Upstream of the throat of an isentropic convergingdiverging...

Upstream of the throat of an isentropic convergingdiverging nozzle at section $(1), V_{1}=150 \mathrm{m} / \mathrm{s}, p_{1}=100 \mathrm{kPa}(\mathrm{abs})$ and $T_{1}=20^{\circ} \mathrm{C} .$ If the discharge flow is supersonic and the throat area is $0.1 \mathrm{m}^{2},$ determine the mass flowrate in $\mathrm{kg} / \mathrm{s}$ for the flow of air.

Upstream of the throat of an isentropic convergingdiverging nozzle at section $(1), V_{1}=150 \mathrm{m} / \mathrm{s}, p_{1}=100 \mathrm{kPa}(\mathrm{abs})$
and $T_{1}=20^{\circ} \mathrm{C} .$ If the discharge flow is supersonic and the throat area is $0.1 \mathrm{m}^{2},$ determine the mass flowrate in $\mathrm{kg} / \mathrm{s}$ for the flow of air.

Key Concepts

Isentropic Flow

An isentropic process is one in which the fluid undergoes changes with no heat transfer and without irreversible losses, implying that entropy remains constant throughout. This assumption simplifies the analysis of compressible flows by providing fixed relationships between pressure, temperature, and density, which are essential when applying conservation laws in nozzle problems.

Converging-Diverging Nozzle

A converging-diverging nozzle is designed to accelerate a fluid from subsonic speeds to supersonic speeds. The converging section accelerates the flow until it reaches sonic speed at the throat, while the diverging section allows the supersonic expansion of the flow. This configuration is critical in applications where high-speed exhaust streams are needed, such as in rocket engines.

Choked Flow

Choked flow occurs when the flow velocity at the narrowest point (the throat) of a nozzle reaches the speed of sound, which limits the mass flow rate. In such a condition, further decreases in downstream pressure do not lead to an increase in the mass flow rate, because the rate is controlled by the sonic conditions at the throat.

Mass Flow Rate Calculation in Compressible Flow

The mass flow rate in compressible flows is determined by multiplying the local density, velocity, and cross-sectional area. Under isentropic conditions and at the throat where choked flow occurs, these values are interconnected through the gas dynamic relations. This calculation typically involves using the speed of sound and the thermodynamic properties of the gas to express density and velocity in terms of measurable parameters.

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