The critical pressure ratio of a convergent nozzle is defined as

Explanation:

Critical Pressure Ratio in a Convergent Nozzle

• The critical pressure ratio of a convergent nozzle is defined as the ratio of the outlet pressure to the inlet pressure of the nozzle when the flow through the nozzle reaches its maximum mass flow rate per unit area. This ratio is a critical parameter in the design and operation of convergent nozzles, particularly in applications involving compressible fluid flow, such as in turbines, jet engines, and various industrial processes.

Convergent nozzle:

• A convergent nozzle is a device used to accelerate a fluid by decreasing its cross-sectional area. When a compressible fluid, such as a gas, flows through a convergent nozzle, the velocity of the fluid increases as the cross-sectional area decreases, in accordance with the principle of conservation of mass and energy. The relationship between the pressure, velocity, and density of the fluid is governed by the isentropic flow equations for compressible fluids.
• The critical pressure ratio occurs when the flow at the throat of the nozzle (the point of smallest cross-sectional area) reaches the speed of sound, also known as the Mach number equal to 1. At this point, the flow is said to be "choked," and the mass flow rate through the nozzle becomes maximum for the given inlet conditions. Any further decrease in the outlet pressure below this critical value will not increase the mass flow rate.

The critical pressure ratio of a convergent nozzle is the ratio at which the flow becomes choked and mass flow rate per unit area is maximum.

At this point, further decrease in outlet pressure does not increase the mass flow rate.

\( \left(\frac{P_2}{P_1}\right)_{\text{critical}} = \left(\frac{2}{\gamma + 1} \right)^{\frac{\gamma}{\gamma - 1}} \)

Where \( \gamma \) is the ratio of specific heats, \( P_1 \) is inlet pressure, and \( P_2 \) is outlet pressure.