Hong Life CalculatorsUnderstanding the continuity equation and pipe flow velocity calculator — concept and principle
The continuity equation says that the volume of liquid entering a pipe each second must leave it each second. From a flow rate and two inside diameters, the calculator gives the mean velocity in each section and the flow in m³/h. It is used to size pipes, check that velocities stay in a sensible range, and find the speed at a reducer, nozzle or narrow section.
How it is calculated
- Cross-sectional area of a round pipe: A = π(d/2)² = πd²/4, with d converted from mm to m.
- Flow converted from L/s to m³/s: Q ÷ 1000.
- Mean velocity in each section: v = Q ÷ A (m/s).
- Flow in m³/h: Q (L/s) × 3.6, because 1 L/s = 3,600 L/h = 3.6 m³/h.
Because the same Q passes both sections, A₁v₁ = A₂v₂, so v₂ = v₁ × (d₁/d₂)².
The principle
Mass is conserved. In steady flow nothing accumulates inside the pipe, so the mass passing any cross-section per second, ρAv, is the same everywhere. For a liquid, density is practically constant, which leaves Av = constant. Area depends on the square of diameter, so velocity scales with 1/d²: halve the diameter and the area becomes one quarter, so the velocity becomes four times larger. The velocity here is the average over the section; the real profile is faster in the centre and zero at the wall.
Worked example
- Inputs: Q = 5 L/s, d₁ = 50 mm, d₂ = 25 mm.
- Q = 5 ÷ 1000 = 0.005 m³/s.
- A₁ = π × 0.025² = 0.0019635 m²; v₁ = 0.005 ÷ 0.0019635 = 2.546 m/s (shown as 2.546479089).
- A₂ = π × 0.0125² = 0.00049087 m²; v₂ = 0.005 ÷ 0.00049087 = 10.186 m/s (shown as 10.18591636), exactly 4 × v₁.
- Flow: 5 × 3.6 = 18 m³/h.
| Diameter 1 (mm) | Velocity 1 (m/s) |
|---|---|
| 12.5 | 40.74366543 |
| 25 | 10.18591636 |
| 37.5 | 4.527073937 |
| 50 | 2.546479089 |
| 62.5 | 1.629746617 |
| 75 | 1.131768484 |
| 100 | 0.6366197724 |
How to read the result
Many design guides keep water velocity in pressure pipes roughly between 1 and 3 m/s: lower speeds mean larger, costlier pipes and possible sediment, higher speeds mean more friction loss, noise, erosion and stronger water hammer. In the example, 2.5 m/s is reasonable for a 50 mm pipe, while 10 m/s in the 25 mm section is only acceptable over a very short nozzle or restriction.
Common mistakes
- Using the outer or nominal pipe size instead of the actual inner diameter.
- Forgetting to square the diameter ratio; halving d multiplies v by 4, not 2.
- Mixing L/s, L/min and m³/h; 1 L/s = 60 L/min = 3.6 m³/h.
- Applying Av = constant to a gas with large pressure changes, where density varies.
Frequently asked questions
How do I find the pipe diameter for a target velocity?
Rearrange to d = √(4Q ÷ (πv)). For 5 L/s at 2 m/s, d = √(4 × 0.005 ÷ (π × 2)) ≈ 0.056 m, about 56 mm.
Does the continuity equation work for branching pipes?
Yes. At a junction the flow in equals the sum of the flows out: Q₁ = Q₂ + Q₃.
Why does putting a thumb over a hose make the water go further?
Reducing the outlet area raises the exit velocity for a similar flow, so the jet travels further, although the restriction also reduces the flow somewhat.
Sources: F. M. White, Fluid Mechanics, McGraw-Hill (conservation of mass) · B. R. Munson et al., Fundamentals of Fluid Mechanics, Wiley