Considering the dispersive interactions in the COSMO-SAC model for more. accurate predictions of fluid phase behavior. Abstract

Transcription

1 Consderng the dspersve nteractons n the COSMO-SAC model for more accurate predctons of flud phase behavor Cheh-Mng Hseh, a,* Shang-Ta Ln, b Jadran Vrabec a a Thermodynamcs and Energy Technology, Unversty of Paderborn, Paderborn, Germany b Department of Chemcal Engneerng, Natonal Tawan Unversty, Tape, Tawan Abstract A term to consder the contrbuton of the dspersve nteractons to the non-dealty of mxtures s ntroduced nto the COSMO-SAC model on the bass of molecular smulaton data from classcal model force felds. Ths dsperson term s a one-constant Margules equaton, where the constant s determned from the molecular dsperson parameter of the components. Furthermore, an atomc contrbuton method s proposed to calculate the dsperson parameter for a gven molecule. For bnary systems contanng molecules consstng of C, H, N, O, F and Cl atoms, a total of 13 global parameters s ntroduced wth the COSMO-SAC-dsp model. These parameters are obtaned from regresson to a large tranng set of bnary vapor-lqud equlbrum (VLE) data from experment. The overall devatons for VLE calculatons on ths tranng set are reduced by 25% n terms of the vapor pressure and 12% n terms of the vapor phase mole fracton. Ths dsperson term can provde a sgnfcant mprovement for nfnte dluton actvty coeffcent predctons, where the accuracy was ncreased by around 33%. Keywords: Phase equlbra, nfnte dluton actvty coeffcent, predcton, dspersve nteracton, COSMO-SAC 1

2 *To whom correspondence should be addressed: Emal Hghlghts A dsperson contrbuton to the actvty coeffcent calculaton s ntroduced nto the COSMO-SAC model Ths contrbuton s based on molecular smulaton data for classcal model force felds All parameters n the proposed term are adjustment to VLE data and the overall devatons are reduced by 25% n terms of the vapor pressure and 12% n terms of the vapor phase mole fracton The accuracy of nfnte dluton actvty coeffcent predctons s mproved sgnfcantly, the overall devaton s reduced by about 33% 2

3 1. Introducton Informaton on the dstrbuton of chemcal speces between coexstng phases s of fundamental mportance n the chemcal ndustry, envronmental engneerng and the pharmaceutcal ndustry [1-5]. Whle such nformaton can be obtaned from expermental work, t s crucal to have relable predctve thermodynamc models because of the sgnfcant tme and cost assocated wth measurng expermental data. Many researchers have thus expended sgnfcant efforts n establshng methods to predct the thermodynamc propertes and the phase behavor of pure fluds and mxtures [3-6]. Due to the advances n computng power, methodologcal effcency and the development of accurate force felds (whch descrbe the nter- and ntramolecular nteractons), numercal smulaton methods (Monte Carlo or molecular dynamcs) can be used to study many dfferent problems, ncludng the estmaton of thermophyscal propertes and phase equlbra [6-10]. Molecular smulaton methods are applcable wth very few constrants, as both equlbrum or non-equlbrum condtons can be studed and both statc and dynamc thermodynamc data can be sampled [11]. Furthermore, wth a good descrpton of the molecular nteractons through force felds, molecular smulaton can provde very accurate predctons of the thermophyscal propertes ncludng phase equlbra for pure fluds and mxtures. In addton to accurate results, molecular smulaton also provdes an nsght nto the mechansms on the mcroscopc level and detaled nformaton about the effect of dfferent nteractons. However, such smulatons requre a large computatonal effort (whch translates to hours for a typcal vapor-lqud equlbrum state pont on today s computer equpment) to obtan results [12]. The scope of predctve thermodynamc models can range from smple cubc 3

4 equatons of state (EOS), excess Gbbs energy (G ex ) models, and ther combnatons through G ex -based mxng rules [3-5]. Through research over the past two decades, a new class of predctve methods has emerged that utlzes the results from computatonal chemstry. These COSMO-based methods (such as COSMO-RS [13-15] and ts varants COSMO-SAC [16-19], COSMO-RS(Ol) [20] or COSMO-vac [21]), determne the lqud-phase non-dealty on the bass of the molecular surface charge dstrbuton derved from frst-prncples solvaton calculatons [22,23]. Ths type of models does not contan any speces-dependent parameters and flud phase equlbra can be predcted wth the molecular structure as the only nput. Thus, COSMO-based methods do not suffer from the problem of mssng parameters. Wth the advantage of relyng only on a few global and atomc parameters, these methods have become more mportant and are broadly appled to the predcton of the thermophyscal propertes and the phase behavor of pure fluds and mxtures [1,24]. Wth a combnaton of cubc EOS and G ex -based mxng rules, the applcaton of COSMO-based models can be extended to hgh pressure systems ncludng supercrtcal fluds [25-30]. In precedng works, the electrostatc nteractons of the COSMO-SAC model were revsed to provde good predctons for phase equlbra of organc mxtures [19,31]. However, the neglect of the dspersve nteractons n actvty coeffcent calculaton for mxtures remans unsolved n the COSMO-based models. The dspersve nteracton s the most general attractve ntermolecular nteracton n the condensed phase. It s caused by spontaneous temporary polartes n ndvdual molecules that polarze ther neghbors. The net effect of dspersve nteractons s a short-range and weak attracton (compared wth other molecular nteractons) whch, however, domnates n non-polar lquds. Ths nteracton s expressed as a functon of molecular propertes (e.g., exposed surface area or surface tenson) n some 4

5 thermodynamc models [13,32-34], whereas the composton of the solutes and solvents s taken nto account n other thermodynamc models [35]. In the COSMO-based methods, the dspersve nteracton s consdered as the functon of exposed surface area and atomc type of surface segments. Furthermore, ths nteracton s taken nto account only for the determnaton of vaporzaton data (such as vapor pressure or enthalpy of vaporzaton of pure fluds) [16,17,36]. For mxtures, dspersve nteracton was neglected untl now n the development of the COSMO-SAC model [16]. In ths study, a new dsperson term to consder the contrbuton of the dspersve nteracton to the non-dealty of mxtures s ntroduced nto the COSMO-SAC model on the bass of molecular smulaton data from classcal model force felds. It s termed COSMO-SAC-dsp model. 2. COSMO-SAC model In the orgnal publcaton of the COSMO-SAC model [18], the actvty coeffcent of molecule n mxture S can be determned from * chg * chg G / S G / SG ln / S ln / S, (1) RT where s the solvaton chargng Gbbs energy of molecule n solvent j (j= for chg G * / j pure solvent and j=s for mxture S). The Staverman-Guggenhem (SG) combnatoral term [37,38] was used to consder molecular sze and shape effects SG z ln / S ln q ln l x j jl j, (2) x 2 x wth xq, x q j j j xr z, l ( r q ) ( r 1), where z = 10 s the x r 2 j j j 5

6 coordnaton number, whle q and r are the normalzed surface area and volume of component (the standard surface area and volume for normalzaton were Å 2 and Å 3 ). Accordng to Ben-Nam s defnton [39], the solvaton Gbbs energy G *sol s the Gbbs energy dfference of a molecule (solute) n ts deal gas phase state and n a real soluton (solvent) at the same temperature and pressure. The solvaton Gbbs energy s usually determned from two contrbutons: cavty formaton Gbbs energy G *cav and chargng Gbbs energy G *chg. These are based on a hypothetcal solvaton process: frst, a hard solute molecule s nserted nto the solvent and then the nteractons between solute and solvent are turned on. Accordng to the prevously developed solvaton model [17,33,40-42], the chargng Gbbs energy of solute n the mxture solvent S s calculated from four Gbbs energy contrbutons: deal solvaton (s), chargng-averagng correcton (cc), restorng (rst) and dsperson (dsp) G G G G G. (3) * chg * s * cc * rst / S / S * dsp / S accounts for the energy dfference of molecule n the deal conductor and n the s G * deal gas state. consders the energy shft due to the charge-averagng process cc G * for the molecule. The charge-averagng correcton s requred because parwse nteractons between ndependent segments are assumed n the COSMO-SAC model [17,18]. Snce these two terms are obtaned from quantum mechancal calculatons consderng only molecule, they are pure component propertes and cancel out n the actvty coeffcent calculaton. Then, Eq. (1) becomes * rst * rst * dsp * dsp G / S G / G / S G / SG ln / S ln / S, RT RT res dsp SG ln / S ln / S ln / S. (4) 6

7 The resdual term ln manly consders the permanent electrostatc res / S nteractons between molecules n the mxture and s obtaned by ln G G A a, OH, OT t t t t t p ( )ln S m / ( ) m / * rst * rst nhb res / S / t / S ( m ) RT eff t m, (5) where A s the molecular surface area and a eff s the surface area of a standard surface segment. The -profle p ) s a hstogram of surface area normalzed by ( m molecular surface area wth a screenng charge densty of m for molecule. In the COSMO-SAC model, for a better descrpton of the hydrogen-bondng (hb) nteractons, the molecular surface segments are categorzed nto three types: non-hydrogen-bondng (nhb), hydroxyl groups (OH) and other hydrogen bondng groups (OT),.e. O, N, F and H bound to N and F. Therefore, the -profle of molecule nhb OH OT s the summaton of these three contrbutons, p ( ) p ( ) p ( ) p ( ) [19]. The segment actvty coeffcent ( ) of a segment wth surface charge / j m densty m n soluton S (j = S) or n ts pure flud state (j = ) s determned by nhb,oh,ot t s t t s s s s W ( m, n ) ln / j ( m ) ln p j ( n ) / j ( n )exp, (6) s n RT t s where W (, ) descrbes the electrostatc nteracton between surface segment of m n type t wth screenng charge densty m and surface segment of type s wth screenng charge densty n. It s calculated by t s 2 t s t s 2 c (, )( t s W (, ) c, (7) m n ES m n hb m n m n ) wth the electrostatc nteracton coeffcent c ES /(kcal mol -1 Å 4 e -2 ) = /(T/K) 2 and hydrogen bondng nteracton coeffcents 7

8 t s c hb (, m )/(kcal mol -1 Å 4 e ) n t f s t OH and t f s t OT and s s t f s OH, t OT and otherwse. m m n n m 0 0 s n 0 (8) The resdual contrbuton was taken from precedng work on COSMO-SAC wthout modfcaton or parameter optmzaton. Detals can be found n Ref. [19]. In the development of the COSMO-SAC model, the dsperson contrbuton to the chargng Gbbs energy for solute n solvent j was estmated by usng a dsp G * / j frst-order mean feld approxmaton to consder all possble parwse nteractons between the atoms n dfferent molecules [17]. Because the dfference of ths contrbuton n ts pure flud state (/) and n a mxture (/S) s small, the dsperson contrbuton to the actvty coeffcent ln s also small and was assumed to be zero dsp / S untl now n the COSMO-SAC model. However, t s shown n the present study that ths small dsperson contrbuton does nfluence the accuracy of flud phase behavor predcton sgnfcantly. In ths study, the molecular smulaton tool for thermodynamc propertes ms2 [12] was used to study the contrbuton of the dspersve nteractons to the actvty coeffcent. The role of the dspersve nteracton was quantfed by a seres of molecular smulatons for bnary mxtures composed of Lennard-Jones model fluds (summarzed n Table 1). Ths type of model mxtures was chosen because t contans only repulsve and dspersve nteractons (the resdual term becomes zero). As lsted n Table 1, all Lennard-Jones fluds consdered n ths study have the same sze and shape (spheres wth a constant sze parameter ) n order to study the dsperson contrbuton n the COSMO-SAC (2010) model (the combnatoral terms become 8

9 zero). Dfferent bnary Lennard-Jones model mxtures, where the rato of energy parameters 1 / 2 was vared from 1.2 to 1.8, were smulated to study the effect of dfferent strengths of the dspersve nteracton on the non-dealty of the mxture. The molecular smulaton results for the excess Gbbs energy for bnary mxtures of Lennard-Jones fluds are shown n Fgure 1. These smulaton results can be descrbed by the one-constant Margules equaton G ex Ax x 1 2, (9a) RT wth constant A determned from [43,44] 1 A w , (9b) where w = 0.275; 1 and 2 are the nteracton energy parameters of the two Lennard-Jones model components as summarzed n Table 1. Snce the one-constant Margules equaton can descrbe these model systems that are domnated by dspersve nteracton, t was used to consder the contrbuton of the dspersve nteracton to the actvty coeffcent n the COSMO-SAC model. Only bnary mxtures were regarded n ths work, but t s straghtforward to extend the Margules equaton to multcomponent mxtures [3,5,45]. For bnary mxtures contanng the components 1 and 2, the actvty coeffcent due to the dsperson contrbuton for substances 1 and 2 are then ln Ax, (10a) dsp ln Ax, (10b) dsp where x s the mole fracton of component n the lqud phase and the constant A was determned from the molecular dsperson parameters 1 and 2 va Eq. (9b) wth 9

10 , f (a) water hb - only - acceptor (b) COOH nhb or hb - donor - acceptor w (11) (c) water COOH 0.275, otherwse, where hb-only-acceptor denotes substances that are able to form a hydrogen bond by acceptng a proton from ts neghbor, such as ethers, esters, ketones and ntro compounds; hb-donor-acceptor s substance that s able to form hydrogen bonds by ether provdng a proton or acceptng a proton from ts neghbors, such as alcohols and amnes; COOH ndcates substances wth a carboxyl group, such as carboxylc acds and benzoc acds. The molecular dsperson parameter Molecule for real molecules was determned from an atomc contrbuton approach n 1 N, (12) Molecule N Atom j1 j j where j s the dsperson parameter of atom j, N j the number of type j atoms, n the total number of atoms n the molecule and N Atom the total number of atoms for whch the dsperson parameter s not zero. Currently, only hydrogen atoms were excluded n N Atom, f they are not bound to oxygen, ntrogen and fluorne atoms. Ths treatment s smlar to the coarse-graned modelng n molecular smulatons where the effect of a hydrogen atom s taken nto account through other heaver atoms connected to t. The average of the atomc dsperson parameter over all atoms of a molecule, nstead of the sum of t, s used to determne the molecular dsperson parameter for a molecule because we wsh not to nclude the sze effects, whch has already been taken nto account by the Staverman-Guggenhem combnatoral term. For example, the molecular dsperson parameter of n-hexane would be twce larger than that of n-propane when the sum of the dsperson nteractons was used. Ths dsperson term 10

11 was then too large and deterorated the predcton accuracy. Our choce of the one-constant Margules equaton, whch s approprate for equally szed molecules, also helps exclude the effects from molecular sze and shape dfferences n the dsperson term. The values of the atomc dsperson parameter j, as summarzed n Table 2, were optmzed here to a large set of expermental vapor-lqud equlbrum (VLE) data of bnary mxtures. Detal dscusson s gven below. 3. Computatonal detals and parameter optmzatons The computatonal detals and all parameter values for the resdual and combnatoral contrbutons are the same as those for the COSMO-SAC (2010) model, detals are gven n Ref. [19]. The freely avalable cosmo-fle database (VT-database) contanng over 1400 compounds from Lu s group at Vrgna Tech [46,47] was used n ths work. However, the cosmo-fles of some compounds n the VT-database were updated by Sandler s group at the Unversty of Delaware and were used n ths work. It should be noted that the VT-database was establshed usng the quantum mechancal program DMol3 n Materals Studo, ncludng the COSMO solvaton calculaton [22,23]. Dfferent quantum mechancal packages provde dfferent cosmo-fles (output from COSMO calculatons) and often the accuracy of the predctons may become worse when cosmo-fles from other quantum mechancal packages are used [48-50]. Therefore, t s recommended that a re-optmzaton of the COSMO-SAC parameters may be necessary n such cases. The calculaton of the dsperson contrbuton to the actvty coeffcent was as follows: Frst, the molecular dsperson parameter Molecule was obtaned from Eq. (12). 11

12 Then, the actvty coeffcent due to the dsperson contrbuton was determned for both components n the bnary mxture from Eqs. (9) to (11). In ths study, only substances composed of carbon, hydrogen, ntrogen, oxygen, chlorne and fluorne atoms were consdered. Because an atomc contrbuton approach was used to estmate the molecular dsperson parameter for real molecules, a total of 13 global parameters was ntroduced n the COSMO-SAC-dsp model. In addton, the type of hybrdzaton was consdered for carbon, ntrogen and oxygen atoms. Hydrogen atoms were taken nto account only when they are bound to oxygen, ntrogen or fluorne atoms,.e. f they are capable to form a hydrogen bond. These 13 global parameters were obtaned from optmzaton to expermental VLE data usng the followng objectve functon 2 M calc expt 1 P j Pj calc expt 2 Obj y1, 1, expt j y j, (13) M 1 j Pj where M s number of VLE data ponts, P the vapor pressure and y 1 the vapor phase mole fracton of component 1 n the mxture; the superscrpts calc and expt ndcate the calculated results and the expermental data, respectvely. All expermental VLE data were retreved from the DECHEMA Chemstry Data Seres [51-53]. The parameter optmzaton of these 13 global parameters was carred out as follows: Frst, the ten dsperson parameters for carbon, ntrogen, oxygen, chlorne and fluorne were optmzed to expermental VLE data for 219 non-hydrogen-bondng (nhb) bnary systems. Once these parameters were optmzed, they were fxed n the subsequent optmzaton of the remanng parameters. Second, the dsperson parameters of H(OH) and H(NH) were optmzed to expermental VLE data for 170 bnary systems whch contan hydrogen-bondng (hb) nteractons, but no water or carboxylc acds. Fnally, 52 bnary VLE systems contanng water and carboxylc 12 1/ 2

13 acds were used to optmze the dsperson parameters of H(water/COOH). All bnary systems used n ths optmzaton are summarzed n the Supplementary Materal and the values of these global atomc dsperson parameters are lsted n Table Results and dscussons In ths study, VLE data and nfnte dluton actvty coeffcent data were used to valdate the COSMO-SAC-dsp model. VLE data for a total of 441 bnary mxtures, contanng 1308 sotherms, n the temperature range from K to K and pressure range from kpa to 6.87 MPa, were consdered. These data were retreved from the DECHEMA Chemstry Data Seres [51-53]. The nfnte dluton actvty coeffcent data were taken from the lterature [54-66]. They consst of 2385 data ponts (966 bnary mxtures) n the temperature range from K to K. These systems can be categorzed nto two types: nhb and hb systems. Systems where one or both components have at least one hydroxyl (OH) group (ncludng water and carboxylc acds) or amne (NH and NH 2 ) group were consdered as hb systems; the others are nhb systems. It should be noted that exclusvely VLE data were used n the optmzaton of the dsperson parameters. 4.1 Vapor-lqud equlbrum results The ntroduced dsperson parameters were obtaned from fttng to all VLE data n ths study, but t s nonetheless nterestng to dscuss the descrptve performance of the COSMO-SAC-dsp model. Its accuracy was evaluated usng the average absolute relatve devaton n vapor pressure (AARD-P) and average absolute devaton n vapor phase mole fracton (AAD-y 1 ) 13

14 N M calc expt 1 1 P, j P, j AARD - P (%) 100%, (14a) expt 1 N j1 M P, j N M 1 1 calc expt AAD - y 1 (%) y1,, j y1,, j 100%, (14b) 1 N j1 M where M s the number of data ponts on an sotherm of a bnary mxture; N s the number of sotherms of a bnary mxture for whch expermental data were consdered. For some bnary systems, expermental data were avalable for numerous sotherms n the database, such as for benzene + cyclohexane or water + methanol. Averagng over all sotherms for a bnary mxture was done to avod the overrepresentaton of such systems n the overall devaton. In ths work, the modfed UNIFAC model [67] was used as a baselne reference. Table 3 summarzes the overall devatons wth respect to VLE data from three dfferent models. Modfed UNIFAC s the most accurate, wth the lowest AARD-P and AAD-y 1 of 3.44% and 1.53%, but only 411 out of 441 bnary VLE systems were consdered because of the mssng parameter ssue. The overall AARD-P and AAD-y 1 from the COSMO-SAC-dsp model are 5.11% and 2.12%, respectvely, whch are 25% and 12% lower than the overall AARD-P and AAD-y 1 of the COSMO-SAC (2010) model. Ths sgnfcant mprovement was observed for both nhb and hb systems. As lsted n Table 3, the overall AARD-P and AAD-y 1 for nhb systems from the COSMO-SAC-dsp model are 3.51% and 1.45%, respectvely, those from the COSMO-SAC (2010) model are 4.33% and 1.56%. For hb systems, the overall AARD-P and AAD-y 1 from the COSMO-SAC-dsp model are 6.68% and 2.78%, respectvely, those from the COSMO-SAC (2010) model are 9.18% and 3.26%. Overall, the results from the COSMO-SAC-dsp model are sgnfcantly better than those from the COSMO-SAC (2010) model for both nhb and hb systems. Fgure 2 shows VLE phase dagrams of four exemplary bnary mxtures, whch 14

15 cover a wde varety of substances: alkanes, ketones, aldehydes, acetates, aromatcs and fluoro-compounds. In these cases, the COSMO-SAC (2010) model underestmates the non-dealty of the mxtures. After consderng the dsperson contrbuton, the COSMO-SAC-dsp model can descrbe these systems very well and often has a smlar accuracy as the modfed UNIFAC model. The COSMO-SAC-dsp model s clearly more accurate than the COSMO-SAC (2010) model, especally for systems contanng fluoro-compounds. As summarzed n Table 4, the overall AARD-P and AAD-y 1 from the COSMO-SAC-dsp model were sgnfcantly reduced n comparson to the COSMO-SAC (2010) model. It s worth mentonng that modfed UNIFAC has a severe ssue of mssng parameters for ths type of systems (only three out of 15 bnary mxtures can be descrbed). Therefore, the COSMO-SAC-dsp model can be partcularly useful for ths type of mxtures. For hb systems, also a sgnfcant mprovement was acheved as llustrated n Fg. 3. In analogy to the fndngs above, the COSMO-SAC (2010) model usually underestmates the non-dealty of mxtures whch can be mproved by consderng the dspersve nteractons n mxtures. However, t should be noted that the COSMO-SAC (2010) model sometmes overestmates the non-dealty of mxtures, especally for systems contanng water or small carboxylc acds [three types of systems wth w = n Eq. (11)]. A possble explanaton for ths phenomenon s that the molecular structure and molecular surface screenng charges from quantum mechancal and COSMO calculatons cannot represent these molecules n these mxtures or under certan concentraton or temperature condtons. For example, t has been expermentally shown that acetc acd forms dmers or chan fragments n the lqud phase [68,69], so usng only acetc acd monomers n VLE calculatons s not suffcent. Chen et al. [70] proposed a theoretcally based route to better descrbe 15

16 systems contanng acetc acd. In the present study, we propose an emprcal way, consderng dspersve nteracton wth w = for these complex systems. For mxtures of acetc acd + nhb compounds, an mprovement was acheved as shown n Fg. 3(c) wthout sgnfcantly nfluencng predctons for mxtures of acetc acd + hb compounds as llustrated n Fg. 3(d). 4.2 Infnte dluton actvty coeffcent predctons The nfnte dluton actvty coeffcent usually represents the hghest devaton from the deal soluton and s mportant for chemcal processes desgn. Ths type of data s thus a good canddate to evaluate the predctve power of the COSMO-SAC-dsp model. The overall devatons of the predcted results from the expermental data were calculated as follows Error N 1 1 N ln ln, (15),calc,expt where N s the number of data ponts. The modfed UNIFAC model [67] was agan used as the baselne reference for comparson. As lsted n Table 5, the modfed UNIFAC model provdes the lowest devatons for nhb systems, but the COSMO-SAC-dsp model exhbts the lowest devatons for hb systems. It has been shown n pror work [71] that the modfed UNIFAC model has larger devatons for nfnte dluton actvty coeffcents of both water n alkanes (ncludng cyclc alkanes) and alkanes n water. Fgure 4 compares the predcted ln from the COSMO-SAC-dsp model and the COSMO-SAC (2010) model. For both nhb and hb systems, the COSMO-SAC (2010) model provdes good predctons, but on average t underestmates ln, especally for hgher values of ln, cf. Fg. 4(b). The COSMO-SAC-dsp model leads to slghtly hgher values for ln n case of nhb 16

17 systems and reduces the overall devatons by 25%. In case of hb systems, the predcted ln from the COSMO-SAC-dsp model are shfted up f w = n Eq. (11) or shfted down f w = The overall devatons for hb systems were reduced by 45%. Furthermore, as lsted n Table 5, a total of 1621 ln data (861 bnary mxtures) were consdered as a valdaton dataset snce these bnary mxtures were not ncluded n the tranng set for parameter optmzaton. For nhb systems, the accuracy of COSMO-SAC-dsp for the valdaton dataset and for the whole dataset s smlar; for hb systems, the accuracy of COSMO-SAC-dsp for the valdaton dataset s slghtly worse than that for the whole dataset. However, a sgnfcant mprovement was stll observed, a reducton of 26 % n overall error when comparng wth that of COSMO-SAC (2010). Ths agan supports that consderng the dspersve nteractons n the COSMO-SAC-dsp model can mprove ts predctve accuracy. It s useful to quantfy the relatve contrbuton to the actvty coeffcent from dfferent terms of the COSMO-SAC-dsp model. Some examples are lsted n Table 6: The expermental ln values ncluded n ths study range from (methylethylketone n chloroform) to (1-octadecanol n water) and the contrbuton from the dsperson term ranges from to For bnary mxtures of alkanes + alkanes, the most mportant contrbuton s from the combnatoral term, whle the dsperson term has a contrbuton of zero. The resdual contrbuton domnates for most other bnary mxtures, except for those havng relatvely small ln values wthn the range of the dsperson term. Ths s consstent wth the physcs that the dspersve nteracton s usually weaker than the electrostatc nteracton. 17

18 5. Conclusons A contrbuton consderng the dspersve nteractons n mxtures was ntroduced nto the COSMO-SAC (2010) model, denoted as COSMO-SAC-dsp model, to mprove ts accuracy for VLE and the nfnte dluton actvty coeffcent. Ths dsperson term s based on molecular smulaton data from classcal model force felds for bnary mxtures, where the dspersve nteractons were vared. For bnary mxtures, dsperson was descrbed wth the one-constant Margules equaton where the constant s determned from the molecular dsperson parameter of the components. Furthermore, an atomc contrbuton method s proposed to calculate the dsperson parameter for a gven molecule. A total of 13 global parameters was ntroduced nto the COSMO-SAC-dsp model and all of them were obtaned from regresson to a large set of bnary VLE data from experments. Ths dsperson term sgnfcantly mproves the accuracy of predctons of the nfnte dluton actvty coeffcent for both non-hydrogen-bondng and hydrogen-bondng systems. The extenson of ths method to multcomponent systems s straghtforward and s underway. Acknowledgements The authors are grateful for the fnancal support by the Alexander von Humboldt Stftung. We acknowledge Dr. Stanley I. Sandler at the Unversty of Delaware for sharng the quantum mechancal and COSMO calculaton results from hs group. We wsh to thank Frederk Zysk for hs efforts on performng part of molecular smulatons n ths work. 18

19 Supplementary Materal verson. Supplementary materal assocated wth ths artcle can be found n the onlne 19

20 Tables Table 1. Summary of smulated bnary mxtures of Lennard-Jones model fluds wth 1 /k B = 100 and 1 = 2 and the parameter A of the one-constant Margules equaton 2 /k B (K) 1 / 2 A

21 Table 2. Values of the global atomc dsperson parameters of the COSMO-SAC-dsp model Atom type Atom /k B (K) C (sp3) C (sp2) C (sp) O =O N (sp3) N (sp2) N (sp) F Cl H (OH) H (NH) H (water/cooh)

22 Table 3. Comparson of overall devatons of vapor lqud equlbrum predctons from dfferent predctve methods COSMO-SAC-dsp COSMO-SAC (2010) Mod. UNIFAC (1998) N a AARD-P (%) AAD-y 1 (%) AARD-P (%) AAD-y 1 (%) N a AARD-P (%) AAD-y 1 (%) nhb b hb b Overall b a. Number of bnary mxtures consdered n ths study. It should be noted that for a gven bnary mxture data on several sotherms may have been used. b. Fewer bnary mxtures were consdered n case of modfed UNIFAC due to mssng parameters. 22

23 Table 4. Comparson of overall devatons of vapor lqud equlbrum predctons for systems contanng fluoro-compounds N a AARD-P (%) AAD-y 1 (%) COSMO-SAC-dsp COSMO-SAC (2010) Mod. UNIFAC (1998) 3 b a. Number of bnary mxtures consdered n ths study. It should be noted that for a gven bnary mxture data on several sotherms may have been used. b. Fewer bnary mxtures were consdered n case of modfed UNIFAC due to mssng parameters. 23

24 Table 5. Comparson of overall devatons of nfnte dluton actvty coeffcent predctons COSMO-SAC-dsp nhb hb N a Error b N a Error b 1411 (651) c (210) c (728) (238) COSMO-SAC (2010) 1923 (728) (238) Mod. UNIFAC (1998) 1923 (728) (224) d a. Number of nfnte dluton actvty coeffcent data ponts consdered n ths study. The number of bnary mxtures s gven n parentheses. b. Error was determned from Eq. (15). c. Number of nfnte dluton actvty coeffcent data ponts for bnary mxtures that are not ncluded n parameter optmzaton. d. Fewer nfnte dluton actvty coeffcent data were consdered n case of modfed UNIFAC due to mssng parameters. 24

25 Solute Solvent T (K) Table 6. Comparson of varous contrbutons to ln ln,expt ln,calc ln,calc COSMO-SAC COSMO-SAC -dsp (2010) ln % res ln % comb ln % n-pentane n-heptane % % % n-pentane n-tetracosane % % % n-pentane benzene % % % n-pentane 1,2-dchloroethane % % % 1,1-dchloroethane 1,1,1-trchloroethane % % % n-pentane acetontrle % % % n-pentane acetone % % % benzene trethylamne % % % 1-pentene ethylbenzoate % % % soprene methylethylketone % % % toluene ethylacetate % % % chloroform n-hexane % % % acetone 1-butanol % % % n-pentane ethanol % % % benzene water % % % n-pentane water % % % 1-octadecanol water % % % n-heptane acetc acd % % % dsp 25

26 Fgure Capton Fgure 1. Comparson of the dmensonless excess Gbbs energy from molecular smulaton ( : 1 / 2 = 1.2; : 1 / 2 = 1.4; : 1 / 2 = 1.6; : 1 / 2 = 1.8) and the one-constant Margules equaton wth a constant A as determned by Eq. (9b) (- -: 1 / 2 = 1.2; - -: 1 / 2 = 1.4; ---: 1 / 2 = 1.6; : 1 / 2 = 1.8). Fgure 2. Comparson of vapor-lqud equlbra from COSMO-SAC-dsp ( ), COSMO-SAC (2010) (- -) and modfed UNIFAC (---) for non-hydrogen-bondng systems: (a) n-butane (1) + acetone (2), (b) butanal (1) + n-heptane (2), (c) methyl acetate (1) + benzene (2) and (d) trfluoromethane (1) + sobutene (2).The results from modfed UNIFAC are not shown n (d) because of the mssng parameter ssue. Fgure 3. Comparson of vapor-lqud equlbra from COSMO-SAC-dsp ( ), COSMO-SAC (2010) (- -) and modfed UNIFAC (---) for hydrogen-bondng systems: (a) n-hexane (1) + ethanol (2), (b) cyclohexylamne (1) + N,N-dmethylformamde (2), (c) cyclohexane (1) + acetc acd (2) and (d) ethanol (1) + acetc acd (2). The results from COSMO-SAC (2010) are not shown n (c) because t faled for ths system. Fgure 4. Comparson of predcton of nfnte dluton actvty coeffcent from COSMO-SAC-dsp ( 〇 ) and COSMO-SAC (2010) ( ) wth expermental data for (a) non-hydrogen-bondng systems and (b) hydrogen-bondng systems. 26

27 Fgures Fgure G ex /RT x 1 (mol. mol -1 ) 27

28 Fgure (a) K P (MPa) K (b) K P (MPa) K (c) P (MPa) K K (d) K P (MPa) x 1, y 1 (mol. mol -1 ) K

29 Fgure (a) K P (MPa) K (b) K K P (MPa) K (c) P (MPa) K (d) P (MPa) K x 1, y 1 (mol. mol -1 ) 29

30 Fgure 4(a) 5 4 ln (cal) ln (exp) Fgure 4(b) ln (cal) ln (exp) 30

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34 Corrgendum Corrgendum to < Consderng the dspersve nteractons n the COSMO-SAC model for more accurate predctons of flud phase behavor > <[Flud Phase Equlbra 367 (2014) ]> < Cheh-Mng Hseh, a,* Shang-Ta Ln, b Jadran Vrabec c > < a Department of Chemcal and Materals Engneerng, Natonal Central Unversty, Jhongl, Tawan b Department of Chemcal Engneerng, Natonal Tawan Unversty, Tape, Tawan c Thermodynamcs and Energy Technology, Unversty of Paderborn, Paderborn, Germany> The authors regret that some errors were found n the above artcle. 1. The value of w s ncorrect and four passages n the text should be revsed: (a) On page 111, thrd lne of the left column: where w = should be revsed to where w = ± (b) On page 111, Eq. (11) should also be revsed to: , f (a) water hb - only - acceptor (b) COOH nhb or hb - donor - acceptor w (11) (c) water COOH , otherwse, (c) On page 113, thrd paragraph: [three types of systems wth w = n Eq. (11)] In the present study, we propose an emprcal way, consderng dspersve nteracton wth w = for these complex systems. should be revsed to: [three types of systems wth w = n Eq. (11)] In the present study, we propose an emprcal way, consderng dspersve nteracton wth w = for these complex systems. (d) On page 114, at the end of the frst paragraph on the left column: In case of hb systems, the predcted ln from the COSMO-SAC-dsp model are shfted up f w = n Eq. (11) or shfted down f w = should be revsed to: In case of hb systems, the predcted ln from the COSMO-SAC-dsp model are shfted up f w = n Eq. (11) or shfted down f w = On page 112, Table 2: H (water/cooh) should be revsed to H (water)

35 3. The overall error of COSMO-SAC-dsp for hb systems n predctng the nfnte dluton actvty coeffcent s ncorrect and two should be revsed: (a) On page 114, Table 5: The overall error of COSMO-SAC-dsp for hb systems should be revsed from to (b) On page 114, last sentence of the frst paragraph on the left column: The overall devatons for hb systems were reduced by 45%. should be revsed to: The overall devatons for hb systems were reduced by 22%. The authors would lke to thank Andreas Klamt at COSMOlogc GmbH & Co. KG and Jürgen Rarey at DDBST GmbH for pontng out these errors. The authors would lke to apologse for any nconvenence caused. DOI of orgnal artcle: < /j.flud > <Correspondng author: C.-M. Hseh, Tel.: x34220.> <E-mal address: hsehcm@cc.ncu.edu.tw >