What Is the Coefficient of Linear Expansion?

Many materials, especially metals, physically expand when heated, due to the increased kinetic energy of the atoms. This expansion moves outward in all three dimensions, although not necessarily to the same degree. The coefficient of linear expansion is the value that correlates the difference in length of an object to the difference in temperature of the object when the two length measurements were taken. A larger value means that the material expands more over a set temperature rise than a material with a lower coefficient.

Strictly speaking, the coefficient of linear expansion is also a function of temperature, but for most materials, it can be considered a constant for the range of 32° to 212°F (0° to 100°C). Liquids also expand, and the three-dimensional equivalent, the coefficient of cuboidal expansion at a given temperature, is used in calculations of volume changes. Gases expand to fill any container they are placed in. As their volume is fixed, gases increase in pressure as temperature rises.

Tables of these values are available in engineering handbooks. The values are given in units of 10,000a’ or in 10^-6 m/m K or in 10^-6in/in °F. The symbol a’ is used in the standard American measurement system. Evaluating an example will help make these units clear.

This value is expressed in length units. The coefficient of linear expansion for brass wire is listed at 18.7 x 10^-6 m/m K and 10.4 x 10^-6in/in °F. The calculation for the expansion in length of a brass wire that is 10 feet (3.048 m) long at 70°F (21.1°C) and is heated to 80°F (26.6°C) is:

10 feet is 120 inches. The brass wire will expand 10.4 x 10^-6 inches per inch of initial length per Fahrenheit degree of temperature rise. 120 + (10.4 x 10^-6) x 120 x 10 = 120.0125 inches.

In metric units, the calculation is 3.048 m + (18.7 x 10^-6) x 3.048 x 10x5/9 = 3.048316 m, which equals 120.0124 inches. The 5/9 in the equation converts a degree Fahrenheit to a Celsius degree.

This difference in length may seem trivial, but when designing elements like power cables that are hundreds of miles or kilometers long and which will experience temperature differences of 150° or more, however, thermal expansion must be taken into consideration. Parts with very tight tolerances, such as in optical devices, must also be protected from temperature variations or accommodate the non-uniform expansion of parts made from differing materials.