At turbulent flow you choose a hydraulic diameter that leads to equal average wall shear stress and pressure drop in the completely filled round tube with dH and the noncircular cross section. The equilibrium of forces for the round tube and the noncircular cross section describe the following equations:
The same average speed leads to the same pressure drop in both systems. The ratio of average wall shear stress to average speed is approximately equal in both systems, since the velocity profile is rather flat in turbulent flow.
The volume flow has to be converted. With the same average speed we derive:
For laminar flow good approximation equations exist for numerous noncircular cross sections that describe the volume flow as a function of pressure drop. [Berker 1963] lists closed form solutions for the speed profile of several noncirular cross sections.
For a rectangular channel with BxH (B>H) we can write:
: volume flow η: dynamic viscosity L: channel length
For B>>H this equation converges towards the solution for two parallel plates, also called Couette Flow (see appendix for more details):
The hydraulic diameter is derived as follows: