Compute transient conduction in a body that is initially at a uniform temperature and is suddenly exposed to convection, using the one-term approximation behind the Heisler charts. Vary the Biot and Fourier numbers to see the centre temperature and the temperature at any position of a plane wall or a sphere update in real time.
Parameters
Body shape
The eigenvalue equation and spatial factor depend on shape
Biot number Bi
Surface convection vs internal conduction, Bi = hL/k
Fourier number Fo
Dimensionless time, Fo = αt/L²
Dimensionless position
x/L for the wall, r/r0 for the sphere (0 = centre, 1 = surface)
Results
—
First eigenvalue ζ₁
—
Coefficient C₁
—
Centre temperature θ₀*
—
Temperature at position θ*
—
Biot number Bi
—
Fourier number Fo
—
Body cross-section & temperature profile — Fo sweep
The dimensionless temperature profile θ*(position) is overlaid on the body cross-section. The centre changes slowest and the convective surface (edge) fastest. The colour gradient shows the temperature level; the vertical line marks the current position.
Centre dimensionless temperature θ₀* and the dimensionless temperature θ* at the chosen position. C₁ is the coefficient, ζ₁ the first eigenvalue, Fo the Fourier number. The spatial factor f is cos(ζ₁·x/L) for the wall and sin(ζ₁·r/r0)/(ζ₁·r/r0) for the sphere.
The transcendental equation that fixes the first eigenvalue ζ₁. The Biot number Bi compares surface convection with internal conduction. The one-term form approximates the transient solution with this first term only and replaces the classic Heisler charts.
What is the Heisler Chart Simulator?
🙋
Heat-transfer class threw "Heisler charts" at me. What are those hard-to-read graphs actually for?
🎓
Put simply, they are a tool for reading "how hot the centre is when you suddenly cool something hot" off a single graph. Take a quenched steel bar dropped into water: the surface cools right away, but the centre lags badly. The Heisler chart lets you read that "time versus centre temperature" relationship straight off the graph instead of computing it. Heisler compiled them back in 1947.
🙋
Reading off a graph — that's the trace-with-a-ruler thing, right? Honestly I can never read it accurately.
🎓
Exactly, that is the weak point. So today we compute it directly with the "one-term approximation", as this tool does. The exact solution of transient conduction is actually an infinite series, but once the Fourier number Fo passes 0.2 the second term and beyond almost vanish. So you keep only the first term — that is the one-term approximation. The chart is just a picture of that first term, so computing it gives the same answer more accurately than reading the chart.
🙋
Got it. But the "Biot number" and "Fourier number" on the left — what is the difference? Both look like heat to me.
🎓
Good question. Think of the Biot number Bi as a "ratio of spaces" and the Fourier number Fo as "time". Bi = hL/k is the ratio of how hard heat escapes at the surface (convection resistance) to how hard heat travels inside (conduction resistance). Small Bi means a nearly uniform interior; large Bi means a big gap between centre and surface. Fo = αt/L² is dimensionless time — raise Fo and the whole body drifts toward the ambient temperature. Move both sliders and watch the curve shape change in the chart above.
🙋
There's also a "first eigenvalue ζ₁". What is that one?
🎓
ζ₁ is the number that fixes the "shape and decay rate" of the temperature distribution. For a wall it is the solution of the transcendental equation ζ₁·tan(ζ₁) = Bi, which you cannot solve by hand, so this tool finds it numerically by bisection. The larger ζ₁ is, the faster exp(−ζ₁²·Fo) decays — that is, the faster it cools. Increasing Bi increases ζ₁: with more heat escaping at the surface, the centre also cools faster, and that physics is built right into the equation.
🙋
So in practice, what is it nice to learn from this?
🎓
Things like "how many minutes to heat a chunk of meat until the centre reaches a safe temperature" or "how many seconds until the centre of a steel part reaches the prescribed quench temperature". The centre lags the most, so if the centre meets the condition the whole body does. Conversely, if you only want to heat-treat the surface, you watch the surface Fo. This tool outputs both the centre θ₀* and the temperature θ* at any position, so you can use it directly for that kind of design decision.
Frequently Asked Questions
A Heisler chart is a classic nomograph for reading off transient conduction in a body that is initially at a uniform temperature and is suddenly exposed to convection. The horizontal axis is the Fourier number Fo, the vertical axis is the centre dimensionless temperature θ0*, and the parameter is the inverse Biot number. This tool computes the one-term approximation behind the chart directly, returning numerical results instead of requiring you to read the graph.
The exact solution of transient conduction is an infinite series, but once the Fourier number Fo exceeds about 0.2 the higher modes have decayed rapidly and the first term alone represents the temperature almost exactly. This is the one-term approximation. For Fo below 0.2 — the early stage just after heating or cooling begins — the higher terms still matter, so the tool shows a warning when Fo < 0.2. The charts themselves also assume Fo ≳ 0.2.
The Biot number Bi = hL/k is the ratio of internal conduction resistance to surface convection resistance. When Bi is small (≲0.1) the interior temperature is nearly uniform and the lumped-capacitance method applies; when Bi is large, large temperature gradients form inside the body. The Fourier number Fo = αt/L² is dimensionless time, indicating how far heat has penetrated into the body. The larger Fo is, the closer the body is to steady state (equilibrium with the surroundings).
θ0* is the dimensionless temperature at the centre of the body (the mid-plane of a wall or the centre of a sphere), given by θ0* = C1·exp(−ζ1²·Fo). θ* is the dimensionless temperature at a position away from the centre, equal to θ0* multiplied by a position-dependent spatial factor. The spatial factor is cos(ζ1·x/L) for a wall and sin(ζ1·r/r0)/(ζ1·r/r0) for a sphere. The centre changes most slowly while the convective surface approaches the ambient temperature fastest.
Real-World Applications
Heat treatment and quenching: In quenching, annealing and tempering of steel, quality depends on whether the centre of the part has reached the prescribed temperature, or whether the centre has passed the martensite transformation temperature during cooling. The centre responds the most slowly, so the Heisler chart (one-term approximation) is used to estimate the centre-temperature history and set hold times and cooling rates. A more aggressive quench (large Bi) makes the centre follow faster, but the temperature gap between surface and centre grows, causing thermal stress and cracking.
Food heating and sterilisation: In retort sterilisation of canned food or roasting, the slowest-heating "cold point" near the centre must reach the safe temperature or the required F-value. The food is approximated as a sphere or a slab, and the Fourier number is converted back to time to predict when the centre reaches the condition. The principle that "if the centre meets the condition, the whole body does" is a basic way of taking a safety margin in food engineering.
Transient response of electronic parts and thermal masses: The tool gives a quick estimate of how many seconds a part that is suddenly subjected to heat generation or an ambient-temperature change takes to bring its interior to equilibrium. A part with small Bi (thin, high-conductivity) has a nearly uniform interior and approaches the lumped-capacitance method; a thick part with large Bi needs the one-term approximation that accounts for the internal gradient.
Pre-study and verification for CAE transient thermal analysis: Before running a finite-element or finite-volume transient conduction analysis, the one-term approximation lets you size up the order of magnitude of the centre temperature. If the detailed analysis differs from this estimate by an order of magnitude, it is a sanity check that points to errors in the initial temperature, the convective boundary condition or the material properties. Conversely, agreement with the approximation gives confidence in the mesh and time step.
Common Misconceptions and Pitfalls
The biggest pitfall is using the one-term approximation in the early stage when Fo is small. The tool's calculation keeps only the first term of the series, and this is accurate only when the Fourier number Fo is roughly 0.2 or above. For smaller Fo — right after cooling or heating begins — the second and third higher modes are still large, and the first term alone underestimates (or overestimates) the temperature. The tool shows a warning for Fo < 0.2, but use it in that region only with the understanding that the error can reach several to more than ten percent.
Next, confusing the characteristic length L of the Biot number. The L used in the wall's Bi and Fo is the "half-thickness" (half of the plate thickness), not the plate thickness itself. For a sphere the radius r0 is used. By contrast, the characteristic length used in the lumped-capacitance method (the Bi ≲ 0.1 test) is "volume / surface area", which becomes the half-thickness for a wall or r0/3 for a sphere — a different definition from the L used in the Heisler chart. Even with the same name "Biot number", a different characteristic length shifts the value, so always check which definition of Bi you are using.
Finally, θ* is not the actual temperature itself, but a dimensionless temperature. The θ0* and θ* this tool outputs are the dimensionless quantity θ* = (T−T∞)/(Ti−T∞): closer to 0 means it has reached the ambient temperature T∞, closer to 1 means it is still at the initial temperature Ti. To convert back to the actual temperature you need T = T∞ + θ*·(Ti−T∞). The tool also assumes ideal conditions: constant material properties (conductivity and thermal diffusivity), a uniform initial temperature and a constant convective heat-transfer coefficient. Phase change, temperature-dependent properties or a non-uniform initial condition fall outside its scope and require numerical analysis.
How to Use
Enter Biot number (Bi) by defining h/(k/L) where h is convection coefficient (W/m²·K), k is thermal conductivity (W/m·K), and L is characteristic length (m).
Set Fourier number (Fo) as αt/L² where α is thermal diffusivity (m²/s) and t is elapsed time (s).
Specify normalized position x/L between 0 (centre) and 1 (surface) to extract temperature profile; simulator returns first eigenvalue ζ₁, coefficient C₁, centre temperature θ₀*, and local temperature θ* using analytical Heisler solutions.
Worked Example
Aluminum slab (k=237 W/m·K, α=9.7×10⁻⁵ m²/s) of thickness L=0.05 m initially at 500 K exposed to ambient air at 300 K with h=50 W/m²·K. At t=60 s: Bi=50×0.05/237≈0.0106 (lumped capacitance valid), Fo=9.7×10⁻⁵×60/(0.05)²≈2.33. At x/L=0.5 (mid-plane), centre temperature θ₀*≈0.15 (cooling to ~390 K), with C₁≈1.008 and ζ₁≈0.1028 rad from chart interpolation.
Practical Notes
For Bi<0.1, use lumped capacitance method instead; Heisler charts assume finite internal resistance (Bi>0.1).
Verify Fo convergence: single-term approximation becomes accurate when Fo>0.2; below this, retain multiple eigenvalues.
Position x=0 retrieves centre temperature; x=L retrieves surface temperature (θ*≈1−C₁ζ₁ for semi-infinite geometry).