Question

An ideal gas flows through a passage that first converges and then diverges during an adiabatic, reversible, steady-flow process. For subsonic flow at the inlet, sketch the variation of pressure, velocity, and Mach number along the length of the nozzle when the Mach number at the minimum flow area is equal to unity.

Short answer

Question: Sketch the variation of pressure, velocity, and Mach number along the length of a converging-diverging nozzle when the Mach number at the throat is 1 during an adiabatic, reversible, steady-flow process. Answer: In a converging-diverging nozzle with a subsonic inlet and Mach number equal to 1 at the throat, the pressure will continuously decrease along the nozzle's length, the velocity will increase from the inlet to the exit, and the Mach number will increase from subsonic values at the inlet to supersonic values in the diverging section.

Step-by-step solution

Understand the given information

We are given that an ideal gas flows through a passage that first converges and then diverges during an adiabatic, reversible, steady-flow process. The Mach number at the minimum flow area (throat) is 1, and subsonic flow enters the nozzle inlet. We need to sketch the variation of pressure, velocity, and Mach number along the nozzle.

Understand what happens in a converging-diverging nozzle

In a converging-diverging nozzle, the flow accelerates in the converging section and reaches the speed of sound at the throat when the Mach number is 1. In the diverging section of the nozzle, the flow further accelerates and becomes supersonic if the process is isentropic (i.e., adiabatic and reversible). The pressure and temperature decrease as the flow accelerates, while the Mach number increases.

Sketch the pressure variation along the nozzle

As the flow accelerates in the converging section from the inlet, the pressure decreases continuously until it reaches a minimum at the throat. In the diverging section, the pressure continues to decrease as the flow becomes supersonic. Therefore, sketch a curve that represents the pressure decreasing continuously from the inlet to the throat, and a further decrease in the diverging section.

Sketch the velocity variation along the nozzle

The velocity of the flow increases along the convergent section as the flow accelerates, and it reaches the maximum at the throat where the Mach number is 1 (speed of sound). In the diverging section, the velocity continues to increase as the flow becomes supersonic. Sketch a curve that shows the velocity increasing continuously from the inlet to the throat, and further increase in the diverging section.

Sketch the Mach number variation along the nozzle

The Mach number increases as the flow accelerates. In the converging section, the Mach number increases from subsonic (less than 1) to 1 at the throat. In the diverging section, the Mach number continues to increase as the flow becomes supersonic (greater than 1). Sketch a curve that shows the Mach number increasing from subsonic at the inlet to 1 at the throat, and then continues to increase beyond 1 in the diverging section. In summary, the variation of pressure, velocity, and Mach number along the length of a converging-diverging nozzle with a subsonic inlet and Mach number equal to 1 at the throat can be sketched by showing: (1) decreasing pressure throughout the nozzle, (2) increasing velocity from inlet to exit, and (3) increasing Mach number from subsonic values at the inlet to supersonic values in the diverging section.

Key concepts

Adiabatic Process

An adiabatic process is one in which no heat is transferred into or out of the system. For gases flowing through nozzles, this implies that all the work done by or on the gas appears as a change in the gas's internal energy, which affects its temperature and pressure. In the context of a converging-diverging nozzle, the adiabatic process means that as the gas flows and accelerates, its temperature decreases if the flow is expanding and increases if the flow is compressing, without any external heat exchange. This is crucial for understanding how the properties of the gas such as pressure, temperature, and velocity change along the passage of the nozzle.

When the gas is expanding, like in the diverging portion of the nozzle, its pressure and temperature decrease because it's doing work on the surroundings. Conversely, when the gas is compressing in the converging part, its pressure and temperature increase. This behavior is key in determining the quantity and speed of gas that can pass through the nozzle.

Mach Number

The Mach number is a dimensionless quantity representing the ratio of the speed of an object moving through a fluid and the local speed of sound within that fluid. For gases flowing through a nozzle, the Mach number is a critical parameter because it determines whether the flow is subsonic (Mach number < 1), sonic (Mach number = 1), or supersonic (Mach number > 1).

In a converging-diverging nozzle scenario, when the Mach number at the throat of the nozzle (the narrowest part) equals 1, the flow speed there is equal to the speed of sound — a condition known as 'choked flow.' This is a critical point because it signifies the transition between subsonic and supersonic flow regimes. At the nozzle's entrance (inlet), if the flow is subsonic, the Mach number will increase towards unity as it reaches the throat, and should the conditions allow, continue to increase to supersonic speeds in the diverging section.

Steady-Flow Process

A steady-flow process is one in which the fluid properties at any given point do not change over time. This implies that variables such as pressure, temperature, velocity, and density at a fixed point in the system remain constant, even though they may vary from point to point within the flow field. This concept is essential for analyzing many engineering systems like nozzles, turbines, and compressors.

In the context of a converging-diverging nozzle, the assumption of steady-flow allows us to focus on how these properties change spatially, not temporally, as the gas moves from the inlet to the outlet of the nozzle. Steady-flow is a simplifying assumption that makes it feasible to predict the behavior of the flowing gas without accounting for the complexities of time-dependent changes.

Isentropic Flow

Isentropic flow refers to a flow in which entropy remains constant; this indicates a reversible and adiabatic process. In real-world applications, a perfectly isentropic flow is idealized because some level of friction and heat transfer typically exists, causing entropy to increase. For the flow in a converging-diverging nozzle, an isentropic process means that no shock waves are present within the flow, which would otherwise disrupt the orderly transition from subsonic to supersonic speeds.

In an isentropic flow through a converging-diverging nozzle, as the flow accelerates and the Mach number increases towards and beyond 1, the pressure and temperature decrease while the velocity increases. The assumption of isentropic flow allows for the creation of efficiency benchmarks and the establishment of relationships between the flow properties, which are used to design optimal nozzles and assess their performance.