Thermal Expansion Calculator

What the calculator works out

Almost every solid grows a little when it is heated and shrinks when it is cooled, because its atoms vibrate harder and settle slightly further apart. The size of the effect is described by the coefficient of linear thermal expansion (CLTE, often shortened to CTE): the fractional change in length for each kelvin of temperature change.

This calculator has three modes. Linear expansion gives the change in length of a bar, rod, rail or pipe. Volumetric expansion gives the change in volume of a block or a filled container. Density change tells you how much a material's density falls when it warms up, since the same mass now occupies more space.

How to use it

1. Pick the mode: linear expansion, volumetric expansion or density change.
2. Enter the coefficient. The default unit is ppm/K (millionths per kelvin); switch to /K if your source gives a value such as 0.000012.
3. Enter the initial and final temperatures. Each has its own unit: K, °C, °F, °R or °Re, so you can mix scales if your data comes that way.
4. Enter the starting size: a length, a volume or a density, together with its unit.
5. Press Calculate. You get the change and the final value in the same unit you entered. Cooling gives a negative change.

Only the temperature difference matters, not the temperatures themselves. A rise of 80 K is the same as a rise of 80 °C, 144 °F or 144 °R, and the calculator converts all of these before it multiplies. For small changes in a length the result also appears in a smaller unit in brackets, such as millimetres for a change given in metres or inches for a change given in feet, and a small volume change is also shown in cubic millimetres, converted from whichever volume unit you chose.

The formulas

For a modest temperature change the strain is proportional to the temperature difference, so a straight-line model is accurate enough for most design work.

ΔL = α × L₀ × ΔTα is the linear coefficient, L₀ the starting length and ΔT = T₂ − T₁. The final length is L₀ + ΔL.

ΔV = γ × V₀ × ΔT, with γ ≈ 3αFor an isotropic solid each of the three dimensions grows by the same fraction, so the volumetric coefficient is about three times the linear one.

ρ₁ = ρ₀ ÷ (1 + γ × ΔT)Mass stays the same while volume grows, so density falls by the same factor.

The factor of three is a first-order approximation. Compounding the three growths exactly gives (1 + αΔT)³ − 1, which is slightly larger than 3αΔT; with α = 23 ppm/K and ΔT = 80 K the two differ by only about 0.001 percentage points (0.5530% against 0.5520%).

Worked examples

An aluminium bar. A 10 m aluminium bar (α ≈ 23 ppm/K) is heated from 20 °C to 100 °C, so ΔT = 80 K. The growth is 10 × 23 × 10⁻⁶ × 80 = 0.0184 m, or 18.4 mm, and the final length is 10.0184 m.

A steel rail. A 30 m rail (α ≈ 12 ppm/K) goes from −10 °C on a winter night to 40 °C in summer sun, a swing of 50 K. The rail grows 30 × 12 × 10⁻⁶ × 50 = 0.018 m, which is 18 mm. This is why railways leave expansion gaps or weld rails under controlled tension.

A volume. Two litres of the same aluminium, heated by 80 K, change by 2 × (3 × 23 × 10⁻⁶) × 80 = 0.01104 L. The final volume is 2.01104 L, and the small change is also shown in brackets as 11,040 mm³.

A density. Aluminium at 2,700 kg/m³ and the same 80 K rise: 2,700 ÷ (1 + 0.00552) = 2,685.18 kg/m³, a drop of about 14.82 kg/m³.

Typical expansion coefficients

The values below are rounded figures for temperatures near room temperature. Real alloys, plastics and concretes vary with composition, so use a supplier's data sheet for anything critical. The last column shows what each coefficient means for a 10 m bar heated by 80 K.

Most plastics expand several times more than metals: rigid PVC is commonly quoted at 50 to 80 ppm/K and polyethylene can exceed 100 ppm/K. Invar and fused silica sit at the other extreme, which is why they are used for precision instruments and optical mounts.

Limits and common mistakes

• Coefficients are not constant. They creep upward as a material gets hotter, so one value is only reliable across a limited range.
• Phase changes break the model. Melting, boiling, glass transitions and some crystal changes cause jumps that a single coefficient cannot describe.
• Gases follow different rules. Their expansion depends on pressure; use the ideal gas law calculator instead.
• Anisotropic materials such as wood, composites and some crystals expand differently along each axis, so the three-times rule does not hold.
• Mixing up units. A coefficient of 12 in ppm/K is 0.000012 in /K. Selecting the wrong unit changes the answer by a factor of a million.
• Forgetting that holes grow too. An opening in a plate expands as if it were filled with the same material, so a hot ring slips more easily over a shaft.

The calculator gives the free, unconstrained change. If the part is clamped so it cannot grow, the expansion turns into stress instead, which needs the elastic modulus to evaluate.

Where expansion matters in practice

Bridges carry expansion joints, pipelines include loops or bellows, and long fences and metal roofs use sliding fixings so they can lengthen without buckling. In electronics, mismatched coefficients between a chip and its board can crack solder joints over repeated heating cycles. Even in a liquid-filled thermometer or a fuel tank, volumetric expansion decides how much room to leave for a warm day. For converting between sizes in different units before you start, the length converter and volume converter can help.