Flow Rate Calculator

Calculate volumetric flow rate, fluid velocity, or pipe area using Q = A·v and the continuity equation.

💧 Fluid Flow📐 Q = Av🔧 Pipe Flow
Area (A) m²
Velocity (v) m/s

Circle area: A = π·r² = π·d²/4 (d in meters)

⚠️ Enter valid positive numbers.

What Is Flow Rate?

Volumetric flow rate Q (m³/s) is the volume of fluid passing a cross-section per unit time: Q = A·v, where A is the cross-sectional area (m²) and v is the average fluid velocity (m/s). The continuity equation for incompressible fluids (constant density) states A₁v₁ = A₂v₂ — what flows in must flow out, so narrower pipes mean faster flow.

Flow rate is measured in various units: m³/s (SI), L/s (1 L/s = 0.001 m³/s), L/min, GPM (gallons per minute; 1 GPM = 6.31×10⁻⁵ m³/s), and CFM (cubic feet per minute; 1 CFM = 4.72×10⁻⁴ m³/s). For a circular pipe of diameter d: A = π(d/2)² = πd²/4. The flow in a 100 mm (4-inch) pipe at 2 m/s: Q = π×(0.05)²×2 = 0.0157 m³/s = 15.7 L/s.

The continuity equation A₁v₁ = A₂v₂ is a direct consequence of mass conservation. Garden hose thumb: decreasing A at the tip increases v dramatically. Blood flow: healthy arteries (wide A) have low v; stenotic (narrowed) arteries have high v, causing disturbed flow patterns detectable by Doppler ultrasound.

Mass flow rate (ṁ) = ρ·Q = ρ·A·v (kg/s). For compressible fluids (gases), mass flow is conserved but volume flow varies with density: ρ₁A₁v₁ = ρ₂A₂v₂. This is essential for aircraft engine inlets and combustion system design.

Formula Reference Table

Solve ForFormulaNotes
Flow rateQ = A · vm³/s; 1 m³/s = 1000 L/s
Velocity from Qv = Q / AAverage flow velocity
Area from QA = Q / vCross-section needed
ContinuityA₁v₁ = A₂v₂Mass conservation; incompressible
Circular pipeA = π(d/2)² = πd²/4d = diameter
Mass flow rateṁ = ρ · Qkg/s; ρ = density

3 Worked Examples

Example 1
Pipe Flow Rate

Water pipe: diameter 100 mm, velocity 3 m/s.

  • A = π × (0.05)² = 7.854×10⁻³ m²
  • Q = A × v = 7.854×10⁻³ × 3 = 0.02356 m³/s
  • = 23.56 L/s = 1,413 L/min
✓ Q = 23.6 L/s (1,413 L/min)
Example 2
Continuity — Pipe Narrows

Water at 2 m/s in 200 mm pipe. Narrows to 50 mm. Find new velocity.

  • A₁ = π×(0.1)² = 0.03142 m²
  • A₂ = π×(0.025)² = 1.963×10⁻³ m²
  • v₂ = A₁v₁/A₂ = 0.03142×2 / 1.963×10⁻³ = 32 m/s
✓ v₂ = 32 m/s (16× faster in ¼ diameter pipe)
Example 3
River Flow Rate

River channel 8 m wide, average depth 1.5 m, average velocity 0.4 m/s.

  • A = width × depth = 8 × 1.5 = 12 m²
  • Q = A × v = 12 × 0.4 = 4.8 m³/s
  • = 4,800 L/s = 288,000 L/min — significant flow rate
✓ Q = 4.8 m³/s (4,800 L/s)

Real-World Applications

🚰
Plumbing Design
Water supply pipe sizing uses Q = Av to ensure adequate flow at safe velocities (0.5–3 m/s for domestic water to minimize noise and erosion). Undersized pipes cause pressure drops and insufficient flow to fixtures.
🏗️
HVAC Air Handling
Duct sizing balances flow rate (m³/s of conditioned air) with velocity limits. High velocity increases noise and pressure drop. Wider ducts at lower velocity reduce energy waste in large commercial buildings.
⚕️
Medical Blood Flow
Echocardiography measures blood flow (Q = A × v) through heart valves. Stroke volume = Q × time = A_valve × v_mean × t_systole. Stenotic valves have higher v (continuity), detectable by Doppler ultrasound as a high-velocity jet.
🌊
Hydrology
River discharge Q = Av is fundamental to flood forecasting. Stream gauging stations measure water level (which gives A via a rating curve) and velocity (electromagnetic or acoustic methods) to calculate Q in real time.
⚙️
Pump & Turbine Selection
Flow rate Q and head (pressure) determine pump/turbine selection. Pump affinity laws: Q ∝ RPM, head ∝ RPM², power ∝ RPM³. Engineers match pump curves (Q vs. head) to system curves to find operating point.

Common Mistakes to Avoid

⚠️
Using diameter instead of radius for area

A = π·r² = π(d/2)² = πd²/4. A 100 mm pipe has r = 50 mm = 0.05 m → A = π×0.0025 = 0.00785 m². Using diameter directly gives A that is 4× too large.

⚠️
Confusing flow rate with velocity

Q (m³/s) is total volume flow; v (m/s) is local velocity. A large-diameter pipe at low v can carry much more Q than a narrow pipe at high v.

⚠️
Wrong unit conversions

1 m³/s = 1,000 L/s = 16,667 L/min = 264 GPM. Always convert to consistent SI units (m³/s and m²) before using Q = Av.

⚠️
Assuming uniform velocity profile

Q = Av assumes a uniform (plug) velocity profile. Real pipe flow has parabolic (laminar) or turbulent profiles — v is the cross-sectional average. Actual centerline velocity is 2× average for laminar flow.

⚠️
Applying incompressible formula to gases at high speed

For compressible flow (gas at M > 0.3), density changes with velocity. Use mass flow rate ṁ = ρAv = constant rather than Q = Av.

Frequently Asked Questions

What is the difference between flow rate and velocity? ▾
Flow rate Q (m³/s) = total volume passing per second through the entire cross-section. Velocity v (m/s) = speed at a specific point. Q = A × v_average. A large river can have slow velocity but huge flow rate due to its large cross-sectional area.
How does continuity equation explain garden hose thumb trick? ▾
A₁v₁ = A₂v₂. Thumb restricts the opening area (A₂ << A₁). Since Q = constant, v₂ = A₁v₁/A₂ >> v₁. The flow velocity dramatically increases at the restriction, creating a high-velocity jet with more range. The total flow rate (volume/time) is unchanged.
What is the Reynolds number and why does it matter? ▾
Re = ρvD/μ. Below 2,300: laminar flow (smooth, orderly layers). Above 4,000: turbulent (chaotic mixing). Flow pattern affects pressure drop, heat transfer, and mixing. Q = Av gives average velocity, but the actual velocity profile differs significantly between laminar and turbulent regimes.
How is flow measured in practice? ▾
Mechanical: rotameters (float in tapered tube), turbine meters. Differential pressure: Venturi meters, orifice plates (ΔP → Q via Bernoulli). Electromagnetic: conductive fluid in magnetic field creates EMF proportional to v. Ultrasonic: Doppler or transit-time methods. Thermal mass flow meters for gases.
What is a flow coefficient (Cv)? ▾
Cv (US gallons per minute at 1 psi pressure drop) rates valve capacity. Q = Cv√(ΔP/SG) where SG = specific gravity and ΔP in psi. Engineers select control valves by matching required Q to valve Cv curves. Metric equivalent: Kv = 0.865 × Cv.
How do engineers size water supply pipes? ▾
Select v = 1–2 m/s (low noise, no erosion). Find required Q from fixture units or demand analysis. A = Q/v. d = √(4A/π). Round up to nearest standard pipe size. Check pressure drop at this flow rate using Darcy-Weisbach or Hazen-Williams equations.
What is hydraulic diameter? ▾
For non-circular ducts, hydraulic diameter D_h = 4A/P (P = wetted perimeter). A 100×200 mm rectangular duct: A = 0.02 m², P = 0.6 m, D_h = 4×0.02/0.6 = 0.133 m. Using D_h in place of d in pipe flow formulas gives reasonable approximations for non-circular cross-sections.
How does blood flow rate change in the body? ▾
Total cardiac output ≈ 5 L/min at rest; up to 20–25 L/min during intense exercise. The aorta (d ≈ 25 mm) carries all output. Aortic velocity = Q/A ≈ 5×10⁻³/π×(0.0125)² ≈ 1.0 m/s. By continuity, the billions of capillaries (d ≈ 8 μm) have very low individual velocity (≈0.001 m/s) but enormous total cross-sectional area ≈ 4,500 cm².

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Interpretation: This relationship connects pressure, velocity, density, viscosity, geometry or transport in a fluid system. Assumption: Check whether flow is steady, incompressible, laminar, fully developed or one-dimensional. Reynolds and Mach regimes determine whether simplified formulas are valid.

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