🌊 Continuity Equation Calculator
Calculate fluid velocity, cross-sectional area, or volumetric flow rate using the continuity equation — the mathematical expression of mass conservation in fluid flow. Supports both incompressible (liquids) and compressible (gases) flow with step-by-step solutions.
What Is the Continuity Equation?
The continuity equation is the mathematical expression of the law of conservation of mass applied to fluid flow. It states that the mass flow rate entering a control volume must equal the mass flow rate leaving it, assuming no accumulation or leakage.
In its most general form:
ρ₁A₁V₁ = ρ₂A₂V₂
where ρ = fluid density (kg/m³), A = cross-sectional area (m²), V = flow velocity (m/s) at sections 1 and 2
For incompressible fluids (liquids and low-speed gases), density is constant, so the equation simplifies to:
A₁V₁ = A₂V₂ = Q
where Q is the volumetric flow rate (m³/s)
Physical Meaning
The continuity equation reflects a simple but profound idea: what goes in must come out. When a pipe narrows, the fluid must speed up to allow the same amount of mass to pass through the smaller cross-section per unit time. This is why:
- Garden hose nozzle: Water sprays faster when you cover part of the opening.
- River rapids: Water flows fastest where the river channel is narrowest.
- Blood flow: Blood moves fastest in the aorta and slowest in capillaries (despite their tiny individual diameters, the total cross-sectional area is enormous).
The Velocity-Area Relationship
For circular pipes, the area is A = πD²/4, so the velocity ratio is:
V₂ / V₁ = (D₁ / D₂)²
This means that halving the diameter quadruples the velocity. This inverse-square relationship is a direct consequence of mass conservation.
Incompressible vs. Compressible Flow
The simplified form A₁V₁ = A₂V₂ applies only when density is constant:
- Incompressible flow: Liquids (water, oil) and gases at low speeds (Mach < 0.3). Density changes are negligible.
- Compressible flow: Gases at high speeds (Mach > 0.3), where density changes significantly. The full form ρ₁A₁V₁ = ρ₂A₂V₂ must be used.
Why Use This Calculator?
- Solve for any variable: Calculate V₂, V₁, A₂, A₁, or Q depending on what's unknown.
- Supports both flow types: Incompressible (liquids) and compressible (gases) with density input.
- Pipe visualization: See the flow speed-up or slowdown at a glance.
- Step-by-step solution: See exactly how the calculation is performed.
- Free & private: No registration, no data storage — all calculations are done in your browser.
❓ Continuity Equation Calculator FAQ
What is the continuity equation?
The continuity equation states that mass is conserved in fluid flow: the mass flow rate entering a system equals the mass flow rate leaving it. For incompressible flow, it simplifies to A₁V₁ = A₂V₂.
What is the formula for the continuity equation?
General form: ρ₁A₁V₁ = ρ₂A₂V₂. For incompressible flow: A₁V₁ = A₂V₂ = Q, where Q is the volumetric flow rate.
When can I use the simplified form A₁V₁ = A₂V₂?
Use the simplified form when the fluid is incompressible — that means liquids (water, oil) and gases flowing at Mach < 0.3 (less than about 100 m/s in air). For faster gas flows, use the full form with density.
Why does velocity increase when a pipe narrows?
Because mass must be conserved. The same amount of fluid must pass through a smaller cross-section per unit time, so it must speed up. For circular pipes, halving the diameter quadruples the velocity: V₂/V₁ = (D₁/D₂)².
What is volumetric flow rate (Q)?
Volumetric flow rate is the volume of fluid passing through a cross-section per unit time, typically measured in m³/s or L/s. It equals area × velocity: Q = A × V.
What is mass flow rate (ṁ)?
Mass flow rate is the mass of fluid passing through a cross-section per unit time, measured in kg/s. It equals density × area × velocity: ṁ = ρ × A × V.
Does the continuity equation apply to open channels?
Yes, the continuity equation applies to any flow system where mass is conserved — including open channels, rivers, and partially filled pipes. The cross-sectional area used is the wetted area of the flow.
What is the relationship between velocity and area?
For incompressible flow, velocity is inversely proportional to cross-sectional area: V = Q / A. As area decreases, velocity increases, and vice versa.
How accurate is this calculator?
This calculator uses the exact continuity equation. Accuracy depends on the precision of the input values. For real piping systems with friction losses, additional calculations (e.g., Bernoulli with head loss) may be needed.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required.