**
****The Hydraulic Gradient versus the Line Speed**

**The hydraulic gradient i**_{m} equals the pressure gradient divided by the density of the carrier liquid ρ_{l} and the gravitational constant g. The pressure gradient is the pressure Δp_{m} required to pump a mixture over a certain length of pipeline ΔL divided by this length ΔL.

**So: i**_{m}=Δp_{m}/(ρ_{l}·g·ΔL) and Δp_{m}=i_{m}·(ρ_{l}·g·ΔL)

**The advantage of the hydraulic gradient i**_{m} versus the line speed v_{ls} graph is that it relates directly to dredging practice and can easily be combined with pump curves in order to determine the working point of a pump/pipeline system. The disadvantage is that it depends strongly on the volumetric concentration C_{vs} or C_{vt} and the relative submerged density R_{sd}, so for different concentrations and/or relative submerged densities different graphs have to be made. Multiplying the hydraulic gradient i_{m} with a factor 1000 gives the mixture pressure loss for a ΔL=100 m pipe with water as the carrier liquid.

**Two graphs are shown, first the graph for constant spatial volumetric concentration C**_{vs} curves (often laboratory conditions) and second a graph for constant transport volumetric concentration C_{vt} curves (dredging practice).

**Constant Spatial Volumetric Concentration Curves**

**The basic DHLLDV Framework is based on constant spatial volumetric concentration C**_{vs} and consists of 4 main flow regimes, which may be subdivided into a larger set of sub-regimes. The regimes are the fixed bed regime (FB) without or with sheet flow, the sliding bed regime (SB), the heterogeneous regime (He) and the homogeneous regime (Ho). Only at high particle diameter to pipe diameter ratios (>0.015) a 5^{th} regime may occur, the sliding flow regime (SF).

**Constant Transport Volumetric Concentration Curves**

**The constant transport volumetric concentration C**_{vt} curves are derived from the constant spatial concentration C_{vs} curves and a holdup function or slip factor relation. This holdup function depends strongly on the Limit Deposit Velocity.

**The Relative Excess Hydraulic Gradient versus the Line Speed**

**The relative excess hydraulic gradient E**_{rhg} equals the difference between the mixture hydraulic gradient i_{m} and the liquid hydraulic gradient i_{l} divided by the volumetric concentration C_{vs} or C_{vt} and the relative submerged density R_{sd}.

**So: E**_{rhg}=(i_{m}-i_{l})/(C_{vs}·R_{sd})

**The advantage of using the relative excess hydraulic gradients is that for the constant spatial volumetric concentration C**_{vs} case this parameter is almost independent of parameters like the concentration and the relative submerged density. However for the constant transport volumetric concentration case it depends strongly on the concentration below the Limit Deposit Velocity.

**The advantage of plotting this parameter versus the line speed is that the value can be related to dredging practice where the line speed is a known.**

**Two graphs are shown, first the graph for constant spatial volumetric concentration C**_{vs} curves (often laboratory conditions) and second a graph for constant transport volumetric concentration C_{vt} curves (dredging practice).

**Constant Spatial Volumetric Concentration Curves**

**The basic DHLLDV Framework is based on constant spatial volumetric concentration C**_{vs} and consists of 4 main flow regimes, which may be subdivided into a larger set of sub-regimes. The regimes are the fixed bed regime (FB) without or with sheet flow, the sliding bed regime (SB), the heterogeneous regime (He) and the homogeneous regime (Ho). Only at high particle diameter to pipe diameter ratios (>0.015) a 5^{th} regime may occur, the sliding flow regime (SF).

**Constant Transport Volumetric Concentration Curves**

**The constant transport volumetric concentration C**_{vt} curves are derived from the constant spatial concentration C_{vs} curves and a holdup function or slip factor relation. This holdup function depends strongly on the Limit Deposit Velocity.

**The Relative Excess Hydraulic Gradient versus the Hydraulic Gradient**

**The relative excess hydraulic gradient E**_{rhg} equals the difference between the mixture hydraulic gradient i_{m} and the liquid hydraulic gradient i_{l} divided by the volumetric concentration C_{vs} or C_{vt} and the relative submerged density R_{sd}.

**So: E**_{rhg}=(i_{m}-i_{l})/(C_{vs}·R_{sd})

**The advantage of using the relative excess hydraulic gradients is that for the constant spatial volumetric concentration C**_{vs} case this parameter is almost independent of parameters like the concentration and the relative submerged density. However for the constant transport volumetric concentration C_{vt} case it depends strongly on the concentration below the limit deposit velocity.

**The advantage of plotting this parameter versus the hydraulic gradient of the carrier liquid makes the graph almost dimensionless.**

**Two graphs are shown, first the graph for constant spatial volumetric concentration C**_{vs} curves (often laboratory conditions) and second a graph for constant transport volumetric concentration C_{vt} curves (dredging practice).

**Constant Spatial Volumetric Concentration Curves**

**The basic DHLLDV model is based on constant spatial volumetric concentration C**_{vs} and consists of 4 main flow regimes, which may be subdivided into a larger set of sub-regimes. The regimes are the fixed bed regime (FB) without or with sheet flow, the sliding bed regime (SB), the heterogeneous regime (He) and the homogeneous regime (Ho). Only at high particle diameter to pipe diameter ratios (>0.015) a 5^{th} regime may occur, the sliding flow regime (SF).

**Constant Transport Volumetric Concentration Curves**

**The constant transport volumetric concentration C**_{vt} curves are derived from the constant spatial concentration C_{vs} curves and a holdup function or slip factor relation. This holdup function depends strongly on the Limit Deposit Velocity.