# Fanno Flow and Mach 1

I'm currently taking a course on compressible flow, and although I have a pretty firm grasp of the mathematics, the way that flows behave still seems confusing to me. When considering Fanno flow, I don't understand why subsonic flows are accelerated to Mach 1. I have sufficient background in biology and chemistry to be familiar with the natural tendency of systems to maximize entropy and minimize energy, but I'm failing to see how Mach 1 is the maximum entropy state. What about Mach 1 makes it have the maximum number of microstates for a flow?

The first law and the second law are listed below. $$dU=\delta Q -dW$$ $$dS =\frac{\delta Q}T$$
Conceptually, when the flow is subsonic, $dW$ is greater than $\delta Q$ because the work does not only include doing frictional work but also include accelerating/expanding the system. Therefore, $dU$ decreases and the temperature decreases. However, this doesn't prevent $dS$ from increasing due to increasing in volume (more microstate in space though less micro velocity (i.e. low temperature)). This will last until it can no longer accelerate/expand the system, which is Mach 1.