Answer
$\Delta t = 841.17s= 14.02 min$
Work Step by Step
The energy balance for this system is:
$Q_{rad}+Q_{lost}-W_{fan}=\Delta U $
$(q_{rad}-q_{lost}+w_{fan})\times \Delta t = mc_v\Delta T$
For an ideal gas:
$PV=mRT$
Plugging in the values of $P_1=101.325\ kPa, T_1=10°C=283.15K, V_1=140\ m³, R=0.2870\ kJ/kg$
We get to $m =174.56\ kg$
Plugging into the energy balance $q_{rad}=10000\ kJ/h=2.78\ kW, q_{lost} = 5000\ kJ/h = 1.39\ kW,$
$w_{fan} = 0.1\ kW, \Delta T=10°C, c_v= 0.718\ kJ/kg.K$:
We get: $\Delta t = 841.17s= 14.02 min$
