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Darcy–Weisbach head loss

The Darcy–Weisbach equation estimates major head loss in fully developed pipe flow using friction factor and geometric ratio.

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Гидравлика

Формула

$$h_f = f\frac{L}{D_h}\frac{v^2}{2g}$$
chart Pipe friction loss

Direct dependence of head loss on length, speed, and friction characteristics.

Longer, rougher, faster pipe segments produce greater head losses.

Обозначения

$h_f$
Major head loss due to friction, m
$f$
Darcy friction factor, -
$L$
Pipe length, m
$D_h$
Hydraulic diameter, m
$v$
Average velocity, m/s
$g$
Gravitational acceleration, m/s^2

Условия применения

  • Steady, incompressible, single-phase flow.
  • Constant diameter section in which f can be approximated as average.
  • Losses are dominated by wall friction in the considered reach.

Ограничения

  • Local losses (bends, valves, entrance, exit) are not included.
  • Requires valid friction factor correlation (different for laminar/turbulent and smooth/rough pipes).
  • Not directly applicable to open-surface free-flow without adaptation.

Подробное объяснение

It expresses pressure drop as a frictional head dissipation term proportional to length ratio and dynamic head.

Как пользоваться формулой

  1. Compute Reynolds number and friction factor f for the operating regime.
  2. Compute dynamic head v²/(2g).
  3. Multiply by f and length-to-diameter ratio.

Историческая справка

Darcy and Weisbach forms are long-standing in hydraulics for quantifying major pipeline energy loss.

Пример

L=100 m, D_h=0.1 m, v=2.0 m/s, f=0.02, g=9.81 -> h_f = 0.02·100/0.1·2²/19.62 = 2.04 m.

Частая ошибка

Using a friction factor in absolute value range without ensuring compatibility (Darcy vs Moody-style definitions).

Практика

Задачи с решением

Calculate head loss

Условие. L=250 m, D_h=0.15 m, v=1.5 m/s, f=0.018, g=9.81.

Решение. h_f = 0.018·(250/0.15)·(1.5²/(2·9.81)) = 4.13 m.

Ответ. h_f ≈ 4.13 m.

Back-calculate friction factor

Условие. h_f=3 m, L=120 m, D_h=0.12 m, v=2.0 m/s, g=9.81.

Решение. f = h_f D_h 2g / (L v²) = 3·0.12·19.62 / (120·4) = 0.0147.

Ответ. f ≈ 0.015.

Дополнительные источники

  • White, F. M. (2011). Fluid Mechanics, 7th ed.
  • Fox, R. W., Pritchard, P. J., & McDonald, A. T. (2011). Introduction to Fluid Mechanics, 8th ed.

Связанные формулы

Инженерия

Reynolds number

$Re = \frac{\rho v D_h}{\mu} = \frac{v D_h}{\nu}$

Reynolds number indicates the ratio of inertial to viscous forces in a fluid and is used to determine flow regime.

Инженерия

Bernoulli equation (basic)

$\frac{p_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{v_2^2}{2g} + z_2$

The basic Bernoulli equation links pressure head, velocity head, and elevation head along a streamline for frictionless, steady, incompressible flow.

Инженерия

Pump power

$P = \frac{\rho g Q H}{\eta}$

Pump input power equals hydraulic power raised by the pump divided by pump efficiency.