Ind. Eng. Chem. Res. 1990,29, 1968-1976
1968
Phillips, K. L. Proposed Explanation for Apparent Dependence of Liquid Phase Mass Transfer Coefficients on Pressure. Can. J. Chem. Eng. 1973,51,371-374. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Sawant, S. B.; Joshi, J. B. Critical Impeller Speed for the Onset of Gas Induction in Gas-Inducing Types of Agitated Contactors. Chem. Eng. J. 1979, 18, 87-91. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Sridhar, T.; Potter, 0. E. Interfacial Area in Gas-Liquid Stirred Vessels. Chem. Eng. Sci. 1980, 35, 683-695. Suresh, A. K.; Sridhar, T.;Potter, 0. E. Mass Transfer and Solubility
in Autocatalytic Oxidation of Cyclohexane. AIChE J. 1988, 34, 55-68. Teramoto, M.; Tai, S.; Nishi, K.; Teranishi, H. Effects of Pressure on Liquid-Phase Mass Transfer Coefficients. Chem. Eng. J. 1974, 8, 223-226. Topiwala, H. H. Surface Aeration in a Laboratory Fermenter at High Power Inputs. J. Ferment. Technol. 1972,50, 668-675. Yoshida, F.; Arakawa, S. Pressure Dependence of Liquid Phase Mass Transfer Coefficients. AIChE J . 1968, 14, 962-963.
Received for review November 27, 1989 Revised manuscript received April 10, 1990 Accepted April 27, 1990
Determination of the Macroscopic Structure of Heavy Oils by Measuring Hydrodynamic Properties Rick L. Nortz and Ruth E. Baltus* Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699
Parviz Rahimi C A N M E T , Energy, Mines and Resources Canada, Ottawa, Ontario K1A OG1,Canada
The molecular weight, bulk-phase diffusivity, and intrinsic viscosity of fractionated samples of Athabasca tar sand bitumen vacuum bottoms and of a blend of vacuum resid from Western Canadian crudes has been measured. The diffusion coefficient and intrinsic viscosity of samples of each material of similar molecular weight were found to be in general agreement. Multisubunit structural models are proposed to describe the hydrodynamic characteristics of these materials. These models are rigid structures assembled from identical spherical subunits. The experimental diffusivity values were found to be in excellent agreement with those predicted for multilayer structures containing 6 2 4 subunits. The results indicate that the effective size of these materials appears to be considerably smaller when measuring intrinsic viscosity than when measuring solute diffusivity. We propose that association/dissociation phenomena are responsible for these observations.
Introduction The petroleum industry has been interested in recent years in developing catalytic processes capable of converting heavy petroleum feedstocks and residua into more valuable, lower molecular weight components (Speight, 1984). Heavy oils are arbitrarily defined in terms of API gravity (>20) and sulfur content (>3%). Residua are those materials that do not volatilize when a crude oil is subjected to atmospheric or vacuum distillation. Both materials are characteristically more difficult to process than lighter fractions because of their large molecular size, polarity, and heteroatom and metals content. An understanding of the diffusional resistances influencing catalytic upgrading reactions with heavy oil feedstocks requires knowledge of the transport and equilibrium properties of these materials in small pores. These properties are dependent on the size and conformation of the constituents of the material. The size and shape of heavy oils also influence their elution characteristics in gel permeation chromatography, an analytical technique that is commonly used in characterizing petroleum-derived materials. Therefore, knowledge of the size and shape of the constituents of these materials is also necessary for the accurate interpretation of information obtained from this chromatographic process. For several years, our efforts have been directed toward utilizing hydrodynamic measurements as a means of elucidating information about the size and conformation of the constituents of heavy petroleum fractions from a variety of sources. These measurements involve the determination of bulk-phase diffusivities and the intrinsic 0888-5885190/ 2629- 1968$02.50/ 0
viscosity of narrow size range fractions obtained by gel permeation chromatography. Such.measurements have commonly been used in studies of proteins and polymers. However, the use of these measurements in the characterization of petroleum-derivedmaterials has been limited. The bulk-phase diffusivity, D,, of a macromolecular solute is related to the size of the solute through the Stokes-Einstein equation:
where k is Boltzmann's constant, T is the absolute temperature, f is the frictional coefficient, v0 is the solvent viscosity, and ad is the effective solute radius determined from diffusion. The intrinsic viscosity of a macromolecular solution, [VI, is defined by considering the concentration dependence of the solution viscosity: qs/fl0
= 1 + [TIC
+ KC2 + ...
(2)
where 9 is the viscosity of (s) solution and (0)solvent. For low concentrations, the series expansion in c can be truncated. With this simplification, eq 2 can be rearranged to yield (3)
where qe is the specific viscosity. Equation 3 predicts that a plot op?,p/c versus c should yield a straight line where the intercept at c = 0 is defined as the intrinsic viscosity. 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1969 The intrinsic viscosity depends on both the size and the shape of the constituents of the solution. For spherical solutes, the intrinsic viscosity is related to the solute radius and molecular weight by the following expression: (4)
where NA is Avogadro’s number and M is the solute molecular weight (Hiemenz, 1986). The symbol a, is used to represent the solute radius determined from the intrinsic viscosity, assuming the solute is a sphere. Equations 1and 4 show that bulk-phase diffusivity and intrinsic viscosity depend differently upon solute size. One can take advantage of this difference by combining the results from both measurements. A comparison of the size parameters ad and a,, as defined by eqs 1 and 4, can provide an indication of the conformation of the solute being investigated. If the solute is spherical, ad = a,. For solutes that are rigid spheroids (oblate or prolate) and for h e a r polymers, a, > ad. For solutes modeled as assemblages of spheres, a, C ad. Although the rheological properties of petroleum-derived materials have been of interest to a number of researchers, there has been only a limited number of reports in which the measured solution hydrodynamic properties were related to the molecular size and shape of the constituents. Sakai and co-workers measured the diffusion coefficient and intrinsic viscosity of carbon pitch samples. They found that the pitch molecules could be represented by oblate spheroids with an axial ratio in the range 1-3 (Sakai et al., 1981, 1983). Reerink (1973) investigated the hydrodynamic properties of asphaltene fractions by measuring intrinsic viscosity and sedimentation velocities. The asphaltene molecules were also modeled as oblate spheroids; however, the axial ratio was found to be in the range 7-9. In both of these investigations, stepwise solvent precipitation was used to fractionate the material, a method that has been found to separate more by chemical structure than by molecular size. Results from intrinsic viscosity and bulk-phase diffusion coefficient measurements for narrow size range fractions of Cold Lake vacuum bottoms have previously been reported by our laboratories (Kyriacou et al., 1988a,b Baltus et al., 1988). These measurements were performed in tetrahydrofuran solvent at 10, 20, and 30 OC and in 1methylnaphthalene solvent at 50 “C. The intrinsic viscosity was determined by using a Ubbelohde viscometer, and the procedure was the same as that used in the work reported here. The diffusion coefficient values were determined by measuring solute flux through Nuclepore polyester membranes. This procedure was not used in the work reported here. Multisubunit structural models were proposed to describe the hydrodynamic properties of the constituents of each fraction. These models are rigid structures assembled from identical spherical subunits. The experimental intrinsic viscosity results were found to agree with the properties of model complexes of a planar geometry. This structure is consistent with the chemistry of these materials, which is generally considered to be aromatic and cycloparaffinic. The experimentally measured diffusion coefficients were found to be considerably smaller than those predicted for the model structures that were found to characterize the experimental [v] values. The explanation that was proposed for these observations was that the constituents of these materials associated in solution during the diffusion experiments. Thus,their effective size increased over that observed during the viscosity mea-
surements. The solutions were subjected to shear forces during the viscosity measurements, and these shear forces may have been sufficient to inhibit any tendency toward association. Surface interactions between the residual materials and the polyester membranes were considered as another possible explanation for these observations. However, it was impossible to confirm this hypothesis. The present study was undertaken in order to better understand the structure and properties of heavy oil feedstocks. In this study, intrinsic viscosity and bulk-phase diffusivity for size fractions from two different source materials were measured. The bulk-phase diffusivity of each fraction was determined by measuring Taylor dispersion in a long capillary tube. This technique was chosen because it yields D, values that are less uncertain than those determined by measuring transport through a porous membrane (because the determination of D, involves the measurement of fewer parameters such as pore size and number of pores). This technique also eliminates the possibility of polymer-oil residual interactions that introduced some uncertainty in the interpretation of our previous results with polymeric membranes. In addition, during the Taylor dispersion measurements, solutions were subjected to shear forces, although these forces were not as significant as those present in a capillary viscometer. If our hypothesis proposed to explain our previous results is correct, one might expect to observe a difference in the D, values determined by using this different technique. Theory Taylor Dispersion. In the present study, the measurement of the dispersion of a solute slug in a flowing solvent stream was used to determine the solute diffusivity. The theoretical basis for interpreting dispersion experiments was initially presented by Taylor (1953). More rigorous theoretical developments have been presented since this early classic work (Gilland Sankarasubramanian, 1971). A solute slug is introduced into a capillary tube in which a solvent stream is flowing under laminar flow conditions. As the slug flows with the liquid, it disperses because of convective effects caused by the flow and by diffusion brought about by the concentration gradient between the sample slug and the surrounding solvent. When angular and radial velocities are assumed to be zero, axial molecular diffusion is neglected, and the local solute concentration is assumed to be equal to the cross-sectional averaged concentration, the solution to the continuity equation for this system is
The solute is introduced as a delta injection, where c, is the concentration of the injected sample and Q is its volume (Levenspiel and Smith, 1957). The dispersion coefficient, K, is defined by D2.7 2
where R is the capillary radius and D, is the solute bulkphase diffusivity. Comparing eq 5 to that of a normal distribution shows that the concentration distribution with respect to length at fixed t is a normal distribution. However, in a dispersion experiment, the solute concentration at a fiied position is measured as a function of time. A plot of c, vs time at fixed x (x = L) shows that the concentration distribution with respect to time is normal,
1970 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
provided K / & L is sufficiently small (Levenspiel and Smith, 1957). This means that the concentration profile does not significantly change its shape in the time required for the solute peak to pass by a fixed position. Mathematically, specifying K / a J to be small means that ii,t/L must be close to 1 for c, to be nonzero. By using the assumption LZ,t/L = 1, eq 5 can be rearranged to
C,
-(t
Q/(rR2ii,)
c,(t)
-=
(27r(2KL/~I,3))'/~
- L/U,)*
2(2KL/ii,3)
]
(7)
This is a normal distribution having a variance
Equation 8 is the relationship used in this study to calculate the solute diffusivity from the concentration vs time profile measured a t the outlet of the capillary. Hydrodynamic Properties of Multisubunit Complexes. The frictional coefficient and intrinsic viscosity of model multisubunit complexes containing up to eight spherical elements have been presented (Garcia de la Torre and Bloomfield, 1978). In the investigation reported here, we found it necessary to consider model structures containing more than eight elements. The approach used in calculating the hydrodynamic properties of these structures was the same as that of Garcia de la Torre and Bloomfield. A molecule is considered to contain N spherical subunits, each of radius cr. The position of each subunit in a particle-fixed coordinate system is fixed; Le., the molecule is rigid. The frictional coefficient of this molecule is determined by considering the force, F, needed to hold the molecule stationary in a h i d whose velocity at every point would be u if the molecule was not present. The unperturbed velocity, u,and the solvent velocity at the position of subunit i, vi, differ by a term that arises because of hydrodynamic interactions between subunit i and all other subunits of the molecule:
The 9 N equations represented by eq 11 were solved for the 9 elements in each tensor Gj, and the diffusivity was then calculated by using eq 14. In the calculations performed for this investigation, the maximum number of subunits comprising one structure was 24, yielding a system of 216 equations. It was possible to solve these systems exactly by using a Gaussian elimination algorithm in the IMSL library. Calculations were performed for model structures assembled as layers of regular polygons where each vertex in each polygon represents the center of a spherical subunit. The models were assembled such that the distance between the centers of any two adjoining spheres within the polygon was 2a and the distance between each layer was also 2a. Each subunit was positioned directly above a subunit in the layer below. Structures containing up to 24 spheres assembled from 1,2, or 3 layers were considered. The calculations were performed for many different structures; however, we are only reported results for structures that were eventually used in interpreting our experimental data. The intrinsic viscosity of a solution containing macromolecular multisubunit complexes is determined by considering the additional energy dissipation caused by the introduction of solute molecules into a steady shearing flow. When the energy dissipation is averaged over all orientations of the molecule with respect to the flow field, the following expression for [ a ] is obtained:
a#B
CB)
-+
a##
N'
u - vi =
]=1
Z;*F, (i = 1, 2, ..., N )
(9)
where the * indicates the summation over j # i and F; is the force on subunit j . The modified Oseen tensor Ti,is used to relate the hydrodynamic interactions between subunit i and the other subunits in the molecule:
where CY and /3 = x , y, or z (Garcia de la Torre and Bloomfield, 1978). Vectors Ri and Rjdescribe the position of subunits i and j , respectively, in a particle-fixed coordinate system with an arbitrary origin. Vector C is the center of hydrodynamic resistance of the molecule in the same coordinate system. Matrix Si; is defined by
Si, E (B-').. 11
(16)
where B is a 3 N X 3N "supermatrix" created by the 3 X 3 matrices Qip McCammon and Deutch (1976) provide explicit expressions for the elements of B. where Ri. is the vector connecting subunit i with subunit j , Rij is the length of this vector, and I is the idemfactor. Multiplication of eq 9 by 67r77,a and rearranging yield N
(i = 1, 2, ..., N )
(11)
Q;;= di,I - (1 - bi;)67raOaTi,
(12)
CQij.G, = I ;=l
where and the tensor Gj is defined by Fj = 6rv,aG;.u
(13)
The diffusion coefficient of the molecule is a scalar, related to the Gitensors by (14)
Experimental Section Sample Preparation. Athabasca tar sand bitumen vacuum bottoms (ABVB) was obtained from the sample bank of the Synthetic Fuels Research Laboratory of CANMET, and a blend of vacuum resid from Western Canada (IPVB) was obtained from Esso Petroleum Canada in Sarnia. Preparative gel permeation chromatography (GPC) was carried out on these samples by using a Waters preparative LC 500 system equipped with a refractive index detector and a 122 X 2.5 cm column packed with 100-8, Styragel. Tetrahydrofuran (THF) was used as the solvent at 20.7 mL/min and was freshly distilled before use. For ABVB, it was necessary to remove approximately 1-2 w t 70 sand before chromatographic separation. This was done by centrifuging a solution of ABVB in purified THF and then removing the T H F by using a rotary evaporator.
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1971
-e c
e
c
e u 0 c
0
Elution Volume (ml)
Figure 1. GPC elution profiles of ABVB and IPVB samples. ABVB is the sample from Athabasca tar sand bitumen vacuum bottoms. IPVB is the sample from Western Canadian crude vacuum resid blend.
Table I. Molecular Weight of Fractionated Sampleso fraction feed 1 2 3 4 5 6 7 8 9
ABVB sample wt '70 MW 9.87 11.60 14.00 14.76 14.73 12.02 8.10 5.81 9.06
1208 2253 2231 1789 1546 1471 1137 810 664 613
IPVB sample MW
wt%
10.00 12.86 15.35 16.32 15.39 10.42 6.56 4.43 8.6
896 1420 1238 1202 1117 1026 815 771 621 682
ABVB is the sample from Athabasca tar sand bitumen vacuum bottoms. IPVB is the sample from Western Canadian crude vacuum resid blend.
A solution of each feedstock in THF (0.1 g/mL) was prepared, and approximately 3 mL of the solution was injected in the column. In each case, nine fractions were collected as shown in Figure 1. This procedure was repeated until enough material was collected for further experiments. The solvent was then removed by using a rotary evaporator. Material balance calculations revealed a recovery range from 96% to 99%. The weight percent of each fraction for ABVB and IPVB is shown in Table I. Molecular Weight Determination. The molecular weights of the feedstocks and the corresponding nine fractions were determined in toluene at 40 "C by using a Knauer vapor pressure osmometer. The instrument was calibrated with benzil. For each sample, measurements were performed with three solutions with concentrations ranging between 10 and 30 g/kg. The number-average molecular weight of each fraction determined by this method is shown in Table I. The limited amount of sample did not permit duplication of each molecular weight determination. However, measurements were repeated for several samples, and the molecular weight values were shown to be reproducible within f 5 % . Sample preparation and molecular weight determinations were performed at CANMET Laboratories, Ottawa, Ontario, Canada. Diffusion Coefficient Measurements. Taylor dispersion measurements were performed by using an apparatus assembled from chromatographic equipment and a long stainless steel capillary tube wound in a helical configuration. Solvent was pumped through the capillary at
a constant, steady flow rate by using a Waters Model 590 solvent delivery system. Solute injection was performed with a Waters Model U6K injector. The dispersion peak was detected with a Waters Model 481 ultraviolet (UV) spectrophotometer, and the detector output was recorded by using a strip chart recorder. The capillary tube was positioned in a constant-temperature reservoir. The inside diameter of the capillary tube was 0.0508 cm, and its length was 4000 cm. The capillary was wound in a coil 38 cm in diameter. The injection volume of sample was 50 pL, and the solute concentration in the sample was 0.1 g/L. The solvent flow rate was maintained at 5.5 X lo4 cm3/s (linear velocity = 0.27 cm2/s). These experiments were performed at Clarkson University, Potsdam, NY. In order to ensure that the experimental system was consistent with the assumptions made in deriving the theoretical relationship used in interpreting the results (eq 81, it was necessary to carefully consider the design of our apparatus and the operating parameters. The Reynolds number was less than 1, indicating that laminar flow conditions were maintained. The assumption of negligible axial diffusion was satisfied by ensuring that the rate of axial convection was much greater than the rate of axial diffusion; i.e., GJ - Pe >> 48lI2L Dw R _.-
In our experiments, Pe i= lo9 while 481/2L/R = lo6. The elimination of radial concentration gradients was ensured if
L 2 / R 2>> Pe (18) (Baldauf and Knapp, 1983). In our experiments, L2/R2 = 2.5 X 1O'O. The square injection pulse could be treated as a delta injection because the volume of injected sample was less than 1/1000th of the tube volume. The theoretical analysis of Gill and Sankarasubramanian (1971) considers the complete convection/diffusion process, and their results were used in considering the influence of the entrance region where axial molecular diffusion cannot be neglected. Their calculations show that entrance effects have no influence on the measured solute peak if DwL - >> 0.5 l i s 2
In our experiments, D,L/(ii,,R2) = 23. In order to keep the apparatus within manageable proportions with a long capillary tube, it was necessary to coil the capillary. In doing this, the effect of secondary flow as well as the varying path lengths at different radial positions in the capillary had to be considered. Alizadeh and co-workers have presented results from a theoretical analysis of dispersion in a coiled tube (Alizadeh et al., 1980; Alizadeh and Wakeham, 1982). Their results show that secondary flow can be neglected if R Vo Re2- - = De2& I20 (20) Rc PODHere, R, is the radius of the helical coil, De is the Dean number and Sc is the Schmidt number. In our experiments, De2& i= 14.5. In order to neglect the effect of varying path lengths, the residence time in the capillary must be sufficient so that a molecule will sample all radial positions before exiting the tube. The dimensionless residence time is DJ/(ii,,R2),and as noted previously, the value of this parameter was 23, therefore ensuring that the effect of varying the path length could be justifiably neglected.
1972 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
The solvent used for all experiments was l-methylnaphthalene. This was used without further purification. All sample solutions were filtered through a 5-pm filter before injection into the capillary. Also, a helium atmosphere was always maintained over all solutions because it was determined that the ultraviolet absorbance characteristics of this solvent were influenced by exposure to air. Use of a dual-head chromatographic pump ensured that a constant flow rate through the capillary was maintained. Pure solvent was also continuously pumped through the reference cell in the ultraviolet detector by using a syringe pump. Each experiment yielded a plot of UV absorbance vs time. Previous work in our laboratory at Clarkson University indicated that the concentration of the injected samples in this study was well within the Beer's law region for petroleum-derived materials (Kyriacou et al., 1988a,b). Therefore, UV absorbance and solute concentration were assumed to be directly proportional. The peak variance was determined from the width of the peak a t half height, tilz, and the following relation, which is valid for a normal distribution: s =
Table 11. Frictional Coefficient and Intrinsic Viscosity of Model Multisubunit Comdexes" total no. of no. of subunits subunits in in each no. of [VIM/ structure layer layers f/(67rq0u) (8N*03) Q , / Q ~ 2 1.905 6 3 8.422 0.976 4 2 2.113 8 11.992 0.990 5 2.363 2 16.206 0.979 10 12 6 2.566 2 21.147 0.985 15 5 3 2.770 25.811 0.975 6 3 3.000 32.918 0.977 18 24 8 3 3.466 50.325 0.974 Each structure is assembled from multiple layers of a regular polygon where each vertex in the polygon is the center of a sphere of radius u. Each side of the polygon has length 2u.
t1/2
(8 In 2)1/2
The solute diffusivity was then calculated by using eq 8. The apparatus was tested by performing several experiments with monodisperse polystyrene in n-hexane. Experiments were performed with polystyrene samples of molecular weight 2000 and 4000 at 25 and 50 "C. The measured diffusivity values were compared with those determined by another laboratory using a similar procedure (Barooah and Chen, 1985). Comparing our results for polystyrene with those of Barooah and Chen showed agreement to within 13%. Intrinsic Viscosity Measurements. Solution viscosity was performed by using a Ubbelohde capillary viscometer that was surrounded by a jacket to allow for the circulation of constant-temperature water. These experiments were also conducted a t Clarkson University. Solutions of a known concentration of each oil residual fraction in 1-methylnaphthalene solvent were prepared and introduced into the viscometer through a 5-pm filter. The solution was allowed to equilibrate to constant temperature before the flow time through the capillary was measured. Measurements with each sample were repeated three times. With each fraction at each temperature, data were collected for pure solvent and for three samples of different concentrations. Kinetic energy corrections were negligible for the solutions used in these measurements. The solution and solvent densities were assumed to be equal because extremely dilute solutions were used. Therefore, the ratio of solution to solvent viscosity was equal to the ratio of solution to solvent flow times. The specific viscosity was calculated, and a linear plot of qsp/c vs c was prepared. The intrinsic viscosity was determined from the intercept at c = 0, as predicted by eq 3. The amount of sample available for these measurements was limited because of the considerable time required to perform the chromatographic fractionation. Therefore, our intrinsic viscosity measurements with the oil fractions were limited to a single viscometer. The results obtained by using several viscometers with poly(propy1ene glycol) samples of comparable molecular weight showed that the intrinsic viscosity of this material was independent of the viscometer capillary size, indicating that under the conditions of these experiments the shear rate was sufficiently low to allow the solute molecules to experience all orien-
time
Figure 2. Concentration vs time profile from a Taylor dispersion measurement. The solid line is the experimental profile. The dashed line is the profile for a normal distribution (eq 7).
tations with respect to the direction of the flow field (LaPine, 1988). We assumed that the conditions were the same for our experiments with the residual samples.
Results and Discussion Theoretical Results. The frictional coefficient and intrinsic viscosity of multisubunit model complexes are presented in Table 11. The results were obtained for a large number of structures; however, only the results for the model structures that were used in interpreting the experimental results of this study are included in Table 11. The ratio of the effective solute radius for intrinsic viscosity relative to the effective solute radius for diffusion, av/ad,was calculated by using the predictions for f and [ q ] and the definitions of ad and a, in eqs 1 and 4. This ratio is also listed in Table 11. For these structures, this ratio is slightly less than unity and does not vary significantly with the arrangement or number of subunits. Diffusion Coefficients. A representative plot of UV absorbance at 342 nm vs time recorded with the UV detector a t the capillary outlet is shown in Figure 2. The "double maximum" exhibited in the profile in this figure was observed to some extent in all the experiments with the oil residuals. However, this phenomenon was not observed with the polystyrene experiments, which were conducted under the same operating parameters (except temperature). The cause of this double maximum is not understood, although several explanations were considered. Golay and Atwood (1979) have shown that dispersion peaks eluting from short capillary tubes are predicted to have a characteristic double maximum. However, this double maximum is predicted only when the solute dimensionless residence time (D,L/(a,,RZ))is less than one.
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1973
.-+ 0
1
400
1000
2000 M 0 LEC U LAR WE IG H T
4000
a
1
-
r c
W
6 ,
200
Figure 3. Bulk-phase diffusivity as a function of molecular weight. Experimental values: (0) ABVB fractions at 50 "C, (0)IPVB fractions at 50 "C, ( 0 )ABVB fractions at 70 "C, (m) IPVB fractions at 70 "C. The error bars represent 95% confidence limits of selected data. The solid lines are eqs 22 and 23. and Q are the predictions for model structures using u = 5.1 A (Table 111).
*
(In our experiments, this value was about 23.) Also, the double maximum is predicted to appear as a "hump" on the trailing edge of a nonsymmetric elution peak. The elution peaks observed in our experiments were always symmetric. If the eluting sample was sufficiently concentrated so that the Beer's law relationship between UV absorbance and solute concentration was no longer linear, one might expect a "flattening" at the center of the peak where the concentration was highest. However, as noted earlier, the concentration of the injected samples was well within the linear Beer's law regime, and this possibility does not explain the double maximum that was generally observed. Another possible explanation that was considered was that some of the oil residual material was adsorbing to the capillary wall, causing some of the material to be retained for a longer time. In order for the double maximum to be explained by adsorption, the samples would have to be considered to be basically a bidisperse mixture containing only two components with different adsorption characteristics. Such a description of this heterogeneous material is highly improbable. The anomalous behavior did not affect our ability to determine the peak width at half height and therefore calculate a peak variance, which was the important characteristic needed to calculate a D, value from the elution curve. The experimental profile in Figure 2 is compared to a normal distribution curve constructed by using the experimentally measured variance and elution time. With the exception of the deviation at the top of the peak, there is very good agreement between the experimental profile and that predicted for a normal distribution. This illustrates the validity of the experimental design and the simplifications made in the theoretical development leading to eq 8. The experimentally determined bulk-phase diffusivity values are presented in Figure 3 as a log-log plot of D, vs molecular weight. The results show that there is no significant difference between the D, values determined for each source material and that the data follow a linear relationship when plotted as log D, vs log M. A regression analysis of the results for both materials at each temperature yields the following relationships: at T = 50 "C
D, = 3.63 X 10-6M4.50 cm2/s at T = 70 "C
(22)
D, = 5.82 X 10-6M4.61 cm*/s
(23)
400
1000
2000
4000
M 0 LEC U LAR WE IG HT
Figure 4. Effective solute radius from diffusivity as a function of molecular weight. The symbols are defined in Figure 3. The solid line is eq 24.
The lines described by eqs 22 and 23 are included in Figure 3. The effective solute radius, ad, of each fraction at each temperature was determined from the experimental D, values by using eq 1. A log-log plot of a d vs molecular weight is shown in Figure 4. These results show that there is not a significant difference in the effective size of each fraction at 50 and 70 "C. The difference in diffusivity values at the two temperatures shown in Figure 3 is therefore due solely to a difference in temperature and solvent viscosity. The intermolecular forces that lead to association effects with such materials are apparently not affected by this change in temperature. A regression analysis of the ad vs molecular weight results yields the following relationship: ad
= 0.36M"*@'
A
(24)
This equation provides a relationship that describes the molecular size distribution of these materials. To interpret our experimental observations, we again propose model multisubunit structures to describe the hydrodynamic properties of these fractions because models with a rigid, planar macroscopic structure are expected to reasonably represent the hydrodynamic properties of the heavy oils. In using multisubunit structures to describe the hydrodynamic characteristics of these materials, it is necessary to determine the appropriate structure, i.e., the number and arrangement of the subunits, as well as the size of the subunits, u. In our previous study, the constituents of the smallest fraction were assumed to be spherical and the constituents of the larger fractions were described by larger multisubunit structures comprised of spheres having a radius equal to that of the smallest fraction (Kyriacou et al., 1988a,b). This approach may not yield the most appropriate models because the constituents of the smallest fraction are probably not best described as spheres. Therefore, in the present study, a somewhat different approach was used in fitting the experimental observations to the structural modelw. We have assumed that multisubunit models containing spherical units of the same size can be used to describe the properties of heavy oil fractions over a range of molecular weights. That is, the models used to describe constituents with a range of molecular sizes differ only in the number and arrangement of subunits and not in the size of the subunits. We have also assumed that the molecular weight of each subunit is 100. In making this assumption, we are limiting the analysis to structures containing 6-24 units in order to obtain models with molecular weights over the molecular weight range of our materials. If a value smaller than 100 had been chosen, structures with more units
1974 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
would be needed to describe the properties of the fractions investigated. This would have provided us with more flexibility in arranging the subunits and more elaborate structures could have been considered. However, because each fraction contains a wide range of molecules, we did not feel justified in considering the more detailed models that more subunits would have allowed. On the other hand, choosing the subunit molecular weight to be more than 100 would have decreased the number of subunits in each model structure and therefore would have limited the possible arrangements of the subunits. The value of 100 seemed to balance these two considerations. To determine the model structures that are most appropriate for describing the properties of the materials investigated in this study, the following exercise was performed. A series of 4-7 structures, where each structure was comprised of 6-24 subunits, was considered. The value of a that provided the best agreement between the theoretical predictions for the diffusivity of each structure and the correlations of the experimental diffusivity values was determined for this series of structures by using our theoretical results and eqs 22 and 23. A number of different series of structures were considered, including a series of single-layer polygons where the number of units in the polygon was increased as the molecular size increased and several series where the number of units in the polygon remained the same and the larger molecules were formed by adding additional layers. However, the diffusivity values for these structures did not yield the (-0.5) power dependence on molecular weight that was observed experimentally. The series of structures listed in Table I1 where the larger species are formed by both increasing the number of units per polygon as well as increasing the number of layers was found to provide the best fit to the experimental observations. A comparison of the predictions for these structures with the experimental diffusivity values is included in Figure 3. The predicted D, values were determined using a = 5.1 A. This value was found to provide the best fit to the data at both temperatures. The comparison in Figure 3 shows that there is excellent agreement between the theoretical predictions for this series of structures and our experimental data. The macrostructure of asphaltene fractions as determined by small-angle neutron scattering has been reported by Ravey et al. (1988). The data were found to be in agreement with flat sheet structures with diameters of 50-200 A and thicknesses of 6-8 A. The results from this study differ from our results in that it was found that the thickness of the sheet structures did not vary with molecular size. Our results indicate that the larger fractions are formed by increasing the thickness as well as the diameter of the smaller structures. It is impossible to know whether these differences can be attributed to the fact that the study by Ravey et al. involved only the asphaltene fraction and involved material with molecular size considerably larger than the material used in our investigation. Different solvents were also used, which undoubtedly had some influence on the results. A comparison of the results obtained by using both techniques with the same material under the same experimental conditions might provide some further insigb.ts into the macroscopic structure of heavy oil materials as well as some understanding of the relationship between the structure of these materials and the measurement technique used to determine that structure. Intrinsic Viscosity. The experimentally determined intrinsic viscosity values are presented in Figure 5 as a log-log plot of [77] vs molecular weight. The intrinsic
*a 0
t
**
0
*
t
-
E
+
5 . 1 200
400
'
'1000
2000
'
4000-
MOLECULAR WEIGHT
Figure 5. Intrinsic viscosity as a function of molecular weight. The symbols for experimental values are defined in Figure 3. Sr are the predictions for model structures using u = 5.1 8, (Table 111). are the predictions for model structures using u = 3.5 8, (Table 111). The solid line is eq 25.
*
Table 111. Experimentally Determined Ratio of Radii a ./a .$
fraction
ABVB sample 70 "C
50 "C 0.77 0.79 0.69 0.66 0.72 0.66 0.68 0.72 0.79
0.74 0.72 0.62 0.60 0.66 0.62 0.66 0.73 0.82
IPVB sample 50 "C
70 "C
0.81 0.76 0.77 0.67 0.61 0.62 0.78 0.78 0.68
0.57 0.81 0.55 0.60 0.61 0.65 0.75 0.64 0.77
O a , values determined from [ q ] by using eq 4. ad values determined from D, values by using eq 1.
viscosity is independent of temperature and solvent viscosity. Therefore, one expects the intrinsic viscosity of each fraction to be independent of temperature if the size and conformation of the molecules is the same at each temperature. The results presented in Figure 5 indicate that, in general, the intrinsic viscosity, and therefore the effective size, is the same at each temperature. Again, this change in temperature is apparently not sufficient to lead to any significant change in the association forces in these materials. A regression analysis of the [ q ] vs molecular weight data yields the following relationship: [T]
= 0.16fl.44 cm3/g
(25)
The effective radius of a spherical solute, a,, as defined in eq 4 was calculated for each fraction at each temperature and the ratio av/ad was computed. These values are presented in Table 111. The general agreement between the values for each fraction a t each temperature is simply another way of illustrating the fact that temperature is not significantly affecting the constituents of these materials. The ratio av/adranges from ~ 0 . to 6 0.8, which is less than the value listed in Table I1 for the model multisubunit structures that were found to describe the diffusion properties of these materials. This difference indicates that there is a discrepancy between [ q ] predicted for the model structures and our experimental values. Another illustration of this discrepancy is shown in Figure 5 where the experimental intrinsic viscosity values and the theoretical predictions for the model structures listed in Table 11are presented. This comparison is shown for a = 5.1 A, the value found to be in best agreement with the diffusion results and for a = 3.5 A, a value that was found to yield better agreement with the viscosity results. Figure 5 shows that the intrinsic viscosity values predicted with the model
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1975 structures using u = 5.1 8, are significantly greater than the experimentally observed values. The following explanations for these observations were considered. If our approach to modeling these materials is incorrect, then one would not expect the experimental [ q ] values and the [ q ] values predicted using u = 5.1 A to agree. However, a comparison of our experimentally measured ratios of f&/ad to the ratio for a variety of models (oblate spheroids, prolate spheroids, linear polymers) shows that our observed &/ad values differ significantly from those predicted for these other model structures. In fact, the multisubunit structures are the only simple models for which a,/ad is predicted to be less than unity. These structures are also consistent with the known chemistry of these materials. Therefore, we find it unlikely that the discrepancy illustrated in Figure 5 is due to our modeling approach. The molecular weight values for each fraction were used when calculating a, values from the experimental [q ] values (eq 4). Anomalous results might be expected when interpreting the experimental intrinsic viscosity data if there is significant error in the molecular weight values. Such errors could arise from experimental errors during the vapor pressure osmometry (VPO) measurements or due to the different solvent and temperature conditions during the measurement of [ q ] and M. However, a 100% error in M would be needed to explain the discrepancy between theory and experiment illustrated in Figure 5. The experimental error in the VPO measurements was only about 5 % . I t seems quite unlikely that the experimental conditions of the [a] and M measurements are sufficiently different to cause the molecular size to be a factor of 2 different during these measurements. In developing the relationships between the effective solute size and the hydrodynamic properties D, and [a], it is assumed that the solutions contain uniform size solute molecules. This simplification is obviously an idealized one when considering samples of heavy oil residuals. The samples used in this investigation were fractionated by using gel permeation chromatography, which provided material with a narrow size distribution range. However, the samples can certainly not be described as monodisperse. To estimate the effect of sample polydispersity on our determinations of a, and ad, a five-component mixture containing spherical solute molecules was considered. The molecular weight of each component was chosen so that the number-average molecular weight of the mixture was 1044 and the weight-average molecular weight was 1136, yielding a polydispersity index of 1.09. The diffusivity and intrinsic viscosity of the mixture and the ratio av/ad were determined by using relationships presented by Okabe and Matsuda (1983) for polydisperse mixtures. The ratio av/ad was found to be 0.92. This value did not change significantly when a mixture with a broader size distribution or when a mixture containing more than five components was considered. These calculations show that the ratio av/ad is predicted to be less than unity for a polydisperse mixture of spheres; however, the effect of polydispersity is not sufficient to explain our experimental observations. In our previous study of diffusion and intrinsic viscosity using a different source material, the ratio av/adwas found to be ~ 0 . for 2 the experiments conducted using tetrahydrofuran as solvent and ~ 0 . for 4 the experiments conducted using 1-methylnaphthalene as solvent (Baltus et al., 1988). The fact that the ratio av/ad appears to be dependent upon the solvent used as well as the method used to determine solute diffusivity provides support for
our previously proposed hypothesis that the inconsistency between effective size determined by measuring solution viscosity and solute diffusivity is due to a difference in association forces between solute molecules during these measurements. When comparing samples of the same VPO molecular weight, the solute diffusivity determined by measuring the flux through a membrane was found to be about 40% larger than the diffusivity determined by using Taylor dispersion. The effective solute size appears to be largest during membrane transport measurements and smallest during solution viscosity measurements. These observations are consistent with the explanation that shear forces during the measurement of hydrodynamic properties can have a significant influence on the interpretation of results from these measurements. It is generally well accepted that environmental factors can have a strong influence on the effective size of heavy oil materials. Therefore, our observations are not surprising. In fact, it may be possible to use hydrodynamic measurements as a means of determining the extent of association between the components of these materials. Conclusions The intrinsic viscosity and solute diffusivity of fractions obtained from two different residual materials have been measured. The results show that the diffusion coefficients of these materials are in general agreement when samples of similar molecular weight are compared. Empirical relationships for the results at each temperature were derived. Multisubunit structural models were proposed to characterize the diffusion properties of these materials. The diffusion coefficients predicted for rigid, multilayer structures were found to be in excellent agreement with the experimental results. The effective solute size determined from intrinsic viscosity was found to be considerably smaller than the effective size as indicated by the solute diffusivity. Similar observations were made in our previous study. The results reported here are consistent with our previously proposed explanation in which this anomalous behavior was attributed to a difference in the association forces between the constituents of these materials during the two measurements. We are planning to perform additional experiments that will help us to understand quantitatively the influence of solution shear on the effective size of petroleum-derived materials. The objective of this study was to use the measurement of hydrodynamic properties as a means of determining the size and structure of heavy oil materials. Such information is necessary in order to correctly account for diffusional resistances in catalytic upgrading reactions. Our results indicate that the environmental conditions surrounding the solute can strongly influence the effective size of these materials. If this is true, then it is essential that the solute size be determined under reactive conditions in order to accurately quantify reactant diffusion rates in porous catalysts. Acknowledgment Acknowledgment is made to the donors of the Petrolem Research Fund, administered by the American Chemical Society, for support of this research. Nomenclature a d = effective solute radius determined by diffusion, defined -by eq 1, A a, = effective solute radius determined by viscosity, defined by eq 4,
A
B = "supermatrix" of
Qij
elements, eq 16
1976 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 c = solute concentration in viscosity experiments, g/cm3 c, = cross-sectional averaged solute concentration, c, = solute concentration in injected sample, g/cm3
g/cm3
C = center of hydrodynamic resistance of molecule in par-
ticle-fixed reference frame De = Dean number, defined in eq 20 D, = solute bulk-phase diffusivity, cm2/s f = solute frictional coefficient, g/s F = force exerted on solute molecule, N Fi = force exerted on subunit i, N Gi = tensor, defined in eq 13 k = Boltzmann constant, 1.380 X J/K K = Taylor dispersion coefficient, cm2/s L = length of capillary, cm M = solute molecular weight, g/mol N A = Avogadro’s number, 6.022 X 1023/mol Pe = Peclet number, defined in eq 17 Q = volume of injected sample, cm3 Qij = tensor defined by eq 12 R , = vector connecting subunits i and j &, Rj= positions of subunih i and j in particlefixed reference frame R = capillary radius, cm R, = radius of capillary coil, cm R e = Reynolds number s2 = variance of c, vs time profile, s2 Sc = Schmidt number, defined in eq 20 Sij = tensor defined in eq 16 t = time, s t1,2 = width of peak at half height, s T = temperature, K T,= modified Oseen tensor between subunits i and j , s/g a, = average linear velocity, cm/s u = unperturbed solvent velocity, cm/s vi = solvent velocity at subunit i, cm/s x = axial coordhate in laboratory reference frame Greek Symbols 6ij = Kronecker delta vo = solvent viscosity, g/(cm s) )ts = solution viscosity, g/(cm s) [ v ] = intrinsic viscosity, defined by eq 2, cm3/g vsp = specific viscosity, defined by eq 3 p = solvent density, /cm3 u = subunit radius,
1
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Receiued for review November 20, 1989 Accepted April 26, 1990
