Modeling of Asphaltene Transport and Separation in the Presence of

Article pubs.acs.org/EF

Modeling of Asphaltene Transport and Separation in the Presence of Finite Aggregation Effects in Pressure-Driven Microchannel Flow Siddhartha Das,† Rahul Prasanna Misra,† Thomas Thundat,‡ Suman Chakraborty,∥ and Sushanta K. Mitra*,† †

Micro and Nanoscale Transport Laboratory, Department of Mechanical Engineering, and ‡Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada ∥ Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India ABSTRACT: In this paper, we present a perturbation-based theoretical model on the asphatene transport and aggregation in a pressure-driven microchannel flow. The aggregation effect is captured by a first-order reaction kinetics quantified by the appropriate Damköhler number. We find that the rate of average transport, quantified by the corresponding band velocity of a traveling asphaltene plug, is lowered by an increase in the Damköhler number, suggesting that the aggregation event lowers the flow rate of an asphaltene sample. Also, the average spread of the channel centerline concentration, described by a Gaussian profile and dictated primarily by the axial dispersion (and not Taylor dispersion, in light of the small Peclet number) increases with the Damköhler number. The net effect is a distinctly noticeable separation of asphaltene molecules having different rates of aggregate formation.

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flow field. This multiscale behavior is captured by considering the explicit structure of the asphaltene molecules and their specific interactions with water, which, in turn, is modeled through stochastic rotation dynamics. However, such investigations are naturally limited by the demands of an extremely expensive explicit molecular treatment, which foregoes several gross representative features associated with the transport, aggregation, and deposition of asphaltene. For example, issues such as the effect of variation in aggregation dynamics on the average transport and separation of asphaltene moieties cannot be explicated. A simpler method that gives a better understanding about such parameters, therefore, has been proposed, where the asphaltene transport as well as the processes, such as aggregation, deposition, and precipitation, are modeled in a continuum framework.25,26 Such models are easy to reproduce and provide faster information about the asphaltene dynamics (e.g., their average velocity, separation efficiency, etc.) in a background microflow. In these models, the aggregation or deposition dynamics is described by an average macroscopic rate constant. Such an approach, despite coarse graining the expensive molecular details of Boek et al.,12,22,23 successfully explains several important issues of microfluidic asphaltene dynamics. Another advantage of such a gross specification of the asphaltene dynamics is that it helps to avoid controversial issues regarding the exact mechanism of the asphaltene aggregation and deposition (these issues are unavoidable in a molecular treatment) that have received significant attention.27 An even more useful approach would be to be provide a fullscale analytical treatment of the asphaltene transport and dynamics in the presence of effects such as aggregation (modeled through the above-described macroscopic approach).

INTRODUCTION One of the major limitations involved in the heavy oil production and processing is associated with the asphaltene deposition (or precipitation)1−3 on the inner linings of the pipeline. Therefore, there has been a constant effort to understand the manner in which the asphaltene, which is the heaviest and most polar component of the heavy oil, first aggregates (to form nanoscopic-sized aggregates2,4−8) and then precipitates to block the pipelines. The aggregation and precipitation dynamics is often described in terms of relevant equations of state models.9−11 In fact, such aggregation and precipitation may lead to effects such as blocking of rock pores, thereby changing the wetting characteristics of mineral surfaces within the reservoir and hindering oil recovery.12 Depending upon the clustering concentration, the asphaltene aggregation may be either diffusion- or kinetics-driven2 and there has been a plethora of investigations that quantify the corresponding thermodynamic behavior6,13,14 or the related scaling issues.15−18 There have been significant efforts to elucidate the influence of a background flow field in understanding the aggregation and deposition of asphaltene. For example, there have been investigations delineating the deposition behavior of asphaltene in a background flow field in production pipelines as well as metallic capillaries.19 Further, Monteagudo et al.20 applied network models, and Wang and Civan21 applied the Darcyscale deep-bed filtration continuum model to investigate the deposition-induced formation damage in the reservoir during the flow-driven transport of asphaltene. In a relatively recent endeavor, efforts are being made to understand the effect/ occurrence of asphaltene aggregation in the presence of a background flow field in a microscale capillary.12,22−24 Such a test system is specially useful because it allows for an explicit accounting of the multiscale nature of the asphaltene aggregation and deposition in the presence of a background © 2012 American Chemical Society

Received: July 10, 2012 Revised: August 15, 2012 Published: August 16, 2012 5851

dx.doi.org/10.1021/ef3011542 | Energy Fuels 2012, 26, 5851−5857

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Such calculations will provide readily usable quantifications of the concentration profiles (spatial and temporal) of different forms (non-aggregated and aggregated) of asphaltene, thereby demonstrating the manner in which the aggregation reaction will affect the overall transport. In this paper, we provide such an analytical model (the approach mimics the continuum modeling approach proposed in Vargas et al.26) for the microfluidic migration and dynamics of the asphaltene molecules, with explicit consideration of the aggregation dynamics. There have been earlier attempts to analytically model the asphaltene transport and precipitation in the pipelines and porous media.28−32 However, these calculations often lack the explicit accounting of the aggregation kinetics nor do they provide the full two-dimensional temporal distribution of the non-aggregates and aggregates by accounting for appropriate hierarchies in length and time scales. Our analytical formalism, on the other hand, is developed on the basis of a perturbation approach33−36 that uses the fact that one can always associate two different length scales [one micrometer scale (typically the channel height or width) and another a larger scale (typically the channel length)] to a microfluidic system. There are certain key assumptions based on which our analytical model is developed. First, in the background flow field, the aspahltene molecules are assumed to exist in only two possible states: non-aggregated and aggregated sizes. This is a simplified assumption in light of more complicated models that explicitly consider the distribution of the asphaltene population of different sizes,31,32 attributed to different extents of asphaltene aggregation simultaneously. In our study, we do account for such variation in aggregation (by providing results as a function of the Damköhler number); however, they are not considered simultaneously. Second, the size of the aggregates are of the order of only a few nanometers, so that they are too small to affect the background flow field. In this study, we assume a laminar fully developed flow field as the background flow. This is motivated by the fact that we are considering a transport in the microscale, and therefore, it is different from the turbulent background flow atypical in normal pipe transport.32 Third, the aggregation is assumed to be a reaction-controlled process, with the reaction exhibiting a first-order kinetics.26 Fourth, the asphaltene molecules are assumed to be devoid of any charge, so that there are no asphaltene−asphaltene or asphaltene−wall electrostatic interactions. Finally, we neglect all other kinds of asphaltene− asphaltene or asphaltene−wall interactions (e.g., van der Waals interactions). There are two principal results of this study. First, we show that the aggregation kinetics severely affects the average transport of the asphaltene moieties by substantially reducing the band velocity33−36 of the asphaltene plug; a larger rate of aggregation leads to a more reduced band velocity. Second, which is a natural corollary of the first result, the asphaltene moieties can be separated from each other on the basis of their variation of rate of aggregate formation; therefore, we introduce a new paradigm of reaction-rate-driven separation behavior of the asphaltene molecules.

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Figure 1. Schematic showing the transport of the asphaltene plug of length S in a microchannel (of height h and length L) pressure-driven transport (please note that the schematic is not to scale and we have h ≪ S ). Inside the plug, the asphaltene molecules aggregate, with the aggregation being assumed to be a kinetically driven first-order process with rate constant k1.26 In this schematic, the asphaltene aggregation is magnified. In reality, it is around 1000 times smaller than the channel height.

concentration distribution solely in terms of the concentration of nonaggregated species, with appropriate accounting for the aggregationinduced lowering of its concentration. The migration of the plug (introduced at t = 0) is characterized by a band velocity ub and a dispersion coefficient.33−36 The governing equation for the species (or the non-aggregated asphaltene molecules) transport, in a reference frame moving with the velocity ub, can be expressed as ∂c ∂ ∂ ∂ 2c ∂ 2c ((u p − ub)c) + (vpc) = Dx 2 + Dy 2 − k1c + ∂y ∂t ∂x ∂x ∂y (1) where c is the concentration of the non-aggregated molecules, Dx and Dy are the corresponding axial and diffusivities, up and vp are the corresponding axial and migration velocities, and k1 is the rate constant for the aggregation. Also u p = u + urel, x = u ,

vp = v + vrel, y = v

asphaltene transverse transverse asphaltene (2)

where u and v are the background advection velocities and urel,x and vrel,y are the velocities of the non-aggregated asphaltene molecules relative to the flow (both of which are equal to 0). Also, the background flow is assumed to be a fully developed pressure-driven flow, so that there is a finite u but v ≡ 0. Therefore, the species transport equation reduces to ∂c ∂ ∂ 2c ∂ 2c [(u − ub)c] = Dx 2 + Dy 2 − k1c + ∂t ∂x ∂x ∂y

(3)

Equation 3 needs to be solved in the presence of the following no flux boundary condition at the channel wall (y = ±h/2): ⎛ ∂c − ⎜Dy ⎝ ∂y

⎞

∫ k1cdy⎟⎠

=0

−h /2, h /2

(4)

Perturbation Solution. The fact that we are studying the problem in a microfluidic channel, i.e., where the channel height is several orders smaller than the channel length and the plug width (defined later), we can attempt a perturbation-based analytical solution (taking advantage of this distinct demarcation of the length scales), in a manner similar to our previous studies.34−36 We first obtain the dimensionless forms of eq 3 as

THEORY

∂C L ∂ + [(U − Ub)C ] ∂T S ∂X ⎛ DyL ⎞ S2 ∂ 2C D ⎛ DyL ⎞ ∂ 2C kL = x⎜ 2 ⎟ 2 + ⎜ 2 ⎟ 2 2 − 1 C Dy ⎝ S u ̅ ⎠ ∂X u̅ ⎝ S u ̅ ⎠ h ∂Y

Model for the Species Transport. We consider the transport of a plug of asphaltene molecules in a microchannel, as shown in Figure 1. The asphaltene molecules in the plug can exist in either of two forms: non-aggregated and aggregated. We assume that these two states are kinetically connected. We further assume that the aggregation is much more favored than the deaggregation, so that we can express the

(5)

where 5852


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u , u̅

Article

ub , u̅

Ub =

c C= c0

T=

t tu = ̅, t0 L

Y=

y , h

X=

x , S

U (Y ) =

Da ∂C Pe ∂ ∂ 2C 1 ∂ 2C + [(U − Ub)C ] = R 2 + 2 2 − 20 C ∂T ε ∂X ∂X ε ∂Y ε

Ub = −

⎞

∫ Da0CdY ⎠ ⎟

(18)

Pe(U − Ub)cosh( Da0 Y ) = (A1 + B1Y 2)exp( Da0 Y ) + (A 2 + B2 Y 2)exp(− Da0 Y ) (19) where

(7) A1 = A 2 =

⎞ Pe ⎛⎜ 3 − Ub⎟, ⎠ 2 ⎝2

B1 = B2 = − 3Pe

(20)

We therefore consider the particular integral of the following form: (8)

G PI = (a1Y + b1Y 2 + c1Y 3)exp( Da0 Y ) + (a 2Y + b2Y 2 + c 2Y 3) exp(− Da0 Y )

(9)

(21)

Using eqs 13, 19, and 21, we can obtain the constants as follows:

Using eq 9 in eqs 7 and 8 and equating different powers of ε, we obtain the necessary differential equations and the corresponding boundary conditions, which we solve to obtain the parameters describing the asphaltene migration and separation. We illustrate that procedure below. Equation and Boundary Condition (BC) for the Power 1/ε2. ∂Y 2

(17)

The left-hand side (LHS) of eq 13 can be expressed as

Now, we expand the concentration in an asymptotic form, i.e.

C = C0 + εC1 + ε 2C 2 + ...

6 coth( Da0 /2) Da0

G = M cosh( Da0 Y )+particular integral

=0

Y =−1/2,1/2

12 + Da0

Equation 17 gives the analytical solution of the dimensionless band velocity of the asphaltene plug. Solution of the Function G. Solution of G can be obtained by solving the ordinary differential equation (ODE) expressed in eq 13. The general analytical form of G can therefore be expressed as

Similarly, the boundary condition, expressed in eq 4, reduces to

∂ 2C0

(16)

Therefore, from eq 15, we obtain (6)

Here, u̅ = ∫ h/2 −h/2udy/h is the average advection velocity in the microchannel, h is the microchannel height, S is the length of the asphaltene plug (and the characteristic length scale in the axial direction), t0 = u̅/L is the characteristic time scale (where L is the channel length), and c0 is the characteristic concentration scale. It may be noted here that we choose the advection-based time scale t0 as the reference, because we intend to study the manner in which the aggregation reaction becomes affected by the asphaltene advection and vice versa. 2 Using Pe = (uh/D y), L ∼ (S u/D ̅ y), ε = (h/S ) ≪ 1 (here, ε is the ̅ perturbation parameter), R = (Dx/Dy), Da = (k1L/u̅) = (Da0/ε2) (this approximation assumes that the reaction rate, characterized by the Damköhler number Da is large enough to influence the overall transport; here, Da0 is the modified Damköhler number), (L/S ) = (DyL/u ̅ S 2)(u̅h/Dy)(S /h) = (Pe/ε), we can rewrite eq 5 as

⎛ ∂C ⎜ − ⎝ ∂Y

3 − 6Y 2 2

A1 B1 + , 2 Da0 4Da0 3/2 B B1 b1 = b2 = b = − 1 , c1 = − c 2 = c = 4Da0 6 Da0

a1 = − a 2 = a =

(22)

Hence, we can express G as − Da0C0 = 0

⎛ ∂C BC: ⎜ 0 − Da0 ⎝ ∂Y

G = M cosh( Da0 Y ) + exp( Da0 Y )(aY + bY 2 + cY 3)

(10)

∫

⎞ C 0 dY ⎟ =0 ⎠Y =−1/2,1/2

+ exp(− Da0 Y )(− aY + bY 2 − cY 3)

To evaluate the constant M, we use the constraint ∫ 1/2 −1/2GdY = 0, which yields

(11)

Therefore, the necessary analytical solution is C0 = A(X , T )cosh( Da0 Y )

⎡ c M = − coth( Da0 /2)⎢a + (24 + Da0) − ⎢⎣ 4Da0

(12)

where A(X,T) can be defined as the channel centerline (Y = 0) concentration. Equation and BC for the Power 1/ε. d2G Pe(U − Ub)cosh( Da0 Y ) = − Da0G dY 2

+

∫

⎞ G dY ⎟ =0 ⎠Y =−1/2,1/2

(13)

(14)

Da0 Y )dY = 0

(25)

⎛ ∂C BC: ⎜ 2 − Da0 ⎝ ∂Y

(26)

⎞

∫ C2dY ⎟⎠

=0

Y =−1/2,1/2

1/2 ⎡ ⎤ ∫−1/2 (U − Ub)G(Y )dY ⎥ ∂⟨C0⟩ ∂ 2⟨C0⟩ ⎢ = ⎢R − Pe ⎥ 1/2 ∂T ∂X2 ⎢ ∫−1/2 F(Y )dY ⎥⎦ ⎣

1/2

∫−1/2 U (Y )cosh( Da0 Y )dY 1/2

∫−1/2 cosh( Da0 Y )dY

∂C0 ∂ 2C ∂ 2C 2 ∂ + Pe [(U − Ub)C1] = R 20 + − Da0C 2 ∂T ∂X ∂X ∂Y 2

Using C0 = A(X,T)cosh((Da0)1/2Y) = A(X,T)F(Y), C1 = (∂A/∂X) G(Y), ⟨C0⟩ = A(X,T)∫ 1/2 −1/2F(Y)dY, we can integrate eq 25 in the presence of eq 26, to obtain

1/2

⇒ Ub =

2a 3c b + (8 + Da0) − (8 + Da0) Da0 2Da0 2Da0 3/2 (24)

Using these conditions, we can integrate eq 13 to obtain

∫−1/2 (U − Ub)cosh(

2b ⎤ ⎥ Da0 ⎥⎦

Equation and BC for the Power ε0.

[using eq 12 and C1 = (∂A/∂X)G(Y)]. ⎛ ∂G BC: ⎜ − Da0 ⎝ ∂Y

(23)

(15)

We consider a pure pressure-driven transport, so that

(27)

Hence, the dimensionless dispersion coefficient can be expressed as 5853


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D *= R − Pe

∫−1/2 (U − Ub)G(Y )dY 1/2

∫−1/2 F(Y )dY

proposed perturbation-based species transport model by demonstrating how it can be used to reproduce existing experimental findings. Validation of the Perturbation-Based Species Transport Model. The proposed perturbation-based species transport model that uses the disparate axial and transverse length scales in a micro-nanochannel has been used in several studies to reproduce the experimental results. In this subsection, we shall describe some of these studies and establish the manner in which the theoretical formulation in these studies can be obtained by trivial extension of the perturbation-based species transport model used here. This will validate the present model. Pennathur and Santiago37,38 used this perturbation-based model to reproduce the electromigration of charged analytes in nanochannels. With the analytes being charged (but nonreactive), the species concentration equation (see eq 1) will now have an additional term accounting for the electrophoretic flux of the analyte but there will be no reactive term (k1c). Under this condition, they perform a perturbation-based analysis exactly identical to that provided in the Theory section and obtain the analyte band velocity (similar to that described in Figure 3) that matches quantitatively with the experimental results (see panels a and b of Figure 5 in ref 38). Therefore, we can safely conclude that, by simply considering the electrophoretic flux term in our species transport model, we can easily reproduce the experimental results of band velocity of ionic species in 100 and 40 nm nanochannels.38 Khurana and Santiago39 also used this same perturbationbased model to quantify their experimental findings on analyte preconcentration using peak mode isotachophoresis (ITP). Similar to the previous case,37,38 with their species being charged (but non-reactive) as well, the species concentration equation (see eq 1) will be similarly modified and the perturbation-based analysis leads to the modified species transport equation (identical to eq 32), with the additional electrophoretic term (see eq 5 in ref 39). In the process, they obtain the theoretical results for band velocity of the analytes (similar to that described in Figure 3) that match quantitatively with the experimental results (see Figures 2 and 3a in ref 39). This implies that, just like the previous case, here too, we can trivially extend our analytical model to reproduce the experimental findings of the work by Khurana and Santiago.39 We can cite similar other studies,40,41 which use the same perturbation-based species transport model (with additional trivial modifications, depending upon the nature of the species and transport), to reproduce relevant experimental results. All of these studies can be considered as validation of the species transport model presented here. Variation of the Band Velocity. We shall first discuss the variation of the band velocity with the Damköhler number (Da0). The band velocity is the weighted average of the advection velocity experienced by the asphaltene (nonaggregated) molecules, with the weight being the transverse component of the concentration distribution [here, F(Y); see eq 15]. Hence, the band velocity is the average species velocity, averaged over the channel height, and hence, the variation in asphaltene concentration across the channel height (discussed below) will imply that the band velocity is different from the advection velocity. If there had been no variation in the asphalatene concentration across the channel height, the band velocity would have remained identical to the advection velocity. In fact, in Figure 3, we clearly show that, for a small Damköhler number, where the variation of F(Y) across the

= R + D*T (28)

where D*T is the Taylor dispersion expressed as (using eqs 12 and 23) 1/2

D*T = − Pe

∫−1/2 (U − Ub)G(Y )dY 1/2

=−

∫−1/2 F(Y )dY

Pe Da0 (I1 + I2) 2 sinh( Da0 /2) (29)

where p p p⎞ p ⎛p I1 = cosh( Da0 /2)⎜ 1 + 22 + 33 + 44 + 55 ⎟ ⎝k k k k k ⎠ q q q q ⎞ q ⎛q I2 = sinh( Da0 /2)⎜ 1 + 22 + 33 + 44 + 55 + 66 ⎟ ⎝k k k k k k ⎠

(30)

with (using d = 3/2 − Ub) p1 = 2ad −

3c cd − 6a + , 4 2

p3 = 12(cd − 6a) − 60c , q1 = 2Md + bd − 3M − q3 = 8(bd − 3M ) − 72b , q5 = − 576b ,

p2 = 12b − 4(bd − 3M ), p4 = 288b ,

3b , 2

q2 =

p5 = − 1440c ,

15c − 4ad − 3(cd − 6a), 2

q4 = 360c − 24(cd − 6a),

q6 = 2880c

(31) Calculation of A(X,T). Once we have obtained Ub and D*, we can follow Chan and Chauhan33 and Das and Chakraborty,34−36 to express the equation governing ⟨C0⟩ as

∂⟨C0⟩ ∂⟨C0⟩ ∂ 2⟨C0⟩ + uU = DyD* b ̅ ∂t ∂x ∂x 2

(32)

Therefore, for a pulse input (i.e., input at x = 0), the concentration profile at any axial location x = x0 is given by the following Gaussian profile (i.e., conditions for the x direction are identical to those used in previous studies33−36):

⟨C0⟩ =

⎡ (x − uU t )2 ⎤ Q ̅ b ⎥ exp⎢ − 0 4D*Dyt ⎥⎦ ⎢⎣ 4πDyD*t

(33)

where Q is the mass of the solute present in the pulse. Therefore, we can obtain the value of A(X = x0/S ,T = t/t0) as (see eq 12) A(X = x0/S , T = t /t0) =

1 1/2

∫−1/2 F(Y )dY =

1 1/2

∫−1/2 F(Y )dY

⎡ (x − uU t )2 ⎤ Q ̅ b ⎥ exp⎢− 0 4D*Dyt ⎥⎦ ⎢⎣ 4πDyD*t ⎡ (X − U (L /S)T )2 ⎤ 8 b ⎥ exp⎢− 4(R + D*T )T ⎦ D*T ⎣

(34)

1/2

where 8 = Q/(4π) S .

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RESULTS AND DISCUSSION In this section, results characterizing the transport, spread, and separation of the aspahltene molecules will be provided as a function of the modified Damköhler number Da0. A smaller Da0 implies that the transport and separation is dominated by the advection, whereas a larger Da0 signifies a larger importance of the reaction dynamics. However, first, we validate the 5854


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channel height is very small (see Figure 2), Ub ≈ 1; i.e., the band velocity approaches the advection velocity. There are a

in Figure 3. Both Figures 2 and 3 imply that the aggregation dynamics ensure that the non-aggregates are closer to the wall,

Figure 2. Variation of the function F(Y) = cosh((Da0)1/2Y) with Y for different values of Da0.

Figure 3. Variation of the dimensionless band velocity Ub with Da0.

leading to the lowering of the flow rate of the non-aggregated species. We can easily argue that, as Da0 → 0, there will be no variation in the transverse concentration distribution; i.e., we shall have cosh((Da0)1/2Y) = 1, leading to Ub = ∫ 1/2 −1/2(3/2 − 6Y2)dY/∫ 1/2 −1/2dY = 1. In the opposite limit, i.e., when Da0 → ∞, we have [coth((Da 0 ) 1/2 /2)/((Da0 ) 1/2 /2)] Da0 → ∞ → 2/ (Da0)1/2, and hence, eq 17 reduces to (Ub)Da0 → ∞ → −12/ Da0 + 12/(Da0)1/2 → 0. Therefore, for Da0 → 0, we have the band velocity equal to the advection velocity, and for Da0 → ∞, we have the band velocity equal to 0. Variation of the Dispersion Coefficient. The band velocity, described above, delineates the speed of propagation of the band consisting of the aggregated and non-aggregated asphaltene molecules. The transverse spread of this band (manifesting the variation in the velocities of the molecules inside the band) is provided by the Taylor dispersion coefficient D*T. To obtain D*T, we first study the variation of the function G (see Figure 4). Irrespective of the Damköhler

plethora of studies that demonstrate the manner in which a transverse distribution of the analyte (species) concentration lead to a band velocity (or species transport velocity) different from advection velocity.33−37 To obtain the Damköhler number dependence of the band velocity, we first study and interpret the variation of F(Y) across the channel for different Damköhler numbers (see Figure 2). For a given Da0, the concentration [or F(Y)] increases away from the bulk. This can be explained by analyzing the nature of the equations governing the variation of C0 [or F(Y)], i.e., eqs 10 and 11. These equations clearly show that C0 or F(Y) is obtained by solving a diffusion equation with a source term (that varies linearly with the concentration) in the presence of the no flux boundary condition. The nature of both the equation and the corresponding no flux boundary condition are such that there must be an increase in the concentration value from the channel centerline to the wall, as demonstrated in Figure 2. Also, this source term increases linearly with Da0, justifying the increase in F(Y) with Da0 at a given location away from the bulk. Physically, such an increase in the nonaggregated asphaltene molecules close to the wall can be explained from the no flux boundary condition. The no flux condition is satisfied by the balance of the diffusive flux and the reactive flux (see eq 11) of the non-aggregated species. Hence, non-zero reactive flux at the wall will imply a non-zero diffusive flux. With C0 always being positive, the reactive flux has the same sign as that of the coordinate Y. Hence, the diffusive flux (which we define as dC0/dY) is negative for Y < 0 and positive for Y > 0. Hence, the concentration of non-aggregated molecules must increase from the channel centerline to the wall. Please note that such a distinct spatial variation (increase from the bulk to the wall) in the concentration of nonaggregated molecules occurs because of the fact that we only consider a unidirectional reaction, where the non-aggregated species become depleted to form the aggregated species (and there is no reverse reaction of aggregated species decomposing into the non-aggregated species). If that had been the case (i.e., the reaction is bidirectional), this spatial profile would be decided by the ratio of the rate constants of forward (nonaggregates forming aggregates) and reverse (aggregates forming non-aggregates) reactions. The fact that, with the increase in Da0 ensuring that the concentration value has a higher magnitude at locations closer to the wall (where the advection speed is less), the band velocity (which is a weighted average of the advection velocity; see above) must decrease with an increase in Da0, as reflected

Figure 4. Variation of G(Y)/Pe with Y for different values of Da0.

number, G varies linearly with the Peclet number Pe; therefore, all G/Pe curves (see Figure 4) are independent of the Peclet number. Also, it is clearly seen that, close to the wall, G has a positive value, implying an augmentation of the concentration of the non-aggregated molecules, whereas in the bulk, G is negative, implying a lowering of the concentration of the nonaggragted molecules. This is qualitatively very similar to the finding associated with the variation of F(Y). We use this variation of G(Y) to calculate the Taylor dispersion coefficient D*T, plotted in Figure 5. Irrespective of the Damköhler number, D*T varies as Pe2 (also see the study by Chen and Chauhan33); hence, all D*T/Pe2 curves (see Figure 5) are 5855


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Figure 7. Variation of separation resolution (Rs) with Da0 for the same set of parameters as Figure 6.

Figure 5. Variation of D*T/Pe2 with Da0.

independent of the Peclet number. We find that the spread (quantified by D*T) is uniform for smaller Da0 but drastically reduces for larger Da0. This is associated with the fact that, for larger Da0, a major concentration of the non-aggregated molecules is located close to the wall, thereby signifying lesser variation (or dispersion) in the velocities of the non-aggregated asphaltene molecules. Variation of the Separation Resolution. In this subsection, we shall discuss the characterization of the asphaltene transport and migration by quantifying the efficiency of separation as a function of the Damköhler number. To do so, we first need the variation of the channel centerline concentration A(X,T) for different values of Da0. Figure 6

Rs =

Ts − Tid (Ws + Wid)/2

(35)

where Rs is the resolution for separating a species {whose Gaussian [A(X,T)] attains a peak at T = Ts and the spread of the Gaussian is W = Ws} from a species with ideal behavior (i.e., one that attains a peak at T = 0 = Tid and has zero spread; Wid = 0). The result shows that, for larger Da0, the separation resolution (wrt to the ideal species) comes down. This occurs primarily as a result of the very large dispersion (or spread) in the concentration distribution for larger Da0, because the band velocity is larger for smaller Da0, indicating a lesser value of the time difference Ts − Tid (and, hence, a smaller Rs). Physically, Figure 7 conveys a very important message for the microchannel transport of asphaltene molecules. It demonstrates that the asphaltene molecules having different rates of aggregation (despite the fact that the aggregated and non-aggregated states are identical) will be separated in a microchannel, with the separation efficiency being dictated by the difference in the rates of aggregation. Therefore, we describe a new paradigm of separation of asphaltene aggregates on the basis of the corresponding reaction rate that govern their formation. In this context, it is worthwhile to mention that the exact molecular mechanism of asphaltene aggregation is still a matter of large debate and conjecture;2,3,42 however, we can indeed (even if without very precise control) vary the rate of asphaltene aggregation by changing parameters, such as temperature, solvent, initial concentration,2,3 etc. Before concluding, we state that this model, being developed in dimensionless form, would be valid for all types of fluids at all (initial) concentrations of asphaltene for all channel radii as long as the following assumptions hold true: Asphaltene molecules should exist only in two unique states (aggregated and non-aggregated); the asphaltene concentration at any stage should be weak enough to ensure that we can neglect the asphaltene−asphaltene interactions and any alterations in the liquid viscosity (also, the liquid must continue to behave as a Newtonian liquid); the channel dimensions should be large enough relative to the asphaltene molecules, so that we can neglect the asphaltene−wall interactions and any possible disruptions in the background flow field; the channel dimensions should be such that we can employ microscalebased perturbation analysis; and finally, the non-aggregated asphaltene molecules should undergo only a unidirectional process, forming aggregates. At the moment when the system parameters (e.g., the channel dimensions, asphaltene concentrations, liquid properties, flow velocities, etc.) become such that any one or more of these assumptions are not valid, this model would no longer work. Some relevant examples of

Figure 6. Temporal variation of A(X,T) for different Da0 values for a given set of parameters (stated in the figure). We take Pe = 1.

demonstrates the temporal variation of the channel centerline concentration at a L value corresponding to the channel exit. We show results corresponding to Pe = 1. For a microchannel transport, Pe is typically small;33 therefore, we would mostly find Pe ≤ 1. Variation in Pe only affects the Taylor dispersion; therefore, at smaller Pe values, the overall dispersion [which will now be solely governed by the axial diffusion R (here, R = Dx/ Dy represents the dimensionless axial diffusion, because we assume that Dy, which is the diffusion scale, is constant)] will remain unaffected. Therefore, the plots for A(X,T) will be identical for smaller Pe. Variation of A(X,T) (see Figure 6) represents typical Gaussian behavior (eq 34). For smaller Da0, the band velocity is high, ensuring that the peak in the Gaussian profile is attained at a quicker time. The spread is primarily dictated by the flow field (and not the Taylor dispersion, because the latter is substantially small compared to R = 1; see Figure 5), therefore ensuring that the spread in A(X,T) is more for the case of larger Da0. We use this variation of A(X,T) to obtain the corresponding variation of the separation resolution as a function of Da0 (see Figure 7). The resolution characterizing the efficiency of separation (described in more details later) is defined as 5856


Energy & Fuels

Article

(23) Boek, E. S.; Wilson, A. D.; Padding, J. T.; Headen, T. F.; Crawshaw, J. P. Energy Fuels 2010, 24, 2361. (24) Lawal, K. A.; Crawshaw, J. P.; Boek, E. S.; Vesovic, V. Energy Fuels 2012, 26, 2145. (25) Ramirez-Jaramillo, E.; Lira-Galeana, C.; Manero, O. Energy Fuels 2006, 20, 1184. (26) Vargas, F. M.; Creek, J. L.; Chapman, W. G. Energy Fuels 2010, 24, 2294. (27) Porte, G.; Zhou, H.; Lazzeri, V. Langmuir 2003, 19, 40. (28) Kocabas, I.; Islam, M. R.; Modarres, H. J. Pet. Sci. Eng. 2000, 26, 19. (29) Ravi-Kumar, V. S.; Tsotsis, T. T.; Sahimi, M.; Webster, I. A. Chem. Eng. Sci. 1994, 49, 5789. (30) Worakanok, T.; Johansen, T. E.; Hawboldt, K. J. Pet. Sci. Eng. 2009, 64, 11. (31) Eskin, D.; Ratulowski, J.; Akbarzadeh, K.; Andersen, S. AIChE J. 2012, 58, 2936. (32) Eskin, D.; Ratulowski, J.; Akbarzadeh, K.; Pan, S. Can. J. Chem. Eng. 2011, 89, 421. (33) Chen, Z.; Chauhan, A. J. Colloid Interface Sci. 2005, 285, 834. (34) Das, S.; Chakraborty, S. Electrophoresis 2008, 29, 1115. (35) Das, S.; Chakraborty, S. Langmuir 2008, 24, 7704. (36) Das, S.; Chakraborty, S. Langmuir 2009, 25, 9863. (37) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772. (38) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6782. (39) Khurana, T. K.; Santiago, J. G. Anal. Chem. 2008, 80, 6300. (40) Chen, C.-H.; Lin, H.; Lele, S. K.; Santiago, J. G. J. Fluid Mech. 2005, 524, 263. (41) Garcia-Schwarz, G.; Bercovici, M.; Marshall, L. A.; Santiago, J. G. J. Fluid Mech. 2011, 679, 455. (42) Murgich, J. Pet. Sci. Technol. 2002, 20, 983. (43) Bowden, S. A.; Wilson, R.; Parnell, J.; Cooper, J. M. Lab Chip 2009, 9, 828.

experimental systems and the corresponding system parameters (described above), where this model would be likely to work, can be found in the literature.22,43

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CONCLUSION In this paper, we develop a perturbation-based analytical model to study the microchannel pressure-driven transport of asphaltene molecules in the presence of finite aggregation effects. Through closed-form analytical solutions, we demonstrate that the presence of the aggregation substantially retards the average transport (characterized by the corresponding band velocity of the asphaltene plug) of the non-aggregated species, and this results in developing a new paradigm of separating asphaltene aggregates by varying their rate of formation from identical non-aggregated moieties.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) for providing financial support to S.D. in form of the Banting Postdoctoral Fellowship.

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REFERENCES

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