Measurement and Correlation of Binary Diffusion Coefficients in Near

Ind. Eng. Chem. Res. 1994,33, 1828-1837

1828

Measurement and Correlation of Binary Diffusion Coefficients in Near Critical Fluids Owen J. Catchpole. Zndwtrial Research Ltd., Gracefield Research Centre, P 0 Box 31-310, Lower Hutt, New Zealand

Michael B. King School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham BZ5 2TT, England

The binary diffusion coefficients of benzoic acid, oleic acid, a-tocopherol, and glycerol trioleate were measured in near critical carbon dioxide over the temperature range 298-318 K and pressure range 71-251 bar. A correlation based on an extensive literature database was developed which can be used to estimate both self and binary diffusion coefficients in a range of near critical solvents over the solvent reduced density range 1-2.5, with an average error of &lo%. The correlation is not suitable for estimating binary diffusion coefficients a t the binary mixture critical point. The only input data required for the correlation are the solvent molecular weight, reduced temperature, and density, the solute molecular weight, and an estimate of the solute critical volume. Introduction

Theoretical Section

Research into, and industrial usage of, near critical fluid solvents in extraction processes is increasing (Parkinson and Johnson, 1989). Further industrial scale development is hampered by a lack of suitable models for predicting and correlatingmasstransfer processes (Lionget al., 1991b, 1992). The binary diffusion coefficient is a fundamental parameter in mass transfer processes. Both self diffusion (Takahashi and Iwasaki, 1966; Takahashi, 1977; Harris, 1978,1982),and more recently, binary diffusion coefficients (Liong et al., 1991b, 1992;Feist and Schneider, 1982;Swaid and Schneider, 1979; Dahmen et al., 1990b) have been reported in the literature. To complement the literature, correlations are required to estimate self and binary diffusion coefficients at conditions and for solutes where no data are available. Attempts to correlate binary diffusion coefficients at near critical conditions have been reviewed in detail elsewhere (Liong et aL, 1991a;Funazukuri et al., 1991; Catchpole 1991). Briefly, there are two main approachesto correlating diffusion coefficients. The first approach is based on modifications of the StokesEinstein equation (Sassiat et al., 1987; Debenedetti and Reid, 1986; Sun and Chen, 1986,1987; Liong et al., 1992). The correlations require the prediction of the solvent viscosity at given temperature and pressure and an estimate of the solute molar volume at either the boiling point or the critical point (Catchpole, 1991). These correlations are limited to solute molecules much larger than the solvent and to solvent densities greater than the critical density (Catchpole, 1991). The second approach is based on modified Enskog-Thorne theory (Sun and Chen, 1985b; Chen, 1983; Erkey et al., 1990). This correlation method is described in more detail in the Theoretical Section. In this work, experimentally determined binary diffusion coefficients for benzoic acid, oleic acid, a-tocopherol, and glycerol trioleate are reported. Oleic acid, glycerol trioleate, and a-tocopherol are model components of natural triglycerideoils. Knowledge of their diffusion behavior is useful in developing mass transfer models for supercritical packed bed extraction of natural oils. A correlation based on corrected Enskog-Thorne theory for predicting self and binary diffusion coefficients in near critical fluids is also presented. The literature data used in the correlation includes measurements from nine near critical fluids.

The Enskog-Thorne theory for self and binary diffusion of gases (Hirschfelder et al., 1954) relates the binary diffusion coefficient at low pressure derived from kinetic theory, eq 1, to that at high pressure, eq 2, by a contact

correlation function. The equations for self diffusion are obtained by replacing the subscript 12 with 11. Expressions for the contact correlation function gEare given by Chapman and Cowling (Chapman and Cowling, 1970). Equation 2 was found to give inaccurate results at supercritical conditions by Balenovic et al. (Balenovic et al., 1970)and Swaid and Schneider (Swaid and Schneider, 1979). A number of modifications have been proposed, which replace the contact (Enskog) correlation function with a new correlation function, and a ratio of the “smooth hard sphere” diffusion coefficient to the Enskog diffusion coefficient (Tyrell and Harris, 1984; Alder et al., 1970):

(3) (4)

Until recently the ratio DSHSIDEfor binary systems was only obtainable from computer simulations of diffusion (Alder et al., 1970,1974; Easteal et al., 1983; Michels and Trappeniers, 1978). An expression to calculate this ratio was proposed by Sung and Stell (Sung and Stell, 1984). Sun and Chen applied the method for the correlation of binary diffusion coefficients in near critical 2,3-dimethylbutane with some success (Sun and Chen, 1987). However, the method is cumbersome, and so an alternative estimation procedure is required. The ratio of the binary to the self diffusion coefficient is given by eq 5 (Tyrell and Harris, 1984). Here, the ratio

0 1994 American Chemical Society Q888-58~5/94/2633-~828$04.50/0

Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1829

R given by eq 6 is obtained by correlation of literature

data over a range of binary near critical fluid solvent/ solute pairs. To further simplify eq 5 it is assumed that the molecular diameter ratio is proportional to the cube root of the critical volume ratio, viz. (7)

Applying eq 14 at the critical point, and taking the ratio of the self diffusion coefficient to its value at the critical point, D,,eq 15 is obtained:

Lee and Thodos (Lee and Thodos, 1982) have shown that the temperature dependence of the self diffusion coefficient in the near critical region is proportional to T,, rather than T,'l2. Incorporating this temperature dependence into eq 15, we get

Additionally, the particle masses ml and m2 are replaced with the molar masses MI and M2. A size to mass ratio parameter X is defined by

X = (1+ (Vc2/Vc1)1'3)2/(l+ M1/M,)1/2

(8)

The resultant expression for the binary diffusion coefficient is as follows:

D,,= 23/2DllR/X

(9)

The estimation of 0 1 2 requires a predicted value for Dll, the self diffusion coefficient, and R, the ratio defined by eq 6. Self Diffusion Coefficient. A number of correlations for predicting the self diffusion coefficient based on hard sphere theory have been proposed (Dymond, 1972;Tyrell and Harris, 1984;Speedy et al., 19891, Erkey et al., 1990). The correlations of Dymond (Dymond, 1972) and Erkey (Erkey et al., 1990) take the general form

where

Equation 16 is the working correlation for predicting the self diffusion coefficientin eq 9. The constant C was found to be 0.451 by minimizing the least squares error between experimental values of the left hand-side of eq 16 and the calculated right-hand side of eq 16 for a range of near critical fluids, as outlined in the section on correlation of binary diffusion coefficients. Methods for predicting the self diffusion coefficient at the critical point are also outlined in this section.

Experimental Section Capillary EvaporationTechnique (CE). The binary diffusion coefficients of benzoic acid (c.~&os)and naphthalene (CloH8) in near critical carbon dioxide were measured using a capillary evaporation (CE) technique. The capillary evaporation technique used was similar to that reported by Knaff and Schlunder (Knaff and Schlunder, 1987) and has been described in detail elsewhere (Catchpole, 1991; Catchpole et al., 1992). The experimental diffusion coefficient was obtained from eq 17 by measuring the capillary heights La and Lb over extraction time t.

and

(VJo = N*q13/2 112

(12)

(V1)02/3is the

volume of close-packed hard spheres with diameter q,and Q and q are dimensional constants. The hard sphere diameter is estimated from a linear plot of isothermal self diffusion coefficient data versus W 3 . The hard sphere diameter has a weak temperature dependence, which can be ignored over small temperature ranges. The correlation method outlined below is based on eqs 10-12. However, semiempirical procedures are introduced to provide a simple method for predicting self diffusivities from a knowledge of the self diffusion coefficient at the critical point: no plotting of data is required. Equation 10 can be rearranged in terms of thb critical volume by assuming the following:

I=C ( v p

(13)

By limiting the temperature range of the correlation, the weak temperature dependenceof the hard sphere diameter can be neglected. Making the substitution, eq 10 can be rearranged to give

Capillary tubes were constructed from precision bore glass tubing. The tubes with internal diameter of 1.65 or 0.64 mm and length 12 cm were sealed at one end. The tubes were filled with solid solute by placing the empty tubes upside down in a beaker containing a large excess of solid solute, and then heating the beaker and tubes under vacuum in a vacuum oven until the solute melted: the air was sucked from the tubes and replaced with liquid solute. The tubes were then slowly cooled to atmospheric temperature under vacuum so that the solute solidified into large crystals. An initial interface height of 1-3 mm was used. Heights La and Lb were measured either from the weight loss of the diffusion tubes or by measuring the volume of water required to fill the remainder of the capillary using a high-precisionsyringe (Catchpole, 1991). Experiments were performed by placing three to five prepared tubes in a specially designed tube holder which ensured that the tubes were vertically aligned. The tubes and the tube holder were then placed in the extraction vessel V2 (see Figure 1). Near critical carbon dioxide was then passed through the extraction vessel at constant

1830 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994

OIFFIJ'SION APPARATUS

U

Figure 1. Schematic of experimental apparatus.

temperature, pressure, and flow rate for a given time. The rate of flow past the tubes was varied from 0 to 0.8 kg h-' C02 and was found to have no influence on the results. Taylor-Aris Dispersion Technique (TAD). The binary diffusion coefficients of oleic acid (CI~HWOZ), benzoic acid, a-tocopherol ( C ~ ~ H W O and ~ ) glycerol , trioleate (C57Hl~Oe)were measured using the Taylor-Aris dispersion technique. All test solutes were Sigma research grade (99% purity). The Taylor-Arb dispersiontechnique has been widely used for the determination of diffusion coefficients at near critical conditions (Swaid and Schneider, 1979; Feist and Schneider, 1982; Funazukuri et al., 1989,1991; Sun and Chen, 1985a,b; Kopner et al., 1987). The method involves injecting a narrow pulse of solute into a pure solvent stream in steady laminar flow in a tube of circular cross section. At a distance L from the injection point, the fluid is monitored with a suitable detector. The pulse of solute, after a sufficient residence time, becomes normally distributed, and the diffusion coefficient can be obtained from an analysis of the observed peak shape. Aris (Aris, 1956) extended the mathematical analysis of Taylor (Taylor, 1953,1954) to show that the diffusion coefficient could be obtained from eq 18,

providing DMIUL < 0.01 (Levenspiel and Smith, 1957). The dispersion coefficient, DAX,is related to the local variance, 62(x), by eq 19, again providing D d U L < 0.01 (Levenspiel and Smith, 1957). The local variance is S2(x) = 2D,/UL

(19)

expressed in terms of the theoretical plate height H by eq 20. Rearrangement of eq 18 in terms of the plate height

H = a2(x)/L

H gives the following equation for D12:

(20)

2

D12= -u[H 4 f

112

(.?-!$) ]

(21)

In practice, only the negative root is applicable (Liong et al., 1991). His obtained from the response of the detector on a suitable chart recorder or integrator using eq 22. The

experimental apparatus and method was designed to minimize potential sources of error. A comprehensive discussion on the minimization of measurement errors using TAD is given by Alizedah (Alizedahet al., 1980)and Cloete (Cloeteet al., 1976). Stepstakentominimize errors are outlined below. Secondary Flow Due to Coil Curvature. The linear flow velocity U was kept below the criteria expressed by eq 23 (Alizedah et al., 1980; Daskopoulos and Lenhoff, 1988; Nunge et al., 1972). De2& 5 20

(23)

Injection of Solute. The injection of solutes of low volatility and low solubility into a high-pressure fluid presents some difficulties. The apparatus was designed to inject a small volume of solute dissolved in near critical carbon dioxide at the same temperature and pressure as the diffusion measurements. Many workers have injected either the neat solute or the solute dissolved in a liquid solvent into the near critical fluid (Dahmen et al., 1990a; Lauer et al., 1983; Sassiat et al., 1987; Funazukuri et al., 1989, 1991). This involves an implicit assumption that the solute rapidly dissolves in the near critical fluid, and that when additional liquid solvent is used, it has no influence on the measured diffusion coefficients. The binary diffusion Coefficients of solutes in near criticalfluids containing a low concentration of an entrainer are considerably lower in some cases than the pure near critical fluid (Shenai et al., 1993;Olesik and Woodruff, 1991).To

Ind. Eng. Chem. Res.,Vol. 33, No. 7,1994 1831 overcome the influence of the solute disperson due to the method of injection, some workers have used the "two column subtraction technique" whereby the peak variance is obtained a t two points (Dahmen et al., 199Oa-c; Kopner et al., 1987). The final variance is subtracted from the initial variance, to give eq 24. The initial variance 612(x) P(X)

= 6,2(x) - 6,2(x)

250

fi

(24)

is usually obtained after a short length of diffusion tube. This calculation technique can generate errors as the accuracy with which 612(x) can be obtained is diminished, especially if it is calculated according to the same assumptions as eq 19 (Dahmen et al., 1990a). Diffusioncoefficientswere obtained using the diffusion1 extraction apparatus shown in Figure 1. Solute was injected into the diffusion apparatus using a Valco sixport valve (SPV) which connected the extraction apparatus to the diffusion apparatus. An equilibrium mixture of solute in near critical carbon dioxide was prepared in the extraction apparatus by recirculation of COz a t fixed temperature and pressure through vessel V2, which contained the solute [and around the recycle loop RV1, SPV, RV21. A small portion of the soluteICO2 mixture was then injected into the diffusion apparatus, also at the same temperature and pressure, to begin a measurement. The injected pulse then traveled down the diffusion tube (15.22 m length, 2.159 X 10-3 m internal diameter, 0.5 m coil diameter) and through the detector flow cell (Dl), where the response was recorded on the chart recorder. The flow rate in the diffusion apparatus was observed on a low flow rate wet test meter (FT2). The retention time of the peaks was of the order 1-3 h for benzoic acid and 2-4 h for the liquid solutes glycerol trioleate, oleic acid, and a-tocopherol. Three to six peaks were injected per experiment, at intervals of around 40-60 min, which was sufficient to avoid overlapping. The diffusion apparatus was operated for around 1-2 h before injection of the first sample, to achieve steady flow and temperature. The detector wavelength was 240 nm for benzoic acid, 200 nm for glycerol trioleate, 205 nm for oleic acid, and 295 nm for a-tocopherol. The temperature and pressure range for CE and TAD experiments was 298-318 K and 71-251 bar.

Results and Discussion The measured diffusion coefficients and associated errors for benzoic acid, naphthalene, glycerol trioleate, oleic acid, and a-tocopherol using both the capillary evaporation and Taylor-Aris dispersion techniques are presented in Table 1. The measured diffusion coefficients are plotted in the form DlzprXlTJ3 [for computation of R, see eqs 26 and 27) versus prin Figure 2 and are compared with literature data in Figure 3. Also included for comparison is the binary diffusion coefficient correlation, eq 28 from the Correlation of Binary DiffusionCoefficients section. The literature data include linoleic methyl ester (Funazukuri et al., 1991),stearic ethyl ester (Liong et al., 1991b, 19921, oleic acid (Dahmen et al., 1990), benzoic acid (Debenedetti and Reid, 1986; Shenai et al., 1993), and a-tocopherol (Funazukuri et al., 1992). No literature data could be found for glycerol trioleate. Linoleic acid CIS:^) differs from oleic acid (CU:~) by a single double bond and the methyl or ethyl ester from the acid by a methyl or ethyl group. As oleic acid (ClsHadOz) is a large molecule, the difference in size and molecular mass of the acid and esters will be small. It is shown in the Correlation of Binary Diffusion Coefficients section that the size and

i

v

01

I

I

I

1.2

1.8

2.0

1

Reduced Density Figure 2. Binary diffusion coefficients in the form DlzprXITJ? vs Pr: ( 0 )benzoic acid; (0)oleic acid; (v)a-tocopherol; (v)glycerol trioleate, (-) correlation.

Table 1. Experimental Binary Diffusion Coefficients in Near Critical Carbon Dioxide method CE CE CE TAD CE TAD TAD TAD TAD TAD TAD TAD TAD TAD TAD TAD

solute naphthalene benzoic acid benzoic acid benzoicacid benzoic acid benzoic acid benzoic acid benzoic acid oleicacid oleic acid a-tocopherol a-tocopherol glycerol trioleate glycerol trioleate glycerol trioleate glycerol trioleate

press., bar 160 71 102 141 160 161 201 251 201 251 201 251 201 251 201 251

temp, density, K kgma 318.1 761.1 298.8 745.4 317.8 537.2 318.1 724.3 318.7 756.8 318.1 762.8 318.1 814.6 318.1 858.7 308.1 867.3 308.1 902.5 308.1 867.3 308.1 902.5 298.1 915.4 298.1 944.0 308.1 867.3 308.1 902.5

012,

lern2s-1 11.3 i 1.1 13.4 i 1.9 1 9 . 4 i 3.1 14.3* 1.0 13.1 i 1.1 12.6 t 1.2 12.3 t 0 . 4 10.9 t 0.2 5.59 t 0.11 4 . 8 0 i 0.12 5.69 0.16 4.52 & 0.10 3.48 0.07 3.17 0.24 4.64 i 0.16 3.55 i 0.37

*

molecular mass are key parameters in the correlation of binary diffusion Coefficients. Similarly, stearic acid (CU:~) differs from oleic acid by one double bond, with stearic acid being completely saturated. The density dependence of the diffusion coefficients in near critical carbon dioxide is clearly shown in Figures 2 and 3. The density dependence is similar to that found by other works using a variety of near critical fluids as solvents (Sun and Chen, 1986; Dahmen et al., 1990; Kopner et al., 1987a,b; Feist and Schneider, 1982). Oleic acid diffusion data obtained in this work agree well with the fatty acid ester data of Funazukuri et al. (Funazukuri et al., 1992) and Liong et al. (Liong et al., 1992). The data of Dahmen et al. (Dahmen et al., 1990a) for oleic acid compare poorly with all data sets. Benzoic acid diffusion data, obtained both by capillary evaporation and Taylor-Aris dispersion techniques, were markedly different from those of Debenedetti and Reid (Debenedetti and Reid, 19861, who obtained their results from the dissolution of a flat plate of solute contained in a rectangular duct, but agreed well with Shenai et al. (Shenai et al., 1993),who used TAD. Fluid-phase dimerization of benzoic acid was postulated as the cause of the low values obtained by Debenedetti and Reid. Dimerization of benzoic acid in near critical carbon dioxide was observed

1832 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994

n

250

0

rl

I

I

I

0.4

0.8

1.2

1.2

L

I

I

I

2.0

2.4

1.o

I

OO 0 0 I

0

0.8

'0 4 1501

H*

$

0.6

. I

d 0.4

I

Q!!

D

D

0.2

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1.2

1.6

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2.0

Reduced Density

Reduced Density

Figure 3. Binary diffusion coefficientsin the form Dlw,XITJZ vs pr includingliteraturedata: symbolsa in Figure 2, except ( 0 )benzoic acid (Shenai, 1993);(m) benzoic acid (Debenedetti,1986);( 0 ) oleic acid (Dahman,1990a);(e) a-tocopherol (Funazukuri,1992);(A) oleic methyl ester (Funazukuri,1991); (A)stearic ethylester (Liong, 1992).

by Tsugane et al. (Tsugane et al., 1992), but the ratio of dimers to monomers was small a t densities above the critical density. The average size and mass of the diffusing species would be close to the monomeric form of benzoic acid. Dimerization would occur irrespective of the measurement technique. As a further check on the reliability of the measurements in this work, the diffusion coefficient of naphthalene was measured and found to agree within experimental error with published data (Knaff and Schlunder, 1987; Iomtev and Tsekhanskaya, 1964).

Figure 4. Self diffusion coefficients in the form D1lp,/DoTrvs pr: (0)ccP3; (0) SFe; (VI COa; (V) C z b ; (0)CzHe; (D) C3Hs; (-) correlation. Table 2. Self Diffusion Coefficient Measurements experimental range fluid COz

CzHd Cz& C3Hs

CClF3

Correlation of Binary Diffusion Coefficients The correlation and prediction of binary diffusion coefficients in near critical solvents requires a two-step calculation. The first calculation is the prediction of the self diffusion coefficient from a correlation based on eq 16. The second calculation is the prediction of the binary diffusion coefficient from a correlation based on eq 9. Self Diffusion CoefficientQ1. Experimentalvalues of the right-hand side of eq 16 were used to find the best fit value of C (a universal constant). The self diffusion coefficient at the critical point, D,, was estimated from a modified Fuller/Schettler/Giddings (Fuller et al., 1966) empirical diffusion coefficient correlation to give Dll in terms of the critical density (Catchpole, 1991). D, =

4.300 x

10-7~~,0.76

2,2fSP,

1.6

(25)

with Min g mol-l, T,in K, and p, in kg ma. The diffusion volumes 2, are given in Reid et al. (Reid et al., 1987). Equation 25 utilizes the empirical finding of Lee and Thodos (Lee and Thodos, 1982) that the product Dllp a t the critical temperature remains approximately constant up to and including the critical density. Predictions from eq 25 compare well with experimental values of the self diffusion coefficient, which are reviewed elsewhere (Lee and Thodos, 1982). Self diffusion data from a range of nonpolar, near critical solvents is compared with the self diffusion coefficient correlation in Figure 4 using the best fit value for C of

SFe

T* 0.98-1.14 0.90-1.23 0.90-0.97 0.90-1.14 1.05-1.23 0.44-1.05 0.44-1.49 0.90-0.97 0.30-1.22 1.00-1.15 0.44-1.43 0.75-1.16 0.93-1.25 1.014

PI

0.02-1.77 0.001-1.56 1.67-2.12 0.04-2.32 0.10-1.40 0.58-3.15 1.52->3 1.71-2.12 1.71->3 0.52-2.66 1.72->3 0.06-1.28 1.37-2.60 0.10-1.81

ref Takahashi and Iwaaki, 1966 OHem and Martin, 1955 Robinson and Stewart, 1968 Eteeee et al., 1992 Takahashi, 1977 Arends et al., 1982 Greiner-Schmid et al., 1991 Robinson and Stewart, 1968 Greiner-Schmid et al., 1991 Harris, 1978 Ha and Liidemann, 1989 Tison ahd Hunt, 1971 De Zwan and Jonas, 1973 Zykov et al., 1980

0.451 (over the reduced temperature range 0.9-1.25). Sources of the literature data and ranges of reduced temperatures and densities used in the comparison are collated in Table 2. The agreement between the correlation and experimental data is very good over the reduced density range 0.4-2.5 which encompasses most near critical fluid operations. Correlation of Binary Diffusion Coefficients. Binary diffusion coefficients have been reported in the following near critical fluids: carbon dioxide, ethylene, propane, chlorotrifluoromethane, sulfur hexafluoride, 2,3dimethylbutane, hexane, ethanol, and 2-propanol. Literature sources of experimental data used to develop the correlation are presented in Table 3. The critical properties of the solvents are listed in Table 4. The table does not include all reported diffusion data in the near critical region. Data from the region adjacent to and including the mixture critical point (Tsekhanskaya, 1971;Saad and Gulari, 1984; Gulari et al., 1987; Semenov and Berezhnoi, 1980,1981) were excluded, along with data for solutes that are electrolytes (Olake et al., 1991). Data for which fewer than four measurements at different temperatures or pressures were taken were also excluded. The data of Debenedetti and Reid (Debenedetti and Reid, 1987)were omitted, as they were systematically low. The data of Dahmen et al. (Dahmen et al., 1990a) were also omitted, as they were systematically higher than

Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1833 Table 3. Literature Binary Diffusion Data in Near Critical Fluids

T, Pr rep 0.98-1.08 0.60-2.01 6-8,11,12,16,20,23 coz 0.41-1.91 1.03 1 SFe propane 1.04-1.09 1.47-2.03 12 hexane 1.00-1.07 1.35-1.84 9 DMB 1.05-1.10 1.43-1.91 10 ethanol 1.00-1.05 1.48-2.06 4 2-propanol 1.00-1.05 1.53-2.08 5 toluene 92.14 316 1.03-1.10 1.30-2.00 23,24 co2 0.88-1.91 1.03 1 SFe 1.00-1.07 hexane 1.35-1.84 9 DMB 1.43-1.91 1.05-1.10 10 1.00-1.05 ethanol 1.48-2.06 4 2-propanol 1.00-1.05 0.41-2.25 1.53-2.08 5 l,4-dimethylbenzene 120.19 476 0.89-1.06 1 SFs CClFa 1.05 0.69-1.73 1 hexane 1.00-1.07 1.43-1.91 9 1,3,5-trimethylbenzene 120.19 476 1.01-1.10 0.76-2.00 6,7,23 Con 1.03 0.41-1.91 1 SFe 1.00-1.07 hexane 1.35-1.84 9 ethanol 1.00-1.05 1.48-2.06 4 1.00-1.05 2-propanol 1.53-2.08 5 naphthalene 128.1 413 0.95-1.10 0.76-2.37 6,8, 11,13-15,21 Con naphthalene-da 1.01-1.13 0.98-2.38 13,15 c2)4 1.00-1.07 hexane 1.35-1.84 9 DMB 1.05-1.10 1.43-1.91 10 ethanol 1.00-1.05 1.48-2.06 4 1.00-1.05 2-propanol 1.53-2.08 5 phenanthrene 178.2 554 cc02 1.00-1.10 1.09-1.88 6,17 hexane 1.00-1.07 1.35-1.84 9 DMB 1.05-1.10 1.43-1.91 10 ethanol 1.00-1.05 1.48-2.06 4 1.00-1.05 2-propanol 1.53-2.08 5 chrysene 228.3 701b 1.00-1.10 1.10-1.88 6 c02 108.14 m-cresol 309 1.03-1.06 12 0.90-1.83 coz 1.04-1.09 propane 1.47-2.03 12 1,bdibromobenzene 235.9 4456 CClF3 1.05 0.69-1.73 1 caffeine 194.2 1.03-1.09 4886 0.69-1.73 8,11,21 c02 carbon tetrachloride 276 153.8 0.98-1.02 1.54-1.87 16 cos 1.03 0.41-1.91 1 SFe ethylbenzene 106.17 1.03-1.10 374 1.30-2.00 23 coz n-propylbenzene 1.01-1.10 440 120.19 0.76-2.00 7,23 coz 4256 ieopropylbenzene 120.19 1.03-1.10 1.30-2.00 23 coz 341 benzoic acid 122.14 0.98-1.08 1.16-1.84 17,22 CO2 phenol 94.11 1.03 229 0.76-1.69 8 c02 93.9 carbon dioxide 46 1.06-1.23 0.05-1.59 18 C2H4 130.4 ethylene 31.24 0.98-1.15 0.04-1.68 18 coz 2-propanone 1.00-1.10 209 58.08 1.04-1.88 6 c02 1.04 cclF3 0.69-1.65 1 3-pentanone 86.13 336 1.03 1.27-1.73 2 c02 84.12 cyclopentanone 1.03 268 1.27-1.73 2 coz 114.2 4336 3-heptanone 1.03 1.27-1.73 2 c02 112.2 cycloheptanone 1.03 1.27-1.73 3606 2 coz 5-nonanone 145.2 5456 COZ 1.03 1.27-1.73 2 cyclononanone 143.2 456 1.03 1.27-1.73 2 cos 116.2 1.01-1.05 C1:o ethyl ester 400.0 1.28-1.81 3 c02 172.3 Ce:o ethyl eetep 621b 1.01-1.05 1.28-1.81 3 con 1.01-1.05 200.3 Clw ethyl est& 7336 1.28-1.81 3 coz 228.4 C140 ethyl est.& 1.01-1.05 1.28-1.81 9576 3 coz 284.5 1.01-1.05 10696 Clm ethyl esteP 1.28-1.81 3 c02 1.01-1.05 312.5 Cia. ethyl est& 1181b 1.28-1.81 3 co2 368.6 1.01-1.05 C~2:oethyl ester 14Otib 1.28-1.81 3 c02 330.5 1137b 1.01-1.05 C ~ methyl S 1.28-1.81 3 CO2 122g6 1.01-1.05 342.5 Cn:e methyl esterc 1.28-1.81 3 coz 356.5 12856 Cn:e ethyl estef 1.01-1.05 1.28-1.81 3 CO2 294.5 10856 Cls:~methyl est& 1.01-1.08 1.58-1.98 14 co2 152.1 394 vanillin 1.01-1.05 1.28-1.81 19 c02 1.01-1.05 136.1 1405b phenylacetic acid 1.28-1.81 19 0 2 142.3 n-decane 1.00-1.05 603 2-propano1 1.53-2.08 5 198.4 830 n-tetradecane 1.00-1.05 2-propanol 1.53-2.08 5 885.4 glycerol trioleate 0.98-1.01 3200 1.85-2.01 22 coz 1.01-1.03 430.7 14676 a-tocopherol 1.69-1.93 20,22 cos a (1)Kopner, 1987; (2) Dahmen, 1990b,c; (3) Liong, 1991, 1992; (4) Sun, 1986; (5) Sun, 1987; (6) Sassiat, 1987; (7) Swaid, 1979; (8) Feiet, 1982; (9) Sun, 1985a; (10) Sun, 1985b; (11)Lauer, 1983; (12) Olesik, 1991; (13) Iomtev, 1964; (14) Funazukuri, 1987,1991; (15) Lamb, 1989; (16) Erkey, 1990, (17) Smith, 1990; (18) Takahashi, 1982; (19) Wells, 1992; (20) Funazukuri, 1992; (21) Knaff, 1987; (22) this work; (23) S u e z , 1993; (24) Bruno, 1989. Critical volume estimated by method of Joback (see Reid et al., 1987). Cno.:no.= number of carbons, number of double bonds in fatty ester. solute

benzene

M2 78.11

vc2

259

solvent

1834 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1.4

I

2.0

1.2

p: 1.0

0.2 Oa4

I 1.2

2.0

1.6

1

2.4

0.8

I

0.6

1 n'

Reduced Density Figure 5. Correction factor R as a function of pr for several binary systems: (0)napthalene/COz; ( 0 ) Stearic ethyl ester/COz; (v) Linoleic methyl ester/COz; (v)benzene/COz; (0)1,Cdimethylbenzene/SFe;).( CO~/CZHI;(A)naphthaleneKzH4. Table 4. Solvent Critical Properties

M, solvent CZH4

vc,

mol-' cm3 mol-' 28.054 130.4 coz 44.01 93.9 CClF3 104.46 180.4 198.8 SFs 146.05 DMB 86.178 358.0 hexane 86.178 370.0 203 CSH8 44.094 ethanol 46.069 167.1 2-propanol 60.096 220

Tcr ka m 4 K 215.1 282.4 468.7 304.1 579.0 302.0 734.6 318.7 240.7 500.0 232.9 507.5 217.2 369.8 275.7 513.9 273.2 508.3 pC,

Dc Pc, (eq 25) m2 bar s-1 X 108 50.4 6.131 73.8 4.937 38.7 2.937 37.6 2.920 31.3 6.915 30.1 7.227 42.5 6.775 61.4 8.23 47.6 7.53

previous results reported by the same group (Dahmen et al., 1990b,c). The correction factor R for a given binary system was obtained by comparing experimental diffusion data with eq 9 for pr 1 1 and 0.95 < TIC 1.2. The self diffusion coefficient in eq 9 was estimated using eq 16 with C = 0.451. Two trends emerged immediately on examination of the values of R obtained. These trends are discussed in the following text. The correction factor R for a given binary system is approximately independent of temperature, pressure, and density except in the immediate vicinity of the critical point. This is illustrated in Figure 5, where the ratio of the correction factor R to its average value, R,,, is shown for a number of binary systems over a wide reduced density range and 0.95 C TIC 1.2. There is considerable variation in R in close proximity to the mixture critical point, with IT - Tcl < 5 and pr = 1. As the mixture critical point is approached, a number of studies have shown that the binary diffusion coefficient decreases sharply, and is zero at the critical point (Tsekhanskaya, 1971;Saad and Gulari, 1984;Gulari et al., 1987). It has been argued that the extent of the decrease at the critical point is also concentration dependent, and that as the concentration tends to infinite dilution, the phenomenon disappears (Cliffordand Coleby, 1991).There are insufficient data at present to verify these claims, and the use of the correlation is therefore restricted to the region excluding the immediate vicinity of the mixture critical point. The correction factor R for all binary systems except

Figure 6. Correctionfactor R as a function of size to mass parameter X (0) literature data for eq 26; ( 0 )literature data for eq 27. Table 5. Applicability of Correction Factor Correlations to Systems Studied solvents carbon dioxide

solutes (see Table 3)

all aromatics, ketones,

eq 27

and carbon tetrachloride propane, hexane, dimethylall aromatics and (for CClFs) 27 butane, sulfur hexafluoride, 2-propanone, (for SF6) and chlorotrifluoromethane carbon tetrachloride ethanol and 2-propanol all aromatics and aliphatics 26 all aliphatics except ketones 26 carbon dioxide and ethylene and (for CZ&) naphthalene

phenol/carbon dioxide and carbon tetrachloride/carbon dioxide fell in the narrow range 0.7-1.1, as shown in Figure 6. Further analysis shows that the data falls into two broad classes. In a class 1type system R is independent of the size to mass parameter X,and is correlated by eq 26. In R = 1.0 f 0.1,

2