Flamant Head Loss

Introduction

The Flamant equation is an empirical formula widely used in agricultural hydraulics, particularly for micro-irrigation systems (drip and micro-sprinkler irrigation). It is highly accurate for calculating friction head loss in small-diameter pipes (typically under 50 mm) made of smooth materials like PVC and Polyethylene (PE).

The Roughness Coefficient (\(b\))

The \(b\) factor represents the internal roughness of the pipe according to Flamant’s experiments. For smooth plastic pipes (PVC, PE), the standard value is \(0.000135\).

Equations

The base mathematical equation for head loss (\(h_f\)) in the metric system is: \[h_f = 6.107 \cdot b \cdot \frac{L \cdot Q^{1.75}}{D^{4.75}}\]

Where:

  • \(h_f\): Friction head loss (m)
  • \(L\): Pipe length (m)
  • \(Q\): Volumetric flow rate (\(m^3/s\))
  • \(b\): Flamant roughness coefficient (dimensionless)
  • \(D\): Internal diameter (m)

1. Calculating Head Loss

To calculate the friction head loss, use the calc_head_loss_flamant() function. Let’s calculate the expected head loss for a 50-meter PE pipe (\(b = 0.000135\)) with an internal diameter of 0.015 meters (15 mm) carrying a flow of \(0.0002\ m^3/s\) (0.2 L/s).

library(hf)
calc_head_loss_flamant(length = 50, flow = 0.0002, diameter = 0.015)
#> [1] 6.389998

2. Calculating Required Diameter

If you know the maximum allowable head loss for your lateral irrigation line, you can determine the required minimum pipe diameter using calc_diameter_flamant(). Suppose the system allows a maximum head loss of 1.5 meters for a 50m length and a \(0.0002\ m^3/s\) flow rate.

library(hf)
calc_diameter_flamant(loss = 1.5, length = 50, flow = 0.0002)
#> [1] 0.0203516

3. Calculating Flow Rate

To find the maximum flow rate a small pipe can deliver given an available pressure head, use the calc_flow_flamant() function.

library(hf)
calc_flow_flamant(loss = 1.5, length = 50, diameter = 0.015)
#> [1] 8.737103e-05