Modelling of Phase Equilibria for Supercritical Carbon Dioxide System by Hole-Theory Equation of State with Local Composition Model

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1 Modellng of Phase Equlbra for Supercrtcal Carbon Doxde System by Hole-Theory Equaton of State wth Local Composton Model Yusuke Shmoyama*, Yuta Yokozak Department of Chemcal Scence and Engneerng, Tokyo Insttute of Technology, S , Ookayama, Meguro-ku, Tokyo, , Japan. * yshmo@chemeng.ttech.ac.jp Phase equlbra for mxture ncludng supercrtcal carbon doxde are modelled by an equaton of state based on hole-theory wth local composton model. A partton functon n the hole-theory s composed of contrbutons from combnatoral, free volume and molecular nteracton. The local composton usng Non-random two lqud (NRTL) model s nstalled nto the contrbuton from the molecular nteracton. The vapor lqud equlbra for supercrtcal carbon doxde + water or organc solvent bnary system s calculated by the equaton of state. The model proposed n ths work results n the calculaton of the vapor lqud equlbra n case of polar solvent system better than those by the equaton of state wth mean-force feld. The partton coeffcent of sold solute between supercrtcal carbon doxde and water phases s also calculated. The equaton of state wth the local composton model s used for understandng the local soluton structure usng the calculated local composton. INTRODUCTION Supercrtcal carbon doxde has attracted as tuneable solvent for reacton, separaton or materal producton process because of ts unque propertes, hgh solublty, hgh dffusvty, low vscosty and low surface tenson. Many research groups have studed the applcaton of supercrtcal carbon doxde for the organc synthess, sol-gel reacton, extracton, dryng, nanopartcle productons as solvent or ant-solvent system and flm deposton. A fundamental knowledge of the phase equlbra for supercrtcal carbon doxde can be a crucal and useful for the process desgn and operaton optmzaton on the supercrtcal carbon doxde applcatons. Modellng of the phase equlbra for supercrtcal carbon doxde system has been nvestgated very well usng an equaton of state [1-3] and soluton model [4-6]. Cubc equaton of state has been often appled for the calculaton of the vapor lqud equlbra, sold lqud gas equlbra for supercrtcal carbon doxde systems and the solublty n supercrtcal carbon doxde.. The modellng usng the cubc equaton of state could nduce to the lmtaton that few crtcal propertes of the targeted compounds are avalable. Recently, the equaton of state based on statstcal flud assocaton theory (SAFT) s also utlzed for the phase equlbrum modellng for the supercrtcal carbon doxde systems [7,8]. Usng the SAFT equaton of state, many parameters are requred for the phase equlbrum calculaton. A partton functon based on hole-theory can be extended to the equaton of state that s used for the phase equlbrum modellng for the hgh pressure system. Haruk et al. reported the modellng of the phase equlbra for mxtures contanng water at hgh temperatures and pressure [9] and gases compound systems [10]. The phase equlbrum model based on the

2 hole-theory s expected to represent the dramatc densty change of the supercrtcal carbon doxde for both gas- and lqud-lke flud phase. Our prevous researches have been also nvestgated on the modellng of the solublty n supercrtcal carbon doxde system [11,12]. A cluster structure around the solute or solvent molecule by carbon doxde at supercrtcal flud phase has been known well and studed for a long tme. It s beleved that the cluster structure around solute or solvent molecule affects the phase equlbra for the supercrtcal carbon doxde system. The contrbuton from the molecular nteracton n the conventonal phase equlbrum model based on the hole-theory [9-13] s explaned by a mean-force feld system wthout consderng the effect of the clusterng. Ths work demonstrates the development of the phase equlbrum model based on the hole-theory wth the local composton model n consderaton of the cluster structure. The phase equlbrum model s appled for the calculaton of the vapor lqud equlbra for supercrtcal + solvent bnary system and the partton coeffcent of the sold solute between supercrtcal carbon doxde and water phases. The local compostons n the supercrtcal carbon doxde system are also gven and the effect on the phase equlbra s nvestgated for understandng the effect of the local cluster structure on the phase equlbra. EQUATION OF STATE BASED ON HOLE-THEORY A partton functon from hole-theory [13] s gven by the followng equaton. Q = g v f 3 Λ N exp E kt (1) The partton functon s composed of the combnatoral term g, free volume v f and molecular nteracton energy E. Λ means the de Brogle thermal wavelength, k the Boltzmann constant, N the number of molecule and T temperature. The subscrpt denotes the compound. The equaton of state presentng the pressure volume temperature relaton s gven as follows. p = kt lnq V T (2) p and V are the pressure and volume. The equaton of state usng the mean-force fled can be derved from the equatons (1) and (2) as follows. pv RT c = vs =1 S v ref x c v S S = x v v S v ref ln v vs + cvs v v v v S reft r S vt (3) (4) (5)

3 T r = 1 ( ) v S S S ( v ref ) 2 x j v S S ( j x j )u j (6) where v, v S, v S and v ref S mean the molar volume, the sold molar volume of the system, molecule and the reference lattce cell, respectvely. c and u j are the external degree of freedom and the par-nteracton energy between molecules and j. We are applyng the local composton model based on Non-random two lqud (NRTL) model [14] nto the hole-theory equaton of state for representng the cluster structure formed n supercrtcal carbon doxde system. The equaton of state combned wth NRTL local composton s gven as follows. pv RT T r = z 2k vs =1 S v ref x v S v ref j β j = exp z αε j 2 kt j ln v vs + cvs v v v vs T r S vt v S j x j ε j β j v j S x j β j (7) (8) (9) The external degree of freedom c and the sold molar volume of the system v S are calculated n equatons (4) and (5) as well as that n case of the mean-force feld. α s a non-randomness factor, set to The chemcal potental of the molecule s also requred for the phase equlbrum calculaton. The followng equaton gves the chemcal potental derved from the partton functon. µ = kt lnq N T,V,N j (10) The parameters requred for the phase equlbrum calculaton by the equaton of state are the external degree of freedom and the sold molecular volume for molecule, c and v S S and the par-nteracton energy u j and ε j n case of the mean force feld and the local composton model, respectvely. The sold molar volume of molecule s calculated from a quantum calculaton based on conductor-lke screenng model [15]. The external degree of freedom s determned from the data fttng to the vapor pressure for pure compound. The par-nteracton energy u j and ε j n the mean-force feld and local composton model are obtaned from the fttng to the phase equlbrum data for vapor lqud equlbra or partton coeffcent n supercrtcal carbon doxde system. RESULTS AND DISCUSSION Vapor lqud equlbra for supercrtcal carbon doxde + solvent system Vapor lqud equlbra for supercrtcal carbon doxde + ethanol, + dmethyl sulfoxde and (DMSO) + water are calculated by the equaton of state based on the hole-

4 theory combned wth the local composton model. Fgures 1 to 3 shows the calculated results of the vapor lqud equlbra for supercrtcal carbon doxde + ethanol, + DMSO and + water, respectvely. The calculated results are evaluated by the averaged relatve devatons between the expermental and calculated results as follows. δ x = 1 calc, x exp, 1 x 1 d N 100, δ = 1 calc, y exp, 1 y 1 y N 100 d x y x 1 exp, y 1 exp, (11) The superscrpts calc and exp mean the calculated and expermental results. x and y are the mole facton n lqud and vapor phases. The values of the averaged devatons between the expermental and calculated results are also lsted n Table 1. Table 1. Results of vapor lqud equlbra for carbon doxde (1) + solvent (2) system Solvent mean-force feld local composton model δ x δ y δ x δ y ethanol DMSO water The calculated results by the local composton model represent the expermental data more accurate those by the mean-force felds, except for the carbon doxde mole fracton n vapor phase for carbon doxde + DMSO system. Especally, the calculated results for carbon doxde + water system mprove hghly by applyng the local composton model to the hole theory equaton of state. The carbon doxde + water system could form the complex molecular nteracton ncludng the hydrogen bondng of water molecules and the cluster by carbon doxde molecule. The results n Table 1 can explan that the local composton model s effectve for the phase equlbrum calculatons for the system ncludng the polar molecules wth formng cluster, such as water.

5 Fgure 1. Vapor lqud equlbra for carbon doxde (1) + ethanol (2) system. Exp. Data [16-19]. black : 313 K, red : 333 K, blue : 348 K and green : 353 K. Sold lne ; calculate results by local composton model. Fgure 2. Vapor lqud equlbra for carbon doxde (1) + DMSO (2) system. Exp. Data [20,21]; black : 318 K, red : 328 K, blue : 338 K and green : 348 K. Sold lne ; calculate results by local composton model.

6 Fgure 3. Vapor lqud equlbra for carbon doxde (1) + water (2) system. Exp. Data [22-26]; black : 313 K, red : 323 K, blue : 333 K and green : 353 K. Sold lne ; calculate results by local composton model. Partton coeffcent between supercrtcal carbon doxde and water phases Partton coeffcents of caffene and vanlln between supercrtcal carbon doxde and water phases are calculated from the hole theory equaton of state wth local composton models. The sold molar volume of caffene and vanlln are obtaned from the molecular volume by the quantum calculaton wth conductor-lke screenng model [15]. The external degree of freedom for caffene and vanlln are calculated from the correlaton wth the molecular volume as proposed n the lterature [11]. The partton coeffcent of the solute (3) between carbon doxde (1) and water (2) phases s defned by the followng equaton. y 3 K 3 = (12) x3 Fgures 4 and 5 gves the calculated results of the solute mole fracton n water phase and the partton coeffcent between supercrtcal carbon doxde and water phase, respectvely. The calculated results of the mole fracton n water phase by the local composton model are n agreement wth expermental results. On the calculated results of the partton coeffcent, the local composton model represent the expermental results more accurate than that by the mean-force feld verson.

7 Fgure 4. The calculated results of the mole fracton of solute (3) between supercrtcal carbon doxde (1) and water (2) phases at 313 K. caffene (left) and vanlln (rght). Expermental data from lterature [27]. Calculate results; dotted lne (mean-force feld) and sold lne (local composton model). Fgure 5. The calculated results of partton coeffcent of solute (3) between supercrtcal carbon doxde (1) and water (2) phases at 313 K. caffene (left) and vanlln (rght). Expermental data from lterature [27].Calculate results; dotted lne (mean-force feld) and sold lne (local composton model).

8 We are also demonstratng the local mole fracton of the carbon doxde and water around the solute calculated from the local composton model as shown n Fgure 6. The local mole fracton of carbon doxde around vanlln n lqud phase s larger than that around caffene. On the other hand, the local mole fracton of water around vanlln s smaller than that around caffene. The results of the local mole fracton s explan that the vanlln can form the molecular nteracton wth carbon doxde stronger that that wth water. The partton coeffcent of vanlln over than unty would be resulted from the molecular nteracton wth carbon doxde or water molecule n the lqud phase. Fgure 6. Local mole fracton of carbon doxde (1) (left) and water (2) around solute (3) n lqud phase at 313 K. CONCLUSION The local composton model based on NRTL theory s appled for the hole theory equaton of state forward the modellng of the phase equlbra for supercrtcal carbon doxde systems. On the vapor lqud equlbrum calculatons, the local composton model could be effectve on the calculaton accuracy for the system ncludng the polar molecules formng the cluster or specfc molecular nteracton. The calculaton of the vapor lqud equlbra for supercrtcal carbon doxde + water system s mproved by applyng the local composton model nto the equaton of state. The partton coeffcent of the solute between supercrtcal carbon doxde and water phases s also calculated by the equaton of state combned wth the local composton model. It s found that the equaton of state wth the local composton model represent the expermental results of the partton coeffcent more accurate than that calculated by that wth the mean-force feld. The local composton model s also used for the calculaton of the local mole fracton of carbon doxde and water molecules around the solute n the lqud phase. The nformaton of the local mole fracton can assst to not only knowng the molecular structure n the lqud phase for supercrtcal carbon doxde + water two phase system but also understandng the mechansm of the partton of the solute between both phases.

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