s related to the evoluton of SLV behavor n those same systems. Thus, n natural gas t s often of nterest to predct whether a methane-rch stream wth one of the solutes (such as an n- paraffn of carbon number four or hgher; or benzene, or carbon doxde) wll form a sold phase. The reason behnd that s that the presence of a second solvent can consderably change the solublty of the sold solute, f the solute s a hydrocarbon; ths also occurs to a lesser extent f the solute s carbon doxde. On the other hand, the presence of ntrogen as a second solvent reduces the solublty of solds n methane, ethane, and mxtures thereof. Furthermore, the addton of ntrogen to mscble LG systems can nduce mmscblty. The number of ntrogen bnary systems relevant to LG that exhbt LLV mmscblty s few ntrogen-ethane and ntrogenpropane, beng among the most promnent ones. However, LLV phenomenon has been observed at certan condtons n many ternary and hgher realstc ntrogen-rch LG systems snce LLV behavor can and does occur n multcomponent systems even when for none of the consttuent bnares themselves an LLV locus s reported. Another nterestng example s the ternary mxture methane + ethane + n-octane; the speces n the consttuent bnary pars (methane + ethane) and (ethane + n-octane) are too smlar n molecular nature to be LLV mmscble, whle the par methane + n-octane s too dssmlar to be mmscble. If a multcomponent mxture s consdered to be solute plus solvent n the pseudo-component sense, t can be readly seen why the ternary mxture has a regon of LLV behavor. The methane + ethane form a solvent background n whch n-octane s mmscble. The type of the LLV regon dsplayed by a system depends on whether t contans an mmscble bnary or not. For a ternary system wth no consttuent bnary LLV behavor present the three-phase regon s a "trangular" surface n the thermodynamc phase space wth two degrees of freedom, whle ts boundares have one. It s bounded from above by a K-pont locus; from below by a LCST locus and at low temperatures by a Q-pont locus. The systems methane + n-butane + ntrogen (Merrll et al., 1984a) and methane + n-pentane + ntrogen (Merrll et al., 1984b) belong to ths class. A K-pont occurs when a lqud phase and a vapor phase become crtcal n the presence of a heaver noncrtcal equlbrum lqud phase, whereas an L-pont occurs when two lqud

6 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 363 phases becomes crtcal n the presence of a noncrtcal equlbrum vapor phase. These ponts, also called upper crtcal end ponts (UCEPs) and lower crtcal end ponts (LCEPs), are dfferent dependng on ther locaton n the pressure-temperature space. That s, an UCEP s always located at a hgher temperature and pressure than the LCEP, and when there exsts only one crtcal end pont, so that a K-pont s always an UCEP pont. On the contrary, an L- pont can be an UCEP or an LCEP, dependng on the global phase behavor of the system. The system methane + ethane + n-octane does not exhbt mmscblty n any of ts bnary pars. Although mmscblty has been reported n bnary systems of methane + n-hexane and methane + n-heptane, solutes such as n-octane and hgher normal paraffns crystallze as temperature decreases before any mmscblty occurs. On the other hand, wth ethane as solvent, solutes begnnng wth n-c 19 and hgher paraffns demonstrate LLV behavor. Apparently, the addton of modest amounts of ethane to methane creates a solvent mxture exhbtng mmscblty wth n-octane. 3. Modelng of the phase behavor of LG systems The success of the desgn and operaton of separaton processes n the ol and gas ndustry at low temperatures s crtcally dependent on the accurate descrptons of the thermodynamc propertes and phase behavor of the concerned multcomponent hydrocarbon mxtures wth norganc gases. Phase-splt calculatons and phase stablty analyss n natural gas systems smulaton can take up as much as 50 % of the CPU tme. In complcated problems t may take even more. Thus, t s mportant to develop a relable thermodynamc modelng framework (TMF) that wll be able to predct, descrbe and valdate robustly and effcently the complex phase behavor of LG mxtures. The TMF has three man elements: a lbrary of thermodynamc parameters pertanng to pure-substances and bnary nteractons, thermodynamc models for mxture propertes, and algorthms for solvng the equlbrum relatons. Relable pure-component data for the man consttuents of LG systems are avalable expermentally; an equaton of state s usually the prmary choce for the thermodynamc model. Thus, the focal pont of a TMF for phase behavor calculatons of LG systems s the avalablty of robust methods for thermodynamc stablty analyss, and of relable effcent and effectve flash routnes for three phase splt calculatons. 3.1 Computatonal technque 1 Ths frst technque uses an effcent computatonal procedure for solvng the sothermal multphase problem by assumng that the system s ntally monophasc. A stablty test allows verfyng whether the system s stable or not. In the latter case, t provdes an estmaton of the composton of an addtonal phase; the number of phases s then ncreased by one, and equlbrum s acheved by mnmzng the Gbbs energy. Ths approach, advocated as a stagewse procedure (Mchelsen, 1982b; ghem and L, 1984), s contnued untl a stable soluton s found. In ths technque, the stablty analyss of a homogeneous system of composton z, based on the mnmzaton of the dstance separatng the Gbbs energy from the tangent plane at z, s consdered (Baker et al., 1982; Mchelsen, 1982a). In terms of fugacty coeffcents,, ths crteron for stablty can be wrtten as (Mchelsen, 1982a)

7 364 Advances n atural Gas Technology 1 F ξ 1 ln ln ξ h 1 0 ξ 0 (1) where are mole numbers wth correspondng mole fractons as j1 j y, and h ln z ln z 1,..., (2) Equaton 1 requres that the tangent plane, at no pont, les above the Gbbs energy surface and ths s acheved when F ξ s postve n all ts mnma. Consequently, a mnmum of F ξ should be consdered n the nteror of the permssble regon y 1 1, y 0. To test condton 1 for all tral compostons s not physcally possble; t s thus suffcent to test the stablty at all statonary ponts of F ξ snce ths functon s not negatve at all statonary ponts. Here, the quas-ewton BFGS mnmzaton method (Fletcher, 1980) was appled to eq 1 for determnng the stablty of a gven system of composton z at specfed temperature and pressure. Once nstablty s detected wth the soluton at p 1 phases, the equlbrum calculaton s solved by mnmzaton of the followng functon p ( ) ( ) ( ) x P Mn g n ln ( ) n 1 1 P (3) subject to the nequalty constrants gven by and ( ) p1 ( ) 1 n z 1,..., (4) n 0 1,..., ; 1,..., p 1 (5) where z s the mole fracton of the component n the system, n ( ) ( 1,..., ; ( ) 1,..., p 1 ) s the mole number of component n phase per mole of feed, x s the mole fracton of component n phase, T s the temperature, P s the pressure, and P ( p) ( p) s the pressure at the standard state of 1 atm ( kpa). In eq 3 the varables n, x, ( p) and are consdered functons of n. ( ) Equaton 3 s solved usng an unconstraned mnmzaton algorthm by keepng the ( ) varables n nsde the convex constrant doman gven by eqs 4 and 5 durng the search for the soluton. In ths case, a hybrd approach to mnmze eq 3 s used startng wth the steepest-descent method n conjuncton wth a robust ntalzaton suppled from the stablty test to ensure a certan progress from ntalzatons, and endng wth the quas- ewton BFGS method whch ensures the property of strct descent of the Gbbs energy surface. A detaled descrpton of ths approach for solvng the sothermal multphase problem can be found elsewhere (Justo-García et al., 2008a).

8 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? Computatonal technque 2 The second approach to calculate multphase equlbra used here apples a rgorous thermodynamc stablty analyss and a smple and effectve method for dentfyng the phase confguraton at equlbrum wth the mnmum Gbbs energy. The rgorous stablty analyss s exercsed once and on the ntal system only. It s based on the well-known tangent plane crteron (Baker et al., 1982; Mchelsen, 1982a) but uses a dfferent objectve functon (Stateva and Tsvetkov, 1994; Wakeham and Stateva, 2004). The key pont s to locate all zeros ( y ) of a functon y gven as where wth ( y) y) y (6) 1 k 1 k ( 2 k y ln y ln y h 1,..., (7) h ln z ln z 1,..., (8) and assumng k 1( y) k1( y). Therefore, from eqs 68, t follows that mn ( y) 0 when * * * k1( y) k2( y)... k ( y). The zeros of y) ( conform to ponts on the Gbbs energy * hypersurface, where the local tangent hyperplane s parallel to that at z. To each zero y, a * number k (equal for each y, 1,..., of a zero of the functon) corresponds, such that * * * * k ln y ln ( y) h 1,..., (9) Furthermore, the number k *, whch geometrcally s the dstance between two such * hyperplanes, can be ether postve or negatve. A postve k corresponds to a zero, whch * represents a more stable state of the system, n comparson to the ntal one; a negatve k, a more unstable one. When all calculated k * are postve, the ntal system s stable; otherwse, t s unstable. It s wdely acknowledged that the task to locate all zeros of the tangent-plane dstance functon (TPDF), ( y) n ths partcular case, s extremely challengng because a search over the entre composton space s requred. The search s further complcated by the exstence of a number of trval solutons, correspondng wth the number of equlbrum phases present (Zhang et al., 2011). The specfc form of ( y) (ts zeros are ts mnma) and the fact that t s easly dfferentated analytcally, allows the applcaton of a non-lnear mnmsaton technque for locatng ts statonary ponts, and n ther works Stateva and Tsvetkov (1994) and Wakeham and Stateva (2004) used the BFGS method wth a lne-search (Fletcher, 1980). The mplementaton of any non-lnear mnmzaton technque requres a set of good ntal estmates, and the BFGS method s no excepton. All detals of the organzaton and mplementaton of the ntalzaton strategy employed by the stablty analyss procedure are gven elsewhere (Stateva and Tsvetkov 1994) and wll not be dscussed here.

9 366 Advances n atural Gas Technology Thus, as dscussed by Wakeham and Stateva (2004), a method has been created whch leads, n practce, to an "extensve" search n the multdmensonal composton space. It has proved to be extremely relable n locatng almost all zeros of ( y) at a reasonable computatonal cost. The term almost all zeros s used because there s no theoretcallybased guarantee that the scheme wll always fnd them all. If, however, a zero s mssed, the method s self-recoverng. Furthermore, the TPDF s mnmzed once only, whch s a dstnct dfference from the approach that stage-wse methods generally adopt. Snce stablty analyss on ts own cannot determne unequvocally whch s the stable phase confguraton of a system (dentfed as unstable at the gven temperature and pressure), t s suggested to run a sequence of two-phase flash calculatons to determne the correct number of the phases at equlbrum, and the dstrbuton of the components among the phases. 4. Calculaton of K- and L-ponts Ternary systems whch exhbt L1 L2 V behavor but don t exhbt such behavor n ther consttuent bnares have the mmscblty regon bounded by a K ( L1 L2 V )-pont locus, a LCST ( L1 L2 V ) locus, and a Q ( S L1 L2 V )-pont locus. Ternary systems whch have mmscblty n a consttuent bnary can have boundares smlar to those mentoned above, besdes the ntruson of the bnary L 1 L 2 V locus on the ternary L 1 L 2 V regon. The K-pont and LCST loc can ntersect at a trcrtcal pont where the three phases become crtcal;.e., L1 L2 V. eedless to say that t s costly and tme-consumng to determne the K and LCST loc n ternary systems expermentally; thus, the avalablty of approprate algorthms and numercal routnes n the thrd element of the TMF that wll allow the predcton and relable locaton n the thermodynamc phase space of such ponts, s ndspensable n the study of the complex phase behavor of LG model systems. Among the several such algorthms publshed n the open lterature we have chosen to outlne brefly and mplement those of Gregorowcs and de Loos (1996) and Mushrf and Phoenx (2006). In our choce we have been guded by the fact that the above algorthms can successfully predct the K- and L-ponts of bnary and ternary systems. We wll thus test ther robustness and effcency n the locatng crtcal end ponts n LG model systems. 4.1 Gregorowcz and de Loos algorthm In a study on the modelng of the three-phase LLV regon for ternary hydrocarbon mxtures wth the SRK EoS, Gregorowcz and de Loos (1996) proposed a procedure for fndng K- and L-ponts of ternary systems, based on the soluton of thermodynamcs condtons for the K- and L-pont usng the ewton teraton technque and startng ponts carefully chosen. They appled ther procedure to calculate the K- and L-pont loc for two ternary systems, namely, C 2 + C 3 + C 20 and C 1 + C 2 + C 20, n whch the consttuent bnary C 2 + C 20 exhbt mmscblty. Consequently, the extenson of the three-phase LLV regon of these systems s bounded by the bnary L1 L2 V locus of the system C 2 + C 20 and the ternary K-pont and L-pont loc. Brefly, the strategy followed by these authors to fnd the K- and L-ponts of the two ternary systems was the followng: (1) calculaton of the crtcal lne, the K-pont, the three phase lne, and the L-pont for the system C 2 + C 20 usng thermodynamc condtons,

10 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 367 and (2) calculaton of the K- and L-pont loc for the ternary systems by usng as startng ponts the obtaned coordnates of the K- and L-pont calculatons for the bnary system C 2 + C 20. In ths case, to obtan a K- and L-pont for a ternary system, the followng set of sx nonlnear equatons, A A A VV Vx Vx 1 2 D A A A 0 (10) Vx xx xx A A A Vx x x x x * D D D V x x 1 2 D A A A 0 (11) Vx x x x x A A A Vx x x x x c 0 1,2,3 (12) c P 0 (13) P c c c n seven varables, T, V, V, x1, x2, x1, and x2 have to be solved, where desgnates V * for the L-pont or L 2 for the K-pont, D and D are the two determnants that must be satsfed at a crtcal pont, and s the chemcal potental of component. The crtcal crtera gven by eqs 10 and 11 are based on the Helmholtz energy, whch can be expressed as (Baker and Luks, 1980), A nrt V P dv RTn V V 0 0 ln n( U TS ) (14) n RT 1 1 where n s the mole number of component and V s the system volume. Dervatves of the Helmholtz energy are denoted by a subscrpt n eqs 10 and 11 (e.g., A V, A x ) ndcatng the dfferentaton varable (volume V or mole fracton x of component ). Detals to * obtan the elements of determnants D and D are gven n Baker and Luks (1980). It s worth mentonng that Gregorowcz and de Loos (1996) calculated the ternary K- and L- ponts at chosen values of the temperature, whch t s mportant when the experments are carred out sothermally. 4.2 Mushrf and Phoenx s algorthm The second approach to calculate K- and L-ponts was proposed by Mushrf and Phoenx (2006). Ths approach utlzes an effcent crtcal pont solver and a standard phase stablty test wthn a nested-loop structure to drectly locate K- and L-ponts. The algorthm conssts of two nested nner loops to calculate a crtcal-pont temperature and volume at fxed composton z. An outer loop uses the crtcal pont as a test phase, searches for an ncpent phase at a tral composton ˆn, and updates the crtcal composton to teratvely decrease

11 368 Advances n atural Gas Technology the tangent plane dstance of the ncpent phase to zero. The ewton-raphson method wth numercal dervatves was used n both the nner and outer loops. Ths algorthm s smlar to that proposed by Gauter et al. (1999) to calculate crtcal end ponts (CEPs) for ternary systems of carbon doxde as the near-crtcal solvent and two lowvolatle solutes (1-pentanol or 1-hexanol + n-trdecane) wth the PR (Peng and Robnson, 1976) EoS. These authors used the approach of Hedemann and Khall (1980) to follow a crtcal lne n steps of the mole fracton of the carbon doxde, searchng for a CEP along ths lne;.e., searchng for the occurrence of an addtonal phase n zero amount by usng the formulaton of Mchelsen (1982a). The prncpal dfference between the algorthm of Mushrf and Phoenx and that of Gauter et al. s that the former drectly calculate the K- and L-ponts wthout followng the crtcal lnes. In the algorthm proposed by Mushrf and Phoenx (2006), the crtcal crtera are based on the tangent-plane crtera developed by Mchelsen and Hedemann (1988) at specfed temperature and pressure. Mchelsen and Hedemann (1988) also formulated the crtcalpont crtera n terms of the tangent plane dstance based on the Helmholtz energy at fxed temperature and volume. The condton for the stablty of a mxture, wth respect to a tral phase ˆn, s 1 F nˆ ln f f V PP RT (15) 0 0 where F s the tangent plane dstance from the Gbbs energy surface to the hyperplane tangent of the Gbbs energy surface at the composton z. In eq 15 the sgn of F wll determne the stablty of the test phase;.e., f F 0 the test phase s stable, f F 0 the test phase s n equlbrum wth some alternate phase, and f F 0 the test phase s unstable. Mchelsen (1984) developed the crtera for crtcal ponts by expandng the tangent plane dstance functon n a Taylor seres around the test pont as ) F bs cs ds O( s (16) such that F 0 0 and df ds 0 s hold at the test pont; s beng a parameter that defnes the dstance n composton 0 space from the test pont at s 0. As the sgn of the tangent plane dstance functon F determnes the stablty of the test phase, t s necessary 1/2 to fnd the mnmum of ths functon usng scaled mole numbers as X n z z, where z are mole fractons n the test phase and n are mole fractons n any alternate phase. At the test pont nz ˆ, X 0 and F 0. Expressons for the frst g and second dervatves B of F wth respect to X are gven n Mchelsen (1984). j Functon F s mnmzed by varyng X under the constrant that ux s, where u s a vector of unt length. By applyng the method of Lagrange multplers, coeffcent b can be T expressed as b 12uBu, regardless of the choce of vector u. The least possble value of coeffcent b s obtaned by choosng u as the egenvector of B correspondng to the smallest egenvalue mn ;.e., Bu mnu. T

12 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 369 At tral condtons of temperature and volume, matrx B s calculated and mn, u are determned by nverse teraton (Wlknson, 1965), then b mn 2. If b 0, the system s at the lmt of ntrnsc stablty. At a crtcal pont coeffcents b and c n eq 16 are zero for a gven egenvector u of B correspondng to the smallest egenvalue mn. For the evaluaton of coeffcent c, Mchelsen (1984) showed that ths can be determned effcently from nformaton already avalable of u and g. The soluton procedure to calculate a crtcal pont s as follows: snce coeffcent b s to be a zero egenvalue of B, then t must be a sngular matrx wth a zero determnant;.e., T det B 0, wth a vector u satsfyng Bu 0 ( uu 1 ) The crteron of b 0 s met when the matrx B s sngular and s used to fnd the crtcal temperature at a fxed composton and volume. The determnant of matrx B s calculated through a LU decomposton of B ( LU B where L s lower trangular and U s upper trangular);.e., the determnant of B s the product of the dagonal elements of the LU decomposed matrx. Once the teraton to fnd the stablty lmt has converged, the vector u s determned by nverse teraton technque. The mplementaton of the equaton-of-state approach for calculatng crtcal ponts usng ths procedure requres that temperature T and volume V are terated n a nested way. That s, based on an ntal guess of V, the temperature s determned n a nner loop untl the determnant of the matrx B becomes equal to zero; then the convergence crteron for the coeffcent c s checked. If ths coeffcent, evaluated at the stablty lmt, s equal to zero, the calculaton ends; otherwse, a new estmate for the volume s generated n an outer loop and the teraton on T s evaluated agan. Once T and V have been obtaned, the pressure P s evaluated from the equaton of state. After havng calculated the crtcal pont, a noncrtcal equlbrum phase s searched for at constant temperature and pressure condtons. Mushrf and Phoenx (2006) used the stablty test mplemented by Mchelsen (1982a) usng the crtcal composton z as the reference phase and Y ( Y1,..., Y ) T as the unnormalzed tral phase composton wth correspondng mole fractons as y Y Y. j1 j The tangent plane to the Gbbs energy surface at the tral composton s parallel to the reference-phase tangent plane (crtcal phase) when where 1 crt crt ln f ln f ln( y P) ln f 0 (17) ln( Y ) s the dmensonless dstance between the two tangent planes and s the fugacty coeffcent evaluated at composton y. If 0, the tral phase s n equlbrum wth the reference phase; f 0, the tral phase s an ncpent phase, and f 0, the reference phase s unstable. By combnng eq 17 wth the defnton of, the set of equatons to solve for a statonary pont can be wrtten as crt g Y exp ln f ln ( ) ln P y 0 1,..., (18)

13 370 Advances n atural Gas Technology whch can effcently be solved by ewton teraton or through a mnmzaton method. A K- or L-pont s found f 0. When does not meet the convergence crteron (e.g., ), the crtcal-phase composton of the lghtest component (component 1) s updated usng the ewton-raphson method as z ( k) ( k1) ( k) 1 z1 ( k) ( d dz1) (19) where the dervatve d dz1 s approxmated by perturbng the crtcal composton, recalculatng from eq 17 and calculatng the fnte dfference analogue of the dervatve. Mushrf and Phoenx (2006) have ponted out that falure of the algorthm can occur when a crtcal composton s updated to a value where no statonary pont exst other than the trval soluton. To calculate a K- or L-pont usng ths algorthm, t s necessary to provde approprate ntal estmates of composton, temperature, and volume. In ths case, good ntal guesses for crtcal temperatures and volumes were, dependng on the type of calculaton, the same as those used by Hedemann and Khall (1980). The success of the algorthm to locate a K- or L-pont strongly depend on (1) the bnary nteracton parameters used n the equaton of (0) state and (2) the ntal crtcal composton z. However, t would seem that the value of the ntal crtcal composton sgnfcantly affects the successful convergence of the method to locate a K- or L-pont. 5. Thermodynamc models Modelng of the complex phase behavor of LG systems requres a sutable thermodynamc model and a robust and effcent computatonal algorthm for performng phase stablty and multphase flash calculatons nterwoven n the second element of the TMF. Regardng the thermodynamc models, the SRK EoS and the PC-SAFT EoS have receved wde acceptance n the ndustry because of ther ablty to predct accurately the phase behavor of ol-gas systems. 5.1 The SRK equaton of state The explct form of the SRK equaton of state (Soave, 1972) can be wrtten as RT at P v b v v b (20) where constants a and b for pure-components are related to 2 2 c RT RTc a Tr ; b (21) P P where T r s expressed n terms of the acentrc factor as c c

14 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? /2 T T 2 r r (22) For mxtures, constants a and b are gven by and a j s defned as (23) a x x a ; b xb j j 1 j1 1 a 1 k a a ; k k, k 0 (24) j j j j j where k j s an adjustable nteracton parameter characterzng the bnary formed by components and j. Eq 20 can be wrtten n terms of compressblty factor, Z Pv RT, as where A ap RT 2 and B bp RT Z Z ABB ZAB 0 (25). The expresson for the fugacty coeffcent, f yp, s gven by 2 xa b 1 j j A j b B ln Z1lnZB ln 1 b B a b Z (26) 5.2 The PC-SAFT equaton of state In the PC-SAFT EoS (Gross and Sadowsk, 2001), the molecules are conceved to be chans composed of sphercal segments, n whch the par potental for the segment of a chan s gven by a modfed square-well potental (Chen and Kreglewsk, 1977). on-assocatng molecules are characterzed by three pure component parameters: the temperaturendependent segment dameter, the depth of the potental, and the number of segments per chan m. The PC-SAFT EoS wrtten n terms of the Helmholtz energy for an -component mxture of non-assocatng chans conssts of a hard-chan reference contrbuton and a perturbaton contrbuton to account for the attractve nteractons. In terms of reduced quanttes, ths equaton can be expressed as The hard-chan reference contrbuton s gven by res hc dsp a a a (27) hc hs hs 1 a ma x ( m 1)ln g ( ) (28)

15 372 Advances n atural Gas Technology where m m s the mean segment number n the mxture 1 The Helmholtz energy of the hard-sphere flud s gven on a per-segment bass as hs a ln(1 3) 0 (1 3) 3(1 3) (29) 3 and the radal dstrbuton functon of the hard-sphere flud s g hs j 2 dd 2 j dd 2 j d d j 3 d d j (1 ) (1 ) (1 ) (30) wth n defned as n n xmd 6 1 n0,1,2,3 (31) The temperature-dependent segment dameter d of component s gven by exp 3 kt d (32) where k s the Boltzmann constant and T s the absolute temperature. The dsperson contrbuton to the Helmholtz energy s gven by where hc 1 dsp 2 3 hc Z (, ) 1 2(, ) a I m m m Z I m m (33) hc Z s the compressblty factor of the hard-chan reference contrbuton, and 2 3 j 3 m x xj m mj j 1 j1 kt (34) B j 3 j j j 1 j1 B m x x m m (35) kt The parameters for a par of unlke segments are obtaned by usng conventonal combnng rules ; 1 k 1 j j j j (36) j 2 where k j s a bnary nteracton parameter, whch s ntroduced to correct the segmentsegment nteractons of unlke chans.

16 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 373 The terms I 1 (, m) and I 2 (, m) n eq 33 are calculated by smple power seres n densty (37) 0 0 I (, m) a ( m) ; I (, m) b ( m) where the coeffcents a and b depend on the chan length as gven n Gross and Sadowsk (2001). sys The densty to a gven system pressure P s determned teratvely by adjustng the calc sys reduced densty untl P P. For a converged value of, the number densty of molecules, gven n Å -3, s calculated from 1 3 xmd (38) 1 6 Usng Avogadro s number and approprate converson factors, produces the molar densty n dfferent unts such as kmol m 3. The pressure can be calculated n unts of Pa m 2 by applyng the relaton Å 3 10 P ZkT 10 m (39) from whch the compressblty factor Z, can be derved. The expresson for the fugacty coeffcent s gven by res res res a a ln a x k Z 1 ln Z x Tvx,, k1 x (40) k j Tvx,, jk In eq 40, the partal dervatves wth respect to mole fractons are calculated regardless of the summaton relaton x Results and dscusson Expermental data reported by Llave et al. (1987) for the system ntrogen + methane + ethane, by Hottovy et al. (1981) for the system methane + ethane + n-octane, and by Fall and Luks (1988) for the system carbon doxde + ntrogen + n-nonadecane, were used to test and compare the robustness, effcency and relablty of the two computatonal technques and the SRK and PC-SAFT EoS thermodynamc models embedded n the respectve elements of the TMF. The predcton and modelng of the phase behavor of these systems demonstrates n a clear-cut way the usual numercal dffcultes encountered n the process. The bnary nteracton parameters used wth the PC-SAFT equaton were taken from García- Sánchez et al. (2004) and from Justo-García et al. (2008b), whle those used wth the SRK EoS were taken from Knapp et al. (1982). Some of the nteracton parameters were also obtaned

17 374 Advances n atural Gas Technology from the mnmzaton of the sum of squares of the dfferences between expermental and calculated bubble-pont pressures. The bnary nteracton parameters employed are: kc C, k 2 C C, 3 kc C, k C, k C, k C, k 19 CO , and 2 kco C for the SRK equaton, and k 19 C C, k 2 C C, 3 kc C, k C, k C, k C, k 19 CO , and 2 kco C for the PC-SAFT equaton, respectvely. The components physcal 19 propertes requred for the calculatons performed wth the SRK EoS were taken from DIPPR (Rowley et al., 2006) whle the three pure-component parameters (.e., temperature ndependent segment dameter, depth of the potental, and number of segments per chan m ) of these compounds for the PC-SAFT equaton of state were taken from Gross and Sadowsk (2001). 6.1 The ntrogen + methane + ethane system The three-phase VLL regon dsplayed by ths ternary system s bounded from above by a K-pont locus, from below by a lower crtcal soluton temperature LCST locus, at low temperatures by a Q-pont locus, and, due to the fact that ths system contans a bnary par (ntrogen + ethane) whch exhbts LLV behavor, ts LLV space s truncated. In ths case, the partally mscble par ntrogen + ethane spans the LLV space from a poston of the LCST locus to a poston on the Q-pont space. Because methane s of ntermedate volatlty compared wth ntrogen and ethane, t creates a three-phase LLV space whch extends from the bnary LLV locus upward n temperature. The topographcal nature of the regons of mmscblty for the system ntrogen + methane + ethane s shown n Fg. 1. In ths fgure t can be seen that the L-L=V and L=L-V crtcal end-pont loc ntersect at a trcrtcal pont (L=L=V). Fg. 2 presents the expermental and calculated L 1 -L 2 -V phase behavor (n terms of L 1 -L 2 ntrogen mole fracton data) for the ntrogen + methane + ethane system at 135 K and dfferent pressures. Ths fgure shows a reasonable agreement between the expermental values of lqud phases L 1 and L 2 and those predcted wth both models. otwthstandng, although the LLV calculatons performed wth both equatons up to a poston near the K- pont (about bar wth both models), ths pont s away from the expermental one (43.05 bar at 135 K). An attempt to drectly calculate ether the K- or L-pont for ths ternary system usng the algorthm of Mushrf and Phoenx (2006) was carred out. However, the algorthm was not able to gve correct values of these crtcal end ponts. Ths s because the algorthm s strongly ntalzaton dependent and hence gves dfferent values of these ponts, dependng on the ntal guess of the crtcal composton, whch meets the convergence crteron. Our prelmnary results show that the algorthm advocated by Gregorowcz and de Loos (1996) s more stable than that of Mushrf and Phoenx, even f t also depends on the ntal values of the crtcal composton. Fg. 2 also shows that at pressures away from the LCST pont, the PC-SAFT model gves a better representaton of the expermental compostons for lqud phase L 1 whle both

18 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? K (L1-L2=V) L1-L2-V (Bnary 2-C2) LCST (L1=L2-V) 45 Pressure (bar) Temperature (K) Fg. 1. P-T space of the boundares of the three-phase L 1 -L 2 -V regons for the system ntrogen + methane + ethane. Expermental data from Llave et al. (1985, 1987) Expermental L1 Phase Calculated L1 Phase: PC-SAFT EoS Calculated L1 Phase: SRK EoS Calculated V Phase: PC-SAFT EoS Expermental L2 Phase Calculated L2 Phase: PC-SAFT EoS Calculated L2 Phase: SRK EoS Calculated V Phase: SRK EoS 41 Pressure (bar) Mole fracton, trogen Fg. 2. Comparson of L 1 and L 2 compostonal data of ntrogen at 135 K for the system ntrogen + methane + ethane. Expermental data from Llave et al. (1987).

19 376 Advances n atural Gas Technology models agree wth each other for the lqud L 2 phase but represent the expermental data very closely. Snce there are not expermental data below bar, comparsons of the models wth experment n the regon of the LCST are not possble. However, accordng to the predctons wth both models, the estmated LCST pont wth the PC-SAFT model (37.20 bar) seems to be closer to the hypothetcal expermental LCST pont (38.37 bar) n comparson wth the LCST pont estmated from the SRK model (35.45 bar). Of course, these dscrepances can be due to the fact of usng bnary nteracton parameters determned from VL equlbrum data, whch, apparently, led to less accurate results. 6.2 The methane + ethane + n-octane system The three-phase LLV regon dsplayed by the methane + ethane + n-octane ternary system (a surface n the thermodynamc phase space wth two degree of freedom) s bounded from above by a K-pont locus (L-L=V), from below by a lower crtcal soluton temperature LCST locus (L=L-V), and at low temperatures by a Q-pont locus (S-L-L-V). For the three components n ths system, there s no bnary mmscblty. The topographcal nature of the regons of mmscblty for ths system s shown n Fg. 3, where symbols are the expermental data gven by Hottovy et al. (1981) dentfyng the boundares of the three phase LLV regon for ths ternary system. Ths Fgure shows also that the L-L=V and L=L-V crtcal end-pont loc ntersect at a trcrtcal pont (L=L=V) at the upper temperature lmt K (L1-L2=V) Q (S-L1-L2-V) LCST (L1=L2-V) Pressure (bar) Temperature (K) Fg. 3. P-T space of the boundares of the three-phase L 1 -L 2 -V regons for the system methane + ethane + n-octane. Expermental data from Hottovy et al. (1981). Followng the mmscblty regon, a sngle temperature was chosen to test the capabltes of the PC-SAFT wth computatonal technque 1, and the SRK EoS wth

20 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 377 computatonal technque 2, to predct the phase behavor for the system methane + ethane + n-octane. Fg. 4 compares the performance of the two methods at 210 K, and at dfferent pressures on the bass of the expermentally measured and calculated ntrogen mole fractons for the lqud L 1 and L 2 phases by the two thermodynamc models employed. In ths case, the predctons of the PC-SAFT model are closer to the expermental composton values than those of the SRK model. However, t should be mentoned that t was not possble to contnue the calculatons wth ths model to approach ether the K- or L-pont because the three-phase LLV trangles become so very narrow as pressure decreases or ncreases that t s extremely dffcult to determne an approprate global composton able to separate ths mxture nto three-phase LLV equlbra. On the other hand, because the three-phase LLV trangles predcted wth the SRK model are wder than those predcted wth the PC-SAFT model, t was easer to get a good ntal global composton to calculate the three-phase LLV equlbra from the LCST pont (52.80 bar) to the K-pont (59.09 bar) applyng technque 2. An nspecton of ths fgure shows that the predctons of both EoSs don t follow the behavor of the lqud phase L 1 as well as the varaton of lqud phase L 2 as pressure decreases. Also, t s nterestng to note that although there s not a true expermental value of the LCST at the temperature consdered, Fg. 4 ndcates that the estmated LCST pont wth the PC-SAFT model (56.35 bar) s closer to the expermental one (56.64 bar) than that obtaned wth the SRK model (52.80 bar). onetheless, the expermental K-pont (60.19 bar) s closer to the one calculated wth the SRK model Pressure. bar Expermental L1 Phase Calculated L1 Phase: PC-SAFT EoS Calculated L1 Phase: SRK EoS Calculated V Phase: PC-SAFT EoS Expermental L2 Phase Mole Fracton, Methane Calculated L2 Phase: PC-SAFT EoS Calculated L2 Phase: SRK EoS Calculated V Phase: SRK EoS Fg. 4. Comparson of L 1 and L 2 compostonal data of ntrogen at 210 K for the system methane + ethane + n-octane. Expermental data from Hottovy et al. (1981).

21 378 Advances n atural Gas Technology Fg. 4 also shows that at the dfferent pressures the predcted L 1 -L 2 -V regon (n terms of L 1 and L 2 methane mole fracton data) wth the SRK model devates consderably from the expermental one, whle the PC-SAFT model predctons are more reasonable. However, snce the nteracton parameters for the SRK and PC-SAFT models were determned from bnary vapor-lqud equlbrum data, the rather poor ft n ths regon wth ether model s not unexpected. 6.3 The carbon doxde + ntrogen + n-nonadecane system As mentoned n Secton 2, the type of the LLV regon dsplayed by a ternary system depends on whether t contans an mmscble par or not. In ths context, the system carbon doxde + ntrogen + n-nonadecane exhbts mmscblty n the carbon doxde + n- nonadecane bnary par (Fall et al., 1985), so that ts three-phase regon s smlar to that exhbted by the system ntrogen + methane + n-pentane (Merrll et al., 1984b). Therefore, the LLV regon s trangular and s bounded from above by a K-pont locus (L-L=V), at low temperatures by a Q-pont locus (S-L-L-V), and, from a poston of the Q-pont locus to a poston on the K-pont space, by a bnary carbon doxde + n-nonadecane LLV locus. Fg. 5 presents the expermental pressure-temperature dagram of the LLV space dsplayed by the system (Fall and Luks, 1986). An examnaton of the fgure shows that ths system does not have a LCST (L=L-V) locus and that the Q-pont locus termnates at an nvarant K (L1-L2=V) Q (S-L1-L2-V) V-L1-L2 (Bnary CO2-C19) Pressure (bar) Temperature (K) Fg. 5. P-T space of the boundares of the three-phase L 1 -L 2 -V regons for the system carbon doxde + ntrogen + n-nonadecane. Expermental data from Fall et al. (1985) and Fall and Luks (1986).

22 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 379 pont of the S-L-L=V type. Thus, due to the fact that the carbon doxde + ntrogen + n- nonadecane system contans a bnary par (carbon doxde + n-nonadecane) exhbtng LLV behavor, ts trangular LLV regon s a three-sded space wthout a trcrtcal pont. A temperature of 297 K was chosen to study ths ternary system and the results obtaned are presented n Fg. 6. Ths fgure shows that the PC-SAFT model predcts well the expermental carbon doxde compostons of the lqud L 2 and vapor phases down to the lowest measured pressure of bar (.e., the bnary carbon doxde + n-nonadecane data) for ths sotherm. However, the SRK model s superor to the PC-SAFT model n predctng the three phases n equlbrum for ths ternary system. In ths case, the calculated L 1, L 2, and V phases are close to the expermental ones. Furthermore, though the SRK EoS overpredcts the expermental K-pont (87.30 bar) by 4.39 bar the performance of the PC-SAFT EoS n ths partcular case s nferor as t overpredcts by 14.6 bar the expermental pont Mole fracton, Carbon doxde Expermental L1 Phase Calculated L1 Phase: SRK EoS Calculated L1 Phase: PC-SAFT EoS Expermental L2 Phase Calculated L2 Phase: SRK EoS Calculated L2 Phase: PC-SAFT EoS Expermental V Phase Calculated V Phase: SRK EoS Calculated V Phase: PC-SAFT EoS Pressure, bar Fg. 6. Comparson of L 1 and L 2 compostonal data of carbon doxde at 297 K for the system carbon doxde + ntrogen + n-nonadecane. Expermental data from Fall and Luks (1986). evertheless, t should be recalled that all calculatons were performed usng bnary nteracton parameters obtaned from regresson of bnary expermental VL data, many of them measured at temperatures hgher than that studed here. Furthermore, we are confdent that the performance of the correspondng thermodynamc models could be mproved consderably provded the nteracton parameters were obtaned from the regresson of the expermental data at three phases. Stll, f those sets of nteracton parameters are used to predct the phase behavor of a gven system at condtons dfferent from the orgnal ones then there s the rsk that the equlbra predctons and calculatons could ether gve physcal meanngless results or fal altogether.

23 380 Advances n atural Gas Technology We also tred to drectly calculate the K-ponts at the temperature consdered for ths ternary system by usng the algorthm of Mushrf and Phoenx (2006); however, once agan, the algorthm predcted dfferent values of the crtcal end ponts, dependng on the ntal crtcal composton. In ths case, the strategy to fnd a crtcal temperature close to the temperature of study was to use a seres of ntal compostons. Unfortunately, none of the compostons produced a K-pont smlar to those obtaned from the VLL calculatons wth both models. Fnally, t should be ponted out that for the sake of comparson both equatons of state were nterwoven nto computatonal procedure 1 and 2, respectvely, and that a seres of multphase flash calculatons were carred out at dfferent temperatures and pressures for the three ternary systems studed obtanng the same results for the specfc EoS, rrespectvely of the computatonal procedure utlzed. The results obtaned showed that there are not any essental dfferences between, or partcular advantages of any of two computatonal procedures, ether n ther effcency, effectveness, robustness or n ther convergence behavor. Thus, t can be sad that both procedures 1 and 2 can be used to predct the phase behavor of a wde varety of multcomponent nondeal systems over wde ranges of temperature and pressure. 7. Conclusons Though there has been much progress and advance n two-phase stablty and two-phase splt calculatons wth EoSs, there s stll not much progress n three-phase splt calculatons n natural gas systems despte the large number of publcatons devoted to the subject. The reason behnd that s that the dffcultes and challenges are domnatng over the easy to perform calculatons, f any. Thus, t can be accepted that the algorthms and numercal methods advocated are not robust enough for ncorporaton n a process smulator. In vew of ths, the further development of a relable, robust and effcent TMF and ts subsequent approbaton on model systems, typcal representatves of LG, s of consderable nterest both to scentsts and engneers. On the example of three ternary systems (ntrogen-methane-ethane, methane-ethane-noctane, and carbon doxde-ntrogen-n-nonadecane) the capabltes of a TMF, advocated by us, to predct and model complex phase behavor of systems of mportance to LG processng are demonstrated. The TMF employs two numercal technques whch embed dfferent thermodynamc models and stablty analyss routnes. The results obtaned show that there are many and dfferent challenges and dffcultes that are not always possble to overcome completely. For example, the two technques for multphase flash calculatons cannot always assure steady and non-oscllatory convergence wth no tendency towards a strong attracton to the trval soluton, partcularly n cases close to the crtcal lnes. Besdes, t s known that the phase equlbrum equatons are often dffcult to converge n the crtcal regon and that the use of napproprate ntal estmates can lead to the trval soluton. In vew of ths, the avalablty n a TMF of a robust stablty analyss routne that wll provde good set of ntal estmates for the compostons of possble equlbrum phases and wll guarantee steady convergence of the flash routnes s of great mportance.

24 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 381 Regardng the predcton of K- and L-ponts wth the Gregorowcz and de Loos and Mushrf and Phoenx s methods, t s clear that the both algorthms are strongly ntalzaton dependent. To overcome ths problem, for example, Mushrf and Phoenx suggest that the crtcal phase composton s updated based on the values of parameter, calculated from equlbrum phase calculatons usng the ewton-raphson teraton. However, durng ths process of upgradng, the composton may change to a value where there s no phase n equlbrum, partcularly when dffers sgnfcantly from teraton to teraton. The procedure of Gregorowcz and de Loos to calculate K- and L-ponts seems to be a better ftted method to carry out ths task. However, the evaluaton of the determnants for solvng the condtons of crtcalty requres the second dervatves of the Helmholtz energy wth respect to volume and composton, whch makes dffcult ther evaluaton, partcularly when these dervatves have to be found analytcally applyng the PC-SAFT EoS. Stll, of course, a possble soluton to ths problem s to evaluate these dervatves numercally. Fnally, both the PC-SAFT EoS and the RKS cubc EoS are capable of representng wth a reasonable accuracy the expermentally observed phase behavor of the ternary systems studed. 8. Acknowledgments Ths work was partally supported by the Mexcan Petroleum Insttute under Research Project D One of the authors (D.. J.-G.) gratefully acknowledges the atonal Polytechnc Insttute of Mexco for ther fnancal support through the Project SIP References Baker, L. E. & Luks, K. D. (1980). Crtcal Pont and Saturaton Pressure Calculatons for Multcomponent Systems. Socety of Petroleum Engneers Journal, Vol. 20, o. 1, pp Baker, L. E.; Perce, A. C. & Luks, K. D. (1982). Gbbs Energy Analyss of Phase Equlbra. Socety of Petroleum Engneers Journal, Vol. 22, o. 5, pp , Chen, S. & Kreglewsk, A. (1977). Applcatons of the Augmented Van der Waals Theory of Fluds. I. Pure Fluds. Berchte der Bunsengesellschaft für physkalsche Cheme, Vol. 81, o. 10, pp Fall, J. L. & Luks, K. D. (1986). Effect of Addtve Gases on the Lqud-Lqud-Vapor. Immscblty of the Carbon Doxde + n-onadecane Mxture. Journal of Chemcal and Engneerng Data, Vol. 31, o. 3, pp Fall, D. J.; Fall, J. L. & Luks, K. D. (1985). Lqud-Lqud-Vapor Immscblty Lmts n Carbon Doxde + n-paraffn Mxtures. Journal of Chemcal and Engneerng Data, Vol. 30, o. 1, pp Fletcher, R. (1980). Practcal Methods for Optmzaton. Vol. 1. Unconstraned Optmzaton, John Wley & Sons Ltd, ew York. García-Sánchez, F.; Elosa-Jménez, G.; Slva-Olver, G. & Vázquez-Román, R. (2004). Vapor- Lqud Equlbra of trogen-hydrocarbon Systems Usng the PC-SAFT Equaton of State. Flud Phase Equlbra, Vol. 217, o. 2, pp

25 382 Advances n atural Gas Technology Gauter, K.; Hedemann, R. A. & Peters, C. J. (1999). Modelng of Flud Multphase Equlbra n Ternary Systems of Carbon Doxde as a ear-crtcal Solvent and Two Low Volatle Solutes. Flud Phase Equlbra, Vol , pp Green, K. A.; Tffn, D. L.; Luks, K. D. & Kohn, J. P. (1976). Solublty of Hydrocarbons n LG, GL. Hydrocarbon Processng, Vol. 56, o. 5, pp Gregorowcz, J. & de Loos, Th. W. (1996). Modellng of the Three Phase LLV Regon for Ternary Mxtures wth the Soave-Redlch-Wong Equaton of State. Flud Phase Equlbra, Vol. 118, o. 1, pp Gross, J. & Sadowsk, G. (2001). Perturbed-Chan SAFT: An Equaton of State Based on Perturbaton Theory for Chan Molecules. Industral & Engneerng Chemstry Research, Vol. 40, o. 4, pp Hedemann, R. A. & Khall, A. M. (1980). The Calculaton of Crtcal Ponts. Amercan Insttute of Chemcal Engneers Journal, Vol. 26, o. 5, pp Hottovy, J. D.; Kohn, J. P. & Luks, K. D. (1981). Partal Mscblty of the Methane-Ethane-n- Octane. Journal of Chemcal and Engneerng Data, Vol. 26, o. 2, pp Hottovy, J. D.; Kohn, J. P. & Luks, K. D. (1982). Partal Mscblty Behavor of the Ternary Systems Methane-Propane-n-Octane, Methane-n-Butane-n-Octane, and Methane- Carbon Doxde-n-Octane. Journal of Chemcal and Engneerng Data, Vol. 27, o. 3, pp Justo-García, D..; García-Sánchez, F. & Romero-Martínez, A. (2008a). Isothermal Multphase Flash Calculatons wth the PC-SAFT Equaton of State. Amercan Insttute of Physcs Conference Proceedngs, Vol. 979, pp Justo-García, D..; García-Sánchez, F., Díaz-Ramírez,. L. & Romero-Martínez, A. (2008b). Calculaton of Crtcal Ponts for Multcomponent Mxtures Contanng Hydrocarbon and onhydrocarbon Components wth the PC-SAFT Equaton of State. Flud Phase Equlbra, Vol. 265, os. 1-2, pp Knapp, H. R.; Dörng, R.; Oellrch, L.; Plöcker, U. & Prausntz, J. M. (1982). Vapor-Lqud Equlbra for Mxtures of Low Bolng Substances; DECHEMA Chemstry Data Seres, Vol. VI: Frankfurt, Germany. Kohn, J. P. (1961). Heterogeneous Phase and Volumetrc Behavor of the Methane-n- Heptane System at Low Temperatures. Amercan Insttute of Chemcal Engneers Journal, Vol. 7, o. 3, pp Kohn, J. P. & Bradsh, W. F. (1964). Multphase and Volumetrc Equlbra of the Methane-n- Octane System at Temperatures between -10 and 150 C. Journal of Chemcal and Engneerng Data, Vol. 9, o. 1, pp. 58. Kohn, J. P. & Luks, K. D. (1976). Solublty of Hydrocarbons n Cryogenc LG and GL Mxtures. Research Report RR-22, Gas Processors Assocaton, Tulsa, Oklahoma. Kohn, J. P. & Luks, K. D. (1977). Solublty of Hydrocarbons n Cryogenc LG and GL Mxtures. Research Report RR-27, Gas Processors Assocaton, Tulsa, Oklahoma. Kohn, J. P. & Luks, K. D. (1978). Solublty of Hydrocarbons n Cryogenc LG and GL Mxtures. Research Report RR-33, Gas Processors Assocaton, Tulsa, Oklahoma. Kohn, J. P.; Luks, K. D. & Lu, P. H. (1976). Three-Phase Sold-Lqud-Vapor Equlbra of Bnary-n-Alkane Systems (Ethane-n-Octane, Ethane-n-Decane, Ethane-n- Dodecane). Journal of Chemcal and Engneerng Data, Vol. 21, o. 3, pp

26 Phase Behavor Predcton and Modelng of LG Systems wth EoSs What s Easy and What s Dffcult? 383 Llave, F. M.; Luks, K. D. & Kohn, J. P. (1985). Three-Phase Lqud-Lqud-Vapor Equlbra n the Bnary Systems trogen+ Ethane and trogen + Propane. Journal of Chemcal and Engneerng Data, Vol. 30, o. 4, pp Llave, F. M.; Luks, K. D. & Kohn, J. P. (1987). Three-Phase Lqud-Lqud-Vapor Equlbra n the trogen-methane-ethane and trogen-methane-propane. Journal of Chemcal and Engneerng Data, Vol. 32, o. 1, pp Luks, K. D. & Kohn, J. P. (1978). The Topography of Multphase Equlbra Behavor: What Can t Tell the Desgn Engneer. Proceedngs of the 63rd Annual Conventon, Gas Processors Assocaton, Tulsa, Oklahoma. Merrll, R. C.; Luks, K. D. & Kohn, J. P. (1983). Three-Phase Lqud-Lqud-Vapor Equlbra n the Methane-n-Pentane-n-Octane, Methane-n-Hexane-n-Octane, and Methane-n- Hexane-Carbon Doxde. Journal of Chemcal and Engneerng Data, Vol. 28, o. 2, pp Merrll, R. C.; Luks, K. D. & Kohn, J. P. (1984a). Three-Phase Lqud-Lqud-Vapor Equlbra n the Methane-n-Butane-trogen System. Advances n Cryogenc Engneerng, Vol. 29, pp Merrll, R. C.; Luks, K. D. & Kohn, J. P. (1984b). Three-Phase Lqud-Lqud-Vapor Equlbra n the Methane-n-Hexane-trogen and Methane-n-Pentane-trogen Systems. Journal of Chemcal and Engneerng Data, Vol. 29, o. 3, pp Mchelsen, M. L. (1982a). The Isothermal Flash Problem. Part I. Stablty. Flud Phase Equlbra, Vol. 9, o. 1, pp Mchelsen, M. L. (1982b). The Isothermal Flash Problem. Part II. Phase-Splt Calculaton. Flud Phase Equlbra, Vol. 9, o. 1, pp Mchelsen M. L. (1984). Calculaton of Crtcal Ponts and Phase Boundares n the Crtcal Regon. Flud Phase Equlbra, Vol. 16, o. 1, pp Mchelsen, M. L. & Hedemann, R. A. (1988). Calculaton of Tr-Crtcal Ponts. Flud Phase Equlbra, Vol. 39, o. 1, pp Mushrf, S. H. & Phoenx, A. V. (2006). An Algorthm to Calculate K- and L-Ponts. Industral & Engneerng Chemstry Research, Vol. 45, o. 26, pp ghem, L. X. & L, Y.-K. (1984). Computaton of Multphase Equlbrum Phenomena wth an Equaton of State. Flud Phase Equlbra, Vol. 17, o. 1, pp Orozco, C. E.; Tffn, D. L.; Luks, K. D. & Kohn, J. P. (1977). Solds Foulng n LG Systems. Hydrocarbon Processng, Vol. 56, o. 11, pp Peng, D.-Y. & Robnson, D. B. (1976). A ew Two-Constants Equaton of State. Industral & Engneerng Chemstry Fundamentals, Vol. 15, o. 1, pp Rowley, R. L.; Wldng, W. V.; Oscarson, J. L.; Yang, Y. & Zundel,. A. (2006). DIPPR Data Complaton of Pure Chemcal Propertes Desgn Insttute for Physcal Propertes; Brgham Young Unversty, Provo, Utah. Shm, J. & Kohn, J. P. (1962). Multphase and Volumetrc Equlbra of the Methane-n- Hexane System at Temperatures between -10 and 150 C. Journal of Chemcal and Engneerng Data, Vol. 7, o. 1, pp. 28. Soave, G. (1972). Equlbrum Constants from a Modfed Redlch-Kwong Equaton of State. Chemcal Engneerng Scence, Vol. 27, o. 6, pp Stateva, R. P. & Tsvetkov S. G. (1994). A Dverse Approach for the Soluton of the Isothermal Multphase Flash Problem. Applcaton to Vapor-Lqud-Lqud Systems. Canadan Journal of Chemcal Engneerng, Vol. 72, o. 4, pp

27 384 Advances n atural Gas Technology Wakeham, W. A. & Stateva, R. P. (2004). umercal Soluton of the Isothermal Multphase Flash Problem. Revew of Chemcal Engneerng Journal, Vol. 20, os. 1-2, pp Wlknson, J. H. (1965). The Algebrac Egenvalue Problem, Clarendon, Oxford. Zhang, H.; Bonlla-Petrcolet, A. & Rangaah, G. P. (2011). A Revew on Global Optmzaton Methods for Phase Equlbrum Modelng and Calculatons. The Open Thermodynamcs Journal, Vol. 5, pp

28 Advances n atural Gas Technology Edted by Dr. Hamd Al-Megren ISB Hard cover, 542 pages Publsher InTech Publshed onlne 11, Aprl, 2012 Publshed n prnt edton Aprl, 2012 atural gas s a vtal component of the world's supply of energy and an mportant source of many bulk chemcals and specalty chemcals. It s one of the cleanest, safest, and most useful of all energy sources, and helps to meet the world's rsng demand for cleaner energy nto the future. However, explorng, producng and brngng gas to the user or convertng gas nto desred chemcals s a systematcal engneerng project, and every step requres thorough understandng of gas and the surroundng envronment. Any advances n the process lnk could make a step change n gas ndustry. There have been ncreasng efforts n gas ndustry n recent years. Wth state-of-the-art contrbutons by leadng experts n the feld, ths book addressed the technology advances n natural gas ndustry. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: Blanca E. García-Flores, Damler. Justo-García, Roumana P. Stateva and Fernando García-Sánchez (2012). Phase Behavor Predcton and Modelng of LG Systems wth EoSs - What s Easy and What s Dffcult?, Advances n atural Gas Technology, Dr. Hamd Al-Megren (Ed.), ISB: , InTech, Avalable from: InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A Rjeka, Croata Phone: +385 (51) Fax: +385 (51) InTech Chna Unt 405, Offce Block, Hotel Equatoral Shangha o.65, Yan An Road (West), Shangha, , Chna Phone: Fax: