A multi-fluid nonrandom lattice fluid model: General derivation and application to pure fluids

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1 Korean J. Chem. Eng., 3(3), (006) SHORT COMMUNICATION A multi-fluid nonrandom lattie fluid model: General derivation and appliation to pure fluids Moon Sam Shin, Ki-Pung Yoo, Chul Soo Lee and Hwayong Kim Shool of Chemial and Biologial Engineering and Institute of Chemial Proesses, Seoul National University, Seoul 5-744, Korea Department of Chemial and Biomoleular Engineering, Sogang University, Seoul -74, Korea Department of Chemial and Biologial Engineering, Korea University, Seoul 36-70, Korea (Reeived 8 November 005 aepted 3 Deember 005) Abstrat A multi-fluid nonrandom lattie fluid model with no temperature dependene of lose paked volumes of a mer, segment numbers and energy parameters of pure systems is presented. The multi-fluid nonrandom lattie fluid (MF-NLF) model with the loal omposition onept was apable of desribing properties for omplex systems. However, the MF-NLF model has strong temperature dependene of energy parameters and segment numbers of pure systems; thus empirial orrelations as funtions of temperature were represented for reliable and onvenient use in engineering praties. The MF-NLF model without temperature dependene of pure parameters ould not predit thermodynami properties aurately. It was found that the present model with three parameters desribes quantitatively the vapor pressure and the saturated density for the pure fluid. Key words: Multi-fluid, Nonrandom, Lattie, Loal Composition, Equation of State INTRODUCTION Historially two-liquid theory [Wilson, 964] and rigid lattie desription without vaant sites (or holes) of fluid systems [Guggenheim, 95] have been widely utilied to formulate various exess Gibbs free energy expressions suh as NRTL [Renon and Prausnit, 968] and UNIQUAC [Abrams and Prausnit, 975]. However, they an be applied only to liquid mixtures at low pressure. To overome this limitation, various equation-of-state theories have also been proposed by imbedding holes into the rigid lattie desription of fluids. One of the well known examples is the EOS after Sanhe and Laombe [Sanhe and Laombe, 976; Laombe and Sanhe, 976]. The present authors also reently proposed the multi-fluid nonrandom lattie fluid (MF-NLF) equation of state [Shin et al., 995; Yoo et al., 995, 997] and the MF-NLF model was apable of desribing properties for omplex systems. However, the MF-NLF model has strong temperature dependene of energy parameters and segment numbers of pure systems; thus empirial orrelations as funtions of temperature were adopted for reliable and onvenient use in engineering praties. If the MF-NLF model neglets the temperature dependene of these parameters, this model annot predit thermodynami properties aurately. In addition, Neau [Neau, 00] presented a onsistent method for phase equilibrium alulation using the fugaity oeffiients derived from the Sanhe-Laombe equation of state. It was pointed out that the onfigurational hemial potential derived from the onfigurational partition funtion does not verify phase equilibrium property. In this work, a multi-fluid nonrandom lattie fluid model with no To whom orrespondene should be addressed. hwayongk@snu.a.kr temperature dependene of lose paked volumes of a mer, segment numbers and energy parameters of pure systems will be presented. GUGGENHEIM S COMBINATORY AND THERMODYNAMIC PROPERTIES The anonial partition funtion may be written in the lassial approximation of statistial thermodynamis [Prausnit et al., 999]: Q(N, T, V)=Q int (N, T)Q kin (N, T)Z N (N, T, V) () where Q int, Q kin are the internal and kineti part of the anonial partition funtion and Z N is the onfigurational partition funtion. The expression for the anonial Helmholt free energy may be obtained from the internal and kineti part of the Helmholt free energy and the onfigurational Helmholt free energy. βa(n, T, V)=βA int (N, T)+βA kin (N, T)+βA (N, T, V) () The onfigurational Helmholt free energy A may be omposed of an athermal part A, A whih is equivalent to a ombinatorial ontribution in the random array and a residual part A, R whih is due to the residual nonrandom interation energy. βa =βa, A +βa, R (3) The athermal part of the onfigurational Helmholt free energy for a hole-single omponent fluid is given by, βa A, = N i ρ i i= ln + N 0 ln ρ ( ) q --N q ln + -- ρ r where, N 0 and N i are the number of vaant sites whih may be alled holes and the number of moleules of speies i; r i and q i are the segment number and the surfae area parameter, respetively. q = x i q i, r = x i r i, ρ i = N i r i /N r, ρ = ρ i, N q = N 0 + N i q i (4) (5) 469

2 470 M. S. Shin et al. The residual part of the onfigurational Helmholt free energy with two-liquid theory is obtained from the thermodynami relation at onstant volume and omposition, β βa = Udβ + onstant β 0 where the internal energy U is residual part in the nonrandom twofluid theory. If β 0 is taken as a very high temperature, the solution beomes a random mixture. The onstant in Eq. (6) beomes βa, A and the integral beomes βa, R. Thus, Eq. (6) an be rewritten as, βa R β, = U dβ β 0 Now, N ij are evaluated in the two-liquid theory [Wilson, 964], N ij = --N i q j τ ji i N j q i τ ij j k τ ki k τ kj where the nonrandomness fator τ ji is defined as, τ ji =exp[β(ε ji ε ii )] (9) Then, the residual part of the onfigurational Helmholt free energy with two-liquid theory is given by, βa R, = N q ln τ i j ji + βε ii i= j= (6) (7) (8) (0) and ε ji is the absolute value of the interation energy between segments of speies i and j. From the onfigurational Helmholt free energy expression given by Eq. (3), onfigurational expressions for thermodynami properties an be obtained. Sine the volume is represented by, v = ν N r = ν ( N 0 + r i N i ) () The EOS (equation of state) is obtained from the equation of onfigurational Helmholt free energy, P = P = A V = TN, A V TN, () Obviously, the onfigurational pressure obtained in this way is the absolute pressure; P=P. The anonial and onfigurational hemial potentials are expressed from the anonial and onfigurational Helmholt free energy, respetively, µ i = V( NTV,, ) A int ( NT, ) A kin ( NT, ) = N i TN, j N i TN, j µ i = A ( NTV,, ) A ( NTV,, ) (3) (4) The anonial hemial potentials are not equal to the onfigurational hemial potentials sine A int, A kin are funtions of both temperature and number of moleules. The onfigurational hemial potentials an be used for phase equilibrium onditions in the lassial lattie fluid model; thus the Sanhe-Laombe model leads to an erroneous method for phase equilibrium alulations. To overome this problem, phase equilibrium onditions should be alulated by using fugaity oeffiients whih ould be derived diretly from the equation of state beause the onfigurational pressure is equal to the absolute pressure. Fugaity oeffiients were alulated by derivation of the residual Helmholt free energy A r (T, V, N) obtained by integration of the lattie fluid equation of state. ln ϕ i = A r ( TVN,, ) A r V ( TVN,, ) = P EOS AND FUGACITY COEFFICIENT FOR PURE FLUIDS (5) (6) In this work, we present a multi-fluid nonrandom lattie fluid (MLF) model with no temperature dependene of lose paked volumes of a mer (v i ), segment numbers (r ) and moleular interation energy (ε ) of pure systems. From Eq. (), we obtain the expression for equation of state for pure omponents. P -- = ln ( ρ ) + ln (7) Here, all the quantities with the tilde (~) denote the redued variables defined by P T ρ r P = ----, = ----, ρ = ---- N = , ρ = P T ρ N r r v Where the reduing parameters are defined by P v = kt = --ε M and ε M, are defined by ε M = = -- + τ 0i i β i= k τ ki lnz NkT dv V q r --- N q ( q /r )ρ = = 0 N q + ( q /r )ρ ρ ---- (8) (9) (0) () The Residual Helmholt energy, A r (T, V, N) and fugaity oeffiient were alulated given by Eqs. (5)-(6), whih has an idential form of the MF-NLF model. A r = N r -- kt ρ ρ ln( ) -- N r + N ρ r q --- N ln ρ + N q r ( ) + P q ϕ = r -- q r ln ρ Z ln ln ( r ) r q k τ ki + βε l ( τ il τ 0l( r /q )) ln l= m τ ml m=0 () (3) May, 006

3 A multi-fluid nonrandom lattie fluid model: Pure fluids 47 RESULTS AND DISCUSSION There are apparently four moleular parameters in this model for pure fluids:, v, r, ε. We set =0 as used in lattie fluid theories of the same genre [Panayiotou and Vera, 98]. The present model has three pure omponent parameters without temperature dependene of lose paked volumes of a mer (v ), segment numbers (r ) and energy parameters (ε ) of pure systems. Three pure omponent parameters have been regressed by using the present EOS and fugaity oeffiient from the least-square minimiation using differenes in alulated and experimental vapor pressure and liquid density data for 80 real fluids. For eah ompound, Table lists temperature range, v, r, ε /k and perent average absolute deviation (AAD %) in vapor pres- Table. Pure moleular parameters for real fluids Components v /mol m3 r ε /k, K AADP (%) AADρ (%) T range, K Methane Ethane Propane Butane Isobutane Pentane Isopentane Hexane Heptane Otane Nonane Deane Undeane Dodeane Trideane Tetradeane Pentadeane Hexadeane Heptadeane Oatadeane Nonadeane Eiosane Triosane Tetraosane Otaosane Triaontane Dotriaontane Cylopropane Cylobutane Cylopentane Cylohexane Cyloheptane Cylootane Methylylopentane Ethylylopentane Propylylopentane Butylylopentane Methylylohexane Ethylylohexane Propylylohexane Butylylohexane Deylylohexane Butene Pentene Korean J. Chem. Eng.(Vol. 3, No. 3)

4 47 M. S. Shin et al. Table. Continued Components v /mol m3 r ε /k, K AADP (%) AADρ (%) T range, K Hexene Heptene Htene Benene Toluene Ethylbenene Propylbenene Butylbenene M-xylene Dimethyl ether Diethyl ether Diisopropyl ether Aetone Methylethyl ketone Methyl propyl ketone Diethyl ketone Methylaetate Ethylaetate Butylaetate Nitrogen Oxygen Argon Carbon monoxide Chlorine Carbon disulfide Tetrahloroethylene Carbon tetrahloride Chloroform Dihloromethane Methylhloride Hexahloroethane Trihloroethylene Pentahloroethane Vinylhloride Ethylhloride Chlorobenene sure and liquid density. Vapor pressure and liquid density data are from KDB [Kang et al., 00]. The quality of fit for both vapor pressure and liquid density is as good as an be usually expeted for a reasonable, three-parameter lattie fluid equation of state. The present model is ompared with the MF-NLF model, SAFT [Huang and Rados, 990] model to alulate the vapor pressure and the saturated density. The MF-NLF model has strong temperature dependene of energy parameters and segment numbers of pure systems, as shown in Fig. ; thus empirial orrelations as funtions of temperature were adopted for reliable and onvenient use in engineering praties. 6MF-NLF denotes the MF-NLF model with six adjustable parameters and MF-NLF denotes the MF- NLF model with only two parameters having no temperature dependene. The SAFT model has three pure parameters (segment volume, v 00, segment number, m, dispersion energy, u 0 /k) in nonassoiating moleules. For hexane and benene omponents, Table lists perent average absolute deviation (AAD %) in vapor pressure and liquid density. It is shown that the present model is as aurate as the 6MF- NLF and the SAFT models, but the MF-NLF model failed to orrelate the vapor pressure. Therefore, the MF-NLF model without temperature dependene of pure parameters ould not predit thermodynami properties aurately. In order to ompare three models (present model, 6MF-NLF and SAFT) fairly, we use the same experimental data of vapor pressure and liquid density for the ethane omponent. Pure parameters and perent average absolute deviation (AAD %) in vapor pressure and liquid density are listed in Table 3 and Fig.. It is shown that the present model is slightly better than SAFT model. The 6MF-NLF model is equivalent to the present model, but the 6MF-NLF model May, 006

5 A multi-fluid nonrandom lattie fluid model: Pure fluids 473 Fig.. Temperature dependene of energy parameters and segment numbers of hexane system in the MF-NLF model. has more adjustable pure parameters than the present model and SAFT model. Now we fous on the pure parameter behavior aording to molar mass. The segment number r inreases with inreasing molar mass within eah homologous series pratially linearly. The segment number r is plotted versus molar mass for n-alkanes, n-yloalkanes, n-alkylylopentanes, n-alkylylohexanes, and n-alkylbenenes in Fig. 3. It is reassuring to find that a linear relationship holds for eah hydroarbon. The lose paked volume of a moleule r v is a single linear funtion of the molar mass for all the hydroarbons, as shown in Fig. 4. Apart from testing these reassuring trends, a pratial reason for developing a orrelation for r v is to provide the basis for estimating v, if there are no aurate PVT data available, for a given r and molar mass. A similar molar mass orrelation an be developed for the moleular interation energy ε /k, whih is shown in Fig. 5. Unlike r and r v, ε /k is nonlinear with respet to the molar mass. For the ease of estimating, r, r v and ε /k have been regressed as simple funtions of the mass (MM) for many homologous series. Fig.. Calulation results of vapor pressure and liquid density in ethane. For example, r =A ()+A () MM for all hydroarbons (4) Table. Calulation results of vapor pressure and liquid density in hexane and benene omponent Models Hexane [ K] Benene [ K] AADP (%) AADρ (%) AADP (%) AADρ (%) MF-NLF ε /k=05.46, r =0.6 for Hexane ε /k=0.3, r =0.5 for Benene 6MF-NLF MLF (present model) SAFT Table 3. Calulation results of vapor pressure and liquid density in ethane Models AADP (%) AADρ (%) Pure parameters T range, K MF-NLF E a =58.350, E b = 0.06, E = V a =7.737, V b =0.043, V = 6.45 MLF v =7.9, r =6.066, ε /k= SAFT v 00 =.044, m=.46, u 0 /k= Korean J. Chem. Eng.(Vol. 3, No. 3)

6 474 M. S. Shin et al. Fig. 3. Segment number r for hydroarbons, as linear funtions of molar mass. Fig. 5. Moleular interation energy ε /k for n-alkane and other hydroarbon, as smooth but nonlinear funtion of molar mass. Table 4. Correlation of the segment number r for hydroarbons Systems A () A () MM Range n-alkynes Cylohexanes n-alkylylopentane n-alkylylohexane n-alkylbenene Fig. 4. Close paked volume r v for hydroarbons, as single linear funtions of molar mass. r v = MM for all hydroarbons (5) ε /k=a ()+A () ln(mm) for n-alkanes (6) ε /k=a ()+A () MM for other hydroarbons (7) Table 5. Correlation of the moleular interation energy ε /k for hydroarbons Systems A () A () MM Range n-alkynes Cylohexanes n-alkylylopentane n-alkylylohexane n-alkylbenene We note that Eq. (7) is only a linear approximation that is valid up to the MM of a orresponding n-alkane; for higher MM values, we set ε /k to be the same as that for a orresponding n-alkane. The solid lines and urves shown in Figs. 3-5 have been predited from Eqs. ()-(4). Speifi values for the regression oeffiients A (i) are reported in Tables 4 and 5. In this way, we have a useful method for estimating all the equation of state parameters for pure fluids where no aurate data are available and, espeially, for poorly defined pseudo-omponents where only average molar mass and average aromatiity are available. CONSLUSIONS A multi-fluid nonrandom lattie fluid (MLF) model with three moleular parameters (lose paked volumes of a mer, segment numbers, and moleular interation energy) is presented. The MF-NLF model was apable of desribing properties for omplex systems, but has strong temperature dependene of energy parameters and segment numbers of pure systems. The MF-NLF model without temperature dependene of pure parameters ould not predit thermodynami properties aurately. It was found that the present model with three parameters desribes quantitatively the vapor pressure and the saturated density for a pure fluid. May, 006

7 A multi-fluid nonrandom lattie fluid model: Pure fluids 475 ACKNOWLEDGMENT This work was supported by the BK projet of Ministry of Eduation of Korea. NOMENCLATURE A : Helmholt free energy A (i) : regression onstants in Eq. (6), (8), (9) : number of the omponent f : fugaity k : Boltmann s onstant N : number of moleule N r : defined by Eq. (8) N q : defined by Eq. (5) m : effetive number of segments with the moleule for SAFT model MM : molar mass P :pressure P : onfigurational pressure Q : anonial partition funtion q : surfae area parameter r : number of segments per moleule T : temperature u 0 : dispersion energy of interation between segments for SAFT model V : volume v : molar volume v : lose paked volume of a mer v : segment volume for SAFT model 00 : lattie oordination number Z : ompressibility fator : onfigration partition funtion Z N Greek Letters β : reiproal temperature (/kt) Γ ij : quasi-hemial nonrandomness fator ε ij : moleular interation energy ε M : defined by Eq. (7) ϕ : fugaity oeffiient ρ : molar density ρ : lose paked molar density µ : hemial potential τ ij : nonrandomness fator : surfae area fration Supersripts a : athermal solution properties : onfigurational properties r : residual properties ~ : redued properties : harateristi properties Subsripts i : omponent i 0 : vaant sites or holes int : internal part kin : kineti part REFERENCES Abrams, D. S. and Prausnit, J. M., Statistial thermodynamis of liquid mixtures: a new expression for the exess gibbs energy of partly or ompletely misible systems, AIChE J.,, 6 (975). Guggenheim, E. A., Mixtures, Clarendon Press, Oxford, 5 (95). Huang, S. H. and Rados, M., Equation of state for small, large, polydisperse and assoiating moleules, Ind. Eng. Chem. Res., 9, 84 (990). Kang, J., Yoo, K., Kim, H., Lee, J., Yang, D. and Lee, C. S., Development and urrent status of the korea thermophysial properties databank (KDB), Int. J. Thermophys.,, 487 (00). Laombe, R. H. and Sanhe, I. C., Statistial thermodynamis of fluid mixtures, J. Phys. Chem., 80, 368 (976). Neau, E., A onsistent method for phase equilibrium alulation using the sanhe-laombe lattie-fluid equation-of-state, Fluid Phase Equilibria, 03, 33 (00). Panayiotou, C. and Vera, J. H., Statistial thermodynamis of r-mer fluids and their mixtures, Polymer J., 4, 68 (98). Prausnit, J. M., Lihtenhalter, R. N. and de Aevedo, E. G., Moleular thermodynamis of fluid-phase equilibria, Prentie-Hall, Englewood Cliffs, NJ (999). Renon, H. and Prausnit, J. M., Loal ompositions in thermodynami exess funtions for liquid mixtures, AIChE J., 4, 35 (968). Sanhe, I. C. and Laombe, R. H., An elementary moleular theory of lassial fluids. pure fluids, J. Phys. Chem., 80, 35 (976). Shin, M. S., Yoo, K. P., You, S. S. and Lee, C. S., A new nonrandom lattie fluid model and its simplifiation by two-liquid theory for phase equilibria of omplex mixtures, Int. J. Thermophysis, 6, 73 (995). Wilson, G. M., Vapor-liquid equilibrium. XI. A new expression for the exess free energy of mixing, J. Am. Chem. So., 86, 7 (964). Yoo, K. P., Shin, M. S., Yoo, S. J., You, S. S. and Lee, C. S., A new equation of state based on nonrandom two-fluid lattie theory for omplex mixtures, Fluid Phase Equilibria,, 75 (995). Yoo, K. P., Kim, H. and Lee, C. S., Unified equation of state based on the lattie fluid theory for phase equilibria of omplex mixtures. Part I. Moleular thermodynami framework, Korean J. Chem. Eng.,, 77 (995). Yoo, K. P., Kim, H. and Lee, C. S., Unified equation of state based on the lattie fluid theory for phase equilibria of omplex mixtures. Part II. Appliation to omplex mixtures, Korean J. Chem. Eng.,, 89 (995). Yoo, K. P., Shin, H. Y. and Lee, C. S., Approximate nonrandom twofluid lattie-hole theory. general derivation and desription of pure fluids, Bull. Korean Chem. So., 8, 965 (997). Yoo, K. P., Shin, H. Y. and Lee, C. S., Approximate nonrandom twofluid lattie-hole theory. thermodynami properties of real mixtures, Bull. Korean Chem. So., 8, 84 (997). Korean J. Chem. Eng.(Vol. 3, No. 3)