# 8.10: The Resulting Erhg Constant Spatial Volumetric Concentration Curve

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The constant spatial volumetric concentration **E**_{rhg}** **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.

- Determine the fines fraction.
- 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).
- 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.
- Determine the terminal settling velocity and the hindered settling velocity (see chapter 8.3).
- 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). - Determine the sliding bed (
**SB**) curve, a horizontal line with value**μ****sf****E****rhg** - Determine the heterogeneous flow regime (
**He**) curve including the lift ratio influence (near wall lift effect) (see chapter 8.6 and 8.8). - Determine the homogeneous flow regime (
**Ho**) curve including the mobilization factor (see chapter 8.7 and 8.8). - Add up the heterogeneous curve and the homogeneous curve. The result is the
**He-Ho**curve (see chapter 8.8). - If a sliding bed exists (intersection
**FB-SB**at lower line speed than the intersection**FB-He**):- 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. - The intersection line speed of the fixed bed curve and the sliding bed curve is the Limit of Stationary Deposit Velocity (
**LSDV**). - Add up the heterogeneous curve and the homogeneous curve. The result is the
**He-Ho**curve. - The particle diameter to pipe diameter ratio
**d/ D**_{p}**<r**_{d/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. - The particle diameter to pipe diameter ratio
**d/ D**_{p >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/ D**_{p}**=2·r**this results in the_{d/Dp}**SB**curve (see chapter 8.9).

- 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
- If a sliding bed does not exist (intersection
**FB-SB**at higher line speed than intersection**FB-He**):- 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. - The Limit of Stationary Deposit Velocity (
**LSDV**) does not exist in this case. - Both the sliding bed (
**SB**) regime and the sliding flow (**SF**) regime do not exist in this case.

- At line speeds above the intersection line speed of the fixed bed regime (

The resulting **E**** _{rhg} **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

**E**

**curve is determined:**

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