Find the velocity of air flowing at the outlet of a nozzle, fitted to a large vessel which contains air at a pressure of 294.3 N/cm² (abs.) and at a temperature of 30°C. The pressure at the outlet of the nozzle is 137.34 N/cm² (abs.) Take k = 1.4 and R = 287 J/kg K.

Mass Flow Rate Through a Nozzle (Adiabatic Flow)

Problem Statement

Given Data & Constants

Inlet Conditions (Vessel):
Pressure, $P_1 = 294.3 \, \text{N/cm}^2 = 2,943,000 \, \text{N/m}^2$
Temperature, $T_1 = 30^\circ\text{C} = 303.15 \, \text{K}$
Velocity, $V_1 \approx 0$ (since it's a large vessel)

Outlet Conditions (Nozzle Exit):
Pressure, $P_2 = 137.34 \, \text{N/cm}^2 = 1,373,400 \, \text{N/m}^2$

Gas Properties:
Adiabatic index, $k = 1.4$
Gas constant, $R = 287 \, \text{J/kg K}$

Solution

1. Check for Choked Flow

First, we must determine if the flow is choked by comparing the pressure ratio to the critical pressure ratio.

$$ \text{Pressure Ratio, } \frac{P_2}{P_1} = \frac{137.34}{294.3} \approx 0.4667 $$ $$ \text{Critical Pressure Ratio, } \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}} = \left(\frac{2}{1.4+1}\right)^{\frac{1.4}{0.4}} = (0.8333)^{3.5} \approx 0.528 $$

Since the actual pressure ratio (0.4667) is less than the critical pressure ratio (0.528), the flow is **choked**. This means the velocity at the nozzle throat is the speed of sound at the throat conditions.

2. Calculate Throat Temperature and Velocity

For choked flow, the temperature at the throat ($T^*$) is the critical temperature.

$$ T^* = T_1 \left(\frac{2}{k+1}\right) = 303.15 \times \left(\frac{2}{2.4}\right) \approx 252.6 \, \text{K} $$

The velocity at the throat is the speed of sound at this critical temperature.

$$ V_2 = V^* = c^* = \sqrt{kRT^*} $$ $$ V_2 = \sqrt{1.4 \times 287 \times 252.6} \approx \sqrt{101494} \approx 318.58 \, \text{m/s} $$

Final Result:

The velocity of air flowing at the outlet of the nozzle is approximately 318.6 m/s.

Explanation of Choked Flow

Choked flow is a limiting condition that occurs in compressible flow when the velocity at the narrowest point of a passage (the throat of a nozzle) reaches the speed of sound (Mach 1). This happens when the upstream pressure is sufficiently higher than the downstream pressure.

Once the flow is choked, a further decrease in the downstream pressure will not increase the mass flow rate or the velocity at the throat. The flow is essentially "maxed out." In this problem, because the pressure drop is large enough to cause choked flow, the exit velocity is simply the speed of sound under the conditions (pressure and temperature) that exist at the nozzle throat.

Find the velocity of air flowing at the outlet of a nozzle, fitted to a large vessel which contains air at a pressure of 294.3 N/cm² (abs.) and at a temperature of 30°C. The pressure at the outlet of the nozzle is 137.34 N/cm² (abs.) Take k = 1.4 and R = 287 J/kg K.

Problem Statement

Given Data & Constants

Solution

1. Check for Choked Flow

2. Calculate Throat Temperature and Velocity

Explanation of Choked Flow

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