Continuity Equation Calculator
Solve the mass-conservation relation ρ₁·A₁·v₁ = ρ₂·A₂·v₂ for any of its six unknowns — inlet and exit density, cross-sectional area, or velocity — with step-by-step math, a live pipe schematic, and companion mass-flow metrics.
It is mass conservation, not energy conservation
Continuity says that whatever mass enters a control volume must leave it, unless mass is stored inside. For steady flow there is no storage, so ρ₁A₁v₁ = ρ₂A₂v₂ exactly. Note that this is a separate statement from Bernoulli's equation: continuity fixes the relationship between area, density, and speed, while Bernoulli fixes the relationship between pressure, speed, and height. Together they let you solve almost any one-dimensional duct-flow problem.
Smaller area means faster flow
If the density is constant, area and velocity are inversely proportional: A₁v₁ = A₂v₂. Halve the cross-section and the speed doubles. This is the principle behind a garden hose nozzle, a Venturi meter, the contraction in a wind tunnel, and the throat of a jet engine. Because velocity enters the dynamic pressure as v², a 2× contraction quadruples the dynamic pressure drop — which is exactly why a nozzle produces such a strong suction effect.
Density changes break the simple inverse rule
When the fluid is compressed or heated, ρ₂ ≠ ρ₁ and the area–velocity trade-off is no longer a simple inverse. In a gas turbine nozzle, density falls steeply through the passage, so the flow can accelerate even while the area grows — this is why a supersonic nozzle must diverge after the throat. In a heat exchanger, heating a gas at constant area forces the velocity to rise to carry the same mass flow. This calculator handles all of those cases through the full ρ₁A₁v₁ = ρ₂A₂v₂ form.
It sets the mass flow for everything else
Once you know ṁ = ρAv, you know how much fluid the system is moving, and that number drives almost every other calculation: pump and fan power, pressure drop through a pipe network, thrust in a jet engine, fuel-to-air ratio in a combustor, and the Reynolds number used to estimate friction. In a real design the mass flow is usually fixed by the mission, and continuity then dictates the area and velocity at every station along the machine.
How to Use This Calculator
Continuity links six quantities — ρ₁, A₁, v₁ at the inlet and ρ₂, A₂, v₂ at the exit — through the single statement ρ₁A₁v₁ = ρ₂A₂v₂. Give the tool five of them and it will find the sixth.
1. Choose what to solve for
Pick any one of the six variables from the toggle. The calculator rearranges ρ₁A₁v₁ = ρ₂A₂v₂ for whichever you choose and locks that field while you fill in the rest.
2. Set the two stations
Station 1 is the inlet (upstream) and station 2 is the exit (downstream). Each station carries its own density, area and velocity, so the same tool covers an incompressible pipe, a compressible nozzle, and a heat exchanger.
3. Use realistic presets
Density presets cover helium through mercury; area presets run from a 0.005 m² pipe to a 5 m² duct; velocity presets run from 0.5 m/s to 200 m/s. The "= ρ₁" preset on the exit density instantly makes the flow incompressible.
4. Read the result and verdict
The result card shows the solved value in SI plus the equivalent in km/h, knots and ft/s (for velocity), or kg/h and lb/h (for mass flow). A colour-coded verdict identifies whether the passage is converging, diverging, or constant-area.
5. Check the balance and the schematic
The balance bars compare ρ₁A₁v₁ and ρ₂A₂v₂ so you can see the conservation at a glance. The schematic draws the passage, scales the velocity arrows to speed, and prints the mass flow carried by each station.
6. Switch to Advanced for depth
Advanced mode adds an area-ratio vs. velocity-ratio chart, a +1% sensitivity check on each input, and companion metrics including volumetric flow at both stations, the area and velocity ratios, and the mass flux ρv.
What This Tool Offers
Built around the mass-conservation statement ρ₁A₁v₁ = ρ₂A₂v₂, the fundamental link between passage geometry, fluid density, and flow speed.
Six-way solving
Solve for any one of ρ₁, A₁, v₁, ρ₂, A₂ or v₂. The calculator rearranges the continuity relation automatically for whichever variable you choose.
Live passage schematic
A two-station pipe diagram with the passage drawn to scale with √A, velocity arrows scaled to speed, streamlines that follow the contraction or expansion, and a ṁ arrow scaled to mass flow.
Mass-flow balance bars
See at a glance that ρ₁A₁v₁ and ρ₂A₂v₂ are exactly equal, and how much of the total is carried by area, density, and speed at each station.
Passage-type verdict
Each result is classified as a converging passage, a diverging passage, or a constant-area duct, with a note when the density change dominates the behaviour.
Multi-unit hints
Each field shows the value converted to related units — kg/h and lb/h for mass flow, km/h, knots and ft/s for velocity, and diameter for a circular area.
Reference tables
A mass-flow unit converter, a ṁ = ρAv lookup grid, typical continuity values by situation, and a full variable reference table.
Reference Tables & Conversions
A mass-flow unit converter, a ṁ = ρ·A·v lookup grid, typical continuity values by situation, and the full variable list.
Mass flow rate unit converter
Mass Flow ṁ = ρ·A·v Lookup
Mass flow rate in kg/s for a range of cross-sectional areas and velocities, at five fluid densities. Computed from ṁ = ρAv.
| Area (m²) | v = 1 m/s | v = 5 m/s | v = 10 m/s | v = 20 m/s | v = 50 m/s | v = 100 m/s |
|---|
Values shown for air at ρ = 1.225 kg/m³. Multiply by ρ/1.225 for other fluids.
Typical Continuity Values by Situation
Representative two-station values. Mass flow is conserved exactly in every row; the areas, densities and velocities are order-of-magnitude estimates.
| Situation | ρ (kg/m³) | A₁ (m²) | v₁ (m/s) | A₂ (m²) | v₂ (m/s) | ṁ (kg/s) |
|---|
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| ρ₁, ρ₂ | Fluid density at stations 1 and 2 | 0.09 (H₂) – 13 546 (Hg) | kg/m³ |
| A₁, A₂ | Cross-sectional flow area at stations 1 and 2 | 10⁻⁴ – 10² | m² |
| v₁, v₂ | Mean flow velocity at stations 1 and 2 | 0.01 – 400 | m/s |
| ṁ | Mass flow rate (conserved) | 10⁻⁶ – 10⁴ | kg/s |
| Q | Volumetric flow rate A·v (not conserved if ρ changes) | 10⁻⁶ – 10³ | m³/s |
| ρv | Mass flux (mass velocity) | 0.1 – 10⁴ | kg/(m²·s) |
| D | Equivalent diameter of a circular duct, D = √(4A/π) | 0.01 – 10 | m |
What Does ρ₁A₁v₁ = ρ₂A₂v₂ Measure?
The continuity equation is the statement that mass cannot be created or destroyed inside a flow passage. Whatever mass enters the inlet each second must leave through the exit each second, unless the amount stored inside is changing. For steady flow there is no storage, so the mass flow rate ṁ = ρAv is the same at every station along the passage. This one idea, combined with Bernoulli's equation, is enough to analyse the vast majority of one-dimensional flow devices — nozzles, diffusers, ducts, pipes, intakes, and exhausts.
Continuity Equation Formula
The defining relationship between two stations on the same streamtube is:
Defining the mass flow rate ṁ = ρAv, the equation becomes simply:
For an incompressible flow, where ρ₁ = ρ₂, the densities cancel and the familiar area–velocity form remains:
Rearranged to solve for each of the six unknowns in the general case:
ρ₂ = ρ₁A₁v₁ / (A₂v₂) · A₂ = ρ₁A₁v₁ / (ρ₂v₂) · v₂ = ρ₁A₁v₁ / (ρ₂A₂)
Continuity Variables
| Symbol | Meaning | Notes |
|---|---|---|
| ρ | Fluid density | kg/m³; constant for incompressible flow, variable for gases |
| A | Cross-sectional flow area | m²; measured perpendicular to the flow direction |
| v | Mean flow velocity | m/s; averaged over the cross-section |
| ṁ | Mass flow rate | kg/s; conserved exactly in steady flow |
| Q | Volumetric flow rate | m³/s; Q = Av, conserved only if ρ is constant |
| ρv | Mass flux | kg/(m²·s); the momentum flux per unit velocity |
How Each Variable Affects the Result
The continuity equation is a product, so it is linear in each of its six variables: doubling any one of ρ, A, or v at a station doubles the mass flow through that station. That linearity is what makes the equation so easy to work with, and it is also what makes it so unforgiving — a 10% error in the measured area produces exactly a 10% error in the computed mass flow, with no square or exponential factors to soften it.
Area and velocity are the two variables engineers actually manipulate. Density is usually set by the fluid and the thermodynamic state. In an incompressible system the area–velocity trade-off is a pure inverse relationship, which is why a nozzle that halves the area doubles the speed. In a compressible system the density ratio intervenes, and a passage can accelerate the flow while expanding — the supersonic nozzle that every rocket engine uses to convert thermal energy into exhaust velocity.
The most important consequence of continuity is that it fixes the mass flow once and for all. Every station along the machine carries the same ṁ. That single number then determines the required pump or compressor power, the pressure drop, the heat-transfer duty, and the thrust. In design practice the mass flow is usually specified by the mission, and continuity is used backwards to find the area and velocity that will deliver it.
Continuity Calculation Examples
v₂ = ρ₁A₁v₁ / (ρ₂A₂) = 1.225 × 0.50 × 20 / (1.225 × 0.20) = 12.25 / 0.245 = 50.00 m/s
A₂ = ρ₁A₁v₁ / (ρ₂v₂) = 1.225 × 0.50 × 20 / (2.45 × 30) = 12.25 / 73.5 = 0.1667 m²
ṁ = ρAv = 998 × 0.00785 × 2 = 15.68 kg/s ≈ 56.4 t/h
D = √(4A/π) = √(4 × 0.50 / π) = 0.798 m ≈ 798 mm
Frequently Asked Questions
The form used here assumes steady flow (no accumulation of mass inside the control volume) and a well-defined streamtube with a single inlet and a single outlet. It does not require the flow to be incompressible, inviscid, or irrotational — those are Bernoulli's assumptions, not continuity's. That is why continuity remains valid in a boundary layer, in a viscous pipe, and in a shock wave, where Bernoulli does not. The only real requirements are that no mass crosses the streamtube walls and that the flow is steady, so ṁ is the same at every station.
Because the same mass of water must pass through every cross-section each second. If the area shrinks, the fluid has no choice but to speed up so that ρAv stays constant. This is the principle behind putting your thumb over the end of a garden hose: the area drops, the velocity rises, and the jet shoots further. It is also why a river narrows into rapids and why a Venturi meter can measure flow rate from the pressure drop caused by a deliberate contraction.
Whenever the density does not change appreciably between the two stations. For liquids this is almost always true — water is essentially incompressible below about 100 bar. For gases it is true when the Mach number is below roughly 0.3, where density changes are under about 5%. Above Mach 0.3 in air, or across a large pressure drop, or through a heat exchanger where the gas is heated significantly, you must keep the densities in the equation. The "= ρ₁" preset on the exit density sets the incompressible case in one click.
Only if the density is constant. Mass flow ṁ = ρAv is always conserved in steady flow, but volumetric flow Q = Av is conserved only when ρ₁ = ρ₂. In a compressible flow Q changes from station to station: Q₁ = ṁ/ρ₁ and Q₂ = ṁ/ρ₂. A gas heated at constant pressure expands, so its volumetric flow rises even though its mass flow is unchanged. This distinction matters enormously in HVAC and combustion work, where it is easy to confuse the two.
They are two independent statements that together solve most duct-flow problems. Continuity says the mass flow is the same at every station and relates area, density and speed. Bernoulli says the total mechanical energy is the same at every station (in ideal flow) and relates pressure, speed and height. Use continuity to find the velocity at a second station, then use Bernoulli to find the pressure there. Neither equation alone is enough; together they close the system. This calculator handles the continuity half — pair it with the Bernoulli calculator for the pressure half.
A streamtube is a bundle of streamlines forming a closed surface through which no fluid crosses. Because nothing enters or leaves through the sides, all the mass that goes in at station 1 must come out at station 2 — which is exactly what continuity states. Choosing a good streamtube is the first step in almost every fluid-mechanics problem: it converts a complicated three-dimensional flow into a one-dimensional problem with a well-defined inlet and outlet. In this calculator, the passage between station 1 and station 2 is that streamtube.
Apply the equation pairwise. Continuity guarantees that ṁ is the same at every station, so you can write ρ₁A₁v₁ = ρ₂A₂v₂ = ρ₃A₃v₃ = … along the whole passage. Solve for one station at a time, using the known ṁ as the link. If the passage branches, the rule becomes a sum: the mass flow entering the junction equals the sum of the mass flows leaving it, Σṁ_in = Σṁ_out. This calculator handles the simple two-station case, which is the building block for any network.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| rho | Fluid density | kg/m3 |
| A | Cross-sectional area | m2 |
| v | Average velocity | m/s |
What it means
The continuity equation expresses the conservation of mass for a fluid flowing in a conduit. For steady, one‑dimensional flow, it states that the mass flow rate is constant along the flow path: ρ₁ A₁ v₁ = ρ₂ A₂ v₂. If the fluid is incompressible (density constant), this simplifies to A₁ v₁ = A₂ v₂, meaning that the volumetric flow rate is constant. This principle is used to determine velocity changes when the cross‑sectional area changes, e.g., in a converging nozzle or a pipe of varying diameter. It is the basis for many flow measurement devices like venturi meters and orifice plates. The continuity equation is also applied to multi‑component mixtures and reactive flows, where it is extended with source terms. In engineering, it is essential for designing piping networks, sizing pumps, and analysing distribution systems. The equation is a fundamental law of fluid mechanics and is also used in other fields like acoustics and magnetohydrodynamics.
Worked example
Continuity Equation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| ρ | 1000 kg/m³ |
| A₁ | 0.05 m² |
| v₁ | 2 m/s |
| A₂ | 0.02 m² |
| Parameter | Value |
|---|---|
| A₁ | 0.1 m² |
| v₁ | 1 m/s |
| A₂ | 0.05 m² |
Common mistakes
- Density ρ: For incompressible flow, ρ is constant; for compressible, use the appropriate average density.
- Area and velocity: A and v are cross‑sectional area and average velocity normal to that area.
- Mass flow rate: The product ρAv gives mass flow rate (kg/s) – ensure units are consistent.
- Steady flow: The equation assumes steady, one‑dimensional flow; for unsteady or multi‑dimensional, use differential form.
- Different streams: For branched systems, apply the continuity equation at each junction.
Applications
The continuity equation, ρ₁A₁v₁ = ρ₂A₂v₂, embodies the conservation of mass in fluid flow. For incompressible flow, it simplifies to A₁v₁ = A₂v₂, stating that the volumetric flow rate is constant along a pipe. This principle is used to design piping networks, to determine velocities at different cross‑sections, and to size nozzles and diffusers. In chemical engineering, it is essential for designing reactors and separators where multiple streams merge or split. The continuity equation also underpins the analysis of compressible flows, such as in gas pipelines and supersonic nozzles, where density changes must be accounted for. Engineers apply this equation to ensure that mass is conserved in all process calculations, from simple water distribution to complex chemical plant simulations.
- Sizing of pipe systems and ductwork
- Design of mixing and splitting junctions
- Nozzle and diffuser design for flow acceleration/deceleration
- Compressible flow analysis (gas pipelines, jet engines)
- Process simulation and mass balance calculations
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