# How to extract a building's heat isolation and thermal inertia based on temperature curves?

So imagine we have a house with some isolation and some thermal inertia. This house also has a heating system that turns on (starts heating) when the observed temperature curve goes below some predefined temperature threshold.

I figured that in the scenario where the heating system is off (the observed temperature is above the threshold the temperature decrease will give us information on the thermal isolation and inertia. We need of course to take in to account the external temperature, the wind, and the sun irradiation.

I was wondering if there was a way to dissociate the information of the thermal isolation and the information of the thermal inertia. Clearly, if we had information about the thermal power loss it would give an indication but unfortunately we don't.

The data I have : Observed inside temperature and desired inside temperature, outside temperature, wind, humidity from a weather station in the same city (so I guess only the external temperature can be used)

Objective: Determine the conductivity and thermal mass of the house for sales purposes (sell more isolation, help them master energy efficient behavior etc)

• Your question is quite unclear at the moment. Are you trying to infer the isolation and thermal inertia factors based upon ambient conditions and the existing heating system? Are you asking how you might approach that? Please edit your question and provide more detail about the actual problem you're attempting to solve.
– user16
Sep 15 '15 at 16:10
• I think what you're asking is reasonably clear: you want to get separate values for the thermal resistance of the whole house (probably measured in W/K) and the thermal mass of the whole house (J/K). If, as GlenH7 asks, you can add to the question the actual problem you're trying to solve, we can provide better answers. What difference will knowing these numbers make for you? What decisions will be influenced by them? Sep 15 '15 at 17:01
• Do you mean insulation instead of isolation? They're both heat transfer terms, but they have very different meanings. Oct 21 '15 at 11:59
• @Carlton no, isolation is correct in this context, The reference to [thermal] conductivity makes it clear that the question is about thermal isolation (that is, how hard it is for heat to enter or leave the bulding), rather than insulation (energy-efficiency measures to increase the thermal isolation). "Extracting a buildings insulation" would mean physically removing those energy-efficiency measures. Oct 21 '15 at 12:41
• @Carlton no worries. I'm afraid Samy hasn't been around for a month, so we may have lost them. Oct 21 '15 at 13:10

Yes, there are at least two ways to do it.

### Static, deterministic analysis

One is to look at total energy in over a long time period, such that the internal temperature is the same at the start and the end. And if you take the time period to be long enough (30 days or more, for many buildings), then any changes in heat content of the thermal mass will be negligible when you divide out over the whole period. You can then integrate $$T_{internal}-T_{external}$$ over time, to find the total number of Kelvin-seconds of temperature difference. And divide total energy put in by the heating system plus insolation, by your number of Kelvin seconds, to get the house's thermal conductivity in W/K. Once you've got that, you can then look at individual night's cooling curves to estimate the thermal mass of the house, e.g. by using a simple one-box simulation.

### Bayesian, probabilistic time-series analysis

Another method, and one I've had a little involvement in myself, is the Bayesian approach, which makes the most of all the available data, and jointly estimates probability distributions for both parameters simultaneously. This method is harder to implement, but is a lot more powerful: you will probably only need a few days' worth of data; and you will get much more information out than you would from the static method.

You write down the equation for the probability of observing the temperature series $$T_{internal}$$ for arbitrary values of the house's thermal conductivity and thermal mass R, and given external temperature time series $$T_{external}$$; you add in prior distributions for your thermal mass, thermal conductivity, observation errors, and calculate a posterior distribution for thermal mass and thermal conductivity.

The more information you've got, the more elaborate you can make the model. Add in time-series of heating-system energy use first, as that will be the most useful information. Then insolation, and wind speed and direction. Build up the complexity of your model one variable at a time, to ensure you've got sane, plausible results at each stage.

For an example of this Bayesian approach for a single building element, see "Inferring the thermal resistance and effective thermal mass of a wall using frequent temperature and heat flux measurements" by some of my colleagues. There will be more papers in this series - keep an eye out for new papers by the first two of those authors in particular.