Ind. Eng. Chem. Res. 2010, 49, 7697–7700
7697
Revisiting the Liu-Silva-Macedo Model for Tracer Diffusion Coefficients of Supercritical, Liquid, and Gaseous Systems Ana L. Magalha˜es, Sima˜o P. Cardoso, Bruno R. Figueiredo, Francisco A. Da Silva, and Carlos M. Silva* CICECO, Department of Chemistry, UniVersity of AVeiro, 3810-193, AVeiro, Portugal
This work comprises two main purposes: to present the largest database of tracer diffusion coefficients ever published, comprehending 5279 experimental points and 296 binary systems, and provide the necessary Lennard-Jones diameter and energy, and the interaction parameter of the Liu-Silva-Macedo correlation (TLSMd), since it affords reliable and very good results for all systems studied (the global deviation found was 3.89%). For comparison, the well-known equations of Dymond-Hildebrand-Batschinsky, Zhu and coworkers, and Tyn-Calus have been adopted. Dymond-Hildebrand-Batschinsky,3,4,11 Zhu et al.,12 and Tyn-Calus.2,13
Introduction The tracer diffusion coefficients, D12, are fundamental properties for the design of heterogeneous chemical reactors and ratecontrolled separation processes. Its importance is essential for concentrated solutions also, whether binary or multicomponent, where a Maxwell-Stefan (MS) approach is highly recommended. In fact, the necessary crossed MS diffusivities may be estimated from the binary ones at infinite dilution taking into account relations of the Vignes type.1 With respect to modeling, most of the existing expressions to correlate or predict D12 are based on hydrodynamic theory, kinetic theory, absolute-rate theory of Eyring, free volume theory, and models devised for simple idealized fluids such as hard-sphere, square-well, Weeks-Chandler-Andersen, and Lennard-Jones. A detailed description of these approaches may be found elsewhere.2-4 Since 1990, several models for D12 have been published. A very recent paper by Sua´rez-Iglesias et al.5 pointed out that “the models of Ruckenstein and Liu,6 Liu et al.,7 and Dariva et al.8,9 can be considered the most significant proposed between 1990-2000 because of their general validity (applicable over wide ranges of temperature and pressure for non-hydrogenbonding compounds) and the possibility of using them to estimate binary diffusion coefficients of solutes at infinite dilution”. In particular, the predictive equation by Liu, Silva, and Macedo, hereafter denoted by TLSM, published in 1997 in this journal,10 presented an absolute average relative deviation AARD ) 14.77% for 1033 data points and 77 binary systems, while a derived one-parameter correlation, henceforth labeled TLSMd, accomplished and AARD ) 6.57% for the same database. The good results achieved by the TLSMd expression induced us to revisit both models in order to facilitate and expand their application by reporting the implied parameters. Accordingly, this research note comprises two main purposes: to make available the largest database of tracer diffusivities ever compiled up until now (5279 experimental points from 296 binary systems) and present the TLSMd LJ constants and interaction parameter for all systems analyzed. Results are compared with those from the three well-known equations of * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +351 234 401 549. Fax: +351 234 370 084.
TLSM Predictive Model (Zero Parameters) The tracer diffusion coefficient model due to Liu, Silva, and Macedo,10 TLSM, corresponds to the following set of equations: D12(cm2 /s) )
(
21.16 1000RT 2 2M12 F1σ12,eff
(
)
exp -
1/2
)
0.75F*1 0.27862 (1) 1.2588 - F*1 T*12
where subscripts 1 and 2 stand for solvent and solute, respectively, R ) 8.3144 J/mol · K is the universal gas constant, F1 (cm-3) is solvent number density, and T is absolute temperature; the reduced number density of solvent (F*1 ), reduced temperature (T*12), and reduced molar mass of the system (M12, g/mol) are calculated by 3 F*1 ) F1σ1,eff
(2)
T*12 )
T ε12,LJ /kB
(3)
M12 )
M1M2 M1 + M2
(4)
where kB is the Boltzmann constant and σ12,LJ and ε12,LJ are the binary LJ parameters (diameter and energy) computed from the single ones by the following combining rules: 3 3 3 ε12,LJ /kB ) √σ1,LJ (ε1,LJ /kB) × σ2,LJ (ε2,LJ /kB)/σ12,LJ
σ12,LJ )
σ1,LJ + σ2,LJ 2
(5) (6)
The effective binary and single diameters, σ12,eff and σ1,eff, take into account the influence of temperature upon the LJ diameters. An equation of the Boltzmann type has been adopted for their calculation:7,10,14 -1/6 σn,eff ) σn,LJ21/6[1 + √1.3229T*n ]
(7)
where subscript n equals 1 or 12, and T*n ) T/(εn,LJ/kB). The LJ force constants may be taken from the original article7 or, for
10.1021/ie1009475 2010 American Chemical Society Published on Web 07/09/2010
7698
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010
Table 1. Percent Deviations for Each System system
NDP
NS
TLSM
TLSMd
Dymond
Zhu et al.
Tyn-Calus
supercritical liquid gaseous
4256 601 422
168 91 37
16.71 13.91 8.24
3.90 4.67 2.68
3.79 5.35 2.33
37.29 37.18 39.95
16.63 35.87 -
total
5279
296
15.71
3.89
3.85
37.49
19.01
substances not covered in that essay, are calculated by the corresponding states correlations (with critical constants in K and bar):14 εLJ /kB ) 0.774Tc
3 σLJ (Å3) ) 0.17791 + 11.779
()
(8)
()
Tc Tc - 0.049029 Pc Pc
2
(9)
Equation 9 may be adopted successfully for Tc/Pc < 100. For higher values, one may estimate LJ diameter by a relation provided by the principle of corresponding states2,15,16 σLJ(Å) ) 0.809Vc1/3, where Vc is critical volume in cubic centimeters per mole. By doing so, the TLSM and TLSMd equations may be extended to higher molecular weight solutes. Summarily, the prediction of D12 at temperature T and solvent number density F1 may be accomplished by knowing only three input parameters per molecule: molecular weight and LJ diameter and energy. All data are tabled in this research note for the 296 binary systems studied. TLSMd Correlation Model (One Parameter) From the TLSM model, it is possible to generate a oneparameter correlation by inserting the interaction constant k12,d only into the diameter combining rule (eq 6). Accordingly, eqs 5 and 6 become
ε12,LJ /kB )
3 3 8√σ1,LJ (ε1,LJ /kB) × σ2,LJ (ε2,LJ /kB)
(σ1,LJ + σ2,LJ)3
σ12,LJ ) (1 - k12,d)
σ1,LJ + σ2,LJ 2
(10)
(11)
Summarily, D12 can be predicted by TLSMd model as function of T and F1 by knowing just three input parameters per molecule (M, σLJ, and εLJ/kB) and the binary interaction constant k12,d. Once more, all values of interest are tabled in this research note for the 296 binary systems studied. Data Sources and Data for Pure Substances To carry out a strict validation of both TLSM and TLSMd models and extend their application to much more systems than those presented in the original paper,10 the largest database ever used has been compiled. It comprehends globally 296 binary systems performing 5279 experimental diffusivities: 168 supercritical systems with 4256 points, 91 liquid systems with 601 points, and 37 gaseous systems with 422 points. Table S1 (see the Supporting Information) contains the systems studied, along with the data sources, number of data points (NDP), and reduced ranges of temperature, pressure, and density for each system (here the reduction is accomplished with critical constants). The data for pure substances necessary for the
calculations are presented in Table S2 (see the Supporting Information). Calculation Results In Table S1, the individual results achieved with the TLSM and TLSMd models are also compiled, namely the AARDs and the interaction binary parameters k12,d. Some few authors do not report the solvent densities of their data. In these cases, they have been calculated by the correlations of Pitzer and Schreiber17 for CO2 and Hankinson-Brobst-Thomson2,18 for other fluids. Concerning the nonreported viscosities, which are necessary for the Tyn-Calus equation adopted for comparison in this work, they have been estimated by the correlations of Mehrotra,19 for liquid hydrocarbons, and Altunin and Sakhabetdinov,20 for carbon dioxide. Taking into account the constants and properties listed in this paper, it is now possible to estimate easily the tracer diffusivity of a large number of systems using TLSMd correlation. When k12,d is unknown, the predictive model should be adopted; nonetheless, if some data points may be found in the literature, the value of k12,d may be first optimized. In the whole, very good results have been accomplished, as the grand averages shown in Table 1 point out: AARD(TLSM) ) 15.71% and AARD(TLSMd) ) 3.89%. Concerning the equations adopted for comparison, the deviations of the predictive models of Zhu et al.12 and Tyn-Calus2,13 are 37.49% and 19.01%, respectively, and the two-parameter equation of Dymond-Hildebrand-Batschinsky (DHB)3,4,11 presents 3.85%. The binary parameter, k12,d, seems sufficient to achieve good representations of D12 for all systems studied, which emphasizes the reliability of the model. It is worth noting that its introduction has diminished significantly the AARDs of TLSM. On the other hand, despite possessing two parameters, the DHB equation exhibits results comparable to those of TLSMd (AARD ) 3.89% Versus 3.85%). Nonetheless, it is well-known their parameters are physically meaningless, which limits its interest to interpolation purposes only.3,10 It may be emphasized that the good performance achieved for systems whose LJ force constants (energy and diameter) were both calculated using critical constants also estimated. For instance, this is the case of systems containing 1,1′-dimethylferrocene, cobalt(III) acetylacetonate, copper(II) trifluoroacetylacetonate, dibenzo-24-crown-8, ferrocene, γ-linolenic acid ethyl ester, palladium(II) acetylacetonate, squalene, tetrabutyltin, thenoyltrifluoroacetone, triarachidonin, triolein, ubiquinone CoQ10, and vitamins K1 and K3. In the case of these molecules, the unique properties previously known were molecular weight and boiling point. Note that most group contribution methods available to estimate Tc, Pc, and Vc do not comprehend metallic atoms like Co, Fe, Pd, Cu, and Sn. Hence, the critical constants have been calculated by the Klincewicz method.2,21 Even so, the AARDs found were surprisingly small for these systems, in the range of 0.52-6.27%. In Figure 1, the experimental tracer diffusivities of palladium(II) acetylacetonate and β-carotene in carbon dioxide have been plotted against solvent density, along with the results provided by the TLSMd correlation. These systems were selected to illustrate the
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010
7699
Subscripts 1 ) solvent 2 ) solute 12 ) binary property c ) critical property eff ) refers to the effective hard sphere diameter r ) reduced property (using critical constants) Superscripts * ) reduced property (using LJ parameters)
Figure 1. Experimental tracer diffusivities of palladium(II) acetylacetonate and β-carotene in carbon dioxide, plotted against solvent density. The results provided by TLSMd correlation also are shown.
behavior of the model over wide density ranges for systems whose LJ parameters were estimated, and in the severe situation where even the critical constants were also unknown. As Figure 1 points out, results are in good agreement with experimental data. Conclusions In this work, the largest database of tracer diffusion coefficients has been compiled, embodying 296 systems and 5279 points, in order to enlarge the applicability of the previously published TLSM and TLSMd models (with 0 and 1 parameters, respectively). The input data are the molecular weights and Lennard-Jones diameters and energies of the solute and solvent, and one binary interaction parameter of the system. All of them were calculated/optimized in this research note. The deviations found (AARD ) 3.89%) point out that the TLSMd model provides reliable and very good estimates of D12. Acknowledgment A.L.M. wants to acknowledge with thanks the financial support granted by Fundac¸a˜o para a Cieˆncia e a Tecnologia (SFRH/BD/46776/2008). Nomenclature cal AARD ) average absolute relative deviation: (100/NDP) ∑NDP i-1 |D12,i exp exp 2 - D12,i |/D12,i | (cm /s) D ) diffusion coefficient (cm2/s) DHB ) Dymond-Hildebrand-Batschinski k12,d ) binary interaction parameter LJ ) Lennard-Jones TLSM ) Tracer Liu-Silva-Macedo equation M ) molecular weight (g/mol) Na ) Avogadro constant NDP ) number of data points NS ) number of systems P ) pressure (bar) R ) 8.3144J/mol · K, universal gas constant T ) temperature (K)
Greek Letters ε/kB ) Lennard-Jones energy parameter (K) σ ) Lennard-Jones diameter (cm) F ) number density (cm-3)
Supporting Information Available: Database and the detailed results for TLSM and TLSMd models (Tables S1 and S2). Table S1 contains the systems studied, the reduced ranges of temperature, pressure, and solvent density for each system (reduction performed with critical constants), number of data points (NDP), detailed results obtained with TLSM and TLSMd models, and data sources. In Table S2, the name, formula, molecular weight, critical constants (Tc, Pc, and Vc), and LJ force constants (σLJ and εLJ/kB) are listed for all molecules involved in calculations. This material is available free of charge via the Internet at http://pubs.acs.org/. Literature Cited (1) Taylor, R.; Krishna, R. Multicomponent Mass Transfer: John Wiley & Sons, Inc: New York, 1993. (2) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids. 4th ed.; McGraw-Hill Professional: New York, 1987. (3) Silva, C. M.; Liu, H. Modeling of Transport Properties of Hard Sphere Fluids and Related Systems, and its Applications. In Theory and Simulation of Hard-Sphere Fluids and Related Systems; Mulero, A., Ed.; Springer: Berlin/Heidelberg, 2008; Vol. 753, pp 383-492. (4) Dymond, J. H.; Bich, E.; Vogel, E.; Wakeham, W. A.; Vesovic, V.; Assael, M. J. Theory - Dense Fluids. In Transport Properties of Fluids Their Correlation, Prediction and Estimation; Millat, J.; Dymond, J. H.; Nieto de Castro, C. A., Eds.; Cambridge University Press: London, 1996; pp 66-112. (5) Sua´rez-Iglesias, O.; Medina, I.; Pizarro, C.; Bueno, J. L. On predicting self-diffusion coefficients in fluids. Fluid Phase Equilib. 2008, 269 (1-2), 80–92. (6) Ruckenstein, E.; Liu, H. Self-diffusion in gases and liquids. Ind. Eng. Chem. Res. 1997, 36 (9), 3927–3936. (7) Liu, H.; Silva, C. M.; Macedo, E. A. Unified approach to the selfdiffusion coefficients of dense fluids over wide ranges of temperature and pressure - hard-sphere, square-well, Lennard-Jones and real substances. Chem. Eng. Sci. 1998, 53 (13), 2403–2422. (8) Dariva, C.; Coelho, L. A. F.; Oliveira, J. V. A kinetic approach for predicting diffusivities in dense fluid mixtures. Fluid Phase Equilib. 1999, 158-160, 1045–1054. (9) Dariva, C.; Coelho, L. A. F.; Oliveira, J. V. Predicting diffusivities in dense fluid mixtures. Braz. J. Chem. Eng. 1999, 16, 213–227. (10) Liu, H.; Silva, C. M.; Macedo, E. A. New equations for tracer diffusion coefficients of solutes in supercritical and liquid solvents based on the Lennard-Jones fluid model. Ind. Eng. Chem. Res. 1997, 36 (1), 246– 252; corrected in Ind. Eng. Chem. Res. 1998, 37 (1), 308. (11) Dymond, J. H. Corrected Enskog theory and transport coefficients of liquids. J. Chem. Phys. 1974, 60 (3), 969–973. (12) Zhu, Y.; Lu, X.; Zhou, J.; Wang, Y.; Shi, J. Prediction of diffusion coefficients for gas, liquid and supercritical fluid: application to pure real fluids and infinite dilute binary solutions based on the simulation of LennardJones fluid. Fluid Phase Equilib. 2002, 194-197, 1141–1159. (13) Tyn, M. T.; Calus, W. F. Diffusion coefficients in dilute binary liquid mixtures. J. Chem. Eng. Data 1975, 20 (1), 106–109. (14) Silva, C. M.; Liu, H.; Macedo, E. A. Models for self-diffusion coefficients of dense fluids, including hydrogen-bonding substances. Chem. Eng. Sci. 1998, 53 (13), 2423–2429. (15) Chung, T.-H.; Lee, L. L.; Starling, K. E. Applications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity. Ind. Eng. Chem. Fundam. 1984, 23 (1), 8–13.
7700
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010
(16) Chung, T.-H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Generalized multiparameter correlation for nonpolar and polar fluid transport properties. Ind. Eng. Chem. Res. 1988, 27 (4), 671–679. (17) Pitzer, K. S.; Schreiber, D. R. Improving equation-of-state accuracy in the critical region; equations for carbon dioxide and neopentane as examples. Fluid Phase Equilib. 1988, 41 (1-2), 1–17. (18) Hankinson, W. R.; Thomson, H. G. A new correlation for saturated densities of liquids and their mixtures. AIChE J. 1979, 25 (4), 653–663. (19) Mehrotra, A. K. Generalized one-parameter viscosity equation for light and medium liquid hydrocarbons. Ind. Eng. Chem. Res. 1991, 30 (6), 1367–1372.
(20) Altunin, V. V.; Sakhabetdinov, M. A. Viscosity of liquid and gaseous carbon dioxide at temperatures 220-1300 K and pressure up to 1200 bar. Teploenergetika 1972, 8, 85–89. (21) Klincewicz, K. M.; Reid, R. C. Estimation of critical properties with group contribution methods. AIChE J. 1984, 30 (1), 137–142.
ReceiVed for reView April 22, 2010 ReVised manuscript receiVed June 23, 2010 Accepted June 23, 2010 IE1009475
