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 𝒄𝒑 = 1.005 kJ/kg.K and 𝑹 = 0.287 kJ/kg.K, the rate of heat gain by the air from the room is ____________ kW(round off to two decimal places).
Correct Answer :
Correct answer is : 4.96
Solution :
The correct answer is 4.96.
Step 1: Analyze the Cold Air Stream (State 1)
We are given the following properties for the cold air stream entering the mixing chamber:
Temperature:
Using the ideal gas equation of state, we can find the mass flow rate of the cold air stream ():
Step 2: Analyze the Fresh Air Stream (State 2)
We are given that the temperature of the fresh air is:
The mass flow rate of the fresh air is 1.6 times that of the cold air stream:
Step 3: Energy Balance for the Insulated Mixing Chamber (State 3)
Assuming steady-state operation and an insulated mixing chamber, the energy balance equation is:
Using with a constant specific heat , the equation simplifies to:
Substituting the values into the equation:
Step 4: Rate of Heat Gain by the Air from the Room (State 4)
The air enters the room at temperature and exits the room through the exit duct at a temperature of:
The rate of heat gain by the air () is given by the energy balance of the room:
Substituting the values:
Thus, the rate of heat gain by the air from the room is 4.96 kW.
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