Coincidentally, the wind is also quite brisk at my house, and the temperature is near freezing.
Not knowing how to measure air flow leakage through my bathroom fan, I decided to do an experiment.
I used 4 pieces of masking tape, and secured one of those ultra-light weight produce bags over the inside vent opening.
It just kind of pulsated like a plastic lung, so it appeared that I had no net leakage.
Still curious, I replaced the produce bag with a dry cleaning bag, and inflated it with a hair dryer.
It took ≈3 minutes, to evacuate ≈1/2 of the 1.1 m long x 1.24 m circumference tube, yielding a flow rate of 0.000374 m^3/sec. (0.8 cfm)
Which from my calculations gives me a net energy loss from the vent of ≈10 watts.
Which is not quite as high as my 1700 watts lost through my walls and ceilings. (R-13, 1000 ft^2, 8 ft tall walls, diff temp 40°F)
My guess is that there is some type of Bernoulli effect going on at the roof vent, which is where my fan exhausts to.
As far as the costs go, my fan leak costs me ≈90¢/month, and my poor insulation is costing me $150/month.
The conclusion of the following is that a proposed alternative effect of why the bag shrank is negligible, so
you can ignore it:
Another possible explanation for the bag shrinkage may be the back and forth flow of air through the vent. The vent tube runs through the attic.
So I took some measurements:
attic RTD reading: 5.46 kΩ (My house is wired, for science. There's also an RTD in the crawl space.)
attic temp: 43.1 °F, 6.2 °C, 279 K
Ambient bathroom temp: 70 °F, 21.1 °C, 294 K
From the ideal gas law, we know that PV = nRT
Given that P, n, and R are all constant, Volume must be proportional to Temperature: V = kT
Volume = 0.135 m^3 = k * 294
therefore k = 0.135/294 = 0.00046
Volume @ 279 K = 0.00046 * 279 = 0.128 m^3
0.128/0.135 = 0.95
Which tells me that the bag would only shrink 5% if the bags air was being chilled by 15 K. (27 °F)
