Ind. Eng. Chem. Res. 1987,26, 815-819 815

s = constant related to the physical properties of the dis-

T = temperature, O C

t = sound pulse transmission time, s We = Weber number, (D3Wp/uI&”,6 Greek Symbols a = coefficient of volume expansion, O C - l

p = adiabatic compressibility, m2/N or cm2/dyn at = thickness of thermal layer, m y = heat capacity ratio AT = temperature difference, “C r) = shear viscosity, kg/(s m) or CP A = wavelength, m p = density, kg/m3 u = thermal conductivity, kcal/(kg s “C) or cal/(cm s “C) uIT = interfacial tension, N/m or dyn/cm T = constant related to the physical properties of the dis-

persion, kg/(s m) or CP

. . . .

persion C$ = fractional volume dispersed-phase holdup w = angular frequency, rad/s = 27rf

Literature Cited Ahuja, A. S. J. Appl. Phys. 1973, 44(11), 4863. Ballaro, S.; Mallamaci, F.; Wanderleng, F. Phys. Lett. 1980, 774, 198.

Bonnet, J. C.; Jeffreys, G. V. AZChE J . 1985,31(5), 788. Burdett, J. D.; Webb, D. R.; Davies, G. A. Chem. Eng. Sci. 1981,36,

Chaban, I. A. Sou. Phys.-Acoust. (Engl. Transl.) 1974, 19(6), 600. Coulaloglou, C.; Tavlarides, L. L. AZChE J. 1976, 22(2), 289. Ferrarin, V. D. J. Forschungsheft 1970, Nr. 551. Fisher, A. Ferfahrenstechnik 1971, 5, 360. Havlicek, A.; Sovova, H. Collect. Czech. Chem. Commun. 1984,49,

Hoffer, M. S.; Resnick, W. Chem. Eng. Sci. 1975, 30, 473. Jiricny, V.; Prochazka, J. Chem. Eng. Sci. 1980, 35, 2237. Kol’tsova, I. S.; Mikhailov, I. G. Sou. Phys.-Acoust. (Engl. Transl.)

Kuster, G. T.; Toksoz, M. N. Geophysics 1974, 39(5), 587. Mason, W. P.; McSkimin, H. J. Appl. Phys. 1948,19, 940. Misek, T. Collect. Czech. Chem. Commun. 1963, 28, 426. Morse, P. M.; Ingard, K. U. Theoretical Acoustic; McGraw Hill:

Mylnek, Y.; Resnick, W. AZChE J. 1972, 18(1), 122. Tavlarides, L. L.; Bonnet, J. C. U S . Patent App. Serial 914 370, Nov

Weast, R. C. Handbook ofchemistry and Physics, 53rd ed.; Chemical

Wood, A. B. A Textbook of Sound, 3rd ed.; Bell: London, 1955.

Received for review August 20, 1985 Accepted November 3, 1986

1915.

37e.

1976, 21(4), 351.

New York, 1968.

13, 1986.

Rubber Co.: Cleveland, OH, 1973.

Tracer Diffusion in Dense Methanol and 2-Propanol up to Supercritical Region: Understanding of Solvent Molecular Association and Development of an Empirical Correlation

C. K. Jacob Sun and Shaw-Horng Chen* Department of Chemical Engineering, University of Rochester, Rochester, New York 14627

The tracer diffusion coefficients of aromatic hydrocarbon solutes in liquid methanol (Tc = 512.6 K) up to 473.4 K and of aromatic and aliphatic hydrocarbon solutes in dense 2-propanol (Tc = 508.3 K) up to 536.0 K have been determined with the Taylor-Aris dispersion method. It is demonstrated that molecular association phenomena can be revealed for hydrogen-bonded solvents from the tracer diffusion data using the hard-sphere equation established for cyclohexane solvent. The resultant association numbers are shown to be consistent with those determined by more conventional means. In addition, they allow both tracer diffusion and shear viscosity in dense fluids, be they nonassociating or hydrogen-bonded, to be treated successfully within the same conceptual framework. Based on the tracer diffusion data reported here in addition to those published previously, a general correlation equation is developed which predicts tracer diffusion coefficients in liquids as well as supercritical dense gases within *5% of the observed values.

Knowledge of binary diffusion coefficients is needed of the design of chemical reactors and separation processes where mass transfer is involved. In particular, interphase mass transfer occurring in supercritical-fluid (SCF) ex- traction processes depends on the molecular diffusion of an extracted component in a dense fluid under practical conditions bounded by 0.9 < T R < 1.2 and PR > 1.0 (Paulaitis et al., 1983). In recognition of the importance of binary diffusion data in understanding the fundamentals of mass transfer and in equipment and process design for SCF separation, we have been actively pursuing the measurement, interpretation, and correlation of binary tracer diffusivities of a series of aromatic and aliphatic hydrocarbons in solvents of current and potential appli- cations. Until now we have reported our progress in nonpolar solvents including cyclohexane (Sun and Chen,

*Author to whom correspondence should be addressed.

0888-5885/ 87 / 2626-0815$01.50/0

1985a), n-hexane (Sun and Chen, 1985b), 2,3-dimethyl- butane (Sun and Chen, 1985c), and ethanol (Sun and Chen, 1986). In addition to engineering correlations de- veloped for practical applications, a rough-hard-sphere (RHS) theory based on Sung and Stell’s (1984) formalism of the dynamic correlations of molecular motion was shown to be quite successful in representing the diffusion data in dense fluids from the liquid state to and beyond sol- vents’ critical temperatures (Chen et al., 1985; Sun and Chen, 1985~). The success of the RHS model can be at- tributed to the fact that the diffusional process in nonpolar dense fluids is dominated by the short-range order on the molecular scale. I t has to be noted that our observation thus far has been in the fluid region where the “critical anomaly” (Chu, 1972) is absent.

In the present work we have determined the tracer diffusion coefficients of benzene, toluene, mesitylene, naphthalene, phenanthrene, n-decane, and n-tetradecane

0 1987 American Chemical Society

816 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987

Table I. Tracer Diffusivities (10BD12, m2/s) of Aromatic Hydrocarbon Solutes in Liquid Methanol as a Function of Temperature

109D,1 at T (K), P (kg/m3)," 104w (N-s/m2Ib 313.2, 373.2, 423.2, 473.4,

774, 4.50 712, 2.28 651, 1.35 559, 0.889 benzene 3.26 7.23 11.9 20.1 toluene 3.05 6.72 11.1 18.3 mesitylene 2.51 5.57 9.27 15.7 naphthalene 2.61 5.85 9.64 16.3 phenanthrene 2.21 4.92 8.24 14.2

O1 Densities of liquid methanol from Smith (1948). Viscosities of liquid methanol from Raznjevic (1976) and Stephan and Lucas (1979).

in dense 2-propanol (Tc = 508.3 K; pc. = 273.2 kg/m3) at 0.66 I TR 5 1.05 and pR 1 1.53 and in liquid methanol (Tc = 512.6 K; pc = 271.5 kg/m3) up to TR = 0.92, using the Taylor-Aris dispersion method (Sun and Chen, 1985b). The limitation on the temperature range for solvent methanol is primarily due to the possible corrosion of the stainless steel tubing by methanol (Mowery, 1985), which results in blockage of the flow system. This study was motivated by the fact that the long-range interactions between solvent molecules via hydrogen bonding should extend our previous work on nonpolar solvents toward understanding transport properties in molecular dense fluids and generating correlations between tracer diffu- sivity and readily available solute and solvent properties for engineering applications. Specifically, molecular as- sociation in hydrogen-bonded solvents is to be shown to provide an additional degree of freedom by which not only tracer diffusivity but also shear viscosity in both nonpolar and hydrogen-bonded fluids can be treated within a com- mon framework. Furthermore, all the tracer diffusion data determined to date are employed to develop an empirical correlation for nonpolar and hydrogen-bonded solvents.

Experimental Section The principle and practice of determining binary tracer

diffusion coefficients in liquids and supercritical dense gases have been presented elsewhere (Sun and Chen, 1985a,c). The solvents methanol (99.8%, J.T. Baker) and 2-propanol (99%, MCB) were filtered through a 0.5-pm Teflon membrane on an all-glass filtration apparatus (Millipore) before usage. The solutes benzene (99+ %), toluene (99+ %), mesitylene (Le., 1,3,5-trimethylbenzene, 99%), naphthalene (99%), phenanthrene (98+%), n-de- cane (99%), and n-tetradecane (99%) were all used as received from Aldrich Chemical Company. The experi- mental apparatus and procedures were as described by Sun

and Chen (1985~) except that for aliphatic hydrocarbon solutes, a differential refractometer (R401, Waters Asso- ciates, Inc.) was used instead of the UV absorbance de- tector (UV Monitor 111, Milton Roy Company) fixed at 254 nm for the aromatic hydrocarbon solutes.

Results and Discussion The observed values of the tracer diffusivities of benz-

ene, toluene, mesitylene, naphthalene, and phenanthrene in liquid methanol along the vapor-liquid coexistence curve up to 473.4 K and of n-decane and n-tetradecane, in ad- dition to the four aromatics, in 2-propanol from 333.2 to 536.0 K are summarized in Tables I and 11. Each reported value is the mean of three measurements, and the standard deviation is consistently smaller than f2% of the mean. For solvent methanol, the corrosion of the stainless steel tubing as recently reported by Mowery (1985) apparently resulted in solid deposits that blocked the flow restriction imposed between the end of the dispersion tube and the detector to provide a desired pressure via viscous friction. Details of the experimental apparatus have been presented previously (Sun and Chen, 1985b).

The following discussion of the results is divided into two parts: the investigation of solvent molecular associ- ation as a function of temperature in terms of the hard- sphere relationships for shear viscosity and tracer diffusion and then the establishment of a correlation for predictive purposes.

The association in molecular liquids via hydrogen bonding has been treated in the literature from different perspectives, such as modeling of intermolecular interac- tions (Jorgensen and Ibrahim, 1982; Smith and Nezbeda, 1984) or direct experimental observation (e.g. X-ray dif- fraction (Narten and Habenschuss, 1984; Narten and Sandler, 1979) and NMR chemical shift (Sakai et al., 1973)). In the present study an entirely different approach based on the hard-sphere model was devised to unravel the association phenomena across extended ranges of temperature. This was in view of the fact that the hard- sphere model had been shown on a number of occasions (Chen et al., 1985; Sun and Chen, 198513; Jonas, 1984, Chen et al., 1981; Chandler, 1975) to represent quite satisfactorily the dominant role of short-range repulsive forces present in real fluids of nonpolar and somewhat compact mole- cules. A logical extension of these observations is to obtain information on molecular association in alcohols using the tracer diffusion data for essentially noninteracting solutes. One way to accomplish this is to compare the tracer dif- fusion data in alcohols to those in solvents for which the model has been found to be successful. To do this, we started with the hard-sphere tracer diffusion equation

Table 11. Tracer Diffusivities (1O9Dl2, m2/s) of Aromatic and Aliphatic Hydrocarbon Solutes in Dense 2-Propanol up t o the Supercritical Region

lO9DI2 a t T (K), p (kg/m3)," 104p (N.s/m2)b 333.2, 750, 8.00

benzene 2.39 toluene 2.21 naphtha- 1.80

phenan- 1.47

n-decane 1.55 n-tetra- 1.25

lene

threne

decane

373.2, 708, 4.01 5.20 4.74 4.02

3.01

3.06 2.68

423.2, 634, 1.90 9.50 8.97 7.41

6.42

6.17 5.16

473.2, 493.2, 527, 466, 1.08 0.700 20.0 26.0 17.7 23.9 14.8 20.5

12.9 17.4

13.2 18.0 10.3 14.3

508.3, 569,

0.943 21.6 19.7 17.5

14.6

15.3 12.3

508.3, 521.0, 488, 549,

0.693 0.847 26.1 24.9 23.7 22.2 20.2 19.1

17.1 16.4

17.9 17.5 14.4 13.5

521.0, 473,

0.628 29.0 26.3 22.6

19.4

19.9 16.5

521.0, 437,

0.545 33.7 30.8 24.9

20.3

21.6 16.9

536.0, 548,

0.803 25.5 22.7 20.1

16.4

17.6 13.6

536.0, 536.0, 504, 417,

0.674 0.496 29.2 34.4 26.3 31.3 22.1 26.1

18.9 22.4

20.0 23.8 16.7 18.7

a Densities of 2-propanol from Wilhoit and Zwolinski (1973), Ambrose and Townsend (1963), and Bhattacharyya and Thodos (1964). *Viscosities of 2-propanol from Raznjevic (1976) and Stephan and Lucas (1979).

Table 111. Values of vio (i = 1 for Solute, i = 2 for Solvent) Calculated from the Effective Molecular Radius (reg) Determined with Bottcher's Method"

solute V?. m3 / kg-mol solute V?. m3 / ke-mol ~~ ~

benzene 0.0667 naphthalene 0.102

mesitylene 0.106 n-tetradecane 0.199

toluene 0.0782 phenanthrene 0.137 p-xylene 0.0945 n-decane 0.149

solvent V,O, m3/ke-mol solvent V.0, m3/ke-mol cyclohexane 0.0974b ethanol 0.0217c methanol 0.0160' 2-propanol 0.O25Sc

a V,O = 4 / 3 ~ r , f ~ N a , according to eq 23 of Onsager (1936) and p 194 of Bottcher and Bordewijk (1973); reff for solutes from Goro- dyskii et al. (1975). br,ff from p 297 of Bottcher and Bordewijk (1978). crefl from Khimenko et al. (1973).

formulated previously for RHS model fluids (Sun and Chen, 1985a)

in which DFFs is the tracer diffusivity of a solute (1) in a solvent (2), V/ Vo is the molar volume reduced by that a t closest packing of hard spheres, k is Boltzmann's constant, T is the absolute temperature, and m and u are the mo- lecular mass and diameter of a hard sphere, respectively. Equation 1 was established for a RHS model fluid (Sun and Chen, 1985a) using the Enskog-Thorne dense gas theory coupled with molecular dynamics computation re- sults compiled in Table I of Chen et al. (1981). Noting that the tracer diffusion in aromatic hydrocarbon/cyclohexane systems follows the RHS theoretical description quite well (Sun and Chen, 1985a), we used the diffusion data pres- ented there along with the molar volume of cyclohexane taken from Raznjevic (1976), Reamer and Sage (1957), and Khan et al. (1983) at conditions under which tracer dif- fusivities were measured to rewrite eq 1 as

D12Ml'/6M2'/3( V10)1/3 (V20)2/3(RT)1/2

= 2.49 x io-( - 1) (2)

where D12 is the observed tracer diffusivity, R is the ideal gas constant, M and V are molecular weight and molar volume, respectively, and Vi0 is the molar volume of solute or solvent characteristic of their molecular sizes. The values of Vi0 are given in Table 111. They are calculated for the aromatic hydrocarbon solutes and cyclohexane solvent using the effective molecular radii determined with Bottcher's method (Bottcher and Bordewijk, 1973,1978; Khimenko et al., 1973; Gorodyskii et al., 1975). Note that ( V/')1/3 is adopted here as a measure of the molecular size corresponding to ui in eq 1. The goodness of the linear relationship (correlation coefficient = 0.995) as expressed by eq 2 is demonstrated in Figure 1. The success of eq 2 in correlating the tracer diffusivities in liquid cyclohexane has inspired the following definitions of the reduced dif- fusivity DR and volume VR for an associating solvent

DR I (3) D12M11/6(pM2)'/3( V10)1/3

( p V20) 2 /3 (It Tc2) '1' I 7

VR I 2.- P V2O

T TR I - T c 2

(4)

Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 817

Figure 1. 109DR/TR1/2 vs. V, for the tracer diffusion of benzene and phenanthrene in cyclohexane, methanol, ethanol, and 2-propanol; other solutes are excluded for the sake of clarity. Values of p for the three alcohol solvents are predicted with eq 8-10.

The idea behind eq 3 and 4 is to consider a moving unit in the sea of an associated solvent to be an aggregate of p molecules as viewed by a diffusing solute molecule. Hence, eq 2 can be rewritten for both associating (0 > 1) and nonassociating ( p = 1) solvents with eq 6 rearranged as

- - - 2.49 x 10-9(vR - I) DR TR'i2

In what follow eq 6 is employed to examine the tempera- ture dependence of the solvent-association number, p, for methanol, ethanol, and 2-propanol using the tracer diffu- sion data determined here in addition to those in ethanol reported earlier a t 0.61 I T R I 1.06 and pR I 1.44 (Sun and Chen, 1986).

For three alcohol solvents, the values of V,O were found to be 0.0160, 0.0217, and 0.0258 m3/kg-mol for methanol, ethanol, and 2-propanol, respectively, using Bottcher's effective molecular radii determined by Khimenko et al. (1973). The molecular radii determined by Khimenko et al. (1973) are effective values in the sense that the molar volumes of the alcohol solvents characteristic of molecular sizes are related to reff by V20 = 4 / 3 ?rreff3Na. (See eq 23 of Onsager (1936) and p 194 of Bottcher and Bordewijk (1973).) Note that ethanol and 2-propanol above their critical temperatures were treated here essentially as ex- panded liquids. Using the tracer diffusion data in alcohol solvents, we then determined the value of p(T ) with eq 6 rearranged as

The results for p ( T ) are summarized as p(T ) = 3.25 - 0.00387T, methanol (8)

p(T) = 3.44 - 0.00348T, ethanol (9) P ( T ) = 4.31 - 0.00475T, 2-propanol (10)

These equations are plotted in Figure 2. The observed values of tracer diffusivities with p values predicted with eq 8-10 in the three alcohol solvents are shown in Figure 1. The measured values conform to the solid line estab- lished by solvent cyclohexane. The approach taken here and the results are rationalized as follows:

(i) The relative magnitude of the degree of association in the three alcohols studied here is consistent with that observed by IR spectroscopy (Becker, 1959). That the temperature coefficients of association numbers are neg-

818 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987

1.01

T ( O K ) I 290 380 470 560

Figure 2. Association number, @, as a function of absolute tem- perature, T, for methanol, ethanol, and 2-propanol; error bars indi- cate the range of p values fitted from several solutes in a given solvent a t a given temperature.

ative is consistent with the fact that hydrogen bonds tend to be increasingly ruptured by thermal agitation at higher temperatures.

(ii) The values of /3 for methanol and ethanol were found to be 2.12 and 2.42, respectively, by extrapolating eq 8 and 9 to 293.2 K. These values are in fairly good agreement with those determined with X-ray diffraction (Narten and Habenschuss, 1984): 2.78 and 2.82 for methanol and ethanol, respectively. The lower values of p, found from tracer diffusion data, may be attributable to the partial rupture of hydrogen bonding under nonequilibrium con- ditions.

In short, the association of molecular liquids can be revealed by incorporating the idea of moving units into the hard-sphere description of tracer diffusion of an effectively inert solute. Furthermore, it is possible to treat tracer diffusion in both associating and nonassociating solvents within a consistent framework that leads to eq 6. However, it may be argued that what the above treatment amounts to is simply data fitting with one adjustable parameter, p. It seems reasonable to strengthen the basis of our ap- proach by extending the analysis carried out above to the shear viscosities of liquids and supercritical dense gases, using the values of P ( T ) predicted with eq 8-10.

Starting with Dymond's fluidity equation based on computer simulation of a hard-sphere model flujd (Dy- mond, 1974) and using shear viscosity data for cyclohexane from 293.2 to 554.0 K (Khalilov, 1962), we arrived at the relationship

TR1t2

PUR (11) - - - 1.50 X 1 0 s ( V ~ - 1)

in which

and V, and T R are as defined by eq 4 and 5. Note that P = 1 for nonassociating solvents, such as cyclohexane. The results are plotted in Figure 3 for cyclohexane with eq 11 represented by the solid line. The viscosity data for methanol, ethanol, and 2-propanol over the temperature ranges corresponding to those of tracer diffusion mea- surements (Stephan and Lucas, 1979) are also given in

Figure 3. 10-8TR1/2/pR vs. V, for the shear viscosity of cyclohexane, methanol, ethanol, and 2-propanol with values predicted from eq 8-10.

I cw i0-9+) A 0 I0 20 30 40

Figure 4. Tracer diffusivities predicted with eq 13 compared with experimentally determined values.

Figure 3 using the values of @(r) predicted with eq 8-10. The comparison between the experimental data points and the solid line representing eq 11 suggests that shear vis- cosity of both nonassociating and hydrogen-bonded fluids can indeed be treated successfully within the hard-sphere framework using the same parameter values for molecular association number @(T). It is this commonality of /3 values that has justified the present treatment of transport properties in dense fluids in terms of the hard-sphere model incorporating the idea of moving units in associating solvents. It appears that the more extensively tabulated viscosity data should permit p( T ) to be evaluated for an associating solvent, which enables one to predict tracer diffusivity in the same solvent by using eq 6; however, from the practical standpoint, a correlation equation requiring less input information is more desirable for higher pre- diction accuracy.

To attain such a goal we empirically modified the Stokes-Einstein equation to obtain the relationship

DI2 1.23 x 10-l~ _ - - 0.799v 0.490

P2 cl (13)

The exponents of p2 and Vc, resulted from a multiple re- gression on the tracer diffusion data in cyclohexane (Sun and Chen, 1985a), n-hexane (Sun and Chen, 1985b), and ethanol (Sun and Chen, 1986) in addition to those reported in Tables I and 11. A comparison between experiment and prediction with eq 13 is made in Figure 4 in terms of 1oo(D,,,d - Dobsd)/Dobsd vs. Dobsd. The absolute average error of prediction is 5 % ; the maximum error is 17%.

Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 819

Such performance is hardly expected of the existing cor- relations compiled by Reid et al. (1977) a t temperatures removed from the neighborhood of 298 K, as indicated elsewhere (Chen and Chen, 1985).

Conclusions In summary, we have reported the tracer diffusion

coefficients of several aromatic and aliphatic hydrocarbons in liquid methanol and dense 2-propanol up to the su- percritical conditions. As the idea of moving units of solvent molecules is incorporated in the rough-hard-sphere description of binary diffusion, the association behavior in methanol, ethanol, and 2-propanol can be understood from the tracer diffusion data. The significance of this is that transport properties in dense fluids, regardless of the nature of their intermolecular interactions, can be treated within a common framework based on a hard-sphere model. Intended for practical applications, an empirical correlation has also been developed which can be used to predict fairly accurately the tracer diffusivities in a variety of organic solvents across extended conditions.

Acknowledgment

Acknowledgment is made to the Donors of the Petro- leum Research Fund, administered by the American Chemical Society, for the partial support of this work. The other source of support came from the National Science Foundation under Grant CBT-8500974.

Nomenclature D = molecular diffusivity, mz/s k = Boltzmann’s constant, 1.38 X M = molecular weight, kg/kg-mol m = molecular mass, kg Nu = Avogadro’s number, 6.023 X r = molecular radius, m R = ideal gas constant, 8314 J/(K.kg-mol) T = absolute temperature, K V = molar volume, m3/kg-mol Greek Symbols u = hard-sphere molecular diameter, m p = fluid density, kg/m3 p = shear viscosity, N.s/m2 @ = molecular association number Superscripts RHS = rough-hard-sphere theoretical value Subscripts 1 = solute 2 = solvent 12 = binary diffusion of solute in solvent C = critical property eff = effective value m = property evaluated at equilibrium melting point

J/K

molecules/kg-mol

0 = property at closest packing of spherical particles or

pred = predicted value obsd = experimentally observed value

Registry No. CH30H, 67-56-1; (H3C)&HOH, 67-63-0; CeHs, 71-43-2; CsH&H3, 108-88-3; H3C(CH&CH3, 124-18-5; H3C(C- H2Il2CH3, 629-59-4; mesitylene, 108-67-8; naphthalene, 91-20-3; phenanthrene, 85-01-8.

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Received for review August 22, 1985 Accepted November 3, 1986

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