Continuity

Mass is conserved in a flow. Learn mass and volume flow rate, the continuity equation A₁V₁ = A₂V₂, flow in branching pipes, and compressible mass flow.

Put your thumb over a garden hose and the water shoots out faster. Nothing is created or lost — the same amount of water per second must pass through a smaller opening, so it speeds up. That's the continuity equation.

It is the first equation written for every pipe network, duct, nozzle and pump system.

An analogy

Traffic on a motorway narrowing from three lanes to one: if the same number of cars per minute must pass, they have to go faster in the narrow section.

Flow rate

Volume flow rate is area × average velocity. 1 m³/s = 1000 L/s = 3600 m³/h. A domestic tap delivers about 0.1–0.2 L/s.

Velocity and diameter

For round pipes, V ∝ 1/D². Halving the diameter quadruples the velocity. Designers choose pipe size to keep water velocity ≈ 1–3 m/s (noise, erosion, pressure loss vs pipe cost).

Networks

At every junction, flow in = flow out (like Kirchhoff's current law). Combined with energy losses around loops, this solves water-distribution networks (Hardy Cross method).

Differential form and compressible flow

The differential form ∂ρ/∂t + ∇·(ρV) = 0 is one of the Navier–Stokes equations solved by CFD. In compressible nozzle flow, mass flow per area peaks at Mach 1 (choked flow) — limiting flow through valves and orifices.

Formulas

Worked example: Pipe reducer

Given: 100 mm pipe at 2 m/s; Reduces to 50 mm

Find: Flow rate and velocity in the small pipe.

Answer: 15.7 L/s, 8 m/s

Worked example: Branch

Given: Main carries 20 L/s; Branch A takes 12 L/s through 80 mm pipe

Find: Flow and velocity in branch A, flow in branch B.

Answer: 2.39 m/s; 8 L/s in B

Common mistakes

Applications

Key takeaways

Going further