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I Forge Iron

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Chimney Height
First I suggest that you check your local codes.
But the general rule of thumb found throughout the country at one time was as follows.

Chimney height Rule: 10’-2’-3’

This information is from an old file what I had accumulated and used years ago when I was active, but I would bet it is still valid.
The Factory-built fireplace manufacturers specify minimum and maximum chimney height. Woodstove manufacturers may have similar requirements. Both types of appliances and most codes require the 10 foot/2 foot/3 foot rule for minimum chimney height above the roof:

The chimney must extend at least 2 feet above any portion of any structure within 10 feet (measured horizontally) AND must extend at least 3 feet above the highest point of its roof penetration.
The height of the chimney must meet or exceed both of the minimum requirements. If manufacturer requirements exceed code minimum height requirements, they should be followed.
These minimum heights may have to be exceeded, however, for adequate draft and satisfactory performance.

Additional height can help improve draft and may make the difference in marginal situations. It will not, however, cure severe draft problems. There are two situations that call for specific attention to chimney height:

Heated portion of the structure above the top of the chimney: the structure can act like a chimney and compete for the air supply.
This action, called structure stack effect, can result in weakened draft or even smoke spillage into the structure.

High altitude: Your forge may need stronger draft to deliver a greater volume of air at higher altitudes to make up for the lower concentration of oxygen. Recommended sea level chimney height may need to be increased by 2-3% per 1000 feet of elevation.

In general, a minimum venting system height (chimney or for some woodstoves, chimney + connector) of 15 feet is recommended. It should be noted carefully, however, that too tall a system can cause over-drafting problems for wood stoves, but would most likely be desirable for most coal/coke forges.
Just something to consider.

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