A Helmholtz energy equation of state for calculating the thermodynamic properties of fluid mixtures

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1 A Helmholtz eergy equato of state for calculatg the thermodyamc propertes of flud mxtures Erc W. Lemmo, Reer Tller-Roth

2 Abstract New Approach based o hghly accurate EOS for the pure compoets combed at the reduced T, ρ of the mxture reducg parameters for the T,ρ : composto For smple mxtures wth relatvely smple fuctos very accurate represetato For odeal mxtures modfed reducg fuc.(t, ρ)+departure fuc. 0.% desty, % Cv, % bubble pot pressure

3 Abstract Two applcatos of mxture model cocepts developed depedetly USA ad Germay over the same tme perod cludg the developmet of dvdual equato for each bary system geeralzato of the model for a wde varety of mxtures Smlartes ad dffereces

4 Itroducto Dmesoless Helmholtz eergy, α RT Thermodyamc propertes of pure fluds wth hgh accuracy over a wde rage of T, P All thermodyamc propertes from a sgle mathematcal relato A large umber of adjustable coeff. (up to 58 terms reported) Referece equato for the calculato of property tables ad chart accuracy : restrcted oly by the accuracy of avalable measuremets A

5 Itroducto Due to the complexty Equato wth may coeffcets appled oly to pure fluds Hgh-accuracy EOS developed for oly about 30 pure substaces Oly a few attempts to descrbe mxture propertes usg multparameter EOS

6 Itroducto Plocer et al. (978) oe-flud theory appled to modfed Beedct-Webb- Rub(mBWR) EOS wth a mxg rule for the pseudo-crtcal temperature VLE ad Ethalpes at hgh pressures Platzer ad Maurer (993) geeralzed the 20-term Beder equato predcto for multcompoets mxture propertes Huber ad Ely (994) exteded correspodg states model based o mbwr EOS

7 Itroducto Mxture model (pure mxture) based o hgh-accuracy pure flud EOS for the pure compoets pressure-explct EOS smple tegrato of pressure geerally explct Helmholtz eergy adopted for the mxture combato rules for the Helmholtz eergy fucto of mxture compoets (ot mxg rules) ot requred a fxed structure of the pure flud equato sgle mathematcal expresso

8 Itroducto Geeral form of Helmholtz eergy model depedetly developed by both authors durg overlappg perods of tme Tller-Roth (993) focused o developg a accurate formulato for dvdual bary mxtures for a large amout of accurate expermetal data Lemmo (996) focused o developg a geeralzed model capable of accurate property calculato for a large umber of flud mxtures

9 Itroducto Purpose addtoal detal ad bacgroud for the models especally deal mxture dfferece betwee the models

10 Helmholtz eergy EOS for the pure fluds deal part + resdual part reducg parameter T, V : crtcal propertes ( ) ( ) ( ) δ + α δ α δ α,,, r 0 RT A ( ) ( ) [ ] δ+ δ α exp l l l, m a c c c ( ) ( ) + + δ δ + δ δ α r exp, e m m c c V V T T ρ ρ δ,

11 Helmholtz eergy Trasformg to the Helmholtz eergy form by tegratg the relato p ρrt + δ α δ r

12 Mxture Model α of mxture : composto depedece α α 0 + α Ideal gas mxture : aalytcally from fuctos for the deal gas propertes of the pure fluds α 0 α 0 l 0 ( T, V, x) xα ( T, V ) + cosstet wth deal parts of the pure flud equato wth depedet reduced varable, δ r l x l x

13 Mxture Model Resdual Helmholtz eergy α r l r r (, δ, x ) α (, δ) + Δα (, δ, x) x lear combato of the pure flud resdual eergy departure fucto, Δα r teracto of dfferet speces the mxture Correspodg states prcple shape factors : ratos of the crtcal propertes T r T T h T T, r, r T, T r r T T

14 Mxture Model

15 Mxture Model Reducg parameter of the pure flud equato reducg fuctos of the mxture pseudo-crtcal parameter, δ composto depedece as x approaches ( x) T ( x ), δ( x) T T or ( x) T V ( x) ( x), δ( x) δ V V ( x),, V,

16 Lear mxture model Lear mxture model lear combatos of the pure flud parameters T ( x) xt,, V ( x) departure fucto Δα r omtted o adjustable parameters l l x V,

17 Lear mxture model : Applcato,,,2-tetrafluoroethae(R-34a) +dfluoromethae(r-32) close behavor of a deal soluto VLE : excellet results (Fg.a) Desty subcrtcal rage (Fg.2a) vapor, lqud desty : good agreemet Desty supercrtcal rage (Fg.2b) larger devato Isochorc heat capacty ±%(about the expermetal ucertaty)

18 Lear mxture model : Applcato water+ammoa mxture of polar compoets showg large mxg effects, wde two-phase rego VLE (Fg.b) dew curve : well hgh coc. the vapor phasedeal at lower pressures bubble curve : large devato Desty (Fg.2c) -% ~ +3% devato Excess ethalpy (Fg.3c) poor represetato : related to VLE

19 Lear mxture model : Applcato ethae+carbo doxde azeotropc behavor VLE (Fg.c) correct predcto Isochorc heat capacty (Fg.3b) less accurate >2% devato outsde of the crtcal rego

20 Lear mxture model : Applcato

21 Lear mxture model : Applcato

22 Lear mxture model : Applcato

23 Ehacemet to the Mxture Model Modfcato of the reducg fuctos T V ( ) x x x jt ( ), j T T + T j ( x) l l l l j x, j T,j, V,j : adjustable parameter x Effect of varyg T,j, V,j VLE of ethae+carbo doxde (Fg.4) ( T,j > V,j ) Cv :.4% devato (Fg.5) j V V, j, j T, j V, j 2 2, ( V + V ),, jj, jj

24 Ehacemet to the Mxture Model

25 Ehacemet to the Mxture Model

26 Ehacemet to the Mxture Model Modfcato by Tler-Roth supplemet the quadratc expresso wth β T, β V T V 2 2 βt ( x) x T, + x2t,2 + 2x2( x2 ) T,2 2 2 βv ( x) x V, + x2v,2 + 2x2( x2 ) V, 2 effect of expoets o the shape (Fg.6) fluece of T,2 or V,2 asymmetrc wth respect to composto Addtoal flexblty for tug of the mxture model

27 Ehacemet to the Mxture Model

28 Ehacemet to the Mxture Model Modfcato by Lemmo modfed the lear reducg fucto ζ j,ξ j,β j : modfy the shape of the reducg parameter β j, for ethae+co 2, ζ j -67.4K ( T,j 0.89) + + β ξ + ζ +,, ) ( ) ( l l j j j l l l j j j l x x V x V x x T x T j x x ( )( ) ( )( ),,,,,, + ξ + ζ j V j j j T j j V V T T

29 Departure fucto Departure fucto Δα r (, δ, x) x Δα (, δ,, ) x j j x x j oly to mxture propertes, o effect at the pure flud lmts magtude of departure fucto geerally oe order smaller tha the resdual α r modelg : regresso aalyss by Wager l l j + a set of of expermetal data lowest stadard devato ba of terms subset of term for Δα j optmzato : modfed Marquardt-Fletcher algorthm

30 Applcatos by Tller-Roth Itroducto a expoet γ the departure fucto for a bary system Δα r γ ( δ, x ) x ( x ) Δα (, δ, x, x ), 2 2 asymmetrc fluece very effectve for smple system, R-52+R34a reduce the umber of terms the departure fucto appled oly to bary systems to date For water+ammoa, xδ m exp ( e ) ( ) 2 m e δ ad x δ exp δ j j

31 Applcatos by Tller-Roth Results for R-34a+R-32 fve-term departure fucto Fg.7 ( cf. Fg.2b ) desty at lqud & vapor rego : ±0.% desty at crtcal rego : -6 ~ ±0.4% ( ) ( ) ( ) δ δ + δ δ α Δ γ exp,, e m m r c c x x x

32 Applcatos by Tller-Roth

33 Applcatos by Tller-Roth Results for water+ammoa 4-term departure fucto : 9 adjustable parameters Δα x r (, δ, x) γ ( x ) c δ + c 4 x 2 m δ m 4 c δ exp m exp e4 ( δ ) 3 e m e ( δ ) + x c δ exp( δ ) 7 Fg.8 ( cf. Fg.b, Fg.3c ) mproved represetato

34 Applcatos by Tller-Roth

35 Applcatos by Lemmo Geeralzed departure fucto Δα Δα r j (, δ, x) x x F Δα (, δ) (, δ) c,, m : fxed the same for all mxture l j + F j, ζ j, ξ j, β j : adjustable parameters from (p,v,t,x) 0.2% desty, % heat capactes 0 c -2% bubble pot pressure at crtcal temp. δ l 5-0% bubble pot pressure at crtcal temp. father apart m j j j

36 Applcatos by Lemmo Results propae + -butae : Fg.0 less tha 0.04% average absolute devato desty 0.4% average absolute devato bubble pot pressure Ucertaty of the equato reported 0.2% desty % heat capactes Fg. : terary refrgerat mxture

37 Applcatos by Lemmo

38 Applcatos by Lemmo

39 Coclusos EOS for mxture terms of the Helmholtz eergy has bee developed ad appled to several flud systems by two dfferet research groups By Tller-Roth, model of sgle bary systems for whch comprehesve accurate data are avalable By Lemmo, geeralzed model was preseted whch accurately predcts thermodyamc propertes of mxtures Further research for more odeal systems ad complex multcompoet mxtures

PREDICTION OF VAPOR-LIQUID EQUILIBRIA OF BINARY MIXTURES USING QUANTUM CALCULATIONS AND ACTIVITY COEFFICIENT MODELS

PREDICTION OF VAPOR-LIQUID EQUILIBRIA OF BINARY MIXTURES USING QUANTUM CALCULATIONS AND ACTIVITY COEFFICIENT MODELS Joural of Chemstry, Vol. 47 (5), P. 547-55, 9 PREDICTIO OF VAPOR-LIQUID EQUILIBRIA OF BIARY MIXTURES USIG QUATUM CALCULATIOS AD ACTIVITY COEFFICIET MODELS Receved May 8 PHAM VA TAT Departmet of Chemstry,