Helium expands in a nozzle from 150 psia, 900 R, and negligible velocity to 15 psia. Calculate t

step 1 In this question, helium gas is accelerated in our nozzle. For a specified mass flow rate, the t

Helium expands in a nozzle from 150 psia, 900 R, and...

Helium expands in a nozzle from 150 psia, 900 $\mathrm{R}$, and negligible velocity to 15 psia. Calculate the throat and exit areas for a mass flow rate of $0.2 \mathrm{lbm} / \mathrm{s}$, assuming the nozzle is isentropic. Why must this nozzle be convergingdiverging?

Key Concepts

Converging-Diverging Nozzle

A converging-diverging nozzle is designed to first accelerate the flow in a converging section up to sonic conditions at the throat, and then further accelerate or decelerate the flow in the diverging section. For supersonic exit conditions, a mere converging nozzle is insufficient. The diverging section is necessary to expand the already choked flow and allow it to accelerate to supersonic speeds, matching the pressure conditions at the exit.

Area-Mach Number Relationship

The area-Mach number relationship in compressible flow ties the geometry of the nozzle to the flow speed. Through the use of isentropic flow relations, the ratio of the local flow area to the throat area can be related to the local Mach number. This relation is essential for determining the required throat and exit areas for a given mass flow rate under steady, isentropic conditions.

Isentropic Flow

Isentropic flow refers to a process in which the fluid undergoes changes without any entropy production, meaning that it is both adiabatic and reversible. This is a common assumption in idealized compressible flow problems, as it simplifies the analysis by allowing the use of conservation laws and isentropic relations to relate pressure, temperature, and density throughout the flow.

Choked Flow

Choked flow occurs when the fluid velocity reaches the speed of sound at a particular location in the nozzle, typically at the throat. At this point, the flow rate becomes independent of the downstream conditions. This concept is crucial for nozzle design because the mass flow rate is maximized and restricted by the sonic condition, and it dictates the proper sizing of the throat area.

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