Pressure pipe head losses

The problem of sizing a single pipe between two tanks is hydraulically determined by applying the equation of motion:

Y=L·J+ΣPc

For a pipe between two tanks, the upstream head coincides with the free surface elevation in the starting tank, while for the downstream head, the corresponding one of the pipe section before the outlet is considered.

Where::

Y = difference between the total upstream and downstream heads of the pipe;
J = hydraulic gradient;
Q = flow rate;
D = pipe diameter;
ΣPc = sum of the localized head losses along the pipe (curves, elbows, gate valves…).

f the length L is on the order of a few thousand diameters, it is possible to neglect both the velocity heads and the localized head losses in the calculation of the total heads.

The most general expression linking the head loss J per unit length L of a pipe for an incompressible fluid in steady flow is that of Darcy-Weisbach:

having indicated with D the diameter of the pipe, v the mean flow velocity, g the acceleration of gravity, and λ a dimensionless resistance coefficient which is, in general, a function of the relative roughness of the pipe and the Reynolds number:

with ρ = fluid density and μ = dynamic fluid viscosity.

To calculate λ, the Colebrook-White equation is used:

This formula is usually solved through its representation in the Moody logarithmic diagram, where the equation is represented by a family of curves characterized by relative roughness ε / D = const.
The equation is solved by iterations.

In the case of steady turbulent flow, in addition to the Darcy-Weisbach equation, the Chezy formula is frequently used:

J=4·V²/(χ²·D)

χ = resistance coefficient

In design practice, resistance formulas derived from experiments conducted on different types of pipes are often used; generally expressed by monomial laws:

J=k·Qⁿ/D^m

As an example, the formula obtained by Contessini under conditions of purely turbulent flow, for new bituminized steel pipes, is given:

J=0.0012·Q²/D^5.26