A steam radiator is used to heat a 60-m³ room. Saturated steam at 3.0bar condenses in the radiator, and emerges as a liquid at the saturation temperature. Heat is lost from the room to the outside at a rate Q(kJ/h) = 30.0(T – T₀) where T(°C) is the room temperature and T₀ = 0°C is the outside temperature. At the moment the radiator is turned on, the temperature in the room is 10°C. 

(a) Let m_s (kg/h) denote the rate at which steam condenses in the radiator and n (k mol) the quantity of air in the room. Write a differential energy balance on the room air, assuming that n remains constant at its initial value, and evaluate all numerical coefficients. Take the heat capacity of air (Cv) to be constant at 20.8 J/ (mol∙°C).

(b) Write the steady-state energy balance on the room air and use it to calculate the steam condensation rate required to maintain a constant room temperature of 24°C.

(c) Integrate the transient balance to calculate the time required to achieve a temperature of 23°C, assuming that the steam rate is that calculated in part (b).