Cooling Down
How fast does a cup of coffee, a hot water tank or a house cool down — or warm up with a heater on? Enter how much heat it stores and how well it is insulated, and find the temperature after any time, or the time to reach a temperature.
Newton’s Law of Cooling
A hot object loses heat faster the bigger its temperature difference with the surroundings. As it cools, the difference shrinks, so the cooling slows down: the temperature approaches the room temperature along an exponential curve and never quite reaches it.
The pace is set by the time constant τ: the heat the object can store (m·c) divided by how fast it leaks heat (U·A). After one τ the difference has dropped to 37%, after three τ to 5%. A big, well-insulated mass — a hot water tank, a stone house — has a time constant of days.
Temperature over Time
τCooling Formulas
| Quantity | Formula |
|---|
Worked Techniques
Take the logarithm of how many times smaller the temperature difference must become, and multiply by τ.
Measure the temperature twice with a thermometer and you know τ — and from it U·A, how well your tank or house is insulated.
With a heater on, the temperature rises towards the point where the heater exactly replaces the losses — along the same kind of exponential curve, with the same τ.
Sources & Verification
- T. L. Bergman, A. S. Lavine, F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, 2011 — Sec. 5.1 (The Lumped Capacitance Method).
- Y. A. Çengel & A. J. Ghajar, Heat and Mass Transfer: Fundamentals and Applications, 5th ed., McGraw-Hill, 2015 — Ch. 4 (Transient Heat Conduction).