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8.7: The Homogeneous Transport Regime

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    32191
  • The basis of the homogeneous transport regime model is the equivalent liquid model (ELM). In terms of the relative excess hydraulic gradient, Erhg, this can be written as:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\mathrm{i}_{\mathrm{l}}\]

    Talmon (2013) derived an equation to correct the homogeneous equation (the ELM model) for the slurry density, based on the hypothesis that the viscous sub-layer hardly contains solids at very high line speeds in the homogeneous regime. This theory results in a reduction of the resistance compared with the ELM, but the resistance is still higher than the resistance of clear water. Talmon (2013) used the Prandl approach for the mixing length, which is a 2D approach for open channel flow with a free surface. The Prandl approach was extended with damping near the wall to take into account the viscous effects near the wall, according to von Driest (Schlichting, 1968). Miedema (2015) extended the model with pipe flow and a concentration distribution, resulting in the following equations.

    The value of the Darcy Weisbach wall friction factor λl depends on the Reynolds number:

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

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

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

    For the resulting Darcy-Weisbach friction factor ratio this can be approximated by:

    \[\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{\mathrm{1}}{\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}+{\mathrm{1}}\right)^{2}}\]

    The hydraulic gradient il (for a liquid including the fine solids effect in general) is:

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

    The relative excess hydraulic gradient Erhg is now:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{s}}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}-\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}+\mathrm{1}\right)^{2}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}} \cdot\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}+{\mathrm{1}}\right)^{2}}=\alpha_{\mathrm{E}} \cdot \mathrm{i}_{\mathrm{l}}\]

    The relative excess hydraulic gradient Erhg is mobilized at larger line speeds depending on the ratio between the thickness of the viscous sub-layer and the particle diameter:

    \[\ \frac{\delta_{\mathrm{v}}}{\mathrm{d}}=\frac{\mathrm{1 1 . 6} \cdot \mathrm{v}_{\mathrm{l}}}{\mathrm{u}_{*} \cdot \mathrm{d}}=\frac{\mathrm{1 1. 6} \cdot v_{\mathrm{l}}}{\sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}} \cdot {\mathrm{v}_{\mathrm{l s}} \cdot \mathrm{d}}}\]

    Where this ratio can never be larger than 1! This gives for the relative excess hydraulic gradient Erhg:

    \[\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}}}=\mathrm{i}_{\mathrm{l}} \cdot\left(1-\left(1-\frac{1+\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}}-\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+{1}\right)^{2}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}} \cdot\left(\frac{\mathrm{A}_{\mathrm{C_v}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+1\right)^{2}}\right)\left(1-\left(\frac{\delta_{\mathrm{v}}}{\mathrm{d}}\right)\right)\right)\]

    A value of ACv=3 is advised. The hydraulic gradient of the mixture is now:

    \[\ \mathrm{i_m}=\mathrm{i_l}+\mathrm{i}_{\mathrm{l}} \cdot\left(1-\left(1-\frac{1+\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}}-\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+{1}\right)^{2}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}} \cdot\left(\frac{\mathrm{A}_{\mathrm{C_v}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+1\right)^{2}}\right)\left(1-\left(\frac{\delta_{\mathrm{v}}}{\mathrm{d}}\right)\right)\right)\cdot \mathrm{R_{sd}}\cdot \mathrm{C_{vs}}\]

    The pressure difference of the mixture gives:

    \[\ \mathrm{\Delta p_m=\Delta p_l + \Delta p_l} \cdot\left(1-\left(1-\frac{1+\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}}-\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+{1}\right)^{2}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}} \cdot\left(\frac{\mathrm{A}_{\mathrm{C_v}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+1\right)^{2}}\right)\left(1-\left(\frac{\delta_{\mathrm{v}}}{\mathrm{d}}\right)\right)\right)\cdot \mathrm{R_{sd}}\cdot \mathrm{C_{vs}}\]

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