# Losses in Pipes

## This page provides a quick review of piping losses, starting with Bernoulli's Equation

The basic approach to all piping systems is to write the Bernoulli equation between two points, connected by a streamline, where the conditions are known. For example, between the surface of a reservoir and a pipe outlet. The total head at point 0 must match with the total head at point 1, adjusted for any increase in head due to pumps, losses due to pipe friction and so-called "minor losses" due to entries, exits, fittings, etc. Pump head developed is generally a function of the flow through the system, with head rise decreasing with increasing flow through the pump.

## Friction Losses in Pipes

Friction losses are a complex function of the system geometry, the fluid properties and the flow rate in the system. By observation, the head loss is roughly proportional to the square of the flow rate in most engineering flows (fully developed, turbulent pipe flow). This observation leads to the Darcy-Weisbach equation for head loss due to friction: which defines the friction factor, f. f is insensitive to moderate changes in the flow and is constant for fully turbulent flow. Thus, it is often useful to estimate the relationship as the head being directly proportional to the square of the flow rate to simplify calculations.  Reynolds Number is the fundamental dimensionless group in viscous flow. Velocity times Length Scale divided by Kinematic Viscosity.

Relative Roughness relates the height of a typical roughness element to the scale of the flow, represented by the pipe diameter, D.

Pipe Cross-section is important, as deviations from circular cross-section will cause secondary flows that increase the pressure drop. Non-circular pipes and ducts are generally treated by using the hydraulic diameter, in place of the diameter and treating the pipe as if it were round.

For laminar flow, the head loss is proportional to velocity rather than velocity squared, thus the friction factor is inversely proportional to velocity.  Geometry Factor k Square 56.91 2:1 Rectangle 62.19 5:1 Rectangle 76.28 Parallel Plates 96.00

The Reynolds number must be based on the hydraulic diameter. Blevins (Applied Fluid Dynamics Handbook, table 6-2, pp. 43-48) gives values of k for various shapes. For turbulent flow, Colebrook (1939) found an implicit correlation for the friction factor in round pipes. This correlation converges well in few iterations. Convergence can be optimized by slight under-relaxation. The familiar Moody Diagram is a log-log plot of the Colebrook correlation on axes of friction factor and Reynolds number, combined with the f=64/Re result from laminar flow. An explicit approximation  provides values within one percent of Colebrook over most of the useful range.

#### Calculating Head Loss for a Known Flow

From Q and piping determine Reynolds Number, relative roughness and thus the friction factor. Substitute into the Darcy-Weisbach equation to obtain head loss for the given flow. Substitute into the Bernoulli equation to find the necessary elevation or pump head.

#### Calculating Flow for a Known Head

Obtain the allowable head loss from the Bernoulli equation, then start by guessing a friction factor. (0.02 is a good guess if you have nothing better.) Calculate the velocity from the Darcy-Weisbach equation. From this velocity and the piping characteristics, calculate Reynolds Number, relative roughness and thus friction factor.

Repeat the calculation with the new friction factor until sufficient convergence is obtained. Q = VA.

Here's a video discussing the three types of piping problems:

### "Minor Losses"

Although they often account for a major portion of the head loss, especially in process piping, the additional losses due to entries and exits, fittings and valves are traditionally referred to as minor losses. These losses represent additional energy dissipation in the flow, usually caused by secondary flows induced by curvature or recirculation. The minor losses are any head loss present in addition to the head loss for the same length of straight pipe.

Like pipe friction, these losses are roughly proportional to the square of the flow rate. Defining K, the loss coefficient, by allows for easy integration of minor losses into the Darcy-Weisbach equation. K is the sum of all of the loss coefficients in the length of pipe, each contributing to the overall head loss.

Although K appears to be a constant coefficient, it varies with different flow conditions. Factors affecting the value of K include:

• the exact geometry of the component in question
• the flow Reynolds Number
• proximity to other fittings, etc. (Tabulated values of K are for components in isolation - with long straight runs of pipe upstream and downstream.)

Some very basic information on K values for different fittings is included with these notes and in most introductory fluid mechanics texts. For more detail see e.g. Blevins, pp. 55-88.

To calculate losses in piping systems with both pipe friction and minor losses use in place of the Darcy-Weisbach equation. The procedures are the same except that the K values may also change as iteration progresses.