8.6: The Heterogeneous Transport Regime

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Miedema & Ramsdell (2013) derived an equation for the Relative Excess Hydraulic Gradient for heterogeneous transport based on energy considerations. This equation consists of two parts. A first part for the contribution due to potential energy losses and a second part for the kinetic energy losses. The equation is based on uniform sands or gravels, but Miedema (2014) also derived a modified equation for graded sands and gravels. In its basic form the equation looks like:

\[\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}}}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}}{\mathrm{v}_{\mathrm{ls}}}+\left(\frac{\mathrm{v}_{\mathrm{sl}}}{\mathrm{v}_{\mathrm{t}}}\right)^{2}=\mathrm{S}_{\mathrm{hr}}+\mathrm{S}_{\mathrm{rs}}\]

The Settling Velocity Hindered Relative, S_hr, is the Hindered Settling Velocity of a particle v_t·(1-C_vs/κ_C)^β divided by the line speed v_ls. The S_hr value gives the contribution of the potential energy losses to the Relative Excess Hydraulic Gradient. The S_hr is derived for and can be applied to the heterogeneous regime. The Slip Relative Squared S_rs is the Slip Velocity of a particle v_sl divided by the Terminal Settling Velocity of a particle v_t squared and this S_rs value is a good indication of the Relative Excess Hydraulic Gradient due to the solids, since its contribution to the total is 90%-100%. The S_rs value gives the contribution of the kinetic energy losses to the Relative Excess Hydraulic Gradient. The S_rs is derived for and can be applied to the heterogeneous regime.

The potential energy term is explicit and all the variables involved are known, so this term can be solved. The kinetic energy term however contains the slip velocity, which is not known. The kinetic energy term has been derived by Miedema & Ramsdell (2013) based on kinetic energy losses due to collisions or interactions with the pipe wall or the viscous sub layer. This means that the slip velocity used in the above equation is not necessarily the average slip velocity, but it is the slip velocity necessary to explain the kinetic energy losses. The average slip velocity of the particles will probably be larger, but of the same magnitude. The derivation of the slip velocity equation for uniform sands or gravels will be subject of another paper, but the resulting equation for the E_rhg parameter is given here. Giving for the relative excess hydraulic gradient, the E_rhg parameter:

\[\ \begin{array}{left}\mathrm{F r}_{\mathrm{p}}&=\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\\
\mathrm{E}_{\mathrm{r h g}}&=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\mathrm{S}_{\mathrm{h r}}+\mathrm{S}_{\mathrm{r s}}=\frac{\mathrm{v}_{\mathrm{t}} \cdot(\mathrm{1 - \frac { \mathrm { C } _ { \mathrm { v s } } } { \mathrm { \kappa } _ { \mathrm { C } } }})^{\beta}}{\mathrm{v}_{\mathrm{l s}}}+\mathrm{8 .5}^{\mathrm{2}} \cdot\left(\frac{\mathrm{1}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{l s}}}\right)^{2}\\
\mathrm{\kappa}_{\mathrm{C}}&=\mathrm{0 . 1 7 5 \cdot ( 1 + \beta )}\end{array}\]

The equation has been modified slightly since the original article of Miedema & Ramsdell (2013). The derivation is published in Miedema (2015). The hydraulic gradient for the mixture is now:

\[\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\left(\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}}{\mathrm{v}_{\mathrm{l s}}}+\mathrm{8 .5}^{2} \cdot\left(\frac{\mathrm{1}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{l s}}}\right)^{2}\right) \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}\]

This gives for the pressure difference of the mixture:

\[\ \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}}{\mathrm{v}_{\mathrm{l} \mathrm{s}}}+\mathrm{8 .5}^{2} \cdot\left(\frac{\mathrm{1}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{l s}}}\right)^{2}\right) \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{s}}\]