Thermal expansion is calculated using different formulas depending on whether you're dealing with linear, area, or volume expansion. However, all calculations involve the original dimension, the change in temperature, and a coefficient of thermal expansion specific to the material.
Linear Expansion
Linear expansion refers to the change in length of a solid due to a change in temperature. The formula is:
ΔL = αL₀ΔT
Where:
- ΔL is the change in length.
- α is the coefficient of linear expansion (a material property, typically given in units of 1/°C or 1/K).
- L₀ is the original length.
- ΔT is the change in temperature (final temperature - initial temperature).
Example:
Imagine a steel bridge section that is 20 meters long at 20°C. If the temperature rises to 40°C, how much longer will the bridge section be? The coefficient of linear expansion for steel is approximately 12 x 10⁻⁶ /°C.
ΔL = (12 x 10⁻⁶ /°C) (20 m) (40°C - 20°C)
ΔL = (12 x 10⁻⁶ /°C) (20 m) (20°C)
ΔL = 0.0048 m or 4.8 mm
So, the bridge section will expand by 4.8 millimeters.
Area Expansion
Area expansion refers to the change in area of a solid due to a change in temperature. The formula is:
ΔA = γA₀ΔT
Where:
- ΔA is the change in area.
- γ is the coefficient of area expansion (approximately 2α).
- A₀ is the original area.
- ΔT is the change in temperature.
Since γ is approximately 2α, you can also write it as:
ΔA = 2αA₀ΔT
Volume Expansion
Volume expansion refers to the change in volume of a solid, liquid, or gas due to a change in temperature. The formula is:
ΔV = βV₀ΔT
Where:
- ΔV is the change in volume.
- β is the coefficient of volume expansion (approximately 3α for solids).
- V₀ is the original volume.
- ΔT is the change in temperature.
Again, for solids, since β is approximately 3α, you can rewrite it as:
ΔV = 3αV₀ΔT
For liquids and gases, β values are typically provided.
Key Considerations:
- Units: Ensure all units are consistent. For example, if the coefficient of expansion is given in 1/°C, the temperature change should also be in °C. Length units should also match (e.g., meters, centimeters).
- Coefficient of Expansion: The coefficient of thermal expansion is a material property and varies significantly between different materials. Always use the correct coefficient for the material in question. These values are usually found in engineering handbooks or online databases.
- Temperature Range: The coefficient of thermal expansion can vary slightly with temperature. The formulas above assume a relatively small temperature change. For very large temperature changes, more complex calculations may be required.
In summary, calculating thermal expansion involves identifying the type of expansion (linear, area, or volume), using the appropriate formula, knowing the material's coefficient of thermal expansion, and ensuring consistent units.
