Pressurized flow in pipes and closed conduits of other shapes is used for a lot of purposes. Energy input to the gas or liquid is needed to make it flow through the pipe or conduit. This energy input is needed because there are frictional energy losses (also called frictional head losses) due to the friction between the fluid and the pipe wall and internal friction within the fluid. The Darcy Weisbach Equation, which will be discussed in this article, is commonly used for a variety of calculations involving frictional head loss, pipe diameter, flow rate or velocity, and several other parameters.
The Darcy Weisbach Equation applies to fully developed, turbulent flow. Recall that pipe or conduit flow will be turbulent for a Reynold's Number greater than 4000. Fully developed flow will be present in a pipe or conduit beyond the entrance region. The entrance is where the velocity profile is adjusting to the constant profile that is present throughout the fully developed flow region. The diagram at the left illustrates the concept of the entrance region and fully developed flow.
Most pipe and conduit flow of gases and liquids with a viscosity similar to water will be turbulent flow. If the total pipe length is large compared to the entrance length, then the entrance effects are negligible and the total pipe length is used for calculations.
The Darcy Weisbach Equation provides an empirical relationship among several pipe flow variables as shown here:
The equation is: hL = f (L/D)(V2/2g), where
L = pipe length, ft
D = pipe diameter, ft
V = average velocity of fluid (= Q/A), ft/sec
g = acceleration due to gravity = 32.2 ft/sec2
f = friction factor, a dimensionless empirical factor that is a function of Reynold's Number (Re = DVρ/μ) and/or ε/D, where
ε = an empirical pipe roughness, ft
The friction factor (also sometimes called the Moody friction factor) can be determined for known values of Re and ε/D from empirically derived charts and/or equations. A commonly used chart is the Moody friction factor chart, shown in the diagram on the left. Clicking on the chart will give you a larger scale diagram, so you can see it better. This chart helps to illustrate how the friction factor, f, depends upon Re and ε/D. The straight line at the upper left on the diagram represents laminar flow, in which f is independent of ε/D and depends only on Re. The portion of the chart to the right of the dashed line is called the completely turbulent region, in which f depends only on ε/D. In the portion between the dashed line and dark solid line, the transition region, f depends upon both Re and ε/D. The dark solid line represents "smooth pipe turbulent flow", in which f depends only on Re.
There are equations available for friction factor for each of the four regions of the chart identified above as follows.
For laminar flow (Re < 2100): f = 64/Re
For the completly turbulent region: f = [1.14 + 2 log10(D/ε)]-2
For smooth pipe turbulent flow: f = 0.316/Re1/4
For the transition region: f = {-2 log10[(ε/D)/3.7 + 2.51/Re(f1/2)]}-2
Note that the last equation requires an iterative solution to find f for given values of ε/D and Re, or "solver" can be used in the Excel spreadsheet.
The Darcy Weisbach Equation relates the variable, hL, D, L, V, ε, ρ and μ. It's typical use is to calculate hL, D, L, or V, when all of the other parameters are known. Some of these require iterative calculations.