Supersonic airflow enters an adiabatic, constant area...

Supersonic airflow enters an adiabatic, constant area (inside diameter $=1 \mathrm{ft}$ ) 30 -ft-long pipe with $\mathrm{Ma}_{1}=3.0 .$ The pipe friction factor is estimated to be $0.02 .$ What ratio of pipe exit pressure to pipe inlet stagnation pressure would result in a normal shock wave standing at (a) $x=5$ ft, or $(\mathbf{b}) x=10 \mathrm{ft},$ where $x$ is the distance downstream from the pipe entrance? Determine also the duct exit Mach number and sketch the temperature-entropy diagram for each situation.

Supersonic airflow enters an adiabatic, constant area (inside diameter $=1 \mathrm{ft}$ ) 30 -ft-long pipe with $\mathrm{Ma}_{1}=3.0 .$ The pipe friction factor is estimated to be $0.02 .$ What ratio of pipe exit pressure to pipe inlet stagnation pressure would result in a normal shock wave standing at
(a) $x=5$ ft, or $(\mathbf{b}) x=10 \mathrm{ft},$ where $x$ is the distance downstream from the pipe entrance? Determine also the duct exit Mach number and sketch the temperature-entropy diagram for each situation.

Key Concepts

Normal Shock Waves

Normal shock waves are sharp discontinuities that occur in compressible flows when the flow transitions from supersonic to subsonic speeds. Across a normal shock, there are abrupt changes in flow properties including pressure, temperature, density, and Mach number. Understanding the relationships across a shock wave is fundamental for calculating how the shock position affects the flow characteristics in ducts and for determining pressure ratios, as well as other vital parameters for design and analysis in high-speed aerodynamics.

Fanno Flow

Fanno flow refers to the analysis of compressible flow in a constant-area duct where friction is present but heat transfer is absent (adiabatic conditions). This model describes how properties such as Mach number, pressure, and temperature change along the duct due to friction. It is used to determine the conditions for phenomena such as shock waves within a duct and to assess how the friction influences the thermodynamic and fluid dynamic parameters along the length of the duct.

Stagnation Pressure

Stagnation pressure is the pressure a fluid attains when it is isentropically brought to rest from its moving state. It is a critical parameter in compressible flow analysis, as it acts as a reference state for comparing pressure losses or gains throughout the flow. In problems with shocks and friction, such as Fanno flow, indicating how much the stagnation pressure decreases downstream is essential for the complete characterization of the flow.

Adiabatic Duct Flow

Adiabatic duct flow involves the transport of fluid through a duct in which no heat is exchanged with the surroundings. In these flows, although the energy content of the fluid is only altered by work interactions (such as friction), the absence of heat transfer simplifies the energy equation. This concept is key to both Fanno flow analysis and the evaluation of how shocks affect the thermodynamic state along a duct.

Temperature-Entropy (T-s) Diagram

The temperature-entropy diagram is a graphical representation used to depict thermodynamic processes. In the context of compressible flows with shocks and friction, it allows for the visualization of the irreversible processes involved, such as the increase in entropy across a normal shock. By mapping how temperature and entropy change, engineers can better understand the system’s performance, diagnose losses, and conceptualize the behavior of the flow through the duct.

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