8.2: Default Equations Used In This Book

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The relative submerged density R_sd is defined as:

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

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

\[\ \mathrm{v_t}=\frac{10 \cdot v_{\mathrm{l}}}{\mathrm{d}}\cdot \left(\sqrt{1+\frac{\mathrm{R_{sd}\cdot g \cdot d^3}}{100 \cdot v_{\mathrm{l}}^2}}-1 \right) \]

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

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

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

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

When clear water flows through the pipeline, the pressure loss can be determined with the well-known Darcy- Weisbach equation:

\[\ \mathrm{\Delta p_{l}=\lambda_{l} \cdot \frac{\Delta L}{D_{p}} \cdot \frac{1}{2} \cdot \rho_{l} \cdot v_{l s}^{2}}\]

The hydraulic gradient i_w (for pure water) or i_l (for a liquid in general) is meters of liquid per meter of pipeline:

\[\ \mathrm{i}_{\mathrm{w}}=\frac{\Delta \mathrm{p}_{\mathrm{l}}}{\rho_{\mathrm{w}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}} \quad\text{ or }\quad \mathrm{i}_{\mathrm{l}}=\frac{\Delta \mathrm{p}_{\mathrm{l}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}\]

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

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

With the Reynolds number:

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

The relative excess hydraulic gradient as defined and used in this book:

\[\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}} \quad\text{ or }\quad \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}}\]

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

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

For the Equivalent Liquid Model (ELM) this gives for the hydraulic gradient:

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

The relative excess hydraulic gradient is for the ELM:

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

The DHLLDV Framework is calibrated, based on these equations. Other equations, especially for the terminal settling velocity, may give slightly different results.