Many slurry correlation models have been developed. It is important to understand the assumptions and limitations of each, and select the most appropriate model applicable to the situation at hand.

**Blatch (1906)**

A qualitative study.

Blatch, N.S. (1906). Discussion of Works for the purification of the water supply of Washington D.C. Transactions ASCE., 400-409.

**Wilson (1942)**

A qualitative study.

Wilson, K. C. (1942). A qualitative study of slurry flow in pipes. Journal of Applied Mechanics, 9(2), 59-64.

**Worster (1952)**

Worster based his tests on coarse coal and gravel in 75 mm and 150 mm pipes.

Worster, R. C. (1952). The flow of coal and gravel in pipes. Journal of Applied Mechanics, 19(1), 1-8.

**Durand and Condolios (1952)**

Empirical model. The experiments were carried out mostly on sand and gravel with a d_{50} between 0.18 mm and 22.5 mm in pipes with a D_{P} from 40 mm to 580 mm, and volumetric concentrations C_{vt} from 2% to 22%. Durand and Condolios did not perform experiments at low flows. Their experiments were conducted in "medium" pipe diameters. The main issues are: 1) the wrong use of the particle Froude number vs the drag coefficient, 2) the wrong use of the relative submerged density in the particle Froude number, 3) the wrong power of the particle Froude number and 4) the use of the wrong graph for the limit deposit velocity coefficient in the Durant and Condolios equations.

Durand, R., & Condolios, E. (1952). Hydraulic transport of solid material by pipeline. In Proceedings of the 5th World Petroleum Congress, 407-423.

**Durand ****(1953)**

Empirical model.

When Durand published his results in English, he made an error in determining the asymptotic value, leading to a predicted deposity velocity about 1.3 times higher than would have been predicted in the original French article. Many authors have erroneously followed the English article.

Durand, R. (1953). Basic relationships of the transportation of solids in pipes—experimental research. In Proceedings of the Minnesota International Hydraulics Convention, 89-103.

**Worster and Denny (1955)**

Empirical model.

Worster, R. C., & Denny, D. F. (1955). Hydraulic Transport of Solid Material in Pipes. Proceedings of the Institution of Mechanical Engineers, 169(1), 563–586.

Worster, R. C., & Denny, D. F. (1955). The hydraulic transport of solid material in pipes. Journal of Applied Mechanics, 22(1), 1-8.

**Newitt (1955)**

Empirical model. Newitt carried out experiments in a DN25 pipe using sand particle sizes of 0.0965 mm, 0.203 mm, 0.762 mm and a gravel with size 4.5 mm, 3.2-6.4 m. The Newitt model distinguishes a heterogeneous regime and a sliding bed regime. The friction factor for a sliding bed is likely to be different than the figure Newitt presented for his 1" pipe. Be careful of the limitations of an empirical model based on experience within a small pipe!

Newitt, D. M (1955) Hydraulic conveying of solids in horizontal pipes. *Transactions of the Institution of Chemical Engineers Vol 33*, 93-110.

**Gibert (1960)**

Empirical model. Gibert analysed data from Durand and Condolios' 1952 tests and summarised the results. Gibert maintained that for the limit deposit velocity coefficient F_{L} the correction factor of about 1.1 should be used.

Gibert, M. (1960). Transport hydraulique des solides en conduite. La Houille Blanche, 15(6), 641-680.

**Fuhrboter (1961)**

Furhboter, A. (1961) *Uber die Forderung von Sand-Wasser-Gemischen in Rohrleitungen*. Mitteilungen des Franzius-Instituts, H. 19.

Fuhrboter, A. (1961). Untersuchungen über den hydraulischen Transport von Feststoffen in Rohrleitungen. VDI-Forschungsheft, 479, 1-64.

**Condolios and Chapus **

Condiolis and Chupus corrected the error Durand made in his 1953 English translation. Wakefield and Thorvaldsen subsequently revised the Condolios and Chapus graph.

Condolios E., Chapus E.E., 1963. Solids Pipelines, Chemical Engineering, 24 June

Condolios, E., & Chapus, E. (1963). Transport hydraulique des solides par conduite. La Houille Blanche, 18(5), 577-600.

**Jufin and Lopatin (1966)**

Jufin, A. P. and Lopatin, N. A. (1966). O projekte TUiN na gidrotransport zernistych materialov po stalnym truboprovodam. *Gidrotechniceskoe Strojitelstvo, 9., *49-52.

Jufin, L. A., & Lopatin, A. N. (1966). Hydraulic transport of solids by pipeline. Pergamon Press.

**Zandi and Govatos (1967)**

Empirical model. Zandi and Govatos criticised Durand and Condolios' model. Zandi, I., and Govatos, G. (1967). Heterogeneous flow of solids in pipelines. *Proc. ACSE, J. Hydraul. Div., 93(HY3)*., 145-159.

Zandi, I., & Govatos, G. (1967). A model for predicting the frictional pressure drop of slurries in pipes. AIChE Journal, 13(6), 1105-1111.

**Graf and Acaroglu (1968)**

Empirical model.

Graf, W. H., & Acaroglu, E. R. (1968). Sediment transport in conveyance systems, part 1. Bulletin of the International Association of Scientific Hydrology, 13(2), 20-39.

**Traynis (1970)**

Empirical model.

**Babcock (1970)**

Babcock criticised Durand and Condolios' model. Babcock, H. A. (1970). The sliding bed flow regime. *Hydrotransport 1*. Bedford, England: BHRA.

Babcock, G. L. (1970). Hydraulic transport of coarse gravel—A graphical approach. Journal of the Hydraulics Division, 96(10), 2087-2106.

**Zandi (1971)**

Zandi, I. (1971). Hydraulic transport of bulky materials, *Advances in Solid-Liquid Flow in Pipes and its Applications. *(pp. 1-38). Oxford: Pergamon Pres.

**Charles and Stevens (1972)**

Empirical model.

**Wilson and Judge (1976)**

The Wilson and Judge deposit velocity correlation is widely used to predict the deposit velocity in turbulent pipe flow of sand in water slurries. The correlation is limited to medium size particles as the pipe size increases, and so it is generally of limited applicability of viscous slurries in the mining industry (where particle sizes are frequently less than 100 μm. This was expanded in 2015 by Thomas (see Modified Wilson and Judge).

**Babcock (1977)**

Empirical model.

**Wasp (1970, 1977)**

Wasp is the mainstay. At given flow conditions the model determines the degree of heterogeneity of the solid's particles. It then determines the friction losses contributions of the vehicle (supporting pseudo-homogeneous slurry) which suspends the heterogeneous solids. The total friction loss is calculated by summing losses due each. Main downfall is that the model assumes Newtonain behaviour of the vehicle. Multi-layer model. Waste, E. G., Kenny, J. O., Aude, T.C., Seiter, R.H., and Jacques, R. B. (1970). Deposition velocities transition velocities and spatial distibution of solids in slurry pipelines. *Hydro Transport 1, paper H42. *(pp 53-76). Coventry: BHRA Fluid Engineering.

Wasp, E. J., Kenny, J. P., & Gandhi, R. L. (1970). Hydraulic transport of coarse gravel—A graphical approach. Journal of the Hydraulics Division, 96(10), 2087-2106.

Wasp, E. J., Kenny, J. P., & Gandhi, R. L. (1977). Solid-liquid flow slurry pipeline transportation. Trans Tech Publications.

**Turian and Yuan (1977)**

Empirical model.

Turian, R. M., & Yuan, T. (1977). Flow of slurries in pipelines. AIChE Journal, 23(3), 232-248.

**Kazanskij (1978)**

Empirical model.

Kazanskij, V. N. (1980). Hydraulic transport of solid materials in pipes. Journal of Hydraulic Engineering, 106(1), 1-23.

**Thomas (1979)**

Thomas developed a deposit velocity prediction method for particles smaller than the viscous sub-layer, based on the Wilson sliding bed theory.

This correlation is generally not thought to be conservative. It is based on smaller particles than Oroskar and Turian. It is an empirical model.

Sanders et al modified this theory in 2004.

Thomas, A. D. (1979). An empirical model for the prediction of the deposit velocity of slurries. International Journal of Multiphase Flow, 5(5), 463-473.

**Toda (1979)**

This model is subject to a 180 μm lower limit on particle size. It applies to slurries which have a narrow PSD. Note that there are several errors in the mathematical equations.

Toda, M. (1979). Solitons and heat conduction. Physica Scripta, 20(3-4), 424-430.

**Wilson-GIW (1979)**

2-layer model that allows for a stationary or sliding bed layer with a liquid layer above it.

**Oroskar and Turian (1980)**

The Oroskar and Turian correlation is an industry-derived model focussing on particles larger than 100 μm. It applies to slurries which have a narrow PSD.

Oroskar, A. R., & Turian, R. M. (1980). Prediction of solid-liquid slurry flow in pipes. AIChE Journal, 26(3), 454-465.

**Kim (1986)**

This model is subject to a 90 μm lower limit on particle size. It applies to slurries which have a narrow PSD. This model assumes Stokes or intermediate-range hindered settling. There is no consideration of settling under turbulent flow.

Kim, J. H. (1986). A model for the hydraulic transport of sand-water mixtures. International Journal of Multiphase Flow, 12(4), 609-624.

**Doron (1987)**

Doron, P., Granica, D., & Barnea, D. (1987). Slurry flow in horizontal pipes-experimental and modeling. International Journal of Multiphase Flow, 13(4), 535-547.

**Shah (1991)**

Shah, S. N., & Lord, D. L. (1991). Critical velocity correlations for slurry transport with non-Newtonian fluids. AIChE Journal, 37(6), 863-870.

**Bae (1991)**

This model is subject to a 230 μm lower limit on particle size. It applies to slurries which have a narrow PSD. This model has no pipe size limit.

Bae KS, H Lee CG Park and CS Lee. 1991. "Empirical Correlation for the Minimum Transport Velocity of Multidisperse Slurries in Horizontal Pipes." *Korean Journal of Chemical Engineering *8(2):120-124.

**Gillies and Shook (1991)**

This model is subject to a 150 μm lower limit on particle size. It applies to slurries which have a narrow PSD. This model has been developed for distributions containing fine-particle (<74 μm) carrier fluids, but the definition of "fine" is arbitrary. Gillies and Shook introduced the concept of homogeneous and heterogeneous fractions.

Gillies RG, and CA Shook. 1991. "A Deposition Velocity Correlation for Water Slurries." *Canadian Journal of Chemical Engineering *69(5): 1225-1228.

Gillies, R.G. and Shook, C.A., 1991. Modelling high concentration settling slurry flows. The Canadian Journal of Chemical Engineering, 69(2), 173-179.

There is an updated that has been issued (by Gillies and Shook) in 2000.

**Shook and Roco (1991)**

Multi-level model. The model assumes that the suspended solids are distributed uniformly across the entire pipe, and that the lower layer also contains the solids that contribute Coulombic friction.

Shook, C.A. and Roco, M.C., 1991. Slurry Flow: Principles and Practice. Elsevier Science & Technology Books.

**Wilson et al (1992)**

Wilson criticised Durand and Condolios' model.

Wilson, K. C., Addie, G. R., and Clift, R. (1992) *Slurry Transport using Centrifugal Pumps. *New York: Elsevier Applied Sciences.

**Doron and Barnea (1993)**

Doron, P., and Barnea, D. (1993). A three layer model for solid liquid flow in horizontal pipes. *International Journal of Multiphase Flow, Vol 19, No. 6., 1029-1043.*

**Matousek (1997)**

Multi-level model.

Matousek, V (1997). Flow Mechanism of Sand/Water Mixtures in Pipelines, PhD Thesis. Delft, Netherlands: Delft University of Technology.

**WASC (1997)**

Wilson, K. C., Addie, G. R., Clift, R., Sellgren, A. (1997) *Slurry Transport using Centrifugal Pumps. *Glasgow, UK: Chapman & Hall

**Kaushal and Tomita (2002)**

Multi-level model.

Kaushal, D. R., & Tomita, Y. (2002). Prediction of pressure drop in pipeline flow of multilayered slurry using a modified Wasp model. International Journal of Multiphase Flow, 28(9), 1509-1527.

**Jewett (2002)**

This model uses solids loading to calculate apparent viscosity. It applies to slurries which have a narrow PSD. The basis for the Jewett viscosity correlation is a correlation for Newtonian, non-interacting, spherical, dilute, uniform particles. It is based on the Thomas correlation. The Jewett viscosity correlation does not consider how suspending phase and particle chemistry impact bulk rheology.

Jewett, J. R. (2002). A new method for predicting frictional pressure drop in horizontal and near horizontal slurry pipelines. International Journal of Multiphase Flow, 28(9), 1529-1548.

**Wakefield and Thorvaldsen (2002)**

Wakefield and Thorvaldsen provided a revised Condolios and Chapus graph.

Wakefield A.W., Thorvaldsen G.S., 2002. Comparison of two-layer model with correlative method for prediction of head losses in two large dredging pipelines carrying sand and gravel, Hydrotransport 15 Conf., Banff, Canada, 3-5 June.

Wakefield, J., & Thorvaldsen, S. (2002). Comparison of two methods for predicting head losses in two large dredging pipelines. Journal of Hydraulic Engineering, 128(11), 1014-1021 .

**Sanders et al (2004)**

Sanders et al modified Thomas' 1979 viscous sub-layer theory to include the effect of particle size and concentration.

**Wilson (2006)**

Multi-level model. This model is good for slurries with coarse particles. This model is used in AFT Fathom's slurry module.

Wilson, K. C., Addie, G. R., Sellgren, A., & Clift, R. (2006). Slurry transport using centrifugal pumps. Springer.

**Rojas and Saez (2010)**

Multi-level model.

Rojas, G., & Saez, A. (2010). A three-layer model for solid liquid flow in horizontal pipes. International Journal of Multiphase Flow, 36(9), 691-704.

**Gillies and Shook (2010)**

2-layer model, settling slurries. Extends an existing model for applicability to solids concentration >35 C_{v}.

Randall G. Gillies, Clifton A Shook. "Modelling high concentration settling slurry flows", The Canadian Journal of Chemical Engineering volume (2000) 78(4).

**Poloski PNNL (2010)**

This model corresponds well with experimental data for slurries with Archimedes numbers <80.

Adam P. Poloski, Arthur W. Etchells. "A pipeline transport correlation for slurries with small but dense particles". The Canadian Journal of Chemical Engineering (2010) 88(2): 182-189.

Poloski A.P., A.W. Etchells, J. Chun, H.E. Adkins, A.M. Casella, M.J. Minette, and S.T. Yokuda. 2010. A Pipeline Transport Correlation for Slurries with Small but Dense Particles. Canadian Journal of Chemical Engineering 88, no. 2:182-189.

**Rojas and Saez (2012)**

2-layer model. Top layer of a flowing suspension, and a bottom bed of stationary or moving particles. Wide range of particle density and particle size distribution.

Mario R. Rojas and A. Eduardo Saez, Two-Layer Model for Horizontal Pipe Flow of Newtonian and Non-Newtonian Settling Dense Slurries, Ind. Eng. Chem. Res. (2012) 51(20).

Rojas, M. R., & Saez, A. E. (2012). Two-layer model for horizontal pipe flow of Newtonian and non-Newtonian dense slurries. International Journal of Multiphase Flow, 39, 135-144.

**Modified Wilson and Judge (2015)**

Thomas expanded the original Wilson and Judge correlation from 1976. The Modified Wilson and Judge correlation allows for pipes up to 1,000 mm diameter for silica sand particles in water down to 30 μm.

Thomas, A. (2015). A modification of the Wilson & Judge deposit velocity equation, extending its applicability to finer particles and larger pipe sizes. 17th International Conference on Transport & Sedimentation of Solids Particles, Delft, The Netherlands.

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