Calculate how the temperature of a "very thick body" soaks inward with time after its surface temperature is suddenly changed. Adjust the thermal diffusivity, temperatures, depth and elapsed time to see the depth-wise temperature distribution and the thermal penetration depth from the exact error-function solution, in real time.
Parameters
Thermal diffusivity α
m²/s
How fast heat spreads. Steel ≈ 1.2e-5, concrete ≈ 7e-7
Stepped surface temperature T_s
°C
Value the surface is suddenly raised to and held at
Initial temperature T_i
°C
Uniform temperature of the whole body before the step
Evaluation depth x
m
Depth below the surface whose temperature you examine
Elapsed time t
s
Time since the surface was stepped
Results
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Similarity variable η
—
Error function erf(η)
—
Temp. at depth x (°C)
—
Dimensionless temp. ratio
—
Penetration depth (m)
—
Midpoint-temp. time (s)
—
Semi-infinite solid section — advancing temperature profile
The surface is on the left. The temperature profile advances deeper with time and the thermal penetration depth (dashed line) grows. Colour runs from hot (red) to cold (blue).
Temperature profile — distribution through the depth
Temperature field of a semi-infinite solid with a constant surface temperature. T_s: surface temperature, T_i: initial temperature, α: thermal diffusivity. The temperature depends only on the similarity variable η = x/(2√(αt)) — a self-similar solution.
Similarity variable η and thermal penetration depth δ. δ is the depth where the surface temperature change has decayed to about 1%. Both grow with the square root of time.
Temperature at depth x, and the time for that depth to reach the midpoint temperature (erf = 0.5). The time scales with the square of depth and inversely with thermal diffusivity.
What is a Transient Semi-Infinite Solid?
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A "semi-infinite solid" — that's a strange name. There's no such thing as an infinitely large body in reality, is there?
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Fair point. It doesn't mean "truly infinite" — it means "so thick that, for the time we care about, the heat hasn't reached the far side". When sunshine heats a road in summer, only the top metre or two gets hot; the deep earth never notices. So for a short time the ground behaves like a "bottomless" body — a semi-infinite solid.
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I see — only the surface reacts and the depths sit still. So what happens inside if I suddenly change the surface temperature?
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If you step the surface temperature up and hold it there, the heat-conduction equation gives a beautiful exact solution, and the star of it is the "error function, erf". The temperature doesn't depend on depth x and time t separately — it depends on one dimensionless quantity, the similarity variable η = x/(2√(αt)). Move the depth or time sliders on the left and watch η at the top: for the same η, even with different depth and time, the temperature is the same.
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Wait — the same temperature at different depths and times? How does that work?
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Roughly speaking, the temperature at depth x and time t is identical to the temperature at depth 2x and time 4t. Look at the formula: if x doubles but t goes up by four, then √(αt) also doubles, so η is unchanged. That's the nature of diffusion — heat advances not with time, but with the square root of time. So reaching twice as deep takes four times as long, and the thermal penetration depth δ also grows like √t.
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So the thermal penetration depth is the marker of how far the heat has reached. Where exactly do you draw the line?
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There's no sharp boundary, so by convention we take the depth where only about 1% of the surface change has arrived. In terms of the similarity variable that's around η ≈ 1.8, and it gives δ = 3.6√(αt). In practice, if that δ is much smaller than the body's real thickness, you say "treat it as semi-infinite". Once δ approaches the thickness, the far side can no longer be ignored — that's the cue to switch to a finite-thickness analysis.
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This way of thinking sounds useful for things like quenching and ground temperature.
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Exactly. Surface hardening of a large steel ingot, rapid chilling of a thick concrete wall, the daily and yearly temperature waves soaking into the Earth — all of them are semi-infinite-solid problems. An estimate like "after casting an ingot and quenching its surface, how many minutes until a point 5 cm in reaches the midpoint temperature" comes straight out of this single error-function solution. It is also very handy for a first read before a detailed CAE study.
Frequently Asked Questions
A semi-infinite solid is a body so thick that, over the time of interest, the temperature change from the surface has not yet reached any far side. Only the region near the surface responds; the deep interior stays at its initial temperature. The model fits the ground cooled by a thunderstorm, the brief surface quench of a large steel ingot, or the chilling of a thick concrete wall. The check is simple: the thermal penetration depth must be much smaller than the body's actual thickness.
The similarity variable is eta = x / (2*sqrt(alpha*t)), a dimensionless ratio of depth x to the diffusion length scale. When the surface temperature is held constant, the heat-conduction solution does not depend on depth and time separately but collapses onto this single eta (a self-similar solution). As a result the temperature at depth x and time t is identical to the temperature at depth 2x and time 4t. This formula encodes the fact that diffusion advances with the square root of time.
The thermal penetration depth is the depth at which only about 1% of the surface temperature change has arrived, corresponding to a similarity variable of roughly 1.8. This tool computes it as delta = 3.6*sqrt(alpha*t). It grows with the square root of the elapsed time and of the thermal diffusivity, and everything deeper can be treated as not yet disturbed. The semi-infinite model is valid when the real thickness is much larger than delta.
The midpoint temperature (erf(eta) = 0.5) corresponds to a similarity variable eta = 0.4769. Substituting it into eta = x/(2*sqrt(alpha*t)) and solving for time gives t = x^2/(4*alpha*0.4769^2). The time is proportional to the square of the depth and inversely proportional to the thermal diffusivity. Doubling the depth makes the midpoint take four times as long — the square-root scaling that is the signature of all diffusion.
Real-World Applications
Surface hardening and heat treatment of metals: When the surface of a large steel ingot is rapidly cooled or heated, the heat reaches only the region near the surface, and the core behaves like a semi-infinite solid over short times. The error-function solution gives a quick estimate of "what temperature a point so many millimetres in reaches after so many seconds", helping to predict the depth of the hardened layer, residual stresses and the required heating and cooling times.
Ground temperature in civil and structural engineering: The ground is the classic semi-infinite solid subjected to daily and yearly temperature waves. The burial depth of ground-source heat pumps, the frost-penetration depth (foundation depth for frost-heave protection) and the thermal environment of underground structures are all evaluated with this transient-conduction view. The thermal-stress check of thick concrete walls or dam bodies under sudden temperature changes uses the same framework.
Contact temperature and material feel: When two bodies suddenly touch, the contact surface settles instantly to a fixed temperature and each side responds as a semi-infinite solid. The reason metal feels cold and wood feels warm is exactly the difference in thermal properties (more precisely, thermal effusivity). The model is also the basis for estimating contact heat transfer between moulds, rolls or dies and a hot workpiece.
Pre-study and verification for CAE: Before running a detailed transient-conduction FEM analysis, knowing the order of magnitude of the penetration depth and midpoint time from the error-function solution makes the mesh refinement and time-step settings far more accurate. If the FEM result differs from this analytical solution by an order of magnitude, it is a useful sanity check that points to a mistake in the boundary conditions, material properties or time step.
Common Misconceptions and Pitfalls
The biggest pitfall is assuming the semi-infinite model is valid forever. The model relies on the heat not yet having reached the far side. As time passes, the thermal penetration depth δ = 3.6√(αt) keeps growing and eventually reaches the body's real thickness. Once it does, the far boundary can no longer be ignored and the semi-infinite solution carries error. A practical guideline is to switch to a finite-thickness analysis once δ exceeds half the body thickness. Always compare the penetration depth this tool reports against the thickness of your actual part.
Next, forgetting the "constant surface temperature" boundary condition. The error-function solution used here is for the case where the surface temperature changes in a step and is then held fixed at that value (a Dirichlet, or first-kind, boundary condition). In reality, when the surface is exposed to air or a fluid and cools by convection, the surface temperature is not constant — a different solution involving the convective heat-transfer coefficient (and the Biot number) is needed. A prescribed surface heat flux again gives a different formula. Mistaking the type of boundary condition leads to large errors.
Finally, confusing thermal diffusivity α with thermal conductivity λ. What sets the speed at which temperature soaks in is not conductivity itself but the thermal diffusivity α = λ/(ρc). Even with a large λ, a large density ρ or specific heat c makes α small and the temperature change advances slowly. Copper has both a large λ and a large α, but stainless steel has a moderate λ and a surprisingly small α. When comparing penetration depths or midpoint times, always look at α, not λ. Note also that for contact problems it is the thermal effusivity √(λρc), not α, that governs — another easy confusion to avoid.
How to Use
Enter thermal diffusivity α (m²/s) for your material—steel: 1.2×10⁻⁵, concrete: 7.0×10⁻⁷, brick: 4.8×10⁻⁷
Set surface temperature step Ts (°C)—typically 100–500°C for quenching or fire scenarios
Input initial bulk temperature Ti (°C) and depth x (m) where you want transient response
Simulator calculates similarity variable η, error function erf(η), and temperature penetration over time
Worked Example
Steel plate (α=1.2×10⁻⁵ m²/s) initially 20°C, surface suddenly heated to 500°C. At depth x=0.05 m after t=1800 s: η=0.833, erf(η)≈0.745, resulting temperature≈376°C. Penetration depth ≈0.10 m. For concrete (α=7.0×10⁻⁷) same conditions: η=2.65, erf(η)≈0.999, T≈495°C at 0.05 m (slower diffusion means deeper/faster heating).
Practical Notes
Semi-infinite assumption valid when depth x is small relative to total thickness and time is short (Fo < 0.3 typically)
Quenching operations: use high Ts values (300–600°C) to predict thermal stress and phase transformation zones
Fire resistance testing: model gypsum (α≈3.5×10⁻⁷) to verify 30–120 min burn-through times at different depths
Error function approach fails once boundary effects appear; switch to finite-difference FEA for thick walls with both surfaces exposed