Library / Heat transfer and thermal systems / HTS-19
Heat transfer and thermal systems
Temperature under a constant imposed surface flux
Temperature at depth x and time t in a semi-infinite solid heated at a constant surface flux q″, with ξ = x/(2√(αt)).
- Ref
- HTS-19
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- T [K]
With a constant heat flux q″ into the surface of a semi-infinite solid, the temperature at depth x is T = Ti + (q″/k)[2√(αt/π) exp(−ξ²) − x erfc(ξ)], with ξ = x/(2√(αt)). At the surface it gives Ts = Ti + 2q″√(αt/π)/k, rising as √t. An insulated back face must be beyond the affected region.
Temperature at depth x and time t in a semi-infinite solid heated at a constant surface flux q″, with ξ = x/(2√(αt)). Constant properties; an insulated back face must lie beyond the affected region.
Calculator
- Surface temperature, Ti + 2q″√(αt/π)/k356.4 K
- Temperature rise at depth x5.025 K
- Thermal penetration length, √(αt)0.01 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.
- At 0.02 m deep the solid has warmed by 5.025 K, to 305 K, after 100 s.
- The surface itself would be at 356.4 K, which is the temperature a surface sensor would read.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| t Time since heating began | 1 to 1,000,000 s | 300 | 5,842 | 5,542 | +10% | +0.348 % |
| q″ Surface heat flux into the solid | 10 to 1e7 W/m² | 300 | 5,325 | 5,025 | +10% | +0.165 % |
| α Thermal diffusivity | 1e-8 to 0.001 m²/s | 300 | 1,986 | 1,686 | +10% | +0.348 % |
| Ti Initial temperature | 250 to 1,000 K | 255 | 1,005 | 750 | +10% | +9.84 % |
| k Thermal conductivity | 0.1 to 500 W/(m K) | 300 | 400.5 | 100.5 | +10% | −0.15 % |
| x Depth below the surface | 0 to 0.5 m | 300 | 356.4 | 56.42 | +10% | −0.452 % |
Sweep of Depth below the surface
| Depth below the surface (m) | Temperature at depth x (K) |
|---|---|
| 0 | 356.4 |
| 0.0625 | 300 |
| 0.125 | 300 |
| 0.1875 | 300 |
| 0.25 | 300 |
| 0.3125 | 300 |
| 0.375 | 300 |
| 0.4375 | 300 |
| 0.5 | 300 |
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, in K. Raises the temperature everywhere by the same amount. Raising it by 10 % (to 330) moves the result by +9.84 %, from 305 to 335 K.
Surface heat flux into the solid
The constant surface heat flux into the solid, in W/m². Raises the temperature rise in direct proportion. Raising it by 10 % (to 11,000) moves the result by +0.165 %, from 305 to 305.5 K.
Thermal conductivity
The thermal conductivity, in W/(m K). A higher conductivity lowers the temperature rise in inverse proportion. Raising it by 10 % (to 2.2) moves the result by −0.15 %, from 305 to 304.6 K.
Thermal diffusivity
The thermal diffusivity, in m²/s. A higher diffusivity spreads the heat deeper, lowering the surface rise but raising the deeper temperature. Raising it by 10 % (to 1.1×10−6) moves the result by +0.348 %, from 305 to 306.1 K.
Time since heating began
The time since heating began, in s. More time raises the temperature, with the surface rise growing as √t. Raising it by 10 % (to 110) moves the result by +0.348 %, from 305 to 306.1 K.
Depth below the surface
The depth below the surface, in m. A deeper point is cooler, since the heat has not reached it as much. Raising it by 10 % (to 0.022) moves the result by −0.452 %, from 305 to 303.6 K.
Worked example with the values above
- Penetration term: ξ = x/(2√(αt)) = 0.02 / (2 × 0.01) = 1.
- Bracket: 2√(αt/π) exp(−ξ²) − x erfc(ξ) = 0.0041511 − 0.003146 = 0.0010051 m.
- T = 300 + (10,000/2) × 0.0010051 = 305.03 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 step-temperature solution for a surface heated at constant flux.
- Using the formula once the back face is affected.
- Mixing up the surface flux sign: positive q″ heats the solid.
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)