Computational Fluid Dynamics Modelling of Liquid–Solid Slurry Flows in Pipelines: State-of-the-Art and Future Perspectives
Abstract
: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
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
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
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
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 - 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 |
- | drag coefficient | |
kg/(m3 s) | fluid–solid exchange coefficient | |
- | delivered solids concentration | |
- | average spatial volumetric concentration of solids | |
m | particle diameter | |
- | particle size in wall units | |
D | m | pipe diameter |
kg/(m3 s) | fluid–solid exchange coefficient | |
- | normal particle(parcel)-wall restitution coefficients | |
- | inter-particle collision restitution coefficient | |
- | tangential particle(parcel)-wall restitution coefficients | |
- | dimensionless wear function in Equation (81) | |
- | particle shape related constant in Equation (86) | |
- | radial distribution function | |
- | hydraulic gradient | |
Pa2 | second invariant of the deviatoric stress tensor | |
kg m2 | moment of inertia of the particle | |
k | m2/s2 | turbulent kinetic energy |
- | diffusion coefficient in Equation (88) | |
- | material related constant in Equation (86) | |
kg/(m3 s) | fluid–solid exchange coefficient | |
m | kg | mass of a particle |
- | number of particles in the sampling volume | |
Rep | - | particle Reynolds number |
p | Pa | instantaneous pressure |
Pa | collisional solid pressure | |
Pa | frictional solid pressure | |
Pa | kinetic solid pressure | |
P | Pa | locally-averaged/double-averaged pressure |
m2/s3 | volumetric production rate of | |
- | number of particle size classes | |
- | velocity exponent in Equation (86) | |
m3/s | volumetric flow rate | |
- | relative density of particle | |
t | s | time |
s | time scale | |
m/s | modulus of the impact velocity | |
m/s | deposition-limit velocity | |
m/s | slurry bulk-mean velocity | |
W | m3 | sampling volume |
y+ | - | dimensionless distance of the first grid point to the wall |
Vectors and tensors | ||
N | drag force | |
N | forces exerted on the particle | |
N | total force from liquid to solid phase in sampling volume | |
m/s2 | gravitational acceleration vector | |
N/m3 | momentum exchange term in Eulerian–Eulerian formulation | |
N/m3 | generalized drag in the Eulerian–Eulerian formulation | |
N/m3 | turbulent dispersion force | |
m/s2 | pressure-related vector in Equation (97) | |
m/s2 | viscosity-related vector in Equation (97) | |
N/m3 | momentum exchange term in Eulerian–Lagrangian framework | |
N m | torque exerted on the particle | |
m/s | instantaneous velocity vector of the liquid phase | |
, | m/s | fluctuating velocity vector of the liquid phase |
m/s | the solution of Equation (73) without the last term | |
m/s | diffusion velocities in mixture model | |
m/s | instantaneous velocity vector of a particle/solid phase | |
and | m/s | fluctuating velocity vector of the solid phase |
m | position vector | |
Pa | stresses tensor | |
Pa | pseudo-turbulent stress tensors | |
Pa | deviatoric part of the stresses tensor | |
Pa | “Reynolds”-like stresses in the mixture model | |
Pa | diffusion stresses in the mixture model | |
s−1 | angular velocity vector of the particle | |
Greek Symbols | ||
- | empirical coefficient in two-fluid model of Messa et al. [37] | |
- | numerical coefficient -σ two fluid model | |
kg/(m3 s) | rate of energy dissipation due to collision within the solid particles | |
ε | m2/s3 | turbulence dissipation rate |
m | erosion depth | |
- | empirical coefficient in two-fluid model of Messa et al. [37] | |
° | angle of internal friction of particle | |
° | impact angle | |
m2/s2 | granular temperature | |
Pa s | second viscosity coefficient or bulk viscosity | |
Pa s | dynamic viscosity | |
Pa s | effective viscosity | |
Pa s | eddy viscosity | |
Pa s | collisional solid viscosity | |
Pa s | frictional solid viscosity | |
Pa s | kinetic solid viscosity | |
kg/m3 | density | |
- | turbulent Schmidt number for volume fractions | |
s | response time of a particle in the mixture | |
- | particle spherical coefficient | |
kg/(m3 s) | exchange term in Equation (88) | |
kg/(m2 s) | erosion rate intensity | |
kg/(m2 s) | local average particle impact rate | |
- | instantaneous volume fraction of one phase | |
- | fluctuating volume fraction of one phase | |
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 | |
normal to the wall | ||
parallel to the wall | ||
Operators (applied to the generic variable ψ) | ||
+ | transpose of a tensor | |
volume-average | ||
time-average | ||
Favre-average | ||
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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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
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 StyleMessa, 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