8.10: The Resulting Erhg Constant Spatial Volumetric Concentration Curve
- Page ID
- 32216
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\mathrm{E}_{\mathrm{r h g}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}} \quad\text{ and }\quad \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{\Delta} \mathrm{L} \cdot \mathrm{E}_{\mathrm{r h g}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}\]