Transient Heating and Cooling
The lumped-capacitance method, the Biot number, time constants, quenching and warming up a part.
Steady or transient?
The earlier pages assumed steady state: temperatures that do not change with time. Most real problems start with a transient: a part cools after machining, an oven warms up, a processor heats when it is switched on, a pipe freezes overnight. Then thermal capacity matters (how much heat the body stores), as well as the resistance to heat flow. The question is how long it takes.
The lumped-capacitance method
If a body is small or conducts well, its temperature is nearly uniform at any instant and the whole thing can be treated as one lump. A heat balance gives a simple exponential:
T(t) = T∞ + (Ti − T∞) e−t ÷ τ τ = ρ V c ÷ (h A) = ρ c Lc ÷ h
t = τ ln[(Ti − T∞) ÷ (T − T∞)]
The time constant is the product of a thermal capacitance C = ρ V c (J/K) and a thermal resistance R = 1 ÷ (h A) (K/W): τ = R C, exactly like an electrical RC circuit.
The Biot number
The lumped method is accurate only if the heat conducts through the body much faster than it leaves the surface. The ratio of the internal conduction resistance to the surface convection resistance is the Biot number:
Bi = h Lc ÷ k
Small Bi: a small metal part in air. Large Bi: a thick concrete wall, a large piece of meat, anything cooled by water or a fast liquid. For the thick wall below, a 0.2 m concrete slab with h = 10 gives Bi = 0.71 (half thickness 0.1 m), so the lumped method fails.
A cooling ball in air and in water
Worked example: quenching a steel ball
- A 25 mm steel ball (ρ = 7 800, c = 490, k = 45) from 400 °C into surroundings at 20 °C. V = 8.18 cm³, A = 19.63 cm², Lc = D ÷ 6 = 4.17 mm, mass 0.064 kg.
- In still air (h = 25 W/m²K, natural convection and radiation): Bi = 0.0023 (lumped is excellent), τ = 7 800 × 490 × 0.00417 ÷ 25 = 637 s. Time to reach 100 °C: t = τ ln(380 ÷ 80) = 1.558 τ = 16.5 minutes.
- In stirred water (h = 1 000): Bi = 0.093 (still under 0.1, so valid), τ = 15.9 s, and it reaches 100 °C in 25 seconds. Real quenching in boiling water starts with vapor film and is not this simple.
- The energy released is m c (Ti − T) = 9.4 kJ, enough to heat a liter of water by 2.2 K.
Warming a part in an oven
Worked example: heating a steel cube to 800 °C
- A 50 mm steel cube (ρ = 7 850, c = 490, k = 45) goes into an oven at 800 °C with h = 40 W/m²K (mostly radiation). Lc = V ÷ A = 8.33 mm.
- Bi = 0.0074, so the cube is uniform in temperature. τ = ρ c Lc ÷ h = 7 850 × 490 × 0.00833 ÷ 40 = 801 s.
- To get within 5 percent of the oven temperature takes 3 τ: 40 minutes. Heat-treatment soak times add the time for the center of a thick part to catch up, which the lumped model ignores when Bi is larger.
Time scales for thick bodies
For a thick body, heat diffuses in at a rate set by the thermal diffusivity α = k ÷ (ρ c) (m²/s). The time for the temperature to change through a thickness L is of order t ≈ L² ÷ α (the Fourier number Fo = α t ÷ L² is about 1). Metals respond in seconds, concrete in hours.
| Material | Diffusivity α | Time scale for 25 mm | Time scale for 200 mm |
|---|---|---|---|
| Copper | 113.06 mm²/s | 6 s | 0.1 h |
| Aluminum | 97.86 mm²/s | 6 s | 0.1 h |
| Steel | 11.77 mm²/s | 53 s | 0.9 h |
| Concrete | 0.69 mm²/s | 904 s | 16.1 h |
| Water (still) | 0.14 mm²/s | 4 345 s | 77.3 h |
| Wood (pine) | 0.13 mm²/s | 4 870 s | 86.6 h |
| Still air | 21.52 mm²/s | 29 s | 0.5 h |
A 200 mm concrete wall takes about 16 hours to respond fully to a change in temperature on one side, which is why a thick masonry building stays cool through a hot day (thermal mass) and why a house with a lot of mass warms slowly in the morning.
Why metal feels colder than wood
When two bodies at different temperatures touch, the interface settles at a temperature set by their thermal effusivity e = √(k ρ c): Tcontact = (e₁ T₁ + e₂ T₂) ÷ (e₁ + e₂). The body with the higher effusivity dominates.
Worked example: touching a cold surface
- Skin at 37 °C (effusivity about 1 000 J/m²·K·s½) touches objects at 20 °C.
- Aluminum (e = 23 958): the contact temperature is 20.7 °C, almost the metal’s own: it pulls heat from the skin and feels cold. Steel (e = 13 114): 21.2 °C.
- Wood (e = 335): 32.7 °C: the surface warms to near skin temperature and feels comfortable. It is the same temperature as the metal; the difference is how fast heat is drawn away.
- The same idea sets the safe touch temperatures of hot surfaces: a metal handle at 55 °C burns more quickly than plastic at the same temperature.
Related
- Previous: Heat Exchangers: LMTD and NTU. Next: Cooling Electronics and Thermal Design.
- Conduction and Convection for k and h.
- Heat Sink and Thermal Resistance Calculator.
- Heat Transfer: the whole guide.
FAQ
What if Bi is bigger than 0.1?
The surface cools faster than the center, so use the one-dimensional transient solutions (Heisler charts or the series solutions in the heat transfer textbooks), or a finite-difference or finite-element program. The lumped answer would be too fast at the center.
How do I find the time constant of a real part?
Heat or cool it and record the temperature: ln(T − T∞) against time is a straight line with slope −1 ÷ τ. It lumps the thermal mass and the surface resistance without needing either separately.
Does the heat capacity of the surrounding matter?
Yes if the surroundings are not infinite: a small ball in a large bath works, but a hot block in a small insulated tank warms the tank, and the final temperature is the weighted mean, T = (m₁ c₁ T₁ + m₂ c₂ T₂) ÷ (m₁ c₁ + m₂ c₂).
Is h constant during the cooling?
Not exactly: natural convection and radiation coefficients fall as the body cools. Use an average h, or step through the cooling in small intervals.
References
- Incropera, F. P., et al., Fundamentals of Heat and Mass Transfer, Wiley (lumped capacitance, transient conduction, Heisler charts).
- Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford.
- Çengel, Y. A. and Ghajar, A. J., Heat and Mass Transfer, McGraw-Hill.
- ASM Handbook, Volume 4 (heat treating: heating and quenching times).
Last reviewed: October 2026. Property values and coefficients are typical.