2.3: Applications of Dimensionless Numbers

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2.3.1 The Slurry Flow in the Pipe

The Reynolds number of the slurry flow in the pipe is:

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

The Froude number of the slurry flow in the pipe is:

\[\ \mathrm{F r}_{\mathrm{fl}}=\frac{\mathrm{v}_{\mathrm{l s}}}{\sqrt{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}}\]

The Froude number is also used in fluid mechanics as:

\[\ \widehat{\mathrm{F}} \mathrm{r}_{\mathrm{fl}}=\frac{\mathrm{v}_{\mathrm{ls}}^{\mathrm{2}}}{\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}\]

Where each of the terms on the right has been squared. Here we will use the first definition, according to equation (2.3-2).

The Thủy number of the slurry flow in the pipe in terms of the line speed or the friction velocity is:

\[\ \mathrm{Th}_{\mathrm{ls}}=\left(\frac{v_{\mathrm{l}} \cdot \mathrm{g}}{\mathrm{v}_{\mathrm{ls}}^{3}}\right)^{1 / 3} \quad \text{ or }\quad \mathrm{Th}_{\mathrm{fv}}=\left(\frac{v_{\mathrm{l}} \cdot \mathrm{g}}{\mathrm{u}_{*}^{3}}\right)^{1 / 3}\]

2.3.2 The Terminal Settling Velocity of a Particle

The Reynolds number of the terminal settling velocity of a particle is:

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

The Froude number of the terminal settling velocity of a particle is:

\[\ \mathrm{F r}_{\mathrm{p}}=\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}=\frac{\mathrm{1}}{\sqrt{\mathrm{C}_{\mathrm{x}}}}\]

The Froude number is also used in fluid mechanics as:

\[\ \widehat{\mathrm{F}} \mathrm{r}_{\mathrm{p}}=\frac{\mathrm{v}_{\mathrm{t}}^{2}}{\mathrm{g} \cdot \mathrm{d}}\]

Where each of the terms on the right has been squared. Here we will use the first definition, according to equation (2.3-6).

The Archimedes number of a particle is:

\[\ \mathrm{A r}_{\mathrm{p}}=\frac{\mathrm{g} \cdot \mathrm{d}^{3} \cdot \mathrm{R}_{\mathrm{sd}}}{v_{\mathrm{l}}^{2}}\]

The Thủy number of a particle is:

\[\ \operatorname{Th}_{\mathrm{p}}=\left(\frac{v_{\mathrm{l}} \cdot \mathrm{g}}{\mathrm{v}_{\mathrm{t}}^{\mathrm{3}}}\right)^{\mathrm{1} / 3}\]