Flow Rate Calculator

Q = A · v = (π · d² / 4) · v

Q

Volumetric flow rate: the volume of fluid crossing a section of the pipe each second. It is a volume per unit time, not a mass per unit time and not a speed — the mass flow rate needs the fluid's density on top of this, and the speed is already one of your inputs. The panel prints it in litres per second, cubic metres per hour, cfm and US gallons per minute, all four from this one number

A

The pipe's internal cross-sectional area, π·d²/4 for a round bore. Use the internal diameter, not the outside diameter and not the nominal size: a pipe sold as DN50 has a 50 mm nominal bore but a larger outside diameter once the wall thickness is added, and it is the hole the fluid goes through that sets the area. The panel prints it in square millimetres because that is the unit the answer reads best in — a 50 mm bore is a familiar 1963.5 mm², while the same area in square metres is 0.0019635, four digits all after the decimal point

d

The pipe's internal diameter, in millimetres by default; centimetres, metres, inches and feet are in the same box. Diameter enters the formula squared, which is the single most useful thing to remember about this page: doubling the bore does not double the flow rate, it quadruples it. Go from 50 mm to 100 mm at the same velocity and you go from 3.927 litres per second to 15.708, four times as much, because the area went up by four

v

The average velocity of the fluid along the pipe, in metres per second by default, with km/h and ft/s available. It is the mean over the cross-section rather than the speed at the centreline, which is the higher one in real pipe flow. This is the number the page takes as given: the flow rate follows from it, so a velocity you picked out of a design range produces a flow rate with exactly that status. Nothing here checks the fluid, the wall material, the pressure drop or the noise

π / 4

The constant that turns a diameter into an area. The area of a circle is π·r², and writing the radius as half the diameter gives π·d²/4 — the same thing, arranged so you can use the bore directly in millimetres without halving it first. It is where the square in d² comes from, and it is why small changes in pipe size matter more than they look: a 20% larger bore carries 44% more flow at the same velocity

1 L/s

One litre per second is 3.6 cubic metres per hour, 2.119 cfm and 15.850 US gallons per minute — the four conversion factors the panel applies. They are exact except for the gallon, which is defined as 3.785411784 litres; the cfm factor follows from a cubic foot being 0.3048 m cubed exactly. The second table is built from these same factors, so the printed table and the calculator cannot disagree

Use it whenever you have a pipe and a velocity and want to know how much fluid it carries: sizing a pump for a known line, checking what an existing line can deliver, working out how long a tank will take to fill, or converting a velocity your instrument reports into a flow rate your meter or your customer speaks in. It also answers the reverse question in the form people actually ask it — pick a velocity off a design chart, and the page tells you the flow rate that goes with each standard pipe size. The second table does exactly that for twelve nominal bores at 2 m/s, which is the middle of the usual range for pumped water. What it does not do is choose the velocity for you. Acceptable velocity depends on the fluid (water, steam, slurry and compressed air sit in different bands), on whether noise and erosion matter, and on how much pressure drop you are willing to pay for, and none of those are inputs here. Come with a velocity you can defend, and this page turns it into a flow rate.

1. The default: a 50 mm bore at 2 m/s

   1. Internal diameter 50 mm, velocity 2 m/s
   2. Halve the diameter and square it: 0.025² = 0.000625 m²
   3. Multiply by π: 0.000625 × 3.14159 = 0.0019635 m² — that is the cross-sectional area, printed as 1963.5 mm²
   4. Multiply by the velocity: 0.0019635 × 2 = 0.003927 m³/s
   5. Shift the decimal three places for litres: 3.927 L/s
   6. Convert: 3.927 × 3.6 = 14.137 m³/h, 3.927 × 2.119 = 8.321 cfm, 3.927 × 15.850 = 62.244 gpm

   This is the row a reader can check in their head, and it is the one the second table repeats at DN50. The area is worth a second look because it is where the whole answer comes from: 1963.5 mm² is a number you can sanity-check against the pipe in your hand — a 5 cm hole has a cross-section a bit under 20 cm², which is 2000 mm². Everything after that is one multiplication, and the four conversions at the end are the same number read four ways. If you only ever remember one fact from this page, make it the square: the flow rate goes with the square of the diameter, so the four unit conversions above are bookkeeping while a change in bore is a change in the answer.

2. The same velocity through a 100 mm bore

   1. Internal diameter 100 mm — four times the area of the 50 mm pipe, since 100² is 4 × 50²
   2. Area = π × 0.1² / 4 = 0.007854 m², printed as 7854 mm²
   3. At 1.5 m/s the flow is 0.007854 × 1.5 = 0.011781 m³/s
   4. That is 11.781 L/s, 42.412 m³/h, 24.962 cfm and 186.732 gpm

   Two effects pull in opposite directions here and the arithmetic settles it. The bore went up by a factor of two, which multiplies the area — and therefore the flow — by four. The velocity went down from 2 to 1.5 m/s, which multiplies it by 0.75. Four times 0.75 is three, and 3.927 × 3 = 11.781. That kind of estimate is worth doing before you trust any calculator: a factor-of-two change in bore always dominates a modest change in velocity, which is the practical reason pipe sizing is mostly about diameter. The area landing on 7854 mm² is a small gift — it is 2500π, so you can verify the π·d²/4 step without reaching for a calculator.

3. A small bore at a brisk velocity

   1. Internal diameter 25 mm — a quarter of the 50 mm pipe's area, since 25² is 50²/4
   2. Area = π × 0.025² / 4 = 0.0004909 m² = 490.9 mm²
   3. At 3 m/s: 0.0004909 × 3 = 0.001473 m³/s
   4. That is 1.473 L/s, 5.301 m³/h, 3.12 cfm and 23.342 gpm

   Halving the bore and raising the velocity by half still leaves you with less than half the flow of the default case, because the area falls by four while the velocity rises by only 1.5. This is the trade a designer makes when space is tight: a smaller pipe is cheaper and easier to route, and the price is paid in velocity, which brings pressure drop, noise and erosion with it. The page will do that arithmetic for you but it will not tell you where to stop — which velocity is too brisk depends on the fluid and on the standard you are working to, and that judgement is not an input here.

4. A pipe with no flow in it

   1. Internal diameter 50 mm, velocity 0 m/s
   2. The area is still 1963.5 mm² — the pipe is there, the hole is that size, and that does not change when the fluid stops
   3. The flow rate is the area times the velocity, and the velocity is zero: 1963.5 mm² × 0 = 0
   4. All four flow rows print 0.000, and the area row keeps its 1963.5 mm²

   The one case where the answer is zero and the area is not, and it is worth seeing once because it separates the two quantities the panel prints side by side. A cross-sectional area is a property of the pipe; a flow rate is a property of the pipe and the motion together. Zero is a legitimate input rather than an error — a valve closed on a full line is exactly this state — so the page reports zeros instead of complaining. Check a diameter of zero and the reason for the zeros is different again: then the area itself is zero and there is no pipe to speak of. Two different physical situations, the same four zeros, which is why the area row is worth reading before you conclude anything from a zero.

The velocity is an input, so the flow rate is only as good as the velocity you chose. The relationship between the two is exact for a round pipe of known bore, but real pipe flow is not uniform across the cross-section: the fluid moves fastest at the centreline, and the value this page wants is the mean over the section. If your velocity comes from a pitot tube or a flow meter, check whether the instrument reports the centreline value, since that will read high. Bore is the second source of error, and it is the more expensive one, because diameter enters the formula squared: pipes are sold by nominal size, and the internal diameter of a nominal 50 mm pipe depends on its wall thickness and its material, ranging from about 44 mm in a thick-walled steel schedule to 50 mm and beyond in thin plastic. A 12% error in bore becomes a 25% error in flow rate. Nothing here accounts for pressure drop, friction, fittings, or the fact that a real velocity is not something you hold constant along a line — this page converts one velocity at one section into one flow rate. It also assumes the pipe is full: an open channel or a partly drained pipe has a wetted area that depends on the depth of flow, not on the bore, and this formula does not describe that.

What is the flow rate formula?

Q = A·v, where Q is the volumetric flow rate, A is the pipe's internal cross-sectional area and v is the mean velocity of the fluid. For a round pipe the area is π·d²/4, so the whole thing in one line is Q = (π·d²/4)·v. Everything this page does follows from that: two inputs, one multiplication, four unit conversions. The squared diameter is the part worth remembering, because it is where the surprises live — a bore 10% wider carries about 21% more fluid, and this is also why a small amount of scale or a partly closed valve costs more flow than it looks like it should.

How do I use this flow rate calculator?

Enter the pipe's internal diameter and the velocity of the fluid in it. The diameter box defaults to millimetres and also takes centimetres, metres, inches and feet; the velocity box defaults to metres per second and also takes km/h and ft/s. Both are converted to the units the formula needs before the arithmetic runs, so you can measure in whatever your tape and your datasheet use. You get the volumetric flow rate as the main result plus the same figure in cubic metres per hour, cfm and US gallons per minute, and the cross-sectional area underneath. There is no third input and no reverse solve: this page goes from pipe and velocity to flow, and starting from a target flow rate means picking a velocity and iterating.

Why does it print the same answer four times?

Because it is one number in four units, and which unit is natural depends entirely on who is asking. Litres per second is the metric engineering default and what a pump curve is drawn in. Cubic metres per hour is what water meters, meters in building services and equipment nameplates tend to use. cfm, cubic feet per minute, is the North American unit for air, so it dominates ventilation and ductwork. gpm, US gallons per minute, is the North American unit for water and is what a pump is sold in over there. Converting between them is one multiplication each, and the first table lists the factors so you can check any of them by hand. If your answer looks wrong by a factor of about a hundred, the usual cause is a unit slip between cfm and m³/h rather than an arithmetic mistake.

Does the answer depend on which fluid is in the pipe?

No, and that is the point of the volumetric version. A volume per second is a volume per second whether it is water, air, oil or steam: fill the same pipe at the same velocity and the same volume comes out. What does depend on the fluid is whether that velocity is sensible. Water in a pumped line is usually kept somewhere around 1 to 3 m/s, drainage runs slower, compressed air much faster, and hydraulic systems faster still — these are conventions about noise, erosion and pressure drop, not about the geometry. If your question is really about how much mass is moving, or how much energy it carries, you need the fluid's density or specific heat as well, and that is a different calculation.

What velocity should I put in?

One you can defend, because the page takes it as given and does not judge it. For pumped water, 1 to 1.5 m/s on the suction side and 1.5 to 3 m/s on the discharge side is the common engineering range, which is why 2 m/s is the default here and why the second table is built at that velocity. Below roughly 0.5 m/s water can let solids settle and air pockets collect; above roughly 3 m/s noise, erosion and pressure drop start to bite, and the limits tighten further for slurries and for pipework downstream of a control valve. Ductwork and compressed air sit in different bands again. Pick from the range your standard or your project uses, then check the flow rate that comes out against what the system actually needs.

Why is the cross-sectional area shown as well as the flow rate?

Because it is the step between the two inputs and the answer, and it is the one that explains why the answer moves the way it does. A 50 mm bore is 1963.5 mm², which is a number you can compare against the pipe in your hand; a 100 mm bore is 7854 mm², four times as much, and that factor of four is why doubling the diameter quadruples the flow at the same velocity. It also lets you spot the case where the flow is zero and the area is not: a pipe with no flow still has its full cross-section, and seeing 1963.5 mm² next to four zeros makes it obvious that the two rows mean different things. Square millimetres rather than square metres because the numbers stay readable — the same area is 0.0019635 m², where every significant digit sits after the decimal point.

• College Physics 2e, §12.1 — Flow Rate and Its Relation to Velocity: the definition of volumetric flow rate, Q = A·v, with the continuity relation and worked examples — OpenStax
• Volumetric flow rate — the quantity itself, its units and the conversions between them, including the distinction between volumetric and mass flow rate — Wikipedia
• GB/T 3102.3-1993 Quantities and units — Mechanics (equivalent to ISO 31-3:1992): the Chinese national standard for the units used here — length, area, velocity and volume flow — and the basis of the unit names shown in both tables — State Administration for Market Regulation — National Standards Full-Text Public System