The following notes should enable a mechanical engineer to establish basic flow conditions and head losses along pipe routes in which fluids are flowing.   The equations are most relevant to liquids although approximate sizing for gases can be carried out if appropriate correction factors are used,where necessary, and low gas velocities are considered.

Fluid flowing in pipes has two primary flow patterns.   It can be either laminar when all of the fluid particles flow in parallel lines at even velocities and it can be turbulent when the fluid particles have a random motion interposed on an average flow in the general direction of flow.   There is also a critical zone when the flow can be either laminar or turbulent or a mixture.   It has been proved experimentally by Osborne Reynolds that the nature of flow depends on the mean flow velocity (v), the pipe diameter (D), the density (ρ) and the fluid viscosity Fluid Viscosity( μ). A dimensionless variable for the called the Reynolds number which is simply a ratio of the fluid dynamic forces and the fluid viscous forces , is used to determine what flow pattern will occur. The equation for the Reynold Number is

For normal engineering calculations , the flow in pipes is considered laminar if the relevant Reynolds number is less than 2000, and it is turbulent if the Reynolds number is greater than 4000.  Between these two values there is the critical zone in which the flow can be either laminar or turbulent or the flow can change between the patterns...

It is important to know the type of flow in the pipe when assessing friction losses when determining the relevant friction factors

Reference :

The steady flow equation steady flow equation (energy per unit mass )  for a system is identified below...
Reference... Steady Flow

If q = w = 0 and the fluid is incompressible and frictionless and if the variables are converted to measured heads of the fluid , that is the units are per unit weight (ρg) - then the Bernoulli's equation results ..
Reference .. Bernoulli's Equation ideal fluids
..

In real flow systems there are losses due to internal and wall friction which result in increase in the internal energy of the fluid. (q > 0).   Reference Bernoulli's Equation real Fluids .    The bernoulli equation is modified to reflect these losses by adding a term h _f = Head loss due to friction.= (u₂ -u₁ - q)   The modified bernoullis equation is therefore ..

The object of most pipe flow head loss calculations is to determine the friction head loss and allow estimation of the pump /compressor power required to pump the fluid along the piping.  In most fluid transfer cases the fluid is a incompressible (a liquid) and the flow rate (Q) is constant along the pipe run and therefore the velocity at any point can easily be calculated.  The head (z) can also be easily obtained from the pipeline geometry.  The system pressure and head loss are therefore the variables generally subject to the detailed pipeline calculations....

Note:
In the moody chart above (ε /D ) is identified with both numerator and denominator in metres (for consistency with all other equations on this page.   It is probably more convenient to express both in (mm).i.e a 50mm cast iron pipe (ε _mm; /D_mm ) would simply be (0,203 /50 ).

The losses through fittings are generally evaluated by obtaining K = f_T(L/D)

Table of pipe friction values for clean pipe in region of complete turbulence

The K₁₈₀ value for a 180^o bend may be derived from the equivalent K₉₀ which is calculated from the above tables using the equation

The K₁₈₀ = 0,25.π.f_T . r/D + 1,5.K₉₀

For laminar fluids with low Re numbers ( "<" 500) the K values obtained using the above are probably very innaccurate. The table below illustrates how this affects the K values

The losses through these fitting are generally evaluated by first obtaining β = d₂ / d₁

Important Note: the resulting K values as tabled below are based on the flow velocity in the larger pipe if the flow velocity in the small pipe is used to evaluate the head loss then the K values tabled below should be multiplied by ( β ) ⁴ = (d₂ / d₁) ⁴