$-$ A $4-$m $6-$m room is to be heated by an electric resistance heater placed in a short

duct in the room. Initially, the room is at $15C,$ and the local atmospheric pressure is $98$kPa.

The room is losing heat steadily to the outside at a rate of $150k\frac{J}{m}in.$ A $200-W$ fan circulates

the air steadily through the duct and the electric heater at an average mass flow rate of $40$

$k\frac{g}{m}in.$ The duct can be assumed to be adiabatic, and there is no air leaking in or out of the

room. If it takes $20$min for the room air to reach an average temperature of $25C,$ find $($a$)$ the

power rating of the electric heater and $($b$)$ the temperature rise that the air experiences each

time it passes through the heater. $($c$)$ Is the duct really steady? If not, repeat $($b$)$ but without

assuming steady duct with reasonable assumptions.