8.13: The Bed Height

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The bed fraction can be determined with the following equation:

\[\ \zeta=\hat{\mathrm{A}}_{\mathrm{b}}=\frac{\mathrm{A}_{\mathrm{b}}}{\mathrm{A}_{\mathrm{p}}}=\frac{\left(1-\mathrm{\kappa}_{\mathrm{l d v}}\right) \cdot \mathrm{C}_{\mathrm{v t}}+\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \xi}{\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot(1-\xi)}=\frac{\left(1-\mathrm{\kappa}_{\mathrm{l d v}} \cdot(1-\xi)\right) \cdot \mathrm{C}_{\mathrm{v t}}}{\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot(1-\xi)}\]

The bed fraction f is related to the bed angle according to:

\[\ \zeta=\frac{\beta-\sin (\beta) \cdot \cos (\beta)}{\pi}\]

Since this is an implicit equation, the bed angle has to be determined by iteration. Once the bed angle is determined, the bed height can be determined with:

\[\ \beta=\operatorname{acos}\left(\frac{0.5-\frac{\mathrm{r}}{\mathrm{D_{p}}}}{0.5}\right) \Leftrightarrow \frac{\mathrm{r}}{\mathrm{D_{p}}}=\frac{1-\cos (\beta)}{2}\]

Very often in literature the bed height is used in graphs.