An open-source thermodynamic software library

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1 Downloaded rom orbt.dtu.dk on: Nov 0, 018 An open-source thermodynamc sotware lbrary Rtschel, obas Kasper Skovborg; Gaspar, Jozse; Capole, Andrea; Jørgensen, John Bagterp Publcaton date: 016 Document Verson Publsher's PDF, also known as Verson o record Lnk back to DU Orbt Ctaton APA: Rtschel,. K. S., Gaspar, J., Capole, A., & Jørgensen, J. B An open-source thermodynamc sotware lbrary. Kgs. Lyngby: echncal Unversty o Denmark DU. DU Compute-echncal Report-016, No. 1 General rghts Copyrght and moral rghts or the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton o accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and prnt one copy o any publcaton rom the publc portal or the purpose o prvate study or research. You may not urther dstrbute the materal or use t or any prot-makng actvty or commercal gan You may reely dstrbute the URL dentyng the publcaton n the publc portal I 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 An open-source thermodynamc sotware lbrary * DU Compute echncal Report obas K. S. Rtschel, Jozse Gaspar, Andrea Capole, John Bagterp Jørgensen Department o Appled Mathematcs and Computer Scence & Center or Energy Resources Engneerng CERE, echncal Unversty o Denmark, DK-800 Kgs. Lyngby, Denmark Abstract hs s a techncal report whch accompanes the artcle An open-source thermodynamc sotware lbrary whch descrbes an ecent Matlab and C mplementaton or evaluaton o thermodynamc propertes. In ths techncal report we present the model equatons, that are also presented n the paper, together wth a ull set o rst and second order dervatves wth respect to temperature and pressure, and n cases where applcable, also wth respect to mole numbers. he lbrary s based on parameters and correlatons rom the DIPPR database and the Peng-Robnson and the Soave-Redlch-Kwong equatons o state. Keywords: hermodynamc unctons, Sotware, Phase equlbrum 1. Introducton he purpose o ths techncal report s to document the equatons descrbng vapor-lqud enthalpy, entropy and volume o real and deal mxtures and pure components, together wth ther rst and second order dervatves wth respect to temperature and pressure, and or mxture propertes, also wth respect to mole numbers. For completeness, ths techncal report also descrbes logarthmc ugacty coecents together wth rst and second order dervatves. he ugacty coecents are dened by means o resdual propertes that are also descrbed n ths report. However, ther second order dervatves requre thrd order dervatves o the resdual propertes, whch are not descrbed n ths report. Secton presents DIPPR correlatons together wth derved expressons that are necessary n the deal gas and lqud models. Secton 3 presents deal gas propertes and Secton 4 presents deal lqud propertes. Secton 5 presents real mxture propertes based on the Peng-Robnson PR and the Soave-Redlch-Kwong SRK equatons o state.. DIPPR correlatons or pure components hs secton descrbes the DIPPR correlatons homson, 1996 or deal gas heat capacty, vapor pressure and lqud volume together wth necessary ntegrals and temperature dervatves. * hs project s unded partly by Innovaton Fund Denmark n the CIIES project B and n the OPION project , and partly by the nterreg project Smart Ctes Accelerator SCA. he man motvaton or ncludng second order dervatves o ugacty coecents s that they are necessary or ormulatng second order algorthms or dynamc optmzaton o lash processes.

3 Furthermore, we present all dervatves o these correlatons and derved propertes, that are necessary or the rst and second order dervatves n the deal gas and lqud models..1. Ideal gas heat capacty he deal gas heat capacty o the th component, c g P, = cg P,, s c g P, = A + B C snh C + D E cosh E 1 he parameters A, B, C, D, E are specc to each substance and also to the deal gas heat capacty correlaton. hey are provded by the DIPPR database. he unt or the molar deal gas heat capacty s J/kmoles K and the temperature must be n K. he ntegral o the deal gas heat capacty s used n computaton o deal gas enthalpy and s expressed usng the auxlary uncton Γ = Γ 1 0 C E Γ = A + B C coth D E tanh b c g P, d = Γ 1 Γ 0 a he ntegral o the deal gas heat capacty dvded by temperature s used or the deal gas entropy and s expressed usng the auxlary uncton Π = Π 1 0 g cp, d = Π 1 Π 0 C Π = A ln + B coth C E D tanh E ln cosh ln snh E C he only necessary dervatve o the deal gas heat capacty s the rst order temperature dervatve c g P, = C A c g B C P, + E E + D E tanh 4.. Vapor pressure tanh C snh C he vapor pressure or uraton pressure o the th component, P P ln P = exp ln P = A + B + C ln + D E cosh E = P, s he substance specc correlaton parameters A, B, C, D, E are provded by the DIPPR database and are specc to the vapor pressure correlaton. Because both the vapor pressure and ts rst order temperature dervatve appear n the deal lqud model, we present rst, second and thrd 3a 3b 5a 5b

4 order dervatves o the above correlaton. he dervatves o the vapor pressure are expressed through the dervatves o the logarthmc vapor pressure P P 3 P 3 = ln P P ln P = P = P 3 ln P 3 + ln P + 3 ln P ln P + 6a 6b ln P 3 6c he dervatves o the logarthmc vapor pressure correlaton are ln P = 1 C B + D E E ln P = 1 B C + D E E 1 E 3 ln P = 1 6 B C + D E E 1E E 7a 7b 7c.3. Lqud volume he lqud volume o the th component, v l = vl, s 1 C D v l = B1+ 8 A he substance specc correlaton parameters A, B, C, D are provded by the DIPPR database and are specc to ths correlaton. Because both the lqud volume and ts rst order dervatve appear n the deal lqud model, we present rst, second and thrd order temperature dervatves o the lqud volume correlaton. he dervatves are v l D C 1 C D 1 v l = ln B v l = ln B D 1 C D 1 + D 1 C C 1 3 v l = 3 ln B D + D 1 v l C C 1 C C v l D 1 C 1 C ln B D 1 + C C 1 C v l 9a 9b 9c 3. hermodynamc unctons or deal gases hs secton presents enthalpy, entropy and volume o deal gas mxtures and pure components. hese equatons are based on the reerence enthalpy and entropy o ormaton provded by the DIPPR database, together wth the correlaton or the deal gas heat capacty and the deal gas law. 3

5 3.1. Pure component propertes We descrbe the computaton o enthalpy, entropy and volume o a pure component Enthalpy he molar deal gas enthalpy, h g = h g, s a uncton o temperature only h g = h g 0, P 0 Γ 0 +Γ } {{ } ĥ g, 10 It s more ecent to store ĥ g, = h g 0, P 0 Γ 0 rather than storng h g 0, P 0 and recomputng Γ 0 at every evaluaton o h g. he dervatves o the molar enthalpy are h g = cg P, 11a h g g P, = c where the deal gas heat capacty and ts temperature dervatve are gven by 1 and Entropy he molar deal gas entropy, s g = s g, P, s a uncton o temperature and pressure 11b s g = Π R lnp + s g 0, P 0 Π 0 + R lnp 0 } {{ } 1 ŝ g, It s more ecent to store ŝ g, = s g 0, P 0 Π 0 + R lnp 0 than recomputng Π 0 at every evaluaton o s g. he rst order temperature and pressure dervatves o the molar deal gas entropy are s g g P, = c s g P = R P he second order dervatves o the molar deal gas entropy are 13a 13b s g = 1 dc g P, d 1 cg P, 14a s g P = R P 14b s g P = 0 14c where agan, the deal gas heat capacty and ts temperature dervatve are gven by 1 and 4. 4

6 Volume he molar deal gas volume, v g = v g, P, s v g = R P 15 he rst order dervatves o the deal gas molar volume are he second order dervatves are v g = R P v g P = R P 16a 16b v g = 0 17a v g P = R P 3 v g P = R P 17b 17c 3.. Mxture propertes We descrbe the computaton o volume, enthalpy and entropy o an deal gas mxture o N C components usng the molar propertes o each component and the composton o the gas mxture. he mxture contans n = {n } N C =1 moles o each component Enthalpy he deal gas mxture enthalpy, H g = H g, n, s he rst order dervatves are he second order dervatves are H g = n h g 18 H g H g H g = =1 n c g P, = NC =1 = h g k =1 H g = c g n P,k k H g = 0 n l 5 h g n 19a 19b 0a 0b 0c

7 3... Entropy he deal gas mxture enthalpy, S g = S g, P, n, s S g = n s g where the total amount o moles, N, and the vapor mole racton, y, are =1 R n lny 1 =1 y = n N N = =1 n a b he dervatves o the mxture entropy are S g S g P S g = 1 =1 = NR P n c g P, = s g k R lny k 3a 3b 3c he second order dervatves are S g = 1 c n g P, 1 cg P, S g = NR P P S g P = 0 =1 g P,k S g = sg k = c S g = sg k P P = R P S g δkl = R 1 n l N n l 4a 4b 4c 4d 4e Volume he deal gas mxture volume, V g = V g, P, n, s V g = NR P 5 6

8 he rst order dervatves o the deal gas mxture volume are V g V g P V g = NR P = NR P = R P 6a 6b 6c he second order dervatves are V g 4. hermodynamc unctons or deal lquds = 0 7a V g = NR P P 3 7b V g P = NR P 7c V g = R P 7d V g = R P P 7e V g = 0 n l 7 hs secton presents enthalpy, entropy and volume o deal lqud mxtures and pure components. hese are based on deal gas propertes, vaporzaton propertes and pressure correctng terms. he latter two are urther based on the DIPPR correlatons or vapor pressure and lqud volume, together wth ther rst order temperature dervatves Pure component propertes he propertes o pure component lquds are based on deal gas propertes at vaporzaton temperature and pressure, and P = P, as well as vaporzaton propertes. he vaporzaton enthalpy, h vap = h vap, entropy, s vap = s vap, and volume, v vap = v vap, are v vap s vap h vap = R P v l = P vvap = s vap 8a 8b 8c 7

9 where P = P s the vapor pressure 5 and v l = v l s the lqud volume 8. he dervatves o the vaporzaton volume are v vap = R P 1 P P vl 9a v vap = R P P 1 P P + P v l 9b he dervatves o the vaporzaton enthalpy are h vap h vap = s vap = s vap he dervatves o the vaporzaton entropy are s vap s vap = P vvap = 3 P 3 vvap he molar lqud uraton enthalpy, h h s + P + P + svap + s vap v vap v vap + P = h, and entropy, s = h v hvap = s v, P he dervatves o the molar lqud uraton enthalpy are h h v vap = s, are 30a 30b 31a 31b 3a s vap 3b = cg P, hvap = c g P, he dervatves o the molar lqud uraton entropy s s = sv = s v, P + sv P, P he molar lqud enthalpy, h d, P + s v P P, P 33 h vap 34 svap P + s v, P P 35 P s vap 36 = h d, P, entropy, s d = s d, P, at arbtrary pressure are h d s d = h = s + v l vl vl P P 8 P P 37a 37b

10 he molar deal lqud volume s gven by the DIPPR correlaton 8. he rst order dervatves o lqud enthalpy are h d = h h d P = vl vl v l P P v l vl P 38a 38b he second order dervatves are h d = h v l + 3 v l v l P P 3 + P v l vl P 39a h d P = 0 39b h d P = v l 39c he rst order dervatves o lqud entropy are s d = s s d P = vl he second order dervatves are s d 4.. Mxture propertes = s v l v l P P 3 v l v l P P 3 P P vl P 40a 40b 41a s d P = 0 41b s d P = v l 41c We present volume, enthalpy and entropy o an deal gas mxture o N C components usng the molar propertes o each component and the composton o the gas mxture. he mxture contans n = {n } N C =1 moles o each component Enthalpy he deal lqud mxture enthalpy, H d = H d, P, n, s H d = n h d 4 =1 9

11 he rst order dervatves are he second order dervatves are H d H d P H d = NC =1 = NC =1 = h d k H d = H d h d n n v l vl =1 h d n 43a 43b 43c 44a = 0 44b P H d NC P = v l n 44c H d = hd k =1 H d = v l k P n vl k k H d = 0 n l 44d 44e Entropy he entropy o an deal lqud mxture s gven by where the lqud mole racton, x, s S d, P, n = n s d =1 R n ln x 45 =1 x = n N and the total amount o moles, N, s gven by b. he rst order dervatves are 46 S d S d P S d = NC =1 =1 s d n NC = v l n = s d k R lnx k 10 47a 47b 47c

12 he second order dervatves are S d = S d =1 s d n 48a = 0 48b P S d NC P = v l n 48c =1 S d = sd k S d = vl k P S d = R n l δkl n l 1 N 48d 48e Volume he deal lqud mxture volume, V d = V d, n, s V d = n v l 49 =1 50 he rst order dervatves are he second order dervatves are V d = NC =1 n v l 51a V d = v l k 51b V d = =1 V d = vl k V d = 0 n l n v l 5a 5b 5c 5. hermodynamc unctons or real mxtures hs secton presents the enthalpy, entropy and volume o a real vapor or lqud mxture. hese are based on deal gas propertes and resdual propertes. he latter are obtaned rom ether o the cubc equatons o state, Soave-Redlch-Kwong SRK or Peng-Robnson PR 11

13 5.1. Mxture propertes We consder a vapor or lqud phase contanng N C components wth mole numbers n = {n } N C =1. he total amount o moles n the phase, N, and the mole racton o the th component, z, are gven by N = z = n N he molar enthalpy, h = h, P, n, and entropy, s = s, P, n, are =1 n 53a 53b h = h g + h R 54 s = s g + s R 55 where h g = h g, P, n and s g = s g, P, n are molar deal gas enthalpy and entropy, and h R, P, n and s R = s R, P, n are molar resdual enthalpy and entropy. he molar volume, v = v, P, n, s the soluton o ether the Peng-Robnson or the Soave-Redlch-Kwong equatons o state, both o whch are n the cubc orm P = R a m v v + ɛ v + σ where the scalars ɛ and σ are specc to each equaton o state but ndependent o the gven substances. In practce, the equaton o state 56 s solved or the compressblty actor Z = Z, P, n n whch case the molar mxture volume s v = RZ 57 P Appendx A presents a drect and an teratve approach or solvng the cubc equatons o state or the compressblty actor. he mxture parameters a m = a m, n and = n are obtaned wth van der Waals mxng rules where the mxng parameter a j = a j s a m = z z j a j =1 j=1 = z b =1 a j = 1 k j 56 58a 58b â j 59 he parameter â j = â j = a a j s ntroduced or convenence and the substance specc parameters a = a and b are determned rom the crtcal temperature, c,, and crtcal pressure, P c,, o the th component a = α r,, ω Ψ R c, P c, b = Ω R c, P c, 1 60a 60b

14 able 1: Parameters n the Soave-Redlch-Kwong and the Peng-Robnson equatons o state. Eq. ɛ σ Ω Ψ SRK PR where α r,, ω s a uncton o the reduced temperature, r, = / c,, and the acentrc actor, ω, gven by α r,, ω = 1 + mω 1 1/ r, he scalars Ψ and Ω are related to the equaton o state parameters, ɛ and σ, and ther values are shown n able 1. he uncton mω s a second order polynomum n the acentrc actor, ω, or the Peng-Robnson and Soave-Redlch-Kwong equatons o state and s gven by 61 m SRK ω = ω 0.176ω m PR ω = ω 0.699ω 6a 6b he cubc equaton o state 56 s solved or the compressblty actor Z = Z, P, n and s thereore rewrtten usng the thrd order polynomal q = qz q = Z 3 + d m Z m = 0 63 m=0 where the polynomal coecents {d m = d m A, B} m=0 are d = Bɛ + σ 1 1 d 1 = A Bɛ + σ + B ɛσ ɛ σ d 0 = AB + B + B 3 ɛσ 64a 64b 64c he dmensonless quanttes A = A, P, n and B = B, P, n are ntroduced or convenence and are gven by A = Pa m R B = P R 65a 65b 5.. Resdual enthalpy and entropy he molar resdual enthalpy and entropy are gven n terms o the our auxlary unctons = Z, B, g h = g h, n, g s = g s, n and g z = g z Z, B h R = h R, P, n = RZ ɛ σ g h s R = s R, P, n = Rg z + 1 ɛ σ g s 13 66a 66b

15 he auxlary unctons, = Z, B, g h = g h, n, g s = g s, n and g z = g z Z, B, are Z + ɛb = ln Z + σb g h = g s a m g s = 1 a m g z = lnz B 67a 67b 67c 67d he uncton = Z, B depends on the equaton o state parameters, ɛ and σ, whereas g h = g h, n, g s = g s, n and g z = g z Z, B do not. he rst order dervatves o the resdual enthalpy, h R = h R, P, n, are h R h R P h R Z = RZ 1 + R + 1 gh ɛ σ + g h = R Z P + 1 ɛ σ g h P = R Z + 1 ɛ σ gs + g h 68a 68b 68c he second order dervatves are h R h R Z = R + R Z + 1 ɛ σ = R Z P + 1 ɛ σ g h P P h R P = R Z P + R Z P + 1 ɛ σ h R = R Z + R h R P = R Z P + 1 Z + 1 h R = R Z + 1 n l n l ɛ σ g h + g h + g h gh P + g h P g h + g h ɛ σ gh ɛ σ P + g h P g h + g h + g h n l n l n l + g h + g h + g h n l he rst order dervatves o the resdual entropy, s R = s R, P, n, are s R = R g z + 1 gs ɛ σ + g s s R P = R g z P + 1 ɛ σ g s P s R = R g z + 1 gs + g s ɛ σ 69a 69b 69c 69d 69e 69 70a 70b 70c 14

16 he second order dervatves are s R s R = R g z + 1 ɛ σ = R g z + 1 g s + g s ɛ σ g s P gs P s R P = R g z P + 1 ɛ σ s R = R g z + 1 s R P = R g z P + 1 s R n l = R g z n l + 1 ɛ σ + g s P + g s P g s + g s + g s ɛ σ + g s gs ɛ σ P + g s P g s + g s + g s + g s n l n l n l n l 71a 71b 71c 71d 71e Fugacty coecents In ths report we also present an explct expresson or the logarthmc ugacty coecents {ln φ } N C =1 derved rom the resdual propertes 66. he expresson or the ugacty coecents s obtaned usng the auxlary unctons = Z, B, g z = g z Z, B and g φ, = g φ,, n where the auxlary uncton g φ, = g φ,, n s ln φ = Z 1 b g z 1 ɛ σ g φ, 7 1 g φ, = R b z j a j a m 73 j=1 he rst order dervatves are ln φ ln φ P ln φ = Z = Z P b g z b g z P 1 gφ, 1 ɛ σ ɛ σ g φ, P = Z b Z 1 b b m + g φ, g z 1 gφ, d + g φ, ɛ σ dn k 74a 74b 74c 15

17 he second order dervatves are ln φ ln φ P ln φ P = = Z b g z 1 ɛ σ = Z P b g z P 1 g φ, ɛ σ g φ, P Z b g z P P 1 ɛ σ ln φ = Z b Z 1 ɛ σ ln φ = Z b Z P P P 1 ɛ σ b b m gφ, g z + g φ, g φ, + g φ, d + g φ, d dn k gφ, ln φ = Z b n l n l g z n l 5.4. Auxlary unctons b g z b m P d dp + g φ, P Z + Z Z n l n l 1 ɛ σ + g φ, P + g φ, P b b m d + g φ, g φ, + g φ, d + g φ, d n l dn l n l + Z 1 b b m n l + g φ, dn k n l n l 75a 75b 75c 75d 75e 75 Note on nomenclature: In order to keep the dervatve normaton bre we ntroduce the auxlary varables w 1 and w. Each o these varables are a placeholder or ether temperature,, pressure, P, or a mole number, n k. We wll use these auxlary varables n cases where the structure o the dervatve equatons do not depend on the type o varable. he rst order dervatves o = Z, B wth respect to temperature, pressure and mole numbers are expressed through the dervatves wth respect to the compressblty actor Z = Z, P, n and B = B, P, n = Z Z + he second order temperature, pressure and composton dervatves are, w 1 {, P, n k } 76 w = Z 1 Z + = Z w Z + + Z Z w + Z 1 Z + B Z + + Z w w Z w B + Z + Z w Z w w 16 77a 77b

18 where w 1 {, P, n k } and w {, P, n l }. Note: w 1 and w can represent derent mole numbers, n k and n l, respectvely. he rst order dervatves o = Z, B wth respect to the compressblty actor Z and B are he second order dervatves are Z = 1 Z + ɛb 1 Z + σb = ɛ Z + ɛb σ Z + σb 78a 78b he rst order dervatves o g h = g h, n are Z = 1 Z + ɛb a Z + σb = ɛ σ + 79b Z + ɛb Z + σb Z = ɛ Z + ɛb + σ 79c Z + σb g h = g s g h = g s 1 am a m 80a 80b he second order dervatves are g h = g s + g s g h = g s g h = g s 1 n l n l + 1 b m a m n l am + a m + a m n l n l + n l n l 81a 81b 81c he rst order dervatves o g s = g s, n are g s = 1 a m g s = 1 a m 1 b m a m 8a 8b 17

19 he second order dervatves are g s = 1 3 a m 3 g s = 1 3 a m 1 g s = 1 n l b m b m a m [ n l n l a m n l n l am a m 3 ] a m + n m 83a 83b 83c he rst order dervatves o g z = g z Z, B wth respect to temperature, pressure and mole numbers are expressed through the dervatve wth respect to the compressblty actor Z = Z, P, n and B = B, P, n g z = g z Z Z + g z he second order temperature, pressure and composton dervatves are, w 1 {, P, n k } 84 g z w = g z Z + g z 1 Z Z + g z + g z B g z w = g z Z + g z Z w + g z 1 Z Z Z Z + g z + g z Z w w Z w B + g z Z + Z w Z w w 85a 85b where w 1 {, P, n k } and w {, P, n l }. Note: these dervatve equatons are structurally dentcal to the dervatves o he rst order dervatves o g z = g z Z, B wth respect to the compressblty actor Z and B are he second order dervatves are g z Z = 1 Z B g z = 1 Z B 86a 86b g z Z = 1 Z B 87a g z Z = 1 Z B 87b g z Z = 1 Z B 87c 18

20 he rst order dervatves o the auxlary uncton g φ, = g φ,, n are g φ, = 1 1 R a j z j a m b b j=1 m g φ, g φ, = 1 N a k x j a j b am 1 g φ, + 1 R j=1 he second order dervatves are g φ, = 1 1 R a j z j j=1 a m g φ, = 1 gφ, + 1 g φ, + 1 [ g φ, + 1 R N g φ, n l = 1 gφ, b g φ, a k NC j=1 + g φ, + n l n l n l 1 { a m a m 1 1 n l n l b m x j a j + 1 R 1 1 [ N [ am n l a m 1 n l a m 1 x j a j a l a k j=1 bm 88a 88b 89a b ], 89b ] ] + a m }b n l 89c 5.5. Compressblty actor he compressblty actor s mplctly dened by the cubc equaton 63 or a gven temperature pressure and composton. he rst order dervatves o the compressblty actor Z = Z, P, n are gven n terms o the dervatves o the polynomum q = qz gven n Z q q =, w 1 {, P, n k } 90a Z he second order dervatves are 1 Z q w = q 1 Z Z w = q Z 1 + q Z Z q w + q Z Z + q Z Z w + Z q Z Z w + q w Z Z 91a 91b where w 1 {, P, n k } and w {, P, n l }. he rst order dervatves o the cubc polynomum q = qz are expressed through the dervatves o the polynomal coecents {d m = d m A, B} m=0 q Z = 3Z + q = m=0 md m Z m 1 m=1 9a d m Z m, w 1 {, P, n k } 9b 19

21 he second order dervatves are q Z = 6Z + d q Z = q = q w = m=1 m=0 m=0 m d m Z m 1 d m Zm d m w Z m 93a 93b 93c 93d where w 1 {, P, n k } and w {, P, n l } Polynomal coecents he temperature, pressure and composton dervatves o the polynomal coecents are expressed through the dervatves wth respect to A = A, P, n and B = B, P, n. he rst order dervatves o {d m = d m A, B} k=0 are d m = d m A he second order dervatves are A + d m, w 1 {, P, n k } 94 d m w = d m A + d m 1 A A + d m + d m B d m = d m A A + d m w A w + d m B + d m A w A A w + d m 1 A + d m w A + A w w A A w 95a 95b where w 1 {, P, n k } and w {, P, n l }. Note: w 1 and w can represent derent mole numbers, n k and n l, respectvely. he rst order dervatves o the polynomal coecent d = d A, B wth respect to A and B are d A = 0 d = ɛ + σ 1 96b 96a 0

22 he second order dervatves are he rst order dervatves o d 1 = d 1 A, B are d A = 0 97a d = 0 97b d A = 0 97c he second order dervatves are d 1 A = 1 d 1 = ɛ + σ + ɛσ ɛ σb 98b 98a he rst order dervatves o d 0 = d 0 A, B are d 1 A = 0 99a d 1 = ɛσ ɛ σ 99b d 1 A = 0 99c he second order dervatves are d 0 A = B d 0 = A + ɛσb + 3B 100a 100b d 0 A = 0 101a d 0 = ɛσ + 6B 101b d 0 A = 1 101c 5.7. he quanttes A and B he rst order dervatves o A = A, P, n are gven n terms o the dervatves o the mxng parameter a m = a m, n A = a m A P = a m R A = a m P R A P R 1 10a 10b 10c

23 he second order dervatves are A = P a m R 1 a m 3 A A P = 0 A P = a m 1 R A P A = a m P R A A = a m 1 P R A = a m P n l n l R 103a 103b 103c 103d 103e 103 he rst order dervatves o B = B, P, n are gven n terms o the dervatves o the mxng parameter = n he second order dervatves are = P R P = R = P R B = P R 3 B P = 0 B P = R B = P R B = 1 P R B = P n l n l R 104a 104b 104c 105a 105b 105c 105d 105e Mxng parameters he dervatves o the mxng parameters a m = a m, n explot the symmetry o a j = a j = a j. he rst order dervatves are a m a j z z j =1 j=1 = NC 106a

24 a m = N z a k a m =1 106b he second order dervatves are a m = N C =1 j=1 z z j a j a m = a k N z a m =1 a m = 1 a kl + a m a m a m n l N 107a 107b 107c he relevant thrd order dervatves are 3 a m 3 = N C =1 j=1 3 a m = N 3 a m n l = 1 N =1 z z j 3 a j 3 z a k a m akl + a m a m a m 108a 108b 108c he dervatves o the mxng parameter = n are he dervatves o a j â j = â j = b k N n l = b l b k N 109a 109b = a j are gven n terms o the dervatves o the auxlary uncton a j = 1 k j â j â j a j = 1 1 k j â j 3 a j 3 = 1 k j 4 â j â j 1 â j 1 â j â j + â 3 j 3 110a â j 110b 1 â j a j â j â j a j c 3

25 he dervatves o the auxlary uncton â j = â j are gven n terms o dervatves o the pure component propertes a = a â j â j 3 â j 3 = a a a j j + a = a a j + a a j + a a j = 3 a a 3 j + 3 a a j + 3 a a j + a 3 a j 3 111a 111b 111c 5.9. Pure component propertes he pure component parameters a = a are drectly proportonal to α = α r,, ω and as such ther dervatves are he dervatves o α = α r,, ω are a = α ΨR c, P c, a = α c, ΨR P c, 3 a = 3 α c, 3 3 ΨR P c, α = α mω αc, α = 1 α 1 1 α α 3 α = 1 3 α 1 1 α + α 1 α α 1 α α α 1 11a 11b 11c 113a 113b 113c Appendx A. Soluton o cubc equatons here exsts a number o approaches or solvng the cubc equaton o state 56 or the roots when pressure and temperature are gven. hese approaches are ether drect approaches that use explct ormula or computng the roots, teratve approaches that approxmate the roots o nterest or a combnaton o both where the drect soluton s rened by an teratve approach n order to remove mprecson arsng rom roundng errors. In ths work we use an teratve approach as descrbed by Smth et al. 005 and compare to Cardano s approach whch s brely descrbed by Monroy-Loperena 01. he equaton o state 56 s rewrtten n terms o the compressblty actor Z = PV/R Z 3 Z 1 Bɛ + σ 1 Z ɛ + σ Bɛσ ɛ σ A/B B A + B1 + Bɛσ B = 0, 4 A.1

26 where A and B are gven by A = a m, np R B = Pn R he equaton o state A.1 s wrtten compactly qz = Z 3 + d 1 Z + d Z + d 3 = 0 A. A.3 A.4 Cardano s drect approach he number o real roots are determned by the two quanttes Q and R Q = d 1 3d /9 R = d 3 1 9d 1d + 7d 3 /54 A.5 A.6 here are three real roots R Q 3. In that case, the roots are ound by the ormula Z 1 = Q cosθ/3 d 1 /3 Z = Q cosθ + π/3 d 1 /3 Z 3 = Q cosθ π/3 d 1 /3 A.7a A.7b A.7c where θ s computed by θ = arccosr/ Q 3 A.8 I R > Q 3, there s one real root and two complex conjugate roots that are gven by Z 1 = S + d 1 /3 Z = 1/S + d 1 /3 + 3/S Z 3 = 1/S + d 1 /3 3/S A.9a A.9b A.9c where S = sgnr R + R Q 3 1/3 { Q/S S 0 = 0 S = 0 A.10 A.11 In the case o multple roots, the smallest represents the lqud phase compressblty actor, Z l = mn{z 1, Z, Z 3 }, and the largest s vapor phase compressblty actor, Z v = max{z 1, Z, Z 3 }. An teratve Newton approach he approach descrbed here uses Newton teratons to solve the cubc equaton A.1. It s possble to use hgher-order methods as dscussed by Olvera-Fuentes 1993, due to the cubc nature o the equaton. In the Newton approach, an ntal guess, Z 0, s teratvely mproved by Z k+1 = Z k qz k /q Z k q Z k = 3Z k + d 1Z k + d 5 A.1 A.13

27 he teratve sequence s termnated when both o the ollowng crtera are sed Z k+1 Z k < ɛ A.14 Z 3 k+1 + d 1z k+1 + d Z k+1 + d 3 < ɛ A.15 Once the sequence s termnated, a sngle root has been ound. he ollowng ntal estmates are used, dependng on whether the compressblty actor o the vapor phase, Z v, or o the lqud phase, Z l, s sought Z v 0 = 1 Z l 0 = B A.16 A.17 Reerences Monroy-Loperena, R., 01. A note on the analytcal soluton o cubc equatons o state n process smulaton. Industral & Engneerng Chemstry Research 51, Olvera-Fuentes, C., he optmal soluton o cubc equatons o state. Latn Amercan Appled Research 3, Smth, J.M., Van Ness, H.C., Abbott, M.M., 005. Introducton to Chemcal Engneerng hermodynamcs. 7th ed., McGraw-Hll, New York, NY. homson, G., he DIPPR R databases. Internatonal Journal o hermophyscs 17,