How to Measure Fluid Flowrates

The intended audience of this article is anyone with even a tenuous interest in fluid mechanics. Alternatively, if you don't have any experience with fluids, but enjoy math and can't wait to learn about one method for measuring the flow of fluid in pipes, then this article is for you!

How is the flowrate of fluid flowing through a pipe measured? Have you ever considered this question before? If not, take a moment to think about how this could be accomplished. One obvious technique is to fill a bucket or other container of a specified volume (let’s say 5 gallons) and record the time required to fill the bucket. If it takes one minute to fill the five gallon bucket, then your flowrate is 5 gallons per minute (GPM). However, most useful scenarios don’t allow for filling containers and tracking the time. How else can you measure the fluid flowing through a closed pipe?

Let’s be a little more specific with the question before proceeding. The fluid flowrate we are considering is a volumetric flowrate. The units are volume per time. The volumetric flowrate can be calculated by multiplying the fluid velocity by the pipe cross-sectional area:

Where Q is the volumetric flowrate, A is the pipe cross-sectional area, and v is the fluid velocity. The cross-sectional area is almost always known so the question of determining the fluid flowrate is really a question of how to measure the fluid velocity.

Friction in Pipes

It is not possible to understand most forms of fluid flow measurement without comprehending the basics of how friction affects fluids in pipes. Rub your hands together vigorously for five seconds right now! Do you feel the heat? That is friction. Some of the energy you exerted to move your hands back and forth was transformed into heat energy by friction. As fluid flows through pipes, energy is transferred from the fluid to the surroundings by friction (due to the contact with the pipe walls). The loss in energy results in a lower fluid pressure. The rate at which a fluid loses pressure is related to its velocity. Now we’re onto something!

Pressure Drop and Orifice Plates

The relationship between pressure loss and velocity is the key concept to understand from the previous section. The most common industrial technique to measure fluid flow is to measure the pressure before and after an orifice plate (round plate with a hole in the middle), and calculating a velocity based off of the differential pressure (ΔP). The orifice has one purpose, to cause pressure drop due to friction. The advantage of using an orifice with a known hole-size is that the expected pressure drop across the orifice can be determined experimentally. Flow meters can be calibrated to take the measured pressure drop as input and to calculate the flowrate. Orifice meters are used to measure fluid flow rates all over the world and in every imaginable application.

Orifice Meter Diagram

Figure 1 Example of orifice meter in pipe. The pipe diameter is designated ‘D’, and the orifice diameter is ‘d’. The fluid flowing before the orifice is point ‘1’, and the fluid at the orifice is point ‘2’.

The volumetric flowrates are the same at points 1 and 2. Applying equation 1 to this equation we obtain,

Where the subscript represents the respective point as shown in figure 1.

The Fantastic Mr. Bernoulli

Daniel Bernoulli was born in 1700, the son of Johann Bernoulli who was an early pioneer of Calculus and the nephew of Jacob Bernoulli who discovered the theory of probability (thank you Wikipedia). Daniel is famous for the fluid dynamics principle that carries his name, the Bernoulli Principle. The Bernoulli principle is an energy conservation equation for incompressible, frictionless flow.

A fluid has three types of energy:

1. Flow Energy, P/ρ where P is the fluid pressure and ρ is the fluid density
2. Kinetic Energy, (v^2)/2 where v is the fluid velocity
3. Potential Energy, gz where g is the acceleration due to gravity and z is the height of the fluid relative to a reference point.

The Bernoulli equation states that the sum of the flow, kinetic, and potential energies of a fluid remain constant along a streamline during steady flow (Cengel & Cimbala). With that being explained, Bernoulli’s equation can be applied to figure 1 to obtain,

Both potential energy terms can be eliminated because there is no elevation change between points 1 and 2,

Bringing it all together

Equation 4 relates velocity to pressure. Both pressures are measured, but neither velocity is known. However, Equation 2 can be solved for one of the two velocities and then plugged into Equation 4 to limit the number of unknown variables to a single velocity. The first step is to solve Equation 2 for v1 in terms of v2, the diameter of the pipe, and the diameter of the orifice,

Where β is defined as the ratio of the orifice diameter and the pipe diameter for convenience.

Equations 4 and 5 can be combined and rearranged to form an expression for the fluid velocity as a function of the differential pressure, fluid density, orifice diameter, and pipe diameter.

Equations 6 and 1 can be combined to obtain an expression for the fluid flowrate through a pipe. A correction factor, called the discharge coefficient, is required for accuracy. The velocity calculated by Equation 6 assumes no energy losses due to friction (necessary assumption for the Bernoulli equation to be valid). The discharge coefficient corrects for friction losses. When the Reynolds number is greater than 30,000, the discharge coefficient can be assumed to be 0.61 for sharp-edged orifice plates.

Where Cd is the discharge coefficient correction factor.

Well that was fun! If you are still reading at this point, then why not try the practice problem below! Post your answer in the comments.

Practice Problem

Calculate the flowrate in GPM of water at 25 ℃ flowing through a 10 cm inner diameter pipe. The pipe has a sharp-edged orifice with a 3 cm diameter hole. The pressure upstream of the orifice is 500 KPa and the pressure at the orifice is 495 KPa. Assume a discharge coefficient of 0.61.