The converging-diverging nozzle shown in Fig. P3.22 expands...

The converging-diverging nozzle shown in Fig. P3.22 expands and accelerates dry air to supersonic speeds at the exit, where $p_{2}=8 \mathrm{kPa}$ and $T_{2}=240 \mathrm{K}$. At the throat, $p_{1}=$ $284 \mathrm{kPa}, T_{1}=665 \mathrm{K},$ and $V_{1}=517 \mathrm{m} / \mathrm{s} .$ For steady com- pressible flow of an ideal gas, estimate ( $a$ ) the mass flow in $\mathrm{kg} / \mathrm{h},(b)$ the velocity $V_{2},$ and $(c)$ the Mach number $\mathrm{Ma}_{2}$ .

The converging-diverging nozzle shown in Fig. P3.22 expands and accelerates dry air to supersonic speeds at the exit, where $p_{2}=8 \mathrm{kPa}$ and $T_{2}=240 \mathrm{K}$. At the throat, $p_{1}=$ $284 \mathrm{kPa}, T_{1}=665 \mathrm{K},$ and $V_{1}=517 \mathrm{m} / \mathrm{s} .$ For steady com-
pressible flow of an ideal gas, estimate ( $a$ ) the mass flow in $\mathrm{kg} / \mathrm{h},(b)$ the velocity $V_{2},$ and $(c)$ the Mach number $\mathrm{Ma}_{2}$ .

Key Concepts

Mach Number and Supersonic Conditions

The Mach number is a dimensionless quantity representing the ratio of the flow velocity to the local speed of sound. It is a critical parameter in compressible flow as it distinguishes between subsonic, sonic, and supersonic regimes. In supersonic flows, such as at the exit of a converging-diverging nozzle, the dynamics of the flow change significantly, affecting factors like shock waves and expansion fans.

Ideal Gas Assumption in Compressible Flow

The ideal gas assumption simplifies the analysis of compressible flow by relating pressure, temperature, and density through the ideal gas law. This relationship is used to derive key expressions in compressible flow analysis, allowing for the use of isentropic flow relations and other simplifications when calculating properties such as velocity, density, and pressure in high-speed nozzles.

Choked Flow and Throat Conditions

Choked flow occurs in a nozzle when the flow velocity at the narrowest point (throat) reaches the speed of sound, limiting the mass flow rate through the nozzle regardless of further decreases in downstream pressure. Understanding choked conditions is essential for designing nozzle systems, as it defines the maximum mass flow rate and influences the subsequent acceleration in the diverging section.

Steady Compressible Flow

Steady compressible flow refers to a fluid flow in which the fluid density can change (i.e., compressibility is significant), and the flow properties at any given point remain constant over time. This assumption simplifies the analysis of gases moving at high speeds, where variations in pressure, temperature, and density should be taken into account to accurately apply conservation laws.

Converging-Diverging Nozzle

A converging-diverging nozzle is a type of duct that accelerates a compressible fluid to supersonic speeds. The geometry features a converging section that brings the flow to a throat where the area is minimum—often where the flow reaches sonic conditions—and a diverging section where the flow continues to accelerate, typically to supersonic speeds. This configuration is fundamental in propulsion systems and high-speed wind tunnels.

Mass Flow Rate Calculation in Compressible Flow

Mass flow rate in compressible flow is calculated using the conservation of mass principle, typically expressed with the continuity equation. The mass flow rate depends on the local density, velocity, and cross-sectional area of the flow at a given point in the nozzle, and its calculation is essential for analyzing performance parameters such as thrust and efficiency in high-speed applications.

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