Sterling Vortex Pressure-Gain Tube

I remember early on in my time here, I stumbled upon an idea that just so happened to get me into mechanical engineering as a hobby. It called for the use of a Ranque-Hilsch Vortex tube- A neat little geometric marvel that somewhat embodies the concept of "Maxwell's Demon". Compressed air enters in through tangential port(s) within a section of tubing. Most of the flow will travel up the tube and out the "hot" end; the enlarged exhaust outlet with a conic section blocking the very center and allowing only the outer wall of flow out. Conversely, there is a small diameter hole on the other end of the tube, closest to where the tangential linlet ports are located. This very small hole in the center is called the "cold" end.

Whether through acoustic streaming or otherwise (the mechanism is still today not fully understood), somehow this system splits a compressed gas or gasses into very hot and very cold; from an inlet of, say, 70 degree F air at 80psi, one might obtain 200 degree F air out the hot end and -10F out the cold; the geometry effectively separates out the faster moving molecules while transferring maximal amount of energy to them out of the system; leaving a low pressure core of extremely dense, very low energy gas.

That's all well and good, but it does require a source of compressed air; in essence it trades pressure for temperature separation as both hot and cold outlets are only around @ ~9psi with an input of 80psi.

If one were to theorize that whatever means of temperature separation could be utilized in reverse, my idea is for two low pressure turbochargers, one intercooled one not, forcing their air in through the respective cold and hot side while imparting solid body rotation, and yielding very high pressure air out of the tangential ports at an average of the two temps; essentially using the temperature difference through working mechanism of vortices to amplify the pressure.