Library / Heat transfer and thermal systems / HTS-16
Heat transfer and thermal systems
Temperature after a sudden surface-temperature change
Temperature at depth x and time t in a semi-infinite solid that starts uniform at Ti when its surface steps to Ts.
- Ref
- HTS-16
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- T [K]
When the surface of a semi-infinite solid is stepped from Ti to Ts at t = 0, the temperature at depth x is T = Ti + (Ts − Ti) erfc[x/(2√(αt))]. The similarity variable x/(2√(αt)) means the profile just stretches with √t. At x = 0 it gives Ts, and far from the surface it stays at Ti.
Temperature at depth x and time t in a semi-infinite solid that starts uniform at Ti when its surface steps to Ts. Constant properties and one-dimensional heat flow; α is the thermal diffusivity.
Calculator
- Similarity variable, x/(2√(αt))1
- Fraction of the step reached, erfc(…)0.1573
- Depth where only 1 % of the step is felt, 3.64√(αt)0.03643 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- After 100 s, a point 0.02 m deep has reached 315.7 K, 15.7 % of the way to the surface temperature.
- The step is felt to about 0.0364 m; doubling the depth of interest takes four times as long.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ti Initial temperature | 250 to 1,000 K | 273.6 | 905.6 | 632 | +10% | +8.01 % |
| Ts Surface temperature after the step | 250 to 1,000 K | 292.1 | 410.1 | 118 | +10% | +1.99 % |
| x Depth below the surface | 0 to 0.5 m | 300 | 400 | 100 | +10% | −1.19 % |
| t Time since the step | 1 to 1,000,000 s | 300 | 398.9 | 98.87 | +10% | +0.641 % |
| α Thermal diffusivity | 1e-8 to 0.001 m²/s | 300 | 396.4 | 96.43 | +10% | +0.641 % |
Sweep of Time since the step
| Time since the step (s) | Temperature at depth x (K) |
|---|---|
| 1 | 300 |
| 5.623 | 300 |
| 31.62 | 301.2 |
| 177.8 | 328.9 |
| 1,000 | 365.5 |
| 5,623 | 385 |
| 31,620 | 393.7 |
| 177,800 | 397.3 |
| 1,000,000 | 398.9 |
Sweep one variable
Variables and how each one contributes
Temperature at depth x
What the equation gives you. Everything else feeds it.
Initial temperature
The initial temperature of the solid, in K. Raises the temperature at every depth, most far from the surface. Raising it by 10 % (to 330) moves the result by +8.01 %, from 315.7 to 341 K.
Surface temperature after the step
The surface temperature after the step, in K. Raises the temperature in proportion to the erfc fraction. Raising it by 10 % (to 440) moves the result by +1.99 %, from 315.7 to 322 K.
Depth below the surface
The depth below the surface, in m. A deeper point has felt less of the step, so the temperature approaches Ti. Raising it by 10 % (to 0.022) moves the result by −1.19 %, from 315.7 to 312 K.
Thermal diffusivity
The thermal diffusivity, in m²/s. A higher diffusivity carries the step deeper, so the temperature rises. Raising it by 10 % (to 1.1×10−6) moves the result by +0.641 %, from 315.7 to 317.8 K.
Time since the step
The time since the step, in s. More time lets the step reach deeper, so the temperature rises. Raising it by 10 % (to 110) moves the result by +0.641 %, from 315.7 to 317.8 K.
Worked example with the values above
- Similarity variable: x/(2√(αt)) = 0.02 / (2 × √(1×10−6 × 100)) = 1.
- erfc(1) = 0.1573.
- T = 300 + (400 − 300) × 0.1573 = 315.73 K.
Assumptions and limits
- A semi-infinite solid with constant properties, so the back face is not felt over the time considered.
- A one-dimensional geometry, an ideal step or constant flux at the surface and a uniform initial temperature.
Common mistakes
- Using the formula after the thermal wave has reached the back face.
- Entering time in minutes or hours when seconds are needed.
- Taking x as the distance from the wrong face.
Statistics and uncertainty
Thermophysical properties of soils, concrete and foods vary by 10 % to 50 % with moisture and composition, so early-time results are only as good as the diffusivity and effusivity values.
Where it is used
- Quenching, ground and wall heating, and contact burns.
- Interpreting thin-film and heat-flux sensors.
Maths you need
- Thermal diffusivity: α = k/(ρcp)
- Complementary error function: erfc(x) = 1 − erf(x)
- Thermal effusivity: e = √(kρcp)