Despite what people say. My opinion is, that the cooling effect doesn't suck air in.
It might help.
The barrel is, primarily a radiator. And a downdraft channel.
In a downdraft channel, if the gases are hot, they resist the movement of going down, to some extent. Impeding the draw.
Then, if you cool the gases at this point, these get denser, heavier. And are more readily pushed down by lighter hotter gases.
In my
workshop heater. I have used a fuel tank.
http://donkey32.proboards.com/thread/1817/starting-build-220mm-rocket-double
Good pics of the "barrel" tank on page three.
I have pretty much the same surface exposed, as a barrel. Tho, the gas movement is far freer. What i found, is that i feel there is too much "radiator" area. Tho, now that it is starting it's third winter. and that i live above. I like it more. Tho, if i had to redo one of this type. I think, i would take it the other way. A big mass bell above the heat riser. So there is a huge temperature differential between the mass and the hot gases. So they heat more. And if i needed quick heat, i would put a "radiator" metal bell, before the chimney.
What i see in this arrangement:
If it's too hot, and you need heating the mass à bit more. You could bypass the metal bell.
Gases entering this bell are cooler. So the place doesn't overheat as easily. This, allowing you to heat more the mass.
You're less at risk of burning yourself badly with a cooler "radiator"
HTH.