Pipe Friction Losses

Real pipe flow loses energy to viscous friction. The Darcy-Weisbach equation quantifies this as a head loss $h_f$ (metres of fluid):

$h_f = f \frac{L}{D} \frac{v^2}{2g}$

where:

• $f$ — Darcy friction factor (dimensionless)
• $L$ — pipe length (m)
• $D$ — pipe diameter (m)
• $v$ — mean flow velocity (m/s)
• $g = 9.81$ m/s²

Head loss is the height a fluid column would need to fall to supply the same energy. It is proportional to $v^2$, so turbulent flow is expensive.

Friction Factor for Laminar Flow

For laminar flow ($Re < 2300$) the friction factor has an exact analytical value:

$f = \frac{64}{Re}$

For turbulent flow, empirical correlations such as the Moody chart or the Colebrook equation are used instead.

Pressure Drop

Head loss converts to a pressure drop via:

$\Delta P = \rho g h_f$

This is the pressure the pump must provide to maintain the flow.

Your Task

Implement:

• head_loss(f, L, D, v) — returns $h_f$ (m). Use $g = 9.81$ m/s² inside the function.
• friction_factor_laminar(Re) — returns Darcy $f = 64/Re$ for laminar flow
• pressure_drop(rho, h_f) — returns $\Delta P$ (Pa). Use $g = 9.81$ m/s² inside the function.