In fluids engineering, evaluating the pressure loss due to friction is an important aspect of design. To determine this loss, a factor, such as the Darcy friction factor, is used to simplify the calculations. Understanding how to calculate the Darcy friction factor can assist an engineer with applying it, but a simplification for determining the factor, a Moody chart, enables an engineer to reduce the work needed to find the Darcy friction factor.
Darcy Friction Factor
The Darcy friction factor is a dimensionless value that is used to determine the pressure loss in a fluid flow. It was originally proposed by Henry Darcy as a component of the Darcy-Weisbach equation. Calculating the Darcy friction factor involves an understanding of some key principles regarding fluid flow.
The Reynolds number is a measure of the ratio of inertial forces and viscous forces in a fluid flow. It is most used in determining if a flow is laminar or turbulent, with laminar flow being anything with a Reynolds number less than about 3000.
Depending on the value of the Reynolds number, the calculation of the Darcy friction factor varies, as will be discussed. If the flow is considered laminar, the calculation of the Darcy friction coefficient can be reduced to:
where fD is the Darcy friction factor and Re is the Reynolds number.
The relative roughness of the wall of the pipe through which the fluid flows plays a very important role in determining the friction factor. The roughness is measured as the ratio between the average height of irregularities and the diameter of the pipe, as follows:
where ε is the average irregularity height and D is the diameter of the pipe.
If the Reynolds number is very high, which means the flow is exceedingly turbulent, the roughness of the pipe no longer needs to be considered.
The last thing that needs to be considered when calculating the Darcy friction factor is the diameter of the pipe. A pipe with a larger diameter will have less pressure loss due to friction because more of the flow is unaffected by the friction.
Calculating the Darcy Friction Factor
Using the Reynolds number, roughness, and pipe diameter, it is possible to use the following Colebrook-White equation to calculate the Darcy friction factor:
where D is the diameter of the pipe.
As can be seen by the equation, it cannot be solved without iteration or a numerical method. Although several simplifications and conversions have been developed over time, one of the most effective methods for determining the Darcy friction factor is to use a Moody chart.
The Moody chart was developed by Lewis Moody to plot the Darcy friction factor. The chart simplifies the process of determining the friction from performing iterative calculations to simply finding the relationship between the Reynolds number and relative roughness.
Application of the Moody Chart
Using the Moody chart, an engineer who knows the Reynolds number for the flow and the relative roughness of the pipe can trace to a line that indicates what the Darcy friction factor is. Because the Darcy friction factor is a function of both the Reynolds number and the roughness, there are multiple lines on the Moody chart, each one with a specific friction factor.
Limitations of the Moody Chart
Although the values used in generating a Moody chart can be established very specifically, the chart will still be limited by an engineer’s accuracy in tracing the points. Oftentimes, when using an engineering diagram such as a Moody chart, there will need to be some interpolation and/or extrapolation made by the engineer. This means that different engineers may get slightly different results when using the same chart.
However, because the Moody chart is used only to find which Darcy friction factor line to use, the error is often minimized, especially for higher Reynolds numbers, which cause the lines to become more separated.
Moody vs Darcy Friction Factor
Because the Moody diagram was specifically designed to plot the Darcy friction factor, one could say that both are the same thing. However, because the Darcy friction factor is not the only friction factor used to evaluate the pressure loss due to friction, the use of the Moody chart has been expanded to other friction factors.
Another common friction factor is the Fanning friction factor, which is one-fourth of the Darcy friction factor. To take advantage of the simplification for determining the Darcy friction factor by using a Moody chart, oftentimes it is possible to find a Moody chart for the Fanning friction factor. But care should be paid to which friction factor is being plotted on the Moody chart.
If the chart is not labeled with the friction factor it shows, a good way to determine the factor form is to find the value at a Reynolds number of 1000 for the roughness, then look at the friction factor. If the friction factor is 0.064, the chart shows the Darcy friction factor. If the value is 0.016, the Fanning friction factor is shown.