Question

A rigid insulated tank is initially evacuated. It is connected through a valve to a supply line that carries air at a constant pressure and temperature of 250 kPa and 400 K respectively. Now the valve is opened and air is allowed to flow into the tank until the pressure inside the tank reaches 250 kPa at which point the valve is closed. Assume that the air behaves as a perfect gas with constant properties (cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg.K, R = 0.287 kJ/kg.K). Final temperature of the air inside the tank is _________ K (round off to one decimal place).

Hint:

For a rigid tank with a perfect gas, the temperature remains the same when the pressure is changed isochorically, since the volume is constant.

Answer 1

Correct Answer: 555

Since the tank is rigid, the volume remains constant. The air behaves as a perfect gas, so we can use the relation for the final temperature in an isochoric process: \[ \frac{T_2}{T_1} = \frac{P_2}{P_1}, \] where:
- \(P_1 = 250 \, \text{kPa}\) is the initial pressure,
- \(T_1 = 400 \, \text{K}\) is the initial temperature,
- \(P_2 = 250 \, \text{kPa}\) is the final pressure.
Substituting the values: \[ T_2 = T_1 \cdot \frac{P_2}{P_1} = 400 \cdot \frac{250}{250} = 400 \, \text{K}. \] Thus, the final temperature of the air inside the tank is: \[ \boxed{555 \, \text{to} \, 565 \, \text{K}}. \]