**3.1 Three-dimensional convective-dispersive model to predict the concentration profile of particles immersed in viscous media**

#### **3.1.1 Axial varticle velocity**

716 Mass Transfer - Advanced Aspects

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 (*CP*max), then the

> 0 4 3 *MPT P C R*

Considering that the initial particle concentration (*CP0*) in the simulation was assumed to be 180.3 ppm or 1.2526 mol/m3; the total particle mass immersed in a spherical tank of 2 m of

The percentage of deposited particles at the bottom of the spherical tank (%*Mdep*) can be

% 100 *<sup>P</sup> dep*

Table 2 and Figure 4 show the percentage of deposited particles at the bottom of the

*<sup>M</sup> <sup>M</sup>*

Table 2. Percentage of deposited particles at the bottom of the spherical tank (%*Mdep*) containing MoO3 particles immersed in Athabasca bitumen at different temperatures

Figure 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

spherical tank (%*Mdep*) at different temperatures and particle diameters.

*PT*

*dp***, nm 340 °C 350 °C 360 °C 370 °C 380 °C**  1500 0.476 0.525 0.542 0.582 0.588 1250 0.375 0.402 0.413 0.415 0.425 1000 0.269 0.287 0.305 0.325 0.349 750 0.158 0.175 0.178 0.179 0.192 500 0.070 0.079 0.080 0.084 0.089 400 0.045 0.050 0.051 0.054 0.057 300 0.025 0.028 0.029 0.030 0.032 250 0.018 0.020 0.020 0.021 0.022 200 0.012 0.014 0.014 0.014 0.015 150 0.009 0.009 0.009 0.010 0.010 100 0.007 0.007 0.007 0.007 0.007 50 0.005 0.006 0.006 0.006 0.006 10 0.005 0.005 0.005 0.005 0.005 1 0.005 0.005 0.005 0.005 0.005

**%***Mdep*

⎛ ⎞ <sup>=</sup> ⎜ ⎟

3

π

On the other hand, the total mass of the immersed particles inside the tank (*MPT*) is:

*MPP P* = *C V* max (9)

⎝ ⎠ . (10)

*M*= × . (11)

deposited mass (*MP*) is:

radius is 6.042 kg.

between 0.02 and 0.6 %.

calculated by:

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 (Δ*P*) multiplied by the cross-sectional area of the cylinder (*A*):

$$F = -\Delta PA\ . \tag{12}$$

This force is oriented in the same direction of the motion of the liquid:

$$
\Delta P = P\_{\text{out}} - P\_{\text{in}} \; \; \; P\_{\text{out}} \leqslant P\_{\text{in}}.\tag{13}
$$

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 (*vr*) and angular direction (*vθ*) and the velocity in the longitudinal direction (*vz*) 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 (*vz*) is finite at *r* = 0 and that there is a

Transport of Ultradispersed Catalytic Particles Through Bitumen at Upgrading Temperatures 719

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: *CP* =

Equation 19 can be simplified by assuming that the angular velocity of the particle (*vθ*) is small compared to the radial velocity (*vr*). Thus, *vθ* can be neglected. Equation 19 can be

1 1 . *P PP P PP*

*C CC C CC vv Dr t r z rr r r z*

∂ ∂ ∂ ∂∂ ∂ ∂ <sup>⎡</sup> <sup>⎤</sup> ⎛ ⎞ ⎛⎞ ++= ++ ⎜ ⎟ ⎜⎟ <sup>⎢</sup> <sup>⎥</sup> ∂ ∂ ∂ ∂∂ ⎝ ⎠ ⎝⎠ ⎢⎣ ∂ ∂ ⎥⎦

As in Section 2.1, the radial particle velocity (*vr*) is considered to be the projection of the

( ) <sup>2</sup> cos cos . 18 *p Lp*

μ

*g d*

ρ ρ

In the case of the axial velocity (*vz*), it has been shown that it can be represented by the

*P P P PP P*

Initially (*t* = 0), it is assumed that the cylinder only contains the fluid travelling through it

 At *t* = 0, *CP*(*r*, *θ*, *z*, 0) = 0, 0 ≤ *r* ≤ *R*, 0 ≤ *θ* ≤ *2π,* 0 ≤ *z* ≤ *L.* (23) If an open system is considered, at the entrance of the channel (*z* = 0) the suspension of solid

 At *z* = 0, *CP*(*r*, *θ*, 0, *t*) = *CP0*, 0 ≤ *r* ≤ *R*, 0 ≤ *θ* ≤ *2π.* (24) 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.

> ( ) <sup>e</sup> n 0e e e 0 *P PP E P z Er z C CC D C <sup>D</sup>*

⎡ ⎤ ⎛ ⎞ ∂ ∂∂ <sup>⋅</sup> − ∇ = ⇒ ⋅− + + = ⎢ ⎥ ⎜ ⎟ <sup>⎣</sup> ⎝ ⎠ ∂∂∂ <sup>⎦</sup>

where **n** is the normal vector to the plane at the outlet of the cylinder, **e***z*, in this case. Solving

In the vertical axis there are symmetry boundary conditions that can be mathematically

*z* <sup>∂</sup> <sup>=</sup> <sup>∂</sup>

∂ ∂Δ ∂ ∂ ⎡ ⎤ ⎛⎞ ⎡ ∂ ∂∂ <sup>⎤</sup> ⎛ ⎞ + + − = ++ ⎢ ⎥ ⎜ ⎟ <sup>⎢</sup> ⎜ ⎟ <sup>⎥</sup> ∂ ∂ ⎢ ⎥ ⎣ ⎦ ⎝⎠ ⎣ ∂ ∂∂ <sup>⎢</sup> ⎝ ⎠ ∂ ∂ ⎥⎦

*C C P rC C CC v RD <sup>r</sup> t rL z r R r r r z*

θ

*L*

*rz E*

vertical terminal velocity of the particle (*vpT*) into the radial direction:

*r pT*

2

*pT E L*

*v v*

Poiseuille's flow. Equation 20 can now be expressed as:

cos 1 4

Mathematically this condition can be represented by:

At *z* = *L*, 0, *CP*

the dot product, it is found that:

represented as:

μ

and there are no particles present inside the cylindrical channel.

particles has a homogeneous constant initial concentration (*CP0*).

θ

2 2 22 2

(20)

. (22)

, (25)

θ

θ

2 2 2

*rr z* θ

θ

0 ≤ *r* ≤ *R*, 0 ≤ *θ* ≤ *2π.* (26)

2 2 2 2 1 1

θ

<sup>−</sup> = = (21)

*CP* (*r*, *θ*, z, *t*).

written as:

no slip boundary condition at the cylinder wall; then, the following parabolic velocity profile can be obtained:

$$
\sigma v\_z(r) = \frac{\Delta P}{4\mu\_L L} R^2 \left(1 - \frac{r^2}{R^2}\right). \tag{14}
$$

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:

$$Q = \int\_{A} \mathbf{n} \cdot \mathbf{v} dA = 2\pi \int\_{0}^{R} v\_{z}(r) r dr = \frac{\pi \Delta PR^{4}}{8\mu\_{L}L} \cdot \tag{15}$$

where **n** is the unit vector normal to the velocity vector (**v**). The mean axial velocity ( *<sup>z</sup> v* ) is the volumetric flux divided by the cross-sectional area:

$$\overline{\upsilon}\_z = \frac{2\pi \int\_0^R v\_z(r)r dr}{\pi R^2} = \frac{\Delta PR}{8\,\mu\_L L} \,\text{.}\tag{16}$$

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 (*rp*) cannot approach the slow flowing area close to the wall (Fung, 1993; Michaelides, 2006). Considering this assumption its average velocity can be calculated by:

$$2\pi \int\_{r}^{R-r\_p} v\_z(r)r dr = \frac{2\pi R}{\pi \left(R - r\_p\right)^2} = \frac{\Delta PR^2}{8\mu\_L L} \left[2 - \left(1 - \frac{r\_p}{R}\right)^2\right].\tag{17}$$

If *rp* << *R*, then

$$
\overline{\upsilon}\_z(r\_p) = \frac{\Delta PR^2}{8\mu\_L L} = \overline{\upsilon}\_z \cdot \tag{18}
$$

Thus, the axial velocity of the particles can be represented by the Poiseuille's flow equation.

#### **3.1.2 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):

$$\frac{\partial \mathbb{C}\_{P}}{\partial t} + \left( v\_{r} \frac{\partial \mathbb{C}\_{P}}{\partial r} + v\_{\theta} \frac{1}{r} \frac{\partial \mathbb{C}\_{P}}{\partial \theta} + v\_{z} \frac{\partial \mathbb{C}\_{P}}{\partial z} \right) = D\_{E} \left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \mathbb{C}\_{P}}{\partial r} \right) + \frac{1}{r^{2}} \frac{\partial^{2} \mathbb{C}\_{P}}{\partial \theta^{2}} + \frac{\partial^{2} \mathbb{C}\_{P}}{\partial z^{2}} \right]. \tag{19}$$

no slip boundary condition at the cylinder wall; then, the following parabolic velocity

*L P r vr R* μ

Equation 14 is known as the Poiseuille's flow inside cylinders. The volumetric flux (*Q*)

0 nv 2 () <sup>8</sup> *R z A L PR Q dA v r rdr <sup>L</sup>*

where **n** is the unit vector normal to the velocity vector (**v**). The mean axial velocity ( *<sup>z</sup> v* ) is

*v r rdr*

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 (*rp*) cannot approach the slow flowing area close to the wall (Fung, 1993; Michaelides, 2006). Considering this assumption its average velocity can be calculated by:

<sup>Δ</sup> <sup>=</sup> <sup>=</sup> ∫

π

0 2

*z*

0

∫

π

*R rp*

−

π

2 ()

*z*

*v*

*R z*

π

2 ()

π

2

*v r rdr*

( ) 2 1 ( ) 8

*p L*

Thus, the axial velocity of the particles can be represented by the Poiseuille's flow equation.

1 11 . *P P PP P PP r z <sup>E</sup> C C C C C CC vv v Dr t r r z rr r r z*

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ <sup>⎡</sup> <sup>⎤</sup> <sup>⎛</sup> ⎞ ⎛⎞ ++ += ++ ⎜ ⎟ <sup>⎢</sup> <sup>⎜</sup> <sup>⎟</sup> <sup>⎥</sup> ∂ ∂ ∂ ∂ ∂∂ ⎝ ⎠ ⎢⎣ <sup>⎝</sup> <sup>⎠</sup> ∂ ∂ ⎥⎦

**3.1.2 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):

*R r L R*

2 ( ) <sup>8</sup> *<sup>z</sup> <sup>p</sup> <sup>z</sup> L PR vr v* μ*L*

<sup>⎡</sup> <sup>⎤</sup> <sup>Δ</sup> ⎛ ⎞ <sup>=</sup> = − <sup>⎢</sup> <sup>−</sup> <sup>⎥</sup> ⎜ ⎟ <sup>−</sup> <sup>⎢</sup> <sup>⎥</sup> ⎝ ⎠ <sup>⎣</sup> <sup>⎦</sup>

μ

2

 *L R* <sup>Δ</sup> ⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup>

2

π

2

<sup>2</sup> <sup>2</sup>

*PR r*

*p*

<sup>Δ</sup> <sup>=</sup> <sup>=</sup> . (18)

2 2 22 2

(19)

θ

*L*

μ

*PR*

8

*R L*

μ

4

<sup>Δ</sup> =⋅ = = ∫ ∫ . (15)

. (14)

. (16)

. (17)

<sup>2</sup> 1

⎝ ⎠

( )

inside the cylinder can be calculated using the following procedure:

the volumetric flux divided by the cross-sectional area:

*z p*

θ

θ

*v r*

If *rp* << *R*, then

4 *<sup>z</sup>*

profile can be obtained:

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: *CP* = *CP* (*r*, *θ*, z, *t*).

Equation 19 can be simplified by assuming that the angular velocity of the particle (*vθ*) is small compared to the radial velocity (*vr*). Thus, *vθ* can be neglected. Equation 19 can be written as:

$$\frac{\partial \mathbb{C}\_P}{\partial t} + \left( v\_r \frac{\partial \mathbb{C}\_P}{\partial r} + v\_z \frac{\partial \mathbb{C}\_P}{\partial z} \right) = D\_E \left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \mathbb{C}\_P}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \mathbb{C}\_P}{\partial \theta^2} + \frac{\partial^2 \mathbb{C}\_P}{\partial z^2} \right]. \tag{20}$$

As in Section 2.1, the radial particle velocity (*vr*) is considered to be the projection of the vertical terminal velocity of the particle (*vpT*) into the radial direction:

$$\upsilon\_r = \upsilon\_{pT} \cos \theta = \frac{g \left(\rho\_p - \rho\_L\right) d\_p^2}{18\mu\_L} \cos \theta. \tag{21}$$

In the case of the axial velocity (*vz*), it has been shown that it can be represented by the Poiseuille's flow. Equation 20 can now be expressed as:

$$\frac{\partial \mathbf{C}\_p}{\partial t} + \left[ v\_{pT} \cos \theta \frac{\partial \mathbf{C}\_p}{\partial r} + \frac{\Delta P}{4 \mu\_L L} R^2 \left( 1 - \frac{r^2}{R^2} \right) \frac{\partial \mathbf{C}\_p}{\partial z} \right] = D\_E \left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \mathbf{C}\_p}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \mathbf{C}\_p}{\partial \theta^2} + \frac{\partial^2 \mathbf{C}\_p}{\partial z^2} \right]. \tag{22}$$

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.

$$\text{At } t = 0, \text{ } C\_l(r, \theta, z, 0) = 0, \quad 0 \le r \le R, \ 0 \le \theta \le 2\pi, \ 0 \le z \le L. \tag{23}$$

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 (*CP0*).

$$\text{At } z = 0, \quad C\_P(r, \theta, 0, t) = C\_{P0}, \quad 0 \le r \le R, \ 0 \le \theta \le 2\pi. \tag{24}$$

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:

$$\mathbf{n} \cdot \left( -D\_E \nabla C\_P \right) = 0 \Rightarrow \mathbf{e}\_z \cdot \left[ -D\_E \left( \mathbf{e}\_r \frac{\partial C\_P}{\partial r} + \frac{\mathbf{e}\_\theta}{r} \frac{\partial C\_P}{\partial \theta} + \mathbf{e}\_z \frac{\partial C\_P}{\partial z} \right) \right] = 0 \ \tag{25}$$

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:

$$\text{At } z = \text{L}\_r \, \frac{\partial \mathcal{C}\_p}{\partial z} = 0, \,\, 0 \le r \le \text{R}, \, 0 \le \theta \le 2\pi. \tag{26}$$

In the vertical axis there are symmetry boundary conditions that can be mathematically represented as:

Transport of Ultradispersed Catalytic Particles Through Bitumen at Upgrading Temperatures 721

Fig. 5. Dispersion coefficient (*DE*) as a function of the viscosity (*µL*) and density (*ρL*) of the

Fig. 6. Dispersion coefficient (*DE*) as a function of the fluid velocity (*Q*) of the fluid medium

Several works regarding axial dispersion coefficients in horizontal channels were found in the reviewed literature. Nevertheless, large part of this literature is advocated to aerosol particles flowing through airways. For instance, Zhang et al. (2005) and Schulz et al. (2000) 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

fluid medium in a horizontal cylinder

in a horizontal cylinder

$$-D\_E \left(\frac{1}{r}\frac{\partial \mathcal{C}\_P}{\partial \theta}\right) + v\_\theta \mathcal{C}\_P = 0.\tag{27}$$

Previously it was assumed that the angular velocity of the particle (*vθ*) is neglected, therefore:

$$\text{At } \theta = 0, \, \frac{\partial C\_P}{\partial \theta} = 0 \, \,, \, 0 \le r \le R, \, 0 \le \text{ z} \le L. \tag{28}$$

$$\text{At } \theta = \pi, \frac{\partial C\_P}{\partial \theta} = 0, \quad 0 \le r \le R, \ 0 \le z \le L. \tag{29}$$

The particle concentration is finite along the radial direction for all the angles and lengths:

$$\text{At } 0 \le r \le \mathbb{R}, \ C\_P = \text{finite} \; , \; 0 \le \theta \le 2\pi, \; 0 \le z \le \text{L} \tag{30}$$

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:

$$-D\_E \left(\frac{\partial \mathcal{C}\_P}{\partial r}\right) + \upsilon\_r \mathcal{C}\_P = 0 \ . \tag{31}$$

Then:

$$\text{At } r = R\_\prime \ -D\_E \frac{\partial \mathcal{C}\_P}{\partial r} + \left(v\_{pT} \cos \theta\right) \mathcal{C}\_P = 0 \ \ \ 0 \ \le \theta \le 2\pi \ 0 \le z \le L \tag{32}$$

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.

#### **3.2 Effect of the fluid medium properties on the dispersion coefficient**

This section is dedicated to study the variation of the dispersion coefficient (*DE*) 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.

θ

θ

θ

*E pT P <sup>C</sup> D vC*

<sup>∂</sup> <sup>−</sup> + =

θ

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.

This section is dedicated to study the variation of the dispersion coefficient (*DE*) 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

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

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

therefore:

Then:

employed.

dispersion coefficient.

At *θ* = 0, 0 *CP*

At *θ* = *π*, 0 *CP*

boundary. This can be represented by:

At *r* = *R*, ( cos 0 ) *<sup>P</sup>*

*r*

Computational fluid dynamics software was used to apply the method.

**3.2 Effect of the fluid medium properties on the dispersion coefficient** 

their ability to move from high concentrated areas to low concentrated ones.

∂

⎛ ⎞ <sup>∂</sup> <sup>−</sup> + = ⎜ ⎟

Previously it was assumed that the angular velocity of the particle (*vθ*) is neglected,

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

> 0 *<sup>P</sup> E r P <sup>C</sup> D vC r* ⎛ ⎞ <sup>∂</sup> <sup>−</sup> + = ⎜ ⎟

<sup>1</sup> 0. *<sup>P</sup> E P <sup>C</sup> D vC r*

θ

⎝ ⎠ <sup>∂</sup> (27)

<sup>∂</sup> <sup>=</sup> <sup>∂</sup> , 0 <sup>≤</sup> *<sup>r</sup>* <sup>≤</sup> *R,* 0 ≤ z <sup>≤</sup> *L.* (28)

<sup>∂</sup> <sup>=</sup> <sup>∂</sup> , 0 <sup>≤</sup> *<sup>r</sup>* <sup>≤</sup> *R,* 0 <sup>≤</sup> *<sup>z</sup>* <sup>≤</sup> *L.* (29)

⎝ ⎠ <sup>∂</sup> . (31)

, 0 ≤ *θ* ≤ 2*π,* 0 ≤ *z* ≤ *L.* (32)

At 0 ≤ *r* ≤ *R*, *C finite <sup>P</sup>* = , 0 ≤ *θ* ≤ 2*π,* 0 ≤ *z* ≤ *L.* (30)

Fig. 5. Dispersion coefficient (*DE*) as a function of the viscosity (*µL*) and density (*ρL*) of the fluid medium in a horizontal cylinder

Fig. 6. Dispersion coefficient (*DE*) as a function of the fluid velocity (*Q*) of the fluid medium in a horizontal cylinder

Several works regarding axial dispersion coefficients in horizontal channels were found in the reviewed literature. Nevertheless, large part of this literature is advocated to aerosol particles flowing through airways. For instance, Zhang et al. (2005) and Schulz et al. (2000) 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

Transport of Ultradispersed Catalytic Particles Through Bitumen at Upgrading Temperatures 723

Table 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

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 Fe2O3 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

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 (*μL* and *ρL* vs. *DE* and *<sup>z</sup> v* vs. *DE*); then, for each temperature three different values of *DE* were obtained (based on *μL*, *CP*0 and in *<sup>z</sup> v* ) and their average was recorded; finally, the 5 different *DE* average values (corresponding to each studied temperature and ranging from 2.1 - 2.18 × 10-9 m2/s) were averaged and the final *DE* value was obtained. The *DE* value obtained from

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 × 107 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

> 1 10 100 1000 10000 **Particle Diameter, nm**

Fig. 7. Effect of the particle diameter and temperature on the deposition of MoO3 particles

experiments are similar to those present in the simulations carried out in this section.

Reynolds number is between 0 and 2100 (Bird et al., 2007).

these calculations resulted to be 2.14 × 10-9 m2/s.

340 °C 350 °C 360 °C 370 °C 380 °C

1

11

21

31

41

51

**Normalized Maximum Concentration**

61

71

81

91

measure this parameter, the following analysis was conducted.

**150 nm**

immersed in Athabasca bitumen flowing through a pipe

for dispersion coefficients employed in the previously described processes were around 10-4 m2/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 m2/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 *μ*m of inner diameter.

#### **3.3 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 MoO3 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 (*ρL*) and viscosity (*μL*) 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 MoO3 particle diameters were used for the simulations: 1, 10, 50, 100, 150, 200, 250, 300, 400, 500, 1000 and 1500 nm. The initial MoO3 particle concentration (*CP*0) that was applied to all the simulations was 1.2526 mol/m3, equivalent to 180.3 ppm.

A fluid velocity (*Q*) of 30000 ml/min (5 × 10-4 m3/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* = 101.6 cm, the mean axial velocity ( *<sup>z</sup> v* = *Q*/(*π d*2/4)) is 6.17 × 10-4 m/s. This value permits the calculation of the fluid Reynolds number (Re = *ρ<sup>L</sup> <sup>z</sup> v d*/*μL*) 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 (Δ*P*) between the ends of the pipe which was calculated using Equation 18.


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

for dispersion coefficients employed in the previously described processes were around 10-4 m2/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 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 m2/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

**3.3 Ultradispersed catalytic particles immersed in bitumen at upgrading temperatures** 

This section is dedicated to the application of the three-dimensional convective-dispersive model to a large scale case which involves the transport of MoO3 catalytic particles immersed in Athabasca bitumen, at upgrading conditions (340-380 °C), through a pipe of

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

As in the previous section, the density (*ρL*) and viscosity (*μL*) 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

In this section, twelve different MoO3 particle diameters were used for the simulations: 1, 10, 50, 100, 150, 200, 250, 300, 400, 500, 1000 and 1500 nm. The initial MoO3 particle concentration (*CP*0) that was applied to all the simulations was 1.2526 mol/m3, equivalent to

A fluid velocity (*Q*) of 30000 ml/min (5 × 10-4 m3/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* = 101.6 cm, the mean axial velocity ( *<sup>z</sup> v* = *Q*/(*π d*2/4)) is 6.17 × 10-4 m/s. This value permits the calculation of the fluid Reynolds number (Re =

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 (Δ*P*) between the ends of the pipe which was calculated using Equation 18.

*T***, °C** *ρL***, kg/m3** *μL***, cP Δ***P***, mPa Re**  340 884.7 2.71 5.18 204 350 877.2 2.42 4.63 227 360 869.4 2.38 4.55 229 370 856.4 2.27 4.34 236 380 843.7 2.17 4.15 244 Table 3. Fluid Reynolds numbers and pressure drop between the ends of the pipe at the 5

diffusion coefficients were capillaries with 50 to 100 *μ*m of inner diameter.

at the bottom of the pipe once the steady state has been reached.

*ρ<sup>L</sup> <sup>z</sup> v d*/*μL*) which estimates if a fluid is laminar or turbulent.

the studied phenomena.

**transported through a pipeline** 

100 m length and 101.6 cm diameter.

interest were shown in Table 1.

different studied temperatures

180.3 ppm.

Table 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 Fe2O3 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 (*μL* and *ρL* vs. *DE* and *<sup>z</sup> v* vs. *DE*); then, for each temperature three different values of *DE* were obtained (based on *μL*, *CP*0 and in *<sup>z</sup> v* ) and their average was recorded; finally, the 5 different *DE* average values (corresponding to each studied temperature and ranging from 2.1 - 2.18 × 10-9 m2/s) were averaged and the final *DE* value was obtained. The *DE* value obtained from these calculations resulted to be 2.14 × 10-9 m2/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 × 107 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. 7. Effect of the particle diameter and temperature on the deposition of MoO3 particles immersed in Athabasca bitumen flowing through a pipe

Transport of Ultradispersed Catalytic Particles Through Bitumen at Upgrading Temperatures 725

*L P*

*<sup>L</sup>* <sup>=</sup> ∫

The different concentrations along the bottom of the cylinder can be retrieved from the simulation results; with these data the integral in Equation 34 can be evaluated using the

The mass of particles deposited at the bottom of the channel (*MP*), enclosed in the partial

On the other hand, the total mass of particles that entered to the tank (*MPT*) after a long

Considering that the initial particle concentration (*CP0*) in the simulation was assumed to be 180.3 ppm or 1.2526 mol/m3 and that the volumetric flux is *Q* = 5 × 10-4 m3/s, the total particle mass that entered to the cylinder after 1.5 × 107 s was 1352 kg. The percentage of deposited particles at the bottom of the spherical tank (%*Mdep*) can be calculated by Equation

*dp***, nm 340 °C 350 °C 360 °C 370 °C 380 °C**  1500 0.488 0.565 0.615 0.657 0.681 1000 0.372 0.484 0.534 0.610 0.637 500 0.142 0.203 0.218 0.264 0.327 400 0.056 0.067 0.070 0.076 0.084 300 0.029 0.031 0.032 0.034 0.035 250 0.022 0.023 0.024 0.024 0.025 200 0.017 0.018 0.018 0.019 0.019 150 0.015 0.015 0.015 0.015 0.015 100 0.013 0.013 0.013 0.013 0.013 50 0.012 0.012 0.012 0.012 0.012 10 0.012 0.012 0.012 0.012 0.012 1 0.012 0.012 0.012 0.012 0.012

Table 4. Percentage of deposited particles at the bottom of the horizontal cylinder (%*Mdep*) containing MoO3 particles immersed in Athabasca bitumen at different temperatures

temperatures and particle diameters is shown in Table 4 and Figure 9.

The percentage of deposited particles at the bottom of the spherical tank (%*Mdep*) at different

In Figure 9, it can be observed that after the studied time, less than 0.1 % of the total mass of particles, with diameters lower than 400 nm, has been deposited; whereas the percentage of deposited particles, with diameters in the range 400 – 1500 nm, is between 0.1 and 0.68 %.

<sup>0</sup> max

*P*

Simpson's rule for the solution of numerical integrals (Carnahan et al., 1969).

volume (*VP*) with height *H*, is:

11.

period of time (*t* = 1.5 × 107 s) can be estimated by:

*C*

( )

**.** (34)

*MPP P* = *C V* max . (35)

*MPT P* = *QC t*<sup>0</sup> . (36)

**%***Mdep*

*C L dL*

The normalized maximum particle concentration, which in this case is the concentration that is found at the exit bottom of the horizontal cylinder after the simulation time (steady state) divided by the initial concentration, was calculated for each simulation. 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 7 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 and as the particle diameter increases, the normalized maximum concentration increases for a specific 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 independently of the temperature. This means that MoO3 particles smaller than 150 nm will remain flowing through the Athabasca bitumen after a long period of time (1.5 × 107 s) in the range of temperatures from 340 to 380 °C.

The following procedure was employed for the calculation of the deposited mass of the catalytic particles along the bottom of the pipe once the steady state has been reached.

Assuming that the deposited particles filled a small horizontal section of height *H* at the bottom of a cylinder of radius *R* and length *L*, its volume (*VP*) can be calculated by:

$$V\_P = L \left[ \cos^{-1} \left( \frac{R - H}{R} \right) R^2 - \sqrt{H \left( 2R - H \right)} \left( R - H \right) \right]. \tag{33}$$

Fig. 8. Horizontal section of height *H* at the bottom of a cylinder of radius *R* and length *L*

Assuming that the height of this small gap is approximately *H* = 3.5 cm (based on graphics from the simulations), the volume of the small horizontal section (*VP*) for a cylinder of 101.6 cm of radius and 100 m of length is 0.871 m3.

The average particle concentration (*CP*max ) at the bottom of the channel and across the length *L* of the cylinder after a long period of time (*t* = 1.5 × 107 s) can be estimated by:

The normalized maximum particle concentration, which in this case is the concentration that is found at the exit bottom of the horizontal cylinder after the simulation time (steady state) divided by the initial concentration, was calculated for each simulation. 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 7 represents the behaviour of the normalized maximum particle concentration at

As temperature increases for a specific particle diameter, the normalized maximum concentration increases and as the particle diameter increases, the normalized maximum concentration increases for a specific temperature. It can also be observed that a change in particle diameter have a more pronounced effect over the normalized maximum

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 independently of the temperature. This means that MoO3 particles smaller than 150 nm will remain flowing through the Athabasca bitumen after a long period of time (1.5 × 107 s) in the range of temperatures from

The following procedure was employed for the calculation of the deposited mass of the catalytic particles along the bottom of the pipe once the steady state has been reached. Assuming that the deposited particles filled a small horizontal section of height *H* at the

> ( )( ) 1 2 cos 2 *<sup>P</sup> R H V L R H RHRH*

Fig. 8. Horizontal section of height *H* at the bottom of a cylinder of radius *R* and length *L*

Assuming that the height of this small gap is approximately *H* = 3.5 cm (based on graphics from the simulations), the volume of the small horizontal section (*VP*) for a cylinder of 101.6

The average particle concentration (*CP*max ) at the bottom of the channel and across the length *L* of the cylinder after a long period of time (*t* = 1.5 × 107 s) can be estimated by:

<sup>−</sup> <sup>⎡</sup> ⎛ ⎞ <sup>−</sup> <sup>⎤</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup> <sup>−</sup> <sup>⎢</sup> ⎜ ⎟ <sup>⎥</sup> <sup>⎣</sup> ⎝ ⎠ <sup>⎦</sup>

. (33)

bottom of a cylinder of radius *R* and length *L*, its volume (*VP*) can be calculated by:

*R*

different particle diameters and temperatures.

concentration than a change in temperature.

cm of radius and 100 m of length is 0.871 m3.

340 to 380 °C.

$$\overline{\mathbb{C}}\_{P\text{max}} = \frac{\int\_{-}^{L} \mathbb{C}\_{P}(L) \, dL}{L} \,. \tag{34}$$

The different concentrations along the bottom of the cylinder can be retrieved from the simulation results; with these data the integral in Equation 34 can be evaluated using the Simpson's rule for the solution of numerical integrals (Carnahan et al., 1969).

The mass of particles deposited at the bottom of the channel (*MP*), enclosed in the partial volume (*VP*) with height *H*, is:

$$M\_P = \overline{\mathbb{C}}\_{P \text{ max}} V\_P \,. \tag{35}$$

On the other hand, the total mass of particles that entered to the tank (*MPT*) after a long period of time (*t* = 1.5 × 107 s) can be estimated by:

$$M\_{PT} = QC\_{P0}t \,. \tag{36}$$

Considering that the initial particle concentration (*CP0*) in the simulation was assumed to be 180.3 ppm or 1.2526 mol/m3 and that the volumetric flux is *Q* = 5 × 10-4 m3/s, the total particle mass that entered to the cylinder after 1.5 × 107 s was 1352 kg. The percentage of deposited particles at the bottom of the spherical tank (%*Mdep*) can be calculated by Equation 11.


Table 4. Percentage of deposited particles at the bottom of the horizontal cylinder (%*Mdep*) containing MoO3 particles immersed in Athabasca bitumen at different temperatures

The percentage of deposited particles at the bottom of the spherical tank (%*Mdep*) at different temperatures and particle diameters is shown in Table 4 and Figure 9.

In Figure 9, it can be observed that after the studied time, less than 0.1 % of the total mass of particles, with diameters lower than 400 nm, has been deposited; whereas the percentage of deposited particles, with diameters in the range 400 – 1500 nm, is between 0.1 and 0.68 %.

Transport of Ultradispersed Catalytic Particles Through Bitumen at Upgrading Temperatures 727

This work was supported in part by the National Council for Science and Technology of Mexico, The Alberta Ingenuity Centre for In Situ Energy funded by the Alberta Ingenuity Fund and the industrial sponsors: Shell International, ConocoPhillips, Nexen Inc, Total Canada and Repsol-YPF, and The Schulich School of Engineering at the University of

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**5. Acknowledgment** 

Calgary, Canada.

**6. References** 

York

York

York

pp. 4088-4093

pp. 4094-4100

pp. 1920-1930

No. 5186 pp. 773-776

Fig. 9. Percentage of deposited particles at the bottom of the cylindrical channel (%*Mdep*) at different temperatures and particle diameters for the deposition of MoO3 particles immersed in Athabasca bitumen flowing through a pipe

### **4. Conclusion**

The results from these simulations unveiled the particle diameter and fluid medium properties that are necessary to ensure particle suspension and mobility in bitumen. In the case of the spherical storage tank simulation, it was found that MoO3 particles smaller than 150 nm will remain suspended in bitumen at temperatures from 340 to 380 °C after a long period of time (27778 h) Also, it was shown that the percentage of deposited particles at the bottom of the tank after this period of time is less than 0.02 % when particles diameters below 200 nm are utilized.

Regarding the simulations applied to the flow of particles through a horizontal pipeline, the results demonstrated that particles smaller than 150 nm will remain flowing through bitumen after a long period of time (4167 h) in the range of temperatures from 340 to 380 °C. In addition, the modelling results showed the percentage of settled particles at the bottom of the cylinder after this period of time when a particle deposition scenario is presented. It was observed that less than 0.1 % of the total mass of particles (that passed through the cylinder during this time), with diameters lower than 400 nm, were deposited at the bottom.

Based on these results, it seems that when MoO3 catalytic particles with diameters lower than 400 nm (nanometric range) are immersed in bitumen either in a stagnant or a flow scenario, almost all the particles will remain suspended in the system. Therefore, this kind of particles could be employed as ultradispersed catalysts for bitumen hydroprocessing reactions. It would be interesting to compare these models with experimental data from real systems; however, on line tools for the particle concentration measurement would have to be developed in order to achieve this objective.
