Library / Heat transfer and thermal systems / HTS-20
Heat transfer and thermal systems
Contact temperature of two semi-infinite solids
Interface temperature when two bodies at uniform temperatures T1 and T2 are brought into ideal contact, weighted by their thermal effusivities e = √(kρcp).
- Ref
- HTS-20
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- Tc [K]
When two semi-infinite solids at different uniform temperatures are brought into ideal contact, the interface jumps immediately to Tc = (e₁T₁ + e₂T₂)/(e₁ + e₂), where e = √(kρcp) is the thermal effusivity. The body with the larger effusivity dominates, pulling the contact temperature towards its own. The contact stays at this value while both bodies remain effectively semi-infinite.
Interface temperature when two bodies at uniform temperatures T1 and T2 are brought into ideal contact, weighted by their thermal effusivities e = √(kρcp). Valid while both behave as semi-infinite.
Calculator
- Weight of body 1, e₁/(e₁ + e₂)0.8
- Temperature change of body 1 at the surface, Tc − T₁−20 K
- Temperature change of body 2 at the surface, Tc − T₂80 K
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.
- The surfaces settle at 380 K, which is 80 K above body 2 and 20 K below body 1.
- Body 1 has 4 times the effusivity of body 2, so the contact temperature lies much nearer to T₁; this is why metal feels colder than wood at the same temperature.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T1 Initial temperature of body 1 | 250 to 1,000 K | 260 | 860 | 600 | +10% | +8.42 % |
| T2 Initial temperature of body 2 | 250 to 1,000 K | 370 | 520 | 150 | +10% | +1.58 % |
| e1 Effusivity of body 1 | 100 to 50,000 J/(m² K s^½) | 303.2 | 394.3 | 91.11 | +10% | +0.39 % |
| e2 Effusivity of body 2 | 100 to 50,000 J/(m² K s^½) | 319.4 | 399.2 | 79.82 | +10% | −0.413 % |
Sweep of Effusivity of body 2
| Effusivity of body 2 (J/(m² K s^½)) | Contact temperature (K) |
|---|---|
| 100 | 399.2 |
| 6,338 | 365.4 |
| 12,580 | 348.8 |
| 18,810 | 338.9 |
| 25,050 | 332.4 |
| 31,290 | 327.7 |
| 37,530 | 324.2 |
| 43,760 | 321.5 |
| 50,000 | 319.4 |
Sweep one variable
Variables and how each one contributes
Contact temperature
What the equation gives you. Everything else feeds it.
Effusivity of body 1
The effusivity of body 1, in J/(m² K s^½). A larger effusivity pulls the contact temperature towards T₁. Raising it by 10 % (to 13,200) moves the result by +0.39 %, from 380 to 381.5 K.
Initial temperature of body 1
The initial temperature of body 1, in K. Raises the contact temperature in proportion to its effusivity share. Raising it by 10 % (to 440) moves the result by +8.42 %, from 380 to 412 K.
Effusivity of body 2
The effusivity of body 2, in J/(m² K s^½). A larger effusivity pulls the contact temperature towards T₂. Raising it by 10 % (to 3,300) moves the result by −0.413 %, from 380 to 378.4 K.
Initial temperature of body 2
The initial temperature of body 2, in K. Raises the contact temperature in proportion to its effusivity share. Raising it by 10 % (to 330) moves the result by +1.58 %, from 380 to 386 K.
Worked example with the values above
- Weighted sum: e₁T₁ + e₂T₂ = 12,000 × 400 + 3,000 × 300 = 5,700,000.
- Total effusivity: e₁ + e₂ = 15,000.
- Tc = 5,700,000 / 15,000 = 380 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.
- Using the step solution when the surface is heated by convection, which needs a different solution.
- Entering time in minutes or hours when seconds are needed.
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)