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Инженерия / Гидравлика

Continuity equation

For steady incompressible flow in a pipe network or conduit, the mass flow continuity implies that discharge is the same at all sections.

Опубликовано: Обновлено:

ГидравликаНепрерывность

Формула

$$A_1 v_1 = A_2 v_2 = Q$$
diagram Two-section continuity

Pipe with area reduction from A1 to A2; the same discharge Q crosses both sections.

Discharge conservation links velocities and areas.

Обозначения

$A_1$
Area at section 1, m^2
$v_1$
Average velocity at section 1, m/s
$A_2$
Area at section 2, m^2
$v_2$
Average velocity at section 2, m/s
$Q$
Common volumetric discharge, m^3/s

Условия применения

  • Steady flow (no storage change in the control volume).
  • Incompressible fluid over the interval.
  • No branch or leakage between the two control sections.

Ограничения

  • Not valid when fluid density changes significantly with pressure or temperature.
  • Not valid for strongly unsteady transient events such as water hammer.
  • Cannot replace full momentum analysis where acceleration terms dominate.

Подробное объяснение

The product A v is volume per unit time. Conservation of mass requires this product to remain constant along a streamline in steady incompressible flow.

Как пользоваться формулой

  1. Write Q at each section as A·v.
  2. Set A1·v1 equal to A2·v2.
  3. Solve for the unknown section variable.

Историческая справка

The steady incompressible continuity relation is one of the earliest conservation statements in classical fluid mechanics.

Пример

For A1 = 0.2 m^2, v1 = 1.0 m/s, and A2 = 0.1 m^2: Q = 0.2, so v2 = 2.0 m/s.

Частая ошибка

A common mistake is mixing diameter and area values from different sections without unit conversion.

Практика

Задачи с решением

Find downstream velocity

Условие. A1 = 0.18 m^2, v1 = 1.2 m/s, A2 = 0.10 m^2.

Решение. Q = A1 v1 = 0.216 m^3/s, v2 = Q / A2 = 0.216/0.10 = 2.16 m/s.

Ответ. v2 = 2.16 m/s.

Find upstream velocity

Условие. A1 = 0.06 m^2, A2 = 0.15 m^2, v2 = 0.8 m/s.

Решение. Q = A2 v2 = 0.12 m^3/s, v1 = Q / A1 = 2.0 m/s.

Ответ. v1 = 2.0 m/s.

Дополнительные источники

  • Fox, R. W., Pritchard, P. J., & McDonald, A. T. (2011). Introduction to Fluid Mechanics, 8th ed.
  • Mott, R. L. (2006). Applied Fluid Mechanics.

Связанные формулы

Инженерия

Flow rate

$Q = A v$

Flow rate is the volume of fluid passing through a cross-section per unit time. For idealized one-dimensional flow, it is the product of area and average velocity.

Инженерия

Velocity from flow rate

$v = \frac{Q}{A}$

If flow rate and flow area are known, this equation yields the average velocity in the section.

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.