Why does the rate card not move when I drag the downstream diameter?

Because the rate is set at the inlet, and the inlet has not changed. Volume cannot pile up at a step, so whatever arrives each second has to leave each second: the card holds one string, 11.7810 L/s, across the whole of that slider. The speed card is what absorbs the change instead, and it does so exactly reciprocally.

Why do the tracer marks spread apart in the narrower pipe instead of bunching up?

Because they are released at equal time intervals, so the gap between two of them is the local speed times that interval. Multiply a gap by the area it sits in and you get the same volume in both sections, which is the point of the whole page. Equal volume per gap forces the longer gap where the flow is faster, and the caption prints that volume in litres.

Why is there no pressure readout anywhere on this panel?

Because Q = A v is kinematics. It counts the volume crossing a section each second and says nothing about what pushes it, so a pressure card here would be an answer the arithmetic cannot give. The boundary is deliberate rather than an omission, and the fixed line under the drawing states it: no pressure is computed anywhere on this page.

Why does the verdict warn about the speed of sound in water and air but never in glycerine or oil?

Because this site publishes a speed of sound for water and air only, 1480 m/s and 343 m/s, and inventing a figure for the other two fluids would be worse than leaving the gap visible. So the caveat can fire on water and on air and never on glycerine or SAE 30 oil. Read that silence as missing data, not as a clean bill of health.

Why do I have to choose the rate unit myself instead of the lab switching it?

Because an automatic switch would add a rounding boundary and teach nothing. Four buttons make the same rate in four units something you do rather than something you watch: one press moves the card between 0.0117810 m3/s, 11.7810 L/s, 706.858 L/min and 42.4115 m3/h. The ladder row underneath prints all four at once and never changes when you press one.

The fill time changed from seconds to minutes. Which figure should I trust?

Both, because it is one time printed in two units. That card changes rung on the rounded six-figure reading rather than on the raw one, then prints six figures of the converted value: a hundred litres through the 15 mm pipe reads 9.43140 min rather than a rounded number of seconds. Drop every slider to its floor and it climbs again, to 35.3678 h.

What is the caption reading one scale both ways telling me?

That the bores and the length of the pipe are drawn to a single scale, and what one pixel of the picture is worth in millimetres. That scale is set by the larger of the two bores, so it moves as you drag, and the caption is reprinted on every frame. Without it a 10 mm pipe and a 500 mm pipe would look identical on screen.

What does exactly reciprocal mean on the Area down, speed up row?

That the two factors in that sentence are one number used twice. The area falls by some factor and the speed rises by the same factor, because their product is the rate and the rate has not moved. The lab works that factor out once and prints it on both halves of the row, so the two figures cannot drift apart in their sixth digit.

Flow Rate Formula Simulator | Physics Fundamentals

The flow rate formula simulator is a free interactive lab built around the quantity that refuses to move. Set the inlet bore and the inlet mean speed — between them they fix the rate — then drag the downstream bore and watch the rate card hold one string while the speed card sweeps across two orders of magnitude. The pipe steps abruptly rather than tapering, the rate is labelled once on an arrow spanning the step, and no pressure is computed anywhere on this page.

By the PhysicsFundamentals Editorial Team Reviewed for accuracy

Flow Rate Formula Simulator reference card: Q = A1 v1 = A2 v2, with five labelled chips on two lines - Inlet bore d1, 10 to 500 mm; Inlet mean speed v1, 0.10 to 8.00 m/s; Downstream bore d2, 10 to 500 mm; Vessel to fill V, 1 to 1000 L; and Buttons, 4 fluids, 4 units, 7 presets — Quick-reference card for the flow rate formula simulator - the one rate it carries through both bores, the four sliders that set it and the buttons beside them.

What Is the Flow Rate Formula Simulator?

The flow rate formula simulator is a free interactive physics lab that runs in your browser, with nothing to install and no sign-up. It draws a round pipe in side elevation which steps abruptly from one bore to another at mid-length, carries one steady rate through both sections, and labels that rate once, on a gold arrow spanning the step. The placement is the claim: the rate is the quantity that does not change where the bore does.

Four sliders drive it. Inlet diameter d1 and Downstream diameter d2 both run 10 to 500 mm in whole millimetres, Inlet mean speed v1 runs 0.10 to 8.00 m/s in steps of 0.05, and Vessel to fill V runs 1 to 1000 L in whole litres. Four buttons pick the fluid — Water, Air, Glycerine and SAE 30 oil — four more pick the unit the rate is printed in, seven presets load a published state, and Pause flow and Reset finish the panel.

Eight readouts answer. Four cards give Volume flow rate, Mean speed in the downstream section, Mass flow rate and Time to fill the vessel; a compact grid beneath them adds Areas, inlet and downstream, Area down, speed up, The same rate, four ways and a Verdict. On the state the lab opens in they read 11.7810 L/s, 6.00000 m/s, 11.7598 kg/s, 8.48826 s and Narrower bore - the same rate forces the flow faster.

Each of the three cards beside the rate answers to exactly one control that moves nothing else. Drag the downstream bore and the speed card sweeps while the rate card holds one string; press a fluid button and only the mass flow moves; drag the vessel and only the fill time moves. The rate itself answers to the inlet bore and the inlet speed, because those two are what set it.

The ratio row states the reciprocity rather than implying it: at the opening state it reads area 4.00000x smaller, speed 4.00000x faster - exactly reciprocal, and one factor is worked out once and printed twice, so the two figures cannot drift apart. Press Same bore all the way and that row reads same area, same speed while the drawing becomes a single uniform pipe with no step in it at all.

The tracer marks are released at equal intervals, so the same volume of fluid passes between any two of them — which makes their gaps wider in whichever section the flow is faster, never narrower. The caption under the drawing gives that volume in litres on every frame, and it also names the one scale the picture is drawn to, because that scale moves with the larger of the two bores.

What the panel will not do, it says. Q = A v is kinematics, so no pressure, no pressure drop and no pumping power is worked out anywhere here, and extreme slider positions reach speeds no real pipe could sustain: the verdict then reads Past the speed of sound in water - the incompressible rule has gone, or the same sentence naming air. Those are the only two fluids this site publishes a speed of sound for, so glycerine and SAE 30 oil never trigger that caveat — a stated gap rather than a clean bill of health.

The controls of the flow rate formula simulator, and the ranges they move over
Control	Range	Step, default and the card it moves
Inlet bore d1	10 to 500 mm	step 1; opens at 100; sets the rate with the inlet speed
Inlet mean speed v1	0.10 to 8.00 m/s	step 0.05; opens at 1.50; sets the rate with the inlet bore
Downstream bore d2	10 to 500 mm	step 1; opens at 50; moves the speed card and never the rate card
Vessel to fill V	1 to 1000 L	step 1; opens at 100; moves the fill-time card alone
Buttons	4 fluids, 4 units, 7 presets	one press each; the fluid moves the mass flow alone, plus Pause flow and Reset

How to use the flow rate formula simulator

Read the four cards before you touch anything. They are already answering: Volume flow rate 11.7810 L/s, Mean speed in the downstream section 6.00000 m/s, Mass flow rate 11.7598 kg/s and Time to fill the vessel 8.48826 s. Each card carries the closed form it comes from on its own label, so none of them is a mystery.
Find the rate in the drawing before you change it. The gold arrow spans the step and carries Q = 11.7810 L/s once, which is the claim: the rate is what does not change where the bore does. Either side of it the pipe is labelled d1 = 100 mm with v1 = 1.50 m/s, and d2 = 50 mm with v2 = 6.00000 m/s.
Now move Downstream diameter d2 and watch only two things. The speed card sweeps and the rate card does not: at 20 mm it reads 37.5000 m/s, at 200 mm it reads 0.375000 m/s, and 11.7810 L/s stays on screen throughout. That one slider is the whole argument of the page.
Drag it until the two bores match. At 100 mm the step disappears from the drawing entirely, the speed card reads 1.50000 m/s — the inlet speed, exactly — and the verdict reads Same bore all the way - one area, one speed, one rate. A lab that drew a step of zero height here would be drawing something that is not there.
Press a fluid button and watch three cards ignore you. Glycerine takes the mass card to 14.8558 kg/s and SAE 30 oil takes it to 10.3201 kg/s, while the rate, the speed and the fill time keep the strings they had. Volume conservation does not know what the fluid is made of; kilograms per second do.
Drag Vessel to fill V and watch the other three ignore you in turn. It moves the fill card alone. That is why the card is not the rate restated: it goes down as the rate goes up, and up as the vessel grows, and the two directions are easy to confuse until you have seen them.
Choose the unit the rate is printed in — the lab will not choose for you. The four buttons give 0.0117810 m3/s, 11.7810 L/s, 706.858 L/min and 42.4115 m3/h for one pipe. That is deliberate: an automatic switch would add a rounding boundary for no gain, whereas four buttons make the conversion something you perform.
Read the compact grid as a decomposition, not as more evidence. Areas, inlet and downstream shows the two areas the cards were built from, Area down, speed up prints the trade as one factor used twice, and The same rate, four ways is the whole unit ladder at once. The last of those restates the rate card on purpose, and its label says so.
Use Pause flow when you want to look rather than watch. It stops the tracer marks where they are and leaves a clean static frame, which is the right state for reading the gaps between them. Press it again to resume; the drift is deliberately slowed to be read and is not a real-time speed.
Reset is the way back when too many controls have moved to remember. It returns all four sliders, the fluid and the rate unit to their opening values and lights the first preset again. Touching any slider by hand drops the preset highlight, so no button can claim a state the cards do not show.

Dragging is the right way to see which card answers to which control, and a poor way to hit an exact figure. When you already have two of the three quantities and want the third typed out, the flow rate formula calculator solves the same relation for whichever one is missing, with unit menus on every box. For the definition itself, the full unit table and the worked problems, read the guide to the flow rate formula.

Flow rate formula simulator in the state it opens in, with the drawing on the left and the control panel on the right, and the flow paused. In the panel the inlet diameter d1 row reads 100 mm, the inlet mean speed v1 row reads 1.50 m/s, the downstream diameter d2 row reads 50 mm with its thumb to the left of the inlet one, and the vessel to fill V row reads 100 L. Under them Water 998.2 kg/m3 is the highlighted one of four fluid buttons beside Air 1.204, Glycerine 1261 and SAE 30 oil 876; L/s is the highlighted one of the four rate-unit buttons m3/s, L/s, L/min and m3/h, under the heading Rate unit and the note you choose; the flow button reads Play flow because the animation is stopped, with Reset beside it; and The default pipe: 100 mm at 1.50 metres per second is the highlighted one of seven published-state buttons. The drawing is captioned stepped pipe, water flowing left to right on the left and one scale both ways, 1 px = 0.599 mm on the right. A gold arrow spans the whole pipe carrying Q = 11.7810 L/s once. Below it a round pipe runs left to right and steps down abruptly at mid-length from a wide section to one half its height, the two sharing a dashed centreline. The wide section is labelled d1 = 100 mm and v1 = 1.50 m/s, the narrow one d2 = 50 mm and v2 = 6.00000 m/s. Pale vertical tracer marks cross the pipe, closely and evenly spaced along the wide section and far enough apart in the narrow one that only a single mark falls inside it, and the caption under the pipe reads 0.182023 L between marks in both bores - wider gaps where the flow is faster. Four cards under the drawing read Volume flow rate, under Q = A1 v1, 11.7810 L/s; Mean speed in the downstream section, under v2 = Q / A2, 6.00000 m/s; Mass flow rate, under mdot = rho Q, 11.7598 kg/s; and Time to fill the vessel, under t = V / Q, 8.48826 s. A compact grid below them gives Areas, inlet and downstream 0.00785398 m2 -> 0.00196350 m2; Area down, speed up area 4.00000x smaller, speed 4.00000x faster - exactly reciprocal; The same rate, four ways 0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h; and Verdict Narrower bore - the same rate forces the flow faster. A fixed line at the foot gives the five relations Q = A v, A = pi d^2 / 4, A1 v1 = A2 v2, mdot = rho Q and t = V / Q, the four densities at 20 C with the oil at 40, the two sound speeds this site publishes as 1480 m/s in water and 343 m/s in air, the statement that no speed of sound is published here for glycerine or SAE 30 oil, and the note that Q is a volume per second on this page, not a quantity of heat. — The opening state, paused so the marks can be read. One pipe, one rate on the gold arrow — **11.7810 L/s** — and two speeds either side of the step, **1.50 m/s** in and **6.00000 m/s** out. The detail worth staring at is the tracer spacing: the gaps are *wider* in the narrow section, not tighter, because every gap in this frame holds the same **0.182023 L** and the faster flow needs more length to hold it.

Worked example: change one thing at a time

Nothing in the four tables below was worked out here. Each row is a state the lab was put into on this page and then read back from, cell by cell, so every string in them is the panel’s own. The first table holds the inlet still and changes the downstream bore, then makes the bores equal, then changes the fluid.

The four cards at five published states, each differing from the one above it by one thing
What was pressed	Downstream bore	Volume flow rate	Mean speed in the downstream section	Mass flow rate	Verdict
The default pipe: 100 mm at 1.50 metres per second	50 mm	11.7810 L/s	6.00000 m/s	11.7598 kg/s	Narrower bore - the same rate forces the flow faster
A nozzle: 100 mm down to 20 mm	20 mm	11.7810 L/s	37.5000 m/s	11.7598 kg/s	Narrower bore - the same rate forces the flow faster
Widen instead: 100 mm up to 200 mm	200 mm	11.7810 L/s	0.375000 m/s	11.7598 kg/s	Wider bore - the same rate lets the flow slow down
Same bore all the way	100 mm	11.7810 L/s	1.50000 m/s	11.7598 kg/s	Same bore all the way - one area, one speed, one rate
Air in a 200 mm duct	200 mm	157.080 L/s	5.00000 m/s	0.189124 kg/s	Same bore all the way - one area, one speed, one rate

Rows one to three share every control but one, and that is the argument. Same inlet bore, same inlet speed, same fluid, same vessel — and one rate string, 11.7810 L/s, repeated down the column while the speed beside it goes from 6.00000 m/s to 37.5000 m/s to 0.375000 m/s. A hundredfold spread in one card and not a digit moved in the other.

Row four is the row worth pressing yourself. With the bores equal the downstream speed is the inlet speed to six figures, the drawing loses its step, and the verdict changes its wording rather than its arithmetic. It is the state that proves the speed card is reporting a consequence of the bore and not a second opinion about the rate.

Row five changes the fluid and the rate column barely notices. The duct carries 157.080 L/s — more than thirteen times the default pipe — and only 0.189124 kg/s of air with it. Anything metered, charged or burned by mass needs that second figure, and the fluid buttons are the only control that moves it.

The ratio row, and why it is one number printed twice

The Area down, speed up row is where the panel states the reciprocity instead of leaving you to spot it. The factor is worked out once from the two bores and then printed on both halves of the sentence, so the two figures are byte-identical by construction rather than by luck. The hydraulic cylinder force calculator uses the same squared-diameter step on a piston, if you want to see the area arithmetic on its own.

The two areas and the trade between them, at the same four bore settings
What was pressed	Areas, inlet and downstream	Area down, speed up
The default pipe: 100 mm at 1.50 metres per second	0.00785398 m2 -> 0.00196350 m2	area 4.00000x smaller, speed 4.00000x faster - exactly reciprocal
A nozzle: 100 mm down to 20 mm	0.00785398 m2 -> 0.000314159 m2	area 25.0000x smaller, speed 25.0000x faster - exactly reciprocal
Widen instead: 100 mm up to 200 mm	0.00785398 m2 -> 0.0314159 m2	area 4.00000x larger, speed 4.00000x slower - exactly reciprocal
Same bore all the way	0.00785398 m2 -> 0.00785398 m2	same area, same speed

Halving the bore is the trap this row exists for. From 100 mm to 50 mm the area does not halve, it quarters, and the row says area 4.00000x smaller, speed 4.00000x faster. Take the bore down to a fifth, 100 mm to 20 mm, and the factor is 25.0000x: the square in the area does all the work, and most wrong answers in this topic come from forgetting it.

The fill card, and the two rungs it climbs on its own

The fourth card divides the vessel by the rate, and its unit changes as the figure crosses sixty — decided on the rounded six-figure reading rather than on the raw float, so the value is never rounded twice. Rows two and three below are the same pipe with a different vessel, which is the cleanest way to see that only this card is listening.

Four published states that put all three rungs of the fill card on one page
What was pressed or dialled	Vessel	Volume flow rate	Time to fill the vessel
A 500 mm water main	1000 L	392.699 L/s	2.54648 s
A domestic 15 mm pipe	10 L	0.176715 L/s	56.5884 s
The same 15 mm pipe, a bigger vessel	100 L	0.176715 L/s	9.43140 min
Every slider at its floor	1000 L	0.00785398 L/s	35.3678 h

One rate, four buttons

The rate unit is yours to choose, and the ladder row underneath is unmoved by the choice. Press each button in turn at the opening state and the card walks the third column of this table while the fourth stays exactly where it was, because that row is the whole ladder every time. A pump quoted in cubic metres per hour and a meter reading in litres per second are the same pipe, and this is the thirty-second check that says so.

The four rate-unit buttons at the opening state, and what each leaves on screen
Button	What the rate card prints	What the ladder row prints
m3/s	0.0117810 m3/s	0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h
L/s	11.7810 L/s	0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h
L/min	706.858 L/min	0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h
m3/h	42.4115 m3/h	0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h

Formula and symbol reference

The lab prints its five relations on itself, in a fixed line under the drawing: Q = A v, A = pi d^2 / 4, A1 v1 = A2 v2, mdot = rho Q and t = V / Q. The first two are the definition and the geometry; the third is the same definition written at two stations; the last two are readings that follow without new physics.

One warning about the letter before the table. Q here is a volume per second, in cubic metres per second, and the same letter stands for a quantity of heat in watts in Fourier's law of heat conduction. Same symbol, different quantity, and nothing on this panel is an energy. The density that turns the rate into kilograms is the subject of the density formula guide.

Ten rows follow and only four of them are things you drag. Two are readings with no control at all, one is pressed on a button, and the last is a factor the panel works out for itself. For the sliders the right-hand column gives the end stops and the step; for everything else it gives the strings the lab prints and the range it prints them over.

The ten quantities this panel names, and what each is allowed to do here
Symbol	Meaning	SI unit	In this lab
Q	Volume flow rate — the area times the mean speed, and the first card. It is a volume per second on this page	cubic metre per second	six significant figures in whichever unit you pressed: `0.0117810 m3/s`, `11.7810 L/s`, `706.858 L/min`, `42.4115 m3/h`. Across the sliders the rate runs from `0.00785398 L/s` to `1570.80 L/s`.
A1, A2	The two cross-sectional areas, each formed from its own bore as `A = pi d^2 / 4`. Neither has a control of its own	square metre	the grid row prints both at once: `0.00785398 m2 -> 0.00196350 m2` at the opening state, and `0.0000785398 m2` at a 10 mm bore, which is the smallest area the panel can draw.
d1, d2	The two internal diameters, in whole millimetres. The inlet one helps set the rate; the downstream one cannot touch it	metre, set in millimetres	10 to 500 mm each, in steps of 1; they open at `100 mm` and `50 mm`. Every published diameter is a whole number, so the thumb reaches each of them exactly.
v1	Inlet mean speed, the second slider. With the inlet bore it is what sets the rate, and it is a mean across the section	metre per second	0.10 to 8.00 m/s, in steps of 0.05, echoed beside the slider as `1.50 m/s`, `1.00 m/s`, `0.10 m/s`. Every published speed is a multiple of the step.
v2	Mean speed in the downstream section, the second card. This is the reading the downstream bore moves, and it moves nothing else	metre per second	six significant figures: `1.50000 m/s` when the two bores match, `6.00000 m/s` at the opening step, `37.5000 m/s` through the 20 mm nozzle and `0.375000 m/s` when the pipe widens to 200 mm.
mdot	Mass flow rate, the third card: the density times the rate. It is proportional to the rate at a fixed fluid, which the card says rather than hides	kilogram per second	six significant figures: `11.7598 kg/s` for water at the opening rate, `14.8558 kg/s` for glycerine and `10.3201 kg/s` for SAE 30 oil at that same rate.
rho	Fluid density — pressed on a button rather than typed, because the four figures are the ones this site already publishes	kilogram per cubic metre	four fixed values, shown on the buttons themselves: water 998.2, air 1.204, glycerine 1261 and SAE 30 oil 876 kg/m3.
V	The vessel the fourth slider fills. It appears in no other reading on the panel	cubic metre, set in litres	1 to 1000 L, in steps of 1, echoed as `10 L`, `100 L`, `1000 L`. It is the only control that moves the fill card and the only one the other three cards ignore.
t	Time to fill the vessel, the fourth card, as `t = V / Q`. It falls as the rate rises, which is why it is not the rate restated	second	the unit climbs on its own, on the rounded figure: `2.54648 s` down a 500 mm main, `8.48826 s` at the opening state, `9.43140 min` for a hundred litres through a 15 mm pipe, `35.3678 h` with every slider at its floor.
k	The area ratio on the `Area down, speed up` row — worked out once from the two bores and printed on both halves of that sentence	dimensionless	not a control and not a card. `4.00000` at the opening step, `25.0000` through the nozzle, and the row drops the factor altogether for `same area, same speed` when the two bores are equal.

If you read one row twice, make it k. It is the only entry with no control and no card, and it is what keeps the ratio row from being the speed card restated: one factor, two halves of a sentence, and a claim of exact reciprocity that the panel is in a position to make.

The physics: one rate, every bore

The rate is the area times the mean speed, and that is a definition rather than a law. In one second a steady flow sweeps a cylinder of fluid past any fixed section, as wide as the pipe and as long as the distance travelled in that second. Put a step in the pipe and nothing can accumulate at it, so the volume arriving each second is the volume leaving each second.

That is the whole of continuity, and the panel is built so you can watch it rather than take it on trust. From the test harness rather than a card: the downstream slider was swept across all 491 of its whole-millimetre positions and the rate card printed one single string, 11.7810 L/s, at every one of them, while the speed card printed 491 different ones. The rate does not depend on the downstream bore to a tolerance; it does not depend on it at all.

The speed, though, must. If the rate is fixed and the area falls, the speed has to rise by exactly the factor the area fell by, which is why the ratio row prints one number twice. And because the area goes as the square of the diameter, a bore halved is an area quartered: the speed-up is the square of the diameter ratio, not the ratio itself.

The v in all of this is the mean speed across the section, which matters the moment you compare it with anything measured. A real pipe has faster fluid in the middle than at the wall, and the Reynolds number guide is where that distinction, and the character of the flow it belongs to, is set out properly. This panel takes the mean as given and discusses no profile.

Four cards, four different functions of the controls, and the panel is deliberately arranged so that each of the three beside the rate answers to exactly one control that moves nothing else. The downstream bore moves the speed, the fluid buttons move the mass flow, the vessel moves the fill time, and the inlet pair move all four because they are what set the rate. One correspondence is stated rather than hidden: the mass flow is the density times the rate, so at a fixed fluid the two are proportional.

That correspondence is worth one more sentence, because it is the only pair on the panel that could be mistaken for a repeated reading. The ratio of the mass card to the rate card is the density, and from the test harness rather than a card: the four buttons sweep it from 1.204 to 1261, a span of more than a thousandfold. Two cards whose digits are unrelated, carrying different units, with one printing kilograms and the other litres.

What none of it says is what the flow costs. Q = A v is kinematics: it counts volume past a point and is silent about the pressure, the viscosity and the energy needed to push it there, which is why this lab computes no pressure anywhere. Bernoulli's principle is where the pressure side of a moving fluid lives, and the pressure-at-a-throat lab is its panel — a smooth venturi taper read for its pressure drop, where this one is an abrupt step read for its rate.

Flow rate formula simulator with Same bore all the way pressed, so that is the highlighted one of the seven published-state buttons and the flow is paused. The inlet diameter d1 row reads 100 mm and the downstream diameter d2 row now reads 100 mm as well, with the two thumbs at the same position; the inlet mean speed v1 row still reads 1.50 m/s and the vessel to fill V row 100 L. Water remains the highlighted fluid button and L/s the highlighted rate unit, and the flow button reads Play flow. The drawing has lost its step entirely: the caption at the top left now reads one bore all the way, water flowing left to right rather than stepped pipe, and the pipe is a single uniform rectangle of constant height from end to end, crossed by its dashed centreline, with no change of section anywhere along it. The gold arrow above it still carries Q = 11.7810 L/s and the scale caption still reads one scale both ways, 1 px = 0.599 mm. The pipe is labelled d1 = 100 mm with v1 = 1.50 m/s on the left half and d2 = 100 mm with v2 = 1.50000 m/s on the right half. Twelve pale vertical tracer marks cross the pipe at one even pitch for its whole length, and the caption under it reads 0.182023 L between marks - one bore, one speed, evenly spaced. The four cards read Volume flow rate 11.7810 L/s, Mean speed in the downstream section 1.50000 m/s, Mass flow rate 11.7598 kg/s and Time to fill the vessel 8.48826 s. The compact grid below them gives Areas, inlet and downstream 0.00785398 m2 -> 0.00785398 m2; Area down, speed up same area, same speed, with no factor printed at all; The same rate, four ways 0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h; and Verdict Same bore all the way - one area, one speed, one rate. — The identity state, and the one worth pressing to check the lab is honest. With the bores equal the downstream card reads **1.50000 m/s** — the inlet speed to six figures — the ratio row stops printing a factor and simply says **same area, same speed**, and the drawing refuses to draw a step of zero height. The marks fall at one even pitch down the whole pipe, which is the same equal-volume rule as the previous figure with the area ratio set to one.

Where the flow rate formula breaks down

The arithmetic here is exact to the last digit a double can hold, and the panel will answer every position its sliders can reach. What fails is the physics around the edges of that bargain: a relation that counts volume and not cost, a density assumed constant, a mean read as a maximum, and one drawing asked to show two bores at once.

It is kinematics, so some reachable states are arithmetically exact and physically absurd: Nothing on this panel asks what drives the flow, so the sliders can reach a downstream speed that no pump, pipe or duct could sustain. The verdict card refuses to leave that unremarked: push the speed past the figure this site publishes for the fluid and it stops describing the bores and reads Past the speed of sound in air - the incompressible rule has gone, or the same sentence naming water. Treat that string as the panel declining to vouch for the state, not as a result.
One rate at every station is the incompressible statement, and a gas only borrows it: Volume in equals volume out exactly when the density does not change along the pipe. For a liquid that is close enough to exact for any practical bore; for a gas it holds while the pressure and temperature barely move, which in practice means well below the speed of sound. Above that, the volume arriving and the volume leaving are simply different numbers and nothing on this panel applies — including the mass card, which is the one reading that would survive the change if the density were tracked.
Two of the four fluids carry no sound-speed caveat at all, and that is a gap in the warning: This site publishes a speed of sound for water, 1480 m/s, and for air, 343 m/s. It publishes none for glycerine and none for SAE 30 oil, so the verdict can never fire for either of those two, however extreme the bores you set. Inventing a figure to close the gap was refused on purpose; the honest reading of a silent verdict on glycerine is that the threshold is unknown here, not that there is not one.
Every speed on the panel is a mean, and a real section has no single speed: Fluid at the wall moves slower than fluid in the middle, so both the slider and the card are averages across the section rather than anything a probe would read at one point. Feeding a centreline reading into the inlet slider will overstate the rate. The Reynolds number guide is where mean and local speeds are separated, and its lab is where the character of the flow gets a panel of its own.
Where a viscosity and a pressure drop set the rate, a different relation fixes it: In a long narrow tube under a steady pressure difference it is the viscosity that decides how much flows, and the relation that gives the rate there is Poiseuille’s law. The definition does not stop holding there — it will describe that flow perfectly well once a speed exists to put into it — but it has no way of producing the speed in the first place, and this panel carries no viscosity control and no pressure difference to give it one. The viscosity guide names that law and works it; this page does neither.
At a large area ratio one scale cannot show both gaps, and the caption says which: The marks are spaced by equal volume, so the gap in the faster bore is longer by the full area ratio. Past about six to one that gap is longer than the section drawn, and the caption changes from wider gaps where the flow is faster to naming the bore whose gap no longer fits. Drawing the marks closer together there instead would be a second scale, and this picture never has one.
The drawing is to scale in the bores, and the smallest ones hit a floor: Both sections are drawn to one scale set by the larger of the two, so a 10 mm bore beside a 500 mm one is a hairline by rights. It is drawn at a two-pixel minimum instead, because a line of zero width is not a pipe, and that is the one place the picture is generous rather than faithful. The cards and the area row are unaffected; read them, not the thickness.
A fill time assumes a rate that never changes: The fourth card divides the vessel by whatever rate is on screen and then holds that rate fixed for the whole of the fill. That is exact arithmetic about a flow nobody actually gets: the conditions driving a real fill drift while it is happening, and they drift in the direction that makes it take longer rather than shorter. Read the card as a lower bound on the time, not as a prediction of it, and the reference notes at the foot of this page say which three effects do the lengthening.
Both bores are internal diameters, and the panel cannot tell when they are not: A nominal pipe size is a label and an outside diameter includes the wall, and because the area follows the square, a bore five per cent out puts the rate about ten per cent out. The error lands in the answer rather than in the input you can see. Take the bore from the pipe schedule you are actually buying, and nothing here is a round section’s substitute for a rectangular duct or a part-full pipe — the Manning equation calculator is the open-channel tool.
Two engines agree readout for readout, and Safari was never run at all: From the cross-browser gate rather than a card: 174 readout values were put side by side in Chrome and in Firefox over five slider positions at each of four viewport widths, and every one matched. There is a hole in that, and it is Safari: the WebKit build on this box refuses to start without privileges the harness does not hold, so Safari was never exercised and nothing here is evidence about it. If your browser ever prints a figure this page does not, trust the panel in front of you and tell us.

Where the flow rate formula is actually used

Settling the halve-the-bore argument in two button presses: This is the misconception the lab exists for. Press The default pipe: 100 mm at 1.50 metres per second and then A nozzle: 100 mm down to 20 mm: the rate card never changes its string while the speed card goes from 6.00000 m/s to 37.5000 m/s. Two presses, ten seconds, and the square in the area stops being something to remember and becomes something you watched.
Sizing a domestic pipe against a main, on the same pair of axes: Press A domestic 15 mm pipe and then A 500 mm water main: 0.176715 L/s against 392.699 L/s. The bore grew by a factor of about 33 and the speed only doubled, yet the rate grew by more than two thousandfold. That is the shape of every pipe-sizing conversation, and it is why a bore is chosen before a speed is argued about.
Estimating a fill or a drain before anyone orders a pump: The fill card answers the question most people actually arrive with. A hundred litres through the default pipe is gone in 8.48826 s; the same hundred litres through the 15 mm kitchen pipe takes 9.43140 min; a thousand litres through a 10 mm bore at a crawl takes 35.3678 h. Those are floors on the time, and they are usually enough to settle whether a plan is sensible.
Working in kilograms when what is paid for is mass: Press Air in a 200 mm duct: 157.080 L/s of air, and 0.189124 kg/s of it. Then press The default pipe, carrying glycerine and watch the mass card alone move to 14.8558 kg/s. Combustion air, refrigerant charges and anything sold by the tonne are specified in the second column, and only the fluid buttons touch it. The density calculator is where a density comes from if the fluid is not one of these four.
Reading a rate in the unit the invoice is written in: A pump is sold in litres per minute or cubic metres per hour, a meter reads in litres per second, and a textbook works in cubic metres per second. The ladder row puts one rate in all four at once, 0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h, which is the quickest way to catch the factor-of-sixty slip that this topic is famous for.
Choosing a nozzle, or deliberately choosing not to: Narrowing the exit is how a jet is made: the same 11.7810 L/s leaves a 20 mm nozzle at 37.5000 m/s. Widening does the opposite, and Widen instead: 100 mm up to 200 mm drops the same rate to 0.375000 m/s, which is what a settling tank or an intake screen is for. One slider, two pieces of design intent, and no new physics between them.
Getting the mean speed a flow-regime question needs before it can start: Almost every next question about a pipe — laminar or turbulent, what the pressure drop will be, whether a probe is reading the right thing — wants a mean speed first. This panel is the quickest way to turn a bore and a rate into one. The Bernoulli equation calculator is where that speed becomes a pressure, and the guide to pressure in physics is the other thing a cross-section does.

Flow rate formula simulator with A nozzle: 100 mm down to 20 mm pressed, so that is the highlighted one of the seven published-state buttons and the flow is paused. The inlet diameter d1 row still reads 100 mm and the inlet mean speed v1 row 1.50 m/s, but the downstream diameter d2 row now reads 20 mm with its thumb almost at the left-hand end of its track, and the vessel to fill V row still reads 100 L. Water is the highlighted fluid and L/s the highlighted rate unit, and the flow button reads Play flow. In the drawing the pipe steps down far harder than in the opening state: a wide section runs from the left to mid-length and then drops to a thin sliver about a fifth of its height, the two sections sharing the dashed centreline. The gold arrow above still carries Q = 11.7810 L/s unchanged, and the scale caption still reads one scale both ways, 1 px = 0.599 mm, because the scale is set by the larger bore and that has not moved. The wide section is labelled d1 = 100 mm with v1 = 1.50 m/s and the thin one d2 = 20 mm with v2 = 37.5000 m/s. The pale tracer marks cross the wide section at the same close even pitch as before, and the thin section carries no mark at all in this frame; the caption under the pipe has changed to 0.182023 L between marks - that gap is longer than the 20 mm section drawn. The four cards read Volume flow rate 11.7810 L/s, Mean speed in the downstream section 37.5000 m/s, Mass flow rate 11.7598 kg/s and Time to fill the vessel 8.48826 s. The compact grid gives Areas, inlet and downstream 0.00785398 m2 -> 0.000314159 m2; Area down, speed up area 25.0000x smaller, speed 25.0000x faster - exactly reciprocal; The same rate, four ways 0.0117810 m3/s = 11.7810 L/s = 706.858 L/min = 42.4115 m3/h; and Verdict Narrower bore - the same rate forces the flow faster. — One slider moved from the opening state, and three of the four cards did not notice. The rate is still **11.7810 L/s**, the mass flow still **11.7598 kg/s** and the fill time still **8.48826 s**, while the speed card has gone to **37.5000 m/s** and the ratio row to **25.0000x** — a bore cut to a fifth, an area cut to a twenty-fifth. This is also where the one scale gives up honestly: the equal-volume gap is now longer than the 20 mm section is drawn, so the caption names the bore instead of crowding the marks into a second scale.

Where to go next

A panel can show you which readings a control is allowed to move, and that is the end of what a panel is for. Flow Rate Formula (Q = Av) is where the subject itself is written out: what the rate is, the unit table in full, the area step, the mass flow rate, the fill time and the problems worked through from their own figures. The calculator that takes those figures typed heads the list of related tools below.

Three neighbouring subjects are deliberately absent from this panel, and each is written up on its own terms. Bernoulli's principle is where the pressure side of the story lives, the Reynolds number guide is where the mean speed meets laminar and turbulent flow, and the viscosity guide is where a pressure drop and a viscosity set the rate instead.

Further out, the density formula guide is where the mass card gets its density, the guide to pressure in physics and the Pascal law and hydraulics guide are the other two things an area does, and Fourier's law is where the other Q lives. Beyond that there is the whole simulation library to work through, an index of calculators for when typing beats dragging, and the blog for everything written out at length.

Flow Rate Formula Simulator