# Computational Fluid Dynamics Modelling of Liquid–Solid Slurry Flows in Pipelines: State-of-the-Art and Future Perspectives

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## Abstract

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## 1. Introduction

#### 1.1. Engineering Aspects and Physical Features of Slurry Pipe Transport

#### 1.2. Modeling of Slurry Pipe Flows

#### 1.3. The Potential of Computational Fluid Dynamics

^{−7}, whereas particle–particle interactions (either direct or indirect) play a role for ${\mathsf{\Phi}}_{\mathrm{s}}$ above 5 × 10

^{−4}, and, finally, the flow becomes dominated by particle–particle contacts above for ${\mathsf{\Phi}}_{\mathrm{s}}$ above 10

^{−1}. However, several issues arise when trying to employ this classification for the simulation of slurry pipe flows. In fact, if ${\mathsf{\Phi}}_{\mathrm{s}}$ were replaced with any of the two bulk solids concentrations ${C}_{\mathrm{vd}}$ and ${C}_{\mathrm{vi}}$, then most slurry pipe transport processes would be regarded as dense, contact-dominated flows. Nonetheless, that would not be a correct approach, since the flow velocity affects the transport mechanisms of slurry flow even if the concentration is the same. For instance, long-lasting inter-granular contacts take place if a bed of particles forms at the bottom of the pipe, which typically occurs at low velocity (Figure 1b,c). However, as the flow velocity increases for a given concentration, the particles become re-suspended and thus the flow is governed either by quick particle–particle collisions (Figure 1b) or by the interactions between the particles and the turbulent flow (Figure 1a,b). The considerations here above indicate that the relation between the flow patterns of slurry pipe transport in Figure 1 and the flow regimes of particle-laden flows in Figure 3 is not direct to find out. Probably, the only reasonable claim is that fully-stratified flows are contact dominated, whereas for all other flow patterns (from homogeneous to heterogeneous) there does not exist a clear match with the regimes in Figure 3. Additionally, values of local solid concentration lower than about 10

^{−4}are typical for air–solid flows, in which the density ratio between the solids and the carrier fluid is higher, but they are almost impossible to handle in slurry flow experiments. Referring to the limits of Elghobashi [29] would imply that, in practice, no slurry flow could by simulated ignoring particle–particle interactions, but this is not the typical approach in the engineering practice. For instance, slurry erosion simulations are often performed under the assumption of a one-way coupling regime even up to solid concentrations of about 1%, as will be discussed later in Section 3 and Section 4.2. The considerations drawn so far suggest that, indeed, the type of interactions between the phases and the key flow mechanisms driving particle transport must be correctly captured by the CFD model in order to provide a satisfactory simulation of the two-phase flow. However, the criteria of Elghobashi [29] might not be directly employed for slurry flow simulation because of their local nature and because they involve extremely low values of volumetric concentration, appearing “too conservative”. In this regard, the validity of the modelling assumptions concerning the coupling regime might not be regarded as an absolute, but dependent upon the scope of the numerical study.

#### 1.4. Scope and Structure of This Paper

## 2. CFD Modelling Approaches

#### 2.1. Eulerian–Lagrangian Modelling

#### 2.1.1. One-Way Coupled Slurry Flows

#### 2.1.2. Two and Four-Way Coupled Slurry Flows

#### 2.1.3. Boundary Conditions

#### 2.2. Eulerian–Eulerian Modelling (Two-Fluid Modelling)

#### 2.2.1. Fundamental Conservation Equations

#### 2.2.2. Constitutive Equations

#### 2.2.3. Interfacial Momentum Transfer

#### 2.2.4. Modelling of Turbulent Flows

#### 2.2.5. Boundary Conditions

#### 2.2.6. Multi-Fluid Modelling

#### 2.3. Mixture Modelling

#### 2.3.1. Fundamental Conservation Equations

#### 2.3.2. Closure Equations

**-**${\epsilon}_{\mathrm{m}}$ turbulence model. Other models evaluate ${\mathit{\tau}}_{\mathrm{m}}^{\mathrm{t}}$ from the contributions of the individual phases, employing differential or algebraic turbulence models for both liquid and solids. Finally, from the theory guide of Ansys Fluent, it seems that the mixture model embedded in that code does not include the term $\nabla \xb7{\mathit{\tau}}_{\mathrm{m}}^{\mathrm{t}}$ in the momentum equation for the mixture. Nonetheless, an eddy viscosity still appears in the same equation because, in Ansys Fluent, the diffusion component of the relative velocity is not neglected in the diffusion stress term (therefore, the full Equation (74) is solved) and, as already said, such diffusion component depends on an eddy viscosity. Thus, the mixture model equations must be solved coupled with a turbulence model.

#### 2.3.3. Boundary Conditions

## 3. Sources of Uncertainty in the CFD Modelling of Slurry Flows

#### 3.1. Numerical Features Producing Uncertainty

#### 3.2. Modelling Features Producing Uncertainty

#### 3.2.1. Modelling Sources of Uncertainty of Eulerian–Lagrangian Models

#### 3.2.2. Modelling Sources of Uncertainty of Two- and Multi-Fluid Models

#### 3.2.3. Modelling Sources of Uncertainty of the Mixture Model

#### 3.2.4. Summary and Recommendations

## 4. Review of Previous Investigations

#### 4.1. Overview of Published Literature

- Firstly, regarding their topic. A total of 69 articles, corresponding to about 80% out of the total 86, concern the modelling of particle transport in slurry pipelines, generally at a high solid volume fraction. Conversely, the remaining 20% are focused on slurry erosion of pipeline components, typically pipe bends, at moderate solid concentration.
- Secondly, regarding the software used. As shown in Figure 15a, more than half of the investigations were performed using the Ansys Fluent code, which embeds all types of models described in Section 2; other commercial codes used were PHOENICS and Ansys CFX. A significant fraction (≈17%) of the numerical studies were performed using in-house codes, which mostly applies to the pioneering investigations. The category field “other” in Figure 15b include papers in which the use was made of a combination of different open source or commercial software, as well as those where no information was provided.
- Thirdly, regarding the modelling approach. As shown in Figure 15b, Eulerian–Lagrangian models, either including or ignoring particle–particle interactions, were used in around 30% of the total published articles, indicating that the Eulerian approach is the preferred one for the modelling of slurry pipe flows. However, the data must be interpreted in the light of the topic of the study: in fact, almost all papers concerning slurry erosion used the Eulerian–Lagrangian models, whereas the slurry transport at high concentrations has been rarely simulated using this approach. It is also interesting to underline the relation between the type of modelling approach and the used software: for instance, almost all investigations using two-fluid models based on KTGF and those using the Mixture Model were performed with Ansys Fluent. Conversely, although particle–particle interaction models are embedded in most commercial codes, CFD-DEM simulations have been usually run by coupling different codes, or with a single in-house package.

#### 4.2. Studies Using Eulerian–Lagrangian Models for Predicting Particle Transport

^{2}. Although two particle diameters (2.4 and 3.5 mm) were reported to be tested, most results are related to the largest particle size. The investigated solid concentrations were 0.05 and 0.10. The CFD-DEM simulations were solved using an in-house code. The transition from the stationary bed flow, through the moving bed flow, to the heterogeneous suspension flow was presented and the simulated pressure drops were consistent with the published experimental data, but no quantitative comparison was made. During the transition from stationary bed flow to the heterogeneous flow the particle–particle and particle–wall forces firstly decreased and then remained nearly constant. The paper suggests that a RANS-based CFD-DEM can be used to predict the transition between slurry flow regimes, but further research and an extensive validation study is needed to give strength to this claim.

^{®}(DCS Computing GmbH, Linz, Austria) and CFDEM

^{®}coupling Open Source CFD-DEM model (DCS Computing GmbH, Linz, Austria) working on Open Foam platform, which models the turbulent fluid flow by solving for the instantaneous Navier–Stokes equations (DNS approach). Such methodology was applied by Zheng et al. [48] to simulate the slurry of 2 mm glass beads in Glycerol solution for Reynolds numbers below 10,000 in a horizontal periodic pipe. The study presents a throughout validation in terms of concentration profile, and then the model was extensively used to find out difficult-to-measure features of the flow for the same conditions.

#### 4.3. Studies Using Eulerian–Lagrangian Models for Predicting Slurry Erosion in Pipeline Systems

#### 4.3.1. The Engineering Problem and Relevant Parameters

#### 4.3.2. Challenges in the CFD Modelling of Slurry Erosion

_{eroded}·m

^{−2}s

^{−1}] is closely related to the concentration of particles (parcels) and particles (parcels) impact angle and velocity:

_{abrasive}·m

^{−2}s

^{−1}], ${v}_{\mathrm{P},\mathrm{imp}}$ and ${\theta}_{\mathrm{P},\mathrm{imp}}$ are the parcel impact velocity and the parcel impact angle (Figure 8a), and $F$ is a dimensionless wear function.

^{−3}] is the sample material density.

^{®}(Ansys Inc., Canonsburg, PA, USA), with a $k$-$\epsilon $ turbulence model along with the Euler–Lagrange model for particle motion. After obtaining the sets of data (${\mathsf{\Phi}}_{\mathrm{m}}$, ${v}_{\mathrm{P},\mathrm{imp}}$, and ${\theta}_{\mathrm{P},\mathrm{imp}}$ as a function of the angular coordinate of the cylinder, $\tilde{\beta}$) from CFD they combined them with the experimental erosion data (${\mathsf{\Phi}}_{e}$ as a function of $\tilde{\beta}$) to determine the dimensionless wear function $F$ in Equation (81). After introducing their own formulation to fit the erosion data:

#### 4.3.3. Critical Evaluation of Previous Investigations

#### 4.4. Studies Using Two- and Multi-Fluid Models Based on KTGF Closures

#### 4.4.1. KTGF Closure Equations

#### 4.4.2. Literature Review

#### KTGF Modelling—Performance Evaluation

#### Verification and Validation of Eulerian KTGF Models

#### 4.4.3. Challenges and Limitations

#### 4.5. Studies Using Two-Fluid Models Based on Empirical Closures

#### 4.5.1. Pioneering Studies by Roco and Co-Workers during the 1980s

#### 4.5.2. The $\beta $-$\sigma $ Two-Fluid Model for Fully Suspended Flow

#### 4.6. Studies Using the Mixture Model

## 5. Concluding Remarks and Recommendations

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Units | Description |

${C}_{d}$ | - | drag coefficient |

${C}_{\mathrm{ls}}$ | kg/(m^{3} s) | fluid–solid exchange coefficient |

${C}_{\mathrm{vd}}$ | - | delivered solids concentration |

${C}_{\mathrm{vi}}$ | - | average spatial volumetric concentration of solids |

${d}_{\mathrm{p}}$ | m | particle diameter |

${d}_{\mathrm{p}}^{+}$ | - | particle size in wall units |

D | m | pipe diameter |

${D}_{\mathrm{ls}}$ | kg/(m^{3} s) | fluid–solid exchange coefficient |

${e}_{\mathrm{n}}$ | - | normal particle(parcel)-wall restitution coefficients |

${e}_{\mathrm{ss}}$ | - | inter-particle collision restitution coefficient |

${e}_{\mathrm{t}}$ | - | tangential particle(parcel)-wall restitution coefficients |

$F$ | - | dimensionless wear function in Equation (81) |

${F}_{\mathrm{s}}$ | - | particle shape related constant in Equation (86) |

${g}_{0,\mathrm{ss}}$ | - | radial distribution function |

${i}_{\mathrm{m}}$ | - | hydraulic gradient |

${I}_{2D}$ | Pa^{2} | second invariant of the deviatoric stress tensor |

${I}_{\mathrm{p}}$ | kg m^{2} | moment of inertia of the particle |

k | m^{2}/s^{2} | turbulent kinetic energy |

${k}_{\mathsf{\Theta}s}$ | - | diffusion coefficient in Equation (88) |

$K$ | - | material related constant in Equation (86) |

${K}_{\mathrm{ls}}$ | kg/(m^{3} s) | fluid–solid exchange coefficient |

m | kg | mass of a particle |

${N}_{\mathrm{p}}$ | - | number of particles in the sampling volume $W$ |

Re_{p} | - | particle Reynolds number |

p | Pa | instantaneous pressure |

${\tilde{p}}_{\mathrm{s},\mathrm{coll}}$ | Pa | collisional solid pressure |

${\tilde{p}}_{\mathrm{s},\mathrm{fric}}$ | Pa | frictional solid pressure |

${\tilde{p}}_{\mathrm{s},\mathrm{kin}}$ | Pa | kinetic solid pressure |

P | Pa | locally-averaged/double-averaged pressure |

${P}_{{k}_{\mathrm{l}}}$ | m^{2}/s^{3} | volumetric production rate of ${k}_{\mathrm{l}}$ |

${n}_{\mathrm{d}}$ | - | number of particle size classes |

$n$ | - | velocity exponent in Equation (86) |

$Q$ | m^{3}/s | volumetric flow rate |

${S}_{\mathrm{p}}$ | - | relative density of particle |

t | s | time |

$T$ | s | time scale |

${v}_{\mathrm{imp}}$ | m/s | modulus of the impact velocity |

${V}_{\mathrm{dl}}$ | m/s | deposition-limit velocity |

${V}_{\mathrm{m}}$ | m/s | slurry bulk-mean velocity |

W | m^{3} | sampling volume |

y+ | - | dimensionless distance of the first grid point to the wall |

Vectors and tensors | ||

${\mathit{F}}_{\mathrm{D}}$ | N | drag force |

${\mathit{F}}_{\mathrm{p}}$ | N | forces exerted on the particle |

${\tilde{\mathit{f}}}_{\mathrm{l}\to \mathrm{s}}$ | N | total force from liquid to solid phase in sampling volume |

$\mathit{g}$ | m/s^{2} | gravitational acceleration vector |

$\mathit{m}$ | N/m^{3} | momentum exchange term in Eulerian–Eulerian formulation |

${\mathit{m}}^{\mathrm{d}}$ | N/m^{3} | generalized drag in the Eulerian–Eulerian formulation |

${\overline{\tilde{\mathit{m}}}}_{}^{\mathrm{td}}$ | N/m^{3} | turbulent dispersion force |

$\mathit{P}$ | m/s^{2} | pressure-related vector in Equation (97) |

$\mathit{S}$ | m/s^{2} | viscosity-related vector in Equation (97) |

${\mathit{S}}_{\mathrm{l}}$ | N/m^{3} | momentum exchange term in Eulerian–Lagrangian framework |

${\mathit{T}}_{\mathrm{p}}$ | N m | torque exerted on the particle |

$\mathit{u}$ | m/s | instantaneous velocity vector of the liquid phase |

${\mathit{u}}^{\prime}$, ${\mathit{u}}^{\u2033}$ | m/s | fluctuating velocity vector of the liquid phase |

${\mathit{U}}_{\mathrm{ls}0}$ | m/s | the solution of Equation (73) without the last term |

${\mathit{U}}_{\mathrm{mk}}$ | m/s | diffusion velocities in mixture model |

$\mathit{v}$ | m/s | instantaneous velocity vector of a particle/solid phase |

${\mathit{v}}^{\prime}$ and ${\mathit{v}}^{\u2033}$ | m/s | fluctuating velocity vector of the solid phase |

$\mathit{x}$ | m | position vector |

$\mathit{\sigma}$ | Pa | stresses tensor |

${\mathit{\sigma}}_{\mathrm{pt},\mathrm{k}}$ | Pa | pseudo-turbulent stress tensors |

$\mathit{\tau}$ | Pa | deviatoric part of the stresses tensor |

${\mathit{\tau}}_{\mathrm{m}}^{\mathrm{t}}$ | Pa | “Reynolds”-like stresses in the mixture model |

${\mathit{\tau}}_{\mathrm{Dm}}$ | Pa | diffusion stresses in the mixture model |

${\mathit{\omega}}_{\mathrm{p}}$ | s^{−1} | angular velocity vector of the particle |

Greek Symbols | ||

${\alpha}_{\mathrm{pm}}$ | - | empirical coefficient in two-fluid model of Messa et al. [37] |

$\beta $ | - | numerical coefficient $\mathrm{in}$-σ two fluid model |

${\gamma}_{\mathrm{s}}$ | kg/(m^{3} s) | rate of energy dissipation due to collision within the solid particles |

ε | m^{2}/s^{3} | turbulence dissipation rate |

$\eta $ | m | erosion depth |

$\left[\eta \right]$ | - | empirical coefficient in two-fluid model of Messa et al. [37] |

$\theta $ | ° | angle of internal friction of particle |

${\theta}_{\mathrm{imp}}$ | ° | impact angle |

${\mathsf{\Theta}}_{\mathrm{s}}$ | m^{2}/s^{2} | granular temperature |

$\lambda $ | Pa s | second viscosity coefficient or bulk viscosity |

$\mu $ | Pa s | dynamic viscosity |

${\mu}^{\mathrm{eff}}$ | Pa s | effective viscosity |

${\mu}^{\mathrm{t}}$ | Pa s | eddy viscosity |

${\mu}_{\mathrm{s},\mathrm{coll}}$ | Pa s | collisional solid viscosity |

${\mu}_{\mathrm{s},\mathrm{fr}}$ | Pa s | frictional solid viscosity |

${\mu}_{\mathrm{s},\mathrm{kin}}$ | Pa s | kinetic solid viscosity |

$\rho $ | kg/m^{3} | density |

$\sigma $ | - | turbulent Schmidt number for volume fractions |

${\tau}_{\mathrm{p}}$ | s | response time of a particle in the mixture |

$\phi $ | - | particle spherical coefficient |

${\phi}_{\mathrm{ls}}$ | kg/(m^{3} s) | exchange term in Equation (88) |

${\mathsf{\Phi}}_{\mathrm{e}}$ | kg/(m^{2} s) | erosion rate intensity |

${\mathsf{\Phi}}_{\mathrm{m}}$ | kg/(m^{2} s) | local average particle impact rate |

$\mathsf{\Phi}$ | - | instantaneous volume fraction of one phase |

${\mathsf{\Phi}}^{\prime}$ | - | fluctuating volume fraction of one phase |

$\omega $ | s^{−1} | specific turbulent dissipation rate |

Subscripts and superscripts | ||

before | just before a particle–wall impact occurs | |

after | just after a particle–wall impact has occurred | |

i | interface | |

k | generic phase | |

l | liquid phase | |

m | mixture | |

p | physical particles | |

P | computational particles in the Lagrangian framework | |

s | solid phase in the Eulerian framework | |

t | target material in erosion modelling | |

@p | at particle position in Eulerian–Lagrangian modelling | |

@P | at parcel position in Eulerian–Lagrangian modelling | |

$\perp $ | normal to the wall | |

$\parallel $ | parallel to the wall | |

Operators (applied to the generic variable ψ) | ||

+ | transpose of a tensor | |

$\tilde{\psi}$ | volume-average | |

$\overline{\psi}$ | time-average | |

$\langle \psi \rangle $ | Favre-average | |

$\mathsf{\Psi}$ | generic averaged (or double averaged) variable | |

Acronyms | ||

CFD | Computational Fluid Dynamics | |

DDPM | Dense Discrete Particle Model | |

DEM | Discrete Element Method | |

DNS | Direct Numerical Simulation | |

EL | Eulerian–Lagrangian model (or approach) | |

GCI | Grid Convergence Index | |

IPSA | Inter-Phase Slip Algorithm | |

KTGF | Kinetic Theory of Granular Flow | |

LES | Large Eddy Simulation | |

RANS | Reynolds-Averaged Navier–Stokes | |

RSM | Reynolds Stress Model | |

SEC | Specific Energy Consumption | |

U-RANS | V-Unsteady Reynolds-Averaged Navier–Stokes |

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**Figure 1.**Different patterns of settling slurry flow in horizontal pipes (

**a**) pseudo-homogeneous flow (

**b**) heterogeneous flow (

**c**) fully-stratified flow. The colors of the particles identified their basic transport mechanism, namely, interaction with the turbulent fluid (grey), quick collision with other particles (green), and long-lasting contacts with nearby particles (red).

**Figure 2.**Pipe characteristic curves for flow of settling slurry at different spatial volumetric concentrations (the plot was built based on experimental data from [12]).

**Figure 3.**Classification of flow particle-laden flows according to the coupling regime and of the key physical mechanisms driving the flow. The distinction between disperse and dense flows has been proposed by Loth [28], whereas that between dilute and dense flows, including the limits to the solid volume fraction, is obtained from the map of Elghobashi [29].

**Figure 4.**Modelling of slurry flow based on (

**a**) Eulerian–Lagrangian approach and (

**b**) the Eulerian–Eulerian approach.

**Figure 6.**Main modelling approaches in the Eulerian–Lagrangian framework: (

**a**) point-particle approach (

**b**) fully-resolved approach.

**Figure 8.**(

**a**) impact of a particle against a wall; (

**b**) deviation between a trajectory calculate under the point-particle assumption and the actual particle trajectory in the proximity of a wall.

**Figure 10.**A summary of significant “numerical” sources of uncertainty in the CFD modelling of slurry flows. For slurry pipe flow simulations in which the inlet–outlet boundary condition scheme is employed, the length of the computational domain for fully developed flow conditions to be achieved shall be added.

**Figure 13.**Normal and tangential restitution coefficients of particle–wall impacts as a function of the impact angle ${\theta}_{\mathrm{P},\mathrm{imp}}$ according to (

**a**) the formulas of Forder et al. [82] and (

**b**) the stochastic rebound model Grant and Tabakoff [83], for which the dotted lines are the mean values and the shaded areas correspond to the standard deviation.

**Figure 14.**A summary of significant “modeling” sources of uncertainty in the CFD simulation of slurry flows.

**Figure 15.**Classification of published research articles on the CFD modelling of slurry flows in pipes according to: (

**a**) the used software; (

**b**) the type of modelling approach.

**Figure 18.**Overview of the Eulerian–Eulerian KTGF implementation strategies found in the peer-reviewed literature.

**Figure 19.**Effect of the near-wall cell size on the predicted hydraulic gradient using (

**a**) the preliminary formulation of the two-fluid model, presented in Messa et al. [37]; (

**b**) the final version of the model, or the $\beta $-$\sigma $ two fluid model, which is reported in both Messa and Matoušek [57] and Messa et al. [38]. In the two figures, ${i}_{\mathrm{m}}^{\mathrm{ref}}$ is the value of ${i}_{\mathrm{m}}$ corresponding to ${y}^{+}$ = 100. In the legend, $C$ stands for the average volumetric concentration of solids.

Approach | Typical Application | Key Advantagesmain Strengths | Main Limitations |
---|---|---|---|

CFD-DEM modelling with p–p interactions | Particle transport in pipes | A lot of information at the particle and sub-particle scales Deep physical insight | High computational cost |

EL modelling ignoring p–p interactions | Slurry erosion of pipeline components | Information at the particle scale Affordable computational cost | Low concentration only Uncertainty due to erosion model and other modelling features |

Two-fluid modelling based on KTGF | Particle transport in pipes | Strong physical basis Affordable computational cost | Several difficult-to-set coefficients, sub-models, and parameters |

Two-fluid modelling not based on KTGF | Particle transport in pipes | Simple mathematical structure Computationally efficient | Weak physical basis Limited applicability |

Mixture modelling | Particle transport in pipes | Low computational cost Multiple solid phases allowed | Stringent assumptions on flow (local equilibrium approximation) |

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Messa, G.V.; Yang, Q.; Adedeji, O.E.; Chára, Z.; Duarte, C.A.R.; Matoušek, V.; Rasteiro, M.G.; Sanders, R.S.; Silva, R.C.; de Souza, F.J.
Computational Fluid Dynamics Modelling of Liquid–Solid Slurry Flows in Pipelines: State-of-the-Art and Future Perspectives. *Processes* **2021**, *9*, 1566.
https://doi.org/10.3390/pr9091566

**AMA Style**

Messa GV, Yang Q, Adedeji OE, Chára Z, Duarte CAR, Matoušek V, Rasteiro MG, Sanders RS, Silva RC, de Souza FJ.
Computational Fluid Dynamics Modelling of Liquid–Solid Slurry Flows in Pipelines: State-of-the-Art and Future Perspectives. *Processes*. 2021; 9(9):1566.
https://doi.org/10.3390/pr9091566

**Chicago/Turabian Style**

Messa, Gianandrea Vittorio, Qi Yang, Oluwaseun Ezekiel Adedeji, Zdeněk Chára, Carlos Antonio Ribeiro Duarte, Václav Matoušek, Maria Graça Rasteiro, R. Sean Sanders, Rui C. Silva, and Francisco José de Souza.
2021. "Computational Fluid Dynamics Modelling of Liquid–Solid Slurry Flows in Pipelines: State-of-the-Art and Future Perspectives" *Processes* 9, no. 9: 1566.
https://doi.org/10.3390/pr9091566