Viscous flow through a real pipe irreversibly converts mechanical energy into internal energy. For steady flow in a straight pipe, the associated distributed pressure loss is commonly modeled by the Darcy-Weisbach equation.

For a pipe of length $L$, diameter $D$, mean speed $V$, and fluid density $\rho$,

$$\boxed{\Delta p_f=f\frac{L}{D}\frac{\rho V^2}{2}},$$

where $f$ is the dimensionless Darcy friction factor.

Expressed as head loss,

$$\boxed{h_f=f\frac{L}{D}\frac{V^2}{2g}},$$

because

$$h_f=\frac{\Delta p_f}{\rho g}.$$

The quantity $V^2/(2g)$ is the velocity head. The factor $fL/D$ determines how much of that scale is dissipated by wall friction over the pipe length.

What the friction factor represents

The friction factor packages the effect of wall shear into a dimensionless quantity. Its value depends primarily on flow regime and, for turbulent flow, wall roughness.

Do not confuse the Darcy friction factor with the Fanning friction factor; the Darcy value is four times the Fanning value.

Laminar circular-pipe flow

For fully developed laminar flow in a circular pipe, Hagen-Poiseuille gives

$$\Delta p=\frac{32\mu LV}{D^2}.$$

Darcy-Weisbach gives

$$\Delta p=f\frac{L}{D}\frac{\rho V^2}{2}.$$

Equating the two expressions,

$$\frac{32\mu LV}{D^2} =f\frac{L}{D}\frac{\rho V^2}{2}.$$

After cancellation,

$$f=\frac{64\mu}{\rho VD}.$$

Since the pipe Reynolds number is

$$Re_D=\frac{\rho VD}{\mu},$$

we obtain the exact laminar result

$$\boxed{f=\frac{64}{Re_D}}.$$

For fully developed laminar flow, ideal smoothness does not remove viscous loss; the friction factor is set directly by Reynolds number.

Turbulent flow

In turbulent pipe flow, $f$ is not given by $64/Re$. It depends on both

• the Reynolds number $Re_D$;
• the relative roughness $\varepsilon/D$, where $\varepsilon$ characterizes wall roughness height.

A Moody chart or a suitable correlation is commonly used to determine $f$. In hydraulically smooth turbulent flow, roughness has little effect over some ranges; at sufficiently large Reynolds number in a rough pipe, roughness can dominate the friction factor.

The Darcy-Weisbach equation remains the pressure-loss model while the rule for obtaining $f$ changes with regime.

Worked example

Water flows through a straight pipe with

$$L=50,\mathrm m,$$

$$D=0.050,\mathrm m,$$

$$V=1.5,\mathrm{m/s},$$

and suppose the appropriate Darcy friction factor is

$$f=0.025.$$

The head loss is

$$h_f =(0.025)\frac{50}{0.050}\frac{(1.5)^2}{2(9.81)}.$$

Since

$$\frac{50}{0.050}=1000,$$

and

$$\frac{(1.5)^2}{2(9.81)}\approx0.1147,\mathrm m,$$

we get

$$h_f\approx(0.025)(1000)(0.1147) \approx\boxed{2.87,\mathrm m}.$$

For water with density about $1000,\mathrm{kg/m^3}$, the corresponding pressure loss is

$$\Delta p_f=\rho gh_f \approx(1000)(9.81)(2.87) \approx\boxed{28.2,\mathrm{kPa}}.$$

Distributed versus local losses

Darcy-Weisbach describes major or distributed losses along a straight pipe length. Valves, entrances, exits, elbows, contractions, expansions, and other fittings produce additional localized losses that are usually modeled separately.

The key idea is that pressure loss in internal flow is not determined by Bernoulli's inviscid equation alone. Darcy-Weisbach supplies the empirical-mechanical link between pipe geometry, mean speed, flow regime, and irreversible wall-friction loss.