EngineeringFormulaFluid Mechanics
Continuity Equation Calculator
Engineering fluid-flow continuity calculator for incompressible two-section flow, using A1v1 = A2v2 to solve section area or velocity.
The engineering continuity equation expresses conservation of mass. For steady incompressible flow in a duct, pipe, or nozzle with one inlet and one outlet, it reduces to A1v1 = A2v2, often written in compact form as A1V1=A2V2, meaning volumetric flow rate stays constant from one section to the next. This calculator solves the two-section engineering form so you can find an unknown area or velocity when the other three quantities are known. It is an engineering fluid-mechanics tool, not the separate medical continuity-equation workflow used in echocardiography.
A1 · Internal flow area at the first section, measured normal to the local flow direction.
v1 · Average flow speed at section 1.
A2 · Internal flow area at the second section of the same flow path.
This page solves the engineering incompressible-flow continuity equation between two sections of the same streamtube. It is not a medical continuity-equation tool and it does not calculate pressure losses.
Solution
Enter the required values to calculate velocity at section 2.
v2 = A1 v1 / A2
Formula Sheet
- A1Area at Section 1
- v1Velocity at Section 1
- A2Area at Section 2
- v2Velocity at Section 2
- QVolumetric Flow Rate
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| A1 | Area at Section 1 | Internal flow area at the first section, measured normal to the local flow direction. | cm², m², in² |
| v1 | Velocity at Section 1 | Average flow speed at section 1. | m/s, ft/s, mph |
| A2 | Area at Section 2 | Internal flow area at the second section of the same flow path. | cm², m², in² |
| v2 | Velocity at Section 2 | Average flow speed at section 2. | m/s, ft/s, mph |
| Q | Volumetric Flow Rate | Implied volumetric flow rate, constant between sections for steady incompressible flow. | L/s, m³/h, US gpm |
How to Use This Calculator
- 01Choose which variable to solve for first: area or velocity at section 1 or section 2.
- 02Enter the other three section values using consistent units. Areas can differ in unit from velocities; the calculator converts them to coherent SI units automatically.
- 03Use section areas normal to the local flow direction. For circular pipes, use the internal flow area, not the outside pipe area. If you only know pipe diameter, use A = pi d² / 4 or the quick area table below before entering the value.
- 04Apply this page only to steady incompressible engineering flow between two sections of the same flow path.
- 05Do not use this page for the medical aortic-valve continuity equation. This calculator is specifically for engineering fluid flow.
How the Formula Works
Conservation of mass says the mass flow rate entering a control volume must equal the mass flow rate leaving it, provided there is no accumulation. In general form for one-dimensional flow that gives rho A v = constant. When density stays effectively constant, the rho terms cancel and the equation reduces to A v = constant, or A1v1 = A2v2 between two sections.
That inverse area-velocity relationship means fluid speeds up in contractions and slows down in expansions, as long as the incompressible assumption remains reasonable. The calculator solves one unknown from the other three by rearranging the same continuity relation, and it also reports the implied volumetric flow rate associated with the solved section values.
Use the implied flow rate as a consistency check. If section 1 and section 2 imply very different Q values in real measurements, the difference points to leakage, branching, compressibility, or bad measurement data rather than a different continuity equation.
Worked Example 01
Velocity increase through a pipe contraction
Known
- Area at Section 1 (A1): 0.05 m²
- Velocity at Section 1 (v1): 2 m/s
- Area at Section 2 (A2): 0.02 m²
Formula
v2 = A1 v1 / A2
Substitution
v2 = 0.05 x 2 / 0.02
Result
v2 = 5 m/s
Because section 2 has only 40% of the original area, the average fluid velocity rises to 5 m/s to conserve volumetric flow.
Worked Example 02
Required downstream area for a higher target velocity
Known
- Area at Section 1 (A1): 0.03 m²
- Velocity at Section 1 (v1): 4 m/s
- Velocity at Section 2 (v2): 7.5 m/s
Formula
A2 = A1 v1 / v2
Substitution
A2 = 0.03 x 4 / 7.5
Result
A2 = 0.016 m²
To accelerate the same incompressible flow from 4 m/s to 7.5 m/s, the second section must contract to about 0.016 square meters.
Worked Example 03
Section-1 area from section-2 measurements
Known
- Velocity at Section 1 (v1): 2 m/s
- Area at Section 2 (A2): 0.04 m²
- Velocity at Section 2 (v2): 6 m/s
Formula
A1 = A2 v2 / v1
Substitution
A1 = 0.04 x 6 / 2
Result
A1 = 0.12 m²
With section 2 running three times faster than section 1, the section-1 flow area must be three times larger for the same incompressible flow rate.
Applications
- 01Estimating velocity change through pipe contractions, expansions, nozzles, and diffusers
- 02Back-calculating the area needed to hit a target average velocity at a second section
- 03Checking whether two reported section conditions are consistent with incompressible flow continuity
- 04Screening whether a pipe contraction creates a high-velocity section that needs a pressure-loss or erosion check
Circular Pipe Area Reference for Common Internal Diameters
| Internal Diameter | Flow Area |
|---|---|
| 25 mm | 4.91 cm² |
| 50 mm | 19.63 cm² |
| 100 mm | 78.54 cm² |
| 150 mm | 176.71 cm² |
| 200 mm | 314.16 cm² |
Assumptions
- 01Flow is treated as steady and incompressible between the two chosen sections.
- 02Each section uses an average one-dimensional velocity rather than a detailed velocity profile.
- 03Both sections belong to the same engineering flow path, with no leakage, branching, or storage between them.
Where This Model Stops
- 01Not intended for strongly compressible gas-flow area-change problems where density varies significantly between sections.
- 02Does not calculate pressure drop, Reynolds number, losses, or Bernoulli energy changes; it only enforces continuity.
- 03Not for the echocardiography or aortic-valve continuity equation. This page is specifically for engineering fluid mechanics.
References
- [1]Conservation of Mass
NASA Glenn Research Center
Summarizes mass conservation for fluid flow and shows that rho A V stays constant, with the incompressible case reducing to area-velocity continuity.
- [2]Lecture 3: Continuity Equation
MIT Fluid Modules
Derives the continuity equation from mass conservation and identifies the incompressible special case.
- [3]Venturi Theory
NASA Glenn Research Center
Provides an intuitive engineering explanation that, for constant density, a smaller area corresponds to a higher velocity to conserve mass flow.
Frequently Asked Questions
When does A1v1 = A2v2 apply?
It applies to steady incompressible flow when the same fluid stream passes through two sections with no leakage or storage between them. If density changes materially, the more general form rho A v = constant should be used instead.
Should I use inside pipe area or outside pipe area?
Use the internal flow area normal to the fluid path. Outside pipe dimensions are not appropriate unless they have already been converted into the actual flow area inside the pipe.
Is this the same continuity equation used in medical echo or aortic valve calculations?
No. This page is for engineering fluid mechanics. Medical continuity-equation workflows use different measured quantities and clinical assumptions, so they should not be calculated on this page.
How is this different from the Flow Rate Calculator?
The Flow Rate Calculator solves Q = A v for one section. This Continuity Equation Calculator links two sections together with A1v1 = A2v2, so it is better for contractions, expansions, nozzles, and checking how velocity changes when area changes.