How To Calculate Volume Flow Rate

Volume flow rate is a term in physics that describes how much matter – in terms of physical dimensions, not mass – moves through space per unit time. For example, when you run a kitchen faucet, a given amount of water (which you might measure in fluid ounces, liters or something else) passes out of the opening of the faucet in a given amount of time (usually seconds or minutes). This amount is considered the volume flow rate.

The term "volume flow rate" almost always applies to liquids and gases; solids do not "flow," even though they too may move at a steady rate through space.

The basic equation for problems of this sort is

\(Q=AV\)

where ​Q​ is the volume flow rate, ​A​ is the cross-sectional area occupied by the flowing material, and ​V​ is the average velocity of flow. ​V​ is considered an average because not every part of a flowing fluid moves at the same rate. For example, as you watch the waters of a river make their way steadily downstream at a given number of gallons per second, you notice that the surface has slower currents here and faster ones there.

The cross section is often a circle in volume flow rate problems, because these problems often concern circular pipes. In these instances, you find the area ​A​ by squaring the radius of the pipe (which is half the diameter) and multiplying the result by the constant pi (π), which has a value of about 3.14159.

The usual SI (From the French for "international system," tantamount to "metric") flow rate units are liters per second (L/s) or milliliters per minute (mL/min). Because the U.S. has long used imperial (English) units, however, it is still far more common to see volume flow rates expressed in gallons/day, gallons/min (gpm) or cubic feet per second (cfs). To find volume flow rates in units not commonly used for this purpose, you can use an online flow rate calculator like the one in the Resources.

Sometimes, you will want to know not just the volume of fluid moving per unit time, but the amount of mass this represents. This is obviously critical in engineering, when it must be know how much weight a given pipe or other fluid conduit or reservoir can safely hold.

The mass flow rate formula can be derived from the volume flow rate formula by multiplying the entire equation by the density of the fluid, ​ρ​. This follows from the fact that density is mass divided by volume, which also means that mass equals density times volume. The volume flow equation already has units of volume per unit time, so to get mass per unit time, you simply need to multiply by density.

The mass flow rate equation is therefore

\(\dot{m}=\rho AV\)

​ṁ​, or "m-dot," is the usual symbol for mass flow rate.

Say you were given a pipe with a radius of 0.1 m (10 cm, about 4 inches) and were told you needed to use this pipe to drain an entire full water tank in less than one hour. The tank is a cylinder with a height (​h​) of 3 meters and a diameter of 5 meters. How fast will the stream of water need to move through the pipe, in m³/s, in order to get this job done? The formula for the volume of a cylinder is:

\(V=\pi r^2 h\)

The equation of interest is ​Q​ = ​AV​, and the variable you are solving for is ​V​.

First, calculate the volume of water in the tank, remembering that the radius is half the diameter:

\(V=\pi (2.5\text{ m})^2(3\text{ m})=58.9\text{ m}^3\)

Then, determine the number of seconds in an hour:

\(60\text{ s/min}\times 60\text{ min/hr} = 3600\text{ s}\)

Determine the required volume flow rate:

\(Q=\frac{58.9\text{ m}^3}{3600\text{ s}}=0.01636\text{ m}^3\text{/s}\)

Now determine the area ​A​ of your drainage pipe:

\(A=\pi (0.1)^2 = 0.0314\text{ m}^2\)

Thus from the equation for volume flow rate you have

\(V=\frac{Q}{A}=\frac{0.01636\text{ m}^3\text{/s}}{0.0314\text{ m}^2}=0.52\text{ m/s}=52\text{ cm/s}\)

Water must be forced through the pipe with a quick but plausible speed of about half a meter, or a little over 1.5 feet, per second to properly drain the tank.