# Fire -Thermodynamics

Could a steel box lined with Rockwool (interior only) be an adequate shelter during a fire? How can I determine the temperature inside of the box at peak fire temp? How long could someone withstand the peak internal temperature? How could I cool the interior? Would a fire extinguisher explode at peak temperature?

• How big is the box, 1 km on each side? Jan 13, 2019 at 22:30
• You probably want the insulation on the outside of the box. Steel loses mechanical strength at around 900 deg F, so the insulation would protect mechanical strength of the box as well as keep you cool. Jan 14, 2019 at 2:56

Imagine a box of area $$A$$, volume $$V$$, density $$\rho$$, temperature $$T$$ and interior heat capacity $$C$$ surrounded by a fire at temperature $$T_f$$. It has a thickness $$d$$ thermal insulation of thermal conductivity $$k$$. Let's ignore the thermal capacity of the insulator. The heat flow across the area will be $$kA(T_f-T)/d$$ Watt, and hence the internal temperature will grow as $$T' = kA(T_f-T)/\rho CVd.$$ Assuming all the values to the right except $$T$$ are constant the solution to this is $$T(t)=T_f - (T(0)-T_f)\exp(-[kA/\rho CVd]t).$$ If the maximum acceptable temperature is $$T_{max}$$ this will happen after time $$t=-\left[\frac{\rho CVd}{kA}\right]\ln\left(\frac{T_{max}-T_f}{T_f - T(0)}\right).$$
Throwing some random numbers at this. Stone wool has a thermal conductivity around 0.020 W/m K. If we assume a $$V=8$$ cubic meter box containing air, $$\rho=1$$, $$C=1.00$$ kJ/kg.K ( ignoring temperature and pressure dependency!), A=24 square meter. Let's set $$T_f=700$$ K and $$T(0)=300$$K and $$T_{max}=400$$ K. Let's add a meter of rock wool, $$d=1$$. Then I get 4794.7 seconds, or 79 minutes. That doesn't sound too crazy given that it is a pretty mild fire and a lot of insulation. Using 1 cm insulation gives you 47 seconds instead.