C'mon Schmidt... didnt you pay attention in 8th grade science class? I just know the teacher had you write (not copy & paste ) this breif calculation 100 times on the blackboard.
Materials generally change their size when subjected to a temperature change while the pressure is held constant. In the special case of solid materials, the pressure does not appreciably affect the size of an object, and so for solids, it's usually not necessary to specify that the pressure be held constant.
Common engineering solids usually have coefficients of thermal expansion that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, practical calculations can be based on a constant, average, value of the coefficient of expansion.
[edit] Linear expansionThe linear thermal expansion coefficient relates the change in a material's linear dimensions to a change in temperature. It is the fractional change in length per degree of temperature change. Ignoring pressure, we may write:
where L is the linear dimension (e.g. length) and dL / dT is the rate of change of that linear dimension per unit change in temperature.
The change in the linear dimension can be estimated to be:
This equation works well as long as the linear expansion coefficient does not change much over the change in temperature ΔT. If it does, the equation must be integrated.
[edit] Effects on strainFor solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by and defined as:
where is the length before the change of temperature and is the length after the change of temperature.
For most solids, thermal expansion relates directly with temperature:
Thus, the change in either the strain or temperature can be estimated by:
where
is the difference of the temperature between the two recorded strains, measured in degrees Celsius or kelvins, and is the linear coefficient of thermal expansion in inverse kelvins.
[edit] Area expansionThe area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, we may write:
where A is some area of interest on the object, and dA / dT is the rate of change of that area per unit change in temperature.
The change in the linear dimension can be estimated as:
This equation works well as long as the linear expansion coefficient does not change much over the change in temperature δT. If it does, the equation must be integrated.
[edit] Volumetric expansionFor a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written [5]:
where V is the volume of the material, and dV / dT is the rate of change of that volume with temperature.
This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 °C. This is an expansion of 0.2%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 °C, or 0.004% per degree C.
If we already know the expansion coefficient, then we can calculate the change in volume
where ΔV / V is the fractional change in volume (e.g., 0.002) and ΔT is the change in temperature (50 C).
The above example assumes that the expansion coefficient did not change as the temperature changed. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, then the above equation will have to be integrated:
where T0 is the starting temperature and αV(T) is the volumetric expansion coefficient as a function of temperature T.
[edit] Isotropic materialsFor exactly isotropic materials, and for small expansions, the linear thermal expansion coefficient is almost exactly one third the volumetric coefficient.
This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, one-third of the volumetric expansion is in a single axis (a very close approximation for small differential changes). As an example, take a cube of steel that has sides of length L. The original volume will be V = L3 and the new volume, after a temperature increase, will be
We can make the substitutions ΔV = αVL3ΔT and, for isotropic materials, ΔL = αLLΔT. We now have:
Since the volumetric and linear coefficients are defined only for extremely small temperature and dimensional changes, the last two terms can be ignored and we get the above relationship between the two coefficients. If we are trying to go back and forth between volumetric and linear coefficients using larger values of ΔT then we will need to take into account the third term, and sometimes even the fourth term.
Similarly, the area thermal expansion coefficient is 2/3 of the volumetric coefficient.
This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just L2. Also, the same considerations must be made when dealing with large values of ΔT
