Yes, Paul's pebble heater has a wooden box that contains the barrel edges.
However, this is a semi-permanent build in his home; here we are discussing a bomb shelter.
Unfortunately, the Ukrainian people are forced to spend more time in them than they want to.
You can use anything available to hold and seal those long edges. All you need is to keep them in place. Mud, bricks, and rocks come to mind.
Fox brings up a valid point: the risers need (should) to be insulated; it's very accurate, and without it, performance will suffer.
But does this really matter?
I come back to this.
It is for a temporary Bomb Shelter.
The whole idea is to remain as warm as possible with as little wood as possible.
You don't want or need longevity, efficiency, or looks; you just want people to stay alive until they can stop huddling in bomb shelters and return to their homes if they are still standing.
Edit)
I just read your latest post.
I did not understand your needs, especially regarding the need for an engineer to sign off on the project.