What is the flow rate formula?

Volume flow rate Q = A × v, where A is the cross-sectional area and v is the average flow velocity. For a circular pipe, A = π(d/2)². Units are m³/s in SI; common conversions to gpm or L/min depending on industry.

What is the continuity equation?

For incompressible flow, A₁v₁ = A₂v₂. If the pipe narrows, the velocity increases to maintain the same flow rate. This is why a stream of water narrows as it falls (gravity accelerates it) and why pinching a hose makes water shoot faster.

What's a typical flow rate for a kitchen faucet?

US code limits new kitchen faucets to 2.2 gpm (8.3 L/min, 1.4 × 10⁻⁴ m³/s). Bathroom faucets: 1.5 gpm. Older faucets without aerators can flow 4+ gpm. Low-flow models save water but feel less powerful.

How do I convert between flow rate units?

1 m³/s = 15,850 gpm (US) = 60,000 L/min. 1 gpm = 6.309 × 10⁻⁵ m³/s. 1 L/min = 1.667 × 10⁻⁵ m³/s. 1 cfm = 4.719 × 10⁻⁴ m³/s. Specialized units like bbl/d (petroleum) and m³/hr (water) are also common.

What's the Reynolds number?

Re = ρvd/μ characterizes flow regime. Re 4,000: turbulent (chaotic). Most engineering flows are turbulent. Water in residential plumbing typically has Re ~ 10,000-100,000.

Why does my faucet stream narrow as it falls?

Gravity accelerates falling water (v increases). By continuity (Av = constant), area must decrease. The stream narrows continuously until surface tension and air drag overcome smooth flow. Often the stream eventually breaks into droplets (Plateau-Rayleigh instability).

Does flow rate depend on pressure?

Indirectly. Higher pressure drives faster flow through a pipe, but the relationship is set by Darcy-Weisbach or Hazen-Williams equations including friction. For a given velocity (set by pump), flow rate depends only on cross-sectional area. Pressure determines velocity, which combined with A gives Q.

How is river discharge measured?

USGS uses the velocity-area method: measure depth at points across the river, multiply by local velocity (from current meter), sum to get total Q = ΣAᵢvᵢ. Real rivers are non-uniform, so multiple measurement points are needed. Acoustic Doppler Current Profilers (ADCP) automate this.

Flow Rate Calculator

Calculate the volume flow rate of a fluid through a pipe or channel. Uses the continuity equation Q = A × v, where A is the cross-sectional area and v is the flow velocity.

Volume flow rate measures how much fluid passes through a cross-section per unit time. The fundamental formula Q = A × v relates flow rate to the pipe's cross-sectional area and the fluid's average velocity. This relationship — and its incompressible-flow cousin A₁v₁ = A₂v₂ (continuity) — underlies virtually all pipe-system design, from household plumbing to oil pipelines to the human cardiovascular system.

The continuity equation explains everyday observations: water speeds up when you pinch a garden hose, blood flows faster in capillaries despite their narrow individual diameter (total cross-section actually expands), and your shower stream narrows as it falls because gravity accelerates it. In each case, the product Av stays constant for incompressible flow.

Flow rate is reported in different units depending on industry: m³/s and L/s in SI engineering, gallons per minute (gpm) in US plumbing, cubic feet per second (cfs) in hydrology, barrels per day (bbl/d) in petroleum, and liters per minute (L/min) in medicine. The calculator works in SI but the math converts to any system.

Common applications: HVAC duct sizing, plumbing design, irrigation, industrial process piping, oil and gas pipelines, river discharge measurement, cardiovascular hemodynamics, and any fluid-handling system.

Inputs

Pipe Diameter (m)

Flow Velocity (m/s)

Results

Flow Rate

0.003927 m³/s

Flow Rate (L/min)

235.62 L/min

Flow Rate (GPM)

62.24 GPM

Flow Rate Results

Parameter	Value
Pipe Diameter	0.05 m (50.0 mm)
Pipe Radius	0.025000 m
Cross-sectional Area	0.001963 m² (19.6350 cm²)
Flow Velocity	2 m/s
Flow Rate (m³/s)	0.003927 m³/s
Flow Rate (L/min)	235.6194 L/min
Flow Rate (GPM)	62.2440 GPM
Mass Flow (water)	3.9270 kg/s
Formula	Q = A × v

Last updated: May 29, 2026

Formula

**Volume flow rate (continuity for incompressible flow):** Q = A × v Where: - Q = volume flow rate (m³/s) - A = cross-sectional area (m²) - v = average flow velocity (m/s) **For a circular pipe:** A = π × (d/2)² = π × d² / 4 Where d = diameter. **Worked example: garden hose** Diameter 1.27 cm (0.5 inch), water flowing at 2 m/s. A = π × (0.00635)² = π × 4.03 × 10⁻⁵ = 1.27 × 10⁻⁴ m² Q = 1.27 × 10⁻⁴ × 2 = 2.53 × 10⁻⁴ m³/s Convert to gpm: 2.53 × 10⁻⁴ × 60 × 264.17 ≈ 4.0 gpm Convert to L/min: 2.53 × 10⁻⁴ × 60 × 1000 ≈ 15.2 L/min **Continuity equation (incompressible):** A₁ × v₁ = A₂ × v₂ If pipe diameter shrinks by half, area shrinks by ¼, velocity quadruples. **Worked example: Venturi** Wide section: A₁ = 0.01 m², v₁ = 2 m/s → Q = 0.02 m³/s. Throat: A₂ = 0.0025 m² (¼ wide section). v₂ = Q/A₂ = 0.02 / 0.0025 = 8 m/s. Velocity quadruples; by Bernoulli, pressure drops. **Mass flow rate:** ṁ = ρ × Q = ρ × A × v Where ρ = fluid density (kg/m³). Units: kg/s. For water (1000 kg/m³) at Q = 0.001 m³/s: ṁ = 1 kg/s. **Common flow rate units:** | Unit | In m³/s | |---|---| | 1 m³/s | 1 | | 1 L/s | 10⁻³ | | 1 L/min | 1.667 × 10⁻⁵ | | 1 gal/min (US) | 6.309 × 10⁻⁵ | | 1 gal/min (UK) | 7.576 × 10⁻⁵ | | 1 cfm | 4.719 × 10⁻⁴ | | 1 cfs | 0.02832 | | 1 m³/hr | 2.778 × 10⁻⁴ | | 1 bbl/d (oil) | 1.84 × 10⁻⁶ | **Reynolds number — laminar vs turbulent:** Re = ρvd/μ = vd/ν Where μ = dynamic viscosity, ν = kinematic viscosity. - Re < 2,300: laminar (smooth, parallel layers). - 2,300 < Re < 4,000: transitional. - Re > 4,000: turbulent (eddies, mixing). Water in a 5 cm pipe at 2 m/s: Re = 2 × 0.05 / 10⁻⁶ ≈ 100,000 → fully turbulent. **Average vs maximum velocity:** For laminar flow in a pipe (Poiseuille): v_avg = v_max / 2. For turbulent flow: v_avg ≈ 0.8 × v_max (depends on Re). Most calculations use average velocity (Q/A). **Open channel flow:** For rivers and partly-full channels: Q = A × v Where A = wetted cross-section area, v = average velocity. Manning's equation gives v from channel properties. **Cardiac output (medicine):** Cardiac output = heart rate × stroke volume For 70 bpm × 70 mL: CO = 4.9 L/min = 8.17 × 10⁻⁵ m³/s. Aortic cross-section ~3 cm² → average aortic velocity ~0.27 m/s. **Flow regimes (fluid mechanics):** | Regime | Range | Behavior | |---|---|---| | Stokes flow | Re < 1 | viscous-dominated | | Laminar | 1 < Re < 2300 | smooth | | Transitional | 2300-4000 | unstable | | Turbulent | Re > 4000 | mixed, chaotic | | Hypersonic | Mach > 5 | shock-dominated | **Pressure drop in pipes (Darcy-Weisbach):** ΔP = f × (L/d) × ½ρv² Where f = friction factor (from Moody chart or Colebrook equation). Doubling velocity quadruples pressure drop. **Worked HVAC example:** Duct cross-section 0.5 × 0.5 m = 0.25 m², air velocity 5 m/s. Q = 0.25 × 5 = 1.25 m³/s = 75 m³/min = 2,650 cfm **For typical residential**: 100-400 cfm per room. Sufficient for a typical home.

How to use this calculator

Enter pipe diameter in meters.
Enter average flow velocity in m/s.
Calculator returns volume flow rate Q in m³/s.
Convert to gpm: × 15,850 (US gallons per minute).
Convert to L/min: × 60,000.
For non-circular shapes, calculate area directly (A = w × h for rectangles).

Worked examples

Shower head flow rate

**Scenario:** Shower head has 80 small holes each 1.5 mm diameter. Water exits at 5 m/s. Total flow? **Calculation:** Per hole: A = π × (0.00075)² ≈ 1.77 × 10⁻⁶ m². Total area: 80 × A ≈ 1.41 × 10⁻⁴ m². Q = 1.41 × 10⁻⁴ × 5 = 7.07 × 10⁻⁴ m³/s = 42.4 L/min = 11.2 gpm. **Result:** ~11 gpm — high-flow showerhead (US efficiency standard limits to 2.5 gpm). Adjusting to 2.5 gpm requires reducing total area to ~3 × 10⁻⁵ m² (fewer or smaller holes), or reducing velocity (lower water pressure).

River discharge

**Scenario:** Stream 5 m wide, 1 m deep, flowing at 0.8 m/s. Discharge? **Calculation:** A = 5 × 1 = 5 m². Q = 5 × 0.8 = 4 m³/s = 4000 L/s. **Result:** ~4 cumecs (cubic meters per second). Typical small river. For comparison: Mississippi River discharge averages ~16,000 m³/s; Amazon ~209,000 m³/s. USGS uses river velocity meters and area measurements to compute discharge for flood prediction.

Venturi flow meter

**Scenario:** Pipe narrows from 10 cm to 5 cm diameter. Inlet velocity 1 m/s. Throat velocity? **Calculation:** A₁/A₂ = (10/5)² = 4. By continuity: v₂ = 4 × v₁ = 4 m/s. **Result:** Velocity quadruples in the throat. Pressure drops by ½ρ(v₂² − v₁²) = 500 × (16 − 1) = 7,500 Pa. This pressure difference is measured to compute flow rate — basis of Venturi flow meters used in industry.

When to use this calculator

**Use flow rate calculations for:**

- **HVAC design**: duct sizing, fan selection. - **Plumbing**: pipe sizing for fixtures. - **Irrigation**: sprinkler and drip system design. - **Pipeline engineering**: oil, gas, water transmission. - **River hydrology**: discharge measurement, flood prediction. - **Industrial process**: chemical reactors, heat exchangers. - **Medical**: cardiac output, IV drip rates, ventilator flow. - **Aerospace**: fuel system design.

**Pipe sizing rules of thumb:**

- **Residential water**: 1-3 m/s typical; max 5 m/s to avoid noise and erosion. - **HVAC supply ducts**: 5-12 m/s. - **HVAC return ducts**: 4-8 m/s. - **Industrial liquid**: 1-3 m/s pumped, 1-2 m/s gravity. - **Gas pipelines**: 5-15 m/s.

Too slow: oversized pipe, wasted material. Too fast: noise, erosion, high pressure loss.

**Continuity equation applications:**

- **Pinching garden hose**: same Q, smaller A → larger v. - **Capillary network**: total capillary cross-section vastly larger than artery, despite individual capillaries being tiny. - **River narrowing**: speed up at constrictions. - **Faucet stream**: narrows as it falls (gravity accelerates → v increases → A decreases at constant Q).

**Common applications:**

- **Plumbing code**: fixture units determine required pipe size. - **Manufacturing**: cooling water systems. - **Mining**: dewatering pumps. - **Power generation**: cooling tower flow, steam piping. - **Agriculture**: drip vs sprinkler irrigation efficiency. - **Medicine**: IV infusion rates (typically mL/hr), respiratory flow.

**Mass vs volume flow:**

For incompressible (liquids): use volume flow Q. For compressible (gases): use mass flow ṁ — volume changes with pressure and temperature.

Gases at constant ṁ: Q changes with P and T. Liquids: Q ≈ constant (density barely changes).

**Reynolds number significance:**

- **Laminar (Re < 2300)**: predictable, low loss, but heat/mass transfer poor. - **Turbulent (Re > 4000)**: better mixing, higher heat transfer, more pressure loss.

Most engineering flow is turbulent. Examples: aircraft wings, car airflow, building HVAC, blood in aorta.

**Pressure drop and pumping power:**

ΔP × Q = power required (W)

Example: pump 0.001 m³/s against 100,000 Pa pressure → 100 W pump.

For pipe systems, choose pipe size to balance: - Larger pipe: lower velocity, lower pressure drop, higher capital cost. - Smaller pipe: higher velocity, higher pressure drop and pumping cost.

**Software:**

- **EPANET**: water distribution network analysis. - **Pipe-Flo**: industrial fluid system design. - **AFT Fathom**: process and HVAC. - **ANSYS Fluent**: CFD for complex flows.

**Pitfalls:**

- **Mixing units**: gpm vs L/min vs m³/s. Convert carefully. - **Average vs peak velocity**: most calculations use average. - **Density assumptions**: gases vary, liquids don't. - **Forgetting continuity**: A₁v₁ = A₂v₂ always for incompressible. - **Ignoring viscosity**: laminar vs turbulent affects pressure drop. - **Sticking to inches in international work**: confuses units.

Common mistakes to avoid

Mixing units (gpm vs L/min vs m³/s).
Using diameter instead of radius (area = π × r², not π × d²).
Confusing average velocity with maximum velocity (factor 2 for laminar).
Ignoring continuity when pipe changes size.
Forgetting density when calculating mass flow rate.
Assuming flow is laminar when it's turbulent (or vice versa).
Using volumetric flow for compressible gases (need mass flow at varying P, T).
Designing for average instead of peak demand.

Frequently Asked Questions

Sources & further reading

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