Ind. Eng. Chem. Res. 1991,30, 22-28
22
Effects of Solvent and Solute Configuration on Restrictive Diffusion in Hydrotreating Catalysts Chung H.Tsai,’ F.E.Massoth,**tS.Y.Lee,$ and J. D. Seadert Departments of Fuels Engineering and Chemical Engineering, University of Utah, Salt Lake City, Utah 84112
Effective diffusivities of polyaromatic solutes in aliphatic solvents were determined from adsorption measurements a t ambient conditions with three hydrotreating catalysts of pore diameters 6.2-17.5 nm, Solutes studied were disklike (coronene, 9-phenylcarbazole, tetraphenylporphyrin, and tetra(4-biphenyly1)porphyrin) and rodlike (naphthalene, anthracene, benzanthracene, and dibenzanthracene) polyaromatics of critical dimensions ranging from 0.74 to 2.18 nm. A modified Wilke-Chang-type correlation was developed to predict bulk diffusivities of the polyaromatic solutes in aliphatic solvents. The effective diffusivities and bulk diffusivities were then used t o determine restrictive diffusion correlations. The structure of saturated hydrocarbon solvents (n-heptane, isooctane, and cyclohexane) was found to have little effect on restrictive diffusion. The critical solute dimension gave the best correlation for restrictive diffusion of nonspherical molecules in catalyst pores. The resultant correlation, based on the ratio of solute critical dimension to average pore dimension, was found to be in close agreement with hydrodynamic theory. Introduction Hydrotreating of high molecular weight feeds, e.g., petroleum residua, coal, or coal-derived liquids, can be significantly influenced by pore diffusion in catalysts (Guin et al., 1986). Diffusion of large molecules in relatively small catalyst pores results in a lowering of the diffusion rate due to steric exclusion and wall effects (Beck and Schultz, 1970). This phenomenon has been referred to as “restrictive”, “configurational”, or “hindered” diffusion. Restrictive diffusion has been studied extensively on porous materials, e.g., zeolites (Moore and Katzer, 1972), controlled-pore glass (Colton et al., 1975), silica-alumina (Satterfield et al., 1973), membranes (Beck and Schultz, 1972; Baltus and Anderson, 1983; Anderson et al., 1988), and aluminas (Prasher and Ma, 1977; Chantong and Massoth, 1983; Cheng et al., 1990). For restrictive diffusion in catalyst pores, the effective diffusivity, De, is given by De = ( ~ D , / T ) F ( X ) (1) where Db is the bulk diffusivity in free solution, t is the catalyst porosity, T is the catalyst tortuosity, and F(X) is the restrictive factor, which is a function of A, the ratio of diffusing solute molecule size, d,, to pore size, d,. Renkin (1954) derived the following expression for F(X): F(X)= (1 - A)2(1- 2.104X + 2.089X3 - 0.948X5), for A I0.5 (2)
Beck and Schultz (1970) presented a simplified form of eq 2 as F(X) = (1 - XIz, where 2 = 4 for X 5 0 . 2 (3) These equations can also be approximated by the following empirical expression F(A) = exp(-bX) (4) where b is a constant. Values of b from 3.9 to 4.6 have been reported (Satterfield et al., 1973; Colton et al., 1975; Baltus and Anderson, 1983; Chantong and Massoth, 1983). The effect of solvent on restrictive diffusion in zeolites has received some attention. Satterfield and Cheng (1972a)
* To whom
correspondence should be addressed. Department of Fuels Engineering. *Department of Chemical Engineering.
showed that the solute-solvent interaction during counterdiffusion would be expected to affect the diffusion rate of the solute. Steric hindrance of solvent in zeolite pores can also affect the rate of diffusion (Satterfield et al., 1971; Satterfield and Cheng, 1972b). The effect of surface blocking caused by relatively large solvent molecules in 5A zeolite has also been observed (Car0 et al., 1980; Jasra and Bhat, 1987). The hydrodynamic theory for the restrictive diffusion of nonspherical solutes in fine pores remains largely unevaluated (Deen, 1987). Investigators have proposed several choices for the solute dimension: (1) radius based on sphere of equivalent volume (Anderson and Quinn, 1974), (2) molar volume average radius (Prasher and Ma, 1977), (3) critical dimension (Moore and Katzer, 1972; Satterfield et al., 1973; Colton et al., 1975), and (4) mean external length (Giddings et al., 1968; Baltus and Anderson, 1984). Recently, combined A values employing Stokes-Einstein radius and mean external length were used to correlate the restrictive factor (Anderson et al., 1988, Limbach and Wei, 1990). The maximum dimension has apparently not been used. Similar studies on the effect of solvent and solute configuration on restrictive diffusion in hydroprocessing catalysts have not been reported. The present study was undertaken to obtain such information. Experimental Section Materials. Catalyst. Three activated aluminas of different pore dimensions were provided by Alumina Company of America. Before impregnation, the supports were calcined at various temperatures in excess of 400 “C to obtain the desired average pore sizes for the diffusion studies. Catalysts, containing 1.5% nickel and 4% molybdenum, were prepared by incipient wetness impregnation. The 3.2 X 10-3-m (1/8-inJdiameter alumina spheres were coimpregnated with appropriate solutions of nickel nitrate and ammonium paramolybdate in 3% hydrogen peroxide and adjusted to a pH of 2.1 by dilute nitric acid. The impregnated catalysts were held for 24 h at ambient conditions to ensure even distribution of the active phases throughout the particle, then oven dried at 110 OC for 6 h, and finally calcined at 400 “C for 18 h. Electron microprobe scans of cross sections of several catalyst particles confirmed essentially uniform distributions of Ni and Mo throughout the particles, with the exception of catalyst
0888-5885/91/2630-0022$02.50/00 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30,No. 1, 1991 23 Table I. Physical Properties of Solutes empirical V:, solute cmg/mol ds: nm d-,'nm formula NAP 0.74 0.88 ClOH8 157.2 ANT 0.74 1.12 CllHlO 214 BAN 0.74 1.36 Cl8HlZ 246 DBA 0.85 1.57 C22H14 302.4 1.11 COR 1.18 CZIHl* 294.6 1.13 9PC 1.15 Cl8H13N 273.6 TPP CUHNNd 1.57 1.90 711.4 TBP 2.18 2.72 CgSHIBNd 1065.8
dlw,dnm 0.79 0.88 0.92 0.98 0.98 0.95 1.31 1.50
9-Phenylcarbazole (9PC)
Coronene
(COR)
Naphthalene (NAP)
a V,, molar volume a t normal boiling point, as estimated by Le Bas correlation (Reid et al., 1987). *d., critical dimension. minimum length parallel to the ring system. cd-, maximum dimension. dd,vc, molar volume average diameter, d,,, = [2(3V,/ ~ " Y I 107.
NiMo-325, which exhibited about 20% higher Mo concentrations near the outer surface. Solutes. The eight aromatic compounds studied were naphthalene, NAP (Analab); anthracene, ANT (Baker); 2,3-benzanthracene, BAN (Aldrich); 1,2,5,6-dibenzanthracene, DBA (Aldrich); coronene, COR (Aldrich); 9-phenylcarbazole, 9PC (Aldrich);tetraphenylporphyrin, TPP (Midcentury); and tetra(4-biphenylyl)porphyrin, TE3P (Midcentury), all of the highest purity available. The structures of these molecules are shown in Figure 1, and their physical properties are given in Table I. The critical and maximum dimensions indicated in Table I were calculated from bond lengths, bond angles, and van der Waals radii (Weast, 1980). Solvents. Three nonpolar hydrocarbon solvents employed in this study were cyclohexane (HPLC grade, 99.9+%), n-heptane (HPLC grade, 99.7+%), and isooctane (spectrograde, 99.3+% ), all obtained from Aldrich. Their viscosities at 25 "C are 0.88,0.40,and 0.39 cP, respectively (Reid et al., 1987). The solvents were pretreated with calcined 5A molecular sieves to remove water and other impurities. Apparatus and Procedure. BET surface areas were measured by nitrogen sorption with a Micromeretics AccuSorb 2100E analyzer. Skeletal densities were obtained from the same instrument by helium displacement. Average micropore diameters were calculated from mercury porosimetry data obtained from a Quantachrome Autoscan 60 porosimeter. Equilibrium solute uptakes for the catalysts were determined as described by Chantong and Massoth (1983). The catalyst particles and solution were mixed in a bottle with a Teflon-coated micromagnetic stirrer for 2 days and then put in a shaker for an additional 7 days before analysis. Diffusion runs were carried out at ambient conditions with 2.1 X 104-4.2 X 1O44-m (35-65 Tyler mesh) crushed samples of catalysts. This particle size range was chosen to assure that external mass transport was negligible (Chantong and Massoth, 1983). Sieving was repeated several times to reduce the fines and irregular-shaped particles. A stirred-tank vessel was used to determine the effective diffusivity (Chantong and Massoth, 1983). To avoid fracture of the particles, the catalyst particles were contained in a 1.3-cm-diameter X 5.1-cm-high cylindrical wire-gauze basket, which was hooked underneath the propeller. Solute initial concentrations were 30 mg or less per liter of solvent and, therefore, were considered to be infinite dilute solutions. Before the calcined catalyst particles were contacted with solution, they were flushed with solvent to remove fines and avoid contact with moisture in the air. The stirring speed of the stirred-tank diffusion unit was maintained at 750 rpm. Samples were
Anthracene (ANT) 5,10,15,20-Tetraphenyl. porphyrin (TPW
2.3-Benzanthracene (BAN)
-8
m
W
~,10,15,20.Tetra(4-biphenylyl)porphyrin (TBW
1,2,5,6-Dibenzanthracene (DBA)
Figure 1. Molecular structure of solutes.
taken periodically to follow the change in solution concentration with time with a Beckman Model 25 UV spectrophotometer, using predetermined calibration curves. The amount of solute taken up is approximately proportional to the surface area of the catalyst (Johnson et al., 1986). To confine the depletion of the solution concentration between 40% and 6070, the amount of catalyst employed per run varied from 0.1 to 1.0 g, except for very dilute TBP solutions, in which 20-40 mg were used. Data Treatment. The effective diffusivity, De, was determined from concentration decay time data of diffusion experiments. When applying a sorption technique, the determination of De depends on the adsorption isotherm (Crank, 1975). In order to calculate De, dimensionless time and dimensionless concentration relationships need to be developed to back-calculate the ratio of dimensionless time for the nonlinear isotherm case, ON, to real time, t , viz.,
De = (0,/t)Cr2
(5)
where r is the particle radius. The modeling equations have been described previously (Chantong and Massoth, 1983; Limbach and Wei, 1990). The resulting coupled partial differential equations were solved with the DSS/P finite-difference program (Schiesser, 1976). The effective diffusivity can also be calculated from the ratio of dimensionless time for the linear isotherm case, bL, to real time and the adsorption constant, K, the ratio of equilibrium uptake to final concentration, by assuming a linear adsorption isotherm, viz., De = ( O , / t ) ( e + pK)r2 (6) Details of the calculation have been given elsewhere
24 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 Table 11. Physical Properties of Catalysts catalyst properties NiMo-125 NiMo-225 BET surface area, m2/g 112 204 skeletal density, ps, g/cm3 3.44 3.30 particle density: p, g/cm3 1.28 1.32 pore volume,b V,, cm3/g 0.49 0.45 porosity, t 0.627 0.594 av micropore diameter: nm 17.5 8.5 Determined by Hg displacement. porosimetry data. a
NiMo-325 291 3.17 1.30 0.45 0.585 6.2
0.4
1
-c
NiMo-125
V, = 1 / p - l / p w From Hg
Table 111. Bulk Diffusivity Data Used T o Develop (7). Aromatics at Infinite Dilution in Aliphatic Solvents 106Db, solute solvent cm2/s type temp, K benzene n-hexane 47.0' Mb 298.2 298.2 46.4 M 299.2 46.6 M 47.6 Tb 298.2 toluene 299.2 n-hexane 43.8 M 298.2 42.1 M p-xylene 299.2 n-hexane 40.7 M mesitylene n-hexane 34.5 M 299.2 naphthalene M n-hexane 37.0 299.2 tetralin 32.7 M n-hexane 298.2 anthracene M 31.5 n-hexane 299.2 phenanthrene n-hexane 30.8 M 298.2 BAPb 26.6 T n-hexane 298.2 benzene 18.9 T cyclohexane 298.2 19.0 M 298.2 toluene 297.1 cyclohexane 16.3 M 16.5 T 298.2 p-xylene cyclohexane 16.5 T 298.2 mesitylene 14.0 T cyclohexane 298.2 naphthalene cyclohexane 14.5 T 298.2 phenanthrene cyclohexane 12.0 T 298.2 benzene n-heptane 39.2 M 298.2 34.0 M 298.2 298.2 38.6 M benzene n-octane M 32.5 298.2 toluene n-heptane 37.2 M 298.2 toluene n-decane 20.9 M 298.2 to 1u en e n-dodecane 13.8 M 298.2 toluene n-tetradecane 10.2 M 298.2
...
ref 1
2 3 4 3
1000
100
10
10000
100000
PORE RADIUS, A Figure 2. Pore-size distribution of NiMo catalysts obtained from Hg porosimetry.
1
3 3 3 2 3 2 4 10 7 6 10 10
10 10 10 7 8 9 9 I 1 1 1
a In case of duplicated data, averaged value was used. Symbols: BAP = benzo(a]pyrene, M = mutual diffusion, T = tracer diffusion. Reference: 1 = Wilke, C. R.; Chang, P. AIChE J. 1955, I , 264. 2 = Bidlack, D. L.; et al. J . Chem. Eng. Data 1969,14, 342. 3 = Dymond, ?J. H. J. Phys. Chem., 1981,85, 3292. 4 = Dymond, J. H.; Woolf, L. A. Chem. SOC. J.,London 1982, 78, 991. 5 = Haluska, J. L.; Colven, C. P. Ind. Eng. Chem. Fundam. 1971, 10,610. 6 = Grushka, E.; Kikta, E. K. J. Phys. Chem. 1974, 78, 2297. 7 = Sanni, S. A.; e t al. J. Chem. Eng. Data 1971, 16, 424. 8 = Calus, W. F.; Tyn, M. T. J. Chem. Eng. Data 1973, 18, 377. 9 = Hassis, K. R.; et al. J. Phys. Chem. 1970, 74, 3518. 10 = Sun, C. K. J.; Chen, S. H. AIChE J. 1985, 31, 1510.
(Chantong, 1982; Johnson et al., 1986). Results and Discussion
Properties of Catalysts. The properties of the calcined catalysts are given in Table 11, and the pore-size distributions of the catalysts are shown in Figure 2. Pore volumes were determined from density differences to account for the "cut-off" by mercury intrusion. Average micropore diameters were calculated from logarithmic mean values of the distribution curves, which were determined from mercury porosimetry data in the micropore regime, assuming a log-normal size distribution below the cut-off. The catalysts possess about the same density, pore volume, and porosity. Bulk Diffusivity. No experimental values were available for the bulk diffusivities of the polyaromatic
0
0
10
20
30
40
50
Db x
106 (cm2/s), EXPERIMENTAL Figure 3. Comparison of predicted and experimental bulk diffusivities for aromatic solutes in aliphatic solvents a t ambient conditions.
solute-aliphatic solvent systems used in this study. To estimate these values, a correlation was developed based on 21 experimental bulk diffusivities from the literature for 9 aromatic compounds in 7 nonpolar hydrocarbon solvents (see Table 111). The literature values were correlated with their physical properties based on a WilkeChang (W-C)-type equation (Wilke and Chang, 1955). Nonlinear regression analysis of these data led to the following specific equation, applicable to aromatic solutes in aliphatic solvents a t ambient conditions:
where Mbis the solvent molecular weight, g/mol, V, is the molar volume of solute at its boiling point, cm3/mol, and Pb is the solvent viscosity, cP. The correlation of the literature data using eq 7 is shown of 0.99, with in Figure 3. A correlation coefficient (R2) a mean error of f7% in bulk diffusivity was obtained. It is clear from Figure 3, that diffusivities predicted by the original W-C equation are consistently lower than the literature values. The estimated D b values for the so-
Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 25 Table IV. Estimated Bulk Diffusivity ( 106Db,cm2/s) Based on a Modified Wilkdhang-Type Equationa solvent solute cyclohexane n-heptane isooctane COR 10.3 (10.9)* 21.0 22.3 9PC 10.7 (10.0) TPP 6.34 (5.98) 13.0 13.8 TBP 5.07 (4.97)
2o
15
k
I
= 18.7 X 10-8(Mb)0%"/rbo."V~B, cmz/s. * ( ) values obtained from experimental data by using eq 9.
Table V. Comparison of Effective Diffusivity Obtained from Linear- and Nonlinear-Adsorption-Isotherm Fits nonlinear-isotherm fit linear-isotherm fit catalyst ma na 106D,,cm2/s K b 106D,,cmz/s NiMo-125 39.6 2.10 4.25 451 3.94 2.98 1101 3.10 NiMo-225 67.8 2.08 NiMo-325 1.58 2.18 1305 2.27 171.4
lute-solvent combinations used in this study, based on this modified W-C equation, are listed in Table IV. Effective Diffusivity. Similar to studies with alumina supports (Chantong and Massoth, 1983), results from equilibrium adsorption experiments of coronene in cyclohexane with the three catalysts showed that equilibrium solute uptakes, qes, are nonlinearly dependent on solution final concentrations, C,, as illustrated in Figure 4. These nonlinear relationships were found to fit well a Freundlich isotherm, viz.,
m(Ceq)l'n
10
30
20
EQUILIBRIUM CONCENTRATION, mglL
Figure 4. Equilibrium adsorption isotherms for coronene in cyclohexane a t 25 OC with NiMo catalysts: ( 0 ) 125, ( 0 )225, (A)325.
Constants in eq 8. *Constant in eq 6.
qeq
0
(8)
where m and n are constants. The latter values are given in Table V. On the basis of the values of m and n, the effective diffusivities of coronene in cyclohexane in the three catalysts were numerically solved by using eq 5. Adsorption constants, K, and effective diffusivities obtained by using eq 6 are also given in Table V. The variation of resulting De values for each catalyst was between 4% and 7%. Although the nonlinear fit stands on a better theoretical basis, it requires complete isotherms for each solute-solvent system. The isotherms for the other solute-solvent combinations were not obtained. However, the OL versus t plots obtained from diffusion runs showed good linearity. The effective diffusivities in this study were, therefore, calculated on the basis of linear isotherms. A similar approach has been applied by Prasher and Ma (1977). Effects of Solvents. Three structurally different aliphatic hydrocarbon solvents, namely, cyclohexane (cyclic), n-heptane (straight chain), and isooctane (branched), were
employed to assess their effect on solute diffusivity. The measured effective diffusivities of various solutes in these solvents are listed in Table VI. The results show that the effective diffusivity of the solute varies with the solvent employed. However, the characteristic diffusion constrictivity factor, 6 = De/tDb(Van Brake1 and Heertjes, 1974), which accounts for differences in bulk diffusivity, was essentially the same for each solute-solvent combination with each catalyst. Thus, solvent structure does not appear to influence restrictive diffusion. This is in contrast to results found from zeolite studies (see Introduction), in which the pore dimension is close to the size of the solvent molecule. The pore sizes of the catalysts in the present study are a t least 1 order of magnitude larger than the solvent molecules; therefore, the blocking effect caused by nonadsorbing solvent molecules should be minimal. The solvent is thus considered as a continuous phase inside the pore. Since there is very little interaction between solvent and solute in these solutions, it is not the shape but the viscosity of the solvent and, thus, the bulk diffusivity that influence the diffusion of the solute. Restrictive Diffusion Correlations. To check the Db values obtained from eq 7 with the experimental data, eqs 1 and 4 were combined to give In ( D e / € )= -bd,(l/d,) + In ( D b / T ) (9) Straight-line fits of the data plotted as In ( D e / € )versus l / d for the solutes in cyclohexane are shown in Figure 5, wiere the intercepts are equal to Db/r. Such an analysis, which is not possible with the combination of eqs 1 and
Table VI. Effective Diffusivity and Diffusion Constrictivity Factor of Disk-Shaped Solutes in Solvents on NiMo Catalysts WD,, cmz/s (6) solute catalyst A" cyclohexane n-heptane isooctane averaged d COR NiMo-125 0.063 3.94 (0.610)* 8.34 (0.633) 7.97 (0.570) 0.60 f 0.03 NiMo-225 0.125 3.10 (0.507) 6.48 (0.519) 6.39 (0.482) 0.50 f 0.02 C 5.10 (0.391) 0.38 0.01 NiMo-325 0.179 2.27 (0.377) 9PC NiMo-125 0.065 3.78 (0.563) NiMo-225 0.127 2.94 (0.463) NiMo-325 0.182 2.49 (0.398) TPP NiMo-125 0.090 2.00 (0.503) 3.89 (0.477) 4.12 (0.476) 0.49 f 0.01 NiMo-225 0.176 1.35 (0.358) 3.02 (0.397) 3.10 (0.378) 0.38 h 0.02 NiMo-325 0.253 1.01 (0.272) 2.13 (0.280) 2.04 (0.253) 0.27 f 0.02 TBP NiMo-125 0.125 1.46 (0.459) NiMo-225 0.245 0.63 (0.209) NiMo-325 0.352 0.42 (0.142)
*
OBased on critical dimension. bValues in ( ) are diffusion constrictivity factor, 6 = D,/cDb. cData obtained were not reproducible.
' 1
26 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991
0
(a) critical dimension, 213.7f0.3
-11
-12
c,
-
COR
-
-0.5
.
-0.4
-0.2
-0.3
.0.1
0.0
0
(b) maximum dimension, 212.8f0.3
'
-15 u.00
Lo 0.05
0.10
0.15
I
0.20
e
I
-'
1Idp Figure 6. Determination of tortuosity and bulk diffusivity of solutes in cyclohexane from eq 9.
-6
-0.6
Table VII. Effective Diffusivity and Diffusion Constrictivity Factor of Rodlike Solutes in Cyclohexane on NiMo Catalysts catalyst solute lo8&, cm*/s lo6&, cm2/s 6 NiMo-125 NAP 14.5 5.19 0.571 NiMo-125 ANT 12.00 4.56 0.606 NiMo-225 3.78 0.530 NiMo-325 3.11 0.443 NiMo-125 BAN 11.4b 4.39 0.614 NiMo-125 DBA 10.2b 4.30 0.612 NiMo-225 3.39 0.559 OData taken from Sun and Chen (19851, assuming Db(anthracene/cyclohexane) = Db(phenanthrene/cyclohexane). Calculated from eq 6.
3, avoids the need to select a pertinent solute dimension . calculate Db,T needs to in order to determine D ~ / TTo be determined first. This requires data from diffusion experiments with a solute having a known value of Db For this reason, anthracene was chosen because it has a relatively large molecule size, is easily analyzed, and has a known value of D b The resulting effective diffusivities for anthracene in cyclohexane solvent with the three catalysts, as shown in Table VI, together with a measured Db of 1.2 X 10" cm2/s (Sun and Chen, 1985) yielded from eq 9 a T of 1.37. By applying the same T to the other solutes, values of Db were obtained (Table IV), which are within 10% of the calculated values from eq 7. The disk-shaped solutes studied have somewhat different lengths in the two dimensions parallel to the ring system. To determine which dimension is more valid in correlating the restrictive factor, diffusivity measurementa were carried out on three additional polyaromatic solutes (NAP, BAN, and DBA) having rodlike shapes with significantly different lengths in the two dimensions parallel to the ring system. The effective diffusivities obtained and corresponding 6 values are listed in Table VII. Values of A were calculated for these and previous solutes from the appropriate average catalyst pore radii (Table 11) and solute dimensions (Table I). The dimensions considered were the critical (minimum), maximum, and molar volume average. Applying each of the three solute dimensions of Table I to the computation of A, correlations of the data, nc-
-0.5
-0.4
-0.3
-0.2
-0.1
0
R 2 -0.83
0.0
I
-I
-0.3
-0.2
-0.1
0.0
In( 1-1) Figure 6. Comparison of 2 factors obtained from different solute molecular dimensions, where = disk-shaped, = rod-shaped solutes.
cording to the combination of eqs 1 and 3 in logarithmic form to give In 6 = 2 In (1- A) - In
T
(10)
are shown in Figure 6. It is seen that the overall data fit is best (R2= 0.95) when using the critical dimension. Also, the use of this dimension gives a value of 2 = 3.7 f 0.3, which is close to the value of 4.0 from eq 3, based on the hydrodynamic theory. Both maximum and average dimensions give unrealistic 2 values. Dimensions also considered were the Stokes-Einstein diameter and mean projected length. The former is generally smaller than the critical dimension or the molar volume average diameter, and neither correlated the data well. Therefore, the use of critical dimension for the range of A values employed in this study appeared to be the best choice for solute dimension in computing A. To determine an average tortuosity factor using all the measured effective diffusivities from Tables VI and VII, eqs 1 and 4 were again combined and recast into the following equation in terms of the definition of 6 to give In (D,/eDb) = In 6 = -bX - In T (11) A semilogarithmic plot of In 6 versus A, using critical di-
Ind. Eng. Chem. Res., Vol. 30,No. 1, 1991 27 1.o
0.8
0.6 h
5 LL
0.4
0.2
0.0
u.0
0.1
0.2
0.3
0.4
0.5
.1
0.0
0.1
0.2
0.3
0.4
h Figure 7. Determination of tortuosity factor and b from eq 11.
mensions of the solutes (Table I), is given in Figure 7. A statistical fit gave T = 1.27f 0.03 and b = 4.30 f 0.3 with a correlation coefficient, R2, of 0.93. The value of b is within the range found by other investigators as discussed in the Introduction section. An alternative method involved the use of eq 3 rather than eq 4 for F(X) to obtain eq 10. From the plot of Figure 6a, based on all the data of Tables VI and VII, a tortuosity of 1.34 f 0.03 was obtained. Another alternative method for determining the average tortuosity factor from all the data was to combine eqs 1 and 2 to give 6 = F(X)/r (12) where F(h) is given by eq 2, applicable up to X I0.5. A plot of 6 versus F(X) yielded a straight line (R2= 0.95)with a slope equal to 1/7,giving a T value of 1.41. These tortuosity factors of 1.27,1.34, and 1.41compare well with a recent NMR study (Cheng et al., 1990) of restrictive diffusion in y-alumina, a similar base material, which found tortuosity factors between 1.07 and 1.56 for liquid-phase diffusion. The above tortuosity values are somewhat lower than those of 1.59-1.71predicted by Wakao and Smith (19621, based on a random-pore model. It has been suggested that the concept of 7 is more complicated than the analysis based on the random-pore model (Van Brake1 and Heertjes, 1974) and that T may change with varying X (Smith, 1986). Therefore, reanalysis of the data was undertaken, treating T as functions of X by both linear and power-law expressions, viz., T = 70 + (13) 7 = To fixr (14)
+
where T~ is the tortuosity factor as A approaches zero, and a,fi, and y are constants. Equation 12 was solved for the parameters of eq 13 or eq 14 by nonlinear regression analysis. Both a and /3 had negative values, indicating that 7 decreases with increasing A; Le., there is less restrictive diffusion effect in the larger pore. Because this is opposite to expectation, the determination of an average tortuosity, by the three methods discussed above, for all three catalysts within the range of A studied (up to A = 0.35) appeared to be justified. Finally, a comparison of the experimental F(X) values for the three literature correlations, as given by eqs 2-4,
Figure 8. Comparison of experimental restrictive-factor data with literature correlations, for an average tortuosity factor of 1.34.
using solute critical dimensions and T = 1.34,the average of the three tortuosity factors determined from the application of the three equations, is given in Figure 8. It can be seen that, within the X range studied, all three correlations give a satisfactory fit to the experimental data. Conclusions 1. An improved correlation has been developed for estimation of bulk diffusivities of polyaromatic compounds in aliphatic hydrocarbon solvents. 2. Effective diffusivities in catalysts were found to be independent of the structure of aliphatic hydrocarbon solvents when bulk diffusivity is accounted for. 3. Restictive diffusion correlations were best when using the critical dimension of the solute. 4. For diffusion of aromatic compounds in hydrotreating catalysts at ambient, nonreactive conditions, the restrictive factor was best correlated by F(h) = (1- XI3.', in reasonable agreement with hydrodynamic theory. Acknowledgment
We thank the Department of Energy for financial support of this research (Contract DE-FG22-87PC79933). Alumina supports provided by Alumina Company of America are also gratefully appreciated. Nomenclature b = constant in eq 4 C , = equilibrium concentration of bulk solution, mg/cm3 Db = bulk diffusivity, cm2/s De = effective diffusivity, cmz/s d,, = molar volume average solute diameter (see footnote of Table I), nm d, = maximum solute dimension (parallel to ring system), nm d, = average micropore diameter, nm d, = critical solute dimension (minimum length parallel to ring system), nm F(X)= restrictive factor, Der/cDb K = adsorption constant, cm3/g Mb = molecular weight of solvent, g / m d m = constant in Freundlich isotherm, eq 8 N = Avogadro's number n = constant in Freundlich isotherm, eq 8 qss = amount of solute adsorbed by catalyst at equilibrium, mg/g
28 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991
R2 = correlation coefficient r = average particle radius, cm 7' = temperature, K t = time, s V, = molar volume of solute at its boiling point, cm3/mol V , = pore volume, cm3/g Z = constant in e q 3
Creek L e t t e r s = constant in eq 1 3 /3 = constant in eq 14 6 = diffusion constrictivity factor, D,/cDb c = particle porosity y = constant in eq 14 pb = viscosity of solvent, CP X = ratio of solute critical dimension to average pore diameter OL = dimensionless time defined b y eq 6 ON = dimensionless time defined b y eq 5 p = particle density, g/cm3 ps = skeletal density, g/cm3 r = tortuosity factor T,, = tortuosity factor when X approaches zero CY
Registry No. COR, 191-07-1; 9PC, 1150-62-5;NAP, 91-20-3; TPP, 917-23-7; ANT, 120-12-7; BAN, 92-24-0; TBP, 81566-83-8 DBA, 53-70-3; Ni, 7440-20-2; Mo, 7439-98-7; heptane, 142-82-5; isooctane, 26635-64-3; cyclohexane, 110-82-7.
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Receiued for review April 17, 1990 Accepted July 17, 1990
