Question

An air-conditioning system provides a continuous flow of air to a room using an intake duct and an exit duct, as shown in the figure. To maintain the quality of the indoor air, the intake duct supplies a mixture of fresh air with a cold air stream. The two streams are mixed in an insulated mixing chamber located upstream of the intake duct. Cold air enters the mixing chamber at 5°C, 105 kPa with a volume flow rate of 1.25 m³/s during steady state operation. Fresh air enters the mixing chamber at 34°C and 105 kPa. The mass flow rate of the fresh air is 1.6 times of the cold air stream. Air leaves the room through the exit duct at 24°C. 

Assuming the air behaves as an ideal gas with \(c_p = 1.005 \, \text{kJ/kg.K}\) and \(R = 0.287 \, \text{kJ/kg.K}\), the rate of heat gain by the air from the room is ________ kW (round off to two decimal places).

Hint:

For mixed streams, the mass flow rate of each stream can be calculated, and then the heat gained or lost can be found by multiplying by the specific heat capacity and the temperature difference.

Answer 1

Correct Answer: 4.9

The volume flow rate \( \dot{V} \) is given as 1.25 m³/s for cold air. The mass flow rate of cold air is: \[ \dot{m}_{\text{cold}} = \frac{\dot{V} \cdot P}{R \cdot T} = \frac{1.25 \times 105000}{0.287 \times (5 + 273)}. \] The mass flow rate of fresh air is: \[ \dot{m}_{\text{fresh}} = 1.6 \times \dot{m}_{\text{cold}}. \] The heat gained by the air is: \[ Q = \dot{m} \cdot c_p \cdot (T_{\text{exit}} - T_{\text{inlet}}). \] The heat gained by the air from the room is: \[ \boxed{4.90 \, \text{to} \, 5.10 \, \text{kW}}. \]