A Moody chart is commonly used by engineers to calculate the Darcy-Weisbach friction factor, which is then in turn used to calculate head/pressure loss due to friction in pipes. On the Moody chart the friction factor is shown on the left hand y-axis, the Reynolds number is shown on the x-axis and the relative roughness is shown on the right hand y-axis.
If any two of these three variables are known then the third may be found using this chart, although in practice it’s the Reynolds number and the relative roughness that are known and the friction factor that is desired.
The first step, as is usual in fluid mechanics problems, is to calculate the Reynolds number of the flow. The Reynolds number is simply the ratio of momentum forces to viscous forces within the flow and is calculated using the dynamic viscosity and density (μ,ρ) fluid properties, the mean fluid velocity and the hydraulic diameter. In pipe flow the hydraulic diameter is equal to the pipe ID and if the bulk fluid velocity is unknown then it can be calculated by dividing the volume flow rate by the pipe cross sectional area.
If the Reynolds number indicates that the flow is laminar, i.e. it is below 2 × 103, then the straight line in the top left quadrant of the Moody chart can be used to find the friction factor. To interpolate the friction factor, a line should be drawn vertically upwards from the value of the Reynolds number on the x-axis, until it meets the 64/Re line. A second line should then be drawn horizontally from this point and where this horizontal line intersects the left hand y-axis the friction factor value can be read off.
For example, if the Reynolds number is 1000, then a line is drawn upwards until it meets the 64/Re line and then across, which gives a value of 0.064 for the friction factor.
Transitional and Turbulent Flows
If the calculated Reynolds number indicates a transitional or turbulent flow, then the relative roughness of the pipe has to be established. The relative roughness is a dimensionless quality equal to the pipe absolute roughness divided by its ID. The absolute roughness (ε) of a pipe is a measure of surface roughness and depends upon the material, manufacturing method, coatings and corrosion status of the pipe. For example, a typical PVC pipe has an absolute roughness of 0.0015 mm, whereas a corroded cast iron pipe could have an absolute roughness as high as 2.5 mm.
With the relative roughness established, a vertical line can be drawn upwards from the value of the Reynolds number, extending until it meets the curved line that represents the value of the relative roughness given on the right hand y-axis. A second horizontal line can now be drawn across from this intersection, until it meets the left hand y-axis where the friction factor value can be read off.
As an example, we can apply this method to a pipe with an ID of 15 mm, manufactured with an absolute roughness of 0.0015 mm with a steady state flow, Reynolds number 3×105, running through it. The relative roughness of this pipe is the absolute roughness (ε) divided by diameter which yields 0.0015/15 = 0.0001. Running a line vertically from the 3×105 Reynolds number until it meets the line corresponding to the 0.0001 relative roughness and then across to the friction factor gives a value of 0.016 for the friction factor.
It’s important to ensure that units are used consistently when calculating non-dimensional quantities such as the Reynolds number and relative roughness. In our example the absolute roughness and pipe ID are both in units of millimeters, but if the units for the division don’t match then the values should be converted so they do.
Another important point is that because of the numerous lines, interpolation errors can occur. To avoid this, it’s important to mark the lines on the diagram when interpolating so that errors are avoided.
The Moody chart is also only applicable to fully developed flow within a pipe; if this condition is not met by the flow then other methods have to be used.
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