6.1: Introduction
 Page ID
 29220
In August 2012 the author was approached by a dredging company with the question which head loss model to use for a project with a cutter dredge and a discharge length of 35 km. This raised the following questions:

What did the company want to know?

How many booster stations to use?

What should be the locations of the booster stations?

What were the real issues?

What should be the total pump pressure to avoid plugging the line?

Where to locate the booster stations to avoid cavitation at the entrance of each pump?

How does this depend on the particle size distribution?
These questions and many others triggered a study in to the existing head loss models. With the knowledge that the main Dutch and Belgium dredging contractors use the Durand & Condolios (1952) and Fuhrboter (1961) models in a modified form, while companies in the USA and Canada often use the Wilson (1992) model in a modified form or the SRC model, the study started with a comparison of these models. Other models that were investigated were the Newitt et al. (1955) model, the Doron & Barnea (1987) model, the Matousek (1997) model and others. Also later models like the 4 component Sellgren & Wilson (2012) model and the 2LM and 3LM models of Wilson (19792015), Matousek (19972016) and SRC (19912016) were investigated.
Usually the models perform well in the neighborhood of the parameters used during the experiments, especially the pipe diameter (small) and the particle diameter, but for real life conditions (large pipe diameters) the models deviate and it's not clear which model matches these conditions. Another issue is that most models are derived for transport (delivered) volumetric concentrations as input and not the spatial volumetric concentrations. The research into the existing models gave some answers but not all.
For the determination of the pressure losses of a solidswater slurry flow many equations, theories and data are available, like Blatch (1906), Howard (1938), Siegfried (Durepaire, 1939), O’Brien & Folsom (1939), Durand & Condolios (1952) and Durand (1953), Gibert (1960), Worster & Denny (1955), Zandi & Govatos (1967), Newitt et al. (1955), Fuhrboter (1961), Jufin & Lopatin (1966), Turian & Yuan (1977), Doron et al. (1987) and Doron & Barnea (1993), Wilson et al. (1997) and Matousek (1997). Some models are based on semiempirical equations, others on mechanistic or phenomenological models. A number of these models and experimental data will be analyzed and issues found will be addressed. The book shows many graphs with original or reconstructed data and graphs with derived quantities. The author has tried to show as many original experimental data as possible, with the philosophy that the experimental data are based on nature, while most models and equations are local fit results and may thus not be applicable for other sand, gravels or pipe diameters.
This chapter is divided into 4 main sections:

Early history up to 1948, describing the phenomena more qualitatively. Blatch (1906), Howard (1938), Siegfried (Durepaire, 1939), O’Brien & Folsom (1939), Wilson (1942) and others. Since the original articles were not available, reproduced graphs were digitized. The graphs shown here only have a qualitative value and should not be used quantitatively.

1948 to present, empirical and semiempirical models and equations. Soleil & Ballade (1952), Durand & Condolios (1952) and Durand (1953), Gibert (1960), Worster & Denny (1955), Zandi & Govatos (1967), Newitt et al. (1955), Silin, Kobernik & Asaulenko (1958) & (1962), Fuhrboter (1961), Jufin & Lopatin (1966), Charles (1970) and Babcock (1970), Graf et al. (1970) & Robinson (1971), Yagi et al. (1972), A.D. Thomas (1976) & (1979), Turian & Yuan (1977), Kazanskij (1978) and IHCMTI (1998). In general the original articles were retrieved and the original graphs were digitized. So the graphs shown here both have a qualitative and a quantitative value.

1979 to present, physical 2 layer (2LM) and 3 layer (3LM) models. Wasp et al. (1963), (1970) and (1977), Wilson et al.  GIW (1979), (1992), (1997) and (2006), Doron et al. (1987) and Doron & Barnea (1993), SRC  Shook & Roco (1991) & Gillies (1993), Kaushal & Tomita (2002B), Matousek (1997) and Talmon (2011) & (2013). In general the original articles were retrieved and the original graphs were digitized. So the graphs shown here have both a qualitative and a quantitative value.

The Limit Deposit Velocity. The models of Wilson (1942), Durand & Condolios (1952), Newitt et al. (1955), Jufin & Lopatin (1966), Zandi & Govatos (1967), Charles (1970), Graf et al. (1970) & Robinson (1971), Wilson & Judge (1976), Wasp et al. (1977), Thomas (1979), Oroskar & Turian (1980), Parzonka et al. (1981), Turian et al. (1987), Davies (1987), Schiller & Herbich (1991), Gogus & Kokpinar (1993), Gillies (1993), Van den Berg (1998), Kokpinar & Gogus (2001), Shook et al. (2002), Wasp & Slatter (2004), Sanders et al. (2004), Lahiri (2009), Poloski et al. (2010) and Souza Pinto et al. (2014) are discussed and general trends are identified.
Many graphs are made with the Delft Head Loss & Limit Deposit Velocity Framework (DHLLDV) as a reference system. Additional graphs can be found on the website www.dhlldv.com.
6.1.1 Coordinate Systems
In literature many different coordinate systems are used. Some coordinate systems are based on measured quantities, like the hydraulic gradient versus the line speed, but many coordinate systems are based on derived quantities. Some of the most used coordinate systems are explained here.
For the ordinate axis often the hydraulic gradient i is used, which is defined as:
\[\ \mathrm{y}=\mathrm{i}=\frac{\Delta \mathrm{p}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}\]
Durand & Condolios (1952) defined the parameter Φ for the ordinate axis according to:
\[\ \mathrm{y}=\Phi=\left(\frac{\mathrm{i}_{\mathrm{m}}\mathrm{i}_{\mathrm{l}}}{\mathrm{i}_{\mathrm{l}} \cdot \mathrm{C}_{\mathrm{v t}}}\right)\]
Babcock (1970) used this parameter Φ for the ordinate axis, but also this parameter divided by the relative submerged density:
\[\ \mathrm{y}=\left(\frac{\mathrm{i}_{\mathrm{m}}\mathrm{i}_{\mathrm{l}}}{\mathrm{i}_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{vt}}}\right)\]
Wilson et al. (1992) defined the stratification ratio, which is used and named in this book as the relative excess hydraulic gradient E_{rhg}:
\[\ \mathrm{y}=\mathrm{E}_{\mathrm{r h g}}=\left(\frac{\mathrm{i}_{\mathrm{m}}\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}}\right)\]
For the abscissa axis often the line speed vls is used in combination with the hydraulic gradient i on the ordinate axis. To make the abscissa axis dimensionless, the flow Froude number can be used:
\[\ \mathrm{x}=\mathrm{F r}_{\mathrm{f} \mathrm{l}}=\frac{\mathrm{v}_{\mathrm{l s}}}{\sqrt{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}}\]
Durand & Condolios (1952) defined the parameter ψ for the ordinate axis according to:
\[\ \mathrm{x}=\Psi=\left(\frac{\mathrm{v}_{\mathrm{ls}}}{\sqrt{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}}\right)^{2} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}_{\mathrm{5} 0}}}\right)^{1}=\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\sqrt{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}}\right)^{2} \cdot \sqrt{\mathrm{C}_{\mathrm{x}}}\]
Later the relative submerged density was added to this abscissa axis parameter according to:
\[\ \mathrm{x}=\Psi=\left(\frac{\mathrm{v}_{\mathrm{l} \mathrm{s}}}{\sqrt{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}}}\right)^{2} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}_{\mathrm{5} 0}}}\right)^{1}=\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\sqrt{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}}}\right)^{2} \cdot \sqrt{\mathrm{C}_{\mathrm{x}}}\]