Turbulent Flow
The hydraulic diamater d_{H} is calculated
by the assumption that pressure drop and average speed are equal in the completely filled tube
with d_{H} and the noncircular cross section. The volume flow has to
be converted:

 
A : noncircular cross section
U : noncircular circumference 

rectangular channel:  
 
: volume flow in tube with d_{H}
: volume flow in rectangular channel
H : channel height B : channel width 

H / B 
0,2  0,4  0,6 
0,8  1 
/

0,44  0,64  0,74  0,78  0,79 

Laminar Flow

The hydraulic diameter
d_{H} can be determined for several cross sections such that
pressure drop and volume flow are equal in the completely filled round tube with
d_{H} and the noncircular cross section. 

rectangular channel:  
d_{H} = H · f (H/B)  
H: channel height B: channel width, B > H f(H/B): derivation see below 

H/B 
0,2  0,4  0,6 
0,8  1 
f (H/B)  1,96  1,59  1,37  1,22  1,09 

Theory
Turbulent Flow
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 d_{H} 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:
rectangular channel:
Laminar Flow
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:

 laminar pipe flow (Hagen Poiseuille Flow)
(see appendix for more details) 

 laminar flow in rectangular channel 

Hagen Poiseuille and Couette Flow 
Literature

[Berker 1963] Tapis Berker, Encyclopedia Of Physics / Handbuch der Physik,
Edited by S. Flügge, SpringerVerlag 1963, page 70 
