# 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.

This page titled 8.13: The Bed Height is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Sape A. Miedema (TU Delft Open Textbooks) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.