Continuity Equation for Pipe Flow
The continuity equation is the conservation of mass applied to fluid flow. For a steady incompressible fluid, the amount of volume passing one section of a pipe each second must equal the amount passing every other section. If the pipe narrows, velocity rises; if the pipe widens, velocity falls.
This calculator solves the common two-section form of the equation. Enter the upstream area, upstream velocity, and downstream area, and it returns the downstream velocity plus the conserved volumetric flow rate.
What the Inputs Represent
- Input Area (A1): the cross-sectional area at the first pipe section.
- Input Velocity (v1): the average fluid velocity at the first pipe section.
- Output Area (A2): the cross-sectional area at the second pipe section.
The calculator first computes flow rate with $Q = A_1v_1$, then divides that same flow rate by the output area to find $v_2$.
The Formula
Example Calculation
Water flows at $2,\text{m/s}$ through a pipe with a cross-sectional area of $0.5,\text{m}^2$. The pipe narrows to $0.25,\text{m}^2$.
- Calculate flow rate: $Q = 0.5 \cdot 2 = 1.0,\text{m}^3/\text{s}$.
- Divide by the new area: $v_2 = 1.0 / 0.25 = 4,\text{m/s}$.
- Compare velocities: the output velocity is twice the input velocity because the output area is half as large.
The velocity in the narrow section becomes $4,\text{m/s}$. The flow rate stays $1.0,\text{m}^3/\text{s}$ throughout the pipe.
Assumptions and Limits
This simplified form assumes steady flow, a single inlet and outlet, and an incompressible fluid such as water or hydraulic oil. It ignores density changes, leakage, storage, and branch pipes. For compressible gases, use the mass-flow version $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ so density changes are included.