My mother used to work for the Lawrence Berkeley Lab, so I have a bit of an affinity for their research. Plus it’s late at night and I can’t sleep so I am going to have to blog about building envelope models.

After my post on Likelihoods for physical type models, Phil Price posted an example problem about modeling air leakage in buildings. His equations are:

$Q_{ho} = C_{ho} P_{ho}^{n_{ho}} + C_{hg}P_{hg}^{n_{hg}}$

$Q_{ho} = C_{ho} P_{ho}^{n_{ho}} + C_{go}P_{go}^{n_{go}}$

$$Q_{ho}$$ is a flow rate of air into the house via a blower door, and the equations express the fact that both the input into the house leaks out of the house to the outside as well as out of the house into the garage (eqn 1), as well as the fact that the flow into the house leaks out of the house to the outside and also out of the garage to the outside (eqn 2).

Let’s introduce two fixed scales $$P_0$$ and $$Q_0$$, the first one is a typical pressure that is set at the blower door (the kind of thing you write in your testing protocol and that the operator tries to match approximately). Most likely it’s something around a few inches of water pressure which is around 5 to 10 millibar. The second one is some sort of flow rate that is typically demanded by a typical house. We don’t want something we have to estimate from our data, but let’s choose a quantity that maybe represents something in the middle of typical values from *past* experiments. Most likely it’s around a few hundred to a thousand cubic feet per minute (cfm).

Now let’s assume that the house to garage interface is fairly tight so that most of the air leaks out of the house, and rewrite our model equation in nondimensional form as follows:

$\frac{Q_{ho}}{Q_0} = (1+\epsilon_{ho})(\frac{P_{ho}}{P_0})^{n_{ho}} + 2\epsilon_{hg}(\frac{P_{hg}}{P_0})^{n_{hg}} – \epsilon_{go}(\frac{P_{go}}{P_0})^{n_{go}}$

We actually did two things here. First we rewrote the equation in nondimensional form. This is more reasonable mathematically (what are the units of the C values in Phil’s equation? They’re flow rate per pressure to some weird exponent, with a different exponent for each interface. Furthermore, if there is uncertainty in the exponent, then there’s uncertainty in the units of C, weird. Also the values of C will depend on how people measure P, whereas if everyone uses a nondimensional model with the same $$P_0, Q_o$$ converted to their favorite units, then the coefficients are all comparable across everyone’s experiments). But nondimensionalization also lets us make estimates of the size of certain things, and then we can deal with dimensionless quantities such as the $$\epsilon$$ values. Second we combined the fact that the flow into the garage is the flow out of the garage so that $$\epsilon_{hg}(\frac{P_{hg}}{P_0})^{n_{hg}} – \epsilon_{go}(\frac{P_{go}}{P_0})^{n_{go}} = 0$$ and we can add it to the first equation with impunity. (Note, technically we need to be careful about the signs of the pressure measurements, are they the difference between inside and outside or outside and inside??? are they all consistent? but I’m assuming Phil is able to figure out how to be careful about this).

Now, the left hand side is always positive, and typically around 1, so we can give it a lognormal prior with $$\log(LHS) \sim N(0,\log(4))$$. In other words, it might be typically between 1/4 and 4.

If the garage is fairly tight then $$\epsilon_{hg}$$ should be small and the majority of leakage is due to the $$(1 + \epsilon_{ho})$$ which also has to be positive, so $$\epsilon_{ho}$$ can be given a prior on $$[-1,\infty)$$ for purposes of working with JAGS it’s probably better to define a variable $$K_{ho}$$ and give that a prior on $$[0,\infty)$$ a good one might be exponential with mean 1. The prior on the other two epsilon values should be also positive and they should be small, so perhaps exponential with mean 1/10 or something. EDIT: actually the $$\epsilon_{hg}$$ value is small, but $$\epsilon_{go}$$ is not, nevertheless the overall term for the garage-outside interface is small because with a small flow into the garage there is a small pressure differential from the garage to the outside.

Now comes the difficult part: the likelihood. Well, Phil seems happy with starting with a measurement error model using normally distributed errors, so perhaps he can start working one up from this model. This has the advantage of not needing the left hand side to appear twice, and also of being nondimensional with everything scaled so that typical things are about 1.

In the mean time, you can check over at Andrew’s blog to see if the debugging process for the JAGS code is moving along.

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