A New Thermodynamic Function for Phase-Splitting at Constant Temperature, Moles, and Volume

Transcription

1 A New Thermodynamc Functon for Phase-Splttng at Constant Temperature, Moles, and olume Jří Mkyška Dept. of Mathematcs, Faculty of Nuclear Scences and Physcal Engneerng, Czech Techncal Unversty n Prague, Trojanova 3, Prague 2, Czech Republc Abbas Froozabad Yale Unversty, Yale School of Engneerng and Appled Scence, Dept. of Chemcal Engneerng & Envronmental Scence, 9 Hllhouse Avenue, ML 03, New Haven, CT 065 Reservor Engneerng Research Insttute, 595 Lytton Ave., Sute B, Palo Alto, CA 9430 DOI 0.002/ac.2387 Publshed onlne August 3, 200 n Wley Onlne Lbrary (wleyonlnelbrary.com). We ntroduce a new thermodynamc functon for phase-splt computatons at constant temperature, moles, and volume. The new volume functon F ntroduced n ths work s a natural choce under these condtons. Phase equlbrum condtons n terms of the volume functons are derved usng the Helmholtz free energy. We present a numercal algorthm to nvestgate two-phase equlbrum based on the fxed pont teraton and Newton method. We demonstrate usefulness and powerful features of the new thermodynamc functon for a number of examples n two-phase equlbrum calculatons. C 200 Amercan Insttute of Chemcal Engneers AIChE J, 57: , 20 Keywords: two-phase equlbrum, constant volume flash, T-flash, Helmholtz free energy, volume functon Introducton Consder a closed system of constant volume n whch there s a mxture of c components wth mole numbers n,,n c at temperature T. Assumng that the system s n two-phase, we want to establsh compostons and amounts of both phases. Ths s the problem of two-phase phase-splt (the so called flash) under the constant temperature, moles, and volume (T-flash). The motvaton for constant volume flash s the equlbrum calculaton n a PT cell n twophase when two nonequlbrum phases are ntroduced.,2 These cells are used to determne dffuson coeffcents n both phases n two-phase state. We have found out that the use of conventonal methods s based on ad hoc approaches. Correspondence concernng ths artcle should be addressed to A. Froozabad at abbas.froozabad@yale.edu. C 200 Amercan Insttute of Chemcal Engneers The standard problem of constant pressure and temperature (PT-flash) s addressed n many references. 3 6 In ths approach, pressure, temperature, and overall chemcal composton are gven. The phase compostons and molar denstes are computed usng the mnmzaton of the Gbbs energy. Ths approach has the shortcomng that t cannot provde an answer when a sngle component s n two-phase regon at temperature T and saturaton pressure P ¼ P sat (T) because the chemcal potentals of the component n both phases are the same. Therefore, we cannot determne whether the component at these specfc condtons s vapor or lqud or a mxture of both. Althought P and T are the most preferred varables n chemcal engneerng, we see that specfyng pressure and temperature s not suffcent for the unque determnaton of the state of the system n ths case. The ssue can be resolved by reformulatng the problem usng the mnmzaton of the Helmholtz free energy rather than Gbbs free energy. In ths formulaton, the volume of AIChE Journal July 20 ol. 57, No

2 the system, mole numbers, and temperature are gven, and the chemcal compostons, molar denstes of the phases, and pressure n the mxture are computed. The selecton of varables, T, and n,( [{,,c}) s natural for pressureexplct equatons of state. Unlke n PT-flash, the T-flash provdes a unque answer n any physcally admssble stuaton because when s known, P can be evaluated readly from say a cubc equaton of state. Therefore, we develop a new volume-based formulaton of equlbrum thermodynamcs. The presented dervatons lead to a new functon F, called volume functon that plays an analogous role to fugacty that s used n the pressure-based formulaton. We present all dervatons n detals to show that ths approach s much more ft for the pressure-explct equatons of state than the standart development usng the Gbbs free energy that suffers from the nonunqueness of volume at gven pressure. A theoretcal possblty of other state functon-based flash specfcaton (ncludng T-flash) s mentoned n Refs. 6 and 7, where a nested optmzaton approach s proposed. Ths means that n an outer loop we search teratvely for pressure, whch s used n PT-flash n the nner loop to evaluate equlbrum state at that pressure. The goal of the teratons s to fnd a pressure for whch the volume constrant s satsfed. Ths procedure allows to use exstng mplementatons of the PT-flash but, on the other hand, s computatonally expensve as t requres many solutons of PT-flashes before the true pressure s found. In ths artcle, we offer an alternatve formulaton allowng to formulate T-flash drectly wthout usng nested teratons. The artcle s structured as follows. In the frst secton, we ntroduce the new thermodynamc volume functon F of varables, T, and n that wll be useful n descrbng thermodynamc behavor of real mxtures under the constant temperature, volume, and moles. Then, we develop an expresson for the chemcal potental of a component n a mxture n terms of ths new functon. We reformulate the two-phase equlbrum condtons at constant temperature, volume, and moles usng F. Next, we propose a numercal algorthm for computaton of T-flash, and, fnally, we present examples showng results of phase-equlbrum computatons based on the new formulaton for a number of mxtures n two-phase state. olume Functon Coeffcent We wll ntroduce a new thermodynamc volume functon that wll be useful to derve basc expressons for the chemcal potental of a component n a multcomponent mxture from the bulk phase equlbrum thermodynamcs. Our dervaton wll be based on Helmholtz free energy and wll use volume, temperature, and moles as prmary varables. Assumng a pressure-explct equaton of state, t s convenent to descrbe the system usng the Helmholtz free energy A ¼ A(,T,n,,n c ). The general expresson for the Helmholtz free energy of a bulk phase s gven by A ¼ P þ Xc ¼ n l ; () where P ¼ P(,T,n,,n c ) s the pressure gven by an equaton of state, and l ¼ l (,T,n,,n c ) s the chemcal potental of the -th component n the mxture. From Algorthm: Two-Phase T-Flash Usng Fxed Pont Iteraton. Let c, z, z c, and T [ 0 be gven. Evaluate P 0 ¼ P(/c,T,z,,z c ), ntalze K s usng Wlson correlaton, 8.e. ln K ¼ 5:37ð þ x Þ T c þ ln P c T P 0 at the ntal pressure P 0 and set the number of teratons, n ¼ Evaluate a [ (0;) by solvng the Rachford-Rce equaton X c ðk n Þz þðk n ¼ Þa ¼ 0; that can be solved, e.g., by Newton s method. 3. Update chemcal compostons of both phases by x 0 ;nþ ¼ we see that z þðk n Þa ; x00 ;nþ ¼ z K n þðk n Þa ; 4. Use bsecton or other method to fnd S 00 nþ [ (0;) satsfyng S00 nþ P cð aþ ; T; x0 ;nþ ; ; x0 c;nþ ¼ P S00 nþ ; T; x00 ;nþ ; ; x00 c;nþ ; ca and update the other saturaton and molar concentratons by S 0 nþ ¼ S00 nþ ; 5. Update K values by c0 nþ da ¼ SdT Pd þ Xc ; cð aþ ¼ S 00 ; c 00 nþ ¼ ca nþ S 00 ¼ c0 nþ U ð; T; c 00 nþ x00 ;nþ ; ; c00 nþ x00 c;nþ Þ c 00 nþ U ð; T; c 0 n x0 ;nþ ; ; c0 n x0 c;nþ Þ : K nþ ¼ l dn ; l : (2) Assumng that A s a smooth functon of ts varables, the mxed second-order dervatves must be nterchangeable, ; (3) wth approprate varables held constant. Integratng (3) between two volumes and 2, we derve the followng expresson descrbng the change of chemcal potental wth volume at constant temperature and moles l ð 2 ; T; n ; ; n c Þ¼l ð ; T; n ; ; n c Þ Z ð; T; n ; ; n c Þd: nþ 6. Check for convergence. If needed, ncrease n by one and go to step 2. : ð4þ 898 DOI 0.002/ac Publshed on behalf of the AIChE July 20 ol. 57, No. 7 AIChE Journal

3 For an deal gas mxture, the equaton of state P ¼ nrt ; can be ntegrated usng (4) to yeld Xc where n ¼ n ¼ l ð 2 ; T; n ; ; n c Þ¼l ð ; T; n ; ; n c Þ RT ln 2 : (5) For real mxtures a more general equaton than (5) must be used. To smplfy our dervatons, t s convenent to have a smlar form of the expresson for the chemcal potental of a component n a real mxture as n the deal case. For ths purpose, we ntroduce the volume functon of -th component F ¼ F (,T,n,,n c ) by the followng propertes l ð 2 ; T; n ; ; n c Þ¼l ð ; T; n ; ; n c Þ RT ln F ð 2 ; T; n ; ; n c Þ F ð ; T; n ; ; n c Þ ; and F ð; T; n ; ; n c Þ lm ¼ : (7)!þ Furthermore, we defne the volume functon coeffcent by U ð; T; n ; ; n c Þ¼ F ð; T; n ; ; n c Þ : (8) Equaton 7 then amounts to sayng that lm!þ U (,T,n,,n c ) ¼ for gven temperature and moles. The volume functon F and volume functon coeffcent U play analogous roles to fugacty and fugacty coeffcents. Comparng (6) wth (4), we have RT ln F Z ð 2 ; T; n ; ; n c Þ 2 F ð ; T; n ; ; n c Þ ð; T; n ; ; n c Þd: Settng ¼, the last equaton can be rearranged to ln F ð; T; n ; ; n c Þ 2 2 F ð 2 ; T; n ; ; n c Þ ¼ RT Z 2 The above equaton can be wrtten as ln F ð; T; n ; ; n c Þ ¼ Z 2 RT ð; T; n ; ; n c ð; T; n ; ; n c Þ d þ ln F ð 2 ; T; n ; ; n c Þ 2 : Passng 2! þ, the last term on the rght hand sde vanshes because of (7), whch yelds ln U ð; T; n ; ; n c Þ¼ Z þ ð; T; n ; ; n c Þ d: The ntegral on the rght hand sde can be evaluated analytcally usng an equaton of state. It follows from (9) that for an deal gas F (,T,n,,n c ) ¼, and U (,T,n,,n c ) ¼ for any gven temperature, moles, and volume. Thus, the volume functon coeffcent can ndcate the degree of nondealty of a component n the mxture. These propertes are volume-based counterparts of analogous propertes of the fugacty and fugacty coeffcents wth respect to pressure. Comparng the ntegral on the rght hand sde of (9) wth the formula for fugacty coeffcents found n lterature (see e.g., Ref. 3), we note that the relatonshp between the volume functon coeffcent and the fugacty coeffcent s U ¼ Zu ; where u denotes the conventonal fugacty coeffcent and Z s the phase compressblty factor. Z and u must now be expressed n terms of volume, temperature, and moles. In the lterature, the Z and u are usually understood as functons of P, T, and composton. However, when Z and u are to be evaluated for gven P, T, and moles, one has to solve the cubc equaton to get volume, whch may not be unque, and n that case, one has to select one of the roots based on the Gbbs free energy crtera or other methods. In ths artcle, we present an alternatve formulaton, whch uses the volume functon coeffcents rather than fugacty coeffcents. In ths formulaton, the root-selecton problems do not appear because all functons are expressed n terms of volume, temperature, and moles. The formula for the volume functon coeffcent U for the Peng-Robnson equaton 9 s presented n the Appendx. Chemcal Potental of a Component n a Real Mxture In the new framework, we descrbe the dependency of the chemcal potental of a component n a real mxture to the chemcal potental n a pure substance. An deal mxture s the one whch obeys l ð; T; n ; ; n c Þ¼l ð; T; 0; ; 0; n ; 0; ; 0Þ: To smplfy the notaton, we wll denote the chemcal potental of the pure substance as l ð; T; n Þ¼l ð; T; 0; ; 0; n ; 0; ; 0Þ; and the volume functon n the pure component as F ð; T; n Þ¼F ð; T; 0; ; 0; n ; 0; ; 0Þ: Equaton 6 can be wrtten for the mxture as well as for the pure component (9) AIChE Journal July 20 ol. 57, No. 7 Publshed on behalf of the AIChE DOI 0.002/ac 899

4 l ð; T; n ; ; n c Þ ¼ l ð ; T; n ; ; n c Þ RT ln F ð; T; n ; ; n c Þ F ð ; T; n ; ; n c Þ ; l ð; T; n Þ ¼ l ð ; T; n Þ RT ln F ð; T; n Þ F ð ; T; n Þ : Subtractng the second expresson from the frst above, l ð; T; n ; ; n c Þ l ð; T; n Þ ¼ l ð ; T; n ; ; n c Þ l ð ; T; n Þ RT ln F ð; T; n ; ; n c Þ F ð ; T; n ; ; n c Þ RT ln F ð ; T; n Þ F ð; T; n Þ : If the volume s suffcently large, the mxture at volume behaves deally. Passng!þ, one can derve l ð; T; n ; ; n c Þ¼l ð; T; n Þ RT ln F ð; T; n ; ; n c Þ F ð; T; n : Þ (0) Let us assume that we have a mxture at two states ( 0,T,n, 0,nc 0) and (00,T,n 00,,n 00 c). Our goal s to express the dfference of chemcal potentals between these two states n terms of the volume functons. Usng (0), we derve l ð 0 ; T; n 0 ; ; n0 c Þ l ð 00 ; T; n 00 ; ; n00 c Þ ¼ l ð0 ; T; n 0 Þ l ð00 ; T; n 00 Þ RT ln F ð 0 ; T; n 0 ; ; n0 c Þ F ð 00 ; T; n 00 ; ; n00 c Þ þ RT ln F ð0 ; T; n0 Þ F ð00 ; T; n 00 Þ : The frst two terms on the rght hand sde can be rewrtten as l ð0 ;T;n 0 Þ l ð00 ;T;n 00 Þ¼l ð0 =n 0 ;T;Þ l ð00 =n 00 ;T;Þ ¼ RT ln F ð0 ;T;n 0 Þ F ð00 ;T;n 00 ; n00 Þ n 0 where we take advantage of the fact that the chemcal potental s a homogeneous functon of order zero and the volume functon s a homogeneous functon of order one n the volume and moles. Combnng the last two equatons, we obtan the followng key expresson for the dfference of chemcal potentals at two dfferent states wrtten n terms of the volume functons l ð 0 ; T; n 0 ; ; n0 c Þ l ð 00 ; T; n 00 ; ; n00 ¼ RT ln n00 n 0 c Þ F ð 0 ; T; n 0 ; ; n0 c Þ F ð 00 ; T; n 00 ; ; n00 c Þ : ðþ Condtons for Two-Phase Equlbrum Consder a mxture of c components wth mole numbers n,,n c occupyng volume at temperature T. Assumng that the mxture wll splt nto two phases, we want to calculate volumes 0 and 00, and mole numbers of each component n each phase n 0 and n 00 for ¼,,c, and consequently the pressure. The equlbrum state s derved from the mnmzaton of the total Helmholtz energy of the mxture A ¼ Að 0 ; T; n 0 ; ; n0 c ÞþAð00 ; T; n 00 ; ; n00 c Þ; whch s subject to the followng constrants and 0 þ 00 ¼ ; (2) n 0 þ n00 ¼ n ; ¼ ; ; c: (3) Usng the Lagrange multpler method, one can fnd the necessary condtons of the phase equlbra and Pð 0 ; T; n 0 ; ; n0 c Þ¼Pð00 ; T; n 00 ; ; n00 cþ; (4) l ð 0 ; T; n 0 ; ; n0 c Þ¼l ð 00 ; T; n 00 ; ; n00 cþ; ¼ ; ; c; (5) as expected. An equvalent expresson of (5) n terms of the volume functons reads as n 0 F ð 0 ; T; n 0 ; ; n0 c Þ ¼ n 00 F ð 00 ; T; n 00 ; ; n00 c Þ : (6) Numercal Algorthm for Two-Phase Flash Computaton In two-phase, we are nterested to calculate phase compostons, amounts, and also the pressure of the system. Let us rewrte the two-phase flash Eqs. 2 5 n terms of concentratons and compostons of both phases. We ntroduce the overall molar concentraton c ¼ n/, the phase molar concentratons c 0 ¼ n 0 / 0 and c 00 ¼ n 00 / 00, overall mole fractons z ¼ n /n, and phase mole fractons x 0 ¼ n 0/n0 and x 00 ¼ n 00 /n00, and phase volume fractons S 0 ¼ 0 /, and S 00 ¼ 00 /, respectvely. Usng ths notaton, Eq. 2 transforms to S 0 þ S 00 ¼ ; (7) whereas the mole balance Eq. 3 can be rewrtten as c 0 x 0 S0 þ c 00 x 00 S00 ¼ cz ; ¼ ; ; c: (8) As pressure s an ntensve property (homogeneous functon of order zero n varables,n,, n c ), Eq. 4 yelds Pð; T; c 0 x 0 ; ; c0 x 0 c Þ¼Pð; T; c00 x 00 ; ; c00 x 00 cþ: (9) Fnally, the chemcal equlbrum Eq. 6 can be wrtten n terms of the volume functon coeffcents usng () as c 0 x 0 U ð; T; c 0 x 0 ; ; c0 x 0 c Þ ¼ c 00 x 00 U ð; T; c 00 x 00 ; ; c00 x 00 c Þ : (20) To solve these equatons usng the fxed pont teraton (also called succesve substtuton teraton, SSI), t s convenent to ntroduce the K values by K ¼ x00 x 0 : (2) From (2) t follows that x 00 ¼ K x 0, whch can be substtuted nto (8) to obtan 900 DOI 0.002/ac Publshed on behalf of the AIChE July 20 ol. 57, No. 7 AIChE Journal

5 Table. Propertes of the Components for the C nc 5 Mxture Used n Examples and 2 Component x [ ] T crt [K] P crt [MPa] M w [g mol ] C nc The C nc 5 bnary nteracton coeffcent s d C nc 5 ¼ x 0 ¼ z þðk Þa ; z K x00 ¼ þðk Þa ; (22) where a ¼ c 00 S 00 /c s the mole fracton of the double-prmed phase. These equatons can be used to evaluate phase compostons provded that K and a are gven. As compostons of both phases n (22) should sum to one, for a gven set of K values, a can be updated by solvng the Rachford-Rce equaton 0 X c ðx 00 x 0 ðk Þz Þ¼Xc ¼ 0: (23) þðk ¼ ¼ Þa Once a and chemcal composton of both phases are establshed, phase molar concentratons and saturatons must be determned so that the pressures n both phases are the same. The followng system of four equatons c 0 S 0 ¼ cð aþ; c 00 S 00 ¼ ca; S 0 þ S 00 ¼ ; P c ; T; x 0 0 ; ; x0 c ¼ P c ; T; x ; ; x00 c ; for the 4 unknowns (c 0, c 00, S 00, and S 00 ) can be readly reduced nto a sngle equaton for one unknown saturaton S 00 [ (0;) S00 P cð aþ ; T; x0 ; ; x0 c ¼ P S00 ca ; T; x00 ; ; x00 c : (24) For a cubc equaton of state, Eq. 24 s an algebrac equaton of the ffth order. In theory t may have up to fve real roots. In all examples we examnated, there was always only one root n the nterval (0;), whch could be readly approxmated usng the bsecton method. Other methods, lke the Newton method, can be used as well. Fnally, K values are updated usng (20) as K ¼ x00 x 0 ¼ c0 U =c 00 ; T; x 00 ; ; x00 c c 00 U =c 0 ; T; x 0 ; ; : x0 c The key steps of the method are summarzed n the Algorthm. Table 2. Overall Propertes of the Mxture and Resultng Phase Propertes n Two-Phase Flash at Constant Temperature T 5 37 K and olume for Example Property Unt Overall mxture Phase Phase 2 Molar concentraton mol m C mole fracton nc 5 mole fracton Phase volume fracton Table 3. Overall Propertes of the Mxture and Resultng Phase Propertes n Two-Phase Flash at Constant Temperature T K and olume for Example 2 Property Unt Overall mxture Phase Phase 2 Molar concentraton mol m C mole fracton nc 5 mole fracton Phase volume fracton The teratons are stopped whenever max ln Knþ 2f; ;cg Numercal Examples ln K n \ tol ¼ 0 2 : We have tested the algorthm n several examples of bnary and multcomponent mxtures n two-phase. Below we show performance of the method for two bnary mxtures and one four-component mxture. Further, we provde two morecomplex phase-splt computatons for a multcomponent reservor flud. All examples are motvated by experments n the PT cells. In these experments, the total volume s fxed. A part of ths volume s flled by a lqud at some ntal pressure P n. The rest of the volume s flled by a gas at the same ntal pressure. When the two fluds are mxed, the pressure changes. The fnal equlbrum pressure P of the system after mxng results from the T-flash computaton. The correctness of the T-flash results s checked by performng the PTflash at the fnal pressure wth the same overall composton and temperature. The agreement was excellent n all cases. Example In the frst example, we nvestgate two-phase equlbrum for a bnary mxture of methane (C ) and n-penthane (nc 5 ) of total concentraton c ¼ mol m 3, wth mole fractons z C ¼ and z nc5 ¼ at temperature T ¼ 37 K. The condton corresponds to the PT-cell experment n whch C (34.4% of volume) s placed on the top of nc 5 at the ntal pressure P n ¼ 5 MPa. Parameters of the Peng-Robnson equaton of state are presented n Table. The algorthm found a soluton n 46 teratons. Wthn each teraton the Rachford-Rce, Eq. 23, was solved by Newton s method wth the ntal guess a ¼ 0.5. The resultng pressure s P ¼ MPa. The overall mxture and phase-splt results are summarzed n Table 2. The results were verfed by the PT-flash computaton performed at the fnal pressure wth the same overall composton and temperature. The PT-flash converged n 45 teratons. Table 4. Propertes of the Components for the Four-Component Mxture Used n Example 3 Component x [ ] T crt [K] P crt [MPa] M w [g mol ] N C C nc AIChE Journal July 20 ol. 57, No. 7 Publshed on behalf of the AIChE DOI 0.002/ac 90

6 Example 2 Table 5. Bnary Interacton Coeffcents for the Four-Component Mxture Used n Example 3 Component N 2 C C 3 nc 0 N C C nc In ths example, we compute two-phase equlbrum for a mxture of methane (C ) and n-penthane (nc 5 ) of total concentraton c ¼ mol m 3, wth mole fractons z C ¼ and z nc5 ¼ at temperature T ¼ K. The condton corresponds to the PT-cell experment n whch C (65% of volume) s placed on the top of nc 5 at the ntal pressure P n ¼ 0.2 MPa. Parameters of the Peng- Robnson equaton of state are presented n Table. The algorthm found a soluton n 20 teratons. The resultng pressure s P ¼ MPa. The overall mxture and spltphase results are summarzed n Table 3. The agreement wth the PT-flash at the fnal pressure s excellent. The PTflash converged n 9 teratons. Example 3 In Example 3, we compute two-phase equlbrum for a four-component mxture of ntrogen (N 2 ), methane (C ), propane (C 3 ), and n-decane (nc 0 ) at temperature T ¼ K. Here, ntrogen (35% of volume) s placed on the top of a four-component mxture (z N2 ¼ 0.0, z C ¼ 0.29, z C3 ¼ 0.29, and z nc0 ¼ 0.4) at the ntal pressure 3.73 MPa. Parameters of the Peng-Robnson equaton of state are presented n Tables 4 and 5. The overall molar concentraton and overall mole fractons of all components are shown n Table 6. The algorthm found a soluton n 25 teratons. The resultng pressure s P ¼ MPa. Note the pressure ncrease due to vaporzaton. The overall mxture and splt-phase results are summarzed n Table 6. The results agree wth those obtaned usng the PT-flash at the fnal pressure. The PT-flash converged n 25 teratons. Example 4 In Example 4, we compute two two-phase equlbra for a multcomponent ol mxed wth ntrogen (N 2 ), and carbon doxde (CO 2 ). The ol s modelled usng seven components. The composton of the ol and parameters of the Peng-Robnson equaton of state are presented n Tables 7 and 8. The Table 6. Overall Propertes of the Mxture and Resultng Phase Propertes n Two-Phase Flash at Constant Temperature T K and olume for Example 3 Property Unt Overall mxture Phase Phase 2 Molar concentraton mol m N 2 mole fracton C mole fracton C 3 mole fracton nc 0 mole fracton Phase volume fracton Table 7. Composton and Propertes of the Components for the Reservor Flud Used n Example 4 Component z x [ ] T crt [K] P crt [MPa] M w [g mol ] N CO C PC PC PC C 2þ pseudocomponents are defned as PC (H 2 S þ C 2 þ C 3 ), PC 2 (C 4 C 6 ), and PC 3 (C 7 C ). In both experments, 50% of volume of the PT cell s flled by ths ol. The ntal pressure P n ¼ MPa, and the temperature s T ¼ 43.7 K. Under these condtons, the ol s n sngle phase wth molar concentraton c ol ¼ mol m 3. The remanng 50% of volume s flled wth ether N 2 or CO 2 at the same ntal pressure. Addton of gas turns the system nto two phase n both cases. ForthecaseofN 2, the overall propertes of the resultng mxture and phase-splt results are summarzed n Table 9. The fnal pressure s P ¼ MPa. The algorthm found a soluton n 33 teratons. The results agree wth those obtaned usng the PT-flash at the fnal pressure. The PT-flash converged n 34 teratons. Some of the K -values n Table 9 are very dfferent from one; the system s far from the crtcal pont. For the case of CO 2, the overall propertes of the resultng mxture and splt-phase results are summarzed n Table 0. The fnal pressure s P ¼ 3.27 MPa. The algorthm found a soluton n 266 teratons. The results agree wth those obtaned usng the PT-flash at the fnal pressure. The PT-flash converged n 254 teratons. The K -values n Table 0 are closer to one. Unlke n the prevous case, the mxture s near-crtcal, whch explans the ncreased number of teratons that are needed to converge usng the fxed pont teraton method. Summary and Conclusons In ths work, we have ntroduced a new thermodynamc functon to descrbe two-phase equlbrum at constant temperature, volume, and moles. The new volume functon coeffcent replaces the fugacty coeffcents that are used n common formulatons of two-phase equlbrum at constant temperature and pressure. Unlke the conventonal approach, our method can determne unquely the equlbrum state of a pure substance n two-phase state. The volume-based formulaton of two-phase equlbrum n terms of the volume functon coeffcents has been derved for the Peng-Robnson Table 8. Bnary Interacton Coeffcents for the Reservor Flud Used n Example 4 Component N 2 CO 2 C PC PC 2 PC 3 C 2þ N CO C PC PC PC C 2þ DOI 0.002/ac Publshed on behalf of the AIChE July 20 ol. 57, No. 7 AIChE Journal

7 Table 9. Overall Propertes of the Mxture of the Reservor Flud from Table 7 (50% of volume) wth N 2 at the Intal Pressure P n MPa and Resultng Phase Propertes n Two-Phase Flash at Constant Temperature T K and olume (Example 4) Property Unt Overall mxture Phase Phase 2 K -values Molar concentraton mol m N 2 mole fracton CO 2 mole fracton C mole fracton PC mole fracton PC 2 mole fracton PC 3 mole fracton C 2þ mole fracton Phase volume fracton The fnal pressure s P ¼ MPa. Table 0. Overall Propertes of the Mxture of the Reservor Flud from Table 7 (50% of volume) wth CO 2 at the Intal Pressure P n MPa and Resultng Phase Propertes n Two-Phase Flash at Constant Temperature T K and olume (Example 4) Property Unt Overall mxture Phase Phase 2 K -values Molar concentraton mol m N 2 mole fracton CO 2 mole fracton C mole fracton PC mole fracton PC 2 mole fracton PC 3 mole fracton C 2þ mole fracton Phase volume fracton The fnal pressure s P ¼ 3.27 MPa. equaton of state, but the same concept can be used for other pressure-explct equatons of state as well. We proposed a numercal algorthm for computaton of the phase-splt propertes, whch s based on a combnaton of the fxed pont outer teraton and Newton s method n the nner teraton. To show effcency of ths approach we have performed results of numercal computatons of multcomponent mxtures of dfferent complexty. The results ndcate that to acheve the same accuracy, the number of teratons of the T-flash method based on the fxed pont teraton s about the same as when usng the PT-flash under the same physcal condtons. As the fnal pressure s not known a pror when the volume s constant, the computaton of Tflash usng the PT-flash combned wth outer teratons, as suggested Refs. 6 and n 7, s necessarly neffcent. Our method provdes the correct soluton n practcally the same number of teratons as one run of the PT-flash at the fnal pressure. 2. Haugen K, Froozabad A. Mxng of two nonequlbrum phases n one dmenson. AIChE J. 2009;55: Froozabad A. Thermodynamcs of Hydrocarbon Reservors. New York: McGraw-Hll, Mchelsen ML. The sothermal flash problem.. Stablty. Flud Phase Equlbra. 982;9: Mchelsen ML. The sothermal flash problem. 2. Phase-splt computaton. Flud Phase Equlbra. 982;9: Mchelsen ML, Mollerup JM. Thermodynamc Models: Fundamentals and Computatonal Aspects. Holte, Denmark: Te-Lne Publcatons, Mchelsen ML. State functon based flash specfcatons. Flud Phase Equlbra. 999;58: Wlson GM. A Modfed Redlch-Kwong Equaton of State, Applcaton to General Physcal Data Calculaton, paper No. 5C presented at the 969 AIChE 65th Natonal Meetng, Cleveland, Oho, May 4 7, Peng DE, Robnson DB. A new two-constant equaton of state. Ind Eng Chem Fund. 976;5: Rachford HH, Rce JD. Procedure for use of electronc dgtal computers n calculatng flash vaporzaton hydrocarbon equlbrum. Trans Am Insttute Mnng Metallurg Eng. 952;95: Acknowledgments The work was supported by the member companes of the Reservor Engneerng Research Insttute, and by the project Mathematcal Modellng of Mult-Phase Porous Meda Flow 20/08/P567 of the Czech Scence Foundaton. Lterature Cted. Haugen K, Froozabad A. Composton at the Interface between multcomponent non-equlbrum phases. J Chem Phys. 2009;30: , 9. Appendx: olume Functon Coeffcent for Peng-Robnson Equaton of State In ths work, we use the Peng-Robnson equaton of state 9 n the form Pð; T; n ; ; n c Þ¼ nrt B A 2 þ 2B B 2 ; where R s the unversal gas constant, n ¼ P c ¼ n s the total number of moles, and coeffcents A and B are gven by AIChE Journal July 20 ol. 57, No. 7 Publshed on behalf of the AIChE DOI 0.002/ac 903

8 A ¼ Xc X c ¼ j¼ n n j a j ; B ¼ Xc ¼ n b p ffffffffffffff a j ¼ð d j Þ a a j; b ¼ 0:0778 RT ;crt P ;crt a ¼ 0:45724 R2 T;crt 2 p þm ffffffff 2; T r P ;crt ( m ¼ 0:37464 þ :54226x 0:26992x 2 ; for x \0:5; 0:3796 þ :485x 0:644x 2 þ 0:0667x3 for x 0:5: In these equatons, d j denotes the bnary nteracton parameter between the components and j, T,crt, P,crt, and x are the crtcal temperature, crtcal pressure, and accentrc factor of the -th component, respectvely. The volume functon coeffcent for the Peng-Robnson equaton of state can be found analytcally usng (9) as ln U ¼ ln B b n B þ Ab BRT 2 þ 2B B " # 2 Ab Xc pffffff n j a j ln þðþ p ffffff 2 ÞB p 2 BRT 2B j¼ þð ffffff 2 ÞB : Manuscrpt receved Apr. 8, 200, and revson receved July 4, DOI 0.002/ac Publshed on behalf of the AIChE July 20 ol. 57, No. 7 AIChE Journal