Height-Pressure Relationship in a Floating Cover Biogas Digester

A floating cover biogas digester consists of a chamber where biogas accumulates, causing a floating cover to rise. As the gas builds up, the pressure inside the chamber increases, causing a change in the height of the floating cover.

Problem Formulation

Given:

Initial pressure in the chamber, $ P_0 $ (e.g., atmospheric pressure).
Variable pressure in the chamber as gas is produced, $ P $.
Cross-sectional area of the chamber, $ A $, assumed constant.
Height of the floating cover at any pressure $ P $, $ h(P) $.

We need to derive an equation that gives $ h(P) $ as a function of $ P $.

Solution

Assuming an ideal gas behavior, we can relate pressure $ P $, volume $ V $, and height $ h $ as follows.

$$ V = A \cdot h(P) $$

According to Boyle's Law, since temperature is constant:

$$ P_0 V_0 = P \cdot V $$

where $ V_0 $ is the initial volume and $ P_0 $ is the initial pressure. Since $ V = A \cdot h(P) $ and $ V_0 = A \cdot h_0 $, where $ h_0 $ is the initial height:

$$ P_0 \cdot A \cdot h_0 = P \cdot A \cdot h(P) $$

Dividing both sides by $ A $ and solving for $ h(P) $:

$$ h(P) = \frac{P_0 \cdot h_0}{P} $$

Thus, the height $ h(P) $ varies inversely with the pressure $ P $, indicating that as pressure increases, the floating cover height decreases according to the relationship above.

Conclusion

This inverse relationship between height and pressure in a floating cover biogas digester ensures that the digester maintains consistent pressure regulation as biogas production varies.