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Computational fluid dynamics (CFD) simulation of three-phase non-Newtonian slurry flows in industrial horizontal pipelines

Understanding the flow behavior of complex concentrated slurries is of tremendous importance for industrial waste management. In this study, the transport of three-phase oil sands tailings in a horizontal pipeline is simulated via the mixture multiphase model coupled with the kinetic theory of granular flow. The solid particles and bitumen droplets are conveyed via a non-Newtonian carrier fluid in a turbulent regime inside an industrial-scale pipeline. The simulation results showed exceptional agreement with the field data, with errors of<3.5% for velocity distribution and<15% for the pressure drop. A systematic parametric investigation was performed for a wide range of flow conditions, showing that the majority of bitumen droplets reside at the top region of the pipe. Our findings may help design an effective process for the separation of bitumen residues during pipeline transport.


Introduction
Concentrated slurry flows are frequently encountered in many industries such as minerals extraction [1], food and farms [2], and petroleum [3]. Transport of extraction or production wastes, i.e., tailings in horizontal pipelines is one example of where the slurry flow management is relevant. A slurry flow is a multiphase system where solid particles are transported via water as the carrier. In reality, the solid particles in the slurry are different in size, also called as polydisperse particles [1,4]. The presence of fine particles, i.e., particles with sizes less than 44 µm can alter the rheological behavior of the carrier by forming a pseudo-homogeneous mixture with water. The carrier fluid in slurry transport usually behaves as a non-Newtonian fluid, possessing a yield stress and a decrease in viscosity as the shear rate increases, i.e., shear-thinning fluid [1,5]. Considering the relevant flow velocities in the industry, and the pipeline characteristics, the transport process may occur in a turbulent regime. A turbulent slurry flow can be classified as a highly complex system, and development of a predictive model is of great importance for pipeline design and optimization of the transport process.
Several groups of researchers have conducted experimental, modeling, and simulation works to understand the slurry flow characteristics and optimize the process with various configurations and applications [6]. However, these type of studies have been performed mostly for two-phase slurries. Experimental investigations in pilot and lab-scale pipelines have been thoroughly reported in the literature for both Newtonian [7][8][9][10][11][12] and non-Newtonian [5,[13][14][15] carrier fluids. For a Newtonian carrier fluid, the researchers at the Saskatchewan Research Council have reported experimental data for slurry flows in various systems and different flow conditions [7][8][9][10], they extensively investigated the concentration distribution of coarse particles and determined the effect of flow parameters such as mixture velocity on the solids concentration distribution. Moreover, experimental data of concentration distribution and pressure drop has been presented by Kaushal et al. [11] and and Kaushal and Tomita [12] for flow velocities up to 5 m/s and solids volume fraction up to 0.5, where they identified the maximum concentration for smaller particles closer to the bottom region, but closer to the core for coarse particles. Furthermore, Vlasák et al. [16] reported the pressure gradient and solids concentration distribution for turbulent flow inside inclined pipes, with flow velocities up to 2.07 m/s and solids volume 2 fraction up to 0.34, where they identified insignificant changes in solids concentration distribution with relatively small changes in pipe inclination, e.g., less than 30°.
For the non-Newtonian carrier fluid, recently Peník et al. [17] and Zheng et al. [5] reported the concentration distribution and pressure drop in a two-phase flow with a carrier fluid fitted with a Herschel-Bulkley model in a turbulent regime. The solids volume fractions were up to 0.2 and 0.16, and flow velocities were up to 4.51 and 3.67 m/s in the works by Peník et al. [17], and Zheng et al. [5], respectively, and more symmetrical solids concentration distributions in case of higher mixture velocities were identified in both works. Matoušek et al. [18] used a Herschel-Bulkely carrier fluid to drive a solid bed to sliding and measured the pressure gradient and mean delivered solids concentration for flow velocities up to 4.76 m/s, and confirmed the suitability of a proposed two-layer model for the prediction of viscous frictions in a laminar regime. Moreover, Ignatenko et al. [15] reported only the pressure drop data for the Herschel-Bulkley carrier fluid in a turbulent regime with flow velocities up to 5.2 m/s and particles concentration of 10%.
Turian et al. [19] presented experimental data of pressure gradient for both laminar and turbulent slurry flows with solids volume fraction up to 0.252 where the carrier fluid follows the Sisko rheological model. Studying the effect of mixture velocity on the pressure gradient is the common feature of the works by Ignatenko et al. [15], and Turian et al. [19], while the evolution of pressure gradient in full transition of laminar to turbulent flow was reported. For a power-law carrier fluid, Eesa and Marigo [14] measured the solid-phase velocity distribution in a laminar flow with outlet solids volume fraction up to 0.4 and flow velocities up to 125 mm/s. In their earlier study, Eesa and Barigou [13] reported the velocity distribution in a similar system and flow conditions but with an Ellis carrier fluid, where they reported more symmetrical solid velocity distributions in the case of smaller solid particles and higher mixture velocities and solids concentration.
As complement to experimental studies, the development of reliable models validated through a broad range of flow conditions is of great importance for understanding and predication of the slurry flow behavior. The attempts to simulate the slurry flow behavior started with the theoretical models and empirical and semiempirical correlations.
Gillies and Shook [8] conducted systematic experiments to measure the concentration distribution of coarse particles in radial direction, and proposed a method to predict the concentration distribution neglecting the particles interactions. In their other studies, they proposed an approach to predict the head loss called the two-layer model originated by Wilson [20] and extended it to various conditions [7,9]. In Wilson [20]'s model, a stratified solid-liquid flow has been modeled assuming two separate layers in which the mixture was considered in single phase having uniform velocity and solid concentration. Similarly, other groups of researchers have developed Two-Layer [21][22][23] and Three-Layer models [24,25].
In parallel with the efforts to improve experimental results and analytical models, computational fluid dynamics (CFD) is becoming more comprehensive to investigate the basic fundamental understanding of slurry flows in a pipeline. Different multiphase flow modeling approaches such as Eulerian-Eulerian [26][27][28][29], Eulerian-Lagrangian [30][31][32], and mixture model [3,15,33] have been utilized. Recently, Messa and Matoušek [34] and Alobaid et al. [35] presented a comprehensive description and comparison of the multiphase modeling approaches.
In the Eulerian-Eulerian (or Two-Fluid) model, the phases are treated as interpenetrating continua, and governing equations are applied to each phase while the phases can exchange momentum via interphase forces. The kinetic theory of granular flow is usually coupled with the Eulerian-Eulerian model to capture the effects of particles interactions on the flow field. This approach makes the consideration of chemical reactions and mass transfer between the phases possible, and offers a relatively good compromise between the computational cost and accuracy [35]. In the Eulerian-Lagrangian scheme, the particles, droplets or bubbles (based on the type of the secondary phase) are tracked individually, and Newton's law of motion is applied to each particle. This approach is especially applicable when the concentration of the secondary phase is dilute, otherwise the computational cost will be tremendous due to the high number of equations to be solved [35,36]. The mixture model is a simplified Eulerian multiphase model that can be beneficial in modeling of complex multiphase systems , e.g., when the number of secondary phases is high or the interactions between the secondary phases are not well-known [37].
The treatment of the multiphase mixture and lower number of differential equations to be solved, make the mixture model an efficient method for the systems with polydisperse particles and multiple phases [36]. Multiphase slurry systems, cyclone separators 4 [38,39], cavitation processes [40], and nanofluidics [41] are some examples of where the mixture model can be applied and has been used previously by other researchers. For the multiphase slurry systems, although numerous efforts have been made to develop models capable of predicting the flow behavior, the lack of a model which can capture the non-Newtonian behavior of the carrier and effects of particle size distribution (PSD) is sensed. Notably, the effects of an additional secondary phase and mutual effects of other phases and flow conditions on the additional phase is another aspect that is not studied in the open literature.
The oil sands tailings slurry is a byproduct of bitumen extraction, a mixture with three phases consisting polydisperse solid particles ( about 55 wt% ), water, and a minute fraction of residual bitumen (about 1 wt%) [42,43]. The oil sands tailings are transported to large open areas called tailings ponds and stored so the particles settle, the water is recycled, and the land could be reclaimed. However, the residual bitumen in the slurry is not treated and separated, and can pose substantial threats to the environment, including decomposition into greenhouse gases and contribution to the global warming [44]. In this regard, separation of bitumen from the tailings before disposal to the tailings ponds is crucial. Understanding the behavior and the effective parameters of the tailings flow is a significant step toward designing a separation technology. Nevertheless, the complexity of the mixture and the low fraction of bitumen in the slurry makes the separation process difficult. There is no predictive model for the transport characteristics of slurry flow in large-diameter pipes, especially for multiple secondary phase solids along with bitumen droplets. Importantly, the effects of PSD and non-Newtonian viscosity have not been studied thoroughly in a complex multiphase flow system. Therefore, in depth understanding of slurry flow characteristics and operating conditions are important to design a suitable process.
In the present study, a turbulent three-phase tailings slurry flow in an industrialscale pipeline was successfully modeled using the mixture multiphase model coupled with the KTGF. The flow consists of highly concentrated polydisperse solid particles, a non-Newtonian carrier fluid and a small fraction of bitumen droplets. The model was 5 validated against ten sets of field data of pressure gradient and velocity distribution.
Systematic parametric study was performed to investigate the effect of mixture velocity, particle size distribution, non-Newtonian viscosity, and pipe angle on the flow behavior.
To the best of our knowledge, the present study is a first-of-a-kind on highly concentrated slurry transport in a industrial pipeline. The results of this study would contribute to better understanding of three-phase slurry system and would be used for pipeline design and optimization, as well as a guideline for designing a separation process inside the pipeline. For the velocity measurements, a velocity profiler is used. Fig. 1b shows the schematic view of the measurement locations, from VF1 at the top to VF5 at the bottom of the pipe.

Collection of field data
The velocities are measured using a SANDtrac Velocity Profile System (CiDRA) based on a non-invasive technique. This device is an array of five sensors attached to the pipe wall, which track the turbulent eddies inducing pressure disturbance and consequently forces on the wall. This array senses the dynamic strains applied to the pipe by these forces and converts them to electrical signals to be interpreted, and the velocity values are calculated [46]. This technique is well-established and applied commercially for slurry 6 (b) (a)

Mixture model
In this section, the theory and equations of the mixture multiphase model are provided. The flow field of the mixture in the pipeline is predicated by solving the conservation of mass and momentum equations in three-dimensional Cartesian coordinates using mixture model [49] based on finite volume method. The continuity and momentum balance equations are written and solved only for the mixture phase, with an algebraic expression for the slip velocity between the primary and secondary phases.
The equations for conservation of mass and momentum are given in Table 1 (2)a. In this equation, n is the number of phases, F is a body force, and µ m is the viscosity of the mixture (µ m = n k=1 a k µ k ). V dr,k is the drift velocity for secondary phase k, and is defined as the relative velocity between k phase velocity and the velocity of center of mass. The slip velocity (V pq ) is defined as the velocity of a secondary phase (p) relative to the velocity of the primary phase (q), and is connected to the drift velocity as Eq.
(2)c, where c k = α k ρ k ρm . Manninen et al. [49] mixture model assumes a local equilibrium between the phases over a short spatial length scale, and uses an algebraic slip formulation.
In a turbulent regime, the slip velocity can be expressed as Eq. (2)d. In this equation, τ p is the particle relaxation time τ p = ρpd 2 p 18µq , d is the diameter of the particles (or droplets or bubbles) of secondary phase p, a is the secondary phase particle's acceleration. σ t is a Prandtl/Schmidt number set to 0.75 and η t is the turbulent diffusivity.
The drag functions f drag are taken from Gidaspow et al. [50] (fluid-solid) and symmetric (fluid-fluid) model between carrier and secondary phases. For the liquid-solid drag model (the drag model for the carrier fluid and solids), numerous studies in the literature show that the model proposed by Gidaspow et al. [50] gives an accurate expression [27]. Nevertheless, as the number of studies for the three-phase simulation with two fluid phases is limited, no reliable analysis for the choice of drag model between the 9 carrier and bitumen can be found in the literature. In this regard, the available drag models in Ansys Fluent were tested to find the most appropriate drag model in this case.
The volume fraction equation for secondary phase p can be achieved from the continuity equations as shown in Eq. (2)e.

Kinetic theory of granular flow (KTGF)
The kinetic theory of granular flow was implemented to capture the effects of particle-particle interactions on the flow behavior. The equations of frequently used KTGF sub-models are given in Table 2  is the average diameter, and p c,qp is the collisional part of the pressure between phases q and p. In Eq. (3)i, p c,qp = 2 1 + e pq g 0,pq α q α p . When the number of solid phases is greater than 1, the radial distribution function between the p th and q th solid phases is expressed by Syamlal et al.

Turbulence model
In our previous study [27], we highlighted the suitability and performance of the Two-Equation turbulence models (k − and k − ω) for the multiphase slurry flows.
Especially, based on the previous works in the literature and the theory of the turbulence models, we emphasized the acceptable and accurate prediction of the turbulent regieme in a multiphase flow using the k −ω model proposed by Menter [52]. In the present study, at first both k − and k − ω models were examined, and the SST k − ω showed a better numerical stability leading to faster convergence, and excellent agreements between the CFD predictions and field data. Fraction 1 of their study has the closest size distribution to the fine particles in this study.

Rheological model
Therefore, the Casson non-Newtonian rheological model considered in this study through user-defined function (UDF), which is a specific as carrier fluid viscosity: In Equation 4, τ is shear stress, τ y represents yield stress, and µ c is Casson viscosity [53]. In case there is a gradual transition from Newtonian behavior to yield region, the Casson rheological model is appropriate to describe the carrier fluid viscosity [54].
The rheological behavior of tailings slurries has been successfully modeled using Casson rheological model by other researchers [55][56][57]. The flow conditions of the simulated cases such as the mixture velocity, solids fraction, carrier density and bitumen fraction can be seen in Table 4. These cases will be used for validation in section 3.4, and Case 2 will be used as the base case for grid-independence check and further investigations.

Composition of tailings slurry
The PSD information in five different measurements is presented in Fig. 2a. As discussed earlier, a sieving method is used to determine the size distribution for solid particles, with mesh sizes from 1.3 to 2000 µm. Particles with sizes below 44 µm are considered as fines and incorporated in the carrier fluid, and the fraction for particles larger than Slip velocity Manninen et al. [49] 2000 µm is zero in all of the cases. With these considerations, eight bins for coarse solid particles are formed, from d p = 75 to 1000 µm. The compositions data is reported in the form of mass fraction, which is first converted to volume fraction with having the densities of solids and water, and later on, used to calculate the carrier fluid density. It must be noted that Set 1 in Fig. 2 is the PSD information for the base case and used for all of the cases simulated in the validation section. The slurry density of three representative time windows is depicted in Fig. 2b.   Accordingly, for the analysis of the CFD results in the the radial direction, a plane at z = 100 m was used to export the data.

Model validation
To demonstrate the accuracy and validity of the developed model, CFD simulations were compared with field data of velocity distribution and pressure gradient. The accordance of the CFD predictions with field data demonstrates that the physics of the system has been well defined, and the model can be considered a reliable tool to perform a parametric study on a similar system. The comparisons between CFD-predicted and  Table 4. Fig. 8 depicts the comparison between the CFD-predicted velocity distribution and the measured ones at the field for six different cases. The results demonstrate exceptional agreement between the carrier fluid velocity data exported from the CFD simulations and measured at the field. For most of the data points, the CFD and field values overlap, with insignificant discrepancies for some data points.
The accuracy of the model and validity of the physics defined for the system was also examined by the comparison of pressure gradients predicted by CFD and measured pressure at the industrial pipeline. Overall, the comparison was made for 10 distinct cases over 10 different time windows, and the results are presented in Fig. 9. For all of the cases, the predictions are in accord with the field data with an error margin of less than 15% which can be considered highly accurate for an industrial-scale pipeline. The details of the flow conditions, such as the mixture velocity, carrier density, total solids concentration, etc., are presented in Table 4. The frictional pressure drop inside a pipe results from particle-wall and particle-particle interactions [60]. In summary, the model was proven to be accurate and sufficiently sophisticated in physics based on the comparison of CFD-predicted velocity distribution and pressure gradient with field measurements. only parameter that distinguishes the phases in mixture model. In Ansys Fluent, the available drag models are (a) schiller-naumann [61], (b) morsi-alexander [62], (c) universal-drag [63], (d) symmetric [37], (e) ishii-zuber [64], and (f) tomiyama [65].  As bitumen is a dispersed phase in the form of droplet, a diameter needs to be considered for these droplets in the simulation. There are multiple studies in the literature for size characterization of bitumen droplets and aggregates in the flotation column, however, no data has been reported for bitumen droplet size in the flow in a horizontal pipeline. For the size distribution of bitumen droplets in a flotation column, it has been reported that most droplets are less than 800 µm in size [66,67]. On this basis, a comprehensive analysis was performed to find the appropriate diameter for bitumen droplets in which the CFD prediction are in accord with the field data with the lowest error. The maximum diameter considered for the droplet was 800 µm as it was assumed that the droplets can not be larger than the aggregates in the flotation column, and the lower band was considered to be 100 µm. It was found that the flow behavior is almost insensitive to the bitumen droplet size. This was expected as the volume fraction of bitumen is low and the considered range for the size is not very broad. Fig. 11 shows the velocity distribution is in the same form with droplets sizes of 100 µm and 400 µm for three representative cases. However, the diameter of 400 µm overall gave better predictions of velocity and pressure gradient, and was selected as the input for the simulations in this study.  In the first section of Table 5

Solids concentration distribution
One of the most important aspects of the slurry flow behavior is the distribution of solid particles in radial direction. The solids concentration distribution influences other flow parameters and conditions such as pressure drop and velocity distribution directly and is of great importance in pipeline design [26]. As discussed earlier, the solid particles in this study are polydisprese and have been considered as separate secondary phases in the mixture model. This approach makes the investigation of concentration distribution for solid particle with specific sizes possible. In Fig. 13, the contours of concentration distribution of solid particles are shown for Case 2 from the smallest to largest particles, respectively. As depicted in Fig. 13a-h, the particles with smaller sizes (75 to 250 µm) are more dispersed in the flow while larger particles (Fig. 13e-h)   gravitational effect, which makes the larger particles with higher weights to settle, but smaller particles are more prone to dispersion owing to lower weights. Based on these results, the pipe in the cross-sectional view can be divided into three zones as explained by Liu et al. [33]. The bottom section is the first zone where the larger particles settle, the second zone is the central region where the flow is steady and the particles are transported with the flow and are smaller in size, and the top part is the third zone where the small particles are suspended.

Parametric study
In this section, a systematic parametric study is presented to investigate the effect For this parametric study, a base case with the flow conditions described as Case 2 in Table 4 was chosen. The particles size distribution information was provided in

Effect of mixture velocity
The mixture velocity has been proven to have a significant effect on the flow behavior of slurries [27]. In this section, the effect of mixture velocity has been investigated by varying the flow velocity from 3.5 m/s to 6.5 m/s. In the real process which has been used for validation, the mixture velocity is in a range of 5.1 to 5.7 m/s, which leads to a highly turbulent flow considering the pipe diameter. Reducing the flow velocity to 3.5 m/s can give an idea about the flow behavior in a weak turbulent regime, and by increasing the velocity, the role of a stronger turbulence can be better understood in a three-phase flow with a Casson carrier fluid.
In Fig. 14  Similar trend is seen in the work by Zhang et al. [29] but for the solid phase velocity distribution, where the mixture velocity seems to have a trivial effect on the normalized velocity compared to other parameters.
The pressure drop shows a significant dependency on mixture velocity as depicted in Fig. 14 b. As the mixture velocity increases and the flow becomes more turbulent, the particle-particle and particle-wall collisions are enhanced significantly, leading to an  in the lower region of the pipe is seen. This shows that more solid particles are suspended owing to the intensified turbulent regime and its mixing effects, but still turbulence is not able to fully suspend the large particles settled on the pipe invert. Particle-particle interactions also contribute to higher suspension of solids at higher velocities, as these interactions are enhanced with increasing the velocity [26]. These findings are in accord with the results achieved by Kaushal et al. [28] for a two-phase slurry flow with fine particles, they mention that at the higher mixture velocity solid particles tend to migrate to the central part of the pipe due to slip velocity, and higher mixing effect from turbulence. Similar trend was also reported by Li et al. [69] from both experimental and simulation results with the mixture model, where the concentration of solid particles decreases near the pipe invert, and the asymmetry of the concentration curve is reduced at higher velocities.
The effect of mixture velocity on the cross-sectional bitumen concentration distribution is presented in Table 5. As the mixture velocity increases, the droplets are more

Effect of particles size distribution (PSD)
In this section, the effect of PSD is investigated, by keeping the total solids concentration constant and varying the fraction of solid phases (corresponding to different sizes). Fig. 15a depicts PSD information for the four cases simulated for this section.   The effect of PSD on the frictional pressure drop in the pipe is presented in Fig. 16 b.
The pressure drop increases with shifting the maximum fraction to larger particle sizes, with showing a significant rise from Set 3 to Set 4. This increase in pressure drop was expected and is in accord with other results shown in this section. More particles settlements and particle-wall collisions, also intensified particle-particle collisions due to the lower distance between the particles lead to more energy loss and pressure drop. As mentioned before, the pressure drop in a slurry flow is mostly a result of particle-particle and particle-wall interactions, both intensified for the cases with higher fractions of large solid particles. From Set 1 to Set 3 case, the pressure gradient shows a 5% increase, while from Set 3 to Set 4, a 20% shift is seen in the value of pressure gradient. The effect of PSD on the pressure gradient is similar to the effect of particle size in a two-phase flow with monodisperse particles. Several researchers have studied the effect of particle size on the flow behavior, and reported an increase in the pressure gradient as the particle size is raised [27,29,70]. It must be noted that the trend and magnitude of changes in the pressure gradient with respect to PSD is different from slurries with single-sized particles.
In Fig. 16 c, the chord-averaged turbulent kinetic energy is presented to evaluate the effect of PSD on turbulent regime and velocity fluctuations. The results show that for all of the cases, the turbulent kinetic energy shows a minimum at the pipe center, and local maximums in the bottom half of the pipe. For the fourth case (Set 4), the turbulent kinetic energy is approaching to zero near the wall, which is due to a dense solid bed at the pipe invert, damping the turbulence dramatically. This is consistent with the analysis presented by Antaya et al. [71] and Gopaliya and Kaushal [72], which show a lower magnitude in the turbulence modulation where the solids concentration is higher.
However, for the top region of the pipe in the Set 4 case, the turbulent kinetic energy is higher the other cases. This shows that for this case, turbulent is stronger and more active in this region, constantly changing the direction of the velocity vectors. This is in accord with the velocity distribution depicted in Fig. 16 a which shows higher velocities at the top region for Set 4.
The effect of PSD on the bitumen concentration distribution is quantitatively analyzed in Table 5. As shown, with increasing the fraction of larger sizes from Set 1 to Set 4, a slight increase in the bitumen concentration at the top half of the pipe is seen, which is due to less small particles present in the flow to occupy that region along bitumen.
However, the difference for the top 25 % is less pronounced, from 43.08 % for Set 1 to 43.7 % for Set 4 case. Overall, the bitumen concentration distribution in the vertical direction seem to be insignificantly dependent on PSD along with the other parameters. in more suspension of the particles in the carrier fluid (Fig. 17 d). With increasing Casson viscosity, there is a consistent decrease in total solids concentration at the pipe invert.
This observation could be due to the more viscous carrier fluid, having a higher core viscosity which leads to more support of the particles in the flow and more suspension.
Casson plastic viscosity quantifies the internal frictions of a fluid, and it is proven that a more viscous carrier fluid can support more particles in the core region and prohibit the stratification of the flow.
The carrier velocity distributions for the simulated cases are presented in Fig. 18 a.
For case with a higher Casson viscosity, the velocity distribution is more symmetric. In Furthermore, the effect of weakened particle-wall collisions are more effective in reducing the pressure compared to the increase in pressure drop due to more viscous fluid.
In Fig. 18 c, the radial distributions of turbulent kinetic energy for the four cases are depicted, showing an inverse proportion especially in the core region of the pipe between the turbulent kinetic energy and Casson viscosity. The relatively lower turbulent kinetic energy for µ c = 0.01 case at the pipe bottom half may be due to the presence of more solid particles in that region for this case (see Fig. 17 d) similar to our previous studied parameters. Also as mentioned, turbulence is weaker for the cases with higher Casson viscosity.
These results are consistent with the analysis performed by Zheng et al. In the fourth section of There is a decrease in the bitumen ratio due to the coexistence of more solid particles with the bitumen droplets at the top region for the case with the highest Casson viscosity, as shown in Fig 17.

Effect of pipe angle
In this section, two cases for the inclination with angles of +5 and +10°and two cases for declination with angles of -5 and -10°are simulated. Other pipe characteristics such as diameter and length also the flow conditions are kept constant in these cases. Fig. 19a shows the carrier velocity distribution in z-direction. The pipe angle has a significant effect on the velocity distribution as the effect of gravity is changing in the cases with the angle (in y-direction). In an inclined pipe, the flow has a higher velocity at the upper half of the pipe, while the velocity distribution is fairly symmetrical in case of a declined pipe. The effect of pipe angle on the pressure drop is shown in Fig. 19b.
The results demonstrate that the pressure drop has a linear relationship with the pipe angle. It must be noted that the negative sign of the pressure drop in the declined cases comes from the zero gauge pressure defined at the pipe outlet as a boundary condition.
The experimental and model results achieved by Doron et al. [73] confirm this trend in the pressure drop with the pipe tilt. In case of the declined pipe, the gravity acts as an driving force for the flow and can eliminate the energy required from the pumps if the slope is high enough.
The pipe angle was proven to have also a nontrivial effect on the turbulence regime as shown with the analysis of chord-averaged turbulent kinetic energy in the radial direction in Fig. 19c. As shown in Fig. 19c, there are more significant velocity fluctuations and consequently stronger turbulence at the top region for the declined cases as well as at the bottom for the inclined cases. As shown in Fig. 19d, solids concentration distribution is weakly dependent on the pipe angle, with slight differences at the pipe obvert and invert.
This observation is in accord with the experimental results reported by Kesely et al. [74] and Vlasák et al. [16] for two-phase slurries in declined and inclined pipes, which shows when the difference in the pipe angle is not large (less than 20°), the particles distribution is similar. The tendency of solid particles to settle is slightly higher in inclined pipes, which is in accord with previously shown results for velocity distribution and pressure drop. Since the effective flow area of inclined pipes is smaller due to more solid particles at the invert, velocities at the top are higher, while enhanced particle-wall particle-wall interactions result in higher pressure drops.
As shown in Table 5, the pipe angle has a non-uniform effect on the bitumen concentration distribution. In the case of shifting from the negative values of the pipe angle to the positive values of the pipe angle (inclined pipes), the bitumen ratio increases at first, then declines further as the angle increases. However, the magnitude of the changes in bitumen concentration distribution with the pipe angle is insignificant and can almost be ignored.

Conclusions
We developed an efficient three-dimensional computational CFD model to study the flow behavior of a turbulent three-phase non-Newtonian tailings slurry flow in an industrial pipeline. The Gidaspow drag model was selected for carrier-solids, and the sub-models for KTGF were selected from the tested standard models. The SST k-ω turbulence model was used to model highly turbulent flow in a pipeline. The Casson rheological model coupled to capture the non-Newtonian behavior of the carrier fluid in a slurry system. The accuracy of the model was proved by validating the results against collected filed data. The CFD results showed exceptional agreement with the field data, with errors of < 3.5% for velocity distribution and < 15% for the pressure drop. We further extended our CFD model applicability to study a comprehensive parametric study to investigate the influence of mixture velocity, particle size distribution, carrier fluid rheology, and pipe angle on the behavior of the slurry flow, including bitumen concen-