Transient Conduction in a Semi-Infinite Solid Simulator
A tool that calculates how heat soaks into the interior of a solid after the surface temperature suddenly changes. Adjust the material, depth and elapsed time to see the interior temperature, dimensionless temperature, thermal penetration depth and surface heat flux update in real time, and get an intuitive feel for the speed of thermal diffusion.
Parameters
Initial temperature Ti
°C
Temperature of the whole solid before any heat arrives
Surface temperature Ts
°C
Temperature the surface jumps to at time zero
Material
Sets thermal diffusivity α and conductivity k
Depth x
mm
Distance from the surface where you want the temperature
Elapsed time t
s
Time since the surface temperature changed
Results
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Temperature at depth T (°C)
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Dimensionless temp θ
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Similarity variable η
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Penetration depth δ (mm)
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Surface heat flux (kW/m²)
—
Thermal diffusivity α (×10⁻⁶ m²/s)
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Temperature penetration in a semi-infinite solid — animation
The heated surface is on the left. The temperature profile drops from Ts at the surface toward Ti deep inside, and the heat penetrates deeper over time. Colour shows how hot the solid is.
Interior temperature after the surface jumps to Ts at time zero. The dimensionless temperature θ on the left equals the error function erf and depends only on the similarity variable η = x/(2√(αt)). x: depth, t: elapsed time, α: thermal diffusivity.
Thermal penetration depth δ (the depth the temperature change roughly reaches) and surface heat flux qs. k: thermal conductivity. Both δ and qs are governed by √t, and α sets how fast heat diffuses.
What is Transient Conduction in a Semi-Infinite Solid?
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"Transient conduction in a semi-infinite solid" — the name is so intimidating. What kind of situation does this actually describe?
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It is not as hard as the name. Picture a winter morning when you spill hot water on a thick concrete floor. The surface heats up at once, but deep inside the floor it is still cold. That situation — one surface changes temperature while the opposite side is so far away the effect hasn't reached it — idealized, is a semi-infinite solid. "Transient" simply means the temperature keeps changing with time.
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I see. But how do you actually calculate the interior temperature? With both position and time as variables it sounds very complicated.
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This is the beautiful part of the problem. You don't have to track depth x and time t separately — they fold into a single "similarity variable" η = x/(2√(αt)). Then the dimensionless temperature is just θ = erf(η), one error function. When you move the depth and time sliders on the left, the tool evaluates that erf to get the temperature. With the default 50 mm depth and 600 s, η ≈ 0.30 and the temperature is about 141 °C.
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So if θ and η are the same, the temperature state is the same? Then what is the "penetration depth δ"?
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Yes, that is exactly what similarity means. δ ≈ 4√(αt) is a yardstick depth — "the heat has reached this far". Beyond δ the solid is still essentially at its initial temperature. The interesting thing is that δ advances with the square root of time. Wait four times as long and the depth only doubles. So even if you think "a bit more time will heat the core", in reality it barely progresses. Switch to aluminum and watch δ on the slider — it reaches much deeper, much faster than steel.
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The speed really depends on the material. I am also curious about the surface heat flux. Why does it decrease as time goes on?
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It is qs = k(Ts−Ti)/√(παt), nearly infinite in theory at first and then falling off as 1/√t. The reason is simple: just inside a freshly heated surface there is still cold material, so the temperature drop is steep and heat pours in. As time passes the warmed layer thickens, the drop becomes gentle and less heat flows in. This quantity is very important when you estimate, for instance, the cooling rate of a quench.
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One last thing. "Semi-infinite" can't really exist. Can the model not be used for an actual finite body?
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Good question. The trick is: "as long as the penetration depth δ is smaller than the body's actual thickness, you can treat it as semi-infinite". Even for a 300 mm thick concrete wall, a few minutes after heating the heat has only reached near the surface, so the semi-infinite formula applies directly. Once δ starts to exceed the thickness, the far boundary begins to matter, and you must switch to a finite-thickness solution such as a Fourier series.
Frequently Asked Questions
It is the time-varying heat conduction in a solid that has one bounding surface while the opposite side is so far away (the body is thick enough) that it has no influence yet. When the surface temperature suddenly changes to Ts at time zero, the heat diffuses slowly into the interior. The temperature is given with the error function as T(x,t) = Ts + (Ti − Ts)·erf(x/(2√(αt))), where x is the depth from the surface, t is elapsed time and α is the thermal diffusivity. This tool evaluates that analytical solution in real time.
It is a yardstick depth showing how far the surface temperature change has reached into the solid, estimated as δ ≈ 4√(αt). Beyond the depth δ the solid is still essentially at its initial temperature Ti (about 99% unaffected). It advances in proportion to the square root of time, so the penetration depth only doubles when four times as much time passes. The semi-infinite model is valid only while this δ stays smaller than the actual body thickness.
Thermal conductivity k describes how much heat a material lets through in the steady state. Thermal diffusivity α = k/(ρ·cp), on the other hand, describes how fast a temperature change propagates. In transient problems α is the governing property: the larger α is, the sooner the temperature reaches the interior. Aluminum, for example, has an α about eight times that of steel and heats up far deeper in the same time. When you ask about the amount of heat itself, such as the surface heat flux, k is also needed.
The surface heat flux is qs = k(Ts − Ti)/√(παt), infinite at time zero and then decaying as 1/√t. Early in the heating there is still cold material just inside the surface, so a steep temperature gradient stands there; as time passes the warmed layer thickens and the gradient becomes gentle. It is an important quantity when evaluating quench cooling rates or the heat transfer of a brief contact.
Real-World Applications
Metal heat treatment and quenching: In surface hardening of steel, the rate at which temperature falls from the surface inward when a heated part is suddenly cooled governs the microstructure and hardness. The semi-infinite solid model is used to estimate how fast the near-surface region cools early in the quench and when a given depth reaches a target temperature. The penetration depth δ also serves as a guide to the effective case-hardening depth.
Heat in buildings and the ground: Thick concrete walls and soil behave exactly as semi-infinite solids against solar radiation and air-temperature changes. How deep into a wall the daytime surface heating reaches, and how many metres down the seasonal change at the ground surface affects, can both be estimated with this erf solution. It is widely used as a basic calculation for geothermal use and heat-island assessment.
Brief contact and burn assessment: The temperature rise of skin briefly touching hot metal or boiling water can also be treated, for the first few seconds, as a semi-infinite solid problem. At the same surface temperature, touching a metal with a large thermal diffusivity such as aluminum or copper drives heat into the skin faster and is more dangerous. It forms the basis of burn-risk assessment and protective-equipment design.
Verification and first estimates for CAE analysis: Before running a finite-element transient conduction analysis, using this analytical solution to roughly estimate "when the near-surface temperature reaches what value" lets you judge whether the mesh and time step are reasonable. If the FEM result deviates greatly from the erf solution near the surface, it works as a sanity check pointing to a time step that is too coarse or a wrong boundary condition.
Common Misconceptions and Pitfalls
The biggest pitfall is continuing to use the semi-infinite model on a finite-thickness body indefinitely. The model is valid only while the thermal penetration depth δ ≈ 4√(αt) is smaller than the body's actual thickness. Once δ reaches the thickness, the influence of the far boundary appears and the erf solution predicts a higher temperature than reality. When taking long elapsed times, always check δ, and once it exceeds the body thickness, switch to a finite-thickness solution (Fourier series or a numerical solution). This tool shows this caution in its verdict text.
Next, getting the boundary condition wrong. This formula is for a surface temperature instantly fixed at a constant value Ts (a first-kind boundary condition). Real heating may apply a constant heat flux at the surface, or exchange heat with a surrounding fluid by convection (a third-kind boundary condition, described by the Biot number), and each gives a completely different temperature distribution. First confirm whether the surface temperature can really be considered constant. If convection dominates, you need a lumped-capacitance method or a Biot-number-based solution rather than the error function.
Finally, the misconception of confusing thermal diffusivity α with thermal conductivity k. In a transient problem it is α, not k, that sets how fast the temperature reaches a given depth. Concrete has a k roughly ten times that of wood, but the difference in α is not that large, so the difference in how fast temperature penetrates is nowhere near the ratio of k. On the other hand, k acts directly on quantities like the surface heat flux qs that ask about "heat per unit time". Note that which property you use depends on which quantity you want.