I apologize in advance for this very long post. Using just a little more math related to the bees’ metabolizing the sugar in the honey over the winter, it seems that a large amount of ventilation – as much as 10 liters of air through a hive per minute – is key. The side insulation used to prevent heat radiation seems less important, although a significant amount of side insulation should considerably reduce the amount of honey stores needed during the winter. The oxygen/carbon dioxide balance does not seem to be an issue, since the amount of ventilation needed by the hive simply to remove all of the water produced by the bees’ respiration would more that adequately move the carbon dioxide out of the hive.

It would be great if someone could double-check my assumptions and this math below. Even better, I hope someone with a knowledge of thermodynamics and/or fluid flow dynamics would take this analysis to a practical conclusion, so we all know more precisely, ideally how much hive ventilation and insulation to use in various climates.

MOISTURE VENTILATION
1.	Recall from David’s math in the first post of this string that 100 lbs. of honey, when metabolized by the bees produces about 30.7 liters of water.
2.	Based on the rule of thumb that a strong hive in two deeps located in the northern U.S. (I live in eastern MA) requires 60-80 lbs. of honey to get through a typical winter, assume that a hive actually consumes 60 lbs. of honey.
3.	This 60 lbs. of honey generates about 18.4 liters (=18.4 kg) of water over the winter, nearly all of which needs to be removed by way of air ventilation.
4.	According to the relative humidity calculator at https://www.lenntech.com/calculators/humidity/relative-humidity.htm which is expressed graphically by Greg Benson at https://en.wikipedia.org/wiki/Relative_humidity#/media/File:Relative_Humidity.png a kilogram of air (=.78 cubic meters of at 1.275 kg/m3) can hold about 6.3 grams of water inside the hive (at10C and 80% relative humidity), but only about 1.3 grams outside the hive (at 0C and 30% RH), for a difference of about 5 grams per kilogram of air.
5.	So, in order to expel the 18.4 kilograms of water, about 3,680 (=18,400/5) kilograms of air, or about 2,870 cubic meters of air needs to move through the hive during the winter.
6.	Assuming this happens over about 200 days from the beginning of October through the middle of April, or 288,000 minutes, the air flow through the hive needs to be about 10 liters per minute on average.
7.	For someplace further south, the ventilation needed to remove the moisture should be much less. Using the same calculator and math, if the average winter outside temperature down south is 10C at 40% RH and the colony consumes only 30 lbs. of honey, the ventilation need would only be about 1.5 liters per minute.

INSULATION
1.	According to http://book.bionumbers.org/what-is-the-free-energy-released-upon-combustion-of-sugar/ the free energy released from the oxidation of sugar by oxygen is about 3,000 kJ/mol, or about 700 kcal/mol.
2.	Based again on David’s math in the original posting, the 60 lbs. of honey is about 120 mol of sugar, so releases about 84,000 kcal of energy as the bees metabolize it over the winter.
3.	A kcal is the amount of energy needed to raise the temperature of 1 kg of water (or any material) 1 degree C. So, the energy lost by way of the ventilation of the total 3,680 kilograms of 10C air that is replaced by 0C air is roughly 37,000 kcal, or about 44% of the energy produced by the metabolism of the honey.
4.	The surface area of a two-deep Langstroth hive is about 1.3 square meters (just multiplying all of the dimensions).
5.	Assuming that the typical hive is ¾” pine, which has a R value of about 1 (at 1.4 per inch), the pine hive can radiate approximately 10 watts (8.6 kcal/hour) per square meter, or about 11.2 kcal/hour, using the insulation formula at https://en.wikipedia.org/wiki/R-value_(insulation) 
6.	Over the same 200 days above (4,800 hours), this translates to about 54,000 kcal lost through the hive body. Together with the 37,000 lost through the ventilated air, the total is pretty close to the 84,000 kcal produced by the metabolism of the honey given all of the assumptions and rounding in these calculations.
7.	The insulation value of a polystyrene hive would reduce the amount of energy radiated from the hive, though not the energy lost through the air ventilation. I could not find the R value of a polystyrene hive, but I assume it is pretty close to the R=5 value of 1” polystyrene insulation.
8.	As such, the polystyrene hive’s insulation would reduce the energy loss by radiation roughly 43,000 kcal, so the hive should (theoretically) require only about half of the honey otherwise needed to get through the winter, and less ventilation to remove the moisture. This in turn would reduce the heat loss – and required honey – even further.
9.	You would need a dynamic model or maybe a little calculus to figure out the equilibrium honey needed at various combinations of temperature, relative humidity, insulation, ventilation, etc., but this is well beyond my capabilities.

CARBON DIOXIDE
1.	Gene asked about whether carbon dioxide buildup might be a problem.
2.	Back to DaviD’s math, the metabolization of 60 lbs. of honey produces 720 mol of CO2. The CO2 has a molecular weight of 44 grams/mol. At 44 grams/mol, that amount of honey produces 31.7 kg of CO2 over the winter.
3.	One kilogram of CO2 occupies roughly a half cubic meter, so the 31.7 kg of CO2 produced over the winter is about 16 cubic meters that needs to be vented over the 200 days.
4.	Compared to the 2,870 cubic meters of air that needS to move through the hive to remove the water, the amount of CO2 does not seem to be a factor.

John

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