Calculation of Interfacial Properties of Binary Mixtures Containing Polar and Non-polar Components

Transcription

1 Calculaton of Interfacal Propertes of nary Mxtures Contanng Polar and Non-polar Components STRCT Oscar Gabrel Nño mézquta 1, Sabne Enders 1 * 1 TU erln, Ernst Reuter Platz 1, D-157 erln, Germany * Correspondng author: Sabne.Enders@tu-berln.de Carbon doxde (CO ) s a component that has become a central role for the ndustry nowadays. Not only s ths compound of wde use for certan applcatons (.e. supercrtcal extracton processes) but also plays an mportant role n global warmng. Therefore, the understandng of the phenomena related wth ths molecule s of great mportance. Furthermore, the accurate modellng of ts mxtures may not only have a postve effect on techncal modellng, but also on economcal revenue n several applcatons. Ths paper focuses on the understandng of phase and nterfacal behavour of CO mxtures. These mxtures may appear n petrochemcal processes lke the enhanced ol recovery or n carbon capture and storage systems [1]. Recently [, 3,,], a method was suggested to predct the surface tenson of non-polar mxtures wth help of the densty gradent theory [5] n very good agreement wth expermental data. The proposed methodology uses a suted equaton of state (EOS), namely the polar perturbed chan statstcal assocaton flud theory (PCP-SFT EOS) [6], for calculaton or predcton of the phase behavour and nterfacal propertes of a gven mxture. Ths model allows to take nto account the quadrupole moment of CO, resultng n mprovements for the modellng of propertes of pure CO and ts mxtures [, 3, ]. Wth help of ths theoretcal background t s possble to make use of the densty gradent theory for predcton of nterfacal tenson of CO n mxtures. Some examples (CO + methane, CO + benzene and CO + toluene) of these mxtures wll then be presented and compared wth expermental data from the lterature, f avalable. INTRODUCTION esdes the phase behavour the nterfacal propertes play an mportant role n process desgn. Ths s especally true n the feld of ol- and gas gatherng. In order to reduce the global warmng effect carbon doxde (CO ) can pressed nto the ol- or/and gas reservor. From the economcal pont of vew the nterfacal tenson between the carbon doxde and the ol- or/and gas phase, and hence the necessary pressure s essental nformaton for the system. For ths reason the modellng of thermodynamc propertes of mxtures contanng CO are of great nterest [7,, 9]. Prevous works [, 3, ] have suggested a method to predct the surface tenson of non-polar and polar-non polar mxtures based on the densty gradent theory (DGT) [5] n very good agreement wth expermental data. In ths case, the most mportant feature of CO s the quadrupole moment. The present paper ams to further applcaton of the theoretcal framework for the followng CO contanng mxtures: CO + methane, CO + benzene and CO + toluene, where the PCP-SFT [6] s used. THEORY The densty gradent theory was frst proposed by van der Waals [1] and further developed from Cahn-Hllard for pure substances [5] and extended by Poser and Sanchez [11] to bnary 1/1

2 mxtures. The core of the theory les on calculaton of the nterfacal propertes based on bulkpropertes of the bnary mxture and the surface tenson of the pure component. DENSITY GRDIENT THEORY Pure Components The DGT [5] descrbes the thermodynamc propertes of a system where an nterface exsts between two equlbrum phases. On the nterface between lqud and gas phases n thermodynamc equlbrum only the densty vares contnuously from the bulk lqud L V densty,, to the vapour densty,. ssumng that the densty gradent s small compared wth the recprocal of the ntermolecular dstance allows the densty and ts dervatves to handle as ndependent varables. Ths means that the Helmholtz free energy F for a system wth an nterface can be obtaned by expandng the Helmholtz free energy F n a Taylor seres about the equlbrum bulk state. Followng Cahn and Hllard [5], the surface tenson, σ, for a planar surface reads: V σ = κ ω( ) d (1 ) L The nterfacal tenson s thus proportonal to the Helmholtz free energy densty, ω ( ), of homogeneous flud. Ths quantty s also called grand thermodynamc potental. ω can be represented as the vertcal dstance between the curve of Geometrcally, ( ) f ( ) versus and a straght lne touchng f ( ) and at the vapour and lqud denstes, V. ddtonally, the nfluence parameter κ of the pure substance has to be known. The densty profle, ( z) n the drecton perpendcular to the nterface, s descrbed wth help of the followng equaton: L κ z = z + d ( ) ω( ) z and represent an arbtrarly chosen orgn and composton. The z -orgn can be L V arbtrarly located, for example, at a densty of ( ) / +. dstance z may be determned for any lyng between the bulk denstes by means of numercal evaluaton of the ntegral. efore any nterfacal propertes can be computed, t s necessary to fnd the thermodynamc equlbrum denstes between whch the nterface s formed. The denstes of the vapour and the lqud phase can be calculated at a gven temperature and pressure. fter ths, the expressons (Eqs. 1 and ) permt calculaton of surface tensons and densty profles f κ and ω are known. The nfluence parameter, κ, has a molecular theoretcal defnton [5] but s not usually calculable. Thus, t s treated as a parameter of the flud n the same sprt as /1

3 the parameter of sem emprcal models. Equatons of state (PCP-SFT) wll be used to evaluate ω and explaned n a further secton. nary Mxtures The frst step durng the calculaton of any nterfacal propertes s the calculaton of the equlbrum of the two phases, between whch the nterface beng consdered. Such calculatons are carred out by equatng the chemcal potentals and the pressures n both coexstng phases. The Cahn - Hllard Theory [5] descrbes the thermodynamc propertes of a system where an nterface exsts between two flud phases. In contrast to systems of pure components, n bnary mxtures not only the densty but also the composton changes through the nterface. In order to take both effects nto account we defne the partal denstes,. = X ( =, ) (3 ) Usually, the thermodynamc quanttes of bnary systems at constant temperature are functons of densty and composton, expressed by mole fracton (e.g. Helmholtz energy F X. pplyng Eq. 3 the thermodynamc functons can be rewrtten as functons of both ( ), partal denstes (e.g. (, ) F ). Usng a smlar procedure lke Cahn and Hllard [5], to calculate the nterfacal tenson of an planar nterface, Poser and Sanchez [11] worked out a, wthn the nterface. The method that consder the change of two varables ( ) nterfacal tenson between two flud phases n equlbrum reads [11]: II 1/ ' (, ) d I σ = κ ω ( ) The value of κ ' results from the nfluence parameters of the pure components, κ. Same as n ω, s defned as the grand thermodynamc potental. the case of a pure substance ( ) I II The lmts of ntegraton and are the partal denstes n the coexstng phases and can be obtaned by Eq. 3. κ ' s gven by: d d κ ' = κ + κ + κ d d (5 ) The κ -parameters can be ftted to one expermental surface tenson. The parameter κ can be calculated usng geometrcal average of the pure component parameters: κ = κ κ (6 ) Eq. 6 allows the predcton of surface tensons of the mxtures based on the surface tenson of ω, s then: the pure components. The grand thermodynamc potental ( ) 3/1

4 I I (, ) ( ) F(, ) (, ) (, ) I I I = µ (, ) µ (, ) ω = + µ µ + p ( ) I I I I Eq. (7 ) The quantty chemcal potentals, Eq. p s the equlbrum pressure. The Helmholtz free energy, F, and the µ, can be obtaned by an equaton of state, f the quanttes and are known. Therefore, the calculaton of ω (, ) needs the knowledge of to each value of wthn the nterface. Ths dependence conforms to the densty profles through the nterface. The mnmsaton of the nterfacal tenson leads to the followng system of equatons to calculate the denstes at each value of z, the drecton perpendcular to the nterface [11]. ω (, ) = κ κ + z z ( ) ω (, ) = κ κ + z z (9 ) The soluton of ths system of equatons represents the profles of the partal denstes, ( z) and ( z), durng the nterface. The use of the geometrcal mxng rule for κ (Eq. 6 ) shows notceable smplfcatons [11]. In ths case the system of equatons (Eqs. and 9 ) reduce to one algebrac equaton: ω(, ) ω(, ) κ = κ (1 ) The applcaton of Eq. 1 n Eqs. and 9 leads to followng relaton: I I ( (, ) ) = ( (, ) ) κ µ µ κ µ µ (11 ) Ths expresson shows the combnaton of the partal denstes and n one expresson. t constant phase equlbrum temperature Eq. 11 has two varables, and. If the value of s chosen as a samplng pont wthn the nterface, to calculate the ntegral Eq. the correspondng value can estmate n a way, that Eq. 11 s fulflled. fter ths, the grand ω, can be evaluated wth Eq. 7 at every samplng pont. thermodynamc potental ( ) The dfferental rato n Eq. s then gven at each value. Durng the dervaton of Eq. the coordnates transformaton from the locaton space to the densty space was used [11]. The ntegraton of ths transformaton permts the calculaton of the densty: /1

5 ( z) κ ' z = z + d (1 ) ω(, ) ( zo ) whereas κ ' s gven n Eq. 5 and z represents an arbtrarly chosen orgn. nother mportant quantty whch can be dervng from the densty profles s the thckness of the nterface. Ths thckness s thermodynamcally not exactly defned. In order to specfy a physcally value the 1/9-rule, gven by Lekner and Henderson[1] can be used. PCP-SFT-EOS The PCP-SFT-EOS [6] s a further development of the PC-SFT-EOS [13]. In ths model the chan-length dependence of the attractve (dspersve) nteractons s also taken nto account. hard-chan flud serves as a reference for the perturbaton theory, dfferng from sphercal molecules as n the orgnal SFT verson. Molecules are conceved as chans composed of sphercal segments. The par potental for the segment of a chan s gven by a modfed square-well potental. ased on the Werthem's thermodynamc perturbaton theory [1, 15], the attractve nteractons are treated as a perturbaton of the reference system. The equaton of state s gven has then an deal gas contrbuton (d), a hard-chan contrbuton (hc), and a perturbaton contrbuton, whch accounts for the attractve nteractons (dsp), a contrbuton arsng from the self- or cross-assocaton of polar molecules (assoc) and addtonally a contrbuton takng polar nteractons (polar) nto account: d hc dsp assoc polar f = f + f + f + f + f (13 ) For a non- assocatng substance ths EOS contans three pure-component parameters: the segment dameter (σ ), the nteracton energy related to the dsperson nteracton (ε ), and the number of segments ( m ). The parameters can be ftted usng vapour pressure and lqud denstes. More detals can be found n the papers of Gross and Sadowsk [13]. polar The heart of the PCP-SFT-EOS s the new addtonal term f n Eq. 13. In order to derve an expresson for ths quantty the startng pont s the Padé- approxmaton n terms of the Helmholtz energy: polar f f / RT RT RT 1 / quadrupole = = (1 ) 3 wth and 3 as the second-order and thrd-order perturbaton terms, respectvely. Gross [6] suggested for these terms the followng expressons: RT K QQ 3 QQ = K η J,11 = K 3η J3,111 RT 7 ε σ Q 1 ε σ Q 3 * 3 6 * = K 3 3 = 3 6 m1d 1 ( k ) T ( kt ) m1 d1 (15 ) 5/1

6 where Q * Q 1 s the reduced quadrupole moment and reads: 1 Q = (16 ) 19 * km1ε 1σ 1 where the quadrupole moment Q s measured n 1-6 erg 1/ *cm 5/ and k s the oltzmannconstant. QQ QQ The expresson for the calculaton of the quanttes J,11 and J 3,111 occurrng n Eq. 15 can be found n the lterature [6]. For the applcaton of the equatons of state to mxtures made from the component and j mxng rules must be specfed. The perturbaton theory of aker and Henderson makes use of an average radal dstrbuton functon and thus treats the segments of a chan as ndstngushable. Wthn ths concept, a rgorous applcaton of the perturbaton theory to a mxture s n prncple possble. However, the mathematcal expressons are not avalable n analytcal form. Therefore, two mxng rules were used for the applcaton of the PCP-SFT to mxtures, namely: 1 σ = ( σ + σ j ) ε = ( 1 kj ) εε j (17 ) where k j s a bnary nteracton parameter. The bnary nteracton parameter wll be ftted to phase equlbrum data. RESULTS Pure Components Frst the phase equlbrum of the pure substances was calculated and the surface tenson was taken from the lterature []. The ftted nfluence parameters are lsted n Table 1. Table 1: Influence parameter κ for components usng PCP-SFT. Pure component 1 κ [Jm 5 /mol ] Source Methane [] enzene 3.99 Ths work Toluene 31.9 [] Carbon doxde.37 [] The nfluence parameters taken from prevous works [, ] for the gven substances have proved to be able to descrbe the temperature dependence of the surface tenson of pure components. Only n the case of benzene t s necessary to ft ths parameter to one sngle pont at one temperature. Results of the surface tenson curve wth the gven nfluence parameter from Table 1 are shown n Fgure 1. The expermental behavour can be accurately reproduced wth help of the DGT usng PCP-SFT EOS. 6/1

7 5 σ [mn/m] T [K] Fgure 1: Expermental (squares [16]) and predcted surface tenson (sold lne) for benzene usng PC- SFT-EOS wth κ=3.99*1 [Jm 5 /mol ]. nary Mxtures In order to descrbe the phase behavour of a bnary mxture the bnary nteracton parameter k (Eq. 17 ) has to be specfed. Ths s done va the calculaton of the bnary phase dagram j and comparson wth the expermental data gven n the lterature. Usng the ftted k j from the phase equlbrum and n combnaton wth the ftted nfluence parameters of the pure components, κ, the nterfacal tenson of the bnary mxture can be predcted. Prevous works [, ] have shown the ablty of the combnaton of the DGT wth PCP-SFT to descrbe and predct CO mxtures wth n-alkanes and cycloalkanes; however, no aromatc molecules have been taken nto account n the theoretcal framework. Fgure depcts the vapour-lqud equlbrum for the mxture CO + benzene, at one temperature below and two above the crtcal temperature of CO and comparsons to data taken from the lterature [17, 1]. No bnary nteracton parameter s needed n ths case, showng the predctve power of the equaton of state. For a comparson, the prevous verson wthout ncluson of polar nteractons requres a k j value of. [6]. Gross dscussed also a quadrupole moment for benzene and the nteracton of the quadrupole moments of CO and benzene. Ths stuaton requres a k j =. n order to reach agreement wth the data measures by endale et al. [1]. From these results t can be concluded that the ncorporaton of the quadrupole moment for CO s suffcent for an accurate descrpton of the VLE for the system CO + benzene. 7/1

8 1 1 P [MPa] X CO Fgure : Expermental and calculated VLE applyng PCP-SFT wth k j = of the system CO + benzene at T=9.15K (stars [17] + broken lne), at T=313.15K (squares [17] and sold lne) and at 37,5K (trangles [1] and dotted lne). σ [mn/m] X L CO Fgure 3: Expermental (squares [19]) and predcted (sold lne) surface tenson applyng PCP-SFT wth k j =. of the system CO + benzene at T=3K. The next step s the calculaton of the surface tenson for ths system. Fgure 3 shows a comparson between expermental data [19] and calculaton made wth the DGT n combnaton wth PCP-SFT at 3K. The theoretcal framework does a good job n predctng surface tenson of these mxtures made from polar and non-polar substances. For some applcatons the queston about the densty- and the composton gradent wthn the surface arse. The advantage of the modellng tool les n the possblty to calculate nterfacal propertes, lke the densty profles of both components wthn the nterface, whch are not accessble by experments. The densty profle can be calculated usng Eq. 1. Fgure depcts the partal densty of both components wthn the surface at three dfferent lqud /1

9 phase bulk-concentratons and at 3K. t 3K CO has a hgher vapour pressure than benzene. ecause of ths, the densty profle of CO runs through a maxmum n the nterface. Ths phenomenon s called relatve enrchment n the nterface. The densty profle of benzene shows the typcal tanh-shape. Relatve enrchment can play an mportant role as hndrance aganst the mass transfer over the nterface. CO [mol/dm 3 ] benzene for X L CO =. benzene for X L CO =.5 benzene for X L CO =.6 CO for X L CO =. CO for X L CO =.5 CO for X L CO = z [nm] Fgure : Predcted partal densty profles for the system CO + benzene at T=3K usng the densty gradent theory wth PCP-SFT-EOS (k j =). 6 benzene [mol/dm 3 ] d [nm] d [nm] 1,,,,6, X L CO X L CO Fgure 5: Predcted thckness of the nterface for the system CO + benzene at T=3K usng the densty gradent theory wth PCP-SFT-EOS (k j =). ddtonally, the hgher CO content n the mxture leads to a broader nterface, as shown n Fgure 5, because wth ncreasng concentraton one approaches the crtcal pont. Per defnton, the length of the nterface has to become nfnte at ths pont, snce there are no two phases anymore. The DGT has an enormous potental n ndustral processes, snce t can use propertes of the pure components n order to predct nterfacal propertes of mxtures, where expermental 9/1

10 data may be scarce. We center n ths work the predcton of surface tenson for two mxtures: CO methane and CO toluene. For the last mxture values of the bnary nteracton parameter k j are avalable n the lterature [6]. Fgure 6 shows the vapour-lqud equlbrum curve for the system CO + toluene wth k j =.3. greement between calculated and expermental VLE s excellent, as already stated n prevous works from Gross [6]. Wth help of the VLE data t s now possble to make a predcton of the surface tenson of the mxture. Ths s presented n Fgure 7. Expermental data for ths mxture s not avalable n the lterature; however, t s possble to say, based on prevous analyss of the mxtures, that at least a qualtatve and very lkely a quanttatve agreement wth the real surface tenson can be expected. P [MPa] X CO Fgure 6: Expermental (squares []) and calculated (sold lne) VLE of the mxture CO + toluene usng PCP-SFT-EOS wth k j =.3 at 3K. 3 5 σ [mn/m] X L CO Fgure 7: Predcted surface tenson of the mxture CO + toluene usng PCP-SFT-EOS at 3K. 1/1

11 The mxture CO + methane s of great mportance n ndustral processes, snce t can emulate equlbrum of a mxture of carbon doxde wth natural gas (predomnantly methane). Therefore, t s of nterest to see predctons of the nterfacal propertes for ths mxture. s usual, t s frst needed to know the bnary nteracton parameter for the system. Fgure depcts the calculated VLE data. Wth help of ths we can be able to predct the surface tenson of the mxture. Prevous works [] have shown a good agreement for mxtures wth n-alkanes. Therefore, t s expected that these predctons may be accurate n descrbng the behavour of the surface tenson at several temperatures. P [MPa] x methane Fgure : Expermental [1] and calculated vapour-lqud equlbrum of the system CO and methane at dfferent temperatures (squares and sold lne T=3 K, stars and dotted lne T=5K, trangles and broken lne T=7K) usng PCP-SFT wth k j = σ [mn/m] x L methane Fgure 9: Predcted surface tenson of the system CO and methane at dfferent temperature (sold lne: 3K, dotted lne: 5K, broken lne: 7K) usng PCP-SFT. 11/1

12 SUMMRY In summary, the proposed model s able to predct the nterfacal tenson of bnary mxtures made from CO + benzene close to the expermental data, whereas n our model only the quadrupole moment of CO s taken nto account. Unfortunately, for the other two consdered mxtures, namely CO + methane and CO + toluene no expermental data are avalable. Nevertheless, the theoretcal framework can be used to predct the surface tenson as functon of composton, temperature and pressure. ddtonally, the theory gves useful nformaton, lke the relatve enrchment n the nterface, whch s not accessble by experments. Ths effect occur always f the vapour pressures of the two pure components at the system temperature are strong dfferent. The component havng the hgher vapor pressure wll always enrch n the nterface. REFERENCES 1. OLDENURG, C.M., PRUESS, K., ENSON, S.M., Energy Fuels 15, 1, 93.. NINO-MEZQUIT, O.G., ENDERS, S., JEGER, P.T.; EGGERS, R., Ind. Eng. Chem. Res. 9, 1, NINO-MEZQUIT, O.G., ENDERS, S., JEGER, P.T., EGGERS, R., J. of Supercrtcal Fluds 55, 1, 7.. NINO-MEZQUIT, O.G., ENDERS, S., Computer ded Chemcal Engneerng, 1, CHN, J.W., HILLIRD, J.E., J. Chem. Phys., 195, GROSS, J., IChE J. 51, 5, FU, D., LING, L., LI, X.S., YN, S., LIO, T., Ind. Eng. Chem. Res. 5, 6, 36.. FU, D., WIE, Y., Ind. Eng. Chem. Res. 7,, MÜLLER, E.., MEJI,., Flud Phase Equlbra, 9, VN DER WLS, J.D., Z. Phys. Chem. 13, 19, POSER, C.I., SNCHEZ, I.C., Macromolecules 1, 191, LEKNER, J., HENDERSON, J., Physca. 9, 197, GROSS, J., SDOWSKI, G., Ind. Eng. Chem. Res., 1, WERTHEIM, M.S., J. Stat. Phys., 196, WERTHEIM, M.S., J. Chem. Phys. 7, 197, WOHLFRTH, CH, WOHLFRTH,., n M.D. Lechner (Ed.) Numercal Data and Functonal Relatonshps n Scence and Technology, Vol. 16 Surface Tenson of Pure Lquds and nary Lqud Mxtures, n Landoldt-örnsten, New Seres Group IV Physcal Chemstry, Sprnger Verlag erln, Hedelberg, OHGKI, K., KTYM, T., J. Chem. Eng. Data 1, 1976, ENDLE, P.G., ENICK, R.M, Flud Phase Equlb. 9, 199, NGRJN, N., ROINSON, R. L., JR.: J. Chem. Eng. Data 3, 197, FINK, S.D., HERSHEY, H.C., Ind. Eng. Chem. Res. 9, 199, DVLOS, J., NDERSON, W.R., PHELPS, R.E., KIDNY,.J., J. Chem. Eng. Data 1, 1976, 1. 1/1