Fluid Phase Equilibria

Transcription

1 Flud Phase Equlbra 375 (2014) Contents lsts avalable at ScenceDrect Flud Phase Equlbra jou rn al h om epage: wwwelsevercom/locate/flud Calculaton of phase equlbra for mult-component mxtures usng hghly accurate Helmholtz energy equatons of state Johannes Gernert, Andreas Jäger, Roland Span Thermodynamcs, Ruhr-Unverstät Bochum, Unverstätsstr 150, Bochum, Germany a r t c l e n f o Artcle hstory: Receved 30 January 2014 Receved n revsed form 17 Aprl 2014 Accepted 9 May 2014 Avalable onlne 15 May 2014 Keywords: Helmholtz energy model Mxture Phase equlbrum Stablty analyss Tangent plane dstance a b s t r a c t To test the thermodynamc stablty and to determne the equlbrum phase compostons n case the orgnal phase s found unstable s one of the greatest challenges assocated wth calculatng thermodynamc propertes of mult-component mxtures The mnmzaton of the tangent plane dstance functon s a wdely used method to check for stablty, whle dfferent approaches can be chosen to mnmze the Gbbs energy n order to fnd the phase equlbrum Whle these two problems have been appled to several dfferent thermodynamc models, very lttle work has been publshed on such algorthms usng mult-parameter Helmholtz energy equatons of state In ths work, combned stablty and flash calculaton algorthms at gven pressure and temperature (p,t), pressure and enthalpy (p,h), and pressure and entropy (p,s) are presented The algorthms by Mchelsen et al (1982, 1982, 1987) are used as bass and are adapted to mult-parameter Helmholtz energy models In addton, a robust and sophstcated densty solver s proposed whch s necessary for the calculaton of propertes from the Helmholtz energy model at gven state varables other than temperature and densty All partal dervatves necessary to solve the sothermal, senthalpc and sentropc flash problems usng numercal methods based on the Jacoban matrx are derved analytcally and gven n the supplementary materal to ths artcle Results for some mult-component systems usng the GERG-2008 model (Kunz and Wagner, 2012) are shown and dscussed 2014 Elsever BV All rghts reserved 1 Introducton The analyss of the stablty of a mxture at gven condtons and phase equlbrum calculatons were n the focus of research over the past decades and stll contnue to be mportant problems n thermodynamcs To ensure thermodynamc stablty, the total state functons G(T, p, n), A(T, V, n), U(S, V, n), and H(S, p, n) have to be at the global mnmum Hence algorthms are needed to mnmze the state functons for any gven mxture Dependng on the applcaton, dfferent demands may be formulated for such algorthms In general a compromse for the contradctory goals of developng a fast and effcent but lkewse relable and stable algorthm has to be found Varous algorthms have been proposed to solve ths knd of problem, all of them havng advantages and shortcomngs The publshed algorthms may be splt nto two sub-categores: stochastc and determnstc algorthms Determnstc algorthms (see eg [1 4]) utlze a separate Correspondng author Tel: ; fax: E-mal address: ajaeger@thermorubde (A Jäger) stablty analyss and contnue solvng the phase equlbrum problem Stochastc algorthms (eg [5 8]) mnmze the state functon by applyng a global optmzaton method In addton to the dfferent types of algorthms the type of the equaton of state has to be consdered when choosng a soluton method Some of the algorthms proposed have been desgned to smplfy calculatons usng a specfc type of equaton (cubc equatons of state (EOS), g E models, etc) However, only few methods have been desgned and tested for multparameter fundamental EOS explct n the Helmholtz energy [9] Kunz et al [4] descrbed the basc prncples of treatng phase equlbra for mxtures usng Helmholtz EOS and the method of Mchelsen [2,3] n combnaton wth analytcal dervatves needed to solve the phase equlbrum condtons Ths method was taken as a bass n ths work; correspondng algorthms were reformulated n conjuncton wth the development of a new thermodynamc property program lbrary, and extended for sentropc and senthalpc flash calculatons usng analytcal dervatves Furthermore, methods are presented to predct the stablty of mxtures modeled wth Helmholtz EOS based on gven temperature and pressure, pressure and enthalpy, and pressure and entropy / 2014 Elsever BV All rghts reserved

2 210 J Gernert et al / Flud Phase Equlbra 375 (2014) Helmholtz equatons of state Many models for thermodynamc propertes of mxtures may be found n the lterature Most of these models are based on equatons of state for the flud phase(s) of pure substances Cubc equatons of state (eg [10 12]) wth varous modfcatons (eg the CPA [13] or PSRK [14] models) are most commonly used to descrbe phase equlbra For ths knd of equatons, dfferent approaches to model mxtures exst Ether rather smple lnear or quadratc mxng rules may be appled to the parameters of the EOS or more complex mxng rules lke (modfed) Huron-Vdal mxng rules [15] may be chosen However, the models mentoned above have some weaknesses wth regard to the accuracy of calculated thermodynamc propertes [16], partcularly at dense homogeneous states For pure substances these problems may be overcome by usng fundamental equatons of state explct n the reduced Helmholtz energy [17 19] These equatons typcally comprse an deal gas part and an emprcally determned resdual part: a(t, ) = (, ı) = o(, ı) + r(, ı) (1) RT where ı s the reduced densty and s the nverse reduced temperature It s ı = and = T c (2) c T In recent tmes these models have been extended to mxtures Based on the work of Tllner-Roth [20], Lemmon and Tllner-Roth [21], and Lemmon and Jacobson [22], Kunz and Wagner [16] developed the GERG-2008 equaton of state for natural gases and other mxtures The basc dea of ths model s to combne hghly accurate equatons of state n the Helmholtz energy usng an extended correspondng states prncple The equaton for the mxture reads: (, ı, x) = o(t,, x) + r(, ı, x) (3) where ı s the reduced densty and s the nverse reduced temperature accordng to ı = r ( x) and = T r( x) T wth the reducng functons T r and r as functons of the composton The mxng rules read: 1 r ( x) = T r ( x) = (4) N 1 x 2 1 x + c k,km f,km (x k, x m ), wth f,km (x k, x m ) = x k x k + x m 1 m c,k ˇ2 k=1 k=1 m=k+1,km x and c,km = 2ˇ,km,km k + x m 8 N 1 x 2 k T x c,k + c T,km f T,km (x k, x m ), wth f T,km (x k, x m ) = x k x k + x m m k=1 k=1 m=k+1 be used to model mxture propertes wth hgher accuracy or to model complex mxture behavor (for detaled nformaton, see [16] or Appendx A n the supplementary materal to ths artcle) Kunz and Wagner [16] demonstrated that ths type of model can be used for the very accurate and consstent descrpton of mxture propertes However, the consderable gan n accuracy when usng these models comes at the prze of hgh numercal complexty It s known that the evaluaton of such models s demandng In the followng, a stable algorthm for phase equlbrum calculatons based on prevously publshed approaches has been adapted to mxture models based on emprcal multparameter equatons of state New methods for the calculaton of the sothermal (p,t), senthalpc (p,h), and sentropc (p,s) flash are presented 3 Combned stablty analyss and sothermal flash calculaton Gven the overall composton x spec and the temperature T spec and pressure p spec of a mxture, algorthms for property calculaton need to test whether the gven phase s stable or whether t splts n two (or more) phases If the mxture s found to be unstable, flash calculatons are performed subsequently 31 Stablty analyss The phase stablty calculaton algorthm used n ths work s based on the formulaton by Mchelsen [2,3] and [23] and s also descrbed n the GERG-2004 monograph by Kunz et al, Sect 75 [4] It uses the tangent plane condton of the Gbbs energy of mxng as stablty crteron, whch was frst ntroduced by Baker et al [24] The tangent plane dstance functon TPD TPD( w) = w [ ( w) ( x spec )] 0, (8) has to be non-negatve for any tral phase wth the composton w to ensure that the ntal phase wth the composton x spec s stable The expresson above can be transformed to a more convenent reduced form, whch uses the fugacty coeffcents ϕ rather than the chemcal potentals ( 1 1/3 c,k ˇ2 T,km x k + x m and c T,km = 2ˇT,km T,km (T c,k T c,m ) 05 ) /3 c,m (5) The deal part of the Helmholtz energy for a mxture consstng of N components s gven as: o(t,, x) = x ( o o, (T, ) + ln x ) (6) where o o, are the pure flud contrbutons The resdual part of Eq (3) s gven as r(, ı, x) = (x r o, (, ı)) + r(, ı, x) (7) where r are the resdual contrbutons of the pure fluds and o, r(, ı, x) s an emprcal mult-parameter functon whch can tpd( w) = = TPD( w) RT spec w [ln w + ln ϕ ( w) ln x,spec ln ϕ ( x spec )] (9) The relaton between the fugacty and the reduced Helmholtz energy s gven n Appendx A n the supplementary materal (Eq A12) The stablty check for a thermodynamc system at gven T spec and p spec s performed n three steps

3 J Gernert et al / Flud Phase Equlbra 375 (2014) Step 1: Generaton of tral phase compostons Startng wth the assumpton that two coexstng phases are present, ntal estmates of tral phase compostons are generated usng the generalzed Wlson correlaton [25] to calculate K-values for all N components ( ln K = ln p c, p spec (1 + ω ) ) 1 T c,, = 1,, N (10) T spec From the K-values the phase compostons can be calculated usng the relaton ln K = ln x x = ln ϕ ϕ, = 1,, N (11) and the Rachford Rce equaton g: g = (x x ) = 0 (12) In order to solve Eq (12), the followng relatons between the phase compostons x and x, the feed composton x,spec, the vapor fracton ˇ = n /n, where n s the molar amount of substance n the gas phase and n = n + n s the overall molar amount of substance, and the K-values K are used x K x,spec = 1 ˇ(1 K ) and x = x,spec 1 ˇ(1 K ), = 1,, N (13) When combnng Eqs (12) and (13), the Rachford Rce equaton s expressed as a functon of the vapor fracton ˇ and reads: ( ) K g(ˇ) = x 1,spec = 0 (14) 1 ˇ(1 K ) If the rewrtten Rachford Rce equaton (Eq (14)) s evaluated wth the Wlson K-values, the resultng vapor fracton does not necessarly fulfll the condton 0 ˇ 1 Therefore, the followng checks are performed: The Rachford Rce equaton s solved wth the assumpton that the mxture s at ts bubble pont (ˇ = 0) Eq (14) becomes g(0) = x,spec (K 1) or g(0) = x,spec K, wth g(0) = g(0) + 1 (15) If g(0) 0, respectvely g(0) 1 holds, the mxture s assumed to be at ts bubble pont or at a lower temperature, and the phase compostons are calculated accordng to x = x,speck and x g(0) = x,spec, = 1,, N (16) The Rachford Rce equaton s solved wth the assumpton that the mxture s at ts dew pont (ˇ = 1) Eq (14) becomes g(1) = x,spec ( 1 1 K ), respectvely g(1) = x,spec K, wth g(1) = g(1) + 1 (17) If g(1) 0 or g(1) 1 holds, the mxture s assumed to be at ts dew pont or at a hgher temperature, and the phase compostons are calculated accordng to x = x,spec and x = x,spec K g(1), = 1,, N (18) If nether of the frst two tests leads to ntal phase compostons, the mxture s assumed to be n the two-phase regon, and Eq (14) has to be solved teratvely for the vapor fracton ˇ Once ˇ s found, the test phase compostons can be calculated from Eq (13) 312 Step 2: Successve substtuton method Based on the ntal estmates of the phase compostons, three steps of successve substtuton are performed n order to ncrease the accuracy of the estmates The successve substtuton has been ntroduced for the soluton of phase equlbra condtons by Prausntz and Chueh [26] It ncludes the followng three steps: Usng the prevous estmates for the phase compostons, the fugacty coeffcents ϕ and ϕ are calculated from the equaton of state New K-values are calculated usng the fugacty coeffcents and the relaton gven n Eq (11) From the K-values new phase compostons x and x and a new vapor fracton ˇ are calculated by solvng the Rachford Rce equaton as gven n Eq (14) 313 Step 3: Tangent plane analyss If the vapor fracton exceeds the bounds 0 ˇ 1 after the successve substtuton steps, the algorthm suggests a stable phase and contnues wth the tangent plane stablty analyss For 0 ˇ 1, the system s assumed to be unstable and to splt nto two phases In ths case the dfference between the Gbbs energy of the splt phases and the feed phase G nrt = (1 ˇ) x [ln x + ln ϕ ] + ˇ x [ln x + ln ϕ ] x,spec (ln x,spec + ln ϕ ), (19) wll be negatve, wth ln ϕ = ln ϕ (T spec, p spec, x ) and ln ϕ = ln ϕ (T spec, p spec, x ) Wrtten n terms of tangent plane dstances tpd, Eq (19) reads: G nrt = (1 ˇ)tpd + ˇ tpd, (20) where tpd = tpd( x ) = x (ln x + ln ϕ ln x,spec ln ϕ ) (21) and tpd = tpd( x ) = x (ln x + ln ϕ ln x,spec ln ϕ ) (22) are the reduced tangent plane dstance functons for the feed composton, usng the lqud and vapor compostons as tral phases If the change of the Gbbs energy accordng to Eq (19) s negatve, the nstablty of the feed phase s confrmed and the algorthm contnues wth the sothermal flash calculaton Even f the change of the Gbbs energy wth the two tral phases s postve, but one of the tangent plane dstance functons tpd or tpd s negatve, the feed s proven to be unstable [4]

4 212 J Gernert et al / Flud Phase Equlbra 375 (2014) In case both tangent plane dstance functons tpd and tpd of the ntal tral phases and the change n the Gbbs energy G/nRT are postve, the algorthm contnues wth a more detaled stablty analyss In prncple, the whole composton range needs to be checked for negatve tangent plane dstances Snce such a multdmensonal search s mpractcal for mult-component mxtures another more practcal approach was suggested by Mchelsen and Mollerup [23] From the ntal Wlson K-values and the feed composton, heavy and lght tral phase compostons, x tral,h and x tral,l, are calculated from x,tral,h = x,spec and x K,tral,L = x,spec K, = 1,, N (23) Usng each of these ntal tral phase compostons as startng pont, a successve substtuton search for the composton where the tangent plane dstance functon has a mnmum s performed by repeatng the followng steps a gven number of tmes: Calculate the tangent plane dstance of the current tral phase tpd( x tral,n ) In the frst run (n = 1), save the tangent plane dstance value as tpd mn = tpd( x tral,1 ) Compare the values of the tangent plane dstances of the current tral phase tpd( x tral,n ) and the mnmum tral phase tpd mn If the current value s lower, save t together wth the correspondng tral phase composton: tpd mn = tpd( x tral,n ), x mn = x tral,n Calculate the fugacty coeffcents ϕ,tral,n = ϕ (T, p, x tral,n ) Calculate a new tral phase composton from x,tral,n+1 = x,spec ϕ ϕ,tral,n, = 1,, N (24) Here, the ndex n denotes the nth step n the teraton process The teraton process has several break crtera: The maxmum number of teraton steps s reached The value of tpd mn becomes negatve and thus the feed composton s found unstable In ths case, three more teratons are performed n order to create more accurate ntal values for the followng flash calculaton The change of the tral phase compostons x tral,n and x tral,n+1 becomes very small Ths ndcates that a statonary pont of the tangent plane dstance functon s found (whch does not necessarly have to ndcate nstablty of the feed composton) The tral composton x tral,n merges wth the feed composton x Ths ndcates that n the vcnty of the ntal tral phase composton the tangent plane dstance functon has no local mnma and no negatve values In ths case the system s most lkely stable at the orgnal composton Two values for tpd mn are returned from the search for statonary ponts of the tangent plane dstance functon usng both the heavy and the lght tral phase compostons as startng ponts If one or both of these values are negatve, the feed composton s unstable and the algorthm contnues wth a flash calculaton If both values are postve, the feed composton s assumed to be stable 32 The sothermal two-phase flash The calculaton of an sothermal flash (phase equlbrum calculaton at gven T spec and p spec and overall composton) s a common procedure n thermodynamcs It relates drectly to the thermal, mechancal, and chemcal phase stablty condtons at equlbrum, Equalty of temperatures n both phases T = T = T sat (25) Equalty of pressures n both phases p = p = p sat (26) Equalty of chemcal potental n both phases = =, = 1,, N, (27) where denotes the saturated lqud phase and the saturated vapor (or more volatle lqud n case of a lqud lqud equlbrum) The thrd equlbrum condton s commonly replaced by the equvalent expresson n terms of the fugactes f = f = f, = 1,, N (28) For a mxture wth N components at two-phase equlbrum, 2(N 1) unknowns have to be determned, namely the compostons of the frst N 1 components n each phase of the mxture The composton of the Nth component can be determned by the relaton N 1 x = 1 or x N = 1 x, (29) whch has to be fulflled for each phase Ths concept s well establshed, see eg [1] for the soluton of phase equlbra problems From the chemcal equlbrum condton gven n Eq (28), the frst N equatons can be drectly derved F k = ln f (T spec, p spec, x ) ln f (T spec, p spec, x ) = 0, k = = 1,, N (30) Note that the logarthm of the fugactes s used to ncrease the numercal stablty for very small and very large values of f The mssng N 2 equatons can be derved from the materal balance For each component, the materal balance n = n + n has to be fulflled, whch s drectly lnked to the defnton of the molar vapor fracton Snce the vapor fracton has to have the same value, ndependent of the component that s used for ts calculaton, ths relaton can be used to form the mssng N 2 equatons accordng to F k = x,spec x x x x N 1,spec x N 1 x N 1 x = 0, = 1,, N 2, N 1 k = + N (31) The resultng set of 2(N 1) nonlnear equatons F( X) wth the same number of unknowns X can only be solved numercally by teratve methods, lke the Gauß-Newton, Levenberg Marquardt [27,28] or Powells Dogleg method [29], whch shall not be dscussed here However, for the most common numercal methods the Jacoban matrx J F ( X) s needed, whch holds the partal dervatves of the system of equatons wth respect to all unknowns accordng to x x x x 1 N 1 1 N 1 J F ( X) = F 2N 2 x 1 F 2N 2 F 2N 2 x 1 F 2N 2 x N 1 (32) Dependng on the thermodynamc mxture model the dervatves of the fugacty ln f wth respect to the mole fracton x j, whch are ncorporated n the Jacoban, may become qute complex Kunz et al [4] avoded these dervatves by approxmatng the expresson ( ) ln ϕ (33) x j T,p,x k /= j

5 J Gernert et al / Flud Phase Equlbra 375 (2014) wth ) ln ϕ n( n j T,p,n k /= j (34) In ths work the analytcal dervatve gven n Eq (33) was used Note that the partal dervatve wth respect to the mole fracton of component j can only be physcally reasonable, f the mole fractons are ndependent varables, whch s ensured by the transformaton gven n Eq (29) A detaled dervaton of all dervatves needed for the sothermal flash problem s gven n Appendx A n the supplementary materal The dervatves of the last N 2 equatons wth respect to the frst N 2 mole fractons of the lqud and vapor phase are F k x and = x,spec x (x x )2, = 1,, N 2 and k = + N (35) F k x = x,spec x (x x, = 1,, N 2 and k = + N, (36) )2 respectvely The dervatves of the last N 2 equatons wth respect to the mole fracton of component N 1 n the lqud and vapor phase are F k and F k x N 1 = x N 1,spec x N 1 (x N 1 x N 1 )2, k = N + 1,, 2N 2 (37) = x N 1,spec x N 1 (x N 1 x N 1 )2, k = N + 1,, 2N 2, (38) respectvely All other partal dervatves F k / x j are zero 33 Calculaton of dew and bubble ponts For sake of completeness the system of equatons for saturaton pont calculatons and the dervatves needed to solve t wth gradent methods lke the Newton Raphson method are also suppled The calculaton of saturaton ponts s an mportant task n mxture thermodynamcs, to eg construct the phase envelope of a mxture (see eg [30] or [31]) and thus determne the boundary of phase stablty for a mxture The calculaton of saturaton ponts at gven T spec or p spec only requres the determnaton of N unknowns for a mxture wth N components, namely N 1 mole fractons of the ncpent phase and the pressure or temperature, respectvely The system of equatons necessary to solve for the unknowns can be set up from the equalty of the fugactes of each component n both phases as gven n Eq (30) The Jacoban then ncludes the dervatves of the equatons F k wth respect to the frst N 1 mole fractons of the ncpent phase plus the dervatves wth respect to pressure or temperature For the calculaton of a dew pont (wth the saturated lqud phase x beng the ncpent phase) the Jacoban reads x 1 J F ( X) = F N x 1 F N T or p F N T or F N p (39) The dervatves of ln f wth respect to T and p at constant x, whch are needed to set up the Jacoban for Helmholtz-type mxture models, are suppled elsewhere [4] and can be taken from Appendx A n the supplementary materal For the calculaton of a bubble pont the saturated vapor s the ncpent phase Thus, x must be replaced by x n Eq (39) 4 p,h and p,s flash The calculaton of flud propertes of mxtures wth the feed composton x spec, the enthalpy h spec or the entropy s spec, and the pressure p spec as ndependent varables s of specal mportance n process calculatons In these cases none of the natural nput varables T and of the equaton of state s known and thus a two-dmensonal teraton s needed Stll, the system needs to be analyzed for phase stablty Fnally, f the system s found to be unstable, a flash calculaton has to be performed to fnd the compostons of the coexstng stable phases As far as the authors know, these flash calculatons have not been descrbed n detal for Helmholtz EOS n lterature Unpublshed algorthms to perform p,s and p,h flash calculatons are avalable n the commercally avalable software REFPROP 91 [32] and n the GERG-2004XT08 software [33] 41 Stablty analyss The algorthm descrbed n ths secton s based on the work by Mchelsen [34] In a frst step an ntal estmate for the temperature T s generated for the specfed value q spec by solvng the objectve functon: F T = q(t, p spec, x spec ) q spec = 0 (40) wth p spec and x spec held constant over the whole temperature search range and q beng ether the enthalpy h or the entropy s In order to evaluate Eq (40) usng a Helmholtz equaton of state, the densty needs to be calculated frst as a functon of temperature and pressure (see Secton 5) Eq (40) then becomes F T = q(t, Tp, x spec ) q spec = 0, wth Tp = (T spec, p spec, x spec ) (41) For the generaton of the temperature estmate the mxture s treated as homogeneous phase No stablty checks are performed for these calculatons, hence physcally metastable and unstable solutons may occur When calculatng the functon q(t, p spec, x spec ) at constant p spec and x spec, two cases have to be consdered If the pressure s above the maxmum pressure, at whch two phases are present (crcondenbar), the functon q(t, p spec, x spec ) s contnuously ncreasng wth T and dfferentable and the soluton of Eq (40) already represents the correct temperature assocated wth the gven pressure and enthalpy or entropy (see Fg 1 at p = 10 MPa for an example) In case the gven pressure s below the crcondenbar, the functon q(t, p spec, x spec ) passes a regon where the feed phase s unstable and decomposes nto two or more phases Wthn ths regon the functon passes a pont of nconsstency where t jumps to a hgher value, whch corresponds to the pont where the densty jumps from a lqud lke densty to a vapor lke densty (see Fg 1 at p = 2 MPa) At the nconsstency the Gbbs energy of the metastable overall phase s the same for the lqud lke and the vapor lke densty soluton For the soluton of the objectve functon F T n Eq (40) the regula fals method s appled, whch does not need any dervatves of F T Snce ths method s an nterval search algorthm, a range for q has to be defned, whch can be derved from the temperature range of valdty of the property model Ths search method s advantageous for ths applcaton, snce t can handle functons that are not defned over the whole search range In the example (Fg 1, p = 2 MPa) the enthalpy s not defned for values between 58 kj mol 1 and 147 kj mol 1 For specfed enthalpes

6 214 J Gernert et al / Flud Phase Equlbra 375 (2014) Fg 1 The molar enthalpy as a functon of temperature at constant pressure and composton (no phase stablty s consdered) for the system methane ethane (20% methane) at 10 MPa and 2 MPa, calculated wth the GERG-2008 [16] model h spec wthn ths range, the algorthm returns the transton temperature T transton as soluton Once the temperature s found, the system s analyzed for stablty usng the algorthm descrbed n Secton 3 If t s found to be unstable the algorthm contnues wth a flash calculaton, usng the results of the temperature search and the stablty analyss as ntal values 42 Flash calculaton For the calculaton of two-phase equlbra at gven feed composton, enthalpy h spec or entropy s spec, and pressure p spec the number of unknowns to be determned ncludes compostons of the phases and temperature T Thus the resultng number of unknowns s 2N 1 In order to solve the phase equlbrum condton for ths set of unknowns a set of the same number of objectve functons s needed, e an addtonal objectve functon has to be added to the set of equatons gven n Eqs (30) and (31), namely F 2N 1 = q ( T, p spec, x, x ) q spec = 0, (42) where q spec denotes the value of the specfed property (ether h or s) and q(t, p spec, x, x ) denotes the same property calculated from the equaton of state at the current flash condtons accordng to q(t, p spec, x, x ) = ˇ q(t, p spec, x ) + (1 ˇ) q(t, p spec, x ) (43) The system of equatons can be solved agan by applyng teratve methods The Jacoban matrx for ths flash type reads: T J F ( X n) = x 1 F 2N 1 x 1 F 2N 1 x 1 F 2N 1 x 1 x N 1 F 2N 1 x N 1 F 2N 1 T (44) The dervatves needed for the p,s and p,h flash calculatons can be taken from Appendx B n the supplementary materal 5 Densty solver The ndependent varables of the Helmholtz EOS are densty, temperature T and the composton of the mxture x (see Eqs (3) and (4)) The gven varables for flash calculatons are n general T and/or p and the overall (or phase) composton x Thus, a solver for the densty at gven T spec, p spec and x spec s requred when usng equatons of state explct n the Helmholtz energy The mportance of the densty solver for flash calculatons was already ponted out by Kunz et al [4] In ths work a stable and relable routne for the calculaton of densty s presented The functon to be solved has the followng form: F :=p spec p(t spec,, x spec ) ( ) ) r(ı, spec, x spec ) = p spec RT spec (1 + ı = 0, (45) ı, x wth the specfed varables p spec, T spec, and x spec and as unknown varable For cubc equatons of state ths functon can be solved analytcally, but for mult-parameter equatons of state explct n the reduced Helmholtz energy, the objectve functon F has to be solved numercally usng an teratve method (we found the regula fals method useful) An mportant advantage of Helmholtz EOS over cubc EOS s the gan n accuracy for the lqud and lqud lke supercrtcal phase However, the hgher accuracy of Helmholtz models comes at a prze When T, p (and x) are gven, cubc EOS return up to three solutons for densty, where only the outer densty solutons correspond to physcally correct solutons representng a gas and a lqud densty On the contrary, Helmholtz models may have multple loops n the two phase regon yeldng more than three solutons for densty where agan only the outermost solutons correspond to physcally correct denstes Ths behavor results n a much more complex stuaton wth regard to the densty search at gven p and T (see Fg 2): (a) At pressures close to the saturaton pressure p sat, the equaton of state used n ths example returns fve denstes correspondng to one pressure and one temperature The two outermost solutons (I and V) are the two physcally correct phase denstes For p = p sat, densty I and V correspond to the saturated vapor and saturated lqud densty respectvely; for p > p sat only the lqud densty s stable and for p < p sat the vapor densty corresponds to the stable phase (b) For pressures above the saturaton pressure the densty wth the hghest value (III) corresponds to the physcally correct soluton (c) For pressures below the saturaton pressure the densty wth the lowest value (I) corresponds to the physcally correct soluton For pure substances the procedure used to fnd the physcally correct soluton s rather straght forward, as soon as the two outermost solutons for the densty have been found [17] When t comes to mxtures and especally to flash calculatons, fndng the correct (or more lkely) densty becomes much more challengng [4] In case more than one densty root s present at a specfed

7 J Gernert et al / Flud Phase Equlbra 375 (2014) Fg 2 The problem of multple densty roots at gven T and p, usng the example of the reference EOS of CO 2 by Span and Wagner [18], compared to the cubc Peng Robnson equaton (PR EOS) The meanng of the roman numbers s explaned n the body of the text temperature, pressure and composton, the challenge for the densty solver s to fnd and dentfy the physcally most reasonable root If a good ntal estmate s avalable (eg from the prevous teraton step n a flash or from phase envelope calculatons), ths estmate can be used to fnd a soluton to Eq (45) In case no such estmate s avalable, an ntal estmate s generated from cubc EOS In most cases ths estmate s accurate enough to create a densty nterval contanng the correct root Otherwse a more elaborate search procedure s mandatory Snce the regula fals method s an nterval search method, the most crucal part s to determne an approprate search nterval that contans exactly one root In case a vapor densty s needed, ths s the root to the left of the frst maxmum of the sotherm n a p dagram (at the lowest densty), n case of a lqud densty t s the root to the rght of the last mnmum (at the hghest densty, see Fg 3) The advanced root fndng algorthm developed n the course of ths work uses the followng steps to determne the phase densty at gven temperature, pressure and composton: 1 Intal root search: Use an estmated densty value (ether passed by the callng routne or generated usng the SRK equaton of state) to create a densty nterval for an ntal densty search usng the regula fals method In case a root s found, contnue wth step 4 2 Statonary pont search: In case a lqud densty s requred, search for the pressure mnmum at the hghest densty, n case a vapor Fg 3 Determnaton of search ntervals for the densty root fndng algorthm (A) Isotherm wth only one densty root and (B) sotherm wth multple densty roots at gven p and T densty s requred, search for the pressure maxmum at the lowest densty by usng the necessary crteron ( ) p = 0, (46) as the defnton of a statonary pont In order to test the result, the suffcent crtera ( ) 2 p 2 > 0 (47) for a mnmum, and ( ) 2 p 2 < 0 (48) for a maxmum are appled The start values for the maxmum and mnmum search are set to a densty close to zero and a very large densty (eg a multple of r or the nverse of the covolume b of a cubc EOS), respectvely 3 Phase densty root search: Compare the pressures p max and p mn at the statonary ponts to the gven pressure p spec For p spec < p max a vapor phase densty can be calculated at the gven temperature, and for p spec > p mn a lqud phase densty can be calculated The densty at the pressure statonary pont, max or mn (note: mn (T spec, p mn, x spec ) > max (T spec, p max, x spec ), see Fg 3), respectvely, s used as lmt for the search nterval, snce the followng condtons apply: 0 < (T spec, p spec, x spec ) < max (49) for a vapor phase densty, and (T spec, p spec, x spec ) > mn (50) for a lqud phase densty Whle Eq (49) can be used as search nterval for a vapor phase densty wthout further modfcatons, the upper lmt of the search nterval for lqud denstes s not defned by physcal lmts The lmtng factor s here the physcally correct extrapolaton behavor of the equaton of state Therefore, a user defned maxmum densty has to be chosen as lmt Wth the densty search nterval determned by ths method t s guaranteed that only one physcally correct densty root exsts wthn the nterval The algorthm contnues wth the root search usng the regula fals method 4 Thermodynamc tests: A number of tests s performed to make sure that the densty root shows physcally correct characterstcs In case a lqud lke densty was searched for, the followng condtons have to be fulflled: ( ) ( ) p 2 p > 0, and 2 > 0 (51) However, roots on the mddle branch of a typcal subcrtcal sotherm may fulfll these condtons as well, although they are physcally meanngless Only for a correct lqud densty these condtons are also fulflled for all denstes larger than the root Therefore ths check s performed a number of tmes wth ncreasng denstes untl a defned maxmum densty s reached All vapor denstes must fulfll the followng condtons: ( p ) > 0, and ( ) 2 p 2 < 0 (52) Therefore, n order to check f a root s really a vapor densty, the root tself and all denstes on the same sotherm below ths root must fulfll these condtons

8 216 J Gernert et al / Flud Phase Equlbra 375 (2014) Ths root search algorthm returns the physcally correct densty, as long as the numercal search methods converge and as long as a physcally correct root exsts When the densty of a certan mxture at gven composton, temperature and pressure needs to be determned, eg as part of the stablty analyss, the phase n whch ths mxture s stable s usually not known n advance Consequently, the densty solver has to locate all vald densty roots and, n case two roots were found, determne the more lkely soluton The search strategy for the correct (or more stable) densty root ncludes the followng steps: 1 Lqud phase densty search: The advanced root fndng algorthm s used to determne a densty root wth lqud lke characterstcs 2 Vapor phase densty search: The advanced root fndng s used to determne a densty root wth vapor lke characterstcs For both phases, the search may fal at several ponts n the search algorthm: the statonary pont search fals, no lqud densty root exsts or the regula fals method fals, or the densty root does not pass the thermodynamc stablty test In case two denstes were found, the algorthm contnues wth step 4; f only one densty was found due to a falure, the algorthm contnues wth step (3a); f no densty was found, t contnues wth step (3b) 3 (a) Second root search: If only one statonary pont was found (the numercal method faled fndng the second pont) the second statonary pont s determned usng the regula fals method Once the second statonary pont s found, the second phase densty s determned as descrbed above (steps 3 and 4) If ths search also fals, the statonary pont found most lkely corresponds to an nflecton pont and thus only one correct root exsts and the algorthm s completed(b) Sngle root search: In case both searches for the pressure mnmum and maxmum fal, the sotherm of the specfed mxture s assumed to be a monotonously ncreasng functon (e supercrtcal for pure fluds, see Fg 3) In ths case only one densty soluton exsts for a gven set of temperature, pressure and composton and the search nterval for the densty search ncludes the whole densty range up to a user defned maxmum densty The actual densty search s performed, eg, usng the regula fals method 4 Two densty values Gbbs Energy Crteron: In case two denstes were found n the prevous steps, only one densty represents a thermodynamcally stable phase whle the other densty represents a metastable state In ths case the densty correspondng to the lower Gbbs energy s chosen as the more lkely soluton However, snce the overall Gbbs energy needs to be at a mnmum for a stable system, the densty soluton correspondng to the lowest Gbbs energy of the phase does not necessarly need to be the stable soluton for the phase equlbrum 6 Results The algorthm proposed n ths work was mplemented nto computer code and compled nto the thermodynamc property calculaton tool TREND 11 [35] The algorthm and the computer code were developed wth emphass on relablty and stablty n order to enable the calculaton of thermodynamc propertes of a wde range of mxtures and states, wthout pror knowledge of the locaton of the phase boundares Major other software tools that offer the calculaton of thermodynamc propertes of mxtures from Helmholtz energy mxture models are the GERG-2004XT08 software package developed together wth the GERG-2004 and GERG-2008 models [33], and REFPROP [32] The algorthm used n the GERG-2004XT08 software s partly dscussed n the GERG-2004 monograph [4] Whle the stablty analyss and flash calculaton Fg 4 T,s-dagram for the mxture x CO2 = 0997, x N2 = 0001, x O2 = 0001, and x Ar = 0001 wth selected sobars and lnes of constant enthalpy, calculated wth the algorthm proposed n ths work (TREND 11), usng the EOS-CG mxture model [36] The reference state of the pure substances s defned at T = K and p = MPa At these condtons, the densty s calculated by the deal gas law and the entropy and enthalpy calculated at ths densty are set to zero wth temperature and pressure as nput are descrbed n detal, the respectve algorthms usng pressure and enthalpy or pressure and entropy as nput whch were ntroduced n the GERG-2008 extenson have not been publshed The REFPROP software uses a dfferent algorthm, whch to our knowledge has never been publshed, ether Snce the comparson of dfferent software mplementatons has only lmted sgnfcance concernng the valdaton of calculaton speed or robustness of a numercal algorthm, the other software tools were merely used for the verfcaton of the calculaton results from the algorthm proposed n ths work The applcaton of the new algorthm s demonstrated n two examples In the frst example, a T,s-dagram was constructed for a CO 2 -rch mxture wth ntrogen, oxygen and argon as mpurtes (molar composton: x CO2 = 0997, x N2 = 0001, x O2 = 0001, and x Ar = 0001, see Fg 4) In order to show lnes of constant pressure and enthalpy n such a dagram, the calculaton of propertes wth enthalpy and pressure and wth entropy and pressure as nput s necessary ncludng stablty checks, wth a subsequent flash calculaton for ponts n the two-phase regon All ponts on the solnes were calculated ndependently wthout pror nformaton on the phase boundares or ntal estmates from the prevous pont Fg 4 shows sobars and lnes of constant enthalpy, calculated wth the TREND 11 software usng the EOS-CG mxture model [36] The new algorthm succeeds n the calculaton of all ponts and returns smooth, consstent values over a wde temperature and entropy range REFPROP 91 returns values dentcal to those calculated wth TREND 11 (wthn numercal tolerance) In the second example (see Fg 5), a p,t-dagram was constructed for a typcal lquefed natural gas (LNG) mxture (molar composton: x CH4 = 0918, x N2 = 0008, x C2 H 6 = 0057, x C3 H 8 = 0013, x n-c4 H 10 = 0004) Agan, the new algorthm succeeds to calculate all ponts on a selecton of lnes of constant enthalpy and entropy, usng pressure and enthalpy, or pressure and entropy as nput varables for the teratve calculaton of temperature For ths mxture both the GERG-2004XT08 software and REFPROP 91 were

9 J Gernert et al / Flud Phase Equlbra 375 (2014) system of equatons descrbng the sothermal (p,t) flash problem s reformulated n a way that the N 1 frst mole fractons become ndependent varables, and dervatves of the reduced Helmholtz energy wth respect to the composton are developed Algorthms for the senthalpc (p,h) and for the sentropc (p,s) stablty and flash problem proposed by Mchelsen [34] are appled to Helmholtz mxture models All analytcal dervatves necessary to solve the p,h and p,s flash wth numercal methods usng the Jacoban are suppled (see supplementary materal, Appendx B) Fnally, a robust and sophstcated densty solver s presented that addresses the specal requrements and problems assocated wth mult-parameter Helmholtz energy equatons of state Lst of symbols Fg 5 p,t-dagram for the mxture x CH4 = 0918, x N2 = 0008, x C2 H 6 = 0057, x C3 H 8 = 0013, x n-c4 H 10 = 0004 wth selected lnes of constant enthalpy and entropy, calculated wth the new algorthm proposed n ths work (TREND 11), usng the GERG-2008 mxture model used as verfcaton Both calculaton tools return dentcal results to TREND 11 (wthn numercal tolerance) Besde these two examples the algorthm was tested successfully for a large number of bnary and mult-component mxtures, wth a focus on systems wth large dfferences n volatltes and complex phase equlbra Tested systems nclude water gas mxtures (showng complex phase behavor ncludng lqud lqud equlbra), CO 2 -mxtures wth nert gases, and mult-component natural gas mxtures wth large dfferences n volatltes resultng from hgher-order alkanes and eg hydrogen as mxture components The presented algorthm was successfully appled n fttng a CO 2 hydrate model [37] as well 7 Conclusons Durng the last decades many dfferent algorthms have been proposed to determne the phase stablty of mult-component mxtures at varous combnatons of nput varables Stll, no algorthm can be emphaszed as the best soluton for any knd of problem Dependng on the equatons of state used and the knd of applcaton, dfferent demands can be formulated and a sound balance between stablty and speed has to be found However, the choce of a robust algorthm as well as ts careful mplementaton s mandatory to acheve good results n practcal work The combned stablty and flash calculaton algorthm proposed by Mchelsen [2,3] has proven to be one of the most sophstcated and at the same tme practcal algorthms that returns relable results even for complex systems Kunz and Wagner [4,16] adapted computer code suppled by Mchelsen to the requrements of Helmholtz energy mxture models by provdng the analytcal dervatves of the reduced Helmholtz energy necessary to solve the sothermal flash problem However, one of the man dervatves needed to set up the Jacoban, the dervatve of the fugacty wth respect to composton, was replaced by scaled composton dervatves [4] The new algorthm suggested n ths work s essentally based on the same approach but ncludes some mportant new elements The A total Helmholtz energy a molar Helmholtz energy c Y part of the reducng functon EOS equaton of state or equatons of state F objectve functon, weghng factor of the mxture model f fugacty f Y part of the reducng functon G total Gbbs energy g molar Gbbs energy, Rachford Rce equaton H total enthalpy h molar enthalpy J Jacoban matrx K K-value N number of components n the mxture n molar amount of substance p pressure q functon R unversal gas constant S total entropy s molar entropy T temperature TPD tangent plane dstance functon tpd reduced tangent plane dstance functon U total nternal energy V total volume v molar volume VLE vapor lqud equlbrum w vector of mole fractons of the emergng phase w mole fracton of the emergng phase x vector of mole fractons x element of mole fracton vector Y r reducng functon Z compressblty factor Greek letters reduced Helmholtz energy ˇ molar vapor fracton ˇY reducng functon parameter Y reducng functon parameter ı reduced densty chemcal potental densty nverse reduced temperature ϕ fugacty coeffcent ω acentrc factor Subscrpts c crtcal parameter mn pressure mnmum of an sotherm n the metastable lqud regon

10 218 J Gernert et al / Flud Phase Equlbra 375 (2014) max pressure maxmum of an sotherm n the metastable vapor regon o pure flud property p sobarc r reduced varable sat property at saturaton condtons spec property value specfed by the user, j, k component ndces ı,, x dervatve wth respect to the respectve varable Superscrpts E excess property o deal-gas part of a property r resdual part of a property property of the saturated lqud property of the saturated vapor Acknowledgements The authors are grateful to all organzatons whch contrbuted fundng to ths project, namely to - EON Ruhrgas under contract Calculaton of Complex Phase Equlbra, - the federal government of Nordrhen Westfalen n conjuncton wth EFRE for fundng under contract /2-005-WFBO- 011Z, - the European Commsson, under contract Seventh Framework Program, Nr , IMPACTS Appendx A Supplementary data Supplementary data assocated wth ths artcle can be found, n the onlne verson, at do:101016/jflud References [1] GA Iglesas-Slva, A Bonlla-Petrcolet, PT Eubank, JC Holste, KR Hall, Flud Phase Equlb 210 (2003) [2] ML Mchelsen, Flud Phase Equlb 9 (1982) 1 19 [3] ML Mchelsen, Flud Phase Equlb 9 (1982) [4] O Kunz, R Klmeck, W Wagner, M Jaeschke, The GERG-2004 Wde-Range Equaton of State for Natural Gases and Other Mxtures GERG TM15, VDI Verlag, Düsseldorf, 2007 [5] YP Lee, GP Rangaah, R Luus, Comput Chem Eng 23 (1999) [6] YS Zhu, ZH Xu, Flud Phase Equlb 154 (1999) [7] DV Nchta, S Gomez, E Luna, Comput Chem Eng 26 (2002) [8] V Bhargava, S Fateen, A Bonlla-Petrcolet, Flud Phase Equlb 337 (2013) [9] ML Mchelsen, Flud Phase Equlb 158 (1999) [10] O Redlch, JNS Kwong, Chem Rev 44 (1949) [11] G Soave, Chem Eng Sc 27 (1972) [12] DY Peng, DB Robnson, Ind Eng Chem Fundam 15 (1976) [13] GM Kontogeorgs, EC Voutsas, IV Yakoums, DP Tassos, Ind Eng Chem Res 35 (1996) [14] T Holderbaum, J Gmehlng, Flud Phase Equlb 70 (1991) [15] M-J Huron, J Vdal, Flud Phase Equlb 3 (1979) [16] O Kunz, W Wagner, J Chem Eng Data 57 (2012) [17] R Span, Multparameter equatons of state An accurate source of thermodynamc property data, Sprnger, Berln, 2000 [18] R Span, W Wagner, J Phys Chem Ref Data 25 (1996) [19] W Wagner, A Pruß, J Phys Chem Ref Data 31 (2002) [20] R Tllner-Roth, Fundamental Equatons of State, Shaker, Aachen, 1998 [21] EW Lemmon, R Tllner-Roth, Flud Phase Equlb 165 (1999) 1 21 [22] EW Lemmon, RT Jacobsen, Int J Thermophys 20 (1999) [23] ML Mchelsen, JM Mollerup, Thermodynamc Models: Fundamentals & Computatonal Aspects, 1st ed, Te-Lne Publcatons, Ronnebaervej 59, DK-2840 Holte, Denmark, 2004 [24] LE Baker, AC Perce, KD Luks, Soc Petrol Eng J 22 (1982) [25] GM Wlson, A Modfed Redlch Kwong Equaton of State: Applcaton to General Physcal Data Calculatons, n: Paper no 15C AChe 65th Natonal Meetng, Cleveland, OH, May 4 7, 1968 [26] J Prausntz, P Chueh, Computer Calculatons for Hgh-Pressure Vapor Lqud Equlbra, Prentce-Hall, Englewood Clffs, NJ, 1968 [27] DW Marquardt, J Soc Ind Appl Math 11 (1963) [28] K Levenberg, Q Appl Math 2 (1944) [29] MJD Powell, A hybrd method for nonlnear equatons, n: P Rabnowtz (Ed), Numercal Methods for Nonlnear Algebrac Equatons, Brtsh Computer Socety Gordon and Breach Scence Publshers, London, 1970 [30] ML Mchelsen, Flud Phase Equlb 98 (1994) 1 11 [31] DV Nchta, Energy Fuels 22 (2008) [32] EW Lemmon, ML Huber, MO McLnden, REFPROP 91NIST Standard Reference Database 23, Natonal Insttute for Standards and Technology, Boulder, 2013 [33] O Kunz, W Wagner, GERG-2004 XT08 Software Package, Lehrstuhl für Thermodynamk, Ruhr-Unverstät Bochum, Bochum, Germany, 2009 [34] ML Mchelsen, Flud Phase Equlb 33 (1987) [35] R Span, T Eckermann, J Gernert, S Herrg, A Jäger, M Thol, TREND Thermodynamc Reference and Engneerng Data 11, Lehrstuhl für Thermodynamk, Ruhr-Unverstät Bochum, Bochum, Germany, 2014 [36] J Gernert, A New Helmholtz Energy Model for Humd Gases and CCS Mxtures (PhD thess), Ruhr-Unverstät Bochum, Bochum, Germany, 2013 [37] A Jäger, V Vnš, J Gernert, R Span, J Hrubý, Flud Phase Equlb 338 (2013)