An Asphaltene Association Model Analogous to Linear Polymerization

Res. 40, 21, 4664-4672 ... Part 2: Molecular Representations and Molecular Dynamics Simulations ... Monte Carlo Simulation of Asphaltenes and Products...
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Ind. Eng. Chem. Res. 2001, 40, 4664-4672

An Asphaltene Association Model Analogous to Linear Polymerization Mayur Agrawala and Harvey W. Yarranton* Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4

Asphaltene self-association was modeled in a manner analogous to linear polymerization. The key concept in the model is that asphaltene molecules may contain single or multiple active sites (functional groups) capable of linking with other asphaltenes. Molecules with multiple active sites act as propagators and molecules with single active sites act as terminators in polymerization-like association “reactions”. Asphaltenes consist primarily of propagators. Resins, which are known to affect asphaltene association, consist primarily of terminators. The model was tested on existing molar mass data for asphaltenes in different solvents and at different temperatures as well as on new molar mass measurements of mixtures of asphaltene and resins. The model fit the existing experimental data well and predicted the molar mass of asphalteneresin mixtures to within the accuracy of the measurements. Introduction The availability of conventional light oils has declined, as they were the first oil reserves to be put on production and are now depleting. As a result, oil producers are developing heavier crude oil reservoirs and offshore fields. The production and processing of heavy crudes requires diluent addition and/or heating to reduce viscosity. Some commonly used diluents can cause asphaltenes (the heaviest fraction of a crude oil) to precipitate. A drop in pressure can also trigger asphaltene precipitation. Asphaltene precipitation can foul equipment or the reservoir, increasing operating costs and reducing reservoir permeability. While it is desirable to predict the conditions at which asphaltenes precipitate, it has proven difficult to do so because asphaltenes are a solubility class, not a pure component. They consist of many thousands of chemical species and their physical structure in a crude oil is still debated. One possibility is that asphaltenes exist as free molecules in solution with the other oil constituents. Continuum thermodynamics (e.g., an equation of state) is employed to model asphaltene precipitation from this basis.1 With this approach, a constant molar mass distribution of the asphaltene molecules is assumed. However, there is convincing evidence that asphaltenes self-associate and, in effect, the molar mass distribution is not constant. Asphaltene association has been observed with vapor pressure osmometry2-4 (VPO), interfacial tension measurements,4-8 small-angle X-ray and neutron scattering (SAXS,9-12 SANS13,14) measurements, infrared spectrophotometry,15 dielectric spectroscopy,16 differential scanning calorimetry,17 and laser desorption mass spectroscopy.18 Computer-aided molecular modeling19-22 has also indicated that asphaltenes self-associate and that this self-association depends on temperature, pressure, and composition.21 The colloidal theory is an alternative to continuum thermodynamics that can account for some of the observations of asphaltene association. This theory * To whom correspondence should be addressed. Tel.: (403) 220-6529. Fax: (403) 282-3945. E-mail: [email protected].

posits that asphaltenes exist as colloidal particles with resins adsorbed on their surfaces.23 In this case, asphaltene solubility is controlled by the partitioning of the resins between the colloidal surfaces and the surrounding medium. If sufficient resins desorb, the asphaltenes begin to aggregate and can eventually precipitate. The colloidal model accounts for asphaltene self-association through the limited aggregation of colloidal particles upon partial desorption of the resins. The colloidal model is quite complex and requires a large number of parameters. Recent VPO measurements of asphaltenes in toluene or o-dichlorobenzene4 show that asphaltene self-association increases with asphaltene concentration until a limiting value is reached. The limiting value depends on solvent, temperature, and pressure. The nature of the results suggests that asphaltene association can be modeled in a relatively straightforward manner analogous to linear polymerization. Hirschberg and Hermans24 first proposed this concept but to our knowledge it has not been fully developed and tested. A polymerlike model differs from the colloidal model in that resins do not adsorb on asphaltene-colloid surfaces but are part of asphaltene/resin oligomers (polymer-like aggregates of mixed species). The advantages of this approach are that it can fit the molar mass data with few parameters and the association model can potentially be linked with continuum thermodynamics to predict both the asphaltene molar mass distribution and asphaltene solubility as a function of temperature, pressure, and composition. The purpose of the paper is to develop a simple asphaltene association model and test how well it fits the VPO molar mass measurements of asphaltenes in different solvents and at different temperatures. Molar mass data for asphaltene-resin mixtures is also presented and the predictive capability of the model is tested on these data. Materials. Asphaltenes and resins were extracted from a coker feed Athabasca bitumen supplied by Syncrude Canada Ltd. and from a Cold Lake bitumen supplied by Imperial Oil Ltd. Coker feed bitumen is an oil sands bitumen that has been treated to remove most

10.1021/ie0103963 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/18/2001

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4665 Table 1. SARA Analysis of Athabasca and Cold Lake Bitumen present work bitumen fraction saturates aromatics resins C5-asphaltenes C7-asphaltenes toluene insolubles (weight fraction of C5-asphaltenes) toluene insolubles (weight fraction of C7-asphaltenes) a

literaturea

Cold Lake

Athabasca

16.3 39.8 26.4 17.5 13.4 6.5

17.3 39.7 25.8 17.3 14.5

19.4 38.1 26.7 15.8 11.3

7.8

6.3

2.3

Athabasca

Cold Lake 20.7 39.7 24.8 15.3

Peramanu et al.3

of the sand and water and is ready for upgrading. Cold Lake bitumen is produced from an underground reservoir through cyclic steam injection and has been processed to remove sand and water. For asphaltene extractions and SARA fractionation, reagent-grade n-pentane, n-heptane, and toluene were obtained from Phillips Chemical Co. and reagent-grade acetone, methanol, and dichloromethane were obtained from BDH Inc. Attapulgus clay was supplied by Engelhard Corporation. Octacosane and silica gel (grade 12, 28-200 mesh size) were obtained from Sigma Aldrich Co. For the VPO experiments toluene (99.96% purity) was obtained from VWR and sucrose octaacetate was supplied by Jupiter Instrument Co. SARA Fractionation. A modified Clay-Gel Absorption Chromatography25 (ASTM D 2007) technique was used to separate the bitumen into solubility classes (saturates, aromatics, resins, and asphaltenes). The modifications in the ASTM D2007 procedure were for the asphaltene extraction and the modified extraction method is described below. The resins, aromatics, and saturates were recovered following the standard procedure but only resins and asphaltenes were used in this work. The SARA analysis of the Athabasca and Cold Lake bitumens are given in Table 1. Asphaltenes were precipitated from the bitumen with the addition of 40 volumes of n-pentane to 1 volume of bitumen. The mixture was sonicated using an ultrasonic bath for 45 min and left overnight. The next day, the mixture was filtered through Whatman’s No. 2 (8-µm) filter paper. The filter cake was mixed with 4 volumes of solvent, sonicated for 45 min, and left overnight. The mixture was again filtered and subsequently washed with pentane for 5 days until no discoloration of the solvent was observed. The asphaltenes were dried in a vacuum oven at 50 °C until no change in weight was observed. These asphaltenes are referred to as C5asphaltenes because n-pentane (a C5 n-alkane) was used for the extraction. The C5-asphaltenes typically contain some resinous material that is insoluble in n-pentane but may be soluble in a higher n-alkane such as n-heptane. It was desired to test asphaltenes with less resinous material. Therefore, asphaltenes were also extracted from the bitumens using n-heptane. The same procedure was used as the one used for n-pentane and the resulting asphaltenes are referred to as C7-asphaltenes. The amount of C7-asphaltenes in Athabasca and Cold Lake bitumen are reported in Table 1. The dried C5- or C7-asphaltenes were purified to remove non-asphaltenic solids (consisting of clay, sand,

Figure 1. Effect of K on predicted molar mass at constant (T/P)0 ratio. (Athabasca C5-asphaltenes in toluene at 50 °C.)

and some adsorbed hydrocarbons) that coprecipitated along with the asphaltenes. To remove these solids, the asphaltenes were dissolved in toluene, typically at a concentration of 0.01 g of asphaltene/cm3 of toluene. The mixture was centrifuged at 3500 rpm (1300 RCF) for 5 min. The supernatant was removed and dried in a rotary evaporator at 70 °C under vacuum. The nonasphaltenic solids content of Athabasca and Cold Lake asphaltenes are reported in Table 1. All of the results reported here are for asphaltenes that were treated to remove solids. Note that some fine solids may remain in the asphaltenes after this treatment but are present in such small amounts that they do not appear to affect the consistency of the molar mass measurements. Vapor Pressure Osmometry. The principle and application of vapor pressure osmometry are described in detail elsewhere.3,4 The molar mass of asphaltenes is related to a measured voltage difference as follows:

MA )

K0CA ∆V

(1)

where ∆V is the voltage difference, CA is the asphaltene concentration, and K0 is a calibration constant. The molar masses of asphaltenes and asphaltene-resin mixtures in toluene were measured with a Model 833 VPO from Jupiter Instrument Company. This osmometer has a detection limit of 5 × 10-5 mol/L when used with toluene or chloroform. Sucrose octaacetate (679 g/mol) was used to calibrate the instrument and octacosane (395 g/mol) was used to check the calibration. The measured molar mass of octacosane was found to be within 2% of the correct value. During the asphaltene/resin molar mass measurements, there were slight fluctuations in the voltage at any given condition probably because of slight local temperature variations. Therefore, 3-12 readings were taken at each concentration to obtain the voltage response for that concentration. Asphaltene Association Model Yarranton et al.4 performed VPO measurements of Athabasca and Cold Lake C5- and C7-asphaltenes in different solvents and at different temperatures. Asphaltene association was shown to increase with asphaltene concentration until a limiting molar mass was reached, as shown in Figures 1 and 2. The limiting molar mass depended on the temperature and solvent.

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Figure 2. Effect of (T/P)0 ratio on predicted molar mass at constant K. (Athabasca C5-asphaltenes in toluene at 50 °C.)

Figure 4. Oligomerization-like asphaltene association.

Figure 3. Hypothetical asphaltene molecule adapted from Strausz et al.26 (M ) 1800 g/mol, C117H142N2O4S5).

The introduction or removal of resins also affects the degree of association, as will be shown later. A model that qualitatively accounts for all these observations can be derived starting from a hypothetical view of an asphaltene molecule. Consider the hypothetical asphaltene molecule (adapted from Strausz et al.26) shown in Figure 3. The main structure of the molecule consists of a number of polyaromatic ring clusters with attached aliphatic chains. The molecule also contains heteroatoms (N, O, S) in associated functional groups such as acids, ketones, thiophenes, pyridines, and porphyrins (not all shown in Figure 3). There appears to be many ways in which this molecule can link with other similar molecules. These

links can be formed through aromatic stacking,18 acidbase interactions,16 hydrogen bonding,27 or van der Waals interactions.22 Because asphaltenes are a mixture of many thousands of chemical species featuring a variety of functional groups, the type and strength of the potential links may vary considerably from molecule to molecule and from site to site. Also, the number of potential links that can form would likely depend on the solvent and temperature conditions. For example, in a strongly polar solvent like nitrobenzene, the asphaltene-solvent interactions tend to dominate the asphaltene-asphaltene interactions. “Strong” sites could still form asphalteneasphaltene links but “weak” sites may not. A similar argument applies with temperature. The higher the temperature, the fewer sites that are capable of forming links, and the asphaltenes will tend to remain as separate entities. This view of asphaltene association, illustrated in Figure 4, is analogous to polymerization or oligomerization. In this scheme, asphaltenes are treated as free molecules in solution that contain multiple active sites (heteroatoms or aromatic clusters) and can interact with other similar molecules to form aggregates. Hence, asphaltenes can be treated as propagators in a polymerization-like reaction. On the other hand, resins may have a single active site and can link up with just one other molecule. Hence, they can be treated as terminators in a polymerization-like reaction. In fact, the real difference between asphaltenes and resins may be a functional difference; that is, asphaltenes may have multiple sites and self-associate while resins do not have sufficient sites to self-associate. This view is consistent with the similar chemical nature of asphaltenes and resins and the smaller heteroatom content, aromaticity, and molar mass of the resins.

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Note this view of association is a significant departure from the colloidal model where resins are assumed to adsorb on the surface of asphaltene aggregates. With the colloidal model, asphaltenes are treated as colloidal particles dispersed by resins. Precipitation occurs if resins desorb from the particles, allowing flocculation and settling. With the association model, the aggregates are macromolecules of asphaltenes and resins rather than colloids and precipitation is a conventional phase transition. Asphaltene self-association and asphaltene-resin interactions analogous to linear polymerization are assumed. This is likely an oversimplification but the purpose here is to test if this simple model is capable of fitting asphaltene molar mass data. As with linear polymerization, this model is described in terms of propagators and terminators. A propagator is a molecule that contains multiple active sites (functional groups) and is capable of linking with other similar molecules or aggregates. A terminator is a molecule that contains a single active site (functional group) and is capable of linking with propagators or aggregates, thus terminating association. It is proposed that asphaltenes consist mainly of propagators but also contain a small proportion of terminators. Resins on the other hand consist mainly of terminators but also contain a small proportion of propagators. This mixture reflects how terminators and propagators may partition during aggregation. The end result is that any precipitated material can be characterized as a mixture of propagators and terminators. Model Development. A polymerization-type scheme involving propagation and termination is proposed. As the association is not a free radical reaction, there is no initiation step, unlike polymerization reactions. (a) Propagation. These “reactions” involve propagators and aggregates. A propagator monomer, denoted by P1, can link up with another monomer or with an existing aggregate Pk (where k is the number of monomers in the aggregate). These “reactions” are assumed to be first order with respect to both the propagator monomer and the aggregate molecules and are characterized by an association constant, K, which represents equilibrium between forward and reverse association. Also, the concentration of the aggregates can be expressed in terms of the association constant, K, and equilibrium concentration of propagators, [P1]. The association constant is assumed to be the same for any aggregate size. The “reaction” scheme is as follows: K

P1 + P1 798 P2 ..... [P2] ) K[P1]2 K

P1 + P2 798 P3 ..... [P3] ) K[P1][P2] ) K2[P1]3

(2) (3)

terminator monomer and aggregate molecules and are characterized by an association constant. The concentration of the terminated aggregates can be expressed in terms of the association constant, K, and the equilibrium concentration of terminator monomers, [T], and propagator monomers, [P1]. To obtain an analytical solution, the termination of only one site on the aggregate is assumed to stop further association. This assumption gives the following scheme: K

P1 + T 798 P1T ..... [P1T] ) K[P1][T]

(6)

K

P2 + T 798 P2T ..... [P2T] ) K[P2][T] ) K2[P1]2[T] (7) K

P3 + T 798 P3T ..... [P3T] ) K[P3][T] ) K3[P1]3[T] (8) l K

Pn + T 798 PnT ..... [PnT] ) K[Pn][T] ) Kn[P1]n[T] (9) For simplicity, it is assumed that the association constant is the same for all “reactions”. This means that the probability of a propagator or terminator monomer forming a link with an aggregate is the same as that of forming a link with another monomer. Aggregateaggregate associations are not considered. The implications of these and the other model assumptions are discussed in the Conclusions section. (c) Calculation of Equilibrium Concentrations. The “reaction” schemes are solved in the same manner as a polymerization reaction starting with mass balance equations for both propagators and terminators from eqs 2-9. The equilibrium concentration of propagators is given by

[P1] ) {1 + K(2[P1]0 + [T]0) -

x(1 + K(2[P1]0 + [T]0))2 - 4K2[P1]0([P1]0 + [T]0)}/

[2K2([P1]0 + [T]0)] (10)

The equilibrium concentration of terminators, [T], is given by

[T] ) [T]0(1 - K[P1])

(11)

In the above two equations, [P1]0 and [T]0 denote the initial concentrations of propagators and terminators, respectively. (d) Calculation of Average Molar Mass and Molar Mass Distribution. From the equilibrium concentration of each aggregate in solution, the average molar mass, M h is given by

K

P1 + P3 798 P4 ..... [P4] ) K[P1][P3] ) K3[P1]4 (4) l K

P1 + Pn 798 Pn+1 ..... [Pn+1] ) K[P1][Pn] ) Kn[P1]n+1 (5) (b) Termination. The other set of “reactions” involve terminators and aggregates. A terminator monomer can link up with a propagator or an existing aggregate, thus terminating association. Again, these “reactions” are assumed to be first order with respect to both the

(1 + K[T])[P1]Mp M h )

(1 - K[P1])

+ [T]Mt

([P1] + [T])

(12)

The molar mass distribution can be obtained from the mole fraction of each aggregate in solution, eqs 2-9. Model Implementation. The required inputs are the combined concentration of the asphaltenes and resins (equivalent to the combined concentration of terminator and propagator monomers), the molar mass of the terminators, and the molar mass of the propaga-

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tors. The fit parameters are the ratio of terminator to propagator monomers, (T/P)0, and the association constant, K. A monomer molar mass of 1800 g/mol was chosen for the propagators. This value was estimated by the extrapolation of the VPO molar mass curves to zero concentration and taking an average for the various systems. A value of 1800 g/mol is consistent with previous results based on VPO.3,4,28 A molar mass of 800 g/mol was estimated for terminators. This value falls within the range of molar masses observed for resins.29 Note that it is an oversimplification to use an average monomer molar mass to describe terminators and propagators as in reality there would be a distribution of monomer molecules of varying sizes. However, using an average monomer molar mass minimizes the number of model parameters without introducing significant error. The potential variation in monomer molar mass (≈500-2000 g/mol) is relatively small compared with the molar mass of associated asphaltenes (200020000+ g/mol). To use the model, a (T/P)0 ratio is assumed and the mole fraction of terminators and the mole fraction of propagators are determined. The molar volumes of the solvents at the system temperature and pressure are required for this calculation and were obtained from a density correlation.30 Then, the equilibrium concentrations of propagator and terminator monomers are calculated from eqs 10 and 11, respectively. The average molar mass is determined from eq 12. The procedure is repeated over a range of asphaltene/resin concentrations and the calculated average molar masses are compared with the measured molar masses. The (T/P)0 ratio and the association constant are adjusted until the model fits the data. Typically, about three to four iterations were required to obtain the curve fits using the model for each asphaltene/resin system over a wide range of solute concentrations (0.1-70 kg/m3). Note that, at this stage in the model development, the quality of the fits was judged by inspection rather than by any statistical measure. Because VPO measurements were subject to a greater degree of error (as high as 100%)4 at asphaltene concentrations below 3 kg/ m3, the criteria used to fit the curves was a match at asphaltene concentrations above 3 kg/m3. In general, there was about 3-4% error in the VPO measurements in this concentration range. Typical curve fits using model parameters are shown in Figures 1 and 2. The system used is Athabasca C5asphaltenes in toluene at 50 °C. Figure 1 shows the effect of changing K at constant (T/P)0 and Figure 2 shows the effect of changing the (T/P)0 ratio at constant K. It is clear from the two figures that the association constant, K, essentially determines the concentration at which the limiting molar mass is reached and the (T/P)0 ratio determines the value of the limiting molar mass. From Figure 1 it is apparent that increasing K decreases the concentration at which the limiting molar mass is reached. From Figure 2 it is evident that increasing the (T/P)0 ratio reduces the value of limiting molar mass. These trends are consistent with the assumptions in the association model. Results and Discussion Molar mass measurements and model curve fits are presented in Figures 5-9. The experimental data in Figures 5-7 are taken from Yarranton et al.4 The data

Figure 5. Molar masses of Athabasca C5- and C7-asphaltenes in toluene at 50 and 70 °C. Model curve fits with Mp ) 1800 g/mol and Mt ) 800 g/mol: (1) K ) 130000, (T/P)0 ) 0.195; (2) K ) 160000, (T/P)0 ) 0.24; (3) K ) 130000, (T/P)0 ) 0.33; (4) K ) 160000, (T/P)0 ) 0.41.

Figure 6. Molar masses of Athabasca C5- and C7-asphaltenes in o-dichlorobenzene at 75 and 130 °C. Model curve fits with Mp ) 1800 g/mol and Mt ) 800 g/mol: (1) K ) 50000, (T/P)0 ) 0.34; (2) K ) 70000, (T/P)0 ) 0.41; (3) K ) 50000, (T/P)0 ) 0.495; (4) K ) 70000, (T/P)0 ) 0.62.

in Figures 8 and 9 are new. In all cases, the symbols represent experimental data and the lines represent model fits. Effect of Asphaltene Cut. Figures 5 and 6 show the molar mass of C5- and C7-asphaltenes in toluene and o-dichlorobenzene, respectively. The molar mass of the C7-asphaltenes is higher than that of the C5-asphaltenes. Hence, the (T/P)0 ratio that fits the C7-asphaltene data was lower than the ratio that fits the C5-asphaltene data. In other words, C5-asphaltenes contain more terminators than C7-asphaltenes. To understand the difference, consider that resins and asphaltenes form a continuum of increasing molar mass, polarity, aromaticity, and heteroatom content.31 It is expected that the proportion of terminators decreases along this continuum as the larger and more polar molecules have more sites capable of association. Consequently, a bitumen cut that consists of the “high” (asphaltenic) end of the continuum will have fewer terminators than a larger cut that includes more of the intermediate or low end of the continuum. Now, when pentane is used to extract asphaltenes, more material precipitates from the bitumen than when heptane is used. Consequently, the C5-asphaltenes are expected to have a higher (T/P)0

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4669 Table 2. Summary of Association Constant and (T/P)0 Ratios for Various Asphaltene-Solvent Mixtures solvent toluene o-DCB

temp (°C)

association constant

free energy (kJ/mol)

(T/P)0 ratio (C5-asph.)

(T/P)0 ratio (C7-asph.)

50 70 75 130

130000 160000 50000 70000

-31.6 -34.2 -31.3 -37.4

0.330 0.410 0.495 0.620

0.195 0.240 0.340 0.410

recognize that the model is an averaging approach to thousands of individual associations. In a conventional approach, each reaction can be expressed in terms of free energy

{-∆G° RT }

K ) exp Figure 7. Effect of solvent polarity on molar mass of Athabasca C7-asphaltenes at 70 °C in toluene and 75 °C in o-dichlorobenzene (dcb). Model curve fits with Mp ) 1800 g/mol and Mt ) 800 g/mol: (1) K ) 160000, (T/P)0 ) 0.24; (2) K ) 50000, (T/P)0 ) 0.34.

ratio and a lower degree of association than C7asphaltenes. Note that the same value of K (130000) was used to fit both the C5- and C7-asphaltene molar mass data. In the model development, the value of the association constant was assumed constant for all associations between propagators and terminators from the same source in a given solvent at a particular temperature. It therefore follows that all asphaltenes and resins from the same source oil must have the same association constant for the same conditions. This constraint was adhered to for all the modeling presented below. Effect of Solvent and Temperature. Figures 5 and 6 also show the effect of temperature on the molar mass of C5- and C7-asphaltenes in toluene and o-dichlorobenzene, respectively. As discussed previously, higher temperature results in lower molar mass. Hence, for a given asphaltene, a higher (T/P)0 ratio was required to fit the measured molar masses at higher temperature. The increase in the (T/P)0 ratio suggests that some propagators become terminators as the temperature increases. In other words, the weakest association sites on a given asphaltene molecule may not be able to form a link with another molecule as thermal motion increases. The effect of solvent on asphaltene molar mass is shown in Figure 7 for C7-asphaltenes at a temperature of ≈70 °C. Note that 75 °C was the lowest temperature stable VPO measurements could be obtained in odichlorobenzene and 70 °C was the highest temperature stable measurements could be obtained in toluene. o-Dichlorobenzene is a better solvent for asphaltenes than toluene and, as discussed previously, a better solvent reduces the average asphaltene molar mass. Hence, for a given asphaltene, a higher (T/P)0 ratio is required to fit the measured molar mass in o-dichlorobenzene than is required with toluene. This may be because as the “power” of the solvent increases, asphaltene-solvent interactions tend to dominate as compared to asphaltene-asphaltene interactions. Hence, fewer association sites are able to form links with other asphaltenes. Note that the same trends were observed for C5-asphaltenes. The model parameters, K and (T/P)0, from the twoparameter model are summarized in Table 2. As the temperature increases, K and (T/P)0 both increase. As the solvent “power” increases, K decreases and (T/P)0 increases. To explain these trends, it is necessary to

(13)

where ∆G° is the standard Gibb’s free energy of solution, R is the gas constant, K is the association constant, and T is the temperature. Because association is favored, ∆G° is negative, and as the temperature increases, the association constant is expected to decrease. However, in the proposed model the (T/P)0 ratio is increased as the temperature increases and in effect some potential associations are removed from consideration, as some propagators become terminators. Hence, the ∆G° of the averaged reactions can change. Because ∆G° in effect changes with temperature, the value of K does not necessarily decrease with temperature. Note that as the temperature increases, the combined effect of increasing (T/P)0 and K is to decrease the aggregate size, which is consistent with eq 13. The effect of increasing solvent power is to reduce the ∆G° of association, which is accounted for by higher (T/P)0 and lower K. The standard Gibb’s energy, ∆G°, is computed using eq 13 and is reported in Table 2 for the various systems. Note that an average value for the association constant was used to describe all association “reactions” including termination and propagation for any system. This assumption was used so as to reduce the number of parameters in the model. In reality, termination “reactions” probably have a higher association constant24 than propagation “reactions” because resinasphaltene interactions appear to be stronger than asphaltene-asphaltene interactions.27 However, the assumption of an average association constant does not appear to introduce significant error in the estimation of the average molar mass of asphaltenes. Effect of Resins. To determine the effect of adding resins, C5-asphaltenes and resins were mixed in different proportions and the average molar masses of the resulting systems were measured at different concentrations. The results are presented in Figure 8. The asphaltene/resin mixtures provide an opportunity to test the predictive capability of the association model. First, the association model was employed to fit the C5asphaltene and “pure” resin experimental data. An association constant of 130000 was used and (T/P)0 ratios of 0.33 and 4.5 were found for the C5-asphaltenes and “pure” resins, respectively. The (T/P)0 ratios for mixtures of C5-asphaltenes and resins were then calculated directly as a number average and used to generate the model predictions shown in Figure 8. The predicted limiting molar masses were within 16% of the experimentally measured molar masses, that is, within the accuracy of the measurements. Considering the assumptions used to generate the association model, this level of agreement is very good.

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Figure 8. Molar masses of Athabasca C5-asphaltenes and resins in toluene at 50 °C. For mixtures, A:R denotes the mass ratio of asphaltenes to resins. Model curve fits with Mp ) 1800 g/mol, Mt ) 800 g/mol, and K ) 130000: (1) (T/P)0 ) 0.33; (2) (T/P)0 ) 4.5. Model predictions: (A) (T/P)0 ) 0.97; (B) (T/P)0 ) 1.84; (C) (T/P)0 ) 2.73.

Figure 9. Comparison of molar mass of Athabasca (Ath) and Cold Lake (CL) asphaltenes and resins in toluene at 50 °C. Model curve fits with Mp ) 1800 g/mol, Mt ) 800 g/mol, and K ) 130000: (1) (T/P)0 ) 0.195; (2) (T/P)0 ) 0.33; (3) (T/P)0 ) 4.5.

Comparison of Athabasca and Cold Lake Asphaltenes and Resins. The molar masses of C5asphaltenes, C7-asphaltenes, and resins extracted from Cold Lake bitumen were measured in toluene at 50 °C and are shown in Figure 9. The measured molar masses of the Cold Lake fractions are about 13% lower (possibly within the scatter of the data) than the molar masses of the corresponding Athabasca fractions. The results suggest that asphaltenes from similar regions or sources exhibit similar self-association. Hence, the association model can potentially be generalized at least to asphaltenes from the same geographical region. Sensitivity Analysis on Monomer Molar Masses. The monomer molar masses of terminators and propagators are estimated parameters. Their values are based on molar mass measurements of asphaltenes and resins. However, a considerable range of asphaltene monomer molar masses has been reported depending on the experimental technique and on the extent of asphaltene association. In addition, asphaltenes from different sources may have different monomer molar masses. Proposed values range from ≈500 to ≈2000 g/mol.2-4,32,33 Consistent molar masses have been determined for resins, ranging from 400 to 1000 g/mol depending upon the source crude oils.3,22,29

Figure 10. Sensitivity of the best fit values of the T/P ratio and K to the propagator and terminator monomer molar masses (Athabasca C5-asphaltenes in toluene at 50 °C.) Terminator molar mass range from 400 to 1000 g/mol.

Given the uncertainty in the monomer molar masses, a sensitivity analysis was performed. Different combinations of propagator and terminator monomer molar masses were used to fit the experimental data of C7asphaltenes in toluene at 50 °C. The monomer molar mass of terminators was varied from 400 to 1000 g/mol and that of propagators from 800 to 2200 g/mol in steps of 200 g/mol. In all, 32 simulations were performed using the model and the results are summarized in Figure 10. Figure 10 shows the (T/P)0 ratio versus the monomer molar mass of propagators for different monomer molar mass of terminators (400-1000 g/mol in steps of 200 g/mol) for C5- and C7-asphaltenes on the primary axis. The association constant is plotted on the secondary axis. The (T/P)0 ratio and the association constant both change significantly with a change in the propagator molar mass but relatively little with a change in the terminator molar mass. Hence, the model performance can best be improved by obtaining accurate propagator molar masses. The propagator molar mass can best be estimated from asphaltene monomer molar masses. Hence, improved methods for determining asphaltene monomer molar masses are vital for this modeling approach. Molar Mass Distribution. The molar mass distribution of the asphaltenes is the output from the association model that is of most interest for solubility modeling. The predicted molar mass distributions of C7-asphaltenes in toluene at 50 °C are shown in Figure 11. As expected, the distribution changes dramatically over the concentrations where the asphaltene average molar mass increases. After the limiting molar mass is reached, the molar mass distribution does not change appreciably. Note that, for the system in Figure 11, the molar masses range up to 50000 g/mol. At higher temperatures, in better solvents, or upon the addition of resins, there are fewer large aggregates and narrower distributions are predicted. It is interesting to note that the shape of the molar mass distribution obtained from the model is very similar to those obtained from Gel Permeation Chromatography (GPC) measurements.3,34,35 The upper limits of the GPC asphaltene molar mass distributions ranged from 1400034 and 3500035 to 100000.3 In all cases, the shape of the molar mass distributions appeared to be log-normal.

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4671

Figure 11. Predicted molar mass distributions of C7-asphaltenes in toluene at 50 °C at asphaltene concentrations (CA) of 1, 5, 10, and 50 kg/m3.

Conclusions The molecular association of asphaltenes has been modeled using an aggregation mechanism analogous to linear polymerization. The polymerization “reactions” involved have been described in terms of two distinct classes: terminators and propagators. The model requires two fit parameters (the association constant and the molar ratio of terminators to propagators) and two estimated parameters (the monomer molar masses of terminators and propagators). The association constant determines the concentration at which the limiting molar mass is reached and the (T/P)0 ratio determines the limiting molar mass of aggregates. While this approach is just one of several possible approaches, it is simple to implement and fits the experimental molar masses quite well over a range of temperatures and solvents. Molar masses of asphaltene-resin mixtures in toluene were measured with the VPO at 50 °C. Resins reduced asphaltene association and significant reductions in average molar mass occurred with the addition of a small proportion of resins. The predictive capability of the association model was tested on the asphalteneresin mixture data. The model predicted the molar mass of asphaltene-resin mixtures within the scatter of the experimentally measured molar masses. A number of simplifying assumptions were made in the derivation of the association model so that an analytical solution could be obtained. (1) Linear polymerization-like association was assumed whereas three-dimensional branched association is probably more realistic. This approximation is probably reasonable for small aggregates (