Estimate room energy loss from temperature plots What is the method for determining the heat/energy loss from a room, given a temperature plot/delta and a measure of the energy input over time ?
I'm building an extension to our house, and have really doubled down on the insulation. Currently one room is finished and fully insulated. As it's cold outside (-1C) I thought it a good time to measure how effective the insulation it. There is no heat source active in the room, and the room was stable at about 4C.
With some calculations of the insulation thickness and conductivity, I estimate the U-value of the walls to be 0.15 W/m2/K. The wall area is 100m2, Volume ~200m3. The windows have a U value of about 1.0, and are 2m2.
So putting this together, I estimate that for every degree of difference between inside and outside temperature, I'd need 0.15 * 100 + 1 * 2 = 15+2 = 17W of power to maintain this temperature, e.g. with -1C outside and 4C inside, 17 * 5 = 85W.
I turned on a 1KW heater for 2.5 hours and looked at the temperature rise. The results were not what I expected - the room rose by about ~2 degrees over that time. A plot of the temperature started to show the temp rise levelling off at the end of the 2.5 hours, which was surprising.
So the question is - from this information of the heat-source and temperature rise, can the true U value of this room be determined?
A plot of the temperature change is shown here:
The x-axis shows temperature samples every 30 seconds. The series are 3 different temp sensors - there is some sensor to sensor offset, but all are showing the same trend. Between samples ~230 and ~550 this represents 2.5 hours and shows a rise from 6C to 8C, which is when the room had the 1KW heater active. Over the next 9 hours the room cools to 4.7C from 8C.
Outside temperature was constant at -1C, and there outside conditions were calm (no wind chill).

 A: As an approximation you can use Newton's law of cooling:
$$-mc_p\frac{\text{d}T(t)}{\text{d}t}=UA[T(t)-T_e]\tag{1}$$
As your room is made of walls and windows, you'd have to work it as:
$$UA\approx U_{walls}A_{walls}+U_{windows}A_{windows}$$
$m$ is the mass of the content of the room: assuming it's empty that's air.
$(1)$ is a simple ordinary differential equation, at least provided $T_e=\text{constant}$. Firstly we define the so-called characteristic time $\tau$:
$$\frac{1}{\tau}=\frac{UA}{mc_p}$$
So that:
$$-\frac{\text{d}T(t)}{\text{d}t}=\frac{1}{\tau}[T(t)-T_e]\tag{2}$$
$(2)$ solves to:
$$\ln\frac{T(t)-T_e}{T_0-T_e}=-\frac{t}{\tau}$$
which gives you the theoretical evolution of $T(t)$ in time $t$. $T_0$ is the initial ($t=0$) room temperature.
By plotting the $\text{LHS}$ vs. the $t$ in (e.g.) Excel you should obtain a straight line with slope $\frac{1}{\tau}$.
From that value, $U$ can then be calculated.

where:
$$\Theta=\ln\frac{T(t)-T_e}{T_0-T_e}$$
$\Theta$ is sometimes called the reduced temperature.
