RAB,
I looked at making the same circuit and tried to work out how many watts it would need to put out to make any difference.
According to my calculations (done 2 different ways and pretty much agreeing), the cluster will be burning about 20 watts at -10. This ambient temp for the cluster is affected by many things, like convection of cold air up and through the BB, insulation factor of the walls and roof. The size of the cluster obviously affects how much it 'burns' too and, as it seems the cluster changes size accordingly to ambient temp (!) the whole thing gets pretty hard to predict with any accuracy.
Bottom line though, is that 10 - 20 watts is about right for a normal colony in a -10 C ambient temp.
So . . . we only have to strap a small metal box to the mesh floor which is generating say 5 - 10 watts of heat and the convection should bathe the cluster in a gently raised stream of warm air and save them some work. This saved work should equal less honey/syrup consumed, I guess.
A secondary effect might be that normally the warm air will only cover the inside of the box (ignoring any top ventilation for a moment) down to the lowest point that is producing heat. With a cluster, this will mean that the warm air occupies a space down to the level of the bottom of the cluster. Below the cluster would be a layer of cold, unheated air. This leaves the poor bees at the bottom with their backsides in -10 C air (ouch!). If we heat from a box at floor level, this should mean that the air is warmer from that point upwards (i.e. the whole BB) by convection, so the cluster wouldn't face the same extreme. Maybe this would allow them to roam the BB for stores with a lower heat-loss penalty (?) which could give them an even better chance of survival.
There's a little temp sensor that can drive an Op amp and power transistor, plus say 10 parallel R's to burn 20 watts or so easily. My calcs suggested it could run from a deep-cycle battery (48Ahr) for 24 hrs at 12 watts - getting to 50% charge in the battery. A solar charger could extend the time between charges if it was sunny during the day and only cold at night.
I could power this from a shed at the moment, but it might be worth trying the stand-alone version for those with out-apiaries.
Do you want to collaborate on the design and give it a go? The colony I have needs all the care and attention I can give it and is in a fairly cold spot.