Article pubs.acs.org/EF
Association Model for the Phase Behavior of Asphaltenes Paul Painter,*,† Boris Veytsman,‡ and Jack Youtcheff§ †
The Earth and Mineral Sciences (EMS) Energy Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ‡ School of Systems Biology and Computational Materials Science Center, George Mason University, MS 6A12, Fairfax, Virginia 22030, United States § Turner Fairbanks Highway Research Center, McLean, Virginia 22101, United States ABSTRACT: A simple attenuated association model for the aggregation of asphaltenes in solution is described. The model has only two parameters, a single equilibrium constant and the molecular weight of the “monomer” (unassociated asphaltene molecule). Both parameters were estimated from results reported in the literature and account for the variation of the degree of association with the concentration very well. Using Flory’s treatment of reversible association and Prigogine’s result that the chemical potential of an associating species is equal to the chemical potential of the monomeric species, equations for the free energy of mixing and its derivatives are derived. These equations were then used to model the phase behavior of solutions. Good agreement with observed experimental behavior was obtained using solubility parameters for asphaltenes in a narrow range near 20 MPa0.5, consistent with values reported in the literature.
■
that “monomers” with much higher molecular weights (but below 2000) are present. The phase behavior of bitumen and asphaltene solutions has been widely studied and modeled using various theoretical approaches.6−14 One widely used approach uses a Flory− Huggins-type model to account for differences in the average molar volumes of the components of the mixture.8,9,12−14 To use this approach, Yarranton and co-workers9,12−16 and Shirani et al.17 used association models to calculate the average molecular weight or molecular weight distributions of the asphaltene aggregates, hence, their molar volumes. There is a rich literature on association models aimed primarily at understanding the behavior of simple alcohol/ hydrocarbon mixtures18 and more recently hydrogen-bonded polymer blends.19−21 In this paper, we will show how a particularly simple form of these models can be applied to the self-association of asphaltenes and the phase behavior of their mixtures and solutions. Although the form of these models is similar to that used by Yarranton and co-workers9,12−16 and Shirani et al.,17 there are fewer parameters and the link to equations describing the free energy of mixing and phase behavior is handled differently. Although, at first sight, these models can seem somewhat ad hoc or intuitive, they have a distinguished thermodynamic history and give exactly the same result as other, more formal, statistical mechanical models.22,23 Accordingly, for clarity, we will first briefly describe the background on which these models are based before presenting the approach that we will use and describing how it can be applied to the phase behavior of asphaltene solutions or mixtures.
INTRODUCTION Crude oil, bitumen, and asphalts are complex, multicomponent systems, and their phase behavior can only be modeled by making some drastic assumptions. In part, this is because asphaltenes, maltenes, and other components are not types of molecules but are heterogeneous materials that are defined by their solubility characteristics. Asphaltenes correspond to the fraction of oil, asphalt, or bitumen that is soluble in toluene and insoluble in certain alkanes, usually n-pentane or n-heptane. They are a complex mixture of molecules that are characterized by polycyclic aromatic cores and heteroatom content. The soluble material remaining after asphaltene precipitation, the maltenes, are equally complex mixtures. Although asphaltenes apparently have a higher molecular weight and, on average, are more aromatic than maltenes, recent work by Podgorski et al.1 has confirmed the earlier Boduszynski continuum model,2 where crude oil composition is considered to change gradually and continuously with regard to aromaticity, molecular weight, and heteroatom content. Accordingly, it is likely that some material that would be classified as asphaltenes remains in the toluene-soluble fraction, while maltene-like material precipitates with the asphaltenes. However, molecules classified as asphaltenes self-associate and form clusters (see refs 3 and 4 and citations therein). This is considered to be a consequence of the polycyclic aromatic ringlike structure of asphaltenes, where π−π bond interactions are thought to be responsible for the strong forces of association. Mullins3 and Mullins et al.4 have recently contended that asphaltenes consist predominantly of island-like structures with a “monomer” molecular weight of the order of 750. It was argued that higher molecular weight material detected in various previous studies was actually aggregates. However, in just the last few months, McKenna et al.5 have shown that aggregation restricts the asphaltene molecules that can be sampled by mass analyzers to those with island structures, so © 2014 American Chemical Society
Received: February 7, 2014 Revised: March 27, 2014 Published: March 28, 2014 2472
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
■
Article
BACKGROUND: ASSOCIATION MODELS In their classic 1957 book, Prigogine et al.24 stated that, at the time, there was no satisfactory theory of systems where there are strong orientational effects, such as mixtures of molecules that hydrogen bond. Employing the assumption that a chemical equilibrium exists between the “monomolecules” of the associated species, they proposed that such solutions be treated in terms of the “complexes” that are formed. However, in the 1940s, Flory,25 Tobolsky,26 and Tobolsky and Blatz,27 had actually described a theory for treating mixtures where there is a chemical equilibrium between (covalent) polymeric species, such as those formed in condensation polymerizations. In a subsequent note, Flory28 pointed out the generality of his treatment and its applicability to reversible association. He also identified the state where the individual associated i-mers are separate and oriented as the proper standard reference state. This approach was taken up by Kretschmer and Wiebe,29 who obtained an expression for the free energy of mixtures that hydrogen bond by simply writing down Flory’s equations for the entropy of mixing the equilibrium distribution of species found in the mixture relative to the distribution in the pure state. The Kretschmer/Wiebe model has been widely used to model the phase behavior of mixtures where one of the components self-associates through hydrogen bonds.16 The basis for this essentially thermodynamic approach can be summarized as follows: (1) The species or complexes associate in the form of chains (not a necessary assumption). (2) There is an equilibrium distribution of chain lengths that varies with composition (and temperature). (3) Using Flory’s equations, an expression can be written down for the free energy of mixing these species (i.e., monomers, dimers, ..., i-mers) with one another. This is relative to a reference state where the molecules are initially separate and oriented. (4) The same procedure is applied to the species found in the pure state. (5) The overall free energy change that is a result of association is obtained when one is subtracted from the other. (6) Nonspecific interactions between the associated species and other components are then handled using a conventional Flory χ parameter. Essentially, this approach “short-circuits” some of the usual statistical mechanics by assuming prior knowledge of the equilibrium distribution of associated molecules. However, as noted above, it can be shown30 that the results are identical to those involving a more formal statistical mechanical approach using the combinatorics of distributing specific interactions between “donor” and “acceptor” groups.22,23 Here, we test if a simple model reproduces the main features of association and the phase behavior of asphaltene mixtures. We will make the crude assumption that oil, asphalt, or bitumen contains some molecules that are capable of associating and some that are not. These fractions comprise two “pseudocomponents” and roughly correspond to the asphaltene and maltene fractions, respectively. The approach that we previously used in studies of hydrogen bonds involved continuous, linear association models, similar to those previously applied to asphaltene aggregation.9,12−17 An important feature of asphaltene association is the formation of nanoclusters that are limited in size to roughly six or so “monomer” units. The nanocluster size limitation is thought to be a consequence of the aliphatic “hairs” attached to the polycyclic aromatic core that sterically inhibit the addition of molecules as the cluster size grows. Yarranton and co-workers
modeled cluster size limitation as a consequence of the presence of “chain stoppers”,9,12−14 molecules that can associate with a cluster but prevent the addition of further molecules at that site. Limitations on aggregation will be handled differently in the model described here.
■
ATTENUATED ASSOCIATION MODEL: STOICHIOMETRY
We will assume that the structure of the asphaltene molecules themselves is responsible for limiting association and then develop equations for the thermodynamics of mixing. This can be accomplished by describing association in terms of an equilibrium of the form
ϕA + ϕA ⇌ ϕA i
1
(1)
i+1
where ϕAi is the volume fraction of associated i-mers and an equilibrium constant (K A) describing an association that is continuously attenuated as monomers are added to the cluster is defined as18
ϕA KA i i+1 = ϕA ϕA i + 1 i+1 i
(2)
1
This is a dimensionless equilibrium constant that is equal to an equilibrium constant defined in terms of molar concentrations divided by a molar volume per structural unit. As Flory pointed out,25 because this latter quantity is a constant, either equilibrium constant definition is acceptable. By successive substitution it follows that
ϕA = i
K Ai − 1ϕAi
1
(i − 1)!
(3)
The volume fraction of asphaltene molecules present in the mixture (ϕA) is then given by ∞
ϕA =
∞
∑ ϕA i = ∑ i=1
i=1
ϕA (KAϕA )i − 1 1
1
(i − 1)!
∞
=
∑
ϕA (KAϕA )i
i=0
1
1
i!
(4)
The sum is an expansion of an exponential and
ϕA = ϕA e KAϕA1
(5)
1
The number average degree of association, M̅ A, is given by ∞
M̅ A =
∑ i=1
NA ii NA i
=
∑ ϕA
i
∑ (ϕA /i) i
(6)
Hence
M̅ A =
■
ϕA KA (e KAϕA1 − 1)
(7)
ATTENUATED ASSOCIATION MODEL: THERMODYNAMICS
As noted above, the free energy of mixing linearly associated species is simply given by a Flory−Huggins equation for the entropy of mixing heterogeneous polymers (defined as those consisting of chains of different length). Although the species can be transient, the free energy has the same form, as shown by Flory25,28 and derived using the lattice model by Veytsman.22 However, is a Flory−Huggins-type model appropriate for asphaltene-like molecules? Sirota et al.31 have shown that the presence of rigid units requires modification to the entropy of mixing and proposed a “blob” model that interpolates between a regular solution form and the Flory−Huggins form. In this regard, Krukowski et al.32 developed an exact analytical lattice theory and showed that globular molecules are indeed better treated with regular 2473
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
solution models. However, they also found that molecules that are articulated are “remarkably well treated by Flory−Huggins theory”. Accordingly, we will assume that the structure of asphaltenes, condensed aromatic rings with attached alkyl chains of various lengths, are “articulated” in the sense described by Krukowski et al. and use the Flory−Huggins theory. If the subscript A is the associating component (asphaltenes) and B is a solvent or other low-molecular-weight non-associating hydrocarbon (e.g., maltenes), the free energy of mixing is simply given by
ΔG = RT
∑ n A i ln ϕA i + nB ln ϕB + nBϕA χ
μA − μA0 RT
i
= ln ϕA + 1 −
RT
i
μB − μB0
∞
∑ i=1
ϕA
i
∞
i=1
ϕA
i
VA i
=
∞
1 VA
∑ i=1
i
i
=
1 VA
i=1
1
KAVA
+
i
= ln ϕA + 1 −
RT
i
+
(11)
ϕB VB
KArA
1
iVA {exp(KAϕA ) − 1} 1
KAVA
+
+ ϕA ϕBχ
1
KA
+
RT
(19)
(20)
1
1 KAϕA K ϕ0 1 − e A A1) − rϕ (e B KA
= ln ϕB −
1 KAϕA 2 1 − 1) + ϕ + r ϕ χ (e B A A rKA
(21)
(22)
The equations for the spinodal (second derivative equals 0) and critical conditions (third derivative also equals 0) are then easily determined by differentiation of eq 20. The spinodal is given by ∂ 2(ΔG′/RT ) 1 ∂ϕA1 1 = + − 2χ = 0 2 rAϕA ∂ϕA rBϕB ∂ϕA 1
VAϕB VB
1
μB − μB0
(13)
we obtain
{exp(KAϕA ) − 1}
= ln(ϕA /ϕA0 ) − + rAϕB2χ
VAϕB VB
1
RT
(14)
1
ϕA (e KAϕA1 − 1)
μ A − μ A0
(12)
1
= ln ϕA + 1 −
(18)
where rA = VA/Vref, rB = VB/Vref, and r = rA/rB. The chemical potentials become
μA = μ A
RT
(17)
0
Using Prigogine’s result that, in an associated mixture in equilibrium, the chemical potential of a stoichiometric component is equal to the chemical potential of the monomer
ΔμA
+ ϕA + ϕA2χ
In the classic Flory−Huggins treatment of polymer solutions, the molar volume of the solvent defines the lattice cell size. In eq 19, we have implicitly let the molar volume of the non-associating component (B) be the reference volume. For solutions of asphaltenes in lowmolecular-weight solvents, such as toluene, this would be equivalent to Flory’s approach, but if mixtures of asphaltenes with higher molecular weight components are being considered, it can be more useful to consider the approach used to model the phase behavior of polymer blends, where an arbitrary lattice cell size is defined, usually on the basis of the molar volume of the monomer of one of the components of the mixture. Defining an arbitrary reference molar volume will also be useful later when we consider the use of solubility parameters. If we do this, the nBϕAχ term in eq 19 becomes nBsϕAχ, where nBs is the number of B segments defined by the reference volume, Vref. The free energy per mole of lattice sites is then
where VB is the molar volume of the solvent. It follows that Δμ A
1
KAVA
0
Hence
1 = Ṽ
VB{exp(KAϕA ) − 1)}
ϕ ϕ (e KAϕA1 − 1) ΔG′ ΔG Vref = = A ln(ϕA /ϕA0 ) + B ln ϕB − 1 1 RT RT V rA rB KArA
⎫ ⎧∞ ⎛ i i ⎞ ⎪ 1 ⎪ ⎜ K AϕA1 ⎟ ⎬ ⎨∑ ⎜ 1 = − ⎟ ⎪ KAVA ⎪ i ! ⎠ ⎭ ⎩ i=0 ⎝
{exp(KAϕA ) − 1}
(16)
nA VA(e KAϕA1 − 1) + nBϕA χ KAVA
+
⎛ i i ⎞ 1 ⎜ K AϕA1 ⎟ KA ⎜⎝ i! ⎟⎠
∞
∑
VA 2 ϕ χ VB B
V (e KAϕA1 − 1) ΔG = nA ln(ϕA /ϕA0 ) + nB ln ϕB − 1 1 RT KAVA
(10)
ϕA
+
to give
and VA is the molar volume of an asphaltene “monomer”. For simplicity of presentation, the χ term is omitted for now. Using eq 3
∑
VB
ΔG = nA (μA − μA0 ) + nB(μB − μB0 )
(9)
VB
VAϕB
1
KA
The free energy is then obtained from these equations using
ϕB
+
VA i
1
1
= ln ϕB −
RT
where 1 = Ṽ
{exp(KAϕA ) − exp(KAϕA0 )}
The χ interaction term has been restored to this equation. The chemical potential of the non-associating component (B) is given by
(8)
iVA Ṽ
1
+
where n is the number of moles of each species. This equation is incomplete, in that the entropy of disorientation of the molecules is omitted (recall that Flory’s reference state is the separate and oriented i-mers). These terms cancel from our final expressions and are left out for clarity of presentation. It has also been assumed that interactions between the associated species and the solvent or other component are the types of non-specific “weak” interactions that can be described by the Flory−Huggins χ parameter (final term in eq 8). This expression is of limited use without knowledge of the equilibrium distribution of associated species, but in the classical association model approach, the free energy can be written in terms of the concentration of “monomers” present. First, an expression for the chemical potential of the ith species is obtained Δμ A
= ln(ϕA /ϕA0 ) −
(23)
where
(15)
∂ϕA
Recalling that this is relative to Flory’s reference state, an equivalent expression for the chemical potential of the pure asphaltene can be subtracted from this expression to give the change in chemical potential on going from the pure state (0 superscript) to the mixture
1
∂ϕA
=
⎤ ϕA ⎡ 1 1⎢ ⎥ ϕA ⎢⎣ (1 + KAϕA ) ⎥⎦ 1
(24)
Differentiating and putting the third derivative equal to 0, we obtain 2474
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
1 R − 2 =0 ϕB2 rϕA
trial and error for ϕA = 1; values of ϕA, etc., then follow in a straightforward manner using the equations given above). All of the curves approach a number average degree of association of about 6 at the high end of the concentration range shown, consistent with other estimates of the size of nanoaggregates.3,4 Vapor pressure osmometry has been used by various groups to determine the number average molecular weight of asphaltenes as a function of composition. Here, we will use the data reported by Barrera et al.15 and tabulated in the thesis of Barrera.16 A comparison of the experimental results to those obtained using the model for asphaltenes from Athabasca and Peace River bitumen samples is shown in Figures 2 and 3,
(25)
where
R=
(1 + 3KAϕA + (KAϕA )2 ) 1
1
(1 + KAϕA )3
(26)
1
The critical value of χ, χc, is then given by
χc =
⎤ ⎡ 1 ⎢ 1 1 1 ⎥ + ϕB ⎥⎦ 2rB ⎢⎣ rϕA (1 + KAϕA ) 1
(27)
■
RESULTS AND DISCUSSION Calculation of the Molecular Weight of Associating Species. The first test of the model is its ability to reproduce molecular weight data. The number average degree of association is given by eq 7 and can be calculated as a function of composition using eq 5, which describes the stoichiometry of association. This requires a knowledge of the equilibrium constant KA. One of the goals of this work is reproduce the main features of asphaltene association and phase behavior using as few of adjustable parameters as possible. In this regard, Lisitza et al.33 measured the enthalpy and entropy of association of an asphaltene that gave values of the free energy change upon association of the order of 25−26 kJ/mol, corresponding to values of KA in the range of 11 000−18 000 at 50 °C. Calculated values of the number average degree of association as a function of composition are shown in Figure 1, and it can be seen that values of KA in this range do not greatly affect the calculated degree of association (all calculations in this and the following section were performed by simply using a spreadsheet with values of ϕA1 ranging from 0 to the value determined by Figure 2. Comparison of calculated and experimental values of the number average molecular weight of a Peace River asphaltene. Experimental data are from Barrera.16
respectively. If we fix the value of KA (see below), the crucial parameter in the fitting process becomes the molecular weight of the monomer. The smallest experimental value of the molecular weight for Athabasca bitumen was ∼1700, while that of the Peace River bitumen was near 1200. McKenna et al.5 have shown that asphaltenes associate even at extremely low concentrations; therefore, we assumed the average molecular weight of a monomer would be somewhat less than these values. In principle, extrapolating the number average molecular weight values obtained by Barrera16 to zero concentration should provide a value, but the effect of association is such that a linear extrapolation is questionable, even when applied to just the lowest concentration values (where, in any event, experimental errors would inevitably be larger). On the basis of the lowest concentration data, we then somewhat arbitrarily assumed values of 1400 for Athabasca bitumen “monomers” and 850 for Peace River bitumen “monomers”. These values are within the range of currently accepted values for asphaltenes. We assumed that asphaltene molecules that are larger would associate more strongly and used values of KA of 18 000 for the Athabasca asphaltenes and 11 000 for the Peace River asphaltenes, although as we showed above, the calculated values of molecular weight are not particularly sensitive to this
Figure 1. Calculated degree of association (number of “monomers” in a nanocluster) for asphaltenes using that attenuated association model and values of the dimensionless equilibrium constant in the range defined by experimental free energy of association values. 2475
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
Flory−Huggins theory assuming no association (i.e., eq 23, with ϕA1 = ϕA). It was assumed that the molar volume of an average asphaltene “monomer” molecule was 8 times that of the solvent (i.e., r = VA/VB = 8). The plots are in terms of χ versus composition; the values of χ define the solution stability limit at each concentration. Because χ varies with the inverse of the temperature, this plot corresponds to an “upside down” temperature/composition plot, with a minimum corresponding to an upper critical solution temperature. The Flory−Huggins critical value of χ, χc, is close to 0.9. Also shown in Figure 4 are spinodal curves calculated for associated asphaltenes using KA = 11 000 and values of r = 8 and 14 (note that the parameter r should not be confused with the degree of association, which varies with composition and is accounted for in the model). With association, χ is near 0.64 for r = 8. In other words, phase separation at the critical concentration occurs at much lower values of χ; the solution can only tolerate much smaller differences in solubility parameters between asphaltene molecules and solvent (or, for example, maltenes). Increasing the size of the asphaltene molecules (r = 14) decreases χc further but not by a lot to about 0.61. As might be intuitively expected, association reduces solubility because asphaltene molecules have a stronger affinity for one another than for solvent. Furthermore, association results in a large degree of asymmetry in the phase diagram, with critical asphaltene concentrations near volume fractions of 0.1. Regular solution models assume that the phase diagram is symmetric with composition, and this assumption simplifies calculations of the concentration of coexisting phases. The marked asymmetry in the phase diagrams calculated using the attenuated association model precludes this approach. To compare the model to experimental data, it is necessary to calculate the binodal or coexistence curves defining the composition of coexisting phases. These were calculated from chemical potential plots (eq 16) using the Maxwell construction and a program that calculates the area under the curves. An example of the results is shown in Figure 5. It can be seen that there is a very large area of metastable equilibrium between the spinodal and binodal lines, where phase separation proceeds by a process of nucleation and growth. Coupled with the high viscosity of concentrated asphaltene solutions, this is consistent with the results by Maqbool et al.,34 who showed that the time required to precipitate asphaltenes ranges from minutes to months, depending upon the temperature (i.e., 1/χ) and concentration. Many studies of the phase behavior of asphaltenes are conducted at low concentrations. For example, Barrera16 reported the amount of asphaltenes precipitated by heptane in toluene solutions, whose total asphalt concentration was 10 kg/m3. The phase diagram (binodals) in the concentration range of 0−20 kg/m3 for values of r = 8 and 14 are shown in Figure 6 [a density of 1200 kg/m3 was used to convert volume fractions (ϕA) to concentrations in kilograms per meter cubed]. These values of r correspond to the approximate ratios of the molar volumes of asphaltene molecules (monomers) to toluene determined for Peace-River- and Athabasca-derived asphaltenes using the molecular weight plots shown in Figures 2 and 3, respectively. It can be seen from Figure 6 that, at a concentration of 10 kg/m3, phase separation or precipitation occurs at values of χ near 0.76 and 0.66 for r = 8 and 14, respectively. The coexistence curves were then used to calculate the composition
Figure 3. Comparison of calculated and experimental values of the number average molecular weight of an Athabasca asphaltene. Experimental data are from Barrera.16
parameter. The model appears to do a very good job of fitting the data with parameters that are consistent with experimental values reported in the literature. Calculation of the Phase Behavior of Asphaltene Solutions. It is useful to start this section by considering the effect of association on the phase behavior of asphaltene solutions. Figure 4 shows the spinodal calculated using the
Figure 4. Calculated spinodals (plotted as χ versus the volume fraction of asphaltenes) for a model asphaltene using the Flory−Huggins model (ratio of molar volumes, r = 8) compared to calculated spinodals using the attenuated equilibrium constant association model (r = 8 and 14). 2476
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
Hildebrand and Scott35 (the 1950 edition has much more detail than later versions), they were designed to deal with the mixing of molecules with relatively weak, non-specific interactions, mainly dispersion and weak polar forces. Many attempts have been made to extend their use to systems that involve strong, specific interactions, such as hydrogen bonds, by breaking down contributions to molar attraction constants into various components. Although this can provide a useful guide to mixing, it is inappropriate to apply them in this fashion to thermodynamic models. For example, the enthalpy of mixing is always positive in a solubility parameter model, because it depends upon a difference in a solubility parameter term squared, so that molecules that interact strongly (e.g., phenol and pyridine) and have a negative enthalpy of mixing cannot be described without contorting the theory with additional terms and adjustable parameters. In association models, on the other hand, the strong associative forces (not to mention nonrandom contacts) are accounted for in terms that describe association and it is only the weaker interactions between associated species and other molecules that are described using solubility parameters. To use eq 28 in this fashion, we therefore need a method for calculating the solubility parameter of an “average” asphaltene molecule and, more subtly, this method should be based on data that does not include heats of vaporization of molecules, such as ethanol, that self-associate, because an assumption of the model is that solubility parameters are being used to account for dispersion and weak polar interactions only. Functional group molar attraction constants and molar volume constants that fulfilled this condition were determined in these laboratories more than 20 years ago36 and used to account for the miscibility of a wide range of polymer blends.19 However, the distribution of chemical groups (e.g., CH2 and CH3) in materials such as asphaltenes is not known with any precision. There is another approach but subject to larger errors. Following van Krevelen,37 an atomic group contribution method was developed to calculate solubility parameters of coals.38 When applied to the original data set, it resulted in errors of ±1.2 MPa0.5 compared to errors of ±0.8 MPa0.5 for calculations based on functional group contributions. Surprisingly, this gave calculated values of asphaltene solubility parameters that were consistent with other approaches and which appear to do a good job in accounting for phase behavior (mainly because the calculated values are dominated by the atomic H/C ratio and fraction aromaticity). The equation that we used is38
Figure 5. Spinodal and binodal curves (plotted as χ versus the volume fraction of asphaltenes) calculated using the association model.
⎛ ⎛H⎞ ⎛O⎞ δA = 2.045⎜7.0 + 63.5fa + 63.5⎜ ⎟ + 106⎜ ⎟ ⎝C⎠ ⎝C⎠ ⎝ ⎛H⎞ (N + S) ⎞ ⎛ ⎟ /⎜ −10.9 + 12fa + 13.9⎜ ⎟ + 51.8 ⎝C⎠ ⎠ ⎝ C
Figure 6. Binodal curves at a low concentration calculated using the association model with degrees of association r = 8 and 14.
of the solvent-rich and solvent-poor phases as a function of χ. Each value of χ is assumed to correspond to the value given by the difference in solubility parameters between asphaltene molecules and the mixed toluene/heptane solvent, according to χ=
Vs (δA − δs)2 RT
⎛O⎞ (N + S) ⎞ + 5.5⎜ ⎟ − 2.8 ⎟ ⎝C⎠ ⎠ C
(29)
Using the elemental analysis data and fraction aromaticity values reported by Sheremata et al.,39 we calculated a solubility parameter of 20.5 MPa0.5 for an Athabasca asphaltene. Similar data were not found for Peace River asphaltenes, but Calemma et al.40 reported data for a range of asphaltenes. For those with fraction aromatic carbon contents between 0.48 and 0.58, we calculated solubility parameter values ranging from 20.2 to 20.9 MPa0.5. This range of values is consistent with the solubility
(28)
where δA and δs are the solubility parameters of the asphaltene and solvent, respectively, while Vs is the molar volume of the solvent. This brings us to the subject of solubility parameters. As originally used by Hildebrand and described in the book by 2477
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
largest value of δA would precipitate first, followed by materials with lower values of δA. Essentially, the calculated curve should be a convolution of curves calculated for a range of solubility parameters. A comparison of calculated and experimental values of precipitated material for Athabasca asphaltenes (data reported by Barrera16) is shown in Figure 8. For this asphaltene, which
parameters calculated for Athabasca and Peace River asphaltenes by Akbarzadeh et al.41 of about 21 and 20.5 MPa0.5, respectively. Accordingly, in the calculations that follow, we will assume values for the solubility parameter of the asphaltenes ranging between 20 and 21.5 MPa0.5. The following procedure was then used to calculate the weight fraction of asphaltenes precipitated from toluene/ heptane mixtures. First, The solubility parameter and molar volume of the mixed solvent was calculated as a volume fraction weighted average (with δtoluene = 18.2 MPa0.5 and δheptane = 15.3 MPa0.5). Values of χ were then calculated as a function of solvent composition (the amount of heptane in a heptane/ toluene mixture). For a given value of χ, the composition of asphaltene in the asphaltene-rich (precipitated) phase was then determined graphically from the χ/composition phase diagrams. Figure 7 shows the experimentally determined weight fraction of precipitated Peace River asphaltene plotted as a
Figure 8. Comparison of the experimentally determined weight fraction of precipitated Athabasca asphaltene plotted as a function of the volume fraction of heptane in heptane/toluene mixtures (as reported by Barrera16) to the amounts of precipitated material calculated from the model using assumed asphaltene solubility parameters in the range of 20−21.3 MPa0.5 [values used corresponded to solubility parameters of 9.8, 10, 10.2, and 10.4 (cal/cm3)0.5]. The dashed line is to aid the eye and was not obtained by fitting the data.
has a higher molecular weight than Peace River asphaltenes (and is probably more aromatic), precipitation starts at lower values of heptane volume fractions and the data are encompassed by asphaltene solubility parameters that range from 20 to 21.3 MPa0.5. Again, the remarkable feature of this plot is the narrow range of solubility parameter values for asphaltene molecules that defines the phase behavior of these mixtures. If eq 29 is re-examined, however, it is clear that the value of the solubility parameter calculated for asphaltenes will be dominated by the fraction aromaticity and the H/C atomic ratio, as mentioned above. For example, for an asphaltene derived from an Athabasca bitumen,38 these terms account for about 86% of the numerical value of the numerator. It is the aromatic/aliphatic composition of these materials that largely determines the value of the solubility parameter (as defined here), and for the asphaltenes considered here, the aromaticity lies in a fairly well-defined range. Finally, the model was applied to the calculation of the fraction of material precipitated from heavy oil samples, using the data reported by Peramanu et al.40 These authors demonstrated the reversibility of asphaltene precipitation upon the addition and removal of heptane. The data taken from the plots reported by these authors for Peace River heavy oil and Athabasca bitumen are shown in Figures 9 and 10,
Figure 7. Comparison of the experimentally determined weight fraction of precipitated Peace River asphaltene plotted as a function of the volume fraction of heptane in heptane/toluene mixtures (as reported by Barrera16) to the amounts of precipitated material calculated from the model using assumed asphaltene solubility parameters of 20 and 20.5 MPa0.5 [values used corresponded to solubility parameters of 9.8 and 10 (cal/cm3)0.5]. The dashed line is to aid the eye and was not obtained by fitting the data.
function of the volume fraction of heptane in heptane/toluene mixtures (all experiments at a concentration of 10 kg/m3 asphaltene in solvent, as reported by Barrera16). Precipitation commences at volume fractions of heptane just over 0.5. Also shown in this plot are the amounts of precipitated material calculated from the model using assumed asphaltene solubility parameters of 20 and 20.5 MPa0.5. It can be seen that these essentially encompass the experimental values. The theoretical curves defined by calculated points are sharper than the experimental curve, but this is to be expected. Asphaltenes are not defined by specific structures but consist of a range of molecules of different molecular weights and aromaticities. Thus, the material with structures that have the 2478
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
respectively. To a first approximation, the molar volume of the non-associating oil component was assumed to be close to that of the solvent. It can be seen that the model again reproduces the data with assumed values of asphaltene solubility parameters in a narrow range centered near 20 MPa0.5.
■
CONCLUSION A simple attenuated association model accounts for the association of asphaltenes and their phase behavior. The model assumes that there are molecules that can associate and molecules that cannot. These roughly correspond to asphaltenes and maltenes, respectively. The crude assumption that the phase behavior of the mixture can then be modeled as a pseudo-two-component mixture is then made and works surprisingly well, using values of the solubility parameter of asphaltenes in a narrow range near 20 MPa0.5.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
Figure 9. Comparison of the experimentally determined fraction of material precipitated from a Peace River heavy oil sample (data reported by Peramanu et al.40) to the amounts of precipitated material calculated from the model using assumed asphaltene solubility parameters in the range of 20−20.3 MPa0.5. The square experimental data points are for precipitation, and the round experimental data points are for dissolution [values used corresponded to solubility parameters of 9.8 and 9.9 (cal/cm3)0.5]. The dashed line is to aid the eye and was not obtained by fitting the data.
ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Federal Highway Administration under Subcontract SES 12-01. REFERENCES
(1) Podgorski, D. C.; Corilo, Y. E.; Nyadong, L.; Lobodin, V. V.; Bythell, B. J.; Robbins, W. K.; McKenna, A. M.; Marshall, A. G.; Rodgers, R. P. Energy Fuels 2013, 27, 1268−1276. (2) Boduszynski, M. M.; McKay, J. F.; Latham, D. R. Asphalt Paving Technol. 1980, 49, 123−143. (3) Mullins, O. C. Energy Fuels 2010, 24, 2179−2207. (4) Mullins, O. C.; Sabbah, H.; Eyssautier, J.; Pomerantz, A. E.; Barré, L.; Andrews, A. B.; Ruiz-Morales, Y.; Mostowfi, F.; McFarlane, R.; Goual, L.; Lepkowicz, R.; Cooper, T.; Orbulescu, J.; Leblanc, R. M.; Edwards, J.; Zare, R. N. Energy Fuels 2012, 26, 3986−4003. (5) McKenna, A. M.; Donald, L. J.; Fitzsimmons, J. E.; Juyal, P.; Spicer, V.; Standing, K. G.; Marshall, A. G.; Rodgers, R. P. Energy Fuels 2013, 27, 1246−1256. (6) Whitson, C. H.; Brulé M. R. Phase Behavior Monograph; Society of Petroleum Engineers, Inc.: Richardson, TX, 2000; Henry L. Doherty Series, Vol. 20. (7) Pedersen, K. S.; Christensen, P. L. Phase Behavior of Petroleum Reservoir Fluids; CRC Press (Taylor and Francis Group): Boca Raton, FL, 2007. (8) Wang, J. X.; Buckley, J. S.; Burke, N. E.; Creek, J. L. SPE Prod. Facil. 2004, 19, 152−160. (9) Akbarzadeh, K.; Alboudwarej, H.; Svrcek, W. Y.; Yarranton, H. W. Fluid Phase Equilib. 2005, 232, 159−170. (10) Li, Z.; Firoozabadi, A. Energy Fuels 2010, 24, 2956−2963. (11) Gonzalez, D. L.; Hirasaki, G. J.; Creek, J.; Chapman, W. G. Energy Fuels 2007, 21, 1231−1242. (12) Agrawal, P.; Schoeggl, F. F.; Satyro, M. A.; Taylor, S. D.; Yarranton, H. W. Fluid Phase Equilib. 2012, 334, 51−64. (13) Yarranton, H. W.; Fox, W. A.; Svrcek, W. Y. Can. J. Chem. Eng. 2007, 85, 635−642. (14) Agrawala, M.; Yarranton, H. W. Ind. Eng. Chem. Res. 2001, 40, 4664−4672. (15) Barrera, D. M.; Ortiz, D. P.; Yarranton, H. W. Energy Fuels 2013, 27, 2474−2487. (16) Barrera, D. M. Determination and application of asphaltene property distributions for native and refined crude oils. M.Sc. Thesis, University of Calgary, Calgary, Alberta, Canada, 2012; http://www.
Figure 10. Comparison of the experimentally determined fraction of material precipitated from a Athabasca bitumen sample (data reported by Peramanu et al.40) to the amounts of precipitated material calculated from the model using assumed asphaltene solubility parameters near 20.0 MPa0.5. The square experimental data points are for precipitation, and the round experimental data points are for dissolution [values used corresponded to solubility parameters of 9.7 and 9.8 (cal/cm3)0.5]. The dashed line is to aid the eye and was not obtained by fitting the data.
2479
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
Energy & Fuels
Article
ucalgary.ca/ENCH/AER/theses/ 2012%20MSc%20Thesis%20DianaBarrera.pdf. (17) Shirani, B.; Nikizar, M.; Naseri, A.; Mousavi-Dehgani, S. A. Fuel 2012, 93, 59−66. (18) Acree, W. E. Thermodynamic Properties of Non-electrolyte Solutions; Academic Press: New York, 1984. (19) Coleman, M. M.; Graf, J. F.; Painter, P. C. Specific Interactions and the Miscibility of Polymer Blends; Technomic Publishing: Lancaster, PA, 1991. (20) Coleman, M. M.; Painter, P. C. Prog. Polym. Sci. 1995, 20, 1−49. (21) Painter, P. C.; Coleman, M. M. Polymer Blends; Paul, D. R., Bucknall, C. B., Eds.; John Wiley and Sons, Inc.: New York, 2000; Vol. 1, Fundamentals, Chapter 6. (22) Veytsman, B. A. J. Phys. Chem. 1990, 94, 8499−8500. (23) Panayiotou, C.; Sanchez, I. C. J. Phys. Chem. 1991, 95, 8499− 8500. (24) Prigogine, I.; Bellemans, A.; Malthot, V. The Molecular Theory of Solutions; Interscience: New York, 1957. (25) Flory, P. J. J. Chem. Phys. 1944, 12, 425−438. (26) Tobolsky, A. V. J. Chem. Phys. 1944, 13, 379−380. (27) Tobolsky, A. V.; Blatz, P. J. Chem. Phys. 1945, 12, 425−438. (28) Flory, P. J. J. Chem. Phys. 1946, 14, 49−50. (29) Kretschmer, C. B.; Wiebe, R. J. J. Chem. Phys. 1954, 22, 1697− 1700. (30) Painter, P. C.; Veytsman, B.; Coleman, M. M. J. Polym. Sci., Part A: Polym. Chem. 1944, 32, 1189−1199. (31) Sorota, E. B.; Rangwalla, H.; Peczak, P. Macromolecules 2012, 45, 5281−5295. (32) Lisitza, N. V.; Freed, D. E.; Sen, P. N.; Song, Y.-Q. Energy Fuels 2009, 23, 1189−1193. (33) Maqbool, T; Srikiratiwong, P.; Fogler, H. S. Energy Fuels 2011, 25, 10090−10097. (34) Hildebrand, J.; Scott, R. The Solubility of Non-electrolytes, 3rd ed.; Reinhold: New York, 1950. (35) Coleman, M. M.; Serman, C. J.; Bhagwagar, D. E.; Painter, P. C. Polymer 1990, 31, 1187−1203. (36) Van Krevelen, D. W. Coal; Elsevier: Amsterdam, Netherlands, 1981. (37) Painter, P. C.; Graf, J.; Coleman, M. M. Energy Fuels 1990, 4, 379−384. (38) Sheremata, J. M.; Gray, M. R.; Dettman, H. D.; McCaffrey, W. C. Energy Fuels 2004, 18, 1377−1384. (39) Calemma, V.; Iwanski, P.; Nali, M.; Scotti, R.; Montanari, L. Energy Fuels 1995, 9, 225−230. (40) Akbarzadeh, K.; Ayatollahi, S.; Moshfeghian, M.; Alboudwarej, H.; Yarranton, H. W. J. Can. Pet. Technol. 2004, 43, 1−9. (41) Peramanu, S; Singh, C.; Agrawala, M.; Yarranton, H. W. Energy Fuels 2001, 15, 910−917.
2480
dx.doi.org/10.1021/ef500345q | Energy Fuels 2014, 28, 2472−2480
