Fluid Mechanics Lesson 15B: Compressible Flow and Choking in Converging Ducts

Summary

This lesson explores compressible flow in converging ducts, focusing on choking and critical conditions. It explains how fluid velocity, Mach number, pressure, and temperature change in subsonic and supersonic flow scenarios. The concept of choking is introduced, where the Mach number reaches 1, making the flow independent of further decreases in back pressure. The video walks through an example problem, calculating the Mach number, exit temperature, and mass flow rate, showing how these parameters are affected by back pressure. The lesson provides valuable insights into the behavior of compressible flow and its practical applications.

Takeaways

• 😀 Compressible flow in converging ducts involves a pressurized tank, where stagnation conditions and exit parameters like Mach number and pressure are important.
• 😀 Subsonic flow in a converging duct sees the Mach number increase as speed rises, while pressure and temperature decrease.
• 😀 Once the Mach number exceeds 1 (supersonic), the flow behavior reverses, with pressure and temperature rising, making supersonic flow unsustainable in a converging duct.
• 😀 The flow in a converging duct remains subsonic and approaches Mach 1 at the exit plane but does not exceed it, unless certain conditions are met.
• 😀 The flow is considered 'choked' when the Mach number reaches 1 at the exit plane, and further decreases in back pressure (PB) have no effect on the flow.
• 😀 If the back pressure PB is greater than the critical pressure P*, the flow is subsonic; if PB equals P*, the flow becomes sonic (Mach 1), and if PB is less than P*, the flow is choked.
• 😀 Choked flow occurs when PB is lower than P*, where the Mach number at the exit plane is 1, and the pressure at the exit is P* regardless of further changes in PB.
• 😀 Critical or sonic conditions occur when PB equals P*, and the flow reaches a maximum speed (Mach 1), beyond which it cannot go faster even if PB decreases.
• 😀 The flow in the duct remains unaffected by downstream pressure once it becomes choked, due to the sonic barrier at the exit.
• 😀 The mass flow rate through a converging duct can be calculated using a general equation that takes into account parameters like pressure, area, Mach number, and temperature, with specific adjustments for choked flow.

Q & A

What happens to the Mach number in a converging duct for subsonic flow?

For subsonic flow in a converging duct, the Mach number increases as the flow accelerates. The velocity increases while pressure and temperature decrease.

Why can't the flow become supersonic in a converging duct?

The flow cannot become supersonic in a converging duct because once the Mach number exceeds 1, the flow characteristics would contradict each other. The velocity and Mach number must remain subsonic until the exit, where the flow reaches Mach number 1.

What is meant by 'choked flow'?

Choked flow refers to the condition where the Mach number at the exit plane of the duct reaches 1. Once the flow is choked, any further decrease in back pressure has no effect on the flow.

How do back pressure conditions affect the flow in a converging duct?

If the back pressure (PB) is greater than the critical pressure (P*), the flow is subsonic at the exit. If PB equals P*, the flow reaches sonic conditions (Mach number = 1). If PB is less than P*, the flow becomes choked, and Mach number remains 1 at the exit.

What happens to the temperature of the flow as it accelerates in a converging duct?

As the flow accelerates through a converging duct, the temperature decreases. For example, in the given problem, the temperature at the exit (Te) is significantly lower than the stagnation temperature (Tkn).

What is the significance of the Mach number being equal to 1 at the exit plane?

When the Mach number reaches 1 at the exit plane, it signifies that the flow is at critical or sonic conditions. At this point, the flow is choked, meaning that any further decrease in back pressure does not affect the flow.

How is the Mach number at the exit calculated for subsonic flow?

The Mach number at the exit for subsonic flow can be calculated using stagnation equations, considering pressure ratios and temperature ratios between stagnation and exit conditions.

What is the formula for calculating the mass flow rate in a converging duct?

The mass flow rate (Ṁ) through a converging duct is calculated using the equation: Ṁ = ρ * v * A, where ρ is the density, v is the velocity, and A is the cross-sectional area. For choked flow, the equation simplifies, incorporating critical conditions.

What happens when the back pressure is reduced below the critical pressure (P*)?

When the back pressure drops below the critical pressure (P*), the flow becomes choked. At this point, the Mach number remains 1, and any further reduction in the back pressure does not affect the flow conditions.

In the example problem, what happens to the mass flow rate when the flow is choked?

When the flow is choked, the mass flow rate reaches its maximum value, and the flow rate is limited by the critical conditions. In the example problem, the mass flow rate was calculated as 3.54 kg/s, which is the actual flow rate, while the maximum flow rate (Ṁ_max) is 3.64 kg/s under choked conditions.