Continuity Equation Calculator
How fast a fluid speeds up when the pipe narrows, with the flow rate, the areas and the pressure it trades away.
Results
What this tool does
Water does not pile up inside a pipe. Whatever volume enters one end has to leave the other, so when the pipe gets narrower the same flow has to move faster — and because area goes with the square of the diameter, the speeding up is dramatic. Halving the bore quadruples the velocity. Give the two diameters and one speed and this page gives the other, along with the flow rate they share and the static pressure the fluid surrenders to go faster.
Formula
A₁ v₁ = A₂ v₂ · A = π d² ÷ 4 (incompressible fluid, no leaks)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
d1 | Diameter before | mm |
v1 | Speed before | m/s |
d2 | Diameter after | mm |
V2 | Speed after | m/s |
Q | Flow rate | m³/s |
QL | Flow rate in litres | L/min |
A1 | Area before | m² |
A2 | Area after | m² |
AR | How many times smaller the opening is | x |
SR | How many times faster it goes | x |
DP | Static pressure given up for the speed | Pa |
Worked example
- Diameter before100 mm
- Speed before2 m/s
- Diameter after50 mm
- Speed after8.000000 m/s
- Flow rate0.01570796 m³/s
- Flow rate in litres942.4778 L/min
- Area before0.00785398 m²
- Area after0.00196350 m²
- How many times smaller the opening is4.000000 x
- How many times faster it goes4.000000 x
- Static pressure given up for the speed30,000.00 Pa
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
Frequently asked questions
Why does halving the diameter make it four times faster?
Because what has to stay constant is the area, not the width, and area goes with the square of the diameter. Halve the diameter and the opening is a quarter the size, so the same water has to pass through it four times faster. This is why a thumb over a hose end sends the jet so far, and why a small restriction anywhere in a pipe run does far more damage to the pressure than its length would suggest.
What is the pressure drop figure?
How much static pressure the water gives up to buy that extra speed, taken straight from Bernoulli's equation for water at a thousand kilograms per cubic metre. It is not a loss — if the pipe widens back out, the pressure comes back. Real losses from friction and turbulence are a separate matter and need the Darcy-Weisbach calculation. Note that the figure grows with the square of the new speed, so a severe narrowing can drop the pressure enough to boil the water cold, which is what cavitation is.