Answer
a) $\Delta t=146\ s$
b) $\dot{m}=6.02\ kg/s$
Work Step by Step
For the house:
$m=\frac{P_1V_1}{RT_1}$
$P_1=95\ kPa,\ R=0.287\ kJ/kg.K,\ V_1=400\ m^3,\ T_1=287\ K$
$m=461.3\ kg$
$W_e+W_s-Q=\Delta U$
$\Delta t(\dot{W}_e+\dot{W}_s-\dot{Q})=mc_v(T_2-T_1)$
$\dot{W}_e=30\ kW,\ \dot{W}_s=250\ W,\ \dot{Q}=450\ kJ/min,\ c_v=0.718\ kJ/kg.°C,\ T_2=24°C,\ T_1=14°C$
$\Delta t=146\ s$
For the duct:
$\dot{W}_e+\dot{W}_s+\dot{m}h_1=\dot{m}h_2$
$\dot{W}_e+\dot{W}_s=\dot{m}c_p\Delta T$
$c_p=1.005\ kJ/kg.°C,\ \Delta T=5°C$
$\dot{m}=6.02\ kg/s$
