# Flow Measurement

Flow measurement is a measurement of the volume of fluid that passes a given point over time, or in other words the volume flow rate. This article explores the basic concepts of flow measurement and the basic tools used for its measurement.

## Contents |

## Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 381-389.

## Flow Rate

^{[1]}Noting that volume flow rate is given by:

Where:

- A = Cross Sectional Area (m²)
- v = Fluid Velocity (m/s)

It is easy to see that flow measurement can be found by measuring velocity instead. It is often the case then that flow meters don't measure volume flow rate directly, but rather find the fluid velocity and use that to calculate the volume flow rate.

Note also that from continuity, under incompressible flow, the flow rate is constant throughout all pipe (or flow) sections. Hence:

## Flow Measurement Tools

The most basic flow meter is the Pitot tube (or Pitot Static Probe), which uses pressure differences to find the fluid velocity. Similarly, obstruction flow meters such as an Orifice, Venturi and Nozzle flow meters use a similar approach with manometers. The derivations of the calculations involved in finding the flow rate are discussed below.

### Pitot-Static Probe

^{[2]}A Pitot-Static Probe is essentially just an upside down L-shaped tube inserted into a pipe (or wherever a fluid flows). The tube's entrance is faced directly in front of the fluid flow so that the fluid hits the probe straight on where the fluid comes to a full stop (due to the no slip condition). As a result, the pressure at that point is the **stagnation pressure**. Further down the probe are holes where the fluid exerts pressure equal to the **static pressure** (fluid gauge pressure). The result is that the probe measures both stagnation and static pressures for the fluid and the difference between the two yields a relationship to the fluid velocity.

The analysis for such a system is simple (and in fact has already been alluded to, see velocity calculation). If we choose point 1 to be the stagnation pressure point and point 2 to be the static pressure point, then we can write Bernoulli's Equation between the two points:

and noting that at point 1, V = 0 and that z_{1} = z_{2}, we can rearrange to find that the fluid velocity is:

The flow rate is then given by the velocity times the cross sectional area of the pipe (or flow area). The Pitot-Static Probe is simple, inexpensive and reliable as long as the tube is aligned with the maximum fluid velocity (remember that fluid velocity increases from zero at the boundaries to a maximum at some point in the flow).

### Obstruction Flow Meters

^{[3]}Obstruction flow meters take advantage of the continuity equation, noting that the flow rate remains constant regardless of cross sectional flow area since the fluid velocity changes to compensate. Then the analysis is essentially the same as the Pitot-Static tube, except that using the continuity equation we can find an expression for the velocity at one cross section with the velocity at another cross section. That is,

Where:

- D = Diameter of large area
- d = Diameter of smaller area

Then using Bernoulli's Equation and substituting this expression for velocity, we get the result that:

Where:

- β = d/D

Then the flow rate is given by the velocity times the cross sectional area. However since the cross section changes, some of the fluid becomes completely motionless and there are some losses. To compensate, the **discharge coefficient** is introduced (and whose value is found independently) giving the equation:

Note that the pressure difference is found by using pressure gauges or manometers. Note also that while the Orifice, Venturi and Nozzle are essentially all obstruction flow meters, they each have slightly different shapes so that the losses are minimised.

## References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).