Energy Fuels 2010, 24, 1088–1093 Published on Web 12/07/2009
: DOI:10.1021/ef9009444
Asphaltene Aggregates Fractal Restructuring Model, A Population Balance Approach Hassan Rahimi and Ali R. Solaimany Nazar* Chemical Engineering Department, University of Isfahan, Isfahan 81744, Iran Received August 29, 2009. Revised Manuscript Received November 15, 2009
A new population balance model is developed to predict asphaltene fractal aggregates size distribution. Asphaltene aggregates fractal dimension is increased when shear is induced to them. Fractal dimension evolution is a dynamic process. This model predicts aggregate size distribution more accurately in comparison to a previously proposed model, which considered that aggregates have a constant fractal dimension. The fractal dimension increases from an initial value of 1.6 to final steady-state values. The final value depends upon the shear rate and is described by an exponential function. This function contains some constants that are calculated from experimental data. The asphaltene size distribution of Iranian crude oil asphaltene solutions in mixtures of toluene and n-heptane are predicted by the developed model accurately.
structure), and V is the total solid volume of all particles in the aggregate. The subscript on dn indicates the integer value of dn for Euclidean geometry.4 In some cases, the subscript 3 is replaced with f or d3 = df. Sahimi et al.1 proposed 2D fractal models for asphaltene aggregates. They reported fractal dimension in the range of 1.06-2.5. As noted previously,5 one of the most important properties of asphaltene aggregates is their aggregate size distribution (ASD) and its evolution. Modeling of asphaltene aggregation using a population balance approach is a good way to predict ASD. The results of such a model can be very useful to solve asphaltene-contained process problems (e.g., in the modeling of porous media blockage or pipeline choking because of asphaltene aggregation). In a particulate process (e.g., asphaltene aggregates formation and evolution in a petroleum solution), the smallest particles are formed as nucleuses and then nucleuses or other particles collide together and make larger aggregates. These processes are termed aggregation, agglomeration, coagulation, etc. The birth of nucleuses takes place while the solution is supersaturated. The supersaturation condition also increases the particle size because of the growth phenomenon. During growth, solid mass is transferred from solution to the particle solid surface. As the aggregate size increases, it becomes more susceptible to the breakage process. During breakage, fragile aggregates are fragmented.5 The other phenomenon that is possible to take place in particulate processes is restructuring.6,7 Restructuring commonly happens for porous particles, such as asphaltene aggregates. During restructuring, aggregates become more porous or more compact. For example, shear induction in a pipeline or an annulus space between to coaxial rotating cylinders can crumple porous aggregates and decrease their sizes. Alternatively, solvent swelling into aggregates can increase their size and porosity. When the solvent swells into a fractal aggregate the distance between aggregate branches increases. This is due to the
Introduction Asphaltenes are a heavy dark fraction of bitumen, crude oil, and its cuts, which are soluble in some species, such as aromatics (benzene, toluene, etc.), but insoluble in some others, such as paraffins.1 They can exist in petroleum and oily components partly in solid deposit phase, partly in micellar form, and partly in dissolved form.2 Asphaltene causes a lot of difficulties in the petroleum industry. Flocculation and deposition of asphaltene aggregates is a serious problem in heavy petroleum production and its pursuant treatment. The formation of two separate phases in heavy oil, their conflict, and asphaltene sediment particle size distribution are complicated interrelated phenomena. Asphaltene deposition increases catalyst fouling and coke formation during petroleum treatment.3 As noted in the literature,1,4 asphaltene solid particles have a fractal-like nature. Fractal objects are characterized by fractal dimension. Fractal dimensions relate aggregate size to some property in n dimensions, where n = 1, 2, or 3, and dn is the fractal dimension in n dimensions. For example, fractal dimensions of aggregates imbedded in one, two, and three dimensions are P l d1 A l d2 V l d3
or V l df
where l is the characteristic length scale, which can be taken as either the shortest, longest, geometric mean, or equivalent radius (based on area) of an aggregate, P is the aggregate perimeter, A is the aggregate cross-sectional area (i.e., including the space between solid particles in an aggregate *To whom correspondence should be addressed. Telephone: 0098311-7934027. Fax: 0098-311-7934031. E-mail:
[email protected]. (1) Sahimi, M.; Rassamdana, H.; Dabir, B. SPE J. 1997, 2, 157–169. (2) Mansoori, G. A. Proceedings of Controlling Hydrates, Asphaltenes and Wax Conference, London, U.K., Sept 1996; Paper 2, p 27. (3) Nielsen, B. B.; Svreck, W. Y.; Mehrotra, A. K. Ind. Eng. Chem. Res. 1994, 33, 1324–1330. (4) Rahmani, N. H. G.; Dabros, T.; Masliyah, J. H. J. Colloid Interface Sci. 2005, 285, 599–608. r 2009 American Chemical Society
(5) Solaimany Nazar, A. R.; Rahimi, H. Energy Fuels 2008, 22, 3435– 3442. (6) Bouyer, D.; Line, A.; Cockx, A.; Do-Quang, Z. Chem. Eng. Res. Des. 2001, 79, 1017–1024. (7) Spicer, P. T.; Keller, W.; Pratsinis, S. E. J. Colloid Interface Sci. 1996, 184, 112–122.
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Energy Fuels 2010, 24, 1088–1093
: DOI:10.1021/ef9009444
Rahimi and Solaimany Nazar
occupation of space by the solvent. In this case, the aggregate solid body occupies less portion of space in comparison to the case before solvent swelling. Thus, the porosity of aggregates can increase. For fractal aggregates, restructuring changes the fractal dimension. Oles8 and Marsh9 showed that the fractal dimension of polystyrene latex particles increased from 2.1 at the initial times to 2.5 in the later stages of shear induction to aggregates. Jullien and Meakin10 also reported a fractal dimension increase from 1.89 to 2.13 in shear-induced suspensions. Then, fractal dimension changes can be used as a measure of restructuring. In a previously presented work,5 a fractal population balance model had been developed for the prediction of the asphaltene ASD. In that model, a constant fractal dimension of 1.6 was supposed for asphaltene aggregates, without any attention to the restructuring effects. This value of fractal dimension was reported in the literature, which made best fits of ASD prediction with experimental results. Here, another version of the population balance model is presented to predict asphaltene ASD. The restructuring effects of shear induction into the asphaltene aggregates are included. The new model gives more accurate results than the previous model in comparison to experimental data for the asphaltene aggregates of Bangestan crude (Iranian heavy oil) in the mixtures of toluene and n-heptane under shear rates in the range of 1.879-13.15 s-1.
fraction of the fragments of size v braking from the larger aggregates of size v00 . Model Description. A discretized form of eq 1 is used to determine PSD13 i -2 X dNi 1 ¼ 2j -iþ1 Ci -1, j Ni -1 Nj þ Ci -1, i -1 Ni -1 2 2 dt j ¼1
-Ni
j ¼1
imax X
Ci, j Nj -Si Ni þ
j ¼1
imax X
Γi, j Sj Nj
j ¼i þ 1
where Ni is the number of particles of characteristic size Vi in unit volume of suspension or solution (e.g., Ni = number/m3), Ci,j is the aggregation kernel that describes the rate at which particles of volumes Vi and Vj collide and coalesce, Si is the breakup rate function of aggregates of size Vi, and Γi,j is the breakup distribution function defining the volume fraction of the fragments of size Vi breaking from the larger aggregates of size Vj. Aggregation Kernel. The orthokinetic aggregation mechanisms and asphaltene fractal-like aggregate structure approach are considered in modeling. The aggregation kernel of this approach is based on the idea that all collisions form larger aggregates. The aggregation kernel is expressed as14 Ci, j ¼ Ri, j βi, j ð7Þ where Ri,j is the collision efficiency of particles of volumes Vi and Vj. As aggregation proceeds, particles grow as porous objects with highly irregular and open structures. They are usually referred to as fractal aggregates, characterized by their fractal dimension df, defined as follows:15
Theoretical Section
Mi ri df
ð8Þ
where Mi is the aggregate mass, ri is the aggregate radius, and df is the fractal dimension. The fractal dimension is a value that depends upon aggregate size, shear rate, and shear history.16 As noted in the literature,1,4 asphaltene aggregates have a fractallike structure. They have fractal dimensions in the range of df = 1.06-2.0.4,17 For a same value of aggregate mass, the higher the fractal dimension, the smaller the aggregate radius. Fractal dimension cannot be higher than 3, which represents nonporous rigid particles. The large asphaltene aggregates are formed because of collision of the individual particles and small aggregates to each other. The collision efficiency of unity appears realistic for fractal structure aggregates, such as asphaltene.16 For a fractal aggregate with mass equivalent volume of Vi, the collision diameter dc,i is related to the number of primary particles xi of diameter dp in it as follows:18
ð1Þ
where n is the particle number density of size L (or volume v) and is defined as the particle number per unit size in unit volume of suspension or solution [e.g., n = number/(m m3) or number/ (m3 m3)]. BA and DA are11 Z 1 v BA ðvÞ ¼ Cðu, v -uÞnðuÞnðv -uÞdu ð2Þ 2 0 Z ¥ DA ðvÞ ¼ nðvÞ Cðv, v0 Þnðv0 Þdv0 ð3Þ 0
dc, i ¼ dp xi 1=df
0
The function C(v, v ) is the aggregation kernel that describes the rate at which particles of volumes v and v0 collide and coalesce. BB and DB are presented as12 Z ¥ BB ðvÞ ¼ γðv, v00 ÞSðv00 Þnðv00 Þdv00 ð4Þ
ð9Þ
The collision frequency of fractal aggregates is given as follows:16 βi, j ¼ 0:31GVp ðxi 1=df þxj 1=df Þ3
v
DB ðvÞ ¼ -SðvÞnðvÞ
2j -i Ci, j Nj -Ni
ð6Þ
Basic Theory. Simultaneous aggregation and fragmentation can be modeled on the basis of the governing population balance equation (PBE). The unsteady-state model considering both the aggregation and breakage processes may predict the dynamic evolution of the asphaltene ASD in shear-induced petroleum solutions. A simplified form of PBE in a system with constant volume and without supersaturation is as follows: dnðvÞ ¼ BA -DA þ BB -DB dt
i -1 X
ð10Þ
Fragmentation Kernel. The fragmentation kernel for fractal aggregates in dilute suspensions (that viscosity of suspension is
ð5Þ
S(v) is the breakup rate function of aggregates of size v, and γ(v, v00 ) is the breakup distribution function defining the volume
(13) Hounslow, M. J.; Ryall, R. L.; Marshall, V. R. AIChE. J. 1988, 34, 1821–1832. (14) Kusters, K. A.; Wijers, J. G.; Thoenes, D. Chem. Eng. Sci. 1997, 52, 107–121. (15) Somasundaran, P.; Runkana, V. Int. J. Miner. Process 2003, 72, 33–55. (16) Barthelmes, G.; Pratsinis, S. E.; Buggisch, H. Chem. Eng. Sci. 2003, 58, 2893–2902. (17) Rastegari, K.; Svrcek, W. Y.; Yarranton, H. W. Ind. Eng. Chem. Res. 2004, 43, 6861–6870. (18) Matsoukas, T.; Friedlander, S. K. J. Colloid Interface Sci. 1991, 146, 495–506.
(8) Oles, V. J. Colloid Interface Sci. 1992, 145, 351–358. (9) Marsh, P. Effect of shear-induced breakup and restructuring on the size and structure of aggregates. Ph.D. Thesis, University of New South Wales, Sydney, Australia, 2005. (10) Jullien, R.; Meakin, P. J. Colloid Interface Sci. 1989, 127, 265– 272. (11) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes, 2nd ed.; Academic Press: New York, 1988. (12) Zhang, J.; Li, X. AIChE. J. 2003, 49, 1870–1882.
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: DOI:10.1021/ef9009444
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Figure 1. DLP asphaltene aggregate restructuring under shear rate: (a) bend aggregate and (b) aggregate restructuring.1 Figure 2. Fractal dimension trend df0 = 1.6, df¥ = 2.3, and t* = 5 min.
not affected by solid particles) is given by16 0
Si ¼ A G vp q
1=3
dc, i dp
!3=df
dimension. In this work, a dynamic time-dependent fractal dimension function is proposed as
ð11Þ
df ¼ df0 þ að1 - e-bt Þ
0
a ¼
where A and q are the fragmentation parameters, which are determined experimentally. A binary fragmentation (or aggregate splitting) model was accepted in the modeling, which as a distribution form ð12Þ Γi, j ¼ 2 ðVi ¼ Vj =2Þ 0 ðVi 6¼ Vj =2Þ
df¥
-
ð13Þ
df0
where df0 is the initial fractal dimension and df¥ is the steady-state fractal dimension. Parameter b is a constant and can be calculated from an intermediate time fractal dimension. If df* is the fractal dimension of time t*, then b is 2 !3 df 6 6 1 -d 0 7 7 1 6 f 7 7 b ¼ - log6 ¥ ð14Þ 6 df -df0 7 t 4 5
In this breakage distribution function, it is assumed that the breakage product of a particle of volume Vj is two equal volume particles. This simple model is used widely.5,9-16 Restructuring. The occurrence of the restructuring phenomenon changes the physical structure of aggregates. When shear is induced on an aggregate as an external agent, it can break the aggregate or carry it near other aggregates. Thus, they may collide together and an agglomeration process may take place. Moreover, it can bend aggregate slender branches to aggregate inside. When these branches are bent to aggregate inside, the aggregate collapses and its diameter will decrease; however, the total mass of the aggregate does not change. Then, according to eq 8, the aggregate fractal dimension should increase. Figure 1 shows a diffusion-limited particle (DLP) asphaltene aggregate. The original image (Figure 1a) is presented by Sahimi et al.1 This is a famous fractal model of asphaltene aggregates. As projected in this figure, fluid shear forces the branches of the aggregate, such as the marked branch in a, to bend it. This effect of fluid shear induction causes aggregate restructuring in b and increases its fractal dimension. As shear induction to the aggregate continues, the particle becomes more compact and its fractal dimension increases more. During the initial times, the branches of a fractal aggregate are slender and the aggregate is too vulnerable against breakage and restructuring. As the aggregate is restructured because of shear, its slender branches collapse and merge with the aggregate body. Therefore, during the later times, the aggregate potential for more restructuring and the rate of the fractal dimension change are decreased. According to the definition of fractal dimension, it cannot be higher than 3. Then, it is expected that, in an infinite limitation of time, the fractal dimension may tend to a steady-state value not higher than 3. A comparison between a previously presented model results and experimental data shows an ascending error in that constant fractal dimension model results. Then, in the case of constant and continuous shear rate orthokinetic aggregation, it is reasonable to assume a special trend for the aggregate average fractal
The trend of the fractal dimension can be fixed using three points: initial time, steady state, and intermediate time fractal dimensions. This trend can be fixed by fitting the model results on the experimental data. Figure 2 shows the fractal dimension trends for the cases with constant orthokinetic aggregation. In general, the fractal dimension depends upon several parameters, such as asphaltene and solvent composition, aggregation mechanism, and thermodynamic conditions. Then, d 0f , df*, df¥, and t* in foregoing equations depend upon these parameters. Simulated Case Study. Asphaltene solution in shear-induced suspensions was considered in the simulated case study. The experimental data of the evolution of the asphaltene ASD recently presented were used to validate the prediction of the presented model.5 In their work, asphaltene was extracted from a crude sample of a well located in the Bangestan petroleum field of Iran. The API and the asphaltene mass fraction of the crude oil extracted by n-hexane are 25.5 and 0.178, respectively. Asphaltene powder was dissolved in toluene to achieve a stock solution with a concentration of 1 g/L. Then, samples were provided by adding distinct volumes of n-heptane to a toluene-asphaltene solution. The n-heptane solvent is a precipitation agent, which causes asphaltene precipitation and aggregation in solution. Samples were prepared in different values of a toluene/ heptane volume ratio (T/H). The shear rate was induced on the samples previously poured in an annulus cell of an experimental apparatus, while the inner wall of the cell rotates mechanically. An image-processing technique was used to study the evolution of asphaltene ASD in shear-induced suspensions.5 In this method, the images of asphaltene aggregates were analyzed to determine ASD and the average size of aggregates. The values of T/H, average shear rate, and asphaltene solid particles volume fraction used in the performing of experiments are presented in Table 1. Further details about the experimental procedure are presented previously.5 The breakage parameters are considered as 0.00388 m-1 s and 2, respectively.5 1090
Energy Fuels 2010, 24, 1088–1093
: DOI:10.1021/ef9009444
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Figure 3. Aggregates number average diameter prediction with (a) constant fractal dimension and (b) increasing the fractal dimension. df decreases from case 1 to case 3. Table 1. Toluene/n-Heptane Volume Ratio, Shear Rate, and Asphaltene Solid Volume Fraction of Samples in Every Run5 volume ratio of toluene/ n-heptane (T/H)
average shear rate, G (s-1)
asphaltene solid particles volume fraction
1:3 1:4 1:5 1:6 1:9
1.879-13.15 1.879-13.15 1.879-13.15 1.879-13.15 1.879-13.15
6.3797 10-5 7.2845 10-5 9.0669 10-5 9.7136 10-5 8.0938 10-5
Figure 4. Asphaltene ASD evolution with (a) constant fractal dimension (df = 1.6) and (b) increasing the fractal dimension ¥ (df0 = 1.6, d* f = 1.75, df = 2.3, and t* = 5 min). Table 2. Fractal Dimension Trend Parameters for the Cases of Figure 3b case 1 case 2 case 3
Results and Discussion Model Results for the Systems without Breakage. Restructuring of the fractal aggregates reduces their size even without breakage. Figure 3 compares the results of original and new models for the prediction of the asphaltene aggregates number average diameter. The number average diameter is defined as iP max ni dc, i ð15Þ dav ¼ i ¼1i max P ni
d0f
df*
d¥ f
t*
1.6 1.6 1.6
1.75 1.65 1.62
2.3 2 1.8
5 5 5
As shown in Figure 3, in processes without breakage and constant fractal dimension, the number average diameter increases during the time and reaches a maximum and steady-state value, whereas for the variable fractal dimension, the number average diameter reaches a maximum and then declines to the smaller values. This means that restructuring can decrease the size of aggregates and cause the birth of smaller aggregates without breakage. This result is also shown in Figure 4 in the form of ASD. This figure presents the results of both models in the prediction of the aggregate size distribution at several times. The vertical axis shows the relative number frequency (Ni/Ntotal), which is the ratio of the number concentration of particles in the ith bin to the total number concentration of particles in all bins.19 In a constant fractal dimension case, all of the aggregate diameter distributions include only particles bigger than dc,k in the initial distribution, while smaller particles are available in the second model results. This is the effect of the restructuring phenomenon. This kind of decrease in size is not a result of breakage, and it cannot create particles
i ¼1
The population balance equation (eq 6) is solved in nonsteady conditions, and then the accumulations of particles in several discrete size intervals and aggregate size distributions are obtained. Figure 3 shows the number average of ASDs. In these case studies, breakage is neglected to show the effect of restructuring separately. Table 2 indicates the parameters of Figure 3b cases as eqs 13 and 14. Figure 3a shows the results of the original model for constant values of the fractal dimension, whereas Figure 3b presents the results of the new model for three other cases, in which df increases, as shown in eq 13.
(19) Rahmani, N. H. G.; Dabros, T.; Masliyah, J. H. Chem. Eng. Sci. 2004, 59, 685–697.
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: DOI:10.1021/ef9009444
Rahimi and Solaimany Nazar
Figure 5. Number average diameter prediction by the model (df0 = -1 1.6, d ¥ f = 2.5, t* = 5 min, dav,0 = 114.76 μm, and G = 3.757 s ).
Figure 7. Asphaltene number average diameter prediction by the original model (constant fractal dimension, df = 1.6) and the new model (df0 = 1.6, df* = 1.75, df¥ = 2.3, and t* = 5 min). In both cases, T/H = 1:9, dav,0 = 114.76 μm, and G = 5.635 s-1.
the original model, and the experimental results of Bangestan oil asphaltene aggregates. This new model predicts the number average diameter accurately by accounting for the restructuring effects. Figure 8 presents the asphaltene ASD evolution. The model predicts experimental data with good accuracy. Figure 9 shows the third moment evolution. The third moment is defined as m3 ¼ kv
imax X π i ¼1
Figure 6. Asphaltene number average diameter prediction by the original model (constant fractal dimension, df = 1.6) and the new model (df0 = 1.6, df* = 1.75, d ¥ f = 2.3, and t* = 5 min). In both cases, T/H = 1:4, dav,0 = 114.76 μm, and G = 5.635 s-1.
0
dc3, i Ni
ð17Þ
where kv is a constant and is called the third moment shape factor.11 This curve shows that the total volume of asphaltene aggregates (i.e., collision volume and not mass equivalent volume) can reach a maximum value several times higher than its initial amount and its steady-state value can be smaller than its initial value. As mentioned previously,5 the porosity and then the density of fractal aggregates depend upon their size. Bigger aggregates have higher porosity and less density. Because aggregate sets have size distribution, they also have porosity and density distribution even in the constant fractal dimension model. Shear induction to aggregates changes their density and porosity distribution similar to the size distribution dynamically, and then the third moment of aggregates changes. According to the model results in Figure 9, shear induction first increases the size of aggregates and then the size-dependent porosity increases. This means that aggregates occupy the bigger portion of the space in suspension and the third moment ratio increases to its initial value. When the size of aggregates declines again to smaller values beyond a maximum, their porosity decreases and, hence, the occupied space by them descends. Alternatively, the third moment descends again and finally approaches a steady value similar to the aggregates average size. In the case of restructuring, higher values of the fractal dimension cause smaller sizes and smaller porosities in comparison to a constant fractal dimension case. Then, the third moment trend for the restructuring case lies under the constant fractal dimension one. The new model results in an aggregates total volume ratio to its initial amount less than 0.065 for time 60 min. This ratio was 0.464 for the original model. Figure 9 shows that
smaller than a limit size. This limit size is achieved when particles are crumpled completely and the fractal dimension meets 3. The limit diameter is dc¥, k ¼ dc0, k xk ð1=3 -1=df Þ
6
ð16Þ
An increase of df decreases the aggregate size because of restructuring, decreases the aggregation kernel, and then prevents aggregate size growth. This result is deduced from Figure 4 and eq 10. Effect of Restructuring on Aggregation Fragmentation Processes. Figure 5 shows model results in the prediction of the asphaltene number average diameter in three different cases. In all cases, all parameters are the same unless df*. In all cases, the final fractal dimension is 2.5 but some trends contain a more rapid increase of df. An increase of df decreases the aggregation and fragmentation rates, and then aggregation trends are reached later and with smaller maxima. After the maximum average size time, the breakage process becomes more significant than aggregation and then the average diameter declines to smaller values. For the higher values of df*, aggregates become more compact sooner. These new and stronger aggregates are less susceptible to breakage. Then, in these cases, the average size declines to the bigger sizes and these suspensions include a fewer number of particles, because they include the bigger and more compact aggregates. Comparison of New and Original Model Results to Experimental Data. Figures 6 and 7 present the results of this model, 1092
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: DOI:10.1021/ef9009444
Rahimi and Solaimany Nazar
Figure 8. Asphaltene ASD prediction by the original model (constant fractal dimension, df = 1.6, presented by dashed lines) and the new model (d0f = 1.6, df* = 1.75, df¥ = 2.3, and t* = 5 min, presented by solid lines). In both cases, T/H = 1:9, dav,0 = 114.76 μm, and G = 3.757 s-1.
BB = rate of birth because of breakage Ci,j = aggregation kernel of particles of diameters Vi and Vj DA = rate of death because of aggregation DB = rate of death because of breakage dc,i = collision diameter of aggregates in bin i df = fractal dimension d0f = initial fractal dimension d¥ f = steady-state fractal dimension di = diameter of rigid sphere mass equivalent aggregates in bin i dp = diameter of a spherical primary particle (μm) G = shear rate (s-1) imax = number of classes or bins kv = third moment shape factor l = aggregate characteristic length m3 = third moment n = particle number density Ni = number concentration of aggregates in bin i having a characteristic volume Vi (m3) Ntotal = total number concentration of particles in all bins (m-3) q = fragmentation parameter ri = aggregate radius Si = fragmentation or breakup rate of aggregates of size i (s-1) t = time (s) v = volume of particle (m3) Vfa = mean mass equivalent volume Vi = mean characteristic mass equivalent particle volume of bin i (m3) vp = volume of a spherical primary particle (m3) xi = sectional spacing
Figure 9. Asphaltene aggregates third moment prediction by the original model (constant fractal dimension, df = 1.6) and the new model (d0f = 1.6, d*f = 1.75, d¥ f = 2.3, and t* = 5 min). In both cases, T/H = 1:9, dav,0 = 114.76 μm, and G = 5.635 s-1.
restructuring can decrease the aggregates total volume seriously. Conclusions Modeling of the asphaltene aggregate size distribution and its evolution is very important in petroleum industry studies. The presented model can predict asphaltene ASD and average size variations with good accuracy. Aggregates fractal dimension changes during shear induction, and its variations can be described by a time-dependent function. The parameters of the restructuring model (a and b) may depend upon some parameters such as asphaltene and media composition, shear rate, and thermodynamic conditions. In this work, the effects of these parameters could not be found on the parameters of the restructuring model. However, the effect of restructuring on the asphaltene aggregate size evolution is approved.
Greek Letters Ri,j = collision efficiency of particles of diameters Vi and Vj or the fraction of collisions that result in aggregation βi,j = collision frequency of particles of diameters Vi and Vj Γi,j = breakup distribution function γ(v,v0 ) = breakup distribution function defining the volume fraction of the fragments of size v originating from v0 -sized particles
Nomenclature 0
A = fragmentation parameter a = restructuring model parameter ASD = aggregate size distribution BA = rate of birth because of aggregation b = restructuring model parameter 1093
