That is a gigantic J-tube! The dimensions generally scale proportionally, though at such sizes, I don't know if anybody really knows how it will perform. As there is a lower limit around 4" where the combustion to surface area ratio causes excessive losses and reduced temperatures, there may also be an effect at much larger sizes where temperatures get so high that you approach blast furnace conditions, with temperatures far above 2000 F. I would absolutely expect even stainless steel to be degraded quickly, and perhaps even melt.
You will need to use only refractory materials in your core. I don't know how your lava rock behaves, though I suspect it will start melting by 2000 F. Also, temperatures around 2500 F or so cause nitrogen to burn, creating NO2 and similar compounds which are harmful. You really don't want temperatures much over 2000 F. The highest grade firebrick I know of is rated for 2600 F.
Generally, the feed tube is the lowest temperature zone, with the later burn tunnel and lower riser the hottest. I am not sure how air velocity through the core will scale; I don't think it will be fully linear, as in three times faster than in an 8" J-tube. Thus, there may not be an advantage in making a 14' tall riser aside from the chimney draft effect. I don't think the horizontal burn tunnel needs to be fully twice as long as the feed depth. 28" diameter as you show is so huge that I think you would need industrial engineering and materials to construct and operate it safely. Even 12 to 16" square would have a very large capacity and high temperatures for incinerating.
What kinds of material do you want to incinerate? What size is the largest thing you would need to put in? What volume per day would you be burning?
For the riser, it needs to be one single cross section, not split up, as that would introduce more friction and slow the system.
Rather than trying to hold the trash in a drum, I think it would work better to make a firebrick "grate" of closely spaced fins standing up some from the floor, with the slots following the airflow streamlines so they get swept out by the draft. What would be the minimum common size of items in the load? Spacing to hold most of those up would allow good flow around the load. Fins/slots running down the sides of the feed might work well to get airflow around the load without dead spots.