# The Theory, and some Maths (Part 1)

A project log for Everyman's turbomolecular pump

(Maybe) access to the high vacuum environment for the rest of us

Keegan Reilly 10/24/2015 at 14:120 Comments

I don't actually know much about fluid dynamics, and could not hope to do an accurate CFD simulation of this myself. Luckily, in 2008 [Peter Harwood] wrote a very nice paper accomplishing this for bladeless turbines operating as air compressors or blowers, however the applicability of his equations to the molecular flow regime seems to be completely unexplored in the literature. The main equation I got from him is the math governing Pmax, or maximum back pressure. This is in theory the maximum pressure differential that the pump can sustain before flow starts reversing.

With rho = Density of air (1.225 kg/m^3 at atmospheric pressure), omega = angular velocity (in radians/second), and r_o and r_i are the outer and inner radii of the discs respectively.

While this was a good start for me, there seem to be some conceptual issues with applying this equation to vacuum pumps. The first problem is that this yields only a pressure differential, without taking into consideration the intake pressure. Mathematically, we could surmise that:

This is obvious, and was experimentally verified by [Harwood] for his application, with 1x10^5 Pa (Pmax) = 2x10^5 Pa (Exhaust) - 1x10^5 Pa (Intake, atmospheric pressure). For vacuum however, we could arbitrarily set:

Even more dubious, the resultant compression ratio looks like this:

With some numbers plugged in, just to get an idea of realism, this implies that a 22cm diameter disc at 35000 RPM could reach an arbitrarily low vacuum (10^-a billion Torr?) in a single stage. The reason this sounds unrealistic is that the math is wrong. We have not accounted for the variable density of air from intake to exhaust. The previous equation assumed incompressible flow, and for this application we are very much interested in compressible flow, for obvious reasons. More on that in Part 2.