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    Calculates locally in your browser · transient conduction
    Heat transfer / transient

    Transient heating and cooling

    Every other thermal page here is steady state. This one is the transient case, and it splits in two. When the internal resistance is small compared with the surface resistance — Biot below 0.1 — the whole body is at one temperature and the answer is a single exponential. When it is not, a gradient forms and the one-term approximation is used.

    ModelOne-termBi4.17Fo0.34
    Temperature now
    165.7 °C
    after 100 minutes
    Biot number
    4.17
    gradient forms — one-term used
    Time constant
    69.7 min
    lumped value, ρcL/h
    Time to target
    51 min
    to reach 100 °C
    Which model applies
    Biot decides whether the body can be treated as one temperature; Fourier decides whether the series has converged.
    166 °C
    Which model applies
    QuantityValueNote
    Initial temperature20 °Cas entered
    Ambient temperature200 °Cas entered
    Characteristic length50.0 mmradius
    Convection coefficient50 W/m²·Kas entered
    Conductivity0.60 W/m·Kas entered
    Biot number4.167hL/k
    Lumped modelNot validBi below 0.1
    Time constant τ4180 sρcL/h
    Time elapsed100 minas entered
    Fourier number0.344αt/L²
    ζ₁ and C₁2.53, 1.73one-term coefficients
    Temperature165.7 °Ccentre, one-term
    Time to target51.4 min100 °C

    Biot is 4.17, above 0.1, so a temperature gradient forms and the lumped model would be optimistic. The one-term approximation is used, though the Fourier number is below 0.2 where it is least accurate.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Temperature against time
    Exponential once the gradient settles; the early part is where the lumped model is optimistic.
    Temperature against time0.00044.4588.90133.35177.80222.2410 min10 min30 min30 min60 min60 min120 min120 min240 min240 min°C
    Centre against surface
    The gap between them is the internal gradient — the reason Biot matters.
    Centre against surface0.00039.1178.21117.32156.43195.53165.71CentreCentre157.16LumpedLumped°C
    Canonical equations
    The exact relationships implemented by this engine.

    Bi = hL/k · Below 0.1 is lumped.

    θ = e^(−t/τ), τ = ρcL/h · One exponential.

    θ₀ = C₁ e^(−ζ₁² Fo) · Valid for Fo > 0.2.

    Fo = αt/L² · α = k/ρc.

    engineering methodology

    A clear path from inputs to results.

    Every other thermal page here is steady state. This one is the transient case, and it splits in two: below a Biot number of 0.1 the body is at one temperature, and above it a gradient forms.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the initial and ambient temperatures.
    2. Set the convection coefficient and the material properties.
    3. Choose the geometry and the characteristic length.
    4. Read the temperature at the elapsed time and the time to the target.

    The formula

    θ=e−t/τ  or  θ0=C1e−ζ12Fo\theta = e^{-t/\tau} \;\text{or}\; \theta_0 = C_1 e^{-\zeta_1^2 Fo}

    Reference: Lumped capacitance and the one-term approximation of the exact series; coefficients from Incropera & DeWitt Table 4.1.

    Worked example

    A 50 mm sphere of water in 200 °C air

    Inputs
    • 20 °C initial, 200 °C ambient, h = 50.
    • Water: k 0.6, ρ 1,000, c 4,180. Radius 50 mm.
    Calculation path
    1. 1

      The Biot number is 50 × 0.05/0.6 = 4.17, far above 0.1.

    2. 2

      So the lumped model is invalid and the one-term approximation is used.

    3. 3

      The sphere coefficients at Bi 4.17 are ζ₁ = 2.53 and C₁ = 1.73.

    4. 4

      After 100 minutes the Fourier number is 0.344 and the centre reaches about 166 °C.

    5. 5

      The lumped model would have said 200 °C — a 34 K error, which is why the Biot check matters.

    Result

    Bi 4.17 · 166 °C after 100 min · lumped model invalid.

    Interpreting the result

    The Biot number is a ratio of internal to external resistance. Below 0.1 the body conducts heat far faster than the surface can shed it, so it stays uniform; above it the surface runs ahead of the centre and the one-term approximation replaces the exponential.

    Assumptions & limitations

    • Constant properties and a constant convection coefficient.
    • One-dimensional conduction in the chosen geometry.
    • A uniform initial temperature.
    • No radiation, no internal heat generation and no phase change.

    Frequently asked questions

    What does the Biot number mean physically?

    The ratio of internal conduction resistance to external convection resistance. A small Biot means the surface resistance dominates, so the body is nearly isothermal.

    What if the time is very short?

    Neither model is right — the surface responds almost immediately and the centre does not. A semi-infinite solid solution is the appropriate one.

    How do I choose the characteristic length?

    Half-thickness for a plane wall, radius for a cylinder or sphere. Getting it wrong changes the Biot number and therefore the model.

    Why is the one-term approximation limited?

    Because it is the first term of an infinite series. Once the Fourier number exceeds about 0.2 the later terms have decayed and it is accurate to 2%; below that it is not.

    Does this work for food or a casting?

    Yes, with the caveat that the properties change as it heats, and a phase change needs a different treatment entirely.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.