Question

Consider subsonic Rayleigh flow that is accelerated to sonic velocity \((\mathrm{Ma}=1)\) at the duct exit by heating. If the fluid continues to be heated, will the flow at duct exit be supersonic, subsonic, or remain sonic?

Short answer

Answer: If the fluid continues to be heated in a subsonic Rayleigh flow accelerated to a Mach number of 1 at the duct exit, the flow at the duct exit will remain sonic (Mach number = 1) and not become supersonic or subsonic.

Step-by-step solution

Define Rayleigh flow and Mach number

Rayleigh flow is a type of compressible flow in which heat is added or removed from the fluid as it moves through a constant area duct. The Mach number (Ma) is a dimensionless quantity that represents the ratio of an object's speed to the speed of sound in the fluid.

Relate Rayleigh flow with the Mach number

In a Rayleigh flow, the Mach number has an important relationship with the duct length and temperature. If the duct length decreases, the Mach number increases, and the flow accelerates. In contrast, if the duct length increases, the Mach number decreases, and the flow decelerates. For subsonic flow (\({Ma}<1\)), additional heating accelerates the flow, increasing the Mach number.

Assess the effect of heating

When the Mach number reaches 1, the flow becomes sonic, and the fluid's velocity is equal to the speed of sound in the medium. We are given that the fluid has been heated along the Rayleigh flow so that the Mach number reaches 1 at the duct exit. Now, let's consider the case where heating continues.

Determine the behavior of flow

As we continue to heat the fluid in the Rayleigh flow, we need to analyze its effect on the Mach number. For a subsonic Rayleigh flow, additional heating causes the flow to accelerate and the Mach number to increase until it reaches 1. However, in a Rayleigh flow situation, the flow will not exceed the Mach number of 1 even if the heating continues. This is because the addition of more heat to the system causes the pressure to decrease, which in turn reduces the speed of sound in the medium. The flow will eventually reach an equilibrium with the decreased speed of sound, maintaining a Mach number of 1.

Conclusion

If the fluid continues to be heated in a subsonic Rayleigh flow accelerated to a Mach number of 1 at the duct exit, the flow at the duct exit will remain sonic (Mach number = 1) and not become supersonic or subsonic.