Calculating the Cross-Sectional Area of a Pipe

by Jesse Yoder, PhD

With all the sources of uncertainty in flow measurement, the last thing we need is another source of uncertainty based on the geometry of flow. Yet there is such a source of uncertainty that has to do with calculating the cross-sectional area of a pipe. Volumetric flow is defined as the actual volume of fluid that passes a given point in a pipe per unit time. This is determined by the following formula:

Q = Av

Here A is the cross-sectional area of the pipe and v is the average velocity of the fluid.

The cross sectional area A is as follows:

A = πr²

Substituting for A in the formula for Q, we have:

Q = Av = πr² v

Here Q = Volumetric flowrate

A = the cross-sectional area of the pipe

r = the internal radius of the pipe (also = D/2 where D is the internal diameter of the pipe)

v = the average velocity of the flow

It is really not possible to get around the use of π in this calculation because πr² is the mathematical formula for calculating the area of a circle, and most pipes are round. Of course, this introduces another area of uncertainty in flow measurement, since not all pipes are perfectly round. They may be made “out of round,” or they may have build-up on the inside walls that makes them less than round. This is an important consideration for ultrasonic flowmeters, since it is important for ultrasonic flowmeters in computing flowrate to know the distance from one side of the pipe to the other. If a pipe is not exactly round, this may influence that distance.

Most flow engineers are familiar with the use of π in calculating cross-sectional area. The value of π is the value of the ratio of the circumference of a circle to its diameter. While π has been computed to 105 trillion places, no exact value has ever been found, making π an irrational number with a nonrepeating decimal value. In place of π, when a value has to be assigned to it, most engineers use an approximate value, such as 22/7, 3.14, or 3.1416. As a result, they may choose to use a wavy equals sign (≈) to indicate that the two values are approximately, though not exactly, equal. This is exactly what engineers should do, since so far no one has found a way to precisely calculate the value of π.