Heisler Charts
Theory: exact solutions organised by dimensionless groups
Overview
When quenching a steel part, I want to know how many seconds the centre takes to cool. Should I use the Heisler charts from the textbook?
For shapes close to a plate, cylinder or sphere with a roughly constant heat transfer coefficient, it's exactly the right tool. Heisler charts plot the exact (infinite-series) solutions of the conduction equation in terms of two dimensionless groups, the Biot and Fourier numbers. Today you can evaluate the series directly instead of reading charts, but understanding what's inside tells you when a simple formula suffices and when detailed analysis is needed.
Dimensionless temperature, Biot and Fourier numbers
$L_c$ is the half-thickness for a plate and the radius for cylinders and spheres. The Biot number compares internal conduction resistance with surface heat loss; the larger it is, the bigger the centre–surface difference. The Fourier number is dimensionless time. θ = 1 is the initial temperature, 0 the fluid temperature.
Series solution and one-term approximation
$\zeta_n$ are roots of each shape's eigenvalue equation and $C_n$ the resulting coefficients. Higher terms decay faster, so beyond Fo = 0.2 the first term suffices; Heisler charts plot exactly this first term.
A 'chart computer' before computers
Published by Heisler in the US in 1947, these charts replaced the laborious task of solving eigenvalue equations and summing series with simply reading a graph. They were widely used wherever people needed to know when a body's centre reaches a target temperature — steel quenching, food sterilisation, curing concrete. Charts for surface and intermediate positions and for accumulated heat followed, and together with Gröber's refinements they remain in heat-transfer textbooks today.
Worked example: quenching steel
Steel (conductivity 40 W/(m·K), diffusivity 1.17×10⁻⁵ m²/s) quenched in oil (500 W/(m²·K)) and water (5,000 W/(m²·K)). The table gives centre dimensionless temperature θ (1 initial, 0 coolant):
100 mm diameter ball
| Coolant | Bi | Fo | Time | Exact | One-term | Lumped |
|---|---|---|---|---|---|---|
| Oil | 0.62 | 0.05 | 11 s | 0.998 | 1.084 | 0.911 |
| Oil | 0.62 | 0.20 | 43 s | 0.842 | 0.846 | 0.687 |
| Oil | 0.62 | 1.00 | 214 s | 0.225 | 0.225 | 0.153 |
| Water | 6.25 | 0.05 | 11 s | 0.987 | 1.290 | 0.392 |
| Water | 6.25 | 0.20 | 43 s | 0.439 | 0.443 | 0.024 |
With water, the lumped model badly underestimates the centre temperature.
With a Biot number of 6.25 the surface cools fast but internal heat takes time to reach it. The lumped model ignores internal gradients and applies the surface's rapid cooling to the whole body: at 43 s the centre still has 44% of the initial temperature difference, but the lumped model says 2% — a fatal error if you're judging whether quenching hardens the core. The one-term form, by contrast, matches the exact solution from Fo = 0.2 on; at 11 s (Fo = 0.05) it even exceeds 1, so use more series terms at short times.
Plate and bar (Fo = 1.0)
| Shape | Coolant | Bi | Time | Exact | Lumped |
|---|---|---|---|---|---|
| Plate 40 mm | Oil | 0.25 | 34 s | 0.824 | 0.779 |
| Plate 40 mm | Water | 2.50 | 34 s | 0.325 | 0.082 |
| Bar D50 mm | Oil | 0.31 | 53 s | 0.602 | 0.535 |
| Bar D50 mm | Water | 3.12 | 53 s | 0.055 | 0.002 |
Even with oil cooling at Biot numbers of 0.25–0.3, the lumped model differs by 5–7 points. The rule 'lumped is fine below Bi 0.1' comes down to whether that kind of error is acceptable.
Use and limits
When it applies
- Shape: infinite plates and cylinders, and spheres. Finite cylinders and boxes can be approximated by products of plate and cylinder solutions.
- Conditions: uniform initial temperature, constant ambient temperature and heat transfer coefficient, constant properties.
- Time: one-term for Fo > 0.2; for shorter times add terms or use the semi-infinite-solid solution.
Limits
In real quenching, the heat transfer coefficient changes greatly over time as boiling shifts from film to nucleate boiling, and steel properties vary with temperature. Such cases need numerical analysis of the temperature history; Heisler's solution remains useful as a benchmark to verify it.
“Simulation and chart disagree”
Checking a transient thermal analysis against Heisler charts, the centre temperature differed by over 10%. Three common causes: using the wrong characteristic length for Biot and Fourier numbers (half-thickness, not full thickness, for a plate); too coarse a time step or mesh to follow the steep early temperature change; and chart-reading error, especially at small Fourier numbers. Today, computing the series directly and comparing is the reliable route.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Using full thickness | Bi and Fo off by up to 4× | Half-thickness or radius |
| One-term below Fo 0.2 | Large errors | Add series terms |
| Lumped model at large Bi | Centre temperature underestimated | Check Bi, use the series |
| Constant h in quenching | Wrong cooling curve | Numerical analysis with boiling regimes |
| Relying on chart reading | Reading errors | Compute the series directly |
I'd like to compute transient conduction myself.
Try the transient conduction simulator for transient temperature and the 1D heat conduction tool for 1D conduction. Related solutions are covered in the lumped-capacitance model, semi-infinite solids and multidimensional transient conduction, and the meaning of the dimensionless groups in the Fourier number.
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