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?
 A: 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. 
The other questions like how to cool or when an extinguisher explodes have the answer "it depends". 
