Very long technical post
  My quest continues to dissect and pull out as much information out of simple temperature sensor data. An engineer always wishes they had more (reduce the intervals, increase the accuracy) but that always comes at a cost. But I am happy with what I will try to show you. I looked at several clustering models that assumed Tambient = Tinternal (Southwick, Watmough/Camazine, Myerscough, Sumpter, Omholt) and attempted to apply their principals to an insulated hive. Instead of a sphere, I am using hemisphere geometry. You will see from the Temperature distribution profiles, the bees in a single brood box volume assume that shape. By having very well insulated walls and cover, the bees seem to incorporate those as part of their winter cluster. The heat loss off those surface is less than that of the lowest cluster thermal conductivity.  [0.0003 W/cmC]/divided by the thickness of the insulation (R40) Vs [0.01W/cmC] divided by the thickness of the outer cluster mantle (smaller is better).
  I see this same behavior in my double but instead of using the top cover they are using the opposite side wall in the center of the hive. It also seems like the heat off the sideways hemispherical cluster is rising drifting across to the opposite side and then cooling and dropping to the bottom of the hive. The cluster seems to tighten in colder temperatures to the right thus leaving a cold air downward moving chimney.  (1st attached picture).
  Next, I will look at the single brood box cluster isotherms, I am assuming the outer mantle is between 10 and 12C. I summed the squares with temperature greater than 10C, greater than 12C and so on. Spreadsheet allows me to adjust these value as needed. I then use the area to calculate the radius of the T isotherm. Assuming the cluster forms a hemisphere simplifies this. So the bee  thickness is just r10-r12 and for all the intervals I have chosen. Using a density relation p(T) from Myerscough's model (taken from Heinrich) I am easily able to determine bee density (bee number) for that layer and the average weighted temperature for that group. With the radius I am able to determine the surface area, the volume and thermal properties in theory.
  One of my goals was to determine an estimated MR Metabolic Rate. I used Southwicks chart here and ran it for 2 bee numbers (20K and 30K) as that is my bee estimate for that hive. I also ran it for 2 different  temperatures. The average weighted temperature of the cluster (gives a very high Watts/hr) and the average lower temperature that the lower cluster is exposed to (Gives a very realistic value). I also attempted to use Myerscough's approach but it was too difficult. Works well for modelling simulations but not real life data. The challenge is that the heat output of the cluster is not based on one temperature but the local temperature of each bee.
   I have it setup now that I will be able to record these calculations for hourly intervals for the complete data set into a pivot table (radius, surface area, volumes, bee density, bee numbers per isotherms, MR, equivalent honey, excess water generated). The MR estimate using the lower average temperature (T Cluster Out) gives a very realistic value. Due to the conservation of energy law, heat output must equal heat loss. It will allow me to estimate the heat loss off the lower cluster surface using 1st principals (heat transfer coefficient from Heinrich (1981) and compare it to the difference between heat generated minus upper hive heat loss as per previous post. I also plan on modifying my entrance using styrofoam to reduce it to a set size. Before and after measurements should provide some insight. Remember I have a built dead air space below my hives to provides extra air and location for condensation to collect. I will also tape a small IOS IR camera to a small metal rod using a Lightning cable extension (male to female) to get lower cluster video/photos via the lower entrance.
   The last 2 pictures show the cluster profiles (radiuses, T profiles, MR, etc..) for -9C and -32C. By applying the model to real data I am able to show some of the clustering characteristics (cluster Temperature gradients, bee density, heat output, etc...). As D Mitchell mentioned in a couple of his papers. In well insulated hives, bees really don't cluster until at least -10C and even at -32C the bees are in a much reduced "cold" stress situation. In my setup, the clustering behavior is a secondary survival mechanism with the hive enclosure is the main driver. Back to the feral colony/bee tree discussion, the right tree would have the same effect.
  Why am I doing this: I like to know why stuff happens but mostly to continue tweaking my winter setup (size and R-value), manage moisture, optimize food stores. I should also be able to start building relationships between local climate and optimal R-value to maximize benefits (consumption vs moisture generation, cost).
  As always, I am human and I may have some mistakes/errors, I haven't had any of this peer reviewed, but it has been a really good mental exercise in these Covid times.

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