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Compound measures
Converting metric units Conversion of measurement units Volume of a rectangular prism Volume of a cylinder Volume of a coneFlow rate
Here you will learn about flow rate, including how to calculate it and how to use it to solve problems involving volume and capacity.
Students will first learn about flow rate as part of algebra in high school.
What is flow rate?
Flow rate (also called volumetric flow rate) is the term used to describe the rate at which an amount of fluid or substance flows into or out of an object during a specific period of time.
To calculate flow rate, we can use information about the change in capacity (or volume) of the substance and the amount of time taken for that change to occur.
The flow rate formula is
\text{Flow rate}=\cfrac{\text{change in volume}}{\text{time taken}}This flow rate equation is also commonly written as Q=\cfrac{V}{t} with flow rate Q, Volume V, and unit time t.
It is important to look at the unit of time being used and whether the question refers to a volume of fluid. The flow rate could be given as a compound measure such as depth per second or volume per minute.
Typical mathematics questions involve finding the time taken for a shape to be filled or for the depth of a container to reach a specific height.
For example,
A faucet is used to fill a container in the shape of a rectangular prism measuring 1.5\mathrm{~m} by 2\mathrm{~m} by 0.4\mathrm{~m}. The faucet releases water at a flow rate of 5 liters per minute.
Find the time taken for the container to be filled.
What is flow rate?
Common Core State Standards
How does this relate to high school math?
- High School – Algebra – Creating Equations (HS.A.CED.A.1)
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- High School – Algebra – Creating Equations (HS.A.CED.A.2)
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- High School – Algebra – Creating Equations (HS.A.CED.A.4)
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V=IR to highlight resistance R.
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DOWNLOAD FREEHow to solve flow rate problems
In order to solve flow rate problems:
- Use information provided to calculate the flow rate.
\textbf{Flow rate}=\cfrac{\textbf{change in volume}}{\textbf{time taken}} - Use information provided to calculate volume that will be changing.
- Calculate the required value.
Flow rate examples
Example 1: finding the time taken to fill a container
Water flowing from a garden hose fills a 10 -liter bucket in 2 minutes. Find the time it would take to fill a container in the shape of a rectangular prism measuring 2 meters by 1 meter by 50\mathrm{~cm}.
- Use information provided to calculate the flow rate.
\textbf{Flow rate}=\cfrac{\textbf{change in volume}}{\textbf{time taken}}
The flow rate of the water is 10 \div 2=5 liters per minute.
2Use information provided to calculate volume that will be changing.
Volume of the rectangular prism =2 \times 1 \times 0.5=1 \mathrm{~m}^3.
3Calculate the required value.
We can work out the time by dividing the volume by the flow rate.
\text{Time taken}=\cfrac{\text{volume}}{\text{flow rate}}We need to convert the volume so that the units are liters.
1\mathrm{~m^3}=1000\mathrm{~liters} 1000\div 5=200So the time will be 200 minutes or 3 hours 20 minutes.
Example 2: finding the volume after an allotted time
A faucet has been left running. The flow measurement of water from the faucet is 250\mathrm{~cm^{3}}\text { per minute. }
How many liters of water will come out of the faucet in half an hour?
Use information provided to calculate the flow rate.
We have been given the flow rate: 250\mathrm{~cm^{3}}\text { per minute. }
Use information provided to calculate volume that will be changing.
Here we need to find the volume. But we have been given a time – half an hour is 30 minutes.
Calculate the required value.
We need to adapt the flow rate formula:
\text{Volume}=\text{flow rate}\times \text{time taken}
So,
\begin{aligned} \text{Volume}&=\text{flow rate}\times \text{time taken} \\\\ \text{Volume} &= 250\times 30 \\\\
\text{Volume}&=7500\mathrm{~cm}^3 \end{aligned}
Example 3: finding the height after an allotted time
Water flowing from a faucet fills a 500\mathrm{~ml} jug in 80 seconds.
The same faucet will be used to fill this cylindrical container, with a radius of 12\mathrm{~cm}.
Find the height of the water, h, after 3 minutes.
Use information provided to calculate the flow rate.
\textbf{Flow rate}=\cfrac{\textbf{change in volume}}{\textbf{time taken}}
We will be finding the volume of the cylinder in cm^3, so we will convert the flow rate so that the units match.
500\mathrm{~ml}=500\mathrm{~cm}^3
The flow rate of the water is 500 \div 80=6.25 \mathrm{~cm}^3 per second.
Use information provided to calculate volume that will be changing.
The cross-sectional area of the cylinder is a circle.
The volume of the cylinder of water will be \pi \times {{12}^{2}}\times h=144\pi h\,c{{m}^{3}}.
Calculate the required value.
We need to calculate the volume.
The flow rate is in cm^3 per second, so we need to convert the time to seconds to fit.
3 minutes =180 seconds
We need to adapt the flow rate formula:
\text{Volume}=\text{flow rate}\times \text{time taken}
So,
\begin{aligned} \text{Volume}&=\text{flow rate}\times \text{time taken} \\\\ 144 \pi h &= 6.25\times 180 \\\\ h &= \cfrac{6.25\times 180}{144 \pi} \\\\ h&=2.4867… \\\\ h&=2.49 \, \text {cm} \, \text{(to the nearest hundredth)} \end{aligned}
Example 4: finding the time taken for a container to empty
A container full of water is found to have a leak.
The container is in the shape of a trapezoid prism shown below and water is leaking from point P.
4 minutes after being completely full, the water level in the container has dropped by 2\mathrm{~m}.
Assuming the flow rate of the leak is constant, find the total time for all the water to leak out of the container.
Use information provided to calculate the flow rate.
\textbf{Flow rate}=\cfrac{\textbf{change in volume}}{\textbf{time taken}}
As the shape of the container is a trapezoid, use the volume of the water lost to find the flow rate.
The volume lost =10 \times 12 \times 2=240 \mathrm{~cm}^3.
The flow rate of the water is 240 \div 4=60\mathrm{~cm}^3 per minute.
Use information provided to calculate volume that will be changing.
The cross-sectional area of the container is a trapezoid.
The volume of the trapezoid =\cfrac{20+16}{2} \times 10 \times 12=2160 \mathrm{~cm}^3.
This is the volume of the full tank.
Calculate the required value.
2160 \div 60=36 minutes
The tank would take a total of 36 minutes to empty.
Teaching tips for flow rate
- Begin by explaining that flow rate is a key concept in fluid dynamics and fluid mechanics, measuring how much fluid moves through a point in a system during a given time.
- Teach students the relationship between flow rate and flow velocity by explaining that flow rate is determined by multiplying the fluid’s velocity by the cross-sectional area. If the area decreases, the velocity increases, and vice versa.
- Demonstrate how a flow meter measures flow rate in real-world applications, and introduce the SI unit for flow rate, typically cubic meters per second (m^3/s).
- Explain Bernoulli’s equation and how it applies to flow rate, showing that as pressure in a fluid decreases, the fluid velocity increases, and this affects the flow rate.
- Walk students through the derivation of the flow rate formula.
Easy mistakes to make
- Forgetting to convert units of volume or time
It is important to ensure the correct units are being used throughout a problem. If the flow rate is given in cm^3 per minute, it is important to make sure all measurements are converted to centimeters and that time is given in minutes.
Practice flow rate questions
1. A faucet can fill a 500\mathrm{~ml} jug in 8 seconds. Calculate the flow rate giving the units.
0.0625 ml per second
62.5 ml per second
0.016 ml per second
62.5 seconds per ml
Use the formula: \text{Flow rate}=\cfrac{\text{change in volume}}{\text{time taken}}
\text{Flow rate}=\cfrac{500}{8}=62.5
2. Fluid flows from a faucet at a flow rate of 30\mathrm{~cm}^3 per second.
How long will it take to fill a container with a volume of 195\mathrm{~cm}^3?
6.5 seconds
5,850 seconds
0.154 seconds (to the nearest thousandth)
195 seconds
Use the formula \text{Flow rate}=\cfrac{\text{change in volume}}{\text{time taken}}
and rearrange to find the time taken.
\text{Time taken}=\cfrac{\text{Change in volume}}{\text{Flow rate}}=\cfrac{195}{30}=6.5
3. A hose pipe is used to fill a rain barrel. Water flows at a rate of 3.5 liters per minute and fills the rain barrel in 12 minutes and 52 seconds. Calculate the volume of the rain barrel to the nearest liter.
51 liters
44 liters
45 liters
42 liters
Convert the time to decimalized form, then use the formula:
\text{Flow rate}=\cfrac{\text{change in volume}}{\text{time taken}}
and rearrange to find the volume.
\text{Volume}=\text{flow rate}\times \text{time taken}=3.5\times 12\cfrac{52}{60}=45.0333…
The volume will be 45 liters, to the nearest liter.
4. A hose can produce a flow rate of 6 cubic meters per hour. How long would it take the same hose to fill the swimming pool below? The cross-section of the swimming pool is a trapezoid. Give your answer in hours and minutes.
8 hours 45 minutes
9 hours
8 hours 15 minutes
8 hours 25 minutes
The volume of the swimming pool is \text{Volume}=\cfrac{1}{2}\times (4+1.5)\times 6\times 3=49.5.
Using the formula, \text{Flow rate}=\cfrac{\text{change in volume}}{\text{time taken}}
we rearrange to find the time taken.
\text{Time taken}=\cfrac{\text{change in volume}}{\text{flow rate}}=\cfrac{49.5}{6}=8.25
So the time is 8.25 hours or 8 hours 15 minutes.
5. A swimming pool has dimensions shown in the diagram below.
It took 19 hours and 12 minutes to fill the swimming pool with a hose. Find the flow rate of the water from the hose in m^3 per hour.
5\mathrm{~m}^3 per hour
5.02\mathrm{~m}^3 per hour (nearest hundredth)
5.83\mathrm{~m}^3 per hour (nearest hundredth)
5.86\mathrm{~m}^3 per hour (nearest hundredth)
Find the volume of the swimming pool.
\text{Volume}=(7\times 3\times 4)+(\cfrac{1}{2}\times 2\times 3\times 4)=96
Then use the formula
\text{Flow rate}=\cfrac{\text{change in volume}}{\text{time taken}}=96\div 19\cfrac{12}{60}=5.
6. A container in the shape of a cone with radius 6\mathrm{~cm} and height 27\mathrm{~cm} is leaking water from its apex.
The container is originally full and after 20 minutes the depth of water has reduced by 9\mathrm{~cm}. Calculate the flow rate of the leak in cm^3 per minute. Give your answer to 3 significant figures.
39.7\mathrm{~cm}^3 per minute (to 3 sf)
35.8\mathrm{~cm}^3 per minute (to 3 sf)
29.3\mathrm{~cm}^3 per minute (to 3 sf)
24.7\mathrm{~cm}^3 per minute (to 3 sf)
Use similar triangles to find the radius of the reduced cone.
Find the volume of the small cone and the large cone and then subtract the volumes to find the change in volume.
Then we can find the flow rate.
Flow rate FAQs
Flow rate is the volume of fluid that passes through a given point in a system per unit of time.
Flow rate can be calculated using the formula: \text{Flow rate}=\cfrac{\text{Volume}}{\text{Time}}.
Common units for flow rate include liters per second (L/s), gallons per minute (GPM), and cubic meters per hour (m^3/h). The SI unit is cubic meters per second (m^3/s).
Volumetric flow rate measures the volume of fluid passing through a point per unit time (example, liters per second), while mass flow rate measures the mass of fluid passing through a point per unit time (example, kilograms per second).
Flow rate can be affected by factors such as the pipe diameter, fluid viscosity, pressure differences, and the presence of obstructions.
The next lessons are
- Ratio
- Proportion
- Converting fractions, decimals and percents
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