# 8.10: The Resulting Erhg Constant Spatial Volumetric Concentration Curve


The constant spatial volumetric concentration Erhg curve for a single diameter particle forms the basis of the DHLLDV Framework. A short rehearsal of the steps to be taken to construct this curve.

1. Determine the fines fraction.
2. Adjust the pseudo liquid density, kinematic viscosity and relative submerged density, based on the fines content. Adjust the spatial volumetric concentration based on the fines content (see chapter 8.3).
3. If there are fines the following steps are carried out with the pseudo liquid properties, if there are no fines the following steps are carried out with the carrier liquid properties.
4. Determine the terminal settling velocity and the hindered settling velocity (see chapter 8.3).
5. Determine the stationary/fixed bed curve (FB) with the bed shear stress based on a bed roughness equal to the particle diameter and a bed shear stress based on sheet flow. The largest of the two at a certain line speed is the required bed shear stress. This results in a curve with bed shear stress based on a bed roughness equal to the particle diameter for line speeds starting at zero and bed shear stress based on sheet flow above a certain line speed. Usually the transition is at a Shields number around 1. The result is the FB curve (see chapter 8.4).
6. Determine the sliding bed (SB) curve, a horizontal line with value μsf in the Erhg graph (see chapter 8.5).
7. Determine the heterogeneous flow regime (He) curve including the lift ratio influence (near wall lift effect) (see chapter 8.6 and 8.8).
8. Determine the homogeneous flow regime (Ho) curve including the mobilization factor (see chapter 8.7 and 8.8).
9. Add up the heterogeneous curve and the homogeneous curve. The result is the He-Ho curve (see chapter 8.8).
10. If a sliding bed exists (intersection FB-SB at lower line speed than the intersection FB-He):
1. If the fixed bed curve is smaller than the sliding bed curve, take the fixed bed curve. Otherwise take the sliding bed curve. The result is the FB-SB curve.
2. The intersection line speed of the fixed bed curve and the sliding bed curve is the Limit of Stationary Deposit Velocity (LSDV).
3. Add up the heterogeneous curve and the homogeneous curve. The result is the He-Ho curve.
4. The particle diameter to pipe diameter ratio d/ Dp <rd/Dp/2, equation (8.9-5): At line speeds above the intersection line speed of the sliding bed regime (SB) and the heterogeneous regime (He), the He-Ho curve is valid otherwise the FB-SB curve is valid.
5. The particle diameter to pipe diameter ratio d/ Dp >rd/Dp/2, equation (8.9-5): For line speeds above the intersection line speed between sliding bed transport (SB) and heterogeneous transport (He), a weighted average of the sliding bed curve SB and the He-Ho curve has to be determined, equation (8.9-6). The result is the sliding flow (SF) curve. For ratios above d/ Dp =2·rd/Dp this results in the SB curve (see chapter 8.9).
11. If a sliding bed does not exist (intersection FB-SB at higher line speed than intersection FB-He):
1. At line speeds above the intersection line speed of the fixed bed regime (FB) and the heterogeneous regime (He), the He-Ho curve is valid otherwise the FB curve is valid.
2. The Limit of Stationary Deposit Velocity (LSDV) does not exist in this case.
3. Both the sliding bed (SB) regime and the sliding flow (SF) regime do not exist in this case.

The resulting Erhg curve follows from the flow chart Figure 8.1-1 and the steps described above. The hydraulic gradient and the pressure difference for the mixture can be determined with, once the Erhg curve is determined:

$\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\mathrm{E}_{\mathrm{r h g}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}} \quad\text{ and }\quad \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{\Delta} \mathrm{L} \cdot \mathrm{E}_{\mathrm{r h g}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}$

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