Steady-state surface temperature
Heat crosses four layers in series: gas-side convection into the coating, conduction through the thermal-barrier coating (TBC), conduction through the cast-iron wall, then convection and radiation out to the underhood air. Change any parameter below and the solver re-converges instantly.
Steady-state comparison against the autoignition band
Paper default: bare 484.14 °C, TBC 477.50 °C, a 6.64 °C (1.4%) drop; both still above the danger band.
Post-shutdown heat soak
Turn the engine off and forced convection stops. The only things left cooling the surface are weak natural convection and radiation. Two solutions are compared: a fourth-order Runge-Kutta (RK4) numerical integration of the full T⁴ radiation term, and a linearized shortcut that freezes the radiation coefficient at its hottest value so it can be solved by hand.
Surface temperature vs. time since shutdown
RK4 (solid) vs. linearized analytical shortcut (dashed): the shortcut diverges 68–70 °C from the true solution 12–20 minutes in, because it assumes the radiation rate never slows down.
The 311–450 °C band above is a hard cutoff on purpose — real ignition isn't. Hot-surface ignition studies on engine fluids find three zones, not a switch: no ignition well below the range, then increasing likelihood as temperature climbs, then ignition on virtually every contact once you're well into it. For engine oil specifically, published tests put "essentially never" below roughly 300 °C and "essentially always" above roughly 335 °C, with a genuinely probabilistic middle in between (Santrock, NHTSA 98-3588-193, 1998; SAE 2006-01-1013 and 2006-01-1014). This doesn't change any number on this page: the paper's fixed band is the right way to size a conservative safety margin, and every model above still uses it as-is. It's here because a probability curve is the more accurate picture of what's actually happening physically, and the transition zone shown in that research sits comfortably inside this page's own band, not outside it.
Sensitivity, uncertainty, and numerical accuracy
Three separate checks, all run live: how sensitive is the answer to gas-side convection, how much does manufacturing tolerance on the coating move the result, and whether the RK4 solver actually converges at the rate it's supposed to (the one check most simulations skip).
h₁ sensitivity
Sweeping gas-side convection from 80 to 400 W/m²K. This is the lever that actually moves the temperature, more than coating thickness ever will.
Manufacturing tolerance (Monte Carlo)
±10% random variation in TBC thickness and conductivity, matching real coating-process tolerance.
A 4th-order method should quarter its error every time you halve the step size (error ∝ dt⁴, so halving dt should divide the error by 2⁴ = 16, an "observed order" of 4). Real solvers rarely show one clean regime across the whole range. This runs the actual sweep, live, against a reference solution 500×+ finer than any step tested.
Error vs. step size, log–log
Reference: dt = 0.00195s. Comparison point: t = 2048s into the soak (mid-transient, where truncation error is actually visible).
| dt (s) | T(2048s) predicted | |error| vs. reference | observed order |
|---|
Three regimes, not one: dt ≥ 64s is pre-asymptotic (order looks too good; higher-order truncation terms haven't been dominated yet). dt ≈ 32–16s is a crossover anomaly where terms partially cancel. dt = 16→2s is the genuine asymptotic 4th-order regime, highlighted below. Below dt = 1s the error runs into float64 noise (~10⁻¹¹–10⁻¹² °C) and the order estimate degrades again.
A "lumped" model only makes sense if the wall's internal temperature gradient is small enough to ignore, i.e. the wall cools faster by convection off its surface than heat can conduct through it. The standard test (Cengel & Ghajar, Heat and Mass Transfer, ch. 4) is the Biot number, Bi = h·Lc/k: below 0.1, treating the whole body as one temperature typically introduces less than 5% error. This isn't reported in the paper; it's a live check on the paper's own modeling choice, using the current wall thickness and effective outer heat-transfer coefficient (natural convection + linearized radiation at the hottest point in the soak).
Biot number, live