Fluid Mechanics – Friction Factor and Darcy-Weisbach Equation

In fluid mechanics, Darcy-Weisbach equation relates pressure-loss to average flow velocity. It contains dimensionless friction factor known as Darcy-Friction equation. This is also called Darcy-Weisbach friction factor which simply friction factor. Here, fluid is assumed to be incompressible in nature.

Darcy-Weisbach equation

However, Pressure loss in pipe flow is calculated using equation known as Darcy-Weisbach equation. This Darcy-Weisbach equation is given as-

\Del h_{f}=f\frac{L}{D}\frac{V^{2}}{2_{g}}

Where hf is head loss (m), f is friction factor, L is the length of the pipe (m), D is the internal diameter of the pipe (m), V is the average flow velocity (m/s) and g is local acceleration due to gravity (m/s2), f is a dimensionless parameter called the Darcy friction factor, resistance coefficient or simply friction factor.

 F=F\(\frac{pVD}{\mu},\frac{\varepsilon}{D}\)

Where \frac{pVD}{\mu} Reynolds is number Re and  is specific surface roughness of pipe material. See Pressure Scales in Fluid Mechanics

Moody friction chart

In fluid dynamics we have to solve a problem which is use of Darcy-Weisbach friction factor f but flow is steady or transient. In circular pipes this factor can be solved directly with Swamee-Jain equation as well as other. We have to use this factor but for other types it is difficult to solve. In these cases Moody charts are really handy.

This Moody friction chart is most convenient method of getting value of moody friction factor f. But in laminar pipe flow Reynolds number R is less than 2000, so f=64/R because in laminar flows R is head loss of wall roughness. If pipe is not in circular cross-section area, then an equivalent diameter  is used for calculating head loss or friction factor.

Swamee-Jain equation

As per Swamee-Jain equation is used to calculate pipe design diameter directly. Below equation relationship is follows-

D=0.66\quad \varepsilon^{1.25}\quad \(\frac{LQ^{2}}{gh_{f}}\)^{4.75}+vQ^{9.4}+\(\frac {L}{gh_{f}}\)^{5.2}

  • Where D is function of hydraulic diameter and is roughness height (m,ft)
  • Q is volume flow rate and Length is L
  • V is velocity where ‘g’ is gravitational constant and is head loss (m, ft)

How to read Moody Chart

  • Most of fluid mechanics problems involve of Reynolds number. Once Reynolds number is known we can use this chart easily. If there is no velocity given then we have to assume a velocity or an initial friction factor. If your assumption is correct then you will get same answer.
  • We need to use Moody chart if Reynolds number show that flow is laminar. But if flow is turbulent then we have to look through Moody chart.
  • At first, you have to calculate relative roughness of pipe. This value of roughness of pipe is gained divided by diameter of pipe. You need to remember this is unit less, so ensure that roughness and diameter are in same units.
  • If actual roughness is zero then roughness will also be zero. But you will get value for friction factor always and this does not mean that friction factor will be zero.
  • Get line which matches your relative roughness on right side of chart. If you don’t get printed line then think line paralleling to next line of your relative roughness. It may be helpful to sketch in this line.
  • Now follow your line to left as it curves up until to reach line vertical to your flow’s Reynolds Number.
  • Remember mark this point on chart
  • Use scale and follow that point to left parallel to x axis
  • From left of chart read your friction factor
  • Now you can get energy loss after getting friction factor
  • With this help you can calculate new velocity and then Reynolds number
  • Compare your new Reynolds number with your previous values. If Reynolds number is different from your previous value, so repeat calculations with this new Reynolds value. However, it is close to your previous value, your answer has converged and finished.

Example for your understanding Moody chart

Let says we calculate Reynolds Number of 4×10^4. We see that this is in Reynolds Number range for turbulent flow, so we proceed with Moody chart. Let’s imagine we calculate unit less relative roughness of 0.003. Check Bernoulli’s principle from Equation in Fluid Mechanics

Follow curve line in left and we follow this line our Reynolds number from before. From here, we look straight left, shown by orange line, until we hit the left margin of chart. Then you have to mark that point. We got value 0.003. At this point, we add new velocity and new Reynolds Number if needed.

Points to be noted

  • Here all parameters are dimensionless and moody chart is applicable everywhere.
  • There is common interpolation error that you can try to avoid while taking reading
  • This system will work for steady flow only

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