Question

In a converging duct, area and velocity at section P are 1 m\(^2\) and 15 m/s, respectively. The temperature of the fluid is 300 K.
Air flow through the nozzle can be assumed to be inviscid and isothermal. Characteristic gas constant is 287 J/(kg-K) and the ratio of specific heats is 1.4 for air.
To ensure that the air flow remains incompressible (Mach number, \( M \leq 0.3 \)) in the duct, the minimum area required at section Q is _________ m\(^2\) (rounded off to two decimal places).

Hint:

For incompressible flow in a duct, use the Mach number and isothermal flow conditions to compute the required area. Make sure to use the correct value for \( \gamma \) and the gas constant.

Answer 1

Correct Answer: 0.14

To ensure incompressibility (Mach number \( M \leq 0.3 \)), we use the relation between area and Mach number in an isothermal flow:
\[ \frac{A}{A^*} = \frac{1}{M} \left( \frac{2}{\gamma + 1} \left( 1 + \frac{\gamma - 1}{2} M^2 \right) \right)^{\frac{\gamma + 1}{2(\gamma - 1)}}. \]
where
- \( \gamma = 1.4 \) (ratio of specific heats),
- \( M = 0.3 \),
- \( A = 1 \, \text{m}^2 \) (at section P).
From the equation for incompressible flow:
\[ M = \frac{v}{a}, \quad a = \sqrt{\gamma R T}. \]
We calculate the necessary values and find the required area at section Q:
\[ A_Q \approx 0.14 \, \text{m}^2. \]
Thus, the minimum area required at section Q is approximately \( 0.14 \, \text{m}^2 \).