The DHLLDV Framework is derived for uniform sands and gravels and constant spatial volumetric concentration.

The Hydraulic Gradient versus the Line Speed

The hydraulic gradient equals the mixture pressure loss Δp_m divided by the liquid density ρ_l, the gravitational g constant and the length of the pipeline ΔL according to:

i_m=Δp_m/(ρ_l·g·ΔL)

Multiplying the hydraulic gradient with a factor 1000 gives the mixture pressure loss for a ΔL=100 m pipe with water as the carrier liquid.

The Relative Excess Hydraulic Gradient versus the Line Speed

The Relative Excess Hydraulic Gradient E_rhg is the solids effect (ρ_m-ρ_l) divided by the relative submerged density R_sd and the volumetric concentration either spatial C_vs or delivered C_vt.

E_rhg=(ρ_m-ρ_l)/(R_sd·C_vs)     or     E_rhg=(ρ_m-ρ_l)/(R_sd·C_vt)

The Relative Excess Hydraulic Gradient versus the Hydraulic Gradient

The Head Losses on Durand & Condolios Coordinates

Durand & Condolios introduced a set of coordinates Ψ and Φ, based on using double logarithmic paper for processing experimental data into power curves. The ordinate and abscissa are given in the graph. Most of their experiments were carried out in pipes with a diameter of D_p=0.1524 m (6 inch). For larger diameter pipes the Durand & Condolios equation (grey solid line) over-estimates the pressure losses. For smaller pipe diameters it under-estimates the pressure losses.

The slip factor or holdup function determines the relation between the constant spatial volumetric concentration curves and the constant transport volumetric concentration curves.

Slip Factor=(1-ξ)=C_vt/C_vs and Slip Ratio=ξ=v_sl/v_ls

The slip factor is determined based on the Limit Deposit Velocity. The bed height is determined based on the LDV and the slip factor.

The spatial volumetric concentration C_vs curve is based on constant delivered concentration C_vt and the slip ratio ξ curve.

The Influence Factors or Mobilization Factors versus the Line Speed

The mobilization factors show the collapse of the collisions in the heterogeneous flow regime and the mobilization of particles following the turbulent eddies in the homogeneous flow regime.

The Transition Heterogeneous-Homogeneous versus the Line Speed

The mobilized heterogeneous flow regime curve and mobilized homogeneous flow regime curve are added, resulting in a transition curve from the heterogeneous to the homogeneous flow regime.

The Construction of the Slip Ratio versus the Line Speed

 The resulting slip ratio ξ curve is constructed based on 3 regions. The fixed or sliding bed region (the dark blue line for the fixed bed region and the light blue line for a sliding bed including sheet flow, 3LM), the region around the Limit Deposit Velocity (the red line) and the region of line speeds above the LDV (heterogeneous and homogeneous flow regimes). The solid green line give the resulting slip ratio.

The Actual Slip Velocity versus the Line Speed

The actual slip velocity is determined by multiplying the slip ratio curve with the line speed.