SAFT equations of state for complex fluids Model development and applications

Size: px

Start display at page:

Download "SAFT equations of state for complex fluids Model development and applications"

Jessie Smith
6 years ago
Views:

1 SAFT equations of state for complex fluids Model development and applications Joachim Gross InMoTherm, Lyon, 2012

2 Outline Background to fluid theories perturbation theories SAFT model Applications phase behavior of complex fluids Ionic solutions / liquids Liquid Crystals group contribution approaches integrated process and solvent design (CoMT-CAMD) interfacial properties density functional theory transport properties entropy scaling molecular simulations supported by analytical theories parameterizing GCMC simulations adjusting force fields

3 Structure of fluid theories u(r) u vdw (r) r r u dipole-dipole (r) ~ consider a spherically symmetric fluid r 1 r 2 r u(r) = u vdw (r) + u dipole-dipole (r) ~ + u ion-ion (r) +... for additive contributions to the intermolecular potential, we get F(T,,x) = F ig + F vdw + F dipole-dipole + F ion-ion +... Helmholtz energy (equation of state) p(t,, x) 2 F T,x

4 2 routes to equation of state models perturbation theory for a given intermolecular potential u(r) define a suitable reference fluid, the properties of which are known (g ref (r),a ref ) g(r) integral equation approach radial distribution function g(r) g(r 12 ) 1 c(r 12 ) g(r 13 ) 1 c(r 32 )dr 3 A A ref A perturbation r E E ig 1 2 N u(r)g(r)dr

Perturbation Theory perturbation theory F F ref 1 2 1 0 u(r) u a (r) g (r)dr d d u ref (r) u a (r) u(r) = u ref (r) + u a (r) r r expansion of g (r)

5 Perturbation Theory perturbation theory F F ref u(r) u a (r) g (r)dr d d u ref (r) u a (r) u(r) = u ref (r) + u a (r) r r expansion of g (r) around reference fluid g(r) g (r) g ref (r) ref... 2nd order leads to first order perturbation term often hard-sphere or hard chain serves as a reference

6 Wertheim s perturbation theory F association ktv ln M 1 2 M M M 2 assoc assoc g hs (r ) exp 1 kt well-width related to in the limit of infinitly strong association ==> chains (of m segments) chain formation F (m1) ln g hs (r ) Chapman et al. (1988) ktv

7 Molecular Models atomistic molecular model used in molecular simulations molecular model used in theory (SAFT) fused spheres tangent spheres

8 PC-SAFT Equation of state Perturbed-Chain Statistical Associating Fluid Theory EOS one of many members of the SAFT-family F = F id + F hs + F chain + F disp + F assoc + F multipole m=4 : number of segments : segment size parameter : segment energy parameter molecular model of SAFT is coarse grained compared to molecular simulations captures the essential characteristics

9 Applications of PC-SAFT to phase equilibria H 2 S methane at p=3.447mpa (k ij =0.015) polypropylene propane CO2 (w pp =0.03)

10 Applications phase behavior of complex fluids Ionic solutions / liquids Liquid Crystals group contribution approaches integrated process and solvent design (CoMT-CAMD) interfacial properties density functional theory transport properties entropy scaling using analytical theories with molecular simulations parameterizing GCMC simulations adjusting force fields

Mean-Spherical Approximation (np-msa) integral equation theory solvent

11 Ionic systems: Molecular model and parameter PC-SAFT equation of state repulsive, dispersive, and associating interactions & Non-primitive Mean-Spherical Approximation (np-msa) integral equation theory solvent explicitly considered Ion-ion, dipole-ion and dipole-dipole interactions σ s m=1 - σ Ion + MSA

12 / g l -1 (m) Institut für Hamer, Wu 1972 np MSA + PC-SAFT T = 25 C m / mol kg -1 exp. data np MSA + PC-SAFT m / mol kg -1 NaBr + H 2 O 20 C 100 C +- (m) P / bar Hamer, Wu 1972 np MSA + PC-SAFT T = 25 C m / mol kg -1 exp. data np MSA + PC-SAFT 70 C 60 C 50 C 40 C 30 C m / mol kg -1 Herzog et al. I&ECR 2010 exp: Patil, et al. 1991

13 Polyelectrolytes ion dimerization fraction of dimerized ions equilibrium constant (electrostatic potential of ion dimerization) H2O-activity in aqueouspoly(sodium-acrylate) enters in the chain term in the ion term charge

14 Ionic Liquids + non-spherical molecular shape Coulomb-Coulomb interactions from integral equation theory (primitive MSA) P / MPa bmim + & BF 4- CO 2 T = K T = K T = K Modell + Anion-cation dimerization one adjustable binary parameter (dimerisization parameter) x CO2 exp. Data: Kroon et al., 2004

15 Liquid Crystals Solvents with a solubility switch

smectic (LC) nematic (LC) liquid (isotropic) w CO2 Liquid crystals are

16 Liquid Crystals as Process Solvents Liquid crystals show (1 st order) phase transitions Temperature Liquid crystals as process solvents T smectic (LC) nematic (LC) liquid (isotropic) w CO2 Liquid crystals are solvents with a solubility switch small T are sufficient for regeneration of Liquid Crystal

17 Equation of State for Liquid Crystal Mixtures Anisotropic contribution to EOS with Maier-Saupe theory the order parameter s gives the alignment of molecules, as s : 3 cos f 2 3cos 1 2 d cos single-molecule orientational distribution function the anisotropic contribution to the Helmholtz energy is F aniotr N kt 1 2 f d cos Us f ln entropy decrease due to molecular alignment anisotropic intermolecular interactions LC E U 4 kt LC S k

18 Equation of State for Liquid Crystal Mixtures The orientational distribution function minimizes F anisotr, thus f 1 Z exp Us 3cos where Z is the orientational partition function 1 2 Z 3cos 1 exp Us 2 0 d cos iteration of the order parameter s F aniotr N kt ln Z 1 2 Us 2 Maier, Saupe, Z. Naturforsch. A, mixtures: Hino, Prausnitz, Liquid Crystals 1997

19 Correlation of LC phase behavior with PC-SAFT P / MPa PCH-5 + CO T / K 2 pure component LC-parameters from pure component data

20 Equation of State for Liquid Crystal Mixtures P / MPa 3 crystal nematic isotropic 2 PCH-5 + CO 2 (3 mass% CO 2 ) 1 PC-SAFT model with Maier-Saupe term (no binary LC-parameter) T / K

21 Group contribution equation of state

FPE 2004) group contributions pseudohomosegmented mixture CO 2 n-pentane

22 Group-Contribution Equation of States Segment based Equations of state (GC-SAFT, Tamouza et al. FPE 2004) group contributions pseudohomosegmented mixture CO 2 n-pentane hetero-segmented Lymperiadis, Adjiman, Galindo, Jackson 2007 Le Thi, Tamouza, Passarello, Tobaly, de Hemptinne, Ind. Eng. Chem. Res. 2006

23 Integrated process and solvent design Lehrstuhl für Prof. Dr. André Bardow RWTH Aachen University

24 Integrated process and solvent design molecular structure physically-based equation of state analysis, direct problem process (e.g. absorption /desorption) design, inverse problem objective function Lehrstuhl für Prof. Dr. André Bardow RWTH Aachen University

25 Continuous Molecular Targeting intermolecular potential 0 molecular distance assume: all substanceshcan 2 O be described by 2 molecular parameters ( und ) / J (attractive interactions) He Ne parameters of physically-based equations of state - size & shape-parameter - dispersive attraction -H-bonding - dipol- & quadrupole moment - polarisability Ar Kr Lehrstuhl für Prof. Dr. André Bardow / Å (repulsive interactions)

26 Molecular Mapping f(x,p) f (x*,p*) Taylor expansion around optimum * f pf 1 2 pt 2 f p p* p real solvent candidates from databank ranking according to lowest f 8/10 Lehrstuhl für Prof. Dr. André Bardow

27 example pre-combustion CO 2 capture objective function based on primary energy use degrees of freedom make-up x = p flash p = (m,, /k, AB, AB /k) solvent parameters p and process variables x are simultaneously optimized 9/10 Lehrstuhl für Prof. Dr. André Bardow

28 Interfacial properties density functional theory

29 Classical Density Functional Theory (DFT) The grand potential (x) r); T, F( r); T ( dr ( r) the equilibrium density profile across an interface is determined by a minimum of the grand potential Equilibrium ( r); T, F( r); T ( r) x 0 0 equil ( ) ( r) r ( r)

DFT with PC-SAFT treated locally hard-sphere chains non-local contribution for hard spheres non-local contribution for chains (TPT1) Rosenfeld, Phys.Rev.Letter 1989 Roth et al. J. Phys.: Condens.

30 DFT with PC-SAFT treated locally hard-sphere chains non-local contribution for hard spheres non-local contribution for chains (TPT1) Rosenfeld, Phys.Rev.Letter 1989 Roth et al. J. Phys.: Condens. Matter 2002 and Yu and Wu, J. Chem. Phys Tripathi and Chapman, Phys.Rev.Letter 2005 F disp.chain [(r)] van der Waals attraction perturbation theory of 1 st order with g(r) of chain fluids compatible with PC-SAFT

31 DFT with PC-SAFT for non-spherical fluids Helmholtz energy F as a sum of repulsive and attractive contributions attractive chains perturbation theory of 1 st order with g(r) of chain fluids compatible with PC-SAFT treated locally Gloor, Jackson, Blas, del Rio, de Miguel, JCP 2004, JPC 2007

32 Surface tension of various real substances alkanes esters (ethanoates) Gross, JCP 2009 Tang, Gross, JSF 2010

33 Structural properties from DFT a lipid model Jain et al. JCP, 2007,

34 Transport properties entropy scaling

35 Entropy scaling Rosenfeld (1977) postulated for simple fluids: viscosity (s res ) with residual molar entropy s res (T,p) single-variable dependence on s res It was seen, that this scaling holds also for complex fluids currently a vivid field of research Nowak (2011) proposed a segment-based scaling CE seg with residual molar entropy s res (T,p) using PC-SAFT CE seg 5 16 M kt (m ) seg 2 (2,2) segment-based Chapman-Eskog viscosity

$/$$mpa$s$ Example viscosity of ethane 6 0.6$ 5 0.

006$ 0$ 10$ 20$ 30$ 40$ 50$ 60$ p$$/$$mpa$ 1 5 4 3 2 1 0

36 $/$$mpa$s$ Example viscosity of ethane 6 0.6$ $ viscosity ln(/ CE ) $ 0$ 10$ 20$ 30$ 40$ 50$ 60$ p$$/$$mpa$ residuel entropy s res /k viscosity (s res ) shows perfect scaling with residual molar entropy s res (T,p)

37 ! Entropy scaling of viscosity entropy scaling of dodecane group contribution approach? ln CE s res k!!!

38 Molecular simulations supported by fluid theory parameterizing GCMC simulations adjusting force fields

39 Grand canonical Monte Carlo (GCMC) probability P(N) 25 bias potential (N) number of particles N number of particles N accept probability P(N) V exp ( N N 1) min 1, exp N N 1 ( N 1) {,V,T} specified N fluctuates number of particles N U exp ( N 1) ( N ) 3

40 GCMC VLE simulations supported by PC-SAFT estimate chemical potential for coexising phases est. ( T ) PC-SAFT estimate bias potential est. ( N ) GCMC simulation reweighting the result ln P coex. ( N) ln P( N) ( T ) with condition that area of the liquid and vapor region in P(N) are equal est. coex ( T ) N removing the bias

41 Results chemical potential bias potential bias-potential estimated from PC-SAFT: coexisting phases T r =0.7 ethane simulation theory (PC-SAFT) T r = N ~ res ~ ( Tr 0.6) res ( T 0.6) r

42 A time-efficient GCMC implementation 1) Multicanonical GCMC probability P(N) many GCMC-like simulations, each for a small window of N parallel simulations for every N-window error independent of N (Virnau et al. 2004) number of particles N N 2) update bias potential ( N) ln P( N) from Transition Matrix method P( N 1) P( N) unbiased unbiased N N N 1 1 N allows the sampling of the unbiased P(N) and updating the biasing potential, although the acceptance criterion is biased (Fitzgerald, Picard, and Silver 1999) (Errington, 2003)

43 Speed-up: parallel multicanonical GCMC vs. regular GCMC speed-up number of windows

44 Adjusting force fields for molecular simulations supported by PC-SAFT

45 force fields bonded potentials bond-length potential bond angles torsion non-bonded potentials electrostatic potentials charge values and position of partial charges multipoles (static) polarizability or constant effective partial charges VdW Parameter z.b. Lennard-Jones Parameter () obtained from QM adjusted to thermodyn. properties

46 Adjusting force fields for molecular simulations Zielfunktion f(p) Simulation PC-SAFT Parameter p

47 Adjusting force fields approach coarse graining = m group contribution approach =

48 Adjusting 4 parameters to alkanes { CH3, CH3, CH2, CH2 } propane and pentane, both at T r =0.7, 0.8, 0.9 error offset L=0.16Å offset L=0.21Å p sat % L % LV h % p sat % L % LV h % p sat % L % LV h % 0 Iteration st Iteration nd Iteration Lennard-Jones (14-6) potential: propane, butane,hexane, octane at T r = p sat % L % LV h %

49 Conclusion analytic fluid theories... correlate (and predict) phase behavior of complex fluids are used for integrated process and solvent design provide interfacial properties with entropy scaling provides transport properties support molecular simulaions

50 Acknowledgements Dr. Juan Manuel Castillo Sanchez Timo Danner Andrea Hemmen Marina Stavrou Xiaohua Tang Thijs van Westen Financial Support SimTech Center of Excellence Transregio 75 Shell GS NWO

EQUATION OF STATE DEVELOPMENT

EQUATION OF STATE DEVELOPMENT I. Nieuwoudt* & M du Rand Institute for Thermal Separation Technology, Department of Chemical Engineering, University of Stellenbosch, Private bag X1, Matieland, 760, South