Question

Air \((\gamma=1.4)\) flows isentropically through a convergent-divergent nozzle. At a certain section, the Mach number is \(1.0\) (at the throat). If the stagnation temperature is \(720\ \text{K}\), find the static temperature at the throat.

• A. \(720\ \text{K}\)
• B. \(600\ \text{K}\)
• C. \(560\ \text{K}\)
• D. \(864\ \text{K}\)

Hint:

For isentropic compressible flow, \[ \boxed{ \frac{T_0}{T} = 1+\frac{\gamma-1}{2}M^2. } \] At the nozzle throat, \[ \boxed{ M=1. } \]

Answer 1

The Correct Option is B

Step 1: Use the isentropic temperature relation. For isentropic flow, \[ \boxed{ \frac{T_0}{T} = 1+\frac{\gamma-1}{2}M^2, } \] where
• \(T_0\) = stagnation temperature,
• \(T\) = static temperature,
• \(M\) = Mach number.

Step 2: Substitute the given values. Given, \[ T_0=720\ \text{K}, \] \[ \gamma=1.4, \] \[ M=1. \] Hence, \[ \frac{720}{T} = 1+\frac{1.4-1}{2}(1)^2 = 1+0.2 = 1.2. \] Therefore, \[ T = \frac{720}{1.2} = 600\ \text{K}. \] Thus, \[ \boxed{600\ \text{K}} \] is the correct answer. Hence, \[ \boxed{(B)} \] is the correct answer.