How to calculate pressure drops in piping

In any high-pressure system, water does not reach the nozzle with the same energy it had when it left the pump: along the pipes, bends, fittings and valves, part of that energy is dissipated. These are the head losses, and understanding them is the first step to correctly sizing a pump and keeping the system efficient. In this article from the "Pump calculations" series we explain what head losses are, the difference between distributed and localised losses, and which formulas to use to calculate them (Darcy-Weisbach and Hazen-Williams). We close with a practical example applied to a Hawk NPM Series piston pump.

What head losses are

Head loss represents the energy loss of a fluid flowing inside a pipe, caused by the friction between the fluid and the duct walls and by the turbulence generated by every change in the flow path. This lost energy is conventionally expressed either as a pressure loss (in bar or Pascal) or as a head loss in metres of water column (metres, symbol ΔH), two directly related quantities.

In a high-pressure system, head losses occur both on the suction side (from the tank to the pump) and on the delivery side (from the pump to the point of use). They have different but equally important effects: on the suction side they reduce the pressure available at the pump inlet, while on the delivery side they increase the pressure the pump must generate to achieve the desired result at the nozzle. Neglecting them means risking an undersized, unstable system that is prone to premature wear.

Distributed head losses and localised head losses

To calculate head losses correctly, two types must be distinguished, and they add together.

Distributed head losses (also called continuous losses) are those that occur along the straight sections of the pipe, due to the continuous friction between the fluid and the pipe walls. They increase with the length of the section, with the fluid velocity and with the internal roughness of the pipe, while they decrease as the diameter increases. They are generally the dominant component in long lines.

Localised head losses (also called local or accidental losses) are those that occur at a specific point in the system where the flow is diverted, restricted or disturbed. They are typically generated in elements such as:

• bends and elbows, where the fluid changes direction;
• valves, filters and foot valves, which introduce a restriction;
• fittings, reducers and expansions of the cross-section, as well as the inlet and outlet of the pipes.

The total head loss of a system is the sum of the distributed losses of all the straight sections and the localised losses of all the accidental elements present along the path.

The formulas for calculating head losses

There are several methods for calculating head losses in pipes. The two most widely used in system engineering practice are the Darcy-Weisbach formula, of general validity, and the Hazen-Williams formula, specific to water.

The Darcy-Weisbach formula

The Darcy-Weisbach formula is the most rigorous reference for calculating distributed head losses and is valid for any fluid and any flow regime:

ΔH = f · (L / D) · (v² / 2g)

where ΔH is the distributed head loss (m), f is the friction factor (dimensionless), L is the length of the section (m), D is the internal diameter of the pipe (m), v is the mean fluid velocity (m/s) and g is the acceleration due to gravity (9.81 m/s²).

The friction factor f depends on the Reynolds number (Re = v · D / ν, where ν is the kinematic viscosity of the fluid) and on the relative roughness of the pipe. In turbulent flow, the normal condition in high-pressure systems, it is determined using the Colebrook formula or, as a first approximation for smooth pipes, using the Blasius relation (f ≈ 0.316 / Re0.25). For localised losses, the relation ΔH = ΣK · (v² / 2g) is used instead, where each accidental element has its own loss coefficient K (for example, about 0.5 for an inlet, about 0.9 for a 90° bend, 1.5 or more for a foot valve).

The Hazen-Williams formula for water

The Hazen-Williams formula is an empirical formula designed specifically for water at ambient temperature, widely used because it avoids the iterative calculation of the friction factor. In SI units it is expressed as follows:

hf = 10.67 · L · Q1.852 / (C1.852 · D4.87)

where hf is the head loss (m), L the length (m), Q the flow rate (m³/s), D the internal diameter (m) and C the Hazen-Williams coefficient, which depends on the pipe material (roughly around 140-150 for smooth plastic materials and 130 for new steel). The table below summarises the characteristics of the two methods and helps in choosing the most suitable one.

Why head losses matter in pump sizing

Calculating head losses is not an academic exercise: it has direct consequences on the performance and the lifespan of the system. The fluid velocity that appears in all the formulas depends on the pump flow rate and on the pipe diameter: for the same diameter, a higher flow rate increases the velocity and therefore the head losses, which grow with the square of the velocity.

On the delivery side, on the other hand, head losses add to the useful pressure required by the application: the pump must generate a higher pressure to compensate for them, and this translates into an increase in absorbed power. Underestimating head losses therefore leads to choosing an undersized motor or to failing to reach the expected pressure at the nozzle.

How to calculate head losses in a high-pressure system

To calculate the head losses in the pipes of a high-pressure system, three steps are followed. First, the fluid velocity is determined from the pump flow rate and the internal diameter of the pipe. Then the distributed losses are calculated separately, using the Darcy-Weisbach formula (or Hazen-Williams if the fluid is water), together with the localised losses, obtained by summing the K coefficients of all the accidental elements.