KAHIBARO
Table of Contents
What Flow Rate Means
When a fluid moves through a pipe, channel, or any opening, we often want to know how much fluid passes through in a certain amount of time. This idea is called flow rate.
There are two common ways to describe flow rate. One is by volume, and the other is by mass. In many beginner problems, the most common quantity is the volume flow rate.
The volume flow rate tells us how much volume of fluid crosses a surface each second. It is written as
$$
Q = \frac{\Delta V}{\Delta t}
$$
where $Q$ is the volume flow rate, $\Delta V$ is the volume that passes, and $\Delta t$ is the time interval.
Its SI unit is
$$
\text{m}^3/\text{s}
$$
If water flows through a pipe and $0.02 \, \text{m}^3$ passes in $1 \, \text{s}$, then the flow rate is
$$
Q = 0.02 \, \text{m}^3/\text{s}
$$
Volume Flow Rate in a Pipe
To understand flow rate in a pipe, imagine fluid moving with speed $v$ through a pipe of cross sectional area $A$. In a short time $\Delta t$, the fluid moves a distance
$$
\Delta x = v \Delta t
$$
The volume that passes through the pipe is the volume of a cylinder:
$$
\Delta V = A \Delta x = A v \Delta t
$$
Now divide both sides by $\Delta t$:
$$
Q = \frac{\Delta V}{\Delta t} = Av
$$
This is one of the most important formulas in fluid motion.
Important relation for steady flow through a cross section:
$$
Q = Av
$$
where $Q$ is volume flow rate, $A$ is cross sectional area, and $v$ is fluid speed.
This formula shows that a larger pipe can carry more fluid at the same speed, and a faster fluid gives a greater flow rate in the same pipe.
Mass Flow Rate
Sometimes volume is not the best measure, especially when density matters. Then we use mass flow rate, which tells us how much mass passes per second.
It is written as
$$
\dot{m} = \frac{\Delta m}{\Delta t}
$$
where $\dot{m}$ is the mass flow rate.
Since mass equals density times volume,
$$
\Delta m = \rho \Delta V
$$
so
$$
\dot{m} = \rho \frac{\Delta V}{\Delta t} = \rho Q
$$
Using $Q = Av$, we also get
$$
\dot{m} = \rho Av
$$
Its SI unit is
$$
\text{kg}/\text{s}
$$
Mass flow rate formulas:
$$
\dot{m} = \rho Q
$$
and
$$
\dot{m} = \rho Av
$$
Physical Interpretation
Flow rate measures how quickly fluid is transported. A small stream from a tap has a low flow rate. A fire hose has a much larger flow rate.
You can think of flow rate as answering one of these questions:
| Question | Quantity |
|---|---|
| How much volume passes each second? | Volume flow rate, $Q$ |
| How much mass passes each second? | Mass flow rate, $\dot{m}$ |
For liquids like water, density usually changes very little, so volume flow rate and mass flow rate are simply proportional. For gases, density can change more noticeably, so the distinction becomes more important.
Everyday Examples
Suppose a shower head delivers water at
$$
Q = 1.5 \times 10^{-4} \, \text{m}^3/\text{s}
$$
In $10 \, \text{s}$, the total volume delivered is
$$
\Delta V = Q \Delta t = \left(1.5 \times 10^{-4}\right)(10)
$$
$$
\Delta V = 1.5 \times 10^{-3} \, \text{m}^3
$$
If a pipe has cross sectional area
$$
A = 4.0 \times 10^{-4} \, \text{m}^2
$$
and the water speed is
$$
v = 3.0 \, \text{m/s}
$$
then the volume flow rate is
$$
Q = Av = \left(4.0 \times 10^{-4}\right)(3.0)
$$
$$
Q = 1.2 \times 10^{-3} \, \text{m}^3/\text{s}
$$
Interpreting the Formula $Q = Av$
The formula $Q = Av$ is simple, but it is very useful. It connects geometry and motion.
If $A$ increases while $Q$ stays fixed, then $v$ must decrease.
If $A$ decreases while $Q$ stays fixed, then $v$ must increase.
This is why water flows faster through a narrow nozzle than through a wide hose.
The deeper idea of how flow rate stays related at different points in a pipe belongs to the continuity equation, which is treated separately. Here, the key point is just the meaning of flow rate at one cross section and how to compute it.
Units and Conversions
Flow rate can appear in many units in real life. Engineers and everyday devices often use liters per second or liters per minute.
Some useful conversions are:
| Unit | Equivalent |
|---|---|
| $1 \, \text{m}^3$ | $1000 \, \text{L}$ |
| $1 \, \text{L}$ | $10^{-3} \, \text{m}^3$ |
| $1 \, \text{m}^3/\text{s}$ | $1000 \, \text{L/s}$ |
For example, if
$$
Q = 2.0 \times 10^{-3} \, \text{m}^3/\text{s}
$$
then in liters per second,
$$
Q = 2.0 \, \text{L/s}
$$
because
$$
2.0 \times 10^{-3} \, \text{m}^3/\text{s} \times 1000 \, \text{L/m}^3 = 2.0 \, \text{L/s}
$$
Simple Geometric Picture
The idea of flow rate is easiest to see if we imagine a moving plug of fluid crossing an area.
In time $\Delta t$, the shaded fluid moves forward. The volume of that moving section is the area times the distance traveled, which leads directly to $Q = Av$.
Common Mistakes
A common mistake is to confuse speed with flow rate. Speed tells how fast the fluid moves. Flow rate tells how much fluid passes per unit time. A fluid can move quickly in a narrow pipe and still have a smaller flow rate than a slower fluid in a very wide pipe.
Another common mistake is to forget which area to use. In the formula $Q = Av$, the area must be the cross sectional area perpendicular to the flow.
Do not confuse these quantities:
Speed: $v$, unit $\text{m/s}$
Volume flow rate: $Q$, unit $\text{m}^3/\text{s}$
Mass flow rate: $\dot{m}$, unit $\text{kg/s}$
Summary
Flow rate describes how much fluid passes through a surface in a given time. The volume flow rate is
$$
Q = \frac{\Delta V}{\Delta t}
$$
and for fluid moving with speed $v$ through area $A$,
$$
Q = Av
$$
The mass flow rate is
$$
\dot{m} = \frac{\Delta m}{\Delta t} = \rho Q = \rho Av
$$
These relations are the basic language for describing moving fluids and are the starting point for later ideas in fluid dynamics.