**The DHLLDV Framework is derived for uniform sands and gravels and constant spatial volumetric concentration C**_{vs}. But how does the model behave for graded sands and gravels. The **flow chart on this website** **shows how to deal with graded sands and gravels. The resulting hydraulic gradient i**_{m} and relative excess hydraulic gradient E_{rhg} curves depend on the grading of the sand or gravel, the d_{50} of the PSD and the pipe diameter D_{p}. The following graphs are made for a sand with d_{50}/d_{15}=2.718 and d_{85}/d_{50}=2.718. The graphs show that the curves of graded sands and gravels are less steep in the heterogeneous regime, just as predicted by Wilson et al. (1992), but not to the same degree. The more graded the sands or gravels, the less steep the heterogeneous curve. However the steepness also depends on the pipe diameter and the d_{50} of the PSD.

**The Particle Size Distribution (PSD)**

**The PSD graph shows two curves. The original PSD and the remaining PSD. The difference is the amount of fines, assumed to form a homogeneous pseudo liquid with adjusted density and viscosity.**

**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 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 Hydraulic Gradient here is the carrier liquid pressure loss Δp**_{l} divided by the liquid density ρ_{l}, the gravitational g constant and the length of the pipeline ΔL according to:

**i**_{l}=Δp_{l}/(ρ_{l}·g·ΔL)

**The Bed Height/Fraction & the Slip Factor versus the Line Speed**

**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}

**Because of the grading, the slip factor of the graded sand or gravel is slightly larger (the slip is slightly smaller) resulting in a smaller bed fraction (the fraction of the pipe occupied by the bed), compared with the uniform sand or gravel.**