Estimating Darcy Friction Factors for Critical Flow

Question

The Darcy-Weisbach equation is used to calculate the frictional pressure losses in pipes transporting incompressible fluids. This equation uses a dimensionless Darcy friction factor, also known as the Moody factor, to account for the relative roughness of the pipe surface.

This empirical factor was experimentally determined by Moody and is normally read off of the Moody Chart. However I am implementing the pressure drop calculation in software, so I need a non-graphical way to find the Darcy friction factor.

Equations for calculating the Darcy friction factor under laminar (Re < 2320) and turbulent (Re > 4000) flow are readily available. But I haven't been able to find one that is valid for the transitional region which exists between laminar and turbulent flow (2320 < Re < 4000), also known as the 'critical zone'.

I understand that fluid flow is complex and difficult to predict in this region. However, is there a commonly used method that provides reasonable estimates for the friction factor in this critical zone?

I have found a method described in a student paper, but it hasn't been peer reviewed and is limited to smooth pipes only. I'm looking for something more tried and tested.

If no formula is available, what approach do other engineers usually take to mitigate or solve this problem?

Answer 1

As an engineer, sometimes you need an answer, and it might not be the best answer, but you've got to get one.

In this case, there are two ways to handle this. The first would be to extrapolate the turbulent flow backwards to the end of the laminar flow. Since the turbulent flow is always higher than the laminar flow, you'll get a higher than real friction flow in your piping, but since you're mainly using this to size your piping, pumps, etc, in the end you'll just wind up overengineering the pumps, pipes, etc. This isn't a bad thing, but it isn't desired. However, this way you have an answer. If your program states a warning when this happens

"This is in the turbulent zone, so an accurate estimate is difficult. Instead a conservative amount is given. A better design would be in either the laminar or turbulent zones"

And most engineers would be satisfied.

The second way would be to require the user to be in either the laminar region or the turbulent region. This is how most engineers handle this situation - they avoid the region altogether. This way, the system can be accurately and efficiently designed, to avoid overengineering.