Transport of Ultradispersed Catalytic Particles Through Bitumen at Upgrading Temperatures

Mass transfer and deposition of fine particles in cylindrical channels has received considerable attention for a long time due to its practical significance and direct application in industry. For example, this knowledge is helpful in aerosol classification and its deposition under electrical fields, formation of deposits in heat exchangers and pipelines, hydrodynamic field chromatography, thrombus formation in organs and, many other areas (Adamczyk and Van De Ven, 1981). Recently, this phenomenon has gained particular importance on the dispersion of ultradispersed catalysts for heavy crude oil and bitumen hydroprocessing due to its practical significance and direct application (Pereira-Almao et al., 2007; Galarraga and Pereira-Almao, 2010; Loria et al., 2011). Ultradispersed catalysts have been studied for heavy oil and bitumen hydroprocessing as an alternative for typical supported catalysts. An advantage when comparing ultradispersed catalysts to supported ones, in the case of heavy oil and bitumen hydroprocessing, is that the former could be easily incorporated into the reaction media to flow together with the feedstock to be treated, in this manner residence times can be longer than those conventionally used for hydroprocessing (Pereira-Almao et al., 2005; Pereira-Almao, 2007). Recent publications (Loria et al., 2009b, 2009c, 2010) have demonstrated the feasibility of the transport of ultradispersed particles based on their motion through diverse viscous media enclosed in horizontal cylindrical channels. Time-dependent, two and three-dimensional convective-dispersive models, which simulated the transient deposition and suspension of ultradispersed particles immersed in viscous media inside a horizontal cylinder, were developed, solved and experimentally validated. In addition, a study on the effect of the fluid medium properties over the dispersion coefficient was performed. The dispersion coefficient is a proportionality constant that serves to quantify the particle concentration due to convection and dispersion and should be expressed as a function of the properties of the fluid medium (Loria et al., 2010). The solution of the previously mentioned models provides a particle concentration profile along the horizontal channel, as well as information regarding the critical particle size that allows particles to remain suspended in the fluid medium enclosed in this geometry. This knowledge can be applied in the previously referred ultradispersed catalysis of heavy crude oils and bitumen. In these systems, it is important to ensure that catalytic particles remain suspended in the fluid medium in order to make use of their catalytic activity and also, to


Introduction
Mass transfer and deposition of fine particles in cylindrical channels has received considerable attention for a long time due to its practical significance and direct application in industry.For example, this knowledge is helpful in aerosol classification and its deposition under electrical fields, formation of deposits in heat exchangers and pipelines, hydrodynamic field chromatography, thrombus formation in organs and, many other areas (Adamczyk and Van De Ven, 1981).Recently, this phenomenon has gained particular importance on the dispersion of ultradispersed catalysts for heavy crude oil and bitumen hydroprocessing due to its practical significance and direct application (Pereira-Almao et al., 2007;Galarraga and Pereira-Almao, 2010;Loria et al., 2011).Ultradispersed catalysts have been studied for heavy oil and bitumen hydroprocessing as an alternative for typical supported catalysts.An advantage when comparing ultradispersed catalysts to supported ones, in the case of heavy oil and bitumen hydroprocessing, is that the former could be easily incorporated into the reaction media to flow together with the feedstock to be treated, in this manner residence times can be longer than those conventionally used for hydroprocessing (Pereira-Almao et al., 2005;Pereira-Almao, 2007).Recent publications (Loria et al., 2009b(Loria et al., , 2009c(Loria et al., , 2010) ) have demonstrated the feasibility of the transport of ultradispersed particles based on their motion through diverse viscous media enclosed in horizontal cylindrical channels.Time-dependent, two and three-dimensional convective-dispersive models, which simulated the transient deposition and suspension of ultradispersed particles immersed in viscous media inside a horizontal cylinder, were developed, solved and experimentally validated.In addition, a study on the effect of the fluid medium properties over the dispersion coefficient was performed.The dispersion coefficient is a proportionality constant that serves to quantify the particle concentration due to convection and dispersion and should be expressed as a function of the properties of the fluid medium (Loria et al., 2010).The solution of the previously mentioned models provides a particle concentration profile along the horizontal channel, as well as information regarding the critical particle size that allows particles to remain suspended in the fluid medium enclosed in this geometry.This knowledge can be applied in the previously referred ultradispersed catalysis of heavy crude oils and bitumen.In these systems, it is important to ensure that catalytic particles remain suspended in the fluid medium in order to make use of their catalytic activity and also, to obtain the conditions for which these particles will sediment in order to recover and reuse them in recirculation systems.The present study intends to show the application of the convective-dispersive models in cases that involve the storage and transport of catalytic particles immersed in bitumen at upgrading temperatures (340 -380° C) inside cylindrical geometries.The main objective of this paper is to employ the previously developed models as tools for interpretation of cases of interest for the heavy oil and bitumen industry.The modelling parameters of these systems can be based on those employed in the validation experiments for the convective-dispersive models (Loria et al., 2009b(Loria et al., , 2009c(Loria et al., , 2010)), since physical properties from particles and fluids used in the experiments were deliberately chosen to be similar as those that catalytic particles and bitumen would have at upgrading temperatures.The first part of this paper deals with a large scale application of the two-dimensional convective-dispersive model (Loria et al., 2009b(Loria et al., , 2009c)).Since this model predicts the particle concentration profile enclosed in a circumference, it can be applied to the crosssectional part of a spherical storing tank.In this section, the concentration profile of molybdenum trioxide (MoO 3 ) catalytic particles immersed in bitumen enclosed in a 4 m diameter spherical tank is studied.The second section of this paper is related to the three-dimensional convective-dispersive model (Loria et al., 2010).This model simulates the transient deposition and suspension of particles immersed in a fluid travelling through a horizontal cylindrical channel and provides the particle concentration profile along the channel.In this case, simulations involving a pipe of 100 m length and 101.6 cm diameter transporting MoO 3 catalytic particles immersed in bitumen were carried out.The simulations took into account different particle diameters, ranging from 1 to 1500 nm and different temperatures, ranging from 340 to 380° C. The bitumen's physical properties (density and viscosity) vary with respect to the different studied temperatures and are essential inputs of the convective-dispersive model.In order to obtain these properties; a previously developed thermodynamic model for their calculation (Loria et al., 2009a) was employed.

Two-dimensional transport of particles through viscous fluid media
2.1 Two-dimensional convective-dispersive model to predict the concentration profile of particles immersed in viscous media When a particle settles down in a liquid medium, it accelerates until the forces that cause the sedimentation equilibrate with the resistance or drag forces offered by the medium.Once this equilibrium is achieved, the particle has a constant sedimentation velocity called terminal sedimentation velocity (v pT ) which can be represented by (Ramalho, 1983): where g is the acceleration due to gravity, p is the density of the particle, L is the density of the liquid, d p is the diameter of the particle and μ L is the viscosity of liquid medium.Equation 1 is also known as the Stokes' law for the sedimentation of discrete particles.
A continuity equation for the particle concentration in a fluid medium can be obtained by carrying out a mass balance on a differential element of volume.Considering that particle concentration (C P ) varies with time (t), that the effective dispersion coefficient (D E ) is constant, that the particle concentration inside a cylindrical channel is modelled as a function of the position of the particles in the cross-section of the cylinder and the time, C P = C P (r, θ, t), that the angular velocity of the particle (v θ ) is small compared to the radial velocity (v r ) and that this radial velocity can be obtained by projecting the vertical terminal velocity of the particle into the radial direction (v r = v PT cosθ); then the continuity equation in cylindrical coordinates can be written as (Bird et al., 2007): .
D E is called the effective dispersion coefficient because the transport of the particles from high concentrated areas to low concentrated ones is, in this case, due to their mixing with the liquid bulk (Franco, 2008) and it is called effective because includes the motion effects caused by the particles and the gravity force in all directions.It distinguishes from the diffusivity coefficient in the sense that diffusivity is determined by the molecular properties of the particles and the fluid in which they are immersed, whereas the dispersion coefficient is determined by the particle properties and fluid conditions.Generally, in atmospheric transport, dispersive flux dominates the diffusive flux.
In the beginning of the process (t = 0) all particles are well dispersed inside the cross section of the cylindrical channel.This means that particle concentration is uniform everywhere inside the cross-section of the cylindrical channel.This concentration is the initial concentration of the particles (C P0 ).
At t = 0, C P (r, θ, 0) = C P0 , 0 < r < R, 0 < θ < 2 . (3) In the vertical axis there are symmetry boundary conditions that can be represented as: At θ = 0, 0 At θ = , 0 In the centre of the cylindrical channel the particle concentration does not vary with respect to the radius around all the angles: At r = 0, 0 The walls of the channel represent a physical boundary where there is no mass exchange between the interior and exterior of the cylinder.This is an insulation boundary that will prevent any particle from leaving the channel, gathering all settled particles at the bottom.The insulation boundary means that there is no convective or dispersive flux across that boundary.This can be represented by: At r = R, ( ) The convective-dispersive model is a linear second order parabolic partial differential equation; this equation can be solved by a large variety of numerical methods: finite differences, finite element, finite volume, characteristics methods, discontinuous Garlekin methods, etc.A numerical solution based in the finite element method was used to solve this equation.This solution presented good convergence and stability properties due to the regular grid structure and its flexibility with respect to the adaptation to the geometry domain.Computational fluid dynamics software was used in order to apply the method.

Effect of the fluid medium properties on the dispersion coefficient
Two of the parameters which are necessary input data for the numerical solution of the convective-dispersive model are v pT and D E .They can be obtained from experimental physical parameters.The calculation of v pT can be performed directly from equation 1. However; the effective dispersion coefficient of the particles, D E , cannot be obtained directly from any formula, the best way to obtain this value is to perform an adjustment of this parameter using concentration values from experimental data.
In a previous work (Loria et al., 2009c), a study to observe the variation of the dispersion coefficient with respect to the changes of the properties of the fluid medium was performed.
For this study the experimental data collected from Fe 2 O 3 nanoparticles immersed in different mixtures of water and glycerol was used.
Figure 1 shows the behaviour of the dispersion coefficient with respect to an increase of density and viscosity of the fluid medium.It can be observed that as the fluid medium becomes denser and more viscous the dispersion coefficient decreases.The particles gradually decrease their capability to move as the fluid medium becomes denser and more viscous; thus, the particles lose their ability to go from high concentrated areas to low concentrated ones, causing a decrease of the dispersion coefficient.These experimentally obtained dispersion coefficients showed a tendency to increase when the conditions were more favourable for sedimentation, that is, low density and viscosity Literature data related to cross-sectional dispersion coefficients in horizontal channels is scarce.The dispersion coefficient obtained in this research could be compared to the diffusion coefficients obtained by (d'Orlyé et al., 2008).They studied diffusion coefficients of maghemite particles (γ-Fe 2 O 3 ) dispersed in an aqueous solution of nitric acid.They performed diffusion coefficient measurements for nanometric particles (< 10 nm) based on the results of dynamic light scattering experiments in the cross-section of horizontal capillary tube.They obtained diffusion coefficients in the order of 10 -11 m 2 /s, which varied three orders of magnitude from the D E values obtained in this work.This difference is probably due to the use of particles smaller than 10 nm, the employment of capillary tubes of 50 μm of diameter and the use of dynamic light scattering results for the calculation of particle concentrations since dynamic light scattering is a technique primarily designed for particle size measurement.Dispersion coefficients with similar orders of magnitude as the ones obtained in a previous work (Loria et al., 2009c) carried out with submicron particles (10 -8 -10 -9 m 2 /s) were found by (Robbins, 1989;Massabò et al., 2007).However, these results are not comparable with this work since they performed their calculations in a packed column based on concentration measurements that were generated by the computational simulation of a continuous tracer injection experiment in a vertical cylindrical geometry.

Ultradispersed catalytic particles immersed in bitumen at upgrading temperatures stored inside a spherical tank
In this section, the two-dimensional convective-dispersive model is applied to a large scale case which involves the storage of MoO 3 catalytic particles immersed in Athabasca bitumen, at upgrading conditions (340-380 °C), inside a 4 m diameter spherical tank.The objectives of this study are: to find the critical particle diameter for its suspension, and to calculate the deposited mass of the catalytic particles at the bottom of the tank once the steady state has been reached.The density ( L ) and viscosity (μ L ) of the Athabasca bitumen at five different temperatures ranging from 340 to 380 °C were employed in the following simulations.These physical properties were calculated with thermodynamic models for the prediction of density and viscosity of heavy crude oil and bitumen, details of these models were presented by (Loria et al., 2009a).Table 1 shows the densities and viscosities of Athabasca bitumen in the range 340 to 380 °C, these values were obtained with the previously mentioned thermodynamic models.(Perry, 1997)) particle diameters which were used for the simulations ranged from 1 to 1500 nm.In total fourteen different particle diameters were studied: 1, 10, 50,100,150,200,250,300,400,500,750,1000,1250 and 1500 nm.
The initial MoO 3 particle concentration (C P0 ) applied to all the simulations was 1.2526 mol/m 3 .A common concentration unit employed in ultradispersed catalysis for hydroprocessing is ppm (parts per million or g/m 3 ).In this case, 1.2526 mol/m 3 are equivalent to 180.3 ppm.This particle concentration is within the same magnitude order as those that have been proposed for ultradispersed catalysis for hydroprocessing (Pereira-Almao, 2007).The calculation of the dispersion coefficient that was used for these simulations was based on the results that were obtained when the two-dimensional convective-dispersive model was validated with Fe 2 O 3 particles immersed in mixtures of water and glycerol (Loria et al., 2009c).These data were used since particle properties and concentrations in addition to the fluid medium characteristics used in those experiments are similar to those employed in the simulations carried out in this section.
The dispersion coefficient calculation was carried out in the following way: polynomial interpolations were applied to the data from Figure 1 (μ L and L vs. D E ); then, for each temperature a pair of D E values were obtained (one based on μ L and the other in L ) and their average was recorded; finally, the 5 different D E average values (corresponding to each one of the studied temperatures and ranging from 4.9 -5.6 × 10 -8 m 2 /s) were averaged and the final D E value was obtained.The D E value obtained from these calculations resulted to be 5.33 × 10 -8 m 2 /s.A total of 70 different simulations were carried out in this section (based on the 14 different particle diameters and 5 different temperatures).Computational fluid dynamics was used to perform the simulations; each simulation was carried out up to a time of 1 × 10 8 s (27778 h), a time long enough to reach the steady state in each one of them.2508 grid points were employed for the solution of each simulation and their computing time was around 60 s A point of interest in this study is the critical particle diameter for deposition; that is, the particle diameter from which particles with higher sizes will be deposited and particles with lower sizes will remain suspended.In order to measure this new parameter, the following analysis was conducted.
The normalized maximum particle concentration, which is the concentration that is found at the bottom of the circumference after the simulation time (steady state) divided by the initial concentration, was calculated for each simulation.When the normalized maximum concentration tends to the initial one (normalized concentration = 1), it means that there is no particle deposition at the bottom of the circumference.This normalized maximum particle concentration was plotted against the particle diameter in order to observe at which particle diameter the maximum particle concentration becomes significant.Figure 2 represents the behaviour of the normalized maximum particle concentration at different particle diameters and temperatures.
As temperature increases for a specific particle diameter, the normalized maximum concentration increases.Also, as the particle diameter increases, the normalized maximum concentration increases for a constant temperature.It can also be observed that a change in particle diameter have a more pronounced effect over the normalized maximum concentration than a change in temperature.
In the particle diameter axis, there is a zone for values lower than 150 nm where the normalized maximum concentration is equal to the initial one independently of the studied temperature.This means that MoO 3 particles smaller than 150 nm will remain suspended in the Athabasca bitumen after a long period of time (1 × 10 8 s) in the range of temperatures from 340 to 380 °C.In order to calculate the deposited mass of the catalytic particles at the bottom of the spherical tank once the steady state has been reached, the following procedure was employed.Assuming that the deposited particles filled a small gap of height H at the bottom of the spherical tank of radius R, its volume (V P ) can be calculated by: Assuming that the height of this small gap is approximately H = 6.5 cm (based on the simulation results), the volume of the gap (V P ) for a sphere of 2 m of radius is 1.73 ×10 -3 m 3 .
Since particle concentration in this part of the tank after a long period of time is already known from the simulation results, i.e. the maximum particle concentration (C Pmax ), then the deposited mass (M P ) is: On the other hand, the total mass of the immersed particles inside the tank (M PT ) is: Considering that the initial particle concentration (C P0 ) in the simulation was assumed to be 180.3ppm or 1.2526 mol/m 3 ; the total particle mass immersed in a spherical tank of 2 m of radius is 6.042 kg.The percentage of deposited particles at the bottom of the spherical tank (%M dep ) can be calculated by: % 100  4 demonstrates that the percentage of deposited particles at the bottom of the tank is less than 0.02%, when particle diameters below 200 nm are employed; and, when particle diameters in the range 200 -1500 nm are used, the percentage of deposited particles is between 0.02 and 0.6 %.

Axial varticle velocity
The first step to develop the three-dimensional convective-dispersive model is to obtain the flow velocity profile since particle velocity in this direction depends on it.This profile can be obtained from the Navier-Stokes equations.Consider a laminar incompressible fluid that travels inside a horizontal cylinder of radius R and length L. The force (F) pushing the liquid through the cylinder is the change in pressure (ΔP) multiplied by the cross-sectional area of the cylinder (A): This force is oriented in the same direction of the motion of the liquid: If the cylinder is long enough, then the flow through the cylinder is known as fully developed velocity profile, this means that there are not velocity components in the radial (v r ) and angular direction (v θ ) and the velocity in the longitudinal direction (v z ) is only a function of the radial coordinate (r).Assuming a fully developed velocity profile at steady state, that the gravity force is not the force causing the motion and its effect is negligible, that the pressure decreases linearly across the length of the cylinder, that the axial velocity (v z ) is finite at r = 0 and that there is a no slip boundary condition at the cylinder wall; then, the following parabolic velocity profile can be obtained: Equation 14 is known as the Poiseuille's flow inside cylinders.The volumetric flux (Q) inside the cylinder can be calculated using the following procedure: where n is the unit vector normal to the velocity vector (v).The mean axial velocity ( z v ) is the volumetric flux divided by the cross-sectional area: If the particles travelling through a horizontal cylinder are small enough as compared with the size of the cylinder radius, it can be considered that they travel at the same velocity of the fluid.Unlike liquid molecules present in a Poiseulle's flow, a solid spherical discrete particle of radius (r p ) cannot approach the slow flowing area close to the wall (Fung, 1993;Michaelides, 2006).Considering this assumption its average velocity can be calculated by: Thus, the axial velocity of the particles can be represented by the Poiseuille's flow equation.

Analysis of the particle concentration inside a horizontal cylindrical channel
Particle concentration in a fluid medium can be obtained by carrying out a mass balance for particles present in a differential element of the system.In order to develop a convectivedispersive model which simulates the transport of particles travelling through a fluid medium, the mass balance for particles present in a differential element of a cylinder presented in Section 2.1 can be employed.Consider a laminar incompressible flow which contains particles that flow inside a horizontal cylinder of radius R and length L. The mass balance equation that represents this problem is the following (Bird et al., 2007): 22 22 2 11 1 .
The settling of particles affects the distribution of the particle concentration at each point of the horizontal cylinder.Thus, the particle concentration inside the horizontal channel will be modelled as a function of the position of the particles inside the channel and the time: Equation 19 can be simplified by assuming that the angular velocity of the particle (v θ ) is small compared to the radial velocity (v r ).Thus, v θ can be neglected.Equation 19 can be written as: 22 22 2 11 .
As in Section 2.1, the radial particle velocity (v r ) is considered to be the projection of the vertical terminal velocity of the particle (v pT ) into the radial direction: In the case of the axial velocity (v z ), it has been shown that it can be represented by the Poiseuille's flow.Equation 20 can now be expressed as: 2 2 11 cos 1 4 22) Initially (t = 0), it is assumed that the cylinder only contains the fluid travelling through it and there are no particles present inside the cylindrical channel.
If an open system is considered, at the entrance of the channel (z = 0) the suspension of solid particles has a homogeneous constant initial concentration (C P0 ).
At z = 0, C P (r, θ, At the exit of the channel (z = L) the convection dominates the mass transport; this implies that the concentration gradient due to dispersion in a perpendicular direction to this boundary is negligible.This condition eliminates the need of specifying a concentration or a fixed value for the flux at the outlet boundary, since both of them are unknown.Mathematically this condition can be represented by: ( ) e n0 e e e 0 where n is the normal vector to the plane at the outlet of the cylinder, e z , in this case.Solving the dot product, it is found that: In the vertical axis there are symmetry boundary conditions that can be mathematically represented as: 1 0.
Previously it was assumed that the angular velocity of the particle (v θ ) is neglected, therefore: At θ = 0, 0 At θ = , 0 The particle concentration is finite along the radial direction for all the angles and lengths: The walls of the channel represent a physical boundary where there is no mass exchange between the interior and exterior of the cylinder.This is an insulation boundary that will prevent any particle from leaving the channel and gather all them at the bottom.The insulation boundary means that there is no convective or dispersive flux across that boundary.This can be represented by: 0 Then: At r = R, ( ) The convective-dispersive model is a linear second order parabolic partial differential equation.A numerical solution based in the finite element method was used in this work.
Computational fluid dynamics software was used to apply the method.

Effect of the fluid medium properties on the dispersion coefficient
This section is dedicated to study the variation of the dispersion coefficient (D E ) respect to changes in the properties of the fluid medium and the initial concentration of particles.For this study, experimental data collected from a previous work (Loria et al., 2010) was employed.
Figure 5 shows the behaviour of the dispersion coefficient with respect to an increase of density and viscosity of the fluid medium.It can be observed that as the fluid medium becomes denser and more viscous the dispersion coefficient decreases because particles lose their ability to move from high concentrated areas to low concentrated ones.
Figure 6 shows the variation of the dispersion coefficient with respect to a change of the fluid velocity.It can be observed that as fluid velocity increases the dispersion coefficient decreases.In this case an increase in the fluid velocity causes an increase in the axial velocity of the particles, enhancing their ability to remain suspended and reducing their capacity to move from high concentrated areas to low concentrated ones, causing a decrease in the dispersion coefficient.have studied the aerosol particle dispersion coefficients to determine how many and where these particles are deposited in the human respiratory system.Giojelli et al. (2001) have focused their attention to study the dispersion coefficients for aerosol particles separation from biogas produced by sludge coming from wastewaters.The typical orders of magnitude for dispersion coefficients employed in the previously described processes were around 10 -4 m 2 /s, which is not by any means comparable to the ones obtained in the previously referred work (Loria et al., 2010), since totally different materials are involved in the mass transfer of the studied phenomena.
The dispersion of a solute in a laminar Poiseuille's flow to measure diffusion coefficients of proteins and macro-molecules was studied by Bello et al.(1994).Their experimental measurements of the diffusion coefficients gave values that varied from 10 -8 to 10 -11 m 2 /s.Even though these results are closer to the dispersion coefficients calculated the work from Loria et al. (2010), they cannot serve as a standard of comparison, since the diameter of these molecules is reduced to a few nanometers and the vessels where these authors evaluated the diffusion coefficients were capillaries with 50 to 100 μm of inner diameter.

Ultradispersed catalytic particles immersed in bitumen at upgrading temperatures transported through a pipeline
This section is dedicated to the application of the three-dimensional convective-dispersive model to a large scale case which involves the transport of MoO 3 catalytic particles immersed in Athabasca bitumen, at upgrading conditions (340-380 °C), through a pipe of 100 m length and 101.6 cm diameter.The simulations in this section have two different goals.The first is to find the critical diameter to avoid particle deposition.The second is to calculate the deposited particle mass at the bottom of the pipe once the steady state has been reached.As in the previous section, the density ( L ) and viscosity (μ L ) of the Athabasca bitumen at five different temperatures ranging from 340 to 380 °C were employed in the simulations for this section.These physical properties of the Athabasca bitumen at the temperatures of interest were shown in Table 1.
In this section, twelve different MoO 3 particle diameters were used for the simulations: 1, 10, 50, 100, 150, 200, 250, 300, 400, 500, 1000 and 1500 nm.The initial MoO 3 particle concentration (C P0 ) that was applied to all the simulations was 1.2526 mol/m 3 , equivalent to 180.3 ppm.A fluid velocity (Q) of 30000 ml/min (5 × 10 -4 m 3 /s or 272 bbd) was chosen in order to have a large volumetric flux and, also, to maintain a laminar regime inside the pipe.Taking into account that the pipe diameter is d = 101.6cm, the mean axial velocity ( z v = Q/( d 2 /4)) is 6.17 × 10 -4 m/s.This value permits the calculation of the fluid Reynolds number (Re = L z v d/μ L ) which estimates if a fluid is laminar or turbulent.Table 3 shows the different values of the fluid Reynolds number for the Athabasca bitumen transported through a pipe at the 5 different studied temperatures.Also, Table 3 shows the pressure drop (ΔP) between the ends of the pipe which was calculated using Equation 18.  3 reports fluid Reynolds numbers <2100 and small values of pressure drop between the ends of the cylinder which demonstrates that a laminar flow is maintained across the pipe at any of the 5 different studied temperatures.Generally, a fluid is laminar if the Reynolds number is between 0 and 2100 (Bird et al., 2007).The calculation of the dispersion coefficient that was employed for these simulations was based on the results that were obtained when the three-dimensional convective-dispersive model was validated with Fe 2 O 3 particles immersed in mixtures of water and glycerol (Loria et al., 2010).These results were taken into account because particle properties and concentration as well as the fluid medium characteristics and axial velocities used in that experiments are similar to those present in the simulations carried out in this section.The dispersion coefficient calculation was carried out in a similar way as in Section 2.3: polynomial interpolations were carried out with the data from Figure 5 and Figure 6 (μ L and L vs. D E and z v vs. D E ); then, for each temperature three different values of D E were obtained (based on μ L , C P0 and in z v ) and their average was recorded; finally, the 5 different D E average values (corresponding to each studied temperature and ranging from 2.1 -2.18 × 10 -9 m 2 /s) were averaged and the final D E value was obtained.The D E value obtained from these calculations resulted to be 2.14 × 10 -9 m 2 /s.A total of 60 different simulations were carried out in this section (based on the 12 different particle diameters and 5 different temperatures).Computational fluid dynamics was used to perform the simulations; each simulation was carried out up to a time of 1.5 × 10 7 s (4167 h), a time long enough to reach the steady state in each one of them.9878 grid points were employed for the solution of each simulation and their computing time was around 240 s.One subject of interest in this study is the critical particle diameter for deposition.In order to measure this parameter, the following analysis was conducted.

Fig. 1 .
Fig. 1.Dispersion coefficient (D E ) as a function of the viscosity (µ L ) and density ( L ) of the fluid medium in the cross-section of a horizontal cylinder

Fig. 2 .
Fig. 2. Effect of the particle diameter and temperature on the deposition of MoO 3 particles immersed in Athabasca bitumen enclosed in a spherical tank of 4 m diameter

Fig. 3 .
Fig. 3. Gap of height H at the bottom of a spherical tank of radius R

Fig. 4 .
Fig. 4. Percentage of deposited particles at the bottom of the spherical tank (%M dep ) at different temperatures and particle diameters for the deposition of MoO 3 particles immersed in Athabasca bitumen enclosed in a spherical tank of 4 m diameter

Fig. 5 .Fig. 6 .
Fig. 5. Dispersion coefficient (D E ) as a function of the viscosity (µ L ) and density ( L ) of the fluid medium in a horizontal cylinder

Fig. 7 .
Fig. 7. Effect of the particle diameter and temperature on the deposition of MoO 3 particles immersed in Athabasca bitumen flowing through a pipe

Table 2 and
Figure4show the percentage of deposited particles at the bottom of the spherical tank (%M dep ) at different temperatures and particle diameters.

Table 2 .
Percentage of deposited particles at the bottom of the spherical tank (%M dep ) containing MoO 3 particles immersed in Athabasca bitumen at different temperaturesFigure

Table 3 .
Fluid Reynolds numbers and pressure drop between the ends of the pipe at the 5 different studied temperatures Table