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?