8.10: The Resulting Erhg Constant Spatial Volumetric Concentration Curve
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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.
- 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 μsf in the Erhg graph (see chapter 8.5).
- 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/ 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.
- 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).
- 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.
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:
\[\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\mathrm{E}_{\mathrm{r h g}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}} \quad\text{ and }\quad \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{\Delta} \mathrm{L} \cdot \mathrm{E}_{\mathrm{r h g}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}\]