Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom

Every winter the same argument happens in my house. My dad sets the heating timer to come on at 06:00 so that the house is warm when we get up at 07:00, and switches it off at 22:30. My mum thinks the house takes so long to warm up again that it would be cheaper to leave the heating “ticking over” at a low temperature all night. My bedroom is the coldest room — two of its walls face outside — so I decided to settle the argument for my room, with a data logger and mathematics rather than opinions.

When I first plotted how my room cooled, I expected Newton's law of cooling, which we met as an example of a differential equation: the temperature falls exponentially towards the outside temperature. The data did not look like that. The air dropped several degrees in the first hour and then hardly moved for the rest of the night. That shape became the real question of this exploration.

Aim. I will (1) test whether Newton's law of cooling describes my room overnight, (2) build and fit a model with two coupled temperatures — the air and the walls — and solve it exactly using eigenvalues and eigenvectors, (3) check the numerical method I need for the parts that cannot be solved exactly, and test the model on a second night it has not seen, and (4) use it to answer two practical questions: what is the latest time the heating can come on and still reach 19 °C by 07:00, and is ticking over at 16 °C cheaper than switching off?

I used three cheap temperature loggers that record to 0.1 °C. One was on a bookshelf in the middle of the room, away from the radiator (the air temperature, \(A\)); one was taped to the inside surface of the outside wall (the wall temperature, \(W\)); and one was outside on the window sill in a small ventilated box, out of the wind (\(T_{\text{out}}\)). Before using them I put all three in the same drawer for an hour: they agreed to within 0.1 °C, so I did not apply any correction.

Data note: the logs in this exemplar are illustrative, simulated by IB Math Revision to behave like real logger readings; a student must log their own room.

Night 1 was a still, dry night. The heating went off at 22:30 and the timer turned it on again at 06:00; I kept logging until 07:00. With the door closed and the window shut, nothing else in the room changed. Time \(t\) is measured in hours from 22:30 and temperatures in °C.

The outside temperature drifted from about 6.1 °C to 4.7 °C over the night. Every model in this exploration treats it as constant at its mean before 06:00, \(T_{\text{out}} = 5.3\) °C. That is the first assumption I will come back to. The wall behaves very differently from the air: after 06:00 the air warms by several degrees in minutes while the wall barely responds.

Newton's law of cooling says the air cools at a rate proportional to how much warmer it is than outside:

\[ \frac{dA}{dt} = -k\,(A - T_{\text{out}}) \;\Rightarrow\; A = T_{\text{out}} + (A_0 - T_{\text{out}})\,e^{-kt} \]

by separating variables. Taking logarithms, \(\ln(A - T_{\text{out}}) = \ln(A_0 - T_{\text{out}}) - kt\), so if the law holds, a graph of \(\ln(A - T_{\text{out}})\) against \(t\) should be a straight line with gradient \(-k\).

It is not. Regression lines through the first hour and through the part after 01:30 have gradients −0.238 and −0.046 per hour — the early cooling is about 5 times faster. The best single value of \(k\) by least squares is \(k \approx 0.102\) per hour, but Figure 3 (left) shows what that costs: the model is too warm for the first 4.7 hours and too cold for the rest of the night, by up to 1.9 °C — a pattern rather than random scatter (RMSE 1.10 °C). One exponential cannot be fast and slow at the same time.

Physically this makes sense to me. The air in my room holds very little heat, so it cools quickly — but it is surrounded by walls, furniture and a floor that hold far more heat and keep giving it back. So the air is not cooling towards the outside temperature at all in the first hour: it is cooling towards the walls. That suggests a model with two temperatures that affect each other.

Newton's law of cooling does not describe my room overnight: the ln graph bends, and one exponential leaves residuals with a clear pattern. A model with two coupled temperatures, air and walls, fits both logged series to about the loggers' resolution. Its eigenvalues (−0.0429 and −1.431 per hour) show a fast mode with a half-life of about half an hour, in which the air settles relative to the walls, and a slow mode with a half-life of about 16 hours, in which the walls control everything. Using the model, the heating on a mild, still night could come on at about 06:25 instead of 06:00, but on the cold, windy night 2 it needed to come on at 05:54, so 06:00 is close to right for a bad night. Ticking over at 16 °C would run the heater about 89% longer than switching off. So, for my room, switching off is cheaper, and the energy-balance equation shows why: keeping the room warmer for longer always increases the heat lost.

The limitations are mostly in the assumptions. The outside temperature was treated as constant, although it fell by more than a degree during the night; the draught parameter depends on the wind; and a single wall temperature stands for walls, floor and furniture that really warm and cool at different rates (a third temperature would give a third eigenvalue). The loggers' 0.1 °C resolution limits how precisely the fast mode can be measured. Most importantly, I tested the model on only two nights, so I would set the timer for a windy night, not a still one.

What I found most interesting was that the eigenvalues were not just a method for solving the equations: they turned out to be the physics of my room. The fast one explains why warming up is quick, and the slow one explains why the walls decide how cold it gets. My dad was right about switching off, and — to my surprise — his 06:00 timer is about right for a bad night (a few minutes late for the worst one I logged); the real saving would come from a thermostat that chooses its own start time each morning, which is exactly the calculation in Section 8.