Before defining the volumetric flow rate equation and how to apply it, let’s explore the concept of volumetric flow and how the principle is used in engineering and physics.
What exactly is volumetric flow rate?
It can be described as the rate at which a fluid flows. In simple terms, it is a measurement of the fluid volume that passes a specified point per unit of time.
The symbol Q is often used to represent volumetric flow rate (in calculus terms, V̇ is sometimes used to represent the rate of change of volume velocity or volumetric flow).
The SI unit is cubic metres per second (m3/s). In countries that still predominantly use imperial measurements you will commonly see ft3/s or gal./min.
An example of where an engineer may need to use the principle is to measure the volumetric flow rate of oil flowing through a pipeline. They could pick a point in the cross-section of the pipe and measure the volume that passes that point every second.
This is a difficult thing to do in reality as it assumes the constant density of the fluid and it is also difficult to establish a measurement of the precise volume. Therefore, engineers often use a theoretical method to calculate the volumetric flow rate. This is also used in the design of systems that involve fluid flow, e.g. networks of pipelines.
Measuring volumetric flow rate
There are two main ways to practically measure volumetric flow rate.
- Use a bucket and stopwatch. Sounds crude, but it works. For instance, get a 5-litre bucket, mark the 2-litre fill point, then record the time taken to reach the line. This won’t be an entirely accurate measurement, but may be good enough in some circumstances.
- Use a meter. These are often built into pipelines, in an oil plant and water works for example.
Calculating volumetric flow rate using calculus
The above limit defines volumetric flow rate in terms of calculus as the rate of change of volume. It is a scalar quantity as both volume and time are independent of direction.
When considering the volumetric flow rate, it’s important to realise that it refers to the total amount of volume that flows through a section over a given period, not just a simple difference in volume from the initial amount to the final amount, as this would mean that a constant flow would return a value of zero.
Volumetric flow rate equation
The basic volumetric flow rate equation to determine volumetric flow is:
Q = vA
- Q is volumetric flow in m3/s
- v is the flow velocity in m/s
- A is area in m2
A simple example might be water flowing through a rectangular channel that has a 10cm by 5cm cross-section at a velocity of 5 m/s. The area would be: A = 0.1 x 0.05 = 0.005 m2.
Q = vA = 5 x 0.005 = 0.025 m3/s
The simplistic equation above only applies to flat and true plane surfaces. In reality, most fluid carrying vessels such as pipework will be curved in profile. Therefore, a surface integral needs to be used.
The surface integral equation allows for real or imaginary area that is curved, flat, cross-sectional or represents the surface.
Another complication can be if the cross section of the pipe changes, e.g. tapers off along its length. In this case we would assume the density of the fluid is constant and apply the conservation of mass equation. This can also be used to calculate the exit velocity if the inlet velocity and volumetric flow is known.
Applications of volumetric flow rate
The most obvious application of the volumetric flow rate equation is in the oil refining and petroleum industry. They have large networks of pipelines and have to design these with volumetric flow in mind. Extensive calculations are carried out to ensure optimal flow throughout the networks, and systems of meters and gauges are built in to monitor it constantly. They also measure something called mass flow rate in these fluid processing industries, as this can help to assess the effects of chemical injection to improve flow rates.
Volumetric flow rate is also used in the design of internal combustion engines. It helps to decide the optimal valve opening ranges. This is calculated using a complex integral that factors in the time per revolution, the distance from the centre to the tip of the camshaft and the lift, angles and circumference of the valves. This is then compared to volumetric flow rates to assess the design efficiency.