Sizu Riatto
Watt Happens Next · Chapter 9

Flows In, Flows Out

Rate in minus rate out is the rate of change. Step that forward a tenth of an hour at a time and you can call what happens next: the coldest point, the reheat, and whether a heat pump would cost more or less to run.

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

The house: a tank of heat

The thermostat and the boiler

Heat pump against gas: the cost of a kWh of heat

Enter the house, the thermostat and the boiler, then press Calculate.

The formula

  • The balance: C × dT/dt = heat in − heat loss × (T − T_out)
  • Euler's method: T(next) = T + (heat in − heat loss × (T − T_out)) ÷ C × the step
  • The thermostat: by day the boiler runs full when T is below the setting, and holds it there; by night it is off
  • Heat pump: a kWh of heat costs the electricity rate ÷ COP; gas: the gas rate ÷ the boiler's efficiency
  • They cost the same at COP = electricity rate ÷ (gas rate ÷ efficiency)
The book's worked example, Chapter 9. The book's house (300 W per °C, C 10.8 kWh per °C) off from 22:30 on a 4 °C night, stepped every 0.1 h: the coldest point is 16.81 °C; with 12 kW into the rooms from 06:30 it is back at 20 °C at 10:54, 4.4 hours of full-on reheat. A one-box model is slow on reheat: the air warms long before the walls. At the book's tariff a kWh of heat costs 0.0889 from gas; a heat pump matches it at COP 2.925.
Check it. Time your own house after a setback morning with a thermometer in the living room: it will usually beat the model, because the air warms first.

Honest limits. A one-box model with one outdoor temperature: real houses have rooms, sun and wind. Heat pumps, solar, batteries and EV chargers are installed by certified installers (UK: MCS); the arithmetic is yours, the installing is theirs. 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.