Gas conductance through tube at low pressure

[soothsayer]

I have a vacuum chamber of volume V connected to a vacuum pump via a tube of radius R and length L, and I want to design this "tube" such that R can be controlled in order to provide a constant rate of pressure change inside of the chamber of 2.5 mbar/sec, from atmosphere down to ~0.5mbar. Assume that pumping speed is infinite, so my flow rate is entirely limited by the conductance of my adjustable tube.

At the beginning of chamber pumpdown, I feel like I should be able to apply a combination of PV=nRT and Poiseuille's law to convert my 2.5mbar rate of pressure change into a desired flow rate, and then into an effective tube radius. When I do this, I get a result that feels pretty intuitive based on my experience. (Does this seem like a valid calculation to anyone else or have I already made an error here?)

But nearing the end of the pumpdown process, I suspect I can no longer apply Poiseuille's law in the same way since I no longer have laminar gas flow. When I try to apply the same equations the same way I get a resulting effective radius that is much smaller than would make sense. Is there a different equation I can use to calculate a required tube radius in this low pressure regime? Does it even make sense to be trying to calculate this when my desired change in pressure (2.5mbar/s) is larger than my chamber pressure (0.5mbar)? It doesn't feel like I can treat pressure over time as linear here.

Any advice for tackling this problem would be appreciated!

 

[jrmichler]

This sketch is my understanding of your problem:

The Poiseuille equation applies to the incompressible laminar flow of a Newtonian fluid. Air is Newtonian, the Reynolds number needs to calculated to find if the flow is laminar, and this flow is definitely not incompressible.

Your goal is to get a constant rate of pressure decrease in the vacuum chamber. That requires a constant mass flow rate out of the chamber. At the start, the pressure (and the density) in the chamber is high, and the pressure drop is high. Near the end, the pressure (and the density) in the chamber is low, and the pressure drop is low. Regardless of whether the flow in the tube is laminar or turbulent, the mass flow rate will be a function of chamber gas density and pressure drop in the tube. The mass flow rate will be high at the beginning, and decrease as the pressure in the vacuum chamber decreases.

A fixed restriction, whether a tube or an orifice, will not get you a constant rate of pressure decrease in the vacuum chamber. Unless somebody has a better idea, I think you will need some sort of control system. A pressure transmitter, ramp controller, and air flow control valve would do what you want.

 

[FluidDroog]

I have a vacuum chamber of volume V connected to a vacuum pump via a tube of radius R and length L, and I want to design this "tube" such that R can be controlled in order to provide a constant rate of pressure change inside of the chamber of 2.5 mbar/sec, from atmosphere down to ~0.5mbar. Assume that pumping speed is infinite, so my flow rate is entirely limited by the conductance of my adjustable tube.

At the beginning of chamber pumpdown, I feel like I should be able to apply a combination of PV=nRT and Poiseuille's law to convert my 2.5mbar rate of pressure change into a desired flow rate, and then into an effective tube radius. When I do this, I get a result that feels pretty intuitive based on my experience. (Does this seem like a valid calculation to anyone else or have I already made an error here?)

But nearing the end of the pumpdown process, I suspect I can no longer apply Poiseuille's law in the same way since I no longer have laminar gas flow. When I try to apply the same equations the same way I get a resulting effective radius that is much smaller than would make sense. Is there a different equation I can use to calculate a required tube radius in this low pressure regime? Does it even make sense to be trying to calculate this when my desired change in pressure (2.5mbar/s) is larger than my chamber pressure (0.5mbar)? It doesn't feel like I can treat pressure over time as linear here.

Any advice for tackling this problem would be appreciated!

Hi There!

This is a super fun problem. Are you still looking for help?

Check out the PDF by Genick Bar-Meir, "Fundamentals of Compressible Fluid Mechanics"

Let me know if you need help.

 

[DaveE]

I think you will need some sort of control system. A pressure transmitter, ramp controller, and air flow control valve would do what you want.

Yep. This!
You will absolutely have to have a some flow control "valve" that can be adjusted. You will also need a pressure sensor to measure what you ultimately want. The rest is just a (nonlinear) control system to make it work.