# Extended Analytical Turbulent Diffusion Model for Particle Dispersion and Deposition in a Horizontal Pipe: Comparison with CFD Simulation

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## Abstract

A 2D analytical turbulent diffusion model for particle dispersion and deposition at different heights along the pipe flow and circumferential deposition has been developed. This liquid–solid turbulent diffusion model presented in this paper has emanated from an existing gas–liquid turbulent diffusion model. This model can be used as a handy tool for quick estimation one and two-dimensional deposition fluxes of particles in water distribution networks. A comprehensive 3D numerical investigation has been carried out using multiphase mixture model available in “Fluent 6.2” to verify the above analytical model. Different particles sizes and densities were used for 3D numerical investigations. The deposition was studied as a function of particle diameter, density, and fluid velocity. The deposition of particles, along the periphery of the pipe wall and at different depths, was investigated. Both the models findings matched with qualitative phenomena such as deposition of heavier particles at the bottom of the pipe wall were higher at lower velocities and lower at higher velocities. The lighter particles were found mostly suspended with homogeneous distribution. Smaller particles were also suspended with marginal higher concentration near the bottom of the pipe wall. This marginal higher concentration of the smaller particles was found to be slightly pronounced for lower velocity. These analogies of particles are well discussed with the ratio between free-flight velocity and the gravitational settling velocity. Extended analytical model results were compared with the 3D computational fluid dynamics simulation results. Discrepancies in the model results were discussed.

## Keywords

CFD simulation Free-flight velocity Horizontal flow Multiphase Particle deposition and turbulence diffusion## Nomenclature

- \( \vec{a} \)
Secondary-phase particle's acceleration

*C*^{+}Concentration of particles

*C*_{f}Friction coefficient

*D*Pipe diameter

*D*_{f}Fluid diffusivity

*D*_{p}Particle diffusion coefficient

*d*_{p}Diameter of the particles of secondary phase

- \( \vec{F} \)
Body force

*f*_{drag}Drag function

*k*Proportional constant

*k*_{D}Constant

*k*_{n}Eigenvalues

*L*Length scale

*l*Particle mean free path

*m*_{T}Mass transfer

*n*Number of phases

*P*Peclet number

*R*_{D}Deposition flux

*R*_{e}Entrainment flux of the particles

*Re**Reynolds number based on the friction velocity

*Re*_{f}Fluid Reynolds number

*S*Stokes number

*t*_{0}Initial time

*T*_{L}Integral flow time scale

*T*_{P}Particle integral time scale

*U*Velocity scale

*u**The friction velocity

*v*Free-flight velocity

- \( {\vec{v}_{{{\rm{dr,k}}}}} \)
Drift velocity for secondary phase

*V*_{f}Pipe average fluid velocity

- \( v_{\rm{f}}^{\prime } \)
Fluctuating velocity

*v*_{g}Gravitational settling velocity of the particle

- \( {\vec{v}_{\rm{m}}} \)
Mass-averaged velocity of the mixture

- \( {\vec{v}_{\rm{qp}}} \)
Relative velocity

- \( \left\langle {v\prime_{\rm{p}}^2} \right\rangle \)
Particle’s mean square velocity

*λ*_{K}Kolmogorov length scale

*ρ*_{m}Mixture density

*ρ*_{p}Densities of the particle

*α*_{k}Volume fraction of phase

*ε*Kinetic energy dissipation

*ϕ*Angle around the pipe circumference

- γ
_{cross} Crossing trajectories coefficient

- γ
_{inert} Inertial coefficient

*λ*Free-flight/diffusion ratio

*μ*_{m}Viscosity of the mixture

*ν*_{f}Kinematic viscosity

*ρ*_{f}Densities of the fluid

*τ*_{p}Particle relaxation time

*τ*_{qp}Particulate relaxation time

*τ*_{s}Wall shear stress

## References

- 1.Anderson, R. J., & Russell, T. W. F. (1970). Circumferential variation of interchange in horizontal annular two-phase flow.
*Industrial and Engineering Chemistry Fundamentals, 9*, 340.CrossRefGoogle Scholar - 2.Anderson, R. J., & Russell, T. W. F. (1970). Film formation in two-phase annular flow.
*AIChE Journal, 14*, 626–633.CrossRefGoogle Scholar - 3.Binder, J. L., & Hanratty, T. J. (1992). Use of Lagrangian method to describe drop deposition and distribution in horizontal gas–liquid annular flows.
*International Journal of Multiphase Flow*,*18*(403–419).Google Scholar - 4.Csanady, G. T. (1963). Turbulent diffusion of heavy particles in the atmosphere.
*Journal of Atmospheric Science, 20*, 201–208.CrossRefGoogle Scholar - 5.FLUENT. FLUENT INC. http://www.fluent.com/software/fluent/
- 6.Friendlander, S. K., & Johnstone, H. F. (1957). Deposition of suspended particles from turbulent gas streams.
*Industrial and Engineering Chemistry, 49*, 1151.CrossRefGoogle Scholar - 7.Fukano, T., & Ousaka, A. (1989). Prediction of the circumferential distribution of film thickness in horizontal and near-horizontal gas–liquid annular flows.
*International Journal of Multiphase Flow, 15*, 403–419.CrossRefGoogle Scholar - 8.Grainger, C., Wu, J., Nguyen, B. V., Ryan, G., Jayanratne, A., & Mathes, P. (2003).
*Part 1: Settling, re-suspension and transport*. Melbourne: CRC-CFC.Google Scholar - 9.Hossain, A., Naser, J., & Imteaz, M. A. (2010). CFD investigation of turbidity spikes for different velocity and particle load profiles in a horizontal pipe.
*Australian Journal of Water Resources, 14*(1), 63–72.Google Scholar - 10.Hinze, J. O. (1975).
*Turbulence (Chapter 5.7)*. McGraw-Hill.Google Scholar - 11.James, P. W., Wilkes, N. S., Conkie, W., & Burnes, A. (1987). Developments in the modelling of horizontal annular two-phase flow.
*International Journal of Multiphase Flow, 13*, 173–198.CrossRefGoogle Scholar - 12.Kallio, G. A., & Reeks, M. W. (1989). A numerical simulation of particle deposition in turbulent boundary layers.
*International Journal of Multiphase Flow, 15*(3), 433–446.CrossRefGoogle Scholar - 13.Laurinat, J. E., Hanratty, T. J., & Jepson, W. P. (1985). Film thickness distribution for gas–liquid annular flow in a horizontal pipe.
*Physics and Chemistry of Hydrodynamics, 6*, 179–195.Google Scholar - 14.Manninen, M., Taivassalo, V., & Kallio, S. (1996).
*On the mixture model for multiphase flow*. VTT Publications 288, Technical Research Centre of Finland, Finland.Google Scholar - 15.Mols, B., & Oliemans, R. V. A. (1998). A turbulent diffusion model for particle dispersion and deposition in horizontal tube flow.
*International Journal of Multiphase Flow, 24*(1), 55–75.CrossRefGoogle Scholar - 16.Morse, P. M., & Feshbach, H. (1953).
*Methods of theoretical physics (chapter 2)*. New York: McGraw-Hill.Google Scholar - 17.Parash, S. V., & Karabelas, A. J. (1991). Droplet entrainment and deposition in horizontal annular flow.
*International Journal of Multiphase Flow, 17*(4), 455–468.CrossRefGoogle Scholar - 18.Reeks, M. W. (1983). The transport of discrete particles in inhomogeneous turbulence.
*Journal of Aerosol Science, 14*, 729–739.CrossRefGoogle Scholar - 19.Schiller, L., & Naumann, Z. (1935).
*Zeitschrift des Vereins Deutscher Ingenieure, 77*, 318.Google Scholar - 20.Spalart, P., & Allmaras, S. (1992).
*A one-equation turbulence model for aerodynamic flows*. American Institute of Aeronautics and Astronautics.Google Scholar - 21.Swailes, D. C., & Reeks, M. W. (1994). Particle deposition from a turbulent flow. I. A steady-state model for high inertia particles.
*Physics of Fluids, 6*(10), 3392.CrossRefGoogle Scholar - 22.Taylor, G. I. (1921). Diffusion by continuous movements.
*Proceedings of the London Mathematical Society*, 196–212.Google Scholar