It is supposed to have a heat sheild, but I couldn't fit it between the barrel and the wall in the design because the barrel is off scale.
"Of all the beings that exist, man is only one. Among all the Millions of men on Earth, those that live by farming are only a small portion. smaller still are those that have office or wealth and ride in carriages. Of all these, a man on a carriage is nothing but a fleck of hair on a horse's flank."
I think you would be better served by using a "bell" to distribute the exhaust gasses.
The strait line duct will extract less heat into fewer square feet of greenhouse soil.
Matt Walker pioneered building long low bench height bells out of 55 gallon steel barrels cut in half lengthwise.
More recently a similar technique was used to heat the core of a huge mound.
If your green house is say 16' x8', I think you could cover the square footage with 8 barrels.
The larger open spaces within these barrel bells make it easier for exhaust to slow down and stratify, yielding a greater heat transfer over a wider area.
Maybe consider using an opaque, insulated thermal mass as the northern wall of the structure,and include the firebox , chimney and fuel storage within the mass.