How to use this calculator
Pick the geometry, enter the size, the initial and fluid temperatures, h, k and the diffusivity (or density and specific heat), and either the time or the target temperature. Position 0 is the center and 1 the surface.
Formulas
Biot and FourierBi = hL/k, Fo = αt/L²
Centerθ0 = (T0 − T∞)/(Ti − T∞) = C1e−ζ₁²Fo
Eigenvaluewall ζ tan ζ = Bi; cylinder ζJ₁/J₀ = Bi; sphere 1 − ζ cot ζ = Bi
Semi-infinite(T − Ts)/(Ti − Ts) = erf(x/2√(αt))
Worked example
8 cm brass plate (L = 4 cm) at 200 °C cooled in 70 °C oil with h = 500 W/m²·K for 7 minutes (the default)
- Bi = 500 × 0.04/110 = 0.182; ζ1 = 0.4139, C1 = 1.0284.
- Fo = 3.39×10⁻⁵ × 420/0.04² = 8.90.
- θ0 = 1.0284e−0.4139² × 8.90 = 0.2239, so T0 = 70 + 0.2239(130) = 99.1 °C.
Frequently asked questions
When is the one-term solution accurate?
For Fo > 0.2, within about 2%. For shorter times use more terms or the semi-infinite solution.
Why don’t my ζ₁ values match the table exactly?
Tables list rounded values at a few Bi; this calculator solves the eigenvalue equation exactly for your Bi, so it may differ in the last digit.