8.3: The Influence of Fines

The fraction of the sand in suspension, the fines, resulting in a homogeneous pseudo liquid is named X. This gives for the density of the homogeneous pseudo liquid:

\[\ \rho_{\mathrm{pl}}=\rho_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \frac{\mathrm{X} \cdot \mathrm{C}_{\mathrm{vs}} \cdot \mathrm{R}_{\mathrm{sd}}}{\left(1-\mathrm{C}_{\mathrm{vs}}+\mathrm{C}_{\mathrm{vs}} \cdot \mathrm{X}\right)}\]

So the concentration of the fines in the homogeneous pseudo liquid is not C_vs,pl=X·C_vs, but:

\[\ \mathrm{C}_{\mathrm{v s , p l}}=\frac{\mathrm{X} \cdot \mathrm{C}_{\mathrm{v s}}}{\left(\mathrm{1 - C}_{\mathrm{v s}}+\mathrm{C}_{\mathrm{v s}} \cdot \mathrm{X}\right)}\]

This is because part of the total volume is occupied by the particles that are not in suspension. The remaining spatial concentration of solids to be used to determine the hydraulic gradients curve of the solids is now:

\[\ \mathrm{C}_{\mathrm{v s}, \mathrm{r}}=(\mathrm{1}-\mathrm{X}) \cdot \mathrm{C}_{\mathrm{v s}}\]

The dynamic viscosity can now be determined according to Thomas (1965):

\[\ \mu_{\mathrm{pl}}=\mu_{\mathrm{l}} \cdot\left(1+2.5 \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{pl}}+10.05 \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{pl}}^{2}+0.00273 \cdot\left(\mathrm{e}^{16.6 \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{pl}}}-1\right)\right)\]

The kinematic viscosity of the homogeneous pseudo liquid is now:

\[\ v_{\mathrm{pl}}=\frac{\mu_{\mathrm{pl}}}{\rho_{\mathrm{pl}}}\]

One should realize however that the relative submerged density has also changed to:

\[\ \mathrm{R}_{\mathrm{s d}, \mathrm{p l}}=\frac{\rho_{\mathrm{s}}-\rho_{\mathrm{p l}}}{\rho_{\mathrm{p l}}}\]

Over the whole range of Reynolds numbers above 2320 the Swamee Jain (1976) equation gives a good approximation for the Darcy Weisbach friction factor:

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

With the Reynolds number for the pseudo liquid:

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

The equation for the terminal settling velocity in pseudo liquid (in m and m/sec) has been derived by Ruby & Zanke (1977):

\[\ \mathrm{v}_{\mathrm{t}, \mathrm{pl}}=\frac{\mathrm{1 0} \cdot v_{\mathrm{pl}}}{\mathrm{d}} \cdot \left( \sqrt{1+\frac{\mathrm{R}_{\mathrm{sd}, \mathrm{p} \mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{d}^{3}}{\mathrm{1 0 0} \cdot v_{\mathrm{p l}}^{2}}}-1 \right)\]

The general equation for the hindered terminal settling velocity in pseudo liquid according to Richardson & Zaki (1954) yields:

\[\ \mathrm{v}_{\mathrm{th}, \mathrm{pl}}=\mathrm{v}_{\mathrm{t}, \mathrm{pl}} \cdot\left(1-\mathrm{C}_{\mathrm{vs}, \mathrm{r}}\right)^{\beta}\]

According to Rowe (1987) the power in pseudo liquid can be approximated by:

\[\ \beta=\frac{4.7+0.41 \cdot \operatorname{Re}_{\mathrm{p}, \mathrm{pl}}^{0.75}}{1+\mathrm{0 .1 7 5} \cdot \mathrm{R} \mathrm{e}_{\mathrm{p}, \mathrm{p} \mathrm{l}}^{\mathrm{0 . 7 5}}} \quad\text{ with: }\quad \mathrm{R} \mathrm{e}_{\mathrm{p}, \mathrm{p l}}=\frac{\mathrm{v}_{\mathrm{t}, \mathrm{p l}} \cdot \mathrm{d}}{v_{\mathrm{p l}}}\]

With the new homogeneous pseudo liquid density, kinematic viscosity, relative submerged density and volumetric concentration the hydraulic gradient can be determined for the remaining solids, with the adjusted volumetric concentration.

8.3.1 Based on the Pseudo Liquid (A)

When pseudo liquid flows through the pipeline, the pressure loss can be determined with the well-known Darcy- Weisbach equation, this pressure loss is in kPa:

\[\ \Delta \mathrm{p}_{\mathrm{p l}, \mathrm{A}}=\lambda_{\mathrm{p l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\Delta \mathrm{p}_{\mathrm{p l}, \mathrm{B}}=\frac{\lambda_{\mathrm{p} \mathrm{l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p} \mathrm{l}}}{\rho_{\mathrm{l}}} \cdot \Delta \mathrm{p}_{\mathrm{l}}\]

The hydraulic gradient i_pl based on the pseudo liquid is in meters of liquid per meter of pipeline:

\[\ \mathrm{i}_{\mathrm{p} \mathrm{l}, \mathrm{A}}=\frac{\Delta \mathrm{p}_{\mathrm{p l}, \mathrm{A}}}{\rho_{\mathrm{p l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\lambda_{\mathrm{p} \mathrm{l}}}{\lambda_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{l}}\]

The relative excess hydraulic gradient related to the pseudo liquid as defined and used in this book is (which is zero for only the pseudo liquid):

\[\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}} \quad \mathrm{o r} \quad \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p l}} \cdot \mathrm{C}_{\mathrm{v t}, \mathrm{r}}}\]

When mixture flows through the pipeline, the pressure loss can be determined with the well-known Darcy- Weisbach equation for the ELM, based on the pseudo liquid:

\[\ \Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}=\lambda_{\mathrm{p l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \lambda_{\mathrm{p} \mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p} \mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \Delta \mathrm{p}_{\mathrm{p l}, \mathrm{A}}\]

For the Equivalent Liquid Model (ELM) this gives for the hydraulic gradient based on the pseudo liquid:

\[\ \mathrm{i}_{\mathrm{m}, \mathrm{pl}, \mathrm{A}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}}{\rho_{\mathrm{p} \mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{pl}}} \cdot \frac{\lambda_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \mathrm{i}_{\mathrm{pl,A}} \]

The relative excess hydraulic gradient is for the ELM, based on the pseudo liquid:

\[\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p} \mathrm{l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}}=\frac{\frac{\rho_{\mathrm{m}}}{\mathrm{\rho}_{\mathrm{p l}}} \cdot \mathrm{i}_{\mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p} \mathrm{l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}}=\mathrm{i}_{\mathrm{p l}, \mathrm{A}}\]

8.3.2 Based on the Carrier Liquid (B)

Since here the hydraulic gradient and the relative excess hydraulic gradient are based on the pseudo liquid density, these parameters have to be corrected in order to express them in terms of the carrier liquid density and the carrier liquid Darcy Weisbach friction factor according to, this pressure loss is in kPa:

\[\ \Delta \mathrm{p}_{\mathrm{p l}, \mathrm{B}}=\lambda_{\mathrm{p l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\Delta \mathrm{p}_{\mathrm{p} \mathrm{l}, \mathrm{A}}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \Delta \mathrm{p}_{\mathrm{l}}\]

This gives for the hydraulic gradient, carrier liquid based:

\[\ \mathrm{i}_{\mathrm{pl}, \mathrm{B}}=\frac{\Delta \mathrm{p}_{\mathrm{pl}, \mathrm{B}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{pl}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{pl}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\lambda_{\mathrm{pl}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{pl}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{l}}=\frac{\rho_{\mathrm{pl}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{pl}, \mathrm{A}}\]

The relative excess hydraulic gradient related to the carrier liquid as defined and used in this book (which is non-zero for the pure pseudo liquid):

\[\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{B}}=\frac{\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}} \quad$ or $\quad \mathrm{E}_{\mathrm{r h g}, \mathrm{p}, \mathrm{B}}=\frac{\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}}\]

When mixture flows through the pipeline, the pressure loss can be determined with the well-known Darcy-Weisbach equation for the ELM, based on the pseudo liquid:

\[\ \Delta \mathrm{p}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{B}}=\lambda_{\mathrm{p} \mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \lambda_{\mathrm{p} \mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p} \mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}\]

For the Equivalent Liquid Model (ELM) this gives for the hydraulic gradient based on the carrier liquid:

\[\ \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{B}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{p} \mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\rho_{\mathrm{p} \mathrm{l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}=\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \mathrm{i}_{\mathrm{p l}, \mathrm{A}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{p} \mathrm{l}, \mathrm{A}}==\frac{\lambda_{\mathrm{p} \mathrm{l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{l}}\]

The relative excess hydraulic gradient is for the ELM, based on the carrier liquid:

\[\ \mathrm{E}_{\mathrm{r h g}, \mathrm{pl}, \mathrm{B}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\frac{\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{p} \mathrm{l}, \mathrm{A}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\frac{\left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}}-\mathrm{1}\right) \cdot \mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}\]

8.3.3 The Different Flow Regimes

For the different flow regimes, the pressure losses should be determined with the adjusted kinematic viscosity, relative submerged density and terminal settling velocity with the equations that are also used without fines. Based on the pressure losses found, first the hydraulic gradient and the relative solids effect are determined based on the pseudo liquid properties.

This gives for the hydraulic gradient based on the pseudo liquid properties:

\[\ \mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}}{\rho_{\mathrm{p l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}}{\rho_{\mathrm{p l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}} \quad\text{ with: }\quad \Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}=\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}\]

The relative excess hydraulic gradient is, based on the pseudo liquid properties:

\[\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}}\]

The above two equations assume the pseudo liquid is the carrier liquid. In processing experimental data, it is assumed that the real carrier liquid is the carrier liquid (often water). The fraction of fines is often unknown. So to compare model results with experimental results, the hydraulic gradient of the mixture and the relative solids effect have to be related to the real carrier liquid properties and to the hydraulic gradient of the real carrier liquid.

This gives for the hydraulic gradient based on the carrier liquid properties (division by the real carrier liquid density):

\[\ \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{B}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}\]

The relative excess hydraulic gradient or relative solids effect is, based on the carrier liquid properties (deducting the hydraulic gradient of the real carrier liquid in the nominator):

\[\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{B}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p}, \mathrm{B}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}\]