Continuity Equation Calculator

The continuity equation expresses conservation of mass for fluid flow. For steady flow through a pipe or streamtube with no internal sources or sinks, the mass flow rate ṁ = ρAv must be constant. For incompressible liquids (water, oil at moderate pressure changes), density ρ is constant and the volumetric flow rate Q = Av is constant: A₁v₁ = A₂v₂.

Physically: fluid cannot accumulate or disappear inside a sealed pipe — what enters one cross-section per second must leave another. When the pipe narrows (smaller A), the fluid must speed up (larger v) to pass the same volume per second. This is why covering a hose outlet with your thumb produces a fast, narrow jet.

For circular pipes, A = πD²/4. Halving the diameter D reduces area by factor 4 (since A ∝ D²), so velocity increases by factor 4 for the same Q. Example: Q = 0.01 m³/s in D = 0.1 m pipe gives v ≈ 1.27 m/s; reducing to D = 0.05 m gives v ≈ 5.1 m/s.

The average velocity v = Q/A is used in engineering calculations. In laminar pipe flow, the velocity profile is parabolic with v_max = 2v_avg at the center. In turbulent flow, the profile is flatter but continuity still holds for the average.

For compressible gases (air ducts, nozzles), use ṁ = ρ₁A₁v₁ = ρ₂A₂v₂ — density changes with pressure and temperature. At low Mach numbers (ΔP small compared to pressure), air is often treated as incompressible in building HVAC.

Continuity is one of three pillars of pipe flow analysis, together with Bernoulli (energy) and momentum. It explains the Venturi effect when combined with Bernoulli: constriction → higher v → lower P.