Sizu Riatto
Watt Happens Next · Chapter 7

How Fast Your House Cools

Your house cools like a cup of coffee, only slower. One night's log gives its time constant τ: how much a setback can save, and how much heat the walls hold.

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Opens on the book's house. Change any number to your own and press Calculate.

The overnight log: heating off, indoor every hour

The heat store (optional)

Enter the night's readings and the outdoor temperature, then press Calculate.

The formula

  • Newton's law of cooling: T(t) = T_out + (T₀ − T_out) e^(−t/τ): the gap falls by the same fraction each hour
  • τ from two readings t hours apart: τ = t ÷ ln(first gap ÷ last gap)
  • τ from all of them: fit a straight line to ln(gap) against time; τ = −1 ÷ its slope
  • Half-life of the gap = τ × ln 2 = τ × 0.693; after τ hours the gap is 37% of where it began
  • The heat store C = heat loss × τ (kWh per °C): how much heat each degree of the house holds
The book's worked example, Chapter 7. From 20 to 16.5 °C in 9 hours with 4 °C outside: τ = 9 ÷ ln(16 ÷ 12.5) = 36.5 h; the fitted line through all ten says 36.1 h (r² 0.9987). The book uses τ = 36 h: half-life 25 h, and the model says 16.46 °C at 07:00 (the log: 16.5). The heat store is 0.3 × 36 = 10.8 kWh per °C (38.9 MJ, 243 kJ per m² of floor).
Check it. Misread the last thermometer by 0.1 °C either way and τ moves by over a hour: use one good thermometer, and the fitted τ rather than two points.

Honest limits. A one-box model treats the whole house as one temperature; real rooms differ, and wind or sun can move a night. Only test on a night when nobody who needs warmth is home and no pipe can freeze. Never open the boiler, the tank or the meter. The book's house, its meters and its bill were worked out on paper from published figures, not measured by us: read your own meters and log your own night. Tariffs are illustrative, as seen in November 2026 (shaped like the GB price cap of October to December 2026, with US prices beside), shown without a currency sign; put in your own. No saving is promised. Everything stays in your browser: no sign-up, nothing stored.

These calculators go with Watt Happens Next (Make It Add Up, Book 3), where every number on this page is worked by hand.