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Fluid Phase Equilibria 385 (2015) 10–24 Contents lists available at ScienceDirect Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d Correlating and estimating the solubilities of sol
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  Correlating   and   estimating   the   solubilities   of    solid   organic   compoundsin   supercritical   CO 2  based   on   the   modi 󿬁 ed   expanded   liquidmodel   andthe   group   contribution   method HongruLi a, *,DongdongJia b ,Ruihua   Liu a ,BingqianShen a a CollegeofPharmacy,StateKey   LaboratoryofMedicinalChemicalBiologyandTianjinKeyLaboratoryofMolecularDrugResearch,NankaiUniversity,Tianjin    300071,PRChina b SchoolofChemistryandChemicalEngineering&Technology,TianjinUniversityofTechnology,Tianjin300384,PRChina A   R    T   I   C   L    E   I   N   F   O  Article   history: Received   16   August   2014Received   in   revised   form   23   October   2014Accepted   26   October   2014Available   online   31   October   2014 Keywords: Supercritical   CO 2 SolubilityExpanded   liquid   modelCorrelationEstimation A   B   S   T   R    A   C   T Theexpandedliquidmodelwasinvestigatedandamodi 󿬁 edsolutionmodelwasproposedinthispaper.Theresultsshowedthatintheexpandedliquid   modeltheaccuracyofthesolubilityparametersof supercriticalCO 2 (sc-CO 2 )canaffectthechangeofthevariables.Ifthesolubilityparametersofsc-CO 2 areaccurate,thevariablesintheexpandedliquidmodelarenearlytemperature-independent.Iftheinaccuratesolubilityparametersofsc-CO 2  wereused,thevariablesinthemodelwouldbeslightlytemperature-dependent.Whenusingthesolubilityparameterofthesoluteasvariableandcalculatingthesolubilityparametersofsc-CO 2  withaccuratemethodintheexpandedliquidmodel,itisfoundthatthesolubilityparameterofthesoluteisthelinearfunctionoftheratioofthesolubilityparameterandmolarvolumeofsc-CO 2 ( d 1 / v 1 ).Basedonthislinearfunction,theexpandedliquidmodelwasmodi 󿬁 ed.Inthemodi 󿬁 edmodel,theslopeandtheinterceptofthelinearfunctionwereusedasthe 󿬁 ttingparameters.Themodi 󿬁 edmodelwascomparedwiththeothertwo-parameterexpandedliquidmodelsandtheresultsshowedthemodi 󿬁 edmodelcanprovidebettercorrelationresultswiththeaverageabsoluterelativedeviation(AARD)being13.31%.Agroupcontributionmethodwasdevelopedtoestimatetheparametersinthemodi 󿬁 edmodel.Basedonthemodi 󿬁 edmodelandthegroupcontributionmethod,thesolubilitisofmostofthesolutesinsc-CO 2  canbeestimatedintheorderofmagnitude. ã 2014ElsevierB.V.Allrightsreserved. 1.   Introduction Carbon   dioxide   is   often   recognized   as   the   green   solvent   and   it   isoften   used   as   ef  󿬁 cient   solvent   in   reaction,   separation,   crystalliza-tion   and   dyeing   processes   these   days.   In   the   use   of    sc-CO 2 ,   thesolubilities   of    related   compounds   are   the   fundamental   data.Determining   the   solubilities   by   experiments   is   laborious.   So   thesimple   and   quick   solubility   estimation   methods   are   necessaryespecially   in   the   early   process   development   when   a   lot   of    solubilitydata   are   needed.The   compressed   gas   model   is   often   used   in   solubility   prediction[1 – 2]   and   it   is   regarded   as   the   most   promising   model   for   solubilityprediction.   However,   when   using   the   compressed   gas   model,empirical   parameters   are   often   included   in   the   mixing   rules   toimprove   the   correlation   results.   The   empirical   parameters   oftenchange   with   temperature   and   are   dif  󿬁 cult   to   correlate   with   theproperties   and   structures   of    the   related   compounds.   The   mixingrules   based   on   G ex have   better   theoretical   basis,   but   they   are   onlyaccurate   when   the   pressure   is   zero   or   approaches   in 󿬁 nity   [3 – 5].Inaddition   to   the   mixing   rules,   the   compressed   gas   model   has   someother   limitations.   For   example,   the   critical   properties,   ancentricfactors   and   sublimation   pressures   of    related   compounds   arenecessary.   For   many   compounds,   these   properties   have   to   beestimated   and   the   estimation   errors   are   unavoidable.The   expanded   liquid   model   is   often   used   in   solubilitycorrelation.   The   advantage   of    using   the   expanded   liquid   modelis   that   only   the   melting   point   and   melting   enthalpy   of    relatedcompound   are   needed.   These   properties   can   be   easily   determinedby   experiments.   As   the   supercritical   solution   is   not   the   idealsolution,   empirical   parameters   are   often   included   in   activitycoef  󿬁 cient.   By   estimating   the   empirical   parameters   with   appro-priate   method,   the   expanded   liquid   model   can   also   be   used   insolubility   prediction   [6,7].In   this   paper,   the   expanded   liquid   model   is   investigated.   Thecalculation   methods   of    the   solubility   parameters   of    sc-CO 2  andtheir   effect   on   the   variables   in   the   expanded   models   were   studied.A   modi 󿬁 edexpanded   liquid   model   was   proposed   and   itscorrelation   capability   was   compared   with   the   other   two-parameter   expanded   liquid   models.   A   group   contribution   methodwas   developed   to   estimate   the   parameters   in   this   new   model. *   Corresponding   author.   Tel.:   +86   22   23507760;   fax:   +86   22   23507760. E-mail   address:   lihongru@nankai.edu.cn   (H.   Li).http://dx.doi.org/10.1016/j. 󿬂 uid.2014.10.0360378-3812/   ã   2014   Elsevier   B.V.   All   rights   reserved. Fluid   Phase   Equilibria   385   (2015)   10 – 24 Contents   lists   available   at   ScienceDirect Fluid   Phase   Equilibria journal   homepage:   www.else   vie   r.com/locat   e/fluid  Based   on   the   group   contribution   method,   the   modi 󿬁 ed   model   canbe   used   to   predict   the   solubilities   of    most   of    the   compounds   in   theorder   of    magnitude. 2.    Theory   2.1.   Mathematical   model If    the   supercritical   󿬂 uid   is   regarded   as   the   expanded   liquid,thesolubility   of    solid   solute   in   the   supercritical   󿬂 uidcan   be   expressed   as:  y 2  ¼  1 g  2  f  s 2  f  l 2 (1)where    y 2  is   the   solubility   of    the   solute,   g  2  is   the   activity   coef  󿬁 cientof    the   solute.   The   fugacity   ratio   of    pure   solid   solute   and   liquidsolute   can   be   calculated   according   to   Ref.   [8].ln  f  s 2  f  l 2    ! ¼  D H  m RT  m T  m T      1  þ D c   p RT  m T      1      D c   p R  ln  T  m T       (2)where   R   is   the   gas   constant,   T    is   the   temperature,  4 H  m  is   themelting   enthalpy   of    the   solute,   T  m is   the   melting   point   of    the   solute, 4 c   p is   the   difference   of    the   heat   capacity   between   the   liquid   soluteand   the   solid   solute.   The   heat   capacity   terms   are   relatively   smalland   can   be   neglected.   So   the   solubility   of    solid   solute   insupercritical   󿬂 uid   can   be   expressed   as   follows:  y 2  ¼  1 g  2 exp   D H  m RT  m T  m T      1         (3)As   the   solubility   of    solid   solute   in   supercritical   CO 2  is   very   low,the   activity   coef  󿬁 cient   can   be   regard   as   its   in 󿬁 nite   dilution   value.The   in 󿬁 nite   dilution   activity   coef  󿬁 cient   can   be   calculated   usingdifferent   method.   According   to   the   Scatchard – Hildebrand   regularsolution   theory,   the   in 󿬁 nite   dilution   activity   coef  󿬁 cient   can   beexpressed   as   follows:ln   g  1 2     ¼  1 RT v l 2 f 21  d 2     d 1 ð   Þ 2 (4)where   F 1  is   the   volume   fraction   of    the   solvent. f 1  ¼  y 1 v 1  y 1 v 1  þ    y 2 v l 2 (5) v 2  is   the   liquid   molar   volume   of    the   solute,   v 1  is   the   molar   volumeof    sc-CO 2 , d 2  is   the   solubility   parameter   of    the   solute   and   d 1  is   the Nomenclature List   of    symbols  y 2  mole   fraction   of    the   solute   in   supercritical   solution[mol   mol  1 ]   Eq.id=6#(1)  f  s 2  fugacity   of    the   pure   solid   solute   [MPa],   Eq.   (1)  f  l 2  fugacity   of    the   pure   liquid   solute   [MPa],   Eq.   (1) D H  m  melting   enthalpy   of    the   solute   [kJ   mol  1 ]   Eq.id=6#(2) T  m  melting   point   of    the   solute   [K]   Eq.   (2) T    temperature   [K]   Eq.   (2) R   gas   constant   R   =   8.3145    J   mol  1 K  1 Eq.id=6#(2) D C   p  difference   of    the   heat   capacity   between   the   liquidsolute   and   the   solid   solute   [kJ   mol  1 K  1 ]   Eq.id=6#(2) v l 2  liquid   molar   volume   of    the   solute   [cm 3 mol  1 ]   Eq.id=6#(4) v 1  molar   volume   of    sc-CO 2  [cm 3 mol  1 ]   Eq.id=6#(5) D E  2  cohesive   energy   of    the   solute   [J   mol  1 ]  p c   critical   pressure   [atm]   Eq.   (14)  p   pressure   [MPa]   Eq.   (15)  y exp2  experimental   solubility   of    the   solute   in   sc-CO 2  [molmol  1 ]   Eq.id=6#(18)  y cal2  calculated   solubility   of    the   solute   in   sc-CO 2  [molmol  1 ]   Eq.id=6#(18) N    number   of    solubility   data   Eq.   (18)Greek   letters g  2  activity   coef  󿬁 cient   of    the   solute   in   sc-CO 2  Eq.   (1) g  1 2  in 󿬁 nite   dilution   activity   coef  󿬁 cient   of    the   solute   insc-CO 2  Eq.   (4) f volume   fraction   of    the   solvent   Eq.   (4) d 2  solubility   parameter   of    the   solute   [MPa 0.5 ]   Eq.   (4) d 1  solubility   parameter   of    sc-CO 2  [MPa 0.5 ]   Eq.   (4) r density   of    sc-CO 2  [kg   m  3 ]   Eq.id=6#(7) b 12  binary   interaction   parameter   Eq.   (12) r r   reduced   density   of    the   solvent   Eq.   (14) r r   (liq)   reduced   density   of    liquid   Eq.   (14) Fig.   1.   Calculated   solubility   parameters   of    sc-CO 2  using   different   methods   at   313   K. Fig.   2.   Calculated   solubility   parameters   of    phenanthrene   using   Eq.   (6)   versus   thedensity   of    sc-CO 2 .(solubility   parameters   of    sc-CO 2 were   derived   from   Huang ’ s27-parameter   equationand   the   solubilities   were   from   Ref.   [18].) H.   Li   et    al.    /    Fluid   Phase   Equilibria    385   (2015)   10 –  24   11  solubility   parameter   of    sc-CO 2 .   The   solubility   of    solute   can   beexpressed   as:  y 2  ¼   exp  D H  m R 1 T  m  1 T     1 RT v l 2 f 21  d 2     d 1 ð   Þ 2      (6)In   this   model,   the   solubility   parameter   d 2 is   used   as   variable   andit   changes   with   the   density   of    supercritical   󿬂 uid.   Guigard   and   Stiverproposed   two   density   dependent   correlations   (a   linear   󿬁 t   and   apower   󿬁 t)   for   the   solute   solubility   parameter   as   follows   [9]: d 2  ¼   a r þ   b   (7) d 2  ¼   a r n þ   b   (8)Usually,   the   power   󿬁 t   provided   better   correlation   results.The   in 󿬁 nite   dilution   activity   coef  󿬁 cient   can   also   be   calculatedby   regular   solution   model   coupled   with   the   Flory – Hugginsequation. Fig.   3.   Calculated   solubility   parameters   of    phenanthrene   using   Eq.   (6)   versus   thedensity   of    sc-CO 2 .(solubility   parameters   of    sc-CO 2 were   derived   from   PR-EoS   and   the   solubilities   werefrom   Ref.   [18].) Fig.   4.   Calculated   liquid   molar   volumes   of    phenanthrene   using   Eq.   (10)   versus   thedensity   of    sc-CO 2 .(solubility   parameters   of    sc-CO 2 were   derived   from   Huang ’ s27-parameter   equationand   the   solubilities   were   from   Ref.   [18].) Fig.   5.   Calculated   liquid   molar   volumes   of    phenanthrene   using   Eq.   (10)   versus   thedensity   of    sc-CO 2 .(solubility   parameters   of    sc-CO 2 were   derived   from   PR-EoS   and   the   solubilities   werefrom   Ref.   [18].) Fig.   6.   Calculated   binary   interaction   parameters   of    phenanthrene   and   CO 2  usingEq.   (12)   versus   the   density   of    sc-CO 2 .(solubility   parameters   of    sc-CO 2 were   derived   from   Huang ’ s27-parameter   equationand   the   solubilities   were   from   reference   [18].) Fig.   7.   Calculated   binary   interaction   parameters   of    phenanthrene   and   CO 2  usingEq.   (12)   versus   the   density   of    sc-CO 2 .(solubility   parameters   of    sc-CO 2 were   derived   from   PR-EoS   and   the   solubilities   werefrom   Ref.   [18].)12   H.   Li   et    al.    /    Fluid   Phase   Equilibria    385   (2015)   10 –  24  ln   g  1 2     ¼  v l 2 RT   d 1     d 2 ð   Þ 2 þ   1     v l 2 v 1    ! þ   ln  v l 2 v 1    !  (9)where   v l 2  is   the   liquid   molar   volume   of    the   solute,   v 1  is   the   molarvolume   of    sc-CO 2 .   The   solubility   of    solute   can   be   expressed   asfollows:  y 2  ¼   exp  D H  m R 1 T  m   1 T        v l 2 RT   d 1     d 2 ð   Þ 2    1   þ  v l 2 v 1    ln  v l 2 v 1    !   #  (10)where   d 2  ¼   D E  2 = v l 2     0 : 5 and  4 E  2  is   the   cohesive   energy   of    thesolute,   the   liquid   molar   volume   of    solute   is   regarded   as   variableand   it   is   the   function   of    supercritical   󿬂 uid   density   [6].   The   functionis   expressed   as:ln   v l 2     ¼   a ln r þ   b (11)where   the   solubility   parameter   of    the   solute   are   calculatedaccording   to   Fedors   [10].Sometimes,   the   binary   interaction   parameter   b 12  isintroduced   when   calculating   the   in 󿬁 nite   dilution   activitycoef  󿬁 cient   using   regular   solution   model   coupled   with   theFlory – Huggins   equation.  y 2  ¼   exp  D H  m R 1 T  m   1 T     v l 2 RT   d 1     d 2 ð   Þ 2    b 12 h   i    1   þ  v l 2 v 1    ln  v l 2 v 1    !   # (12)The   parameter   b 12  is   independent   of    temperature   and   dependenton   the   density   of    supercritical   CO 2 .   In   the   literature   [11],parameter b 12  was   correlated   with   the   density   of    CO 2  using   a   second   orderpolynomial. b 12  ¼   a 0  þ   a 1 r þ   a 2 r 2 (13)  2.2.   Methods   of    calculating    solubility    parameter    of    supercritical   CO  2 In   the   use   of    expanded   liquid   models,   the   solubility   parametersof    sc-CO 2 are   necessary.   More   than   three   methods   were   used   in   therelated   literatures   to   calculate   the   solubility   parameters   of    sc-CO 2 .Considering   the   accuracy   of    the   solubility   parameters   of    sc-CO 2 may   affect   the   change   of    the   variables   in   the   expanded   liquidmodels,   the   following   calculation   methods   were   compared   and   themost   accurate   one   was   adopted.Gidding   proposed   a   semi-empirical   expression   to   calculate   thesolubility   parameters   for   different   solvent   as   follows   [12]: d ¼   1 : 25  p c1 = 2  r r  r r  ð liq Þ   (14)where    p c  is   in   atmosphere   and   r r  (liq)   is   the   reduced   density   of liquid.   For   CO 2 ,   r r  (liq)   is   equal   to   2.66.Williams   et   al.   calculated   the   Hansen   solubility   parameters   of CO 2  based   on   the   assumption   that   the   total   cohesive   energy   ismade   up   of    the   additive   contributions   from   nonpolar   interactions,polar   interactions   and   hydrogen-bonding   or   other   speci 󿬁 cassociation   interactions   [13].The   solubility   parameters   of    solvents   can   also   be   calculatedaccording   to   the   de 󿬁 nition   of    the   solubility   parameter   [14]: Fig.   8.   Calculated   solubility   parameters   of    phenanthrene   versus   d 1 / v 1 .(solubility   parameters   of    sc-CO 2 were   derived   from   Huang ’ s27-parameter   equationand   the   solubilities   were   from   Ref.   [18].) Fig.   9.   Calculated   solubility   parameters   of    2,2 0 ,4,5,5 0 -pentachlorobiphenyl   versus d 1 / v 1 .(solubility   parameters   of    sc-CO 2 were   derived   from   Huang ’ s27-parameter   equationand   the   solubilities   were   from   Ref.   [19].) Fig.   10.   Calculated   solubility   parameters   of    CI   disperse   orange   3   versus   d 1 / v 1 .(solubility   parameters   of    sc-CO 2 were   derived   from   Huang ’ s27-parameter   equationand   the   solubilities   were   from   Ref.   [20].) H.   Li   et    al.    /    Fluid   Phase   Equilibria    385   (2015)   10 –  24   13
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