Question

For a steam nozzle, if $p_1$ = Inlet pressure; $p_2$ = Exit Pressure and n = Index of isentropic expansion, the mass flow rate per unit area is maximum, when $\frac{p_2}{p_1}$ is equal to:

• A. $\left( \frac{2}{n+1} \right)^{\frac{n-1}{n}}$
• B. $\left( \frac{2}{n-1} \right)^{\frac{n-1}{n}}$
• C. $\left( \frac{2}{n+1} \right)^{\frac{n}{n-1}}$
• D. $\left( \frac{2}{n-1} \right)^{\frac{n}{n-1}}$

Hint:

To remember the exponent: In almost all fluid flow relations, the "Power" involving $n$ usually has $n$ on top and $n-1$ on the bottom when relating pressure to other properties.

Answer 1

The Correct Option is C

1. Maximum Mass Flow Rate (Choking): In a nozzle, as the back pressure ($p_2$) is lowered relative to the inlet pressure ($p_1$), the mass flow rate increases. However, it reaches a

maximum theoretical limit at a specific pressure ratio. Beyond this point, lowering the exit pressure further does not increase the flow rate.

2. Mathematical Derivation: To find the maximum mass flow rate, we express the mass flow as a function of the pressure ratio and differentiate it with respect to $p_2/p_1$, setting the result to zero. For a gas/vapor following the isentropic law $PV^n = C$: $$\frac{p_2}{p_1} = \left( \frac{2}{n+1} \right)^{\frac{n}{n-1}}$$

3. Application and Values: At this ratio, the steam velocity at the throat reaches the

sonic velocity (local speed of sound).

• Saturated Steam ($n = 1.135$): Critical ratio is approx 0.577.

• Superheated Steam ($n = 1.3$): Critical ratio is approx 0.546.

• Air/Diatomic Gas ($n = 1.4$): Critical ratio is approx 0.528.