# 8.8: The Transition Heterogeneous Regime - Homogeneous Regime

## 8.8.1 Introduction

Some general equations that are required:

The Reynolds number in general is:

$\ \mathrm{R} \mathrm{e}=\frac{\mathrm{v}_{\mathrm{l s}} \cdot \mathrm{D}_{\mathrm{p}}}{v_{\mathrm{l}}}$

The Darcy Weisbach friction factor can now be determined with:

$\ \lambda_{\mathrm{l}}=\frac{1.325}{\left(\ln \left(\frac{\varepsilon}{3.7 \cdot \mathrm{D}_{\mathrm{p}}}+\frac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}}=\frac{0.25}{\left(\log _{10}\left(\frac{\varepsilon}{3.7 \cdot \mathrm{D}_{\mathrm{p}}}+\frac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}}$

The friction velocity is:

$\ \mathrm{u}_{*}=\sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}$

## 8.8.2 The Lift Ratio

The near wall lift pushes particles away from the pipe wall. The ratio of the lift force to the sum of submerged weight and momentum of a particle is named the Lift Ratio.

$\ \mathrm{L}_{\mathrm{R}}=\frac{\mathrm{C}_{\mathrm{L}} \cdot \mathrm{u}_{*}^{2}}{\mathrm{d} \cdot\left(\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g}+\frac{1}{2} \cdot \frac{\rho_{\mathrm{s}}}{\rho_{\mathrm{l}}} \cdot \frac{\mathrm{v}_{\mathrm{t h}}^{2} \cdot \mathrm{u}_{*}}{\boldsymbol{\alpha} \cdot \mathrm{1 1 .6} \cdot v_{\mathrm{l}}}\right)} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{C}_{\mathrm{v} \mathrm{b}}}\right)$

## 8.8.3 The Heterogeneous Equation

The heterogeneous losses reduce (collapse) if the near wall lift pushes harder than the sum of the submerged weight and the momentum of a particle, resulting in:

For LR<0.7 the heterogeneous equation becomes:

$\ \mathrm{E}_{\mathrm{rhg}, \mathrm{He}}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{0 . 1 7 5} \cdot(1+\beta)}\right)^{\beta}}{\mathrm{v}_{\mathrm{ls}}}+\mathrm{8 .5}^{2} \cdot\left(\frac{\mathrm{1}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{ls}}}\right)^{2} \cdot\left(1-\mathrm{L}_{\mathrm{R}}^{2}\right)$

When the lift ratio has a value close to 1, theoretically there are no more collisions with the wall. However not all particles will have exactly the same kinetic energy, so even when the lift ratio is larger than 1, still some particles will have collisions. Therefore a smoothing function is proposed for lift ratio’s larger than 70% (ζ=0.5), giving:

$\ \mathrm{E}_{\mathrm{rhg}, \mathrm{He}}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{0 . 1 7 5} \cdot(1+\beta)}\right)^{\beta}}{\mathrm{v}_{\mathrm{ls}}}+\mathrm{8 .5}^{2} \cdot\left(\frac{\mathrm{l}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{ls}}}\right)^{2} \cdot(1-\zeta) \cdot \frac{\zeta}{\mathrm{L}_{\mathrm{R}}^{2}}$

## 8.8.4 The Homogeneous Equation

The mobilization factor of the homogeneous equation is:

$\ \mathrm{m}=\mathrm{e}^{-\mathrm{0} \cdot \mathrm{1} \cdot \frac{\alpha_{\mathrm{sm}}}{\mathrm{C}_{\mathrm{v r}}} \cdot \frac{\mathrm{u}_{\mathrm{*,ldv}}}{\mathrm{u}_{*}} \cdot \frac{\mathrm{v}_{\mathrm{t h}}}{\mathrm{v}_{\mathrm{t h}, \mathrm{ldv}}}}$

Basically this shows the concentration gradient at the center of the pipe. The mobilized homogeneous flow Erhg is now:

$\ \mathrm{E_{rhg,Ho}=m \cdot i_l \cdot\left(1-\left(1-\frac{1+ R_{sd} \cdot C_{vs}-\left(\frac{A_{C_v}}{\kappa}\cdot ln \left(\frac{\rho_m}{\rho_l} \right)\cdot \sqrt{\frac{\lambda_l}{8}}+1 \right)^2}{R_{sd}\cdot C_{vs}\cdot \left(\frac{A_{C_v}}{\kappa}\cdot ln \left(\frac{\rho_m}{\rho_l} \right)\cdot \sqrt{\frac{\lambda_l}{8}}+1 \right)^2} \right) \left(1-\left(\frac{\delta_v}{d}\right) \right)\right)}$

## 8.8.5 The Resulting Relative Excess Hydraulic Gradient

The resulting relative excess hydraulic gradient near the transition area can now be determined by:

$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{H e H o}}=\mathrm{E}_{\mathrm{r h g}, \mathrm{H e}}+\mathrm{E}_{\mathrm{r h g}, \mathrm{H o}}$

The hydraulic gradient resulting from this relative excess hydraulic gradient is now:

$\ \mathrm{i}_{\mathrm{m}, \mathrm{H} \mathrm{e} \mathrm{H} \mathrm{o}}=\mathrm{i}_{\mathrm{l}}+\mathrm{E}_{\mathrm{r} \mathrm{h} \mathrm{g}, \mathrm{H} \mathrm{e} \mathrm{H} \mathrm{o}} \cdot \mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{s}}$

This hydraulic gradient is now valid for the combination of the heterogeneous flow regime and the homogeneous flow regime, but still has to be combined with the stationary bed regime and the sliding bed regime. In the case of d/Dp>0.015 the sliding flow regime will occur and the above equation is not used.

The pressure difference in the transition zone can now be determined with:

$\ \Delta \mathrm{p}_{\mathrm{m}, \mathrm{H} \mathrm{e} \mathrm{H} \mathrm{o}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L} \cdot \mathrm{E}_{\mathrm{r h g}, \mathrm{H} \mathrm{e H o}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}$