I What thermal cycle would describe the action of a candle carousel?

[jsh111]

TL;DR

Could the Brayton cycle describe a candle carousel ( holiday candle powered trinket that is a simple heat engine) when there is no compression at all?

A candle carousel is a popular holiday time trinket that uses wooden if light metal vanes to rotate and assembly by using the heat created by burning a candle directly underneath. It occurred to me that this is a simple, albeit very inefficient heat engine. It resembles a rudimentary gas turbine in operation. I wondered what thermal cycle would best describe its operation.

Since it resembles a gas turbine, the Brayton thermal cycle would seem to best describe it, however the problem I run into is that for this rudimentary machine, there is obviously no compressor involved, and so how on earth would you even draw the PV diagram..... the area of the curve should theoretically be 0.

Similarly, the efficiency for a brayton cycle ( the thermal cycle of gas turbines) is based on the pressure ratio created by the compressor above ambient pressure. Theoretically, the pressure ratio should be 1 since there is no compression, and the device would then not operate. Obviously this is not the case in the actual device.

My question then is, as a mental exercise, where is my flaw in reasoning? I am sure some thermal cycle would describe this, I just don't know what it would be or how the PV diagram would look like. I have attached a picture of the device below:

 

[256bits]

TL;DR: Could the Brayton cycle describe a candle carousel ( holiday candle powered trinket that is a simple heat engine) when there is no compression at all?

there is obviously no compressor involved

Take into consideration the buoyancy of the air as it is heated or cooled.
Try to draw the PV diagram taking into account gravity to account for the buoyancy.