Oxygen Depletion Over TimeSteady-state and t² fire oxygen consumption in a closed room

What this calculates

This sheet estimates the amount of time required for a fire to reduce the amount of oxygen in a closed compartment to below 14%.

How to use it

Enter the dimensions of the room (in feet) in cells D3-D5. Unless this is a special application with elevated oxygen, leave cell D8 as 21%. Choose either a steady state fire (cell C18), or more likely a t-squared fire growth rate (cell C24). If you choose a t-squared fire, adjust the time step in cell D26 so that it provides enough time on the graph for the oxygen level to reach 14%.

Read the graph as follows:

The blue line represents the heat release rate of the t-squared fire that was specified, with time on the bottom and heat release on the left side.

The red line indicates oxygen concentration in the room over time, with the percentage indicated on the right access. By default I have set the graph to start at 14%, assuming anything other than that will result in the end of flaming combustion (although smoldering can continue until 8%).

Variables

W, L, H
Room dimensions [ft / m]
V
Room volume [m³]
O₂
Initial oxygen fraction [%]
Q
Heat release rate (steady state) or growth function [kW]
α
t² growth coefficient [kW/s²]
M_O₂
Molecular weight of O₂ (32 g/mol) [—]
m_O₂
Initial mass of O₂ in room [kg]
ΔH_O₂
Energy released per kg O₂ consumed (≈ 13.1 MJ/kg) [kJ/kg]

Equations

$$ V = W \cdot L \cdot H $$Volume of room
$$ M_{air,O_{2}} = 32 \cdot \text{O}_{2}\% $$Molar mass of O₂ per mole of air
$$ m_{O_{2}} = \dfrac{ 1000 }{ 22.4 } \cdot \dfrac{ M_{air,O_{2}} }{ 1000 } \cdot V $$Initial mass of O₂ in the room
$$ t_{1\%} = \dfrac{ (m_{O_{2}} / \text{O}_{2}\%) \cdot \Delta H_{O_{2}} / 100 }{ Q } $$Time to reduce O₂ by 1 % (steady-state Q)
$$ t_{14\%} = (\text{O}_{2}\% - 14\%) \cdot 100 \cdot t_{1\%} $$Time to reach 14 % O₂ from initial fraction
$$ \text{O}_{2}\%(t) = \text{O}_{2}\%_{0} \cdot \dfrac{ m_{O_{2}} - \int_0^t \alpha \tau^2 / \Delta H_{O_{2}} \, d\tau }{ m_{O_{2}} } $$O₂ % over time for a t² fire (numerical integration)

Discussion

This equation is for closed compartments, no open doors or windows. Typically you will want to use a t-squared fire, but I have provided the steady state in case you have an application where that would be useful.

Worked example

Example

A fire occurs in a bedroom that is 14’ x 12’ with a 7’ ceiling. Based on the contents of the room you determine that the fire growth was most likely a medium t-squared fire. Calculate the time at which the oxygen concentration in the room is reduced to 14%.

Enter the room dimensions in cells D3-D5. Leave the value of D8 at 21%. From the drop down menu in cell D24 choose a Medium growth rate. Set the time step to 5 seconds (anything less will not calculate far enough out).

The oxygen in the room will be depleted down to 14% at 225 seconds. At that time the fire size will be 562.5 kW.

References

  1. Based on the basic chemical properties of oxygen and standard fire-engineering relations for oxygen consumption (≈13.1 MJ per kg O₂ consumed; see SFPE Handbook).