Fluid pressure and flow rate are important considerations when designing a pipe system. In ideal circumstances, there would be no loss of energy within the system to affect the pressure and flow rate. However, in real applications engineers must consider energy lost due to frictional resistance, which is typically referred to as “head loss.”
To calculate head loss, you must first calculate the pipe’s friction factor, which is based on the roughness of the pipe material, pipe diameter, and the type of fluid flow. There are several methods for calculating friction factor that utilize different mathematical methods and offer varying degrees of accuracy.
One of the most common methods for calculating the friction factor for turbulent flow is the Swamee-Jain equation (another is the Haaland Equation).
Defining the Swamee-Jain Equation and Friction Factor
Definition and Importance of the Darcy Friction Factor
The Darcy friction factor correlates to a fluid’s density, viscosity, and flow rate, as well as the diameter and roughness of the pipe it is flowing through. It is often used in the Darcy equation to calculate the head loss in a pipe due to frictional resistance.
Head loss can have a significant impact on pipe system design because it affects fluid flow. The Darcy friction factor is inversely proportional to flow rate and fluid pressure. As the friction factor increases, head loss also increases, which results in decreased flow rate and fluid pressure.
How to Use the Swamee-Jain Equation
If the Reynolds number isn’t already given, it needs to be calculated first to determine what type of flow is being studied. For Re > 4000, the flow is turbulent so the Darcy friction factor can be calculated using the Swamee-Jain equation, which looks like this:
- f = Darcy friction factor (unitless)
- ϵ = absolute roughness (mm)
- d = inside diameter of the pipe (mm)
- Re = Reynolds number (unitless)
Absolute and Relative Roughness
The absolute roughness, ϵ, is a fixed number based on the material of the pipe. It is a measure of the peak-to-valley irregularities in a material’s surface that is measured in millimeters.
Materials like PVC and glass are relatively smooth and have minimal irregularities, so their roughness values are very low. By contrast, materials like concrete and cast iron have visible irregularities and high roughness values. To put it into perspective, the absolute roughness of concrete is 200 to 2000 times greater than the roughness of PVC.
The difference between absolute roughness and relative roughness is their dependence on pipe size. Absolute roughness is based solely on material and is independent of pipe size, whereas relative roughness is a ratio of absolute roughness and size.
Relative roughness is equal to the absolute roughness divided by the pipe diameter, or ϵ/d. As pipe diameter decreases, the relative roughness increases and has a greater impact on head loss.
Exceptions for the Swamee-Jain Equation
The Swamee-Jain equation is only accurate for turbulent flow, meaning it cannot be used when Re < 4000. When Re < 2300, flow is considered laminar and the friction factor can be calculated using the following equation:
When the Reynolds number is between 2300 and 4000, flow is considered transitional and it is often approximated using the Moody diagram.
Other Methods to Calculate the Friction Factor
When considering the accuracy of the Swamee-Jain equation, the results are most often compared to results from the Colebrook-White equation. In the Colebrook-White equation, the friction factor variable appears on both sides of the equation and must be calculated using iterative methods. The equation looks like this:
The variables in this equation represent the same values as those in the Swamee-Jain equation.
There is no specific value for f that needs to be used for the first iteration. For every subsequent iteration, start with the previous result for f. As the number of iterations increases, the results for f should converge to one value.
This method can be more time consuming but is generally regarded as the most accurate method.
The Moody Diagram is another common method for determining friction factor. This is a graphical interpretation where Reynolds numbers are on the X-axis, relative roughness is represented by curved lines starting on the right-hand side of the Y-axis, and friction factor values are on the left side of the Y-axis.
Once you know the relative roughness and Reynolds number, you can trace the appropriate line for the roughness and find where it intersects with the Reynolds number along the bottom of the graph.
Draw a horizontal line between that intersection point and the leftmost axis where the friction factors are listed. That intersection value represents the friction factor of the pipe being evaluated. If the intersection point doesn’t land on an exact value, it can be interpolated using the values directly above and below it.