Question

A multistage axial compressor takes in air at 1 atm, 300 K and compresses it to a minimum of 5 atm.
The mean blade speed is 245 m/s and work coefficient, \( \frac{\Delta c \, \theta}{U} \), is 0.55 for each stage.
For air, use \( C_p = 1005 \, \text{J/(kg-K)} \), \( R = 287 \, \text{J/(kg-K)} \), and \( \gamma = 1.4 \).
If the compression is isentropic, the number of stages required is _________ (rounded off to the next higher integer).

Hint:

In axial compressors, the total work and compression stages can be determined by using the isentropic relations along with given work coefficients and thermodynamic properties of the fluid.

Answer 1

Correct Answer: 1

For isentropic compression in an axial compressor, the following equation relates the pressure ratio to the temperature ratio and work coefficient: \[ \left( \frac{P_2}{P_1} \right) = \left( \frac{T_2}{T_1} \right)^{\frac{\gamma}{\gamma-1}}. \] Given the pressure ratio \( P_2/P_1 = 5 \), we can find the temperature ratio: \[ \frac{T_2}{T_1} = 5^{\frac{\gamma - 1}{\gamma}} = 5^{\frac{0.4}{1.4}} \approx 1.623. \] Now, using the given work coefficient \( \frac{\Delta c \, \theta}{U} = 0.55 \), we use the relation for work done per stage in an axial compressor: \[ \frac{\Delta h}{U} = 0.55 \quad \Rightarrow \quad \Delta h \approx 0.55 \times 245. \] Thus, the total work required for compression leads us to calculate the number of stages required. The final answer will be approximately \( 6 \) stages or greater than \( 6 \) depending on the rounding method.
Thus, the number of stages required is approximately \( 6 \).