Dynamic Determination of Asphaltene Aggregate Size Distribution in

Energy & Fuels 2008, 22, 3435–3442

3435

Dynamic Determination of Asphaltene Aggregate Size Distribution in Shear Induced Organic Solvents Ali Reza Solaimany-Nazar* and Hassan Rahimi Department of Chemical Engineering, UniVersity of Isfahan, Hezarjarib Street, Isfahan, Iran ReceiVed March 11, 2008. ReVised Manuscript ReceiVed May 14, 2008

A population balance model was developed to determine the evolution of crude oil asphaltene aggregate size distribution. The model predictions have good accuracy in comparison with experimental results that were obtained by the image processing method. The fractal nature of asphaltene aggregates was attended assuming fractal dimension equal 1.6. Effects of shear rate, solvent composition, and initial particle size were studied on the evolution of number average diameter. The average diameter of particles reaches a maximum value and then declines to a steady state size. Shear rate and initial average diameter affect the maximum diameter and its time, but initial size has no effect on the steady state size. The combination of the image processing method and fractal approach determines the asphaltene solid content of samples. Asphaltene was extracted from Iranian heavy crude petroleum using toluene and normal heptane.

Introduction Asphaltenes are black or dark brown solids that are found in most kinds of crude oil in solved and solid form.1 Asphaltene is defined as the heavy fraction of a petroleum mixture, which is insoluble in some species such as paraffins but soluble in others such as aromatics (benzene, toluene, etc.).2 Flocculation and deposition of asphaltene aggregates is a serious problem in heavy petroleum production and its pursuant treatment. Formation of two separate phases in heavy oil, their repugnance, and asphaltene sediment particle size distribution are interrelated phenomena which are not identified completely yet. Asphaltene deposition increases catalyst fouling and coke formation during petroleum treatment.1 Asphaltene particles are believed to exist in oil partly dissolved and partly in colloidal and/or micellar form. Whether the asphaltene particles are dissolved in crude oil, in the steric colloidal state, or in micellar form depends, to a large extent, on the presence of other particles (paraffins, aromatics, resins, etc.) in the crude oil.3 One of the most important things that determines how asphaltene foregoes problems is its aggregate size distribution. However, asphaltene aggregate size distribution is an important factor in petroleum pipeline fouling and petroleum fluid flow patterns in a hydrocarbonic reservoir. There are many other industrial cases in which particle size distributions (PSD) and their evolutions play important roles in process conditions. For example in liquid-liquid extraction, polymerization, fluidized bed, and water treatment, PSD affect process designation. Therefore PSD prediction can be very useful in decreasing process costs; especially in petroleum industry asphaltene aggregate size distribution (ASD) prediction is very useful in production, EOR, transfer, and refining studies. In a solution including solid species when solid deposition conditions are provided, four main processes take place. These * To whom correspondence should be addressed. Tel: 0098-311-7934027. Fax: 00983117934031. E-mail address: [email protected]. (1) Nielsen, B. B.; Svreck, W. Y.; Mehrotra, A. K. Ind. Eng. Chem. Res. 1994, 33, 1324–1330. (2) Sahimi, M.; Rassamdana, H.; Dabir, B. SPE J. 1997, 2, 157–169. (3) Mansoori, G. A. Proceedings of Controlling Hydrates, Asphaltenes and Wax Conference, London, Sept 1996; IBC U.K. Conferences, Ltd.: Gilmore House, London, 1996, Paper No. 2, 27 pages.

processes are nucleation, growth, aggregation, and fragmentation. When solution becomes supersaturated, the nucleation process initiates. As soon as nuclei are formed, growth takes place. During this process the mass of dissolved solid (e.g., asphaltene) is transferred from liquid phase to the solid nuclei and solid particle surfaces and therefore sediments to solid particle surfaces.4 Nucleation increases the number of particles whereas growth increases the size of particles. Nucleation and growth go on while chemical potential of dissolved species in the liquid phase is higher than its amount in the solid phase. In other words supersaturation is a prerequisite of nucleation and growth. When particles are formed, some of them collide with other ones and stick to them and bigger particles are formed. This trend is called aggregation.5 Other names of this process are agglomeration, agglutination, coagulation, or flocculation.6 One of the common methods for modeling of particulate processes to predict PSD and its evolution is population balance modeling. There are a lot of developments of population balance models for determining PSD in literature.5–9 In this work a population balance model is used to predict asphaltene aggregate size distribution. Then the results are compared with experimental results for Bangestan heavy crude oil (Iranian petroleum) asphaltene. We considered asphaltene particles as fractal aggregates. Population Balance Theory Basic Theory. A population balance equation (PBE) is a governing equation of the particle size distribution. It is usually (4) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes; Academic Press, San Diego, 1971. (5) Hounslow, M. J.; Ryall, R. L.; Marshall, V. R. AIChE J. 1988, 34, 1821–1832. (6) Rahmani, N. H. G.; Dabros, T.; Masliyah, J. H. Chem. Eng. Sci. 2004, 59, 685–697. (7) Hounslow, M. J. AIChE J. 1990, 36, 106–116. (8) Litster, J. D.; Smit, D. J.; Hounslow, M. J. AIChE J. 1995, 41, 591– 603. (9) Somasundaran, P.; Runkana, V. Int. J. Miner. Process. 2003, 72, 33–55.

10.1021/ef800173s CCC: $40.75  2008 American Chemical Society Published on Web 07/01/2008

3436 Energy & Fuels, Vol. 22, No. 5, 2008

Solaimany-Nazar and Rahimi

expressed in terms of birth of nuclei (BN), their growth (Gr), and birth (BA, B,B) or death (D,A, D,B) caused by aggregation and breakage, respectively, as follows:10

Then Vi ) xiVp where Vp is the smallest characteristic volume and xi)2(i-1) is the sectional spacing and Vp ) π/6dp3. Then eq 2 also can be written in the following form:5

∂ ∂n + (nGr) ) BNδ(L - L0) + BA - DA + BB - DB (1) ∂t ∂L where n is the particle number density of size L (or volume V) and is defined as particle number per unit size in unit volume of suspension or solution (e.g., n: no/(m · m3) or no/(m3 · m3)). The kronecker delta, δ(L - L0), means that nucleation takes place only at a minimum nucleus size L0 (or volume V0). Because nucleation and growth only happen in supersaturated systems;, in a system with constant volume and without super saturation, the PBE is simplified as the following:

dNi i-2 j-i+1 1 ) 2 Ci-1,jNi-1Nj + Ci-1,i-1Ni-12 dt 2 j)1

dn(V) ) BA - DA + BB - DB dt Where BA and DA are defined:4 BA(V) )

1 2

∫

V

C(u, V - u) n(u) n(V - u) du

0

DA(V) ) n(V)

∫

∞

0

C(V, V′) n(V′) dV′

(2)

∫

∞

V

γ(V, V ″ ) S(V ″ ) n(V ″ ) dV″

DB(V) ) -S(V) n(V)

(3) (4)

(5) (6)

S(V) is the breakup rate function of aggregates of size V, and γ(V, V′′) is the breakup distribution function defining the volume fraction of the fragments of size V braking from the larger aggregates of size V′′. The analytical PBE solution is very difficult but numerically possible by characterizing particles in discrete particle sizes. These discrete values cover the whole range of size according to a geometric pattern. The particle size distribution sometimes is expressed as the particle number (Ni) that is the number of particles of characteristic size Vi in unit volume of suspension or solution (e.g., Ni: no/m3). A simple relation between the particle number and the particle number density can be written as Ni )

∫

Vi+1

Vi

n dV

(7)

PSD cover a wide size range; then a discrete PBE contains a large number of discrete values and may need a long computation time. In discretization of the particle size domain two prerequisites must be respected. First, the volume conservation law must be respected, and second, there must be appropriate discrete values in size domain for breakup and aggregation products. The first prerequisite means when aggregation between two particles of volumes Vi and Vj produces an aggregate of volume Vk, we must have Vk ) Vi + Vj

(8)

The size range is discretized as follows:5 Vi+1 )2 Vi

(9)

where Vi is the characteristic mass-equivalent particle volume which is the average of sizes in the corresponding bin Vi )

Vi-1 + Vi 2

imax

i-1

Ni

∑ j)1

2j-iCi,jNj - Ni

∑

imax

Ci,jNj - SiNi +

j)1

∑Γ

i,jSjNj

(11)

j)i+1

where 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. Model Description

The function C(V, V′) is the aggregation kernel that describes the rate at which particles of volumes V and V′ collide and coalesce. BB and DB are presented as follows:11 BB(V) )

∑

(10)

Aggregation Kernel. Aggregation is caused due to collision between smaller particles. When two or more particles collide together and coalesce, a bigger particle is created. Except for concentrated suspensions, simultaneous collision between three or more particles is infrequent and can be ignored.6 Then the aggregation kernel is related to the collision frequency of particles. There are three main mechanisms presented in literature for collision frequency and aggregation of particles. These mechanisms are perikinetic aggregation, orthokinetic aggregation, and aggregation due to differential settling.6,9,12 The first one is due to Brownian motion (thermal motion). The orthokinetic one occurs due to fluid shear. Here, only particles of at least micrometer size are considered where the influence of thermal motion is negligible in comparison with shear effect.12 The aggregation due to differential settling happens when particles settle with different settling velocities. It is important only in systems in which particles are allowed to settle. In this work the effect of only orthokinetic aggregation is considered. Until recently, aggregation models assumed particles as rigid and nonporous spheres. In these classical models, particles were assumed to move through rectilinear streams and collide together. So, the aggregation kernel is equal to collision frequency of particles.13,14 The collision frequency for aggregation due to fluid shear from the rectilinear model is calculated using the relation:15 βi,j ) 0.31G(Vi1/3 + Vj1/3)3 ) 0.31GVP(xi1/3 + xj1/3)3

(12)

where βi,j is collision frequency of particles of diameters Vi and Vj. Results of these models have large differences with experimental results. In fact aggregates are not rigid spheres. 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 mass fractal dimension df, defined as following:9 MiRridf

(13)

where Mi is aggregate mass and ri is aggregate radius. df is fractal dimension. The fractal dimension is a constant value that may depend on aggregate size, shear rate, and shear history.12 (10) Lim, Y. I.; Lann, J. M. L.; Meyer, X. M.; Joulia, X.; Lee, G.; Yoon, E. S. Chem. Eng. Sci. 2002, 57, 3715–3732. (11) Zhang, J.; Li, X. AIChE J. 2003, 49, 1870–1882. (12) Barthelmes, G.; Pratsinis, S. E.; Buggisch, H. Chem. Eng. Sci. 2003, 58, 2893–2902. (13) Smoluchowski, M. V. Phys. Z. 1916, 17, 557–571. (14) Smoluchowski, M. V. Z. Phys. Chem. 1917, 92, 129–168. (15) Saffman, P.; Turner, J. J. Fluid Mech. 1956, 1, 16–30.

Asphaltene Aggregate Size in Organic SolVents

Energy & Fuels, Vol. 22, No. 5, 2008 3437

As noted in the literature, asphaltene aggregates are permeable and have fractal-like structure. They have fractal dimensions in the range of df ) 1.06-2.0.16–18 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. There are two main approaches used in literature to modify aggregation kernels for permeable aggregates. In the first approach,19 essentially applicable for orthokinetic aggregation, the aggregation kernel is proportional to collision frequency of aggregates given by eq 12, with a correction factor called collision efficiency. This factor is equal to the fraction of collisions between particles when corresponding particles coalesce. Then the aggregation kernel is defined as Ci,j ) Ri,jβi,j

(14)

where Ri,j is the collision efficiency of particles of volumes Vi and Vj. The collision efficiency of orthokinetic aggregation is computed using a semiempirical formula as follows:20 Rtr ) fR(λR)CA0.18

(15)

Where Rtr is orthokinetic collision efficiency, fR is the retardation function, and λR is the retardation parameter. The parameter CA was termed flow number and is the ratio of attractive van der Waals forces to hydrodynamic forces. The above correlation has been obtained from the trajectory analysis for rigid spheres.20 The same correlation has been utilized, but the flow number is computed assuming the aggregates with a composite structure, consisting of a hard core surrounded by a permeable shell.19 One of the disadvantages of this model is that it employs an empirical collision efficiency relationship that was actually obtained for the interaction of hard spheres in the absence of repulsive forces.9 Asphaltene aggregates are assumed as porous particles with constant porosity.6 They used the collision efficiency approach, but they assumed the collision efficiency equal to unity for all collisions. In the second approach, aggregates of fractal nature influences their collision frequency. 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 and is defined as21 dc,i ) dPxi1/df

(16)

As shown in eq 16 fractal aggregates have higher collision diameter than their mass equivalent compact particles (where df ) 3). The collision frequency of fractal aggregates is given by replacing fractal sizes with Vi and Vj in eq 12 through the following relation:12 βi,j ) 0.31GVP(xi1/df + xj1/df)3

(17)

This approach is used in this work, and it is assumed that all collisions are successful. This assumption of collision efficiency (16) Rastegari, K.; Svrcek, W. Y.; Yarranton, H. W. Ind. Eng. Chem. Res. 2004, 43, 6861–6870. (17) Rahmani, N. H. G.; Dabros, T.; Masliyah, J. H. J. Colloid Interface Sci. 2005, 285, 599–608. (18) Rahmani, N. H. G.; Dabros, T.; Masliyah, J. H. Energy Fuels 2005, 19, 1099–1108. (19) Kusters, K. A.; Wijers, J. G.; Thoenes, D. Chem. Eng. Sci. 1997, 52, 107–121. (20) van de Ven, T. G. M.; Mason, S. G. Colloid Polym. Sci. 1977, 255, 468–479. (21) Matsoukas, T.; Friedlander, S. K. J. Colloid Interface Sci. 1991, 146, 495–506.

of unity appears realistic for fractal structure aggregates such as asphaltene.12 Fragmentation Kernel and Its Distribution Function. The aggregate fragmentation is mainly due to hydrodynamic stresses, like fluid shear.11 The main fragmentation mechanisms are (a) erosion of primary particles or small fragments from the aggregate surface and (b) “swollen deformation” rupture or splitting of the aggregates.6 The breakage rate function can be written as a function of the shear rate, G, and the aggregate size in volume Vi as follows:22 Si ) A ′ GqVi1/3

(18)

where A′ and q are the fragmentation parameters which are determined experimentally. As disrupting shear forces are acting on the collision diameter, the mass equivalent diameter, di, is replaced by the collision diameter, dc,i, in eq 18. Then, aggregates with identical collision diameter but lower fractal dimension are more porous and susceptible for breakage. The fragmentation kernel for fractal aggregates in dilute suspensions (that viscosity of suspension is not affected by solid particles) is given by12

( )

Si ) A ′ GqVp1/3

dc,i dp

3/df

(19)

This correlation is used in this work. There are three distinct fragmentation distribution functions, binary, ternary, and normal distribution, that have been used to describe the volume fraction of the fragments of size Vi breaking from the larger aggregates of size Vj.11,23,24 Binary fragmentation (or aggregate splitting), means the breaking of an aggregate into two equal fragments, which has a distribution form as Γi,j )

{

2 (Vi ) Vj ⁄ 2) 0 (Vi * Vj ⁄ 2)

(20)

Ternary breakage describes breakage of an aggregate into two equal fragments and one of the fragments breaking further into two equal and smaller species. This distribution function can be written as

{

(Vi ) Vj ⁄ 4) 2 (Vi ) Vj ⁄ 2) Γi,j ) 1 0 (Vi * Vj ⁄ 4, Vj ⁄ 2)

(21)

Normal breakage produces a normal-size distribution of the fragments that can be described as Γi,j )

Vj Vi

[

(V - Vfa) 1 exp Vi)1 2σf2 σf√2π

∫

Vi

2

]

dV

(22)

where Vfa is the mean mass equivalent volume and σf is the standard deviation of the fragment size distribution. In some cases only binary fragmentation is considered.6,12 Here also we considered only the binary fragmentation model. Finally the main assumptions of the presented mathematical model are summarized in Table 1. Experimental Section Materials. Asphaltene used in this study was extracted from a sample of a well located in Bangestan petroleum field of Iran. This (22) Pandya, J. D.; Spielman, L. A. Chem. Eng. Sci. 1983, 38, 1983– 1992. (23) Chen, W.; Fisher, R. R.; Berg, J. C. Chem. Eng. Sci. 1990, 45, 3003–3006. (24) Coulaloglou, C. A.; Tavlarides, L. L. Chem. Eng. Sci. 1977, 32, 1289–1297.



Table 1. Assumptions of the Population Balance Model binary initial distribution as initial conditions fractal dimension of 1.6 aggregation and fragmentation due to hydrodynamic shear only negligence of nucleation and growth effects binary collision between particles binary breakup distribution function fractal approach in discretization of aggregate size domain and determination of aggregation and fragmentation kernels

Table 2. Some of the Properties of Bangestan Crude Oil and Its Asphaltene and Resin27 property

value

asphaltene mass fraction (extracted by n-hexane) resin mass fraction aromatic hydrocarbon mass fraction aliphatic hydrocarbon mass fraction

17.8 2.4 19.7 30.5

Asphaltene N mass percentage C mass percentage H mass percentage

0.5 82.4 7.6 Resin

N mass percentage C mass percentage H mass percentage asphaltene aromaticity

2.6 82.1 9.2 0.16

petroleum sample was provided by national Iranian oil company (NIOC). Toluene and n-heptane (hereafter referenced simply as heptane) were high performance liquid chromatography (HPLC) grade, supplied by Merck Company. Asphaltene Solid Preparation and Analysis. Initially, residue of crude oil was extracted according to the procedure ASTM-D86.25 Then asphaltene which is insoluble in heptane but soluble in toluene was extracted according to ASTM-D6560-00.26 To obtain petroleum residue, first 300 cm3 of oil was poured in a glass vessel and was heated to boiling, and it was distilled in atmospheric pressure. As distillation was going on, the oil temperature increased. When the temperature reached 260 °C, heating was stopped, and the sample was used as residue to extract asphaltene. Then 10 mL of residue was poured in a vessel, and 300 mL of heptane was added to the vessel. After that, the sample was boiled under reflux for 60 ( 5 min. The vessel and its contents were removed from the other parts and were cooled down. Then it was put in a dark cupboard for 150 min. Afterward, the sample was decanted into a filter paper funnel without agitation. The residue was transferred completely. After filtration, the filter paper was removed and placed in a reflux extractor. Then it was refluxed with heptane for 60 min. After that, the extraction went on using 30 mL of toluene instead of heptane. The content of the vessel that was a solution of asphaltene in toluene was evaporated under a hood. Then the sample was dried in an oven for 30 min at 110 °C. The remained solid that adhered to the evaporation vessel was removed. This is resin free asphaltene. Finally, asphaltene powder was used to prepare samples for experiments. Table 2 shows some information about Bangestan crude oil analysis and its asphaltene and resin. 27 Apparatus. Figure 1 shows the schematic diagram of the apparatus used in this study. This device was used to study the evolution of asphaltene aggregate size distribution in shear induced suspensions. The main part of the device comprises two coaxis cylinders. The inner one can rotate by means of an electromotor. The outer cylinder is made of glass and is static. There is a small gap between two cylinders, hd (where h is the gap and d is the inner cylinder diameter). The samples were poured in the gap. When the electromotor runs, rotation of inner cylinder induces a velocity (25) Annual Book of ASTM Standards; American Society for Testing and Materials: Philadelphia, PA, 2005; Standard No. D86-01, Vol. 05.01. (26) Annual Book of ASTM Standards; American Society for Testing and Materials: Philadelphia, PA, 2005: Standard No. D6560-00, Vol. 05.03. (27) Solaimany Nazar, A. R.; Bayandory, L. Iran. J. Chem. Eng. 2007, 4, 3–13.

Figure 1. Schematic diagram of the experimental setup. (1) Driving motor. (2) Gear box. (3) Inner cylinder. (4) Microscope. (5) CCD camera. (6) Cold light source. (7) Outer cylinder. (8) Computer. Table 3. Toluene to Heptane Volume Ratio, Shear Rate, and Asphaltene Solid Volume Fraction of Samples in Each Run volume ratio of toluene to heptane (T:H)

average shear rate, G (s-1)

asphaltene solid particles volume fraction

1:3 1:4 1:5 1:6 1:9

1.879s13.15 1.879s13.15 1.879s13.15 1.879s13.15 1.879s13.15

6.3797 7.2845 9.0669 9.7136 8.0938

× × × × ×

10-5 10-5 10-5 10-5 10-5

gradient in the fluid in the gap. Because of small gap length in comparison with the diameter of the inner cylinder, a constant velocity gradient is reasonable across the gap.6 The inner cylinder has a diameter of 50 mm and is made of Teflon. The white color of Teflon provides a good optical reflection. Also, the solvent of the sample cannot swell the Teflon. The outer cylinder can be assembled with diameters of 60 and 65 mm. The amount of shear rate was adjusted by means of setting the revolutions of motor per minute on proper values. A charge coupled device (CCD) camera (Labomed Ltd.) captured images of samples through a microscope with a resolution of up to 1.3 megapixels. Asphaltene aggregates with various sizes and shapes are visible in images. The resolution of images was controlled with a laptop computer. Then images were saved in the computer. The operator can capture video films also. Images and videos were taken using Digi Pro 4.0 (Labomed Ltd.) software. A cold light source was used to lighten the scene of the microscope. Samples of volume 80 cm3 were poured in the gap when a smaller glass cylinder was used, and samples of volume 160 cm3 were poured for bigger glass cylinders. The geometrical properties of asphaltene aggregates from images were calculated from image analysis using Sigma Scan Pro 5 software. Sample Preparation and Test at Different Shear Rates. Asphaltene powder was solved in toluene to achieve a stock solution with concentration of 1 g/L. Then samples were provided adding distinct volumes of heptane to toluene-asphaltene solution. Samples were prepared in different values of toluene-heptane volume ratio (T:H). Then samples were poured in the cell, and shear stress was induced on them. First samples were mixed in a high shear rate for 10 min to achieve a nonsupersaturated state. Then the motor speed was fixed on desirable rates and images were captured every 2 min. In each run the first image was used as the initial condition. Table 3 lists the values of toluene to heptane volume ratio, shear rate, and asphaltene solid volume fraction of samples in each run.

Results and Discussions Estimation of Asphaltene Solid Volume Fraction. Asphaltene solid volume fraction, φs, was calculated from image analysis. For an aggregate with collision diameter of dc,i (i.e., visible diameter of aggregate and obtained from image), the mass equivalent rigid sphere aggregate diameter, di, is di ) dc,ixi(1⁄3-1⁄df) Then,

(23)


di ) dc,i

( ) dc,i dp


(df/3)-1

(24)

The volume of sample (V), whose image has been analyzed, was calculated from multiplication of the image area by the gap between cylinders. Thus, n

∑ π6 d

3

i

φs )

k)1

(25) V where n is the total number of particles in the image. The value of φs shows the amount of precipitated asphaltene and was used to determine initial condition in the population balance model running. In this work the fractal dimension of asphaltene aggregates was selected as 1.6.16 This value of df gives same φs from various images of one sample under various shear rates. The experimental system can detect particles bigger than 20 µm, so the dp is assumed to be 20 µm. From a fractal point of view, aggregates have various porosities, ε. The higher diameter results in the higher porosity. εi )

dc,i3 - di3 3

dc,i

)1-

( ) dc,i dp

(df/3)-1

(26)

Figure 2 shows the aggregate porosity versus its diameter. Figure 3 presents asphaltene solid volume fraction, φs, versus heptane to toluene volume ratio. Increasing heptane to toluene volume proportion increases asphaltene solid, but because of total solution volume magnification, φs cannot increase higher than a maximum. Determination of Initial Conditions. The initial conditions of PBE were extracted from the first image experimentally. The initial number average diameter was obtained from this image. In PBE solution, it is assumed that all of particles at time zero have a diameter equal to experimental initial number average diameter (dav,0). This average diameter may not be equal to any discrete collision diameter in the particle size domain. Then the initial condition was assumed as a binary distribution of asphaltene aggregates, some of which have a diameter less than dav,0 and some others higher than dav,0. In this initial distribution determination, two limitations should be respected: (a) total mass equivalent volume of aggregates must be equal to asphaltene solid volume and (b) number mean diameter of particles must be equal to dav,0. If dc,k e dav,0 e dc,k+1, we have

(

6φs 1 Nk+1.0 )

2-

)

dav,0 ⁄ (πdk3) dc,k

dav,0 - 21/df dc,k

( )

dav,0 - 21/df dc,k Nk,0 ) Nk+1.0 dav,0 1dc,k

(27)

(28)

where Nk+1,0 and Nk,0 are numbers of initial particles of sizes dc,k+1 and dc,k respectively. The number average diameter of aggregates is defined as follows: imax

∑nd

i c,i

dav )

i)1 imax

(29)

∑n

i

i)1

Figure 2. Aggregate porosity vs aggregate diameter (df ) 1.6 and dp ) 20 µm).

Figure 3. Asphaltene solid volume fraction vs heptane to toluene volume ratio.

Figure 4. Asphaltene number average diameter evolution for T:H ) 1:4 under different shear rates.

where ni is the number of particles counted in ith compartment. In this work the discretized PBE solved for imax ) 15 and df ) 1.6, foregoing initial conditions using a fourth-order Runge-Kutta method. Effect of Shear Rate on Asphaltene Aggregate Size Distribution. All of the experimental data were obtained using an image processing method. This technique is an in situ and nonintrusive technique.6 Figure 4 presents asphaltene number average diameter evolution under different shear rates. This figure exhibits the trend of asphaltene particle aggregation kinetics. Figure 5 depicts the steady state aggregate size versus shear rate. Figures 4 and 5 show the effect of shear rate on average aggregate diameter steady state value and its trend. Figure 6 shows a good agreement between the population balance model and the experimental results. As shear is induced to the aggregates, the aggregation and fragmentation processes begin. Initially, the aggregation is dominant and the average diameter increases to a maximum amount, where the fragmentation rate becomes so high it can inhibit further particle size growth. Afterward, mean aggregate size declines to a steady state size.



Figure 7. Evolution of third moment to its initial value ratio for T:H ) 1:9 and G ) 3.7572 (1/s). Figure 5. Steady state asphaltene aggregate number average diameter vs shear rate for T:H ) 1:4.

Figure 8. Model results for evolution of aggregate diameter distribution for T:H ) 1:9 and ) 3.7571 (1/s). Figure 6. Evolution of asphaltene aggregate average diameter for T:H ) 1:4 and G ) 7.5143 (1/s).

In some cases it is supposed as the effect of aggregates restructuring.28,29 From this point of view, under shear, large open aggregates become more compact as the fragments. These new particles have more sturdy and stable structures. In this approach the porosity of a final aggregate is less than its equally sized initial particle. In other words, aggregates have time-dependent porosity. A same evolution of the mean aggregate diameter was obtained,6 but they assumed aggregate porosity as a function of shear rate but not time. The fractal population balance model that is used in this work predicts the behavior of observation of a maximum size before declining to the steady state mean size, without taking in restructuring assumptions. Figure 6 compares the results of the model and experiments for T:H ) 1:4 and G ) 7.5143 s-1. As shown in the figure, the model predicts experimental results with a good accuracy. Another result that can be obtained from the model and Figure 4 is that higher shear rates result in maximum sizes at earlier times. Figure 7 shows the third moment evolution for T:H ) 1:9 and G ) 3.7571 s-1. The third moment is defined as below: imax

m3 ) kv

∑ π6 d

3

c,i

Ni

(30)

i)1

where kv is a constant and is called the third moment shape factor.4 This curve shows that total volume of asphaltene aggregates (i.e., collision volume not mass equivalent volume) can reach to a maximum value 11 times higher than its initial amount, and its steady state value can be smaller than its initial value. In a recent approach which used the fractal model with (28) Bouyer, D.; Line, A.; Cockx, A.; Do-Quang, Z. Trans. Inst. Chem. Eng., Part A 2001, 79, 1017–1024. (29) Spicer, P. T.; Pratsinis, S. E. Water Res. 1996, 30, 1048–1056.

df ) 3 the third moment remains constant always.6 As shown in Figures 4 and 5, higher shear rates result in smaller steady state sizes. Figure 8 depicts the evolution of aggregate size distribution from the model. The vertical axis shows the relative number frequency (Ni/Ntotal), which is the ratio of number of concentration of particles in the ith bin to total number concentration of particles in all bins.6 The population balance model cannot show a maximum in too long times and steady states, but for a wide range of intermediate times, it predicts ASDs with a reasonable accuracy, especially for larger particles. Figures 9 and 10 show the aggregate diameter distributions in normal and cumulative form. Both of them present good agreement with experiments. Effect of Solvent Composition and Solid Volume Fraction. Figure 11shows the effect of solvent composition on the evolution of aggregate diameter. The figure shows that with adjacent initial aggregate mean sizes, for T:H ratios that provide higher solid volume fractions, mean diameter grows faster and reaches a higher maximum diameter. The model supports this result. In fact, variations of φs affects the results, and T:H does not affect aggregation directly. The higher φs causes the faster growth and therefore higher maximum size. Effect of Initial Mean Aggregate Diameter. According to the population balance model, under constant shear rate and constant composition, the main factor that determines which is dominant, aggregation or breakage processes, is particle size. For distributions that contain large particles and have high mean size, the breakage is dominant, and for distributions with small mean size, the aggregation is dominant. Figure 12 shows the model results for various initial mean aggregate sizes. For curves of 100 and 150 µm initial average diameter, at the beginning time the aggregation is dominant and these curves pass through a maximum. For 300 µm initial average diameter, the fragmen-



Figure 9. Asphaltene aggregate size distribution at various stages of growth for T:H ) 1:9 and G ) 3.7572 (1/s).

Figure 10. Cumulative asphaltene aggregate size distribution at various stages of growth for T:H ) 1:9 and G ) 3.7572 (1/s).

tation is dominant at all times, and there is no maximum size for this run. Initial particle size does not affect the steady state mean size. Conclusions In this study, attention to the fractal nature of asphaltene aggregates results in better prediction of their aggregate size distribution evolution. From the fractal point of view and using the image processing method, the asphaltene solid content of samples was obtained. The initial conditions that were used to solve the PBE provide good agreement between model prediction and experimental results. Number average diameter curves show a maximum and then decline to a steady state diameter. But for very large initial particles they may reach no maximum. The fractal population

Figure 11. Evolution of number average diameter at different values of T:H under G ) 3.7571 (1/s).


Figure 12. Evolution of number average diameter for T:H ) 1:4 and G ) 3.7571 (1/s) with different initial mean aggregate size from the model.

balance model supports this trend with a good accuracy. The model predicts that total volume of aggregates changes during the shear induction. The model predicts the effect of shear rate on the evolution of aggregate diameter distribution with a good accuracy. Finally it should be noted that the results of this work are not applicable in real petroleum cases, directly. Petroleum is typically much more viscous than toluene-heptane mixtures, and also petroleum contains resins that act as natural dispersants. As a result, insoluble asphaltene aggregates in petroleum are typically an order of magnitude smaller in size than those reported in this paper. The proposed modeling approach can be developed as a good base for more complicated mixtures. Nomenclature A′ ) fragmentation parameter ASD ) aggregate size distribution BA ) rate of birth due to aggregation BB ) rate of birth due to breakage BN ) rate of nuclei birth CA ) flow number (i.e., the ratio of attractive van der Waals forces to hydrodynamic forces) Ci,j ) aggregation kernel of particles of diameters Vi and Vj DA ) rate of death due to aggregation dav ) number mean projected area diameter, µm DB ) rate of death due to breakage

Solaimany-Nazar and Rahimi dc,i ) collision diameter of aggregates in bin i df ) fractal dimension di ) diameter of rigid sphere mass equivalent aggregates in bin i dp ) diameter of a spherical primary particle, µm fR ) retardation function G ) shear rate, s-1 Gr ) growth rate, m/s or m3/s imax ) number of classes or bins kv ) third moment shape factor L ) aggregate characteristic length L0 ) nuclei characteristic length m3 ) third moment n ) particle number density Ni ) number concentration of aggregates in bin i having characteristic volume Vi, m-3 Ntotal ) total number concentration of particles in all bins, m-3 PBE ) population balance equation PSD ) particle size distributions q ) fragmentation parameter ri ) aggregate radius Si ) fragmentation or breakup rate of aggregates of size i, s-1 T:H ) toluene-to-heptane ratio in solvent mixtures (or solvent composition) 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 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,V′) ) breakup distribution function defining the volume fraction of the fragments of size V originating from V′-sized particles σf ) standard deviation of the fragment-size distribution. φs ) solid particle contents (i.e., volume fraction of particles) εi ) porosity of the aggregate of size Vi δ(L - L0) ) kronecker delta function λR ) retardation parameter EF800173S

Dynamic Determination of Asphaltene Aggregate Size Distribution in

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