7.1: Introduction
 Page ID
 29227
7.1.1 Considerations
In the last decades many head loss models for slurry transport have been developed. Not just for the dredging industry but also for coal and phosphate transport and in the chemical industries. Some models are based on the phenomena occurring combined with dimensionless parameters, resulting in semiempirical equations (Durand & Condolios (1952), Gibert (1960), Worster & Denny (1955), Jufin Lopatin (1966), Zandi & Govatos (1967), Fuhrboter (1961)), while others are based on physics with 2 and 3 layer models (Newitt et al. (1955), Wasp et al. (1977), Doron & Barnea (1987), Wilson (1979), the SRC model (1991) and Matousek (2009)). The physical models are based on stationary transport in time and space, while the semiempirical models may incorporate non stationary or dynamical processes. An analysis of these models and of data collected from numerous publications for particles with densities ranging from 1.14 ton/m^{3} to 3.65 ton/m^{3}, particle diameters ranging from 0.005 mm up to 45 mm, concentrations up to 45% and pipe diameters from 0.0254 m up to 0.9 m has led to an overall model of head losses in slurry transport, a sort of Framework. The Framework is based on 5 main flow regimes determining the source of energy losses, the fixed or stationary bed regime, the sliding bed regime, the heterogeneous flow regime, the homogeneous flow regime and the sliding flow regime.. One can distinguish viscous friction losses, dry friction losses, potential energy losses, kinetic energy losses, Magnus lift work, turbulent lift work and turbulent eddy work. The losses do not have to occur at the same time. Usually one or two will be dominant depending on the flow regime.
Although sophisticated 2 and 3 layer models exist for slurry flow (here the flow of sand/gravel water mixtures), the main Dutch and Belgium dredging companies still use modified Durand & Condolios (1952) and Fuhrboter (1961) models, while the main dredging companies in the USA and Canada use a modified Wilson et al. (1992) model for heterogeneous transport and sliding bed transport or the SRC model. When asked why these companiesdon’t use the more sophisticated models, they answer that they require models that match their inputs and they feel that the 2 and 3 layer models are still in an experimental phase, although these models give more insight in the physics. Usually the companies require a model based on the particle size distribution or d_{50}, the pipe diameter D_{p}, the line speed v_{ls}, the relative submerged density R_{sd} and the temperature (the viscosity of the carrier liquid ν_{l}). Parameters like the bed associated hydraulic radius are not known in advance and thus not suitable. Usually the dredging companies operate at high line speeds above the Limit Deposit Velocity (LDV) in the heterogeneous or homogeneous regime. This implies that the bed has dissolved and 2 and 3 layer models are not applicable anyway.
Still there is a need for improvement, since the existing models give reasonably good predictions for small diameter pipes, but not for large diameter pipes as used in dredging. Recent projects require line lengths up to 35 km with 5 to 6 booster pumps and large diameter pipes. Choosing the number of booster pumps and the location of the booster pumps depends on the head losses. However it should be considered that the slurry transport process is not stationary. Densities may vary from a water density of 1 ton/m3 to densities of 1.6 ton/m3 and particle size distributions will change over time. This results in a dynamic process where pumps, pump drives and slurry transport interact. The fundamental 2 and 3 layer models require a stationary approach, while the more empirical equations may take the dynamic effects as time and place averaged effects into account. The question is whether a semi empirical approach is possible, covering the whole range of pipe diameters and giving the empirical equations a more physical background, but still using the parameters available to the dredging industry.
Transporting sand with water through a pipeline, in general, results in an increase of the pressure required compared with pumping water or pure liquid. Since pressure times flow equals power and power times time equals energy, this can also be interpreted as an increase of the energy required to pump the solids. Energy or work also equals force times distance or stress times volume. The fact that more power is required to pump a solidliquid mixture compared with just pumping the liquid implies that there are additional energy losses and energy dissipation when pumping the solids. In order to go into detail to the model developed, first the different types of energy dissipation due to the solids effect are discussed.
It is clear that the flow regimes and the magnitude of the relative excess hydraulic gradient depends strongly on the pipe diameter and the particle diameter. In the large pipe a sliding bed will never occur in the constant spatial volumetric concentration case. In the small pipe however it will for particles larger than 0.5 mm. In the small pipe, the larger particles exceed the ratio d/D_{p}>0.015 as set by Wilson et al. (1997), resulting in almost 100% stratified flow, here considered to be the sliding flow regime. In the large pipe this criterion is never met, except when pumping large gravel, cobbles or boulder sized pieces such as cut rock or clayballs.
Figure 7.11, Figure 7.12, Figure 7.13 and Figure 7.14 show the results of the energy approach for 9 sands ranging from d=0.1 mm up to d=10 mm in pipes with diameters of D_{p}=0.1524 m and D_{p}=1 m. For each pipe diameter the constant spatial volumetric concentration curves and the constant delivered volumetric concentration curves are shown.
On the E_{rhg} graph the regimes are clearly distinguishable: the fixed bed is an upwardsloping line on the left, the sliding bed regime is the flat (horizontal) line at about 0.4 (the sliding friction factor), the heterogeneous regime is the downwardsloping line in the middle, and the pseudohomogeneous regime is upwardsloping on the left. For large particles the sliding flow regime is almost flat (similar to sliding bed), in the flow region of the heterogeneous regime. The figures above clearly show how the available regimes change with pipe diameter.
So head losses from experiments in pipes of 0.1524 m can hardly be translated into head losses for a pipe of 1 m as often used in dredging. The physical processes are different. Small pipe sliding bed versus large pipe no sliding bed and small pipe sliding flow versus large pipe no sliding flow. In fact the smaller the pipe diameter, the higher the probability of the occurrence of a sliding bed and sliding flow and the larger the pipe diameter, the lower the probability of the occurrence of a sliding bed and sliding flow. Only if the physical processes involved are similar, is scaling possible.
This explains why many equations and models from literature give good results for small pipe diameters, but deviate for large diameter pipes. The way energy is dissipated in small diameter pipes is often different from the way it is dissipated in large diameter pipes at operational line speeds. It also explains while a lot of research is focused on 2 and 3 layer transport with a sliding bed, which often occurs in small pipes, but much less in large pipes.
7.1.2 Energy Dissipation
When a liquid is transported through a pipeline, energy is dissipated by viscous friction and by turbulence (assuming high Reynolds numbers). When solids are added, there will also be energy dissipation in the form of potential losses, kinetic losses and possibly friction losses and losses due to Magnus and turbulent lift work and turbulence in general.

Potential energy losses. In turbulent flow, because the solids are under the influence of gravity and the turbulence has to keep them floating. The potential energy losses will depend on the terminal settling velocity and be influenced by hindered settling. Since the settling process does not depend on the line speed, at a higher line speed the energy dissipation per unit of time will not change. This implies that the energy dissipation per unit of pipeline length is reversely proportional with the line speed. So at high line speeds the influence of the potential energy losses will diminish.

Kinetic energy losses. During transport, because the particles interact with the wall, with each other and with the turbulent eddies and in all cases they lose part of their kinetic energy. With the kinetic energy losses one may expect that the number of interactions is more or less constant in time, so at higher line speeds the number of interactions per unit of line length will decrease reversely proportional with the line speed, resulting in a decrease of the excess pressure due to the solids. At higher line speeds however the momentum of the particles also increases and it is more difficult to change the direction of the particles. This might decrease the number of interactions with the wall per unit of time. The total losses will be reversely proportional with the line speed to a power higher than 1, let’s say a proportionality with a power between 1 and 2. The proportionality depends on the physical properties and the grading of the solids. Although near wall lift will exist at low line speeds, it is negligible until a certain line speed where the lift force is strong enough to keep the solids away from the wall. At this line speed there are no more interactions with the wall and the excess pressure due to interactions collapses. At about the same line speed the lift forces start driving the solids to the center of the pipe resulting in a more homogeneous flow. The pure heterogeneous regime stops abruptly, because there are no more interactions with the wall, and the pseudo homogeneous regime starts, based on the work of lift forces and turbulence. The transition line speed depends on the particle and the pipe diameter. So the sudden regime change as described will only occur in uniform or very narrow graded sands.

Sliding and rolling friction. Sliding and rolling friction occur if there is a sliding or moving bed. Forces are transmitted directly between particles and the internal and external friction coefficients determine the friction forces. These coefficients are dependent on the type of solids and the particle size distribution.

Magnus lift work. When the thickness of the viscous sublayer is bigger than the particle diameter, particles with rotation due to interactions with the wall will be subjected to Magnus lift forces. This will only occur for the combination of a low line speed and small particles. The Magnus lift forces will carry out work if they actually lift the particles, contributing to the energy losses. When the line speed increases, the thickness of the viscous sublayer decreases and the particles do not fit in the viscous sublayer anymore. The Magnus lift work will diminish when the size of the particles is bigger than the layer thickness. At a higher line speed, the turbulent lift and turbulent eddies will take over.

Turbulent lift and eddy work. At high line speeds the turbulent lift and turbulent eddies becomes important. Since lift force times the distance over which it acts equals the work carried out, this will also result in energy losses. Since the lift force increases with the velocity gradient near the wall, the losses due to the lift force will increase with the line speed. At relatively low line speeds most solids will be transported in the bottom part of the pipeline, resulting in an asymmetrical concentration profile, matching heterogeneous flow. This results in an opposite asymmetrical velocity profile, with the highest flow at the top of the pipeline. Below a certain line speed the lift force on a particle is smaller than the weight of the particle and the lift force will not carry out any work. But above this transition velocity suddenly the particles will be lifted. The lift forces are dependent on the velocity gradient and thus will appear at the full circumference of the pipe, but they will first start pushing the solids upwards from the bottom and thus start to create a more symmetrical concentration and velocity profile. With increasing line speed the concentration and velocity profile will get closer to the symmetrical profiles, matching pseudo homogeneous transport.
Resuming it can be stated that the potential and kinetic losses decrease with an increasing line speed with a power of the line speed between 1 and 2, while the losses due to near wall lift forces increase with an increasing line speed, until the pseudo homogeneous regime is reached. For each combination of particle and pipe diameter, there exists a transition line speed. Below this line speed kinetic losses dominate the excess pressure; above this line speed the work carried out by turbulent lift and eddy forces dominates the excess pressure. For uniform sand, kinetic losses and work carried out by lift forces will not occur at the same line speed. For graded sands a transition region, with respect to the line speed, will occur, the size of which depending on the grading. In the case where the particles are much smaller than the thickness of the viscous sub layer, theoretically there is Magnus lift if the particles are rotating. One may expect that the excess pressure due to the solids will continue decreasing with increasing line speed. In this case the excess pressure will reach zero asymptotically and there is no solids effect at very high line speeds. It is obvious that the collapse of the interactions with the wall, resulting in a collapse of the kinetic losses, due to the lift force, will happen at about the same line speed where the work of the lift forces starts increasing. This is the transition line speed between heterogeneous and pseudo homogeneous transport. It is not possible that the collapse of the kinetic losses appears at a line speed higher than the line speed where the work carried out by the lift forces starts, for uniform sands. It might be possible that this collapse appears at a slightly lower line speed, resulting in a collapse of the excess pressure, but at higher line speeds this will increase again because of the work of the lift forces.
Wilson et al. (1997) introduced the Stratification Ratio R, which in fact equals the relative excess hydraulic gradient E_{rhg}. The higher the Stratification Ratio, the more asymmetrical the concentration and the velocity profile in the pipe. With increasing line speed, the Stratification Ratio decreases with power of 0.251.7, depending on the grading of the sand. However, once the transition line speed between heterogeneous and homogeneous transport is passed, the relative excess hydraulic gradient will increase again, while the stratification decreases. The term Stratification Ratio corresponds with the heterogeneous transport, with potential and kinetic losses, but not with the pseudo homogeneous transport with losses due to lift work. Therefore a new term is introduced, the Slip Relative Squared or S_{rs}, which is the ratio between the slip velocity and the terminal settling velocity squared. Where the slip velocity is defined as; the contribution of the velocity difference between the line speed and the particle velocity to explain for the head losses. Mathematically the Stratification Ratio Solids and the Slip Relative Squared are the same, but physically the Slip Relative Squared tells more about the physics of the heterogeneous hydraulic transport. So the S_{rs} value explains for the kinetic energy losses in the heterogeneous flow regime. The potential energy losses are taken into account by the Settling Velocity Hindered Relative, the S_{hr} value. These potential energy losses are present both in the heterogeneous flow regime and the homogeneous flow regime.
Many graphs in this book and specifically this chapter have the relative excess hydraulic gradient as the ordinate and the hydraulic gradient of pure liquid as the abscissa. Since the relative excess hydraulic gradient equals the mixture hydraulic gradient minus the pure liquid hydraulic gradient, divided by the relative submerged density of the solids and the volumetric concentration, the graph is almost dimensionless. Almost, because there is are still some nonlinear effects of the relative submerged density and the volumetric concentration. The mixture hydraulic gradient minus the pure liquid hydraulic gradient is often called the solids effect, so the increase of the hydraulic gradient due to the presence of solids. The volumetric concentration can be either the spatial or the transport concentration, depending on the measurement method. Most researchers, in their models, assume the mixture hydraulic gradient equals the pure liquid hydraulic gradient plus the solids effect. Only the more physical models, the 2LM and 3 LM models, have a different approach.
Figure 7.15 and Figure 7.16 show the solids effect for the constant spatial volumetric concentration C_{vs} case and the constant transport volumetric concentration C_{vt} case. The solids effect in general decreases with increasing line speed. For low line speeds the C_{vt} case gives a higher solids effect compared with the C_{vs} case due to increasing slip with decreasing line speed.
Figure 7.17 shows a case where the transition velocity is the same for the collapse of the kinetic interactions and the start of the lift work. Figure 7.18 shows a case where the transition velocity of the lift work is higher than the transition velocity for the collapse of the kinetic interactions. The latter results in a collapse of the relative excess hydraulic gradient. In both examples the same solids are used, but in the latter case the pipe diameter is bigger. Other experiments by Clift et al. (1982) with narrow graded 0.42 mm masonry sand, shows exactly the same phenomena.
7.1.3 Starting Points
Before discussing the Delft Head Loss & Limit Deposit Velocity (DHLLDV) Framework in detail, some starting points have to be pointed out. First of all, the Framework is based on a set of 5 submodels for 5 main flow regimes. These submodels are all based on a constant spatial volumetric concentration C_{vs}. Curves for constant volumetric transport concentration C_{vt} are derived from the 5 submodels based on the slip velocity v_{sl}. The slip velocity v_{sl} is defined as the difference between the velocity of the mixture v_{ls} and the velocity of the solids v_{s}:
\[\ \mathrm{v}_{\mathrm{s l}}=\mathrm{v}_{\mathrm{l s}}\mathrm{v}_{\mathrm{s}}=\mathrm{v}_{\mathrm{l s}} \cdot\left(\mathrm{1}\frac{\mathrm{v}_{\mathrm{s}}}{\mathrm{v}_{\mathrm{ls}}}\right)=\mathrm{v}_{\mathrm{l} \mathrm{s}} \cdot\left(\mathrm{1}\frac{\mathrm{C}_{\mathrm{v} \mathrm{t}}}{\mathrm{C}_{\mathrm{v s}}}\right)\]
For a certain control volume the volumetric transport concentration C_{vt} can be determined if the volumetric spatial concentration C_{vs} and the slip velocity v_{sl} are known, given a certain line speed v_{ls}.
\[\ \mathrm{C}_{\mathrm{v t}}=\left(1\frac{\mathrm{v}_{\mathrm{sl}}}{\mathrm{v}_{\mathrm{l s}}}\right) \cdot \mathrm{C}_{\mathrm{v s}}\]
Likewise, for a certain control volume, the volumetric spatial concentration C_{vs} can be determined if the volumetric transport concentration C_{vt} and the slip velocity v_{sl} are known, given a certain line speed v_{ls}.
\[\ \mathrm{C}_{\mathrm{v s}}=\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\mathrm{v}_{\mathrm{l s}}\mathrm{v}_{\mathrm{sl}}}\right) \cdot \mathrm{C}_{\mathrm{v t}}\]
These equations will be used a lot in the following derivations and are considered to be well known. The 5 main flow regimes are:

A Fixed Bed (FB) regime or restricted pipe regime. The behavior of this main flow regime is, the solids form a bed at the bottom of the pipe. This bed is stationary (fixed), so the liquid has to flow through a restricted area above the bed, resulting in higher pressure losses. At higher line speeds it is probable that part of the solids start eroding and be transported heterogeneously above the bed. At the Limit Deposit Velocity, the bed has been eroded completely. As long as the pressure losses correspond with the behavior of flow through the restricted area above the bed, the flow regime is considered to be a fixed bed regime.

A Sliding Bed (SB) regime or sliding friction dominated regime. The behavior of this main flow regime is, the solids form a sliding bed at the bottom of the pipe. The pressure losses are the sum of the losses as a result of the sliding friction of the solids and the viscous friction of the liquid. At higher line speeds it is probable that part of the solids start eroding and be transported heterogeneously above the bed. At the Limit Deposit Velocity, the bed has been eroded completely. At higher concentration it is possible that sheet flow occurs and the sliding bed curve is followed right of the intersection with the heterogeneous transport curve. As long as the pressure losses correspond with the behavior of sliding friction, the pressure loss curves are parallel with the clean water resistance curve in the i_{m} versus v_{ls} plot, the sliding bed regime is considered.

Heterogeneous (He) transport or collision dominated regime. The behavior of this main flow regime is, the solids interact with the pipe wall through collisions. The solids are distributed nonuniformly over the cross section of the pipe with higher concentrations at the bottom of the pipe. This may be due to saltation or to Brownian motions of the particles in turbulent transport. For very small particles this may follow the fixed bed regime directly, for coarse particles this will follow the sliding bed regime.

Homogeneous (Ho) transport. The behavior of this main flow regime is, the particles are uniformly distributed over the cross section of the pipe due to the mixing capability of the turbulent flow. The pressure losses behave according to Darcy Weisbach, but with the mixture density as the liquid density. For very fine particles the viscosity has to be adjusted by the apparent viscosity.

The Sliding Flow (SF) regime. If the ratio between the particle diameter and the pipe diameter is above a certain value and the spatial volumetric concentration is above about 5%, the turbulence is not capable of carrying the particles anymore. This will result in a high speed flow with the characteristics of sliding friction, however the bed concentration decreases with increasing line speed. So it’s named Sliding Flow.
The hydraulic gradient i_{w} (for water) or i_{l} (for a liquid in general) and for a mixture are:
\[\ \mathrm{i}_{\mathrm{l}}=\mathrm{i}_{\mathrm{w}}=\frac{\Delta \mathrm{p}_{\mathrm{l}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}} \quad\text{ and }\quad \mathrm{i}_{\mathrm{m}}=\frac{\Delta \mathrm{p}_{\mathrm{m}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}\]
The Relative Excess Hydraulic Gradient Erhg is the difference between the mixture gradient i_{m} (in meters of carrier liquid column) and the hydraulic gradient i_{l} divided by the relative submerged density R_{sd} and the volumetric concentration C_{vs}. This E_{rhg} will also be referred to as the solids effect. The Slip Relative Squared S_{rs} is the Slip Velocity of a particle v_{sl} divided by the Terminal Settling Velocity of a particle v_{t} squared and this S_{rs} value is a good indication of the excess pressure losses due to the solids in the heterogeneous regime. The Settling Velocity Hindered Relative S_{hr} is the ratio between the hindered settling velocity v_{t}·(1C_{vs}/κ_{C})β and the line speed v_{ls}, divided by the relative submerged density R_{sd} and the volumetric concentration C_{v}. For all regimes the E_{rhg} value is:
\[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v s}}}\]
In the heterogeneous regime the relation between these parameters is:
\[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\mathrm{S}_{\mathrm{h r}}+\mathrm{S}_{\mathrm{r s}}\]
Figure 7.19, Figure 7.110, Figure 7.111 and Figure 7.112 show the 5 main flow regimes for small, medium and large particles in an 0.1524 m (6 inch) pipeline. The abscissa, the horizontal axis, is the line speed v_{ls}. The ordinate, the vertical axis, is the hydraulic gradient of the mixture i_{m}. The red solid line is the constant volumetric spatial concentration C_{vs} line. The green dashed line the constant volumetric transport concentration C_{vt} line. The light brown dashed lines show the sliding bed curves, where the thick line is based on the sliding friction coefficient and the thin lines give a margin of +/ 12.5% of the sliding friction coefficient. The solid blue line is the pure liquid hydraulic gradient, the dashed blue line the ELM (Equivalent Liquid Model) curve and the dark brown dashed line the theoretical homogeneous regime curve. The dotted lines give the Limit Deposit Velocity curves for spatial and transport concentration.
For very fine particles, the fixed bed regime transits directly to the heterogeneous regime, without the occurrence of the sliding bed regime. This can be seen in Figure 7.19 because the intersection point is below the sliding bed curve. The Limit Deposit Velocity is at the transition between the heterogeneous regime and the homogeneous regime. Although there is some slip above the Limit Deposit Velocity, the slip and thus the difference between the constant volumetric spatial concentration C_{vs} lines and the constant volumetric transport concentration C_{vt} lines increases with a decreasing line speed at line speeds below the Limit Deposit Velocity. The intersection point between the fixed bed regime and the heterogeneous regime will be at an increasing E_{rhg} value with an increasing particle diameter.
For medium particles, Figure 7.110, the intersection point between the fixed bed regime and the heterogeneous regime lies above the sliding bed regime curve, meaning that the fixed bed regime is followed by the sliding bed regime, followed by the heterogeneous regime, with increasing line speed. The Limit Deposit Velocity is now somewhere between the intersection of the sliding bed regime with heterogeneous regime and the heterogeneous regime with the homogeneous regime. The larger the particle the closer is the Limit Deposit Velocity to the intersection of the sliding bed regime with heterogeneous regime.
The examples given here are for an 0.1524 m pipe. For other pipe diameters, the sliding bed (constant sliding friction coefficient) and the homogeneous regime curves, will stay at the same position and do not depend on the pipe diameter. The fixed bed curve will move to the right with increasing pipe diameter, while the heterogeneous regime curve will move to the left with increasing pipe diameter. One could also say that both curves move downwards with an increasing pipe diameter.
The transitions between the main flow regimes are not instantaneous, but gradually. Special attention will be given to the transition between the heterogeneous regime and the homogeneous regime.
For large particles, Figure 7.111, the behavior is similar to the medium particles, except for the fact that the Limit Deposit Velocity is at the sliding bed regime, below the intersection point between the sliding bed regime and the heterogeneous regime. This is possible because in reality this transition is not sharp but gradual.
Very coarse particles, Figure 7.112, show sliding flow behavior. Turbulence is not capable anymore to bring the particles in suspension. The behavior is a mix of sliding bed and heterogeneous flow.
7.1.4 Approach
Chapter 7 describes the new Delft Head Loss & Limit Deposit Velocity (DHLLDV) Framework. The DHLLDV Framework is based on uniform sands or gravels and constant spatial volumetric concentration.

An explanation of the Delft Head Loss & Limit Deposit Velocity Framework.

A detailed description of the 8 different flow regimes and 6 scenarios is given. The occurrence of flow regimes
depends on the particle to pipe diameter ratio and on the spatial volumetric concentration. Figure 7.113 gives an example of the different flow regimes occurring depending on the particle diameter and the line speed. Each pipe diameter and each spatial volumetric concentration requires such a graph.

The stationary bed regime without sheet flow and with sheet flow. The stationary bed without sheet flow is based on a 2 layer model for low line speeds and a 3 layer model for higher line speeds. Usually the bed starts sliding when there is sheet flow, however for small particles it is possible that there is a direct transition from the stationary bed regime to the heterogeneous flow regime.

The sliding bed regime. The sliding bed is based on a 3 layer model showing an almost constant relative excess hydraulic gradient equal to the sliding friction coefficient. The sliding bed regime does not always occur. The larger the particles and the larger the volumetric concentration, the higher the probability of the occurrence of a sliding bed.

The heterogeneous regime. The heterogeneous model is based on energy considerations, resulting in a two component model, potential energy losses and kinetic energy losses.

The homogeneous regime. The homogeneous model is based on the equivalent liquid model (ELM) with a correction based on a particle free viscous sub layer.

The sliding flow regime. The sliding flow model assumes a high speed flow with the macroscopic behavior of sliding friction and heterogeneous flow. The porosity of the bed increases with the line speed and particles do not necessarily rest on each other.

A new model for the Limit Deposit Velocity is derived, consisting of 5 particle size regions and a lower limit. This model is based on the ratio of the potential energy of the particles to the total energy in the liquid flow for small particles and on a limiting small bed for large particles.

Based on the LDV a method is shown to construct slip velocity or slip ratio curves from zero line speed to the LDV. Based on the slip ratio, the constant delivered volumetric concentration curves can be constructed. The resulting model is compared with models from literature.

The concentration distribution. Also based on the LDV, in this case the assumption that at the LDV the concentration at the bottom of the pipe equals the bed concentration, a new diffusivity approach is developed. The resulting concentration distributions are compared with experiments.

The transition heterogeneous versus homogeneous in detail. The transition from the heterogeneous regime to the homogeneous regime requires special attention. First of all, the transition line speed gives a good indication of the operational line speed and allows to compare the DHLLDV Framework with many models from literature. Secondly, at this transition collisions disappear due to near wall lift, while homogeneous transport is mobilized due to turbulence.

Knowing the slip ratio, the bed height for line speeds below the LDV can be determined. Since the LDV is defined as the line speed above which a sliding or stationary bed does not exist, below this line speed a bed does exist. New equations are derived for this.

Finally the grading of the Particle Size Distribution (PSD) is discussed. A method is given to construct resulting head loss, slip velocity and bed height curves for graded sands and gravels.

Inclined pipes. In real life often inclined pipes are used. Whether its in the ladder of a CSD, the suction pipe of a TSHD or an upwards or downwards slope, the hydraulic gradient will differ from a horizontal pipe. The effect of an inclined pipe is derived both for the hydraulic gradient and for the LDV.
7.1.5 NomenclatureIntroduction
C_{vs} 
Spatial volumetric concentration 
 
C_{vt}_{ } 
Delivered (transport) volumetric concentration 
 
d 
Particle diameter 
m 
d_{50} 
50% passing particle diameter 
m 
D_{p } 
Pipe diameter 
m 
E_{rhg} 
Relative excess hydraulic gradient 
 
ELM 
Equivalent Liquid Model 
 
g 
Gravitational constant 9.1 m/s^{2} 
m/s^{2} 
i_{l } 
Liquid hydraulic gradient 
m/m 
i_{w } 
Water hydraulic gradient 
m/m 
i_{m } 
Mixture hydraulic gradient 
m/m 
ΔL 
Length of pipe 
m 
LDV 
Limit Deposit Velocity 
m/s 
Δp_{l} 
Pressure difference liquid 
kPa 
Δp_{m} 
Pressure difference mixture 
kPa 
PSD 
Particle Size Diagram 
 
R 
The Wilson stratification ratio 
 
R_{sd} 
Relative submerged density 
 
S_{hr} 
Settling velocity Hindered Relative 
 
S_{rs}_{ } 
Slip velocity Relative Squared 
 
v_{ls } 
Line speed 
m/s 
v_{s } 
Velocity solids 
m/s 
v_{sl} 
Slip velocity 
m/s 
v_{t } 
Terminal settling velocity 
m/s 
ρ_{l } 
Liquid density 
ton/m^{3} 
ρ_{m } 
Mixture density 
ton/m^{3} 
κ_{C } 
Concentration eccentricity hindered settling 
 
λ_{l } 
Darcy Weisbach friction factor 
 
μ_{sf} 
Sliding friction coefficient 
 
\(\ v_{\mathrm{l}} \) 
Kinematic viscosity 
m^{2}/s 