# Oxygen depletion in a fixed volume due to inert gas leak > [!NOTE] > Also see an [[Oxygen depletion in a fixed volume due to inert gas leak with constant recycle|extension of this work]] which incorporates a recycle stream. In a system where nitrogen leakage into a room results in oxygen depletion, the following may be used to determine the leakage rate resulting in a specified oxygen concentration at equilibrium. ![[oxygen_depletion.png|500]] $\dot{n}=NV_{room}\left(1- \frac{C_f}{0.21} \right)\tag{1}$ Where: $\dot{n}$ is nitrogen flow into room, $m^3/h$ $N$ is the no. of room change per hour due to HVAC, $h^{-1}$ $V_{room}$ is room volume, $m^3$ $C_f$ is final room oxygen concentration at equilibrium, (vol frac) >[!Warning] Assumptions >- Assumes instantaneous perfect mixing. Whilst concentration will average out to some degree in large spaces, this method does not account for local oxygen depletion. **This will be significant at the point of release.** >- All flowrates given in $m^3/h$ at atmospheric pressure. >- Air into the system via HVAC unit is a constant volumetric flow and taken as 21 vol% oxygen. # Derivation ![[Oxygen_depletion_system.png|500]] At equilibrium, oxygen in = oxygen out. $0.21\dot{a}=C_f\dot{Q}\tag{2}$ Where: $\dot{a}$ is air flow into the room, $m^3/h$ $C_f$ is final room oxygen concentration at equilibrium, (vol frac) $\dot{Q}$ is vent flow out of the room, $m^3/h$ Since $\dot{a}=\dot{Q}-\dot{n}$, we can substitute to find: $0.21(\dot{Q}-\dot{n})=C_f\dot{Q}\tag{3}$ Re-arranging for $\dot{n}$: $\dot{n}=\dot{Q}\left( \frac{0.21-C_f}{0.21} \right)\tag{4}$ The vent flow is related simply to the number of air changes per hour: $\dot{Q}=NV_{room}\tag{5}$ Where: $N$ is the number of air changes per hour, $h^{-1}$ $V_{room}$ is the room volume, $m^3$ Substitute $\text{(5)}$ into $\text{(4)}$: $\dot{n}=NV_{room}\left(1- \frac{C_f}{0.21} \right)\tag{6}$