Ok, a few things I have questions about. First, from this photo you provided, does the flame pass under a basket, up around the sides of a basket, or through the contents of the basket? And, the bottom looks like the floor seen outside the unit, I hope there is a refractory bottom.
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Unfortunately the burner comes in perpendicular to the circumference, I would like it better if it were tangential as the swirling of the flame path makes any incompletely combusted fume to remain in the heat zone a bit longer.
Incinerators use afterburners because it is hard to get enough air into the incinerator for complete combustion in the time the smoke from the burning takes place before it passes out of the unit as smoke. Generally the type burner you have gives you enough air for the gas you are using in the burner and does not take into consideration the air required to completely combust what you have in there. But with the dimensions you gave, it would be relatively easy to put a closed chamber in the unit and force the smoke from the closed chamber to exit the chamber at the bottom where the fire will ignite it.
I would first try to fabricate a round crucible rest that would be located in front of the flame. The flame would then bounce off the rest and begin to circulate in the unit. Then I would get a crucible that will allow at least 1 inch (2.54 cm) clearance all around for the flame and burning smoke to exit from under the crucible. Then I would cast a heavy lid out of refractory cement to sit on top of the crucible with a hole in the center into which you cement a 1" pipe that is flush with the bottom of the lid and comes up enough for you to thread on a 90 degree elbow. Then, thread in a nipple long enough to direct the fumes to the side of the crucible and into a 90 degree elbow facing down and continue the piping all the way down to the height where the flame comes out of the burner.
Now you place your unburnt sweeps in the crucible and fit the lid so the combustible fumes will be directed back to the flame zone. Now fire it up. The flame should provide heat to cause the sweeps to give off their volatiles and the smoke should be burned in the process of coming back up the side of the crucible after being directed back down into the heat zone by the pipe. A spiraling flame will help the smoke stay in there a little longer. Actually afterburners only need a 1 second dwell time in a properly sized and correct temperature afterburner.
In theory, the sweeps will get hot enough in the crucible to pyrolyze because there is no air in the crucible, or very little. The sweeps will give off all of their volatile gasses and essentially pyrolyze. Then, after the burn is complete, you will have to spread the pyrolyzed sweeps in a shallow tray and light them. They should burn like charcoal and not smoke just like a barbecue. Once the carbon is gone crush, sift and assay.
