4.7: Conclusions

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The equation of Ruby & Zanke (1977) will be used to determine the terminal settling velocity for sands and gravels. The equation of Richardson and Zaki (1954) will be used for hindered settling, with the equation of Rowe (1987) for the power β in the hindered settling equation. The DHLLDV Framework is calibrated based on these equations.

Using different equations will result in slightly different hydraulic gradients and Limit Deposit Velocities, requiring the constants in the DHLLDV Framework to be adjusted.

Particles with different shapes, like spheres or shells, and particles with different relative submerged densities may require different methods.

One of the main issues is that the Richardson & Zaki (1954) hindered settling equation is based on the spatial volumetric concentration Cvs and not on the relative spatial volumetric concentration C_vr=C_vs/C_vb.

\[\ \frac{\mathrm{v}_{\mathrm{t h}}}{\mathrm{v}_{\mathrm{t}}}=\left(\mathrm{1}-\mathrm{C}_{\mathrm{v} \mathrm{s}}\right)^{\boldsymbol{\beta}}\]

So even when the spatial volumetric concentration reaches a concentration where a bed with maximum porosity occurs, for sand at about C_vs=50%, still a hindered settling velocity is determined, while in reality this hindered settling velocity will be zero. Normal sands will have a porosity of about 40%, so C_vb=60%. A fixed bed may have a porosity of 40%, but a sliding bed will have a higher porosity in between 40% and 50%. The porosities mentioned here depend on the type of sand, but are mentioned to give a feeling of the order of magnitude. Since the Richardson & Zaki (1954) equation is based on small concentrations it is better to use a modified equation based on the relative concentration, for example:

\[\ \frac{\mathrm{v}_{\mathrm{t h}}}{\mathrm{v}_{\mathrm{t}}}=\left(\mathrm{1}-\mathrm{C}_{\mathrm{v} \mathrm{r}}\right)^{\boldsymbol{\beta}^{\prime}}\]

Of course the power of this equation will be different from the original equation. An equation that may even work better is:

\[\ \frac{\mathrm{v}_{\mathrm{t h}}}{\mathrm{v}_{\mathrm{t}}}=\mathrm{e}^{-\boldsymbol{\beta} \cdot \mathrm{C}_{\mathrm{v} \mathrm{r}}^{\mathrm{1 . 2 5}}} \cdot\left(\mathrm{1}-\mathrm{C}_{\mathrm{v} \mathrm{r}}^{\mathrm{3}}\right)\]

For small concentrations this equation gives the same result as the original equation, but for concentrations approaching the bed concentration, this equation approaches a zero settling velocity. This would describe the bed behavior much better. So for small concentrations this equation describes hindered settling, while for large relative concentrations approaching 1, the behavior is more close to consolidation. The power β in this equation is equal to the original power β.