# Summary

To determine the pressure loss or flow rate through pipe knowledge of the friction between the fluid and the pipe is required. This article describes how to incorporate friction into pressure loss or fluid flow calculations. It also outlines several methods for determining the Darcy friction factor for rough and smooth pipes in both the turbulent and laminar flow regime. Finally this article discusses which correlation for pressure loss in pipe is the most appropriate.

# Definitions

 : Absolute Roughness : Hydraulic Diameter : Hydraulic Radius : Darcy Friction Factor : Fanning Friction Factor : Pipe Diameter : Relative Roughness : Reynolds Number : Length of Pipe : Head loss due to friction : Average Velocity : Gravity : Pressure : Density

# Introduction

Pressure loss in piping without any size changes or fittings occurs due to friction between the fluid and the pipe walls. There have been a number of methods developed to describe this relationship; generally a friction factor is used to determine the pressure loss. The most important methods of determining this friction factor are described in this article.

The key influences on the pressure drop as a fluid moves through a pipe are Reynolds Number of the fluid and the roughness of the pipe.

## Friction Factors: Fanning and Darcy

There are two common friction factors in use, the Darcy and Fanning friction factors. The Darcy friction factor is also known as the Darcy–Weisbach friction factor or the Moody friction factor. It is important to understand which friction factor is being described in an equation or chart to prevent error in pressure loss, or fluid flow calculation results.

The difference between the two friction factors is that the value of the Darcy friction factor is 4 times that of the Fanning friction factor. In all other aspects they are identical, and by applying the conversion factor of 4 the friction factors may be used interchangeably.

Unless stated otherwise the Darcy friction factor is used in this article.

Head Loss:

Pressure Drop:

# Methods of Determining the Darcy Friction Factor

The Darcy friction factor may be determined by either using the appropriate friction factor correlation, or by reading from a Moody Chart.

The Darcy friction factor is a dimensionless number; the pipe roughness and the pipe diameter which are used to determine the friction factor should be dimensionally consistent (e.g. use roughness and diameter both measured in mm, or both measured in inches)

## Which method should I use to calculate the Darcy Friction Factor?

There are many relationships available to determine the Darcy friction factor. Here we discuss the practicality and accuracy of applying these methods. Different methods of determining the friction factor as used depending on the flow regime of the fluid, as determined by the Reynolds Number.

## Laminar Flow

In the laminar flow regime the Darcy Equation may be used to determine the friction factor (see 2.2 and 4.1).

## Transitional Flow

In the transitional flow regime the inconsistency of the flow patterns make the prediction of friction factor impossible. No relationships are available to adequately describe this flow regime.

## Turbulent Flow Regime

In the turbulent flow regime the Colebrook equation (See 4.2) is widely accepted for describing the Darcy friction factor. The only drawback to using this equation is that it is implicit, and will require iteration to solve. Where iteration is possible and there are no constraints on computation speed, calculation via the Colebrook equation is appropriate.

If calculating by hand calculator or by computer where iteration is difficult Serghide’s equation (See 4.3) is most appropriate as it is explicit and has very low error (less than 0.003%).

It should be noted that more accurate approximations of the Colebrook equation have been proposed but generally the increased accuracy is not required. The error introduced in approximating the Colebrook equation using Serghide’s equation is likely to be many orders of magnitude less than error from other sources (such as uncertainty in pipe roughness or the uncertainty in the original data from which the Colebrook equation was produced).

# Friction Factor Equations

Here we detail some of the most common relationship for the Darcy friction factor for reference. For a discussion of the most appropriate relationships to use see above.

## Darcy Equation

The Darcy equation describes the Darcy friction factor for laminar flow. If this equation is substituted into the Pressure loss equation above it is also known as Poiseuille’s law or the Hagen–Poiseuille law.

## Colebrook’s Equation

Also known as the Colebrook-White Equation. This equation was developed taking into account experimental results for the flow through both smooth and rough pipe. It is valid only in the turbulent regime for fluid filled pipes. It is widely accepted and most of the relationships discussed in this article are merely explicit approximations for this relationship. Due to the implicit nature of this equation it must be solved iteratively. A result of suitable accuracy for almost all industrial applications will be achieved in less than 10 iterations.

The Colebrook equation may be calculated as follows:

## Serghide’s Equation

The Serghides’ equation is an approximation of the Colebrook equation use to solve for the Darcy friction factor explicitly. It is applied to fluid flowing in a filled circular pipe. The equation is presented using 3 intermediate values for simplicity. It provides and explicit approximation for the Colebrook equation that is highly accurate over a wide range of values for both surface roughness and Reynolds number. This method will result in errors of less than 0.003% in the ranges: Reynolds number 4000-1x1010, relative roughness 1x10-7 – 1.

The friction factor from the Serghide's approximation may be calculated from following set of equations:

$$\begin{split} A &= -2 log_{10}\left(\frac{\varepsilon/D} {3.7} + \frac{12} {Re}\right) \\ B &= -2 log_{10}\left( \frac{\varepsilon/D} {3.7} + \frac{2.51A} {Re}\right) \\ C &= -2 log_{10}\left( \frac{\varepsilon/D} {3.7} + \frac{2.51B} {Re} \right) \\ f &= \left(A - \frac{ \left( B-A \right)^2} {C-2B+A}\right)^{-2} \end{split}$$

## Chen’s Equation

Chen’s equation is an approximation of the Colebrook equation used to solve for the Darcy friction factor explicitly. It is applied to fluid flowing in a filled circular pipe.

The friction factor from Chen's approximation may be calculated as follows:

## Zigrang & Sylvester’s Equation

Zigrang & Sylvester’s equation is an approximation of the Colebrook equation use to solve for the Darcy friction factor explicitly. It is applied to fluid flowing in a filled circular pipe.

## Haaland Equation

The Haaland equation is an approximation of the Colebrook equation use to solve for the Darcy friction factor explicitly. It is applied to fluid flowing in a filled circular pipe and may be calculated as follows:

## Swamee-Jain Equation

The Swamee-Jain equation is an approximation of the Colebrook equation used to solve for the Darcy friction factor explicitly. It is applied to fluid flowing in a filled circular pipe and may be calculated as follows:

## Churchill Equation

The Churchill equation combines both the expressions for friction factor in both the laminar and turbulent flow regimes. It is accurate to within the error of the data used to construct the Moody diagram. This model also provides an estimate for the intermediate (transition) region, however this should be used with caution.

The Churchill equation shows very good agreement with the Darcy equation for laminar flow, accuracy through the transitional flow regime is unknown, in the turbulent regime a difference of around 0.5-2% is observed between the Churchill equation and the Colebrook equation.