Heat conduction calculator

What is Fourier’s law of heat conduction?

The rate of heat flow through a slab is P = k × A × (Th − Tc) ÷ d: the thermal conductivity k of the material, times the area A, times the temperature difference between the two faces, divided by the thickness d (OpenStax University Physics, Volume 2, equation 1.9). OpenStax Example 1.10: a polystyrene foam icebox with 0.950 m² of walls, 2.50 cm thick, holds ice at 0 °C in a 35.0 °C car trunk. With k = 0.035 W/(m·°C) the heat flow is 46.6 W, about 4.0 × 10⁶ J in a day, enough to melt 12 kg of ice (the book prints 12.1 kg because it rounds the 46.55 W to 46.6 before multiplying). Pick “Find: heat flow”, the material and the numbers to repeat it in inches and °F.

How do I find the heat flow through a wall with several layers?

Add the R-values of the layers: R = d ÷ k for each one, and the total is their sum, because the same heat has to cross every layer in turn (OpenStax Example 1.11). The flow per unit area is ΔT ÷ R-total, and each layer takes its share of the temperature drop. In Example 1.11 a steel rod (k = 80) and an aluminum rod (k = 220), each 0.25 m long, are joined end to end between 100 °C and 20 °C: R = 3.13 × 10⁻³ and 1.14 × 10⁻³ m²·°C/W, so the 7.85 × 10⁻⁵ m² rods pass 1.47 W and the joint sits at 41.3 °C. Use the Layered wall tab for that, or for a wall built of plasterboard, insulation and sheathing.

How do I convert R-values between US and metric units?

US R-values are in ft²·°F·h/BTU and metric ones (RSI) in m²·K/W: 1 ft²·°F·h/BTU = 0.1761 m²·K/W, so R-13 is 2.29 m²·K/W. The U-value is the inverse, 1 ÷ R: one BTU/(h·ft²·°F) equals 5.678 W/(m²·K) (both factors from NIST Special Publication 811, Appendix B.8), which makes the U-value of an R-13 wall 0.0769 BTU/(h·ft²·°F), or 0.437 W/(m²·K). The R-value converter tab does it in either direction. OpenStax notes that US R-values are usually quoted without units, for example R-11 for 3.5-inch fiberglass batts and R-19 for 6.5-inch batts.

Why doesn’t the R-value I calculate match the label on the insulation?

The conductivities in the table are representative values near 0 °C, and real insulation varies with density, moisture and temperature. For example, 3.5 inches of glass wool at k = 0.042 W/(m·°C) works out to about R-12.0, while OpenStax quotes R-11 for 3.5-inch fiberglass batts. Studs, air gaps and surface air films are not part of the calculation either. For a real wall use the R-value printed on the product; choose “Layer with a known R-value” and type it in.

What does this calculator leave out?

Only conduction through flat layers: no convection or radiation, no air films on the surfaces, and no thermal bridging through studs. The materials are the 22 substances of OpenStax Table 1.5 (wood is a range, 0.08 to 0.16, so both ends are listed); for anything else choose “Other material” and type its k. To estimate the heat loss of a whole house from its R-values use the heat loss calculator; to see what insulation level is recommended, the insulation calculator; and to see how much a given amount of heat warms or melts, the specific heat and latent heat calculators.

Sources: OpenStax University Physics, Volume 2, section 1.6 (equation 1.9, Table 1.5 and Examples 1.10 and 1.11); NIST Special Publication 811, Appendix B.8 (conversion factors for R-value, U-value and conductivity). Related: heat loss calculator, insulation calculator, specific heat calculator and latent heat calculator.