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.