Helium expands in a nozzle from 1 MPa, 500 K, and negligible velocity to 0.1 MPa. Calculate the

step 1 Hi everyone, critical area is defined as mass flow rate upon inlet pressure root of RT01 upon K

Helium expands in a nozzle from 1 MPa, 500 K, and...

Helium expands in a nozzle from $1 \mathrm{MPa}, 500 \mathrm{~K}$, and negligible velocity to $0.1 \mathrm{MPa}$. Calculate the throat and exit areas for a mass flow rate of $0.25 \mathrm{~kg} / \mathrm{s}$, assuming the nozzle is isentropic. Why must this nozzle be convergingdiverging?

Key Concepts

Area-Mass Flow Rate Relationships

The area-mass flow rate relationships in compressible flows are determined by combining the continuity equation with the energy and momentum equations under the isentropic assumption. These relationships allow one to compute the required throat and exit areas for a prescribed mass flow rate by relating the local Mach number, density, and velocity to the cross-sectional area. This analysis is vital for designing nozzles with specified operating conditions.

Converging-Diverging Nozzles (de Laval Nozzle)

A converging-diverging nozzle, often called a de Laval nozzle, is required when the flow must accelerate from subsonic speeds to supersonic speeds. The convergent section accelerates the flow to Mach 1 at the throat (choked condition), and the divergent section further accelerates the flow into the supersonic regime. This design is critical for applications where a significant pressure drop is involved and a supersonic exit velocity is desired, which is why the nozzle in the question must be converging-diverging.

Isentropic Flow

Isentropic flow refers to a thermodynamic process that is both adiabatic (no heat transfer) and reversible, meaning that the entropy remains constant throughout the process. This assumption is useful in analyzing nozzle flows as it simplifies the calculations by allowing the use of well-defined relationships between pressure, temperature, density, and velocity, which are critical in determining the flow properties during expansion.

Choked Flow and Sonic Conditions

Choked flow occurs when the velocity at the narrowest part of the nozzle (the throat) reaches the speed of sound, representing the maximum possible mass flow rate under given upstream conditions. At choked conditions, any further decrease in downstream pressure does not increase the mass flow rate because the flow rate is limited by the sonic condition at the throat. This concept is essential for determining the throat area in the nozzle design.

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