Computing the friction factor of flow through a pipe can be a tricky task. For one, the flow may change as it goes through a pipe, shifting from laminar to turbulent or vice-versa. This is a dimensionless quality that can also be used to define open channel flow.
While a number of formulae have been derived to compute this friction factor, the Haaland equation is the simplest and is best for hand calculations.
The Haaland Equation
One way to find the friction factor of flow through a pipe is the approximation by professor S.E. Haaland first proposed in 1983. Like the Colebrook equation, this formula is used to derive the friction factor for a full-flowing circular pipe. It was constructed using experimental results of both laminar and turbulent flow studies. However, Haaland first derived his formula with the intention of making things simple.
The Haaland equation is better suited for hand calculations than any of the other Colebrook derivations. The formula is as follows:
- Re = Reynolds Number (unitless)
- ε / D = pipe’s relative roughness (unitless)
The Haaland equation is appropriate for both liquid and gaseous flows through pipes, through the formula shown is most appropriate for liquid flows. In the case of gas transmission, both of the terms inside the brackets can be raised to a power of three, and the entire righthand side of the equation divided by three for a more accurate approximation.
While this approximates the Colebrook formula, the differences in accuracy are small and within the acceptable range of data. Indeed, the estimated error using the Haaland formula for liquid flow is only around 1.4%.
At first glance the Haaland equation still seems complicated, and it is, but can be accomplished by “hand” (or at least with a calculator). And, most importantly, it is not an iterative formula! So where does this simplification come from, what exactly is the more complex Colebrook equation?
Darcy Friction Factor
What scientists and engineers are solving for when they study frictional losses is more formally known as the Darcy-Weisbach Friction Factor or the resistance coefficient, or just simply friction factor. The equation is named after Henry Darcy and Julius Weisbach.
This equation solves for “head loss”, also known as pressure loss, due to friction along a specified length of pipe of a fluid moving at an average velocity. As it stands, this formula taken along with Moody diagrams offers an accurate and applicable way to find frictional losses within pipes.
The Colebrook Equation
Finding friction levels within a pipe obviously has huge implications for a number of industries, so it is important it’s calculated as correctly as possible. Using the Colebrook equation and its variations, one can find the friction factor for laminar, turbulent, and transitional flows, and through both rough and smooth pipes.
How rough a pipe is plays a large role in having thin boundary layers. Which variation of the equation is used depends on the properties of the pipe, the accuracy required, and computational power available.
However, each equation will require knowing or understanding the following: solving for f, the Darcy friction factor, Re, the Reynolds Number defined by pipe diameter, fluid velocity and viscosity, and a pipe’s relative roughness to diameter (ε / D).
The Colebrook equation, used for calculating frictional losses in pipes with turbulent flow, is as follows:
Right away we can see how complicated this equation is, full of logarithms and square roots. What we’re solving for is on both sides of the equation!
And what’s more, the Colebrook equation is iterative, meaning it must be calculated a number of times until the value for frictional factor is within an acceptable range. But fear not, because over the years a number of alternative solutions have been derived from this complicated formula to more easily fit particular scenarios.
One such equation is the Haaland Equation, and another is the Swamee-Jain Equation.
Why This is Important
Ultimately, the friction factor of flow through a pipe needs to be accurately approximated because it is one of the largest factors in understanding how much power one needs to pump a fluid throughout a length of pipe.
If pumps are not being used, or if the power amount is already determined, it’s not much more work to work backwards through the Haaland equation and determine appropriate length, diameter, fluid (via Reynolds number), or pipe material and diameter (via roughness approximation).
Energy losses due to friction are an important factor when engineering designs from everything to water and gas lines that support HVAC systems, and other integral pipe networks that flow silently in the background of rooms we enter every single day.