Calculating outside temperature based on indoor measurements

Question

If I take three consecutive measurements $T_0$, $T_1$, $T_2$ at equal intervals, I can end up with two (Newton’s law of cooling) equations where the $e^{-kt}$ term is the same.

$$T_{n+1} = T_{outside} + (T_n - T_{outside}) × e^{-kt}.$$

So, without knowing the temperature constant of the building, can I calculate the outside temperature?

$$T_{outside} = \dfrac{T_0 × T_2 - T_1^2}{T_2 + T_0 - 2T_1}.$$

Have I missed something? Or is this really possible? (Provided no other sources of cooling and heating)

Answer 1

It's worth a try, but I can foresee two problems:

Firstly Newton's law of cooling is a good approximation for convective cooling of objects, but I don't know how well it would apply to a house. I'd have guessed most houses cool due to draughts and it's not clear whether this sort of cooling would obey Newton's law. Having said this, I would guess you still get an approximately exponential fall with time.

The more serious objection is that finding the equilibrium temperature by fitting an exponential decay gives very poor accuracy unless you have a large number of data points spanning a large temperature range. With only two points you're unlikely to get an accurate result.