Computation of Phase Equilibrium and Phase Envelopes

Transcription

1 Downloaded from orbt.dtu.dk on: Sep 24, 2018 Computaton of Phase Equlbrum and Phase Envelopes Rtschel, Tobas Kasper Skovborg; Jørgensen, John Bagterp Publcaton date: 2017 Document Verson Publsher's PDF, also known as Verson of record Lnk back to DTU Orbt Ctaton APA: Rtschel, T. K. S., & Jørgensen, J. B Computaton of Phase Equlbrum and Phase Envelopes. DTU Compute. DTU Compute-Techncal Report-2017, Vol. 11. General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research. You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the publcaton n the publc portal If you beleve that ths document breaches copyrght please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam.

2 Computaton of Phase Equlbrum and Phase Envelopes Abstract In ths techncal report, we descrbe the computaton of phase equlbrum and phase envelopes based on expressons for the fugacty coeffcents. We derve those expressons from the resdual Gbbs energy. We consder 1 deal gases and lquds modeled wth correlatons from the DIPPR database and 2 nondeal gases and lquds modeled wth cubc equatons of state. Next, we derve the equlbrum condtons for an sothermal-sobarc constant temperature, constant pressure vapor-lqud equlbrum process PT flash, and we present a method for the computaton of phase envelopes. We formulate the nvolved equatons n terms of the fugacty coeffcents. We present expressons for the frst-order dervatves. Such dervatves are necessary n computatonally effcent gradent-based methods for solvng the vapor-lqud equlbrum equatons and for computng phase envelopes. Fnally, we descrbe a Matlab program that computes the phase envelope of a mxture. We present the source code and dscuss practcal detals of the mplementaton. Tobas K. S. Rtschel John Bagterp Jørgensen DTU Compute Techncal Report ISSN: Techncal Unversty of Denmark Department of Appled Mathematcs and Computer Scence & Center for Energy Resources Engneerng Kgs. Lyngby December 15, 2017

3 CONTENTS Contents 1 Introducton A note on nomenclature Resdual Gbbs energy of deal vapor-lqud mxtures Gbbs energy of deal gas mxtures Gbbs energy of deal lqud mxtures Resdual Gbbs energy of deal vapor and lqud mxtures Summary Resdual Gbbs energy of nondeal mxtures Resdual Gbbs energy wth pressure-explct equatons of state Cubc equatons of state Soluton of cubc equatons of state Resdual Gbbs energy wth cubc equatons of state Summary Fugacty and fugacty coeffcents Defnton of chemcal potental Chemcal potental of a pure component deal gas Defnton of fugacty Fugacty coeffcents Ideal gas mxture Ideal lqud mxture Nondeal mxture Summary Vapor-lqud equlbrum Equlbrum condtons Soluton of the equlbrum condtons Summary Equlbrum constants Equlbrum constants Ideal vapor-lqud mxture Summary Vapor-lqud equlbrum of deal mxtures Soluton of the Rachford-Rce equaton Summary

4 CONTENTS 8 Computaton of phase envelopes Soluton of the socurve equatons Computaton of ntal guess Summary Dervatves Logarthmc fugacty coeffcents Ideal lqud mxture Nondeal mxture Vapor-lqud equlbrum equatons Ideal vapor-lqud equlbrum equatons Phase envelope equatons An algorthm for computng phase envelopes and an example program Wlson s approxmaton of equlbrum constants Cubc nterpolaton Automatc selecton of specfed varable Automatc step sze selecton Algorthm for computng socurves Example program Dscusson A Dervatve of mxng term 45 B Dervaton of the resdual Gbbs energy for cubc equatons of state 46 C Dervaton of the logarthmc fugacty coeffcents for cubc equatons of state 47 3

5 1. Introducton 1 Introducton Ths techncal report s structured n the followng way: the purpose of Sectons 2-4 s to derve expressons for the fugacty coeffcents whch are relevant to the vapor-lqud equlbrum equatons that we present n Secton 5 and 7. The vapor-lqud equlbrum equatons and therefore also the fugacty coeffcents are central n the phase envelope computatons that we descrbe n Secton 8. In the remander of ths secton, we gve a bref overvew of the content of the sectons. We derve expressons for the resdual Gbbs energy for deal gas and lqud mxtures n Secton 2 and for nondeal gas and lqud mxtures n Secton 3. The resdual Gbbs energy of deal gases s zero by defnton. We obtan the expresson for the resdual Gbbs energy of deal lquds drectly from the Gbbs energy of the deal gas and lqud. We construct expressons for the resdual Gbbs energy of nondeal mxtures from the resdual Helmholtz energy. That s because the partal dervatve of Helmholtz energy wth respect to volume s negatve pressure and because pressure appears explctly n cubc equatons of state. We do not use the expressons for the resdual Gbbs energy n the actual computatons. We only use them to derve expressons for the fugacty coeffcents n Secton 4. In Secton 4, we frst ntroduce the chemcal potental whch s the partal dervatve of Gbbs energy wth respect to a gven mole number. Next, we defne the fugacty based on the chemcal potental of the mxture of nterest and the chemcal potental of a pure component deal gas. We then defne the fugacty coeffcent and derve an expresson for the fugacty coeffcent based on the resdual Gbbs energy. The fugacty and the chemcal potentals are not nvolved n the actual computatons. They only serve to ntroduce the relaton between the fugacty coeffcents and the resdual Gbbs energy. In Secton 5, we derve vapor-lqud equlbrum condtons for an sothermal-sobarc constant temperature, constant pressure vapor-lqud mxture. The second law of thermodynamcs states that the entropy of a closed system s maxmal at equlbrum. An sothermal-sobarc system s not closed, but the combnaton of the sothermal-sobarc system and ts surroundngs s. The condton of maxmal entropy of the sothermal-sobarc system and ts surroundngs s equvalent to a condton of mnmal Gbbs energy for the system alone Callen, We therefore formulate the vapor-lqud equlbrum problem as an optmzaton problem where the objectve functon s the Gbbs energy of the system. The equlbrum condtons are then the frst-order optmalty condtons of ths optmzaton problem. We rewrte these frst-order optmalty condtons n terms of the fugacty coeffcents. In Secton 6, we ntroduce the equlbrum constants sometmes called equlbrum ratos and use the vapor-lqud equlbrum condtons to derve an expresson for the equlbrum constants based on the fugacty coeffcents. In Secton 7, we use the fact that the equlbrum constants for an deal vapor-lqud mxture are ndependent of the composton to solve the vapor-lqud equlbrum problem effcently. In secton 8, we descrbe the equatons that defne what s called the phase envelope. The phase envelope s a collecton of vapor-lqud equlbrum states that have the same vapor fracton,.e. the same total moles of vapor as compared to the total moles of both vapor and lqud. The phase envelope equatons are therefore based on the vapor-lqud equlbrum condtons. The method for phase envelope computatons that we present s descrbed by Mchelsen and Mollerup 2007, 4

6 2. Resdual Gbbs energy of deal vapor-lqud mxtures Chap. 12. The method nvolves a choce of equlbrum constants, temperature, and pressure as ndependent varables,.e. the varables that are solved for. The method constructs the phase envelope n a sequental manner for a gven specfed vapor fracton. It solves the equatons for a specfed equlbrum constant, temperature, or pressure. Next, t uses the senstvtes of the soluton to create an ntal estmate for solvng the equatons wth a slghtly dfferent specfed equlbrum constant, temperature or pressure. We collect the dervatves of the expressons for the fugacty coeffcents, the vapor-lqud equlbrum equatons, and the phase envelope equatons n Secton 9. In Secton 10, we descrbe, n more detal, the algorthm by Mchelsen and Mollerup 2007 for computng the phase envelope. We also descrbe an example of a Matlab program that mplements the algorthm and computes the phase envelope for a hydrocarbon mxture. We present and dscuss the programs Matlab code. 1.1 A note on nomenclature In Sectons 2-4, we descrbe thermodynamc propertes of ether a vapor or a lqud mxture. When we descrbe propertes of vapor mxtures, e.g. the Gbbs energy of an deal gas, we denote the composton n moles by a vector, n v. In that case, we denote the mole fracton of component by y. For lqud mxtures, we denote the composton vector n l and the mole fracton x. When we descrbe the propertes of nondeal mxtures, and when we defne fugacty and fugacty coeffcents, the expressons are the same for both vapor mxtures and lqud mxtures. We therefore denote the composton by n and the mole fracton by z. Later, n Secton 5 to 8, we consder mxtures that exst n both a vapor phase and a lqud phase. Agan, the vapor phase has composton n v and mole fracton y, and the lqud phase has composton n l and mole fracton x. But n that case, n wll denote the composton of the entre mxture,.e. n = n v + nl, and z wll be the correspondng mole fracton. 2 Resdual Gbbs energy of deal vapor-lqud mxtures We derve the Gbbs energy of deal gas and lqud mxtures from the enthalpy and entropy of those mxtures. We use the pure component molar enthalpy and entropy of each component n the mxture to derve expressons for the enthalpy and the entropy of the mxtures. The resdual Gbbs energy of a mxture s, by defnton, the Gbbs energy of that mxture mnus the Gbbs energy of an deal gas at the same state, e.g. temperature, pressure, and composton n moles. The resdual Gbbs energy of an deal gas mxture s therefore zero by defnton. We use the Gbbs energy of an deal gas mxture and an deal lqud mxture to derve an expresson for the resdual Gbbs energy of a lqud mxture. 5

7 2. Resdual Gbbs energy of deal vapor-lqud mxtures 2.1 Gbbs energy of deal gas mxtures We consder an deal gas mxture of N C components. The molar enthalpy and entropy of the th component at temperature T and pressure P are T h g T = hg T 0 + s g T, P = sg T 0, P 0 + T 0 T c g P, T dt, 1a T 0 g c P, T T dt R ln P P 0. 1b The deal gas enthalpy s ndependent of pressure. h g T 0 and s g T 0, P 0 are the deal gas enthalpy and entropy of formaton at reference temperature T 0 and P 0. c g P, T s the deal gas heat capacty at constant pressure. Values of h g T 0 and s g T 0, P 0 and correlatons for c g P, T are avalable n databases such as the DIPPR database 1 Thomson, However, terms that nvolve h g T 0, s g T 0, P 0, and c g P, T wll cancel out n the computatons that we descrbe n ths techncal report. The molar gbbs energy of component s g g T, P = hg T T sg T, P = h g T 0 T s g T 0, P 0 RT ln P 0 + T T 0 c g P, T dt T T T 0 g c P, T T dt + RT ln P = Γ T + RT ln P, 2 where we have ntroduced the auxlary varable T T Γ T = h g T 0 T s g T 0, P 0 RT ln P 0 + c g P, T dt T T 0 T 0 g c P, T T dt. 3 h g T 0, s g T 0, P 0, and c g P, T appear only n the auxlary varable Γ T and t s ths varable that wll cancel out n the computatons n later sectons. The enthalpy and entropy of the deal gas mxture are H g T, n v = n v h g T, 4a S g T, P, n v = n v s g T, P R n v ln y. 4b n v s a vector of compostons n moles. The th component, n v, denotes the number of moles of component. y s the vapor mole fracton of component : y = n v / j nv j. The sums are over all components. The second term n the expresson for the entropy s a mxng term. The Gbbs 1 6

8 2. Resdual Gbbs energy of deal vapor-lqud mxtures energy of the deal gas mxture s G g T, P, n v = H g T, n v T S g T, P, n v = n v h g T T n v s g T, P + RT n v ln y = n v g g T, P + RT n v ln y = n v Γ T + RT ln P + RT n v ln y Gbbs energy of deal lqud mxtures The vaporzaton change n molar enthalpy of component, h vap T, s related to the vaporzaton change n molar entropy, s vap T, by h vap T = T s vap T. 6 We do not present an expresson for s vap T because t wll cancel out n later expressons. The molar deal lqud enthalpy at saturaton s h sat T = h g T hvap T = h g T T svap T. 7 It s necessary to know the saturaton pressure n order to provde expressons for the molar deal lqud entropy at saturaton. We use the followng correlaton from the DIPPR database P sat T = exp ln P sat T, 8a ln P sat T = A + B T + C lnt + D T E. 8b A, B, C, D, and E are component-specfc parameters. They are also specfc to ths partcular correlaton. The deal lqud entropy at saturaton s s sat T = s g The molar deal lqud Gbbs energy at saturaton s sat T, P T s vap T. 9 g sat T = h sat T T s sat T = h g = h g T T svap T T s g sat T T sg T, P T sat T, P T + T s vap T = Γ T + RT ln P sat T. 10 In order to provde expressons for the molar deal lqud enthalpy and entropy at pressures, P, that are dfferent from the saturaton pressure, P sat T, t s necessary to know the lqud volume as a functon of temperature, as well as ts frst order temperature dervatve. We use the followng 7

9 2. Resdual Gbbs energy of deal vapor-lqud mxtures correlaton from the DIPPR database 1 T C D vt l = B A Agan, A, B, C, and D are component-specfc parameters. They are also specfc for ths correlaton,.e. they are dfferent from the parameters n the correlaton for the saturaton pressure. The molar deal lqud enthalpy and entropy are h d T, P = h sat T + vt l T vl P P sat T, 12a T s d T, P = s sat T vl sat P P T. 12b T Terms that nvolve the temperature dervatve wll cancel out n later expressons. The molar deal lqud Gbbs energy s g d T, P = h d T, P T s d = h sat T + = h sat T, P vt l T vl T P P sat T T s sat T + vt l P P sat T = g sat T + vt l P P sat T T T s sat T + T vl T P P sat T = Γ T + RT ln P sat T + vt l P P sat T. 13 The deal lqud mxture enthalpy and entropy are H d T, P, n l = n l h d T, P, 14a S d T, P, n l = n l s d T, P R n l ln x. 14b n l s a vector of compostons n moles,.e. n l s the moles of component. x = n l / j nl j s the lqud mole fracton of component. The second term n the expresson for entropy s a mxng term. The deal lqud mxture Gbbs energy s G d T, P, n l = H d T, P, n l T S d T, P, n l = n l h d T, P T n l s d T, P + RT n l ln x = n l g d T, P + RT n l ln x = n l Γ T + RT ln P sat T + v l T P P sat T + RT n l ln x. 15 8

10 3. Resdual Gbbs energy of nondeal mxtures 2.3 Resdual Gbbs energy of deal vapor and lqud mxtures The resdual Gbbs energy s the dfference between the Gbbs energy of the mxture and the Gbbs energy of the mxture f t was consdered to be an deal gas mxture. The resdual Gbbs energy of an deal gas mxture s therefore zero: G R,g T, P, n v = 0 16 We use the Gbbs energy of deal gas mxtures to derve the expresson for the resdual Gbbs energy of an deal lqud mxture. It s the dfference between the deal lqud Gbbs energy and the deal gas Gbbs energy evaluated at the same temperature, T, pressure, P, and composton, n l : G R,d T, P, n l = G d T, P, n l G g T, P, n l = n l Γ T + RT ln P sat T + vt l P P sat T + RT n l Γ T + RT ln P + RT n l ln x = n l RT ln P sat T + v l P T P P sat T n l ln x 17 It s mportant to note that the deal gas Gbbs energy s evaluated at the lqud composton, n l, and not n v n the above expresson. That s also the reason that x appears n place of y whch means that certan terms cancel out. Unlke the deal gas heat capacty, c g P, T, the saturaton pressure, P sat T, and the lqud volume, v l T, are necessary n the computatons that we present n ths report. 2.4 Summary The resdual Gbbs energy of deal gas and lqud mxtures are G R,g T, P, n v = 0, G R,d T, P, n l = n l 18a RT ln P sat T + v l P T P P sat T. 18b They wll be used n expressons for the fugacty coeffcents n Secton 4. 3 Resdual Gbbs energy of nondeal mxtures In ths secton, we frst present an expresson for the resdual Gbbs energy of a nondeal mxture wthout assumng a specfc equaton of state. Then we descrbe cubc equatons of state wth van der Waals mxng rules. Fnally, we present the expresson for the resdual Gbbs energy of a nondeal mxture based on a cubc equaton of state. The dervaton of the expresson based on the cubc equaton of state s presented n Appendx B. We wll not use the expresson for the Gbbs energy 9

11 3. Resdual Gbbs energy of nondeal mxtures of an deal gas mxture that was derved n Secton 2. Instead, we use the deal gas law for the deal gas mxture. 3.1 Resdual Gbbs energy wth pressure-explct equatons of state In ths secton, we consder a nondeal vapor or lqud mxture. We treat both phases n the same way. We use the resdual Helmholtz energy n the dervaton. We wll consder the Helmholtz energy a functon of temperature, T, volume, V, and composton n moles, n. Ths s only for the sake of the dervaton. Eventually, we descrbe the resdual Gbbs energy as functon of temperature, T, pressure, P, and mole numbers, n. We omt the superscrpt v or l on the composton vector because we treat both phases n the same way. The resdual Helmholtz energy s A R T, V, n = AT, V, n A g T, V, n. 19 AT, V, n s the Helmholtz energy of the mxture and A g T, V, n s the Helmholtz energy of the mxture f t was an deal gas. We use the fact that the volume dervatve of Helmholtz energy s pressure wth a negatve sgn: A R V T, V, n = A V Ag T, V, n T, V, n V = P T, V, n P g T, V, n. 20 P g T, V, n s obtaned from the deal gas law,.e. P g T, V, n = NRT/V, where N = j n j s the total number of moles n the mxture and R s the gas constant. P T, V, n s obtaned from an equaton of state. The dervaton that we present n ths secton s therefore best suted to equatons of state that are explct n pressure. That s the case for the cubc equatons of state that we descrbe later. We obtan an expresson for the resdual Helmholtz energy of the nondeal mxture: V A R T, V, n = P T, V, n NRT dv. 21 V = V The choce of ntegratng from nfnte volume s a matter of convenence. We ntroduce the compressblty factor, Z = P V NRT. We use relatons between resdual propertes to derve an expresson for the resdual Gbbs energy evaluated at T, P, and n Mchelsen and Mollerup, 2007, Table 6: G R T, P, n = G R T, V, n NRT ln Z = A R T, V, n + P V NRT NRT ln Z V = P T, V, n NRT dv + NRT Z NRT NRT ln Z V = V V = P T, V, n NRT dv + NRT Z 1 NRT ln Z. 22 V V = The above expresson only becomes useful when we substtute a specfc equaton of state for P T, V, n. 10

12 3. Resdual Gbbs energy of nondeal mxtures 3.2 Cubc equatons of state In ths secton, we descrbe cubc equaton equatons of state and van der Waals mxng rules. The cubc equatons of state are formulated n terms of the molar volume, v = V/N: P = P T, V, n = = RT a m v b m v + ɛb m v + σb m RT a m V/N b m V/N + ɛb m V/N + σb m. 23 The parameters ɛ and σ are specfc for each cubc equaton of state. Two of the most popular cubc equatons of state were developed by Soave 1972 and by Peng and Robnson The parameters that are specfc to each equaton of state are shown n Table 1. a m and b m are mxng parameters gven by van der Waals mxng rules: a m = a m T, n = b m = b m n = z z j a j j z b 24a 24b z = n /N s the mole fracton of component. We defne the pure component parameter b shortly. The bnary parameters a j are a j = a j T = 1 k j â j. 25 We assume that the bnary nteracton parameters, k j, are gven constants,.e. we do not provde an expresson for them. They can ether be measured expermentally or predcted wth a model. The parameter â j s the product of the pure component parameters a and a j : â j = â j T = a a j. 26 The pure component parameters a and b are a = a T = αt r,, ω Ψ R2 T 2 c, P c,, 27a b = Ω RT c, P c,. 27b T r, = T/T c, s the reduced temperature. T c, and P c, are the crtcal temperature and pressure of component. ω s the acentrcty factor. We use values from the DIPPR database for the crtcal temperature, pressure, and for the acentrcty factor. Ψ and Ω are parameters that are specfc to the equaton of state. Ther values are shown n Table 1. The functon, α, s αt r,, ω = 1 + mω 1 T 1/2 r, mω s a polynomum that s specfc for each equaton of state. Its expresson s shown n Table 1. Ths concludes the descrpton of the cubc equatons of state. Next, we descrbe how to solve the 11

13 3. Resdual Gbbs energy of nondeal mxtures Table 1: Parameters n the Soave-Redlch-Kwong SRK and the Peng-Robnson PR cubc equatons of state. ɛ σ Ω Ψ mω PR mω = ω ω 2 SRK mω = ω 0.176ω 2 cubc equaton of state of state for the compressblty factor Soluton of cubc equatons of state We reformulate the cubc equaton of state as a polynomum n the compressblty factor Z: 2 qz = Z 3 + d m Z m = The polynomal coeffcents are m=0 d 2 = d 2 A, B = Bɛ + σ 1 1, d 1 = d 1 A, B = A Bɛ + σ + B 2 ɛσ ɛ σ, d 0 = d 0 A, B = AB + B 2 + B 3 ɛσ, 30a 30b 30c where A and B are functons of temperature, T, pressure, P, and composton, n: A = AT, P, n = P a m R 2 T 2, B = BT, P, n = P b m RT. 31a 31b The compressblty factors are therefore also functons of temperature, T, pressure, P, and composton, n,.e. Z = ZT, P, n. The polynomum qz = qz; T, P, n wll have ether one or three roots dependng on the gven T, P, and n. When t has three roots, the smallest root s the lqud phase compressblty and the largest root s the vapor phase compressblty. Ths s essentally what dstngushes the resdual Gbbs energy of the vapor phase from the resdual Gbbs energy of the lqud phase. For gven T, P, and n, we use Newton s method to solve for the compressblty factor: Z k+1 = Z k qz k /q Z k, q Z k = 3Z 2 k + 2d 2 Z k + d 1. 32a 32b We use the stoppng crterum qz k+1 < ɛ, 33 12

14 4. Fugacty and fugacty coeffcents for a gven tolerance, ɛ. The ntal guess for the Newton teratons depends on whether we are searchng for a vapor root or a lqud root: Z 0 = 1, Vapor 34a Z 0 = B. Lqud 34b It s mportant to note that n vapor-lqud equlbrum computatons, the vapor and the lqud phases wll have dfferent compostons, n v and n l. It s therefore necessary to solve the polynomum qz; T, P, n v for the vapor compressblty and qz; T, P, n l for the lqud compressblty. The compressbltes are not roots of the same polynomum. 3.3 Resdual Gbbs energy wth cubc equatons of state We nsert the cubc equaton of state nto the expresson for the resdual Gbbs energy. The evaluaton of the ntegral s descrbed n Appendx B. The fnal expresson becomes V G R T, P, n = V = RT V/N b m + NRT Z 1 NRT ln Z Z B = NRT ln Z a m V/N + ɛb m V/N + σb m NRT dv V N a m ln ɛ σ b m Z + ɛb Z + σb = NRT Z 1 NRT lnz B N a m ln ɛ σ b m + NRT Z 1 NRT ln Z Z + ɛb Z + σb. 35 We reterate that the above expresson descrbes the resdual Gbbs energy of both the vapor phase, G R,v T, P, n v, and of the lqud phase, G R,l T, P, n l. The cubc equaton of state s a thrd order polynomum, qz; T, P, n, n the compressblty factor. The vapor phase compressblty, Z v = Z v T, P, n v, s the largest root of qz v ; T, P, n v and the lqud phase compressblty factor, Z l = Z l T, P, n l, s the smallest root of qz l ; T, P, n l. 3.4 Summary The resdual Gbbs energy of a nondeal mxture ether a vapor or a lqud mxture based on a cubc equaton of state s G R T, P, n = NRT Z 1 NRT lnz B 4 Fugacty and fugacty coeffcents N a m Z + ɛb ln. 36 ɛ σ b m Z + σb In ths secton, we ntroduce the chemcal potental n order to defne the fugacty. Next, we ntroduce the fugacty coeffcents whch are the quanttes that we wll actually use n the vaporlqud equlbrum computatons. That s, the chemcal potentals and the fugactes only serve the purpose of ntroducng expressons for the fugacty coeffcents. 13

15 4. Fugacty and fugacty coeffcents 4.1 Defnton of chemcal potental The chemcal potental of component n a mxture s by defnton the partal dervatve of the Gbbs energy of that mxture wth respect to mole number : µ T, P, n = G n T, P, n. 37 The same defnton holds for any mxture,.e. also for deal gas and lqud mxtures: µ g µ d Gg T, P, n = n v T, P, n v, T, P, n = Gd T, P, n l. n l 38a 38b 4.2 Chemcal potental of a pure component deal gas The defnton of the fugacty of component depends on the chemcal potental of a pure component pc deal gas consstng entrely of component. The Gbbs energy of such a pure component deal gas s G pc,g T, P, n v = n v g g T, P The chemcal potental of that pure component deal gas s therefore = n v Γ T + RT ln P. 39 µ pc,g T, P, n v = Gpc,g n v T, P, n v = g g T, P = Γ T + RT ln P. 40 We note that the pure component deal gas chemcal potental s ndependent of the mole number,.e. µ pc,g T, P, n v = µpc,g T, P. 4.3 Defnton of fugacty We defne the fugacty of a mxture vapor or lqud of composton n. Frst, we recall that the Gbbs energy of an deal gas mxture of composton n v n moles s G g T, P, n v = n v Γ T + RT ln P + RT n v ln y. 41 Next, we express the deal gas mxture chemcal potental of component n terms of the chemcal potental of a pure component deal gas that conssts of component. Because we defne fugacty n the same way for both vapor and lqud mxtures, we use n for the composton nstead of n v. Smlarly, we use z = n /N, where N = j n j, for the mole fracton of component. The th 14

16 4. Fugacty and fugacty coeffcents chemcal potental of an deal gas mxture s µ g Gg T, P, n = n v T, P, n = Γ T + RT ln P + RT ln z = Γ T + RT ln P 0 + RT ln z P P 0 = µ pc,g T, P 0 + RT ln z P. 42 P 0 The dfferentaton of the frst sum n the expresson 41 for the Gbbs energy of an deal gas mxture s straghtforward. We descrbe the dfferentaton of the second sum n Appendx A. The fugacty of component, f T, P, n, s defned such that the above expresson s vald for the chemcal potental of mxtures that are not deal gas mxtures. It s therefore mplctly defned by replacng z P wth f T, P, n n the above: µ T, P, n = µ pc,g T, P 0 + RT ln f T, P, n P Note that t s stll the chemcal potental of a pure component deal gas that appears on the rghthand sde of the above expresson. Next, we ntroduce the fugacty coeffcents and use the above expresson to relate the fugacty coeffcents to the resdual Gbbs energy. 4.4 Fugacty coeffcents In ths secton, we ntroduce the fugacty coeffcents and relate them to the resdual Gbbs energy. Frst, we solate the chemcal potental of the pure component deal gas n 42: µ pc,g T, P 0 = µ g T, P, n RT ln z P. 44 P 0 Then we substtute the chemcal potental of the pure component deal gas nto 43: µ T, P, n = µ pc,g T, P 0 + RT ln f T, P, n P 0 = µ g T, P, n RT ln z P + RT ln f T, P, n P 0 P 0 = µ g T, P, n + RT ln f T, P, n. 45 z P We defne the fugacty coeffcent of component as φ T, P, n = f T, P, n. 46 z P 15

17 4. Fugacty and fugacty coeffcents We substtute the fugacty coeffcent nto 45, solate the term that contans the fugacty coeffcent, and use the defnton of the chemcal potentals: RT ln φ T, P, n = µ T, P, n µ g T, P, n = G T, P, n Gg T, P, n n n = GT, P, n G g T, P, n n = GR n T, P, n 47 We thus obtan an expresson for the logarthm of the fugacty coeffcents: ln φ T, P, n = 1 G R T, P, n. 48 RT n It s only the logarthms of the fugacty coeffcents that we use n the vapor-lqud equlbrum computatons. It s therefore not necessary to solate the fugacty coeffcents n the above. Next, we derve expressons for the logarthmc fugacty coeffcents of deal and nondeal gas and lqud mxtures Ideal gas mxture The resdual Gbbs energy of deal gas mxtures s zero. The logarthmc fugacty coeffcents are therefore also zero: ln φ g T, P, nv = 1 G R,g T, P, n RT n = Ideal lqud mxture The resdual Gbbs energy of deal lqud mxtures s lnear n the mole numbers. The dfferentaton s therefore straghtforward. The logarthmc fugacty coeffcents are ln φ d T, P, n l = 1 G R,d RT = 1 RT n l RT ln P sat T + v l P T P P sat T = ln P sat T + vl sat T P P T. 50 P RT The fugacty coeffcents of deal lquds are therefore ndependent of the composton,.e. φ d T, P, nl = φ d T, P. 16

18 5. Vapor-lqud equlbrum Nondeal mxture We present the dervaton of the fugacty coeffcents for nondeal mxtures n Appendx C. The fnal expresson s ln φ T, P, n = Z 1 b b m lnz B 1 ɛ σ 1 2 RT b m j b z j a j a m Z + ɛb ln. Z + σb The above expresson apples to both vapor and lqud mxtures. The logarthmc fugacty coeffcents of vapor mxtures, ln φ v T, P, nv, use the vapor phase compressblty, Z v = Z v T, P, n v, whch s the largest root of the cubc equaton of state, qz v ; T, P, n v. The logarthmc fugacty coeffcents of lqud mxtures use the lqud phase compressblty, Z l T, P, n l, whch s the smallest root of qz l ; T, P, n l. b m 51 Furthermore, several of the quanttes n the above expresson depend on the composton. It s therefore dfferent values of a m T, n, b m n, BT, P, n, and z that appear n the expressons for the vapor and lqud logarthmc fugacty coeffcents. 4.5 Summary The expressons for the logarthmc fugacty coeffcents are ln φ g T, P, nv = 0, ln φ d T, P, n l = ln P sat T P + vl ln φ T, P, n = Z 1 b b m lnz B 1 ɛ σ 52a sat T P P T, 52b RT 1 2 b z j a j a m Z + ɛb ln. RT b m b j m Z + σb The latter expresson apples to both nondeal vapor and lqud mxtures. 52c 5 Vapor-lqud equlbrum In ths secton, we formulate the vapor-lqud equlbrum problem as an optmzaton problem. We then derve the vapor-lqud equlbrum condtons as the frst-order optmalty condtons of ths optmzaton problem. Next, we reformulate the equlbrum condtons n terms of the fugacty coeffcents that we ntroduced n Secton 4. Fnally, we use Newton s method to solve the equlbrum condtons. 5.1 Equlbrum condtons We consder a mxture that exsts n both a vapor phase v and a lqud phase l. The mxture contans N C components. The vapor phase has composton n v, and the lqud phase has composton n l. Both n v and n l are vectors of mole numbers. Both phases have the same temperature, T, and the same pressure, P. The mxture s sothermal and sobarc,.e. the temperature and the pressure are 17

19 5. Vapor-lqud equlbrum specfed constants. The problem s to fnd the vapor-lqud composton n moles at equlbrum. The condton of equlbrum s that Gbbs energy of the mxture s mnmal. We formulate ths condton as the optmzaton problem mn G v T, P, n v + G l T, P, n l, 53a n v,n l s.t. n v + n l = n, = 1,..., N C. 53b The total moles of each component, n, s specfed. The constrant ensures that mass s conserved,.e. that mass s dstrbuted among the vapor phase and the lqud phase. Because the constrant s lnear, we use t to elmnate the lqud moles,.e. n l = n n v. By dong so, we obtan an unconstraned optmzaton problem: mn G v T, P, n v + G l T, P, n n v. 54 n v The frst-order optmalty condtons requre that the dervatves of the objectve functon wth respect to the vapor mole numbers are zero: G v n v T, P, n v Gl n l T, P, n n v = We have used that n G l T, P, n n v = v Gl T, P, n n v. For brevty, we wrte n l nstead of n l n n v n most of the followng equatons n ths secton, but t s mplctly understood that n l has been elmnated. Next, we rewrte the equlbrum condtons n terms of the chemcal potentals: G v n v T, P, n v Gl n l T, P, n l = µ v T, P, n v µ l T, P, n l = µ v T, P, n v µ pc,g T, P 0 µ l T, P, n l µ pc,g T, P 0. We can obtan an expresson for the dfference n each parentheses from the defnton of the fugactes: µ v T, P, n v µ pc,g T, P 0 µ l T, P, n l µ pc,g T, P 0 56 = RT ln f v T, P, nv P 0 RT ln f l T, P, nl P From the defnton of the fugacty coeffcents, we know that f v T, P, nv = φ v T, P, nv y P and f l T, P, nl = φ l T, P, nl x P. We use those expressons to ntroduce the fugacty coeffcents: RT ln f v T, P, nv P 0 RT ln f l T, P, nl P 0 = RT ln f v T, P, n v ln f l T, P, n l = RT ln φ v T, P, n v y P ln φ l T, P, n l x P = RT ln φ v T, P, n v + ln y ln φ l T, P, n l ln x

20 6. Equlbrum constants We recall that we have elmnated the lqud mole numbers,.e. n l = n n v. We solve the vapor-lqud equlbrum problem by solvng the followng nonlnear equatons for the vapor mole numbers, n v : g T, P, n v = ln φ v T, P, n v + ln y ln φ l T, P, n n v ln x = The lqud mole numbers are x = n l / j nl j = n n v / j n j n v j. 5.2 Soluton of the equlbrum condtons We solve the vapor-lqud equlbrum condtons wth Newton s method: 5.3 Summary 1 g n v,k+1 = n v,k n v gt, P, n v,k. 60 The vapor-lqud equlbrum condtons for a vapor-lqud mxture at temperature T, pressure P, and total composton n are g T, P, n v = ln φ v T, P, n v + ln y ln φ l T, P, n n v ln x = We solve the above equlbrum condtons for the vapor composton, n v. The lqud mole numbers are x = n n v / j n n v. We solve the vapor-lqud equlbrum equatons wth Newton s method: 1 g n v,k+1 = n v,k n v gt, P, n v,k. 62 After soluton, the lqud mole numbers are computed by n l = n n v. 6 Equlbrum constants In ths secton, we ntroduce the equlbrum constants and provde an expresson for them n terms of the logarthmc fugacty coeffcents. The equlbrum constants are useful when solvng vapor-lqud equlbrum problems for deal gas and lqud mxtures. We also use them as ndependent varables n the phase envelope computatons n Secton Equlbrum constants The th equlbrum constant sometmes called the equlbrum rato s the rato between the vapor mole fracton and the lqud mole fracton of component : K = y x

21 7. Vapor-lqud equlbrum of deal mxtures We can therefore express the equlbrum condtons, ln φ v T, P, n v + ln y ln φ l T, P, n n v ln x = 0, 64 n terms of the equlbrum constants: ln K + ln φ v T, P, n v ln φ l T, P, n l = At equlbrum, we can therefore obtan the followng expresson for the logarthmc equlbrum constants: ln K T, P, n v, n l = ln φ l T, P, n l ln φ v T, P, n v Ideal vapor-lqud mxture For an deal vapor-lqud mxture, the logarthmc equlbrum constants are equal to the logarthmc fugacty coeffcents of the deal lqud phase: ln K d T, P, n v, n l = ln φ d T, P, n l ln φ g T, P, nv = ln φ d T, P, n l = ln P sat T + vl sat T P P T P RT 67 The equlbrum constants are therefore ndependent of the composton vectors,.e. K dt, P, nv, n l = K d T, P. That can be exploted n the deal vapor-lqud equlbrum computatons. 6.2 Summary The equlbrum constants for deal and nondeal vapor-lqud mxtures are ln K d T, P = ln P sat T + vl sat T P P T, P RT 68a ln K T, P, n v, n l = ln φ l T, P, n l ln φ v T, P, n v. 68b 7 Vapor-lqud equlbrum of deal mxtures In ths secton, we explot the fact that the equlbrum constants of deal vapor-lqud mxtures, K d T, P, are ndependent of composton. We can therefore solve the vapor-lqud equlbrum by 1 evaluatng the equlbrum constants at the specfed temperature, T, and pressure, P, 2 solve for the vapor fracton, and 3 compute the vapor-lqud composton n moles from the equlbrum constants and the vapor fracton. The deal vapor-lqud equlbrum constants are K d T, P = expln K d T, P, 69a ln K d T, P = ln P sat T P + vl sat T P P T. 69b RT 20

22 7. Vapor-lqud equlbrum of deal mxtures We express the vapor mole fractons, y, n terms of the equlbrum constants, K d T, P, and the lqud mole fractons, x : y = K d T, P x. 70 The total moles n the vapor and lqud phases are N v = nv and N l = nl. The total moles n the mxture s N = N v + N l. The vapor fracton s β = N v /N and 1 β = 1 N v /N = N N v /N = N l /N. The total mole fractons are z = n /N. We derve the expressons for x from the mass balance: n v + n l = n, 71a The lqud mole fracton, x, s therefore N v N n v N + nl N = n N, n v N v + N l N n l N l = n N, βy + 1 βx = z, 71b 71c 71d βk d T, P x + 1 βx = z, 71e 1 β + βk d T, P x = z. 71f x = 1 + βk d z T, P Because we solve for β, y and x wll not sum to one durng the soluton procedure. We therefore requre that both y and x sum to one. When 0 < β 1, the condton that together wth z = 1 mples that x = 1, 73 y = We omt the dervaton. It s therefore suffcent to only requre that x = 1 for 0 < β 1. β = 0 mples that x = z such that x = 1 s satsfed ndependent of the equlbrum ratos, K d T, P. In that case, we therefore requre that y = The followng condton s equvalent to x = 1 for 0 < β 1 and to y = 1 for β = 0: y x = That s, y = 1 and x = 1 clearly mply that y x = 0. Because of the way that we compute y and x, y x also mples that y = 1 and x = 1. We nsert the expressons 21

23 7. Vapor-lqud equlbrum of deal mxtures for y and x to obtan an equaton n β: y x = = K d T, P 1 x K d T, P βk d T, P 1z = 0, 77 We therefore solve the deal vapor-lqud equlbrum problem by solvng the scalar nonlnear equaton, fβ = K dt, P βk d T, P 1z = 0, 78a f β = K dt, P βk d z. 78b T, P 1 fβ = 0 s called the Rachford-Rce equaton. Once we have solved the Rachford-Rce equaton for β, we compute the vapor-lqud composton n moles by n v = nv N v N v N N = y βn Next, we compute the lqud mole numbers by n l = n n v. = βk d T, P 1 + βk d T, P 1n Soluton of the Rachford-Rce equaton We solve the Rachford-Rce equaton wth Newton s method: 7.2 Summary We compute the deal vapor-lqud equlbrum constants wth β k+1 = β k fβ k f β k. 80 K d T, P = expln K d T, P, 81a ln K d T, P = ln P sat T P Next, we solve the Rachford-Rce equaton, + vl sat T P P T. 81b RT fβ = K d T, P βk d T, P 1z = 0, 82 for β usng Newton s method, β k+1 = β k fβ k f β k, 83 22

24 8. Computaton of phase envelopes where the dervatve s f β = K dt, P βk d z. 84 T, P 1 We compute the vapor-lqud composton n moles by n v = βk d T, P 1 + βk d T, P 1n, 85a n l = n n v. 85b 8 Computaton of phase envelopes In ths secton, we descrbe the equatons that we solve n order to construct the phase envelope, and we dscuss how to solve them. The phase envelope conssts of two curves; the bubble-pont curve where β = 0 and the dew-pont curve where β = 1. We wll consder the more general case of computng socurves where β [0, 1] s constant. In the method that we present n ths secton, the logarthmc equlbrum constants, ln K, the logarthmc temperature, ln T, and the logarthmc pressure, ln P, are the ndependent varables,.e. the varables that we solve for. The vapor mole fractons, y, and the lqud mole fractons, x, are dependent varables that are functons of the vapor fracton, β, the equlbrum constants, K, and the total mole fractons, z. The vapor fracton, β, and the vector of total mole fractons, z, are parameters n the problem. We ntroduce N C + 2 equatons that defne one pont on the socurve. N C s the number of components n the mxture. We solve these equatons repeatedly n a sequental manner n order to construct the socurve. The frst N C equatons are the vapor-lqud equlbrum condtons formulated n terms of the equlbrum constants,.e. 65. We repeat the equlbrum condtons here: ln K + ln φ v T, P, y ln φ l T, P, x = The mole numbers do not appear explctly n the expresson for the fugacty coeffcents. They only appear mplctly through the mole fractons. We can therefore wrte ln φ v T, P, nv = ln φ v T, P, ynv = ln φ v T, P, y where y = n v / j nv j and smlarly for the lqud fugacty coeffcents. That s why y and x appear n place of n v and n l n the above equlbrum condtons. However, because y and x are dependent varables, they wll not necessarly sum to one durng the soluton of the equatons. We therefore treat them as mole numbers when we evaluate the logarthmc fugacty coeffcents. That s, we evaluate the vapor phase logarthmc fugacty coeffcents, ln φ v T, P, y, as ln φv T, P, nv where n v = y, and smlarly for the lqud fugacty coeffcents. Also, because y and x are functons of K, we cannot explctly solate K n the vapor-lqud equlbrum equatons above. We compute y and x n the same way that we dd for the deal vapor-lqud equlbrum problem: x = z 1 + βk 1, 87a y = K x. 87b 23

25 8. Computaton of phase envelopes However, here we do not use the deal vapor-lqud equlbrum constants, and the equlbrum constants are ndependent varables. The next equaton s the Rachford-Rce equaton 78a. However, we mplement t n the form, y x = 0, 88 where we do not substtute the expressons for the vapor-lqud mole fractons. That s because we need to compute y and x anyway, n order to evaluate the logarthmc fugacty coeffcents. The Rachford-Rce equaton ensures that y = 1 and x = 1 are both satsfed 2. In order to defne the last equaton, we collect the ndependent varables n a vector, X = [ln K; ln T ; ln P ] R NC+2, 89 where ln K = [ln K 1 ; ; ln K NC ]. The last socurve equaton states that one of the ndependent varables should be specfed,.e. the s th component of X should have the value S: X s S = That means that we specfy ether temperature, pressure, or one of the equlbrum constants n order to compute a pont on the socurve. Mchelsen and Mollerup 2007, Chap. 12 dscuss dfferent strateges for selectng the specfed varable. The socurve s constructed by solvng the socurve equatons for a suffcent number of values of S. We wrte the socurve equatons compactly as F X; S = 0, 91 where F = ln K + ln φ v T, P, y ln φ l T, P, x, = 1,..., N C, 92a F NC +1 = y x, 92b F NC +2 = X s S. 92c 8.1 Soluton of the socurve equatons As mentoned, we construct the socurve by solvng the socurve equatons for a number of specfed values of the s th varable. For the m th value of the specfed varable, S m, we solve the socurve equatons, F X m ; S m = 0, wth Newton s method: 8.2 Computaton of ntal guess 1 F X m,k+1 = X m,k F X m,k ; S m. 93 X When we have solved the socurve equatons for one value of the specfed varable, we want to compute the senstvtes of the soluton n order to compute an ntal guess for the subsequent 2 It s not mmedately obvous that ths s true. It s because of the way that we compute y and x. 24

26 8. Computaton of phase envelopes Newton teratons. We dfferentate the equaton F X; S = F XS; S = 0 wth respect to S: F X X S + F S We solate the senstvtes of X wth respect to S: = X F S = F X S. 95 Once we have solved F X m ; S m = 0 for X m, we can compute an ntal guess, X m+1,0, for the soluton of the socurve equatons for the next value of the specfed varable, F X m+1 ; S m+1 = 0: 8.3 Summary X m+1,0 = X m + Xm S m Sm+1 S m. 96 The socurve equatons or phase envelope equatons f β = 0 or β = 1 are F X; S = 0, 97 where the ndependent varables are X = [ln K; ln T ; ln P ]. The equatons are F = ln K + ln φ v T, P, y ln φ l T, P, x, = 1,..., N C, 98a F NC +1 = y x, 98b F NC +2 = X s S, 98c where the vapor-lqud mole fractons are dependent varables: x = z 1 + βk 1, 99a y = K x. 99b We solve the equatons wth Newton s method: 1 F X m,k+1 = X m,k F X m,k ; S m. 100 X We compute ntal guesses for the Newton teratons wth X m+1,0 = X m + Xm S m Sm+1 S m, 101 where the senstvtes are 1 X F S = F X S

27 9. Dervatves 9 Dervatves In ths secton, we provde the dervatves that are necessary to solve the vapor-lqud equlbrum problem for both deal and nondeal mxtures and to compute the socurves.e. curves where β s constant. In order to do so, we also provde the dervatves of the logarthmc fugacty coeffcents. 9.1 Logarthmc fugacty coeffcents We provde the dervatves of the logarthmc fugacty coeffcents for deal lquds and nondeal mxtures. The logarthmc fugacty coeffcents of deal gases are zero, and the dervatves are therefore also zero Ideal lqud mxture The logarthmc fugacty coeffcents of an deal lqud mxture are ln φ d T, P = ln P sat T P + vl The lqud volume s gven by the DIPPR correlaton, 1 T C D The saturaton pressure s gven by the DIPPR correlaton, sat T P P T. 103 RT v l = B A P sat ln P sat = exp ln P sat, 105a = A + B T + C lnt + D T E. 105b Note that the parameters, A, B, C, and D n the DIPPR correlaton for the saturaton pressure are not the same as those n the DIPPR correlaton for the lqud volume. The dervatves of the logarthmc fugacty coeffcents are ln φ d T ln φ d P = ln P sat T = 1 P + vl T RT. The dervatve of the lqud volume s + 1 v l RT T vl T T v l T = ln B D C P P sat T vt l sat P, 106a T 106b 1 TC D 1 v l

28 9. Dervatves The dervatve of the saturaton pressure s Nondeal mxture ln P sat T P sat T = 1 T C B T + D E T E, 108a = P sat sat ln P T. 108b T The logarthmc fugacty coeffcents of nondeal mxtures have a complcated expresson. We therefore ntroduce auxlary functons such that we can wrte t as ln φ = Z 1 b b m g z 1 ɛ σ g φ,f. 109 The auxlary functons are Z + ɛb f = ln, 110a Z + σb g z = lnz B, g φ, = 1 N C b 2 z j a j a m. RT b m b m j=1 110b 110c The dervatves of the logarthmc fugacty coeffcents are ln φ T ln φ P ln φ n k = Z T = Z P = Z n k b b m g z b b m g z gφ, T 1 ɛ σ T f + g f φ, T P 1 ɛ σ g f φ, P, b b m Z 1 b b 2 m The dervatves of the auxlary functon f are b m n k g z n k, 111a 111b 1 gφ, f f + g φ,. 111c ɛ σ n k n k f T = f Z Z T + f B B T, f P = f Z Z P + f B B P, f = f Z + f B, n k Z n k B n k 112a 112b 112c where f Z = 1 Z + ɛb 1 Z + σb, f B = ɛ Z + ɛb σ Z + σb. 113a 113b 27

29 9. Dervatves The dervatves of the auxlary functon g z are g z T = g z Z Z T + g z B B T, g z P = g z Z Z P + g z B B P, g z = g z Z + g z B, n k Z n k B n k 114a 114b 114c where The dervatves of the auxlary functon g φ, are g φ, T = 1 1 N C a j 2 z j T Rb m T a m T j=1 2 N C a k x j a j N g φ, n k = 1 b m j=1 g z Z = 1 Z B, g z B = 1 Z B. b b m b b m g φ,, am 1 b m b m g φ, + 1. n k b m n k n k RT 115a 115b 116a 116b The compressblty factor, Z, satsfes the cubc polynomum qz; T, P, n = 0. dervatves of the compressblty factor wth the nverse functon theorem: Z q T = Z Z q P = Z Z n k = q Z 1 q T, 1 q P, 1 q. n k We obtan the 117a 117b 117c The dervatves of the polynomum are q Z = 3Z2 + q T = q P = q n k = 2 m=0 2 m=0 2 m=0 2 md m Z m 1 m=1 d m T Zm, d m P Zm, d m n k Z m. 118a 118b 118c 118d 28

30 9. Dervatves The dervatves of the polynomal coeffcents are d m T d m P d m n k = d m A = d m A = d m A A T + d m B B T, A P + d m B B P, A + d m B, n k B n k 119a 119b 119c where d 2 A = 0, d 2 = ɛ + σ 1, B 120b d 1 A = 1, d 1 = ɛ + σ + 2ɛσ ɛ σb, B 120d d 0 A = B, d 0 B = A + ɛσ2b + 3B a 120c 120e 120f The dervatves of A are A T = a m T A P = A = a m n k n k a m R 2 T 2, P R 2 T 2 2 T A, P R 2 T a 121b 121c The dervatves of B are B T = b mp RT 2, B P = b m RT, B = b m P n k n k RT. 122a 122b 122c The dervatves of the van der Waals mxng parameter a m are a m T a m n k N C = NC =1 j=1 NC = 2 N z z j a j T, 123a z a k a m. 123b =1 29

31 9. Dervatves The dervatve of the van der Waals mxng parameter b m s The dervatve of a j s b m n k = b k b m N. 124 a j T = 1 k j 2 â j â j T, 125 where the dervatve of â j are â j T = a T a j + a a j T. 126 The dervatve of the pure component parameter a s The dervatve of the functon α s 9.2 Vapor-lqud equlbrum equatons The vapor-lqud equlbrum condtons are a T = α T 2 T ΨR2 c,. 127 P c, α T = α mω. 128 αt Tc, g T, P, n v = ln φ v T, P, n v + ln y ln φ l T, P, n l + ln x, 129 where the lqud mole numbers are functons of the vapor mole numbers,.e. n l = n n v. The dervatves of g T, P, n v are g n v k = ln φv n v k + ln y n v k + ln φl n l k + ln x n l. 130 k The dervatves of the logarthmc mole fractons are ln y n v k ln x n l k = δ k n v k = δ k n l k 1 N v, 1 N l. 131a 131b δ k s one f = k and zero f k. 30

32 9. Dervatves 9.3 Ideal vapor-lqud equlbrum equatons For deal vapor-lqud mxtures, we solve the Rachford-Rce equaton: fβ = K d T, P βk d T, P 1z = The dervatve of the fβ wth respect to the vapor fracton s f β = K dt, P βk d z. 133 T, P Phase envelope equatons The ndependent varables n the phase envelope computatons are X = [ln K; ln T ; ln P ]. 134 The socurve equatons or phase envelope equatons f β = 0 or β = 1 are F = ln K + ln φ v T, P, y ln φ l T, P, x, = 1,..., N C, 135a F NC +1 = y x, 135b F NC +2 = X s S. 135c The vapor-lqud mole fractons are dependent varables gven by x = z 1 β + βk, 136a y = K x. The dervatves of the vapor-lqud mole fractons wth respect to the equlbrum constants are x K k = y K k = βx 2 z = k, 0 k, x x + K K k = k, 0 k. The dervatves of the vapor-lqud equlbrum condtons are therefore 136b 137a 137b F ln φ v = δ k + K k ln K k n v T, P, y y k ln φl k K k n l T, P, x x k, = 1,..., N C, 138a k K k F ln φ v ln T = T T T, P, y ln φl T, P, x, = 1,..., N C, 138b T F ln φ v ln P = P P T, P, y ln φl T, P, x, = 1,..., N C. 138c P 31

33 10. An algorthm for computng phase envelopes and an example program The dervatve of the equaton that corresponds to the Rachford-Rce equaton are F NC +1 K k The dervatves of the last equaton are = K k yk x k. 139 K k K k F NC +2 X = e s, 140 where element s of e s s 1 and all other elements are zero. The dervatve of the last equaton wth respect to the value of the specfed varable s F NC +2 S = An algorthm for computng phase envelopes and an example program In ths secton, we descrbe an algorthm for computng socurves,.e. curves where the vapor fracton, β, s constant. The algorthm s descrbed by Mchelsen and Mollerup 2007, Chap. 12 and also by Mchelsen It features automatc selecton of the specfed varable and the step sze, and t uses cubc nterpolaton to enhance ntal guesses for the Newton teratons. The algorthm sequentally constructs the socurve. It begns at a pont where the equlbrum constants can be approxmated and sequentally computes ponts on the socurve untl t reaches the crtcal pont Wlson s approxmaton of equlbrum constants We approxmate the equlbrum constants at the frst pont of the socurve wth the approxmaton by Wlson 1969: K T, P = P c, P exp ω 1 T c,. 142 T T c,, P c,, and ω are the crtcal temperature, crtcal pressure, and the acentrc factor. We use values of those parameters from the DIPPR database. We could potentally also use the deal vapor-lqud equlbrum constants that we descrbed n Secton Cubc nterpolaton It s essental to have good ntal guesses for the Newton teratons. The algorthm therefore uses cubc nterpolaton to mprove the estmates we descrbed n Secton 8. When we have computed two ponts on the socurve and ther senstvtes, we can determne a cubc polynomal for each of the varables that ntersects these ponts. The polynomal s p S = j a j S j