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8.9: The Sliding Flow Regime

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  • For large particles the turbulence is not capable of lifting the particles enough resulting in a sort of sliding bed behavior above this transition line speed. One reason for this is that the largest eddies are not large enough with respect to the size of the particles. Sellgren & Wilson (2007) use the criterion d/Dp>0.015 for this to occur. Zandi & Govatos (1967) use a factor N<40 as a criterion, with:

    \[\ \mathrm{N}=\frac{\mathrm{v}_{\mathrm{l} \mathrm{s}}^{2} \cdot \sqrt{\mathrm{C}_{\mathrm{D}}}}{\mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}}\]

    At the Limit Deposit Velocity vls,ldv this equation can be simplified by using:

    \[\ \mathrm{F}_{\mathrm{L}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{ld v}}}{\sqrt{\mathrm{2 \cdot g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}}} \approx 1.34 \quad\text{ and }\quad \sqrt{\mathrm{C}_{\mathrm{D}}} \approx 0.6 \text{ for coarse sand}\]


    \[\ \mathrm{N}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l} \mathrm{d} \mathrm{v}}^{\mathrm{2}} \cdot \sqrt{\mathrm{C}_{\mathrm{D}}}}{\mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v t}}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}^{\mathrm{2}}}{\mathrm{2 \cdot g \cdot R _ { \mathrm { s d } } \cdot \mathrm { D } _ { \mathrm { p } }}} \cdot {\frac{\mathrm{2} \cdot \sqrt{\mathrm{C _ { \mathrm { D } }}}}{\mathrm{C _ { v t }}}}=\mathrm{1 . 3 4 ^ { 2 }} \cdot \frac{\mathrm{2 \cdot 0 .6}}{\mathrm{C}_{\mathrm{v t}}}=\frac{\mathrm{2 .3 7}}{\mathrm{C}_{\mathrm{v t}}}\]

    This gives N=2.37/Cvt<40 or Cvt>0.059 for sliding flow to occur. This criterion apparently is based on the thickness of sheet flow. If the bed is so thin that the whole bed becomes sheet flow, there will not be sliding flow, but more heterogeneous behavior. The values used in both criteria are a first estimate based on literature and may be changed in the future. A pragmatic approach to determine the relative excess hydraulic gradient in the sliding flow regime is to use a weighted average between the heterogeneous regime and the sliding bed regime. First the factor between particle size and pipe diameter is determined:

    \[\ \mathrm{f}=\frac{4}{3}-\frac{1}{3} \cdot \frac{\mathrm{d}}{\mathrm{r}_{\mathrm{d} / \mathrm{D} \mathrm{p}} \cdot \mathrm{D}_{\mathrm{p}}}\]


    \[\ \mathrm{r}_{\mathrm{d} / \mathrm{Dp}}=\frac{1}{2} \cdot \frac{\mathrm{d}}{\mathrm{D}_{\mathrm{p}}}=\frac{1}{2} \cdot \frac{3 \cdot \pi}{16} \cdot \frac{\mathrm{C}_{\mathrm{D}}}{\Psi} \cdot \frac{\mu_{\mathrm{sf}} \cdot \mathrm{C}_{\mathrm{vb}} \cdot \mathrm{C}_{\mathrm{vr}} \cdot\left(1-\mathrm{C}_{\mathrm{vr}}\right)}{\left(\sin (\beta)+\alpha_{\tau} \cdot(\pi-\beta) \cdot \mathrm{C}_{\mathrm{vr}}\right)}\]

    Secondly the weighted average hydraulic gradient or relative excess hydraulic gradient is determined:

    \[\ \begin{array}{left}
    \Delta \mathrm{p}_{\mathrm{m}, \mathrm{S F}} &=\Delta \mathrm{p}_{\mathrm{m}, \mathrm{H e H o}} \cdot \mathrm{f}+\Delta \mathrm{p}_{\mathrm{m}, \mathrm{S B}} \cdot(\mathrm{1 - f}) \\
    &=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{\Delta L} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot\left(\mathrm{E}_{\mathrm{r h g}, \mathrm{H e H o}} \cdot \mathrm{f}+\mu_{\mathrm{s f}} \cdot(\mathrm{1 - f})\right)\\
    \mathrm{i}_{\mathrm{m}, \mathrm{S F}}&=\mathrm{i}_{\mathrm{m}, \mathrm{H e H o}} \cdot \mathrm{f}+\mathrm{i}_{\mathrm{m}, \mathrm{S B}} \cdot(\mathrm{1}-\mathrm{f})=\mathrm{i}_{\mathrm{l}}+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot\left(\mathrm{E}_{\mathrm{r h g}, \mathrm{H e H o}} \cdot \mathrm{f}+\boldsymbol{\mu}_{\mathrm{s f}} \cdot(\mathrm{1}-\mathrm{f})\right)\\
    \mathrm{E}_{\mathrm{r h g}, \mathrm{S F}}&=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{S F}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\mathrm{E}_{\mathrm{r h g}, \mathrm{H e H o}} \cdot \mathrm{f}+\mu_{\mathrm{s f}} \cdot(\mathrm{1 - f})

    The concentration at the bottom of the pipe, a bed concentration, decreases with the line speed according to:

    \[\ \mathrm{C}_{\mathrm{vB}}=3.1 \cdot \mathrm{C}_{\mathrm{vb}} \cdot\left(\frac{\mathrm{v}_{\mathrm{ls}, \mathrm{lsdv}}}{\mathrm{v}_{\mathrm{ls}}}\right)^{0.4} \cdot\left(\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{C}_{\mathrm{vb}}}\right)^{0.5} \cdot \mathrm{v}_{\mathrm{t}}^{1 / 6} \cdot \mathrm{e}^{\mathrm{D}_{\mathrm{p}}} \quad\text{ with: }1.1 \cdot \mathrm{C}_{\mathrm{vs}} \leq \mathrm{C}_{\mathrm{vB}} \leq \mathrm{C}_{\mathrm{vb}}\]

    In the case of Sliding Flow, the bottom concentration decreases with increasing line speed and with decreasing spatial concentration. The bottom concentration can be determined with the following equation, where the bottom concentration can never be larger than the maximum bed concentration Cvb and never smaller than the spatial concentration Cvs. The Limit of Stationary Deposit Velocity (LSDV) has to be determined at a concentration of 17.5%, because the equation is calibrated for Cvs=0.175.