Lets suppose we have access to a blast furnace (Bessemer?) and, instead of iron ore or recycled steel, we fill it with pure rust
(Fe2O3…Fe3O4…whatever).
We get the reaction going so that the temp is in fact hot enough to melt iron; and we pump superheated air and/or oxygen in from below to burn off the impurities.
Would this reduce the iron oxide back into iron?
Short answer - yes.
Longer answer - rust ususally consists of a mixture of hydrated iron oxides and hydroxides, which when heated slightly form Fe[sub]2[/sub]O[sub]3[/sub], or haematite. Mineral haematite is the iron ore of preference for blast furnace feed stock.
A “bessmer converter” is different from a blast furnace. It was a 19th-early 20th century device for converting the “pig iron” output of the then-current blast furnaces to a lower carbon, usable steel by blowing air through the molten metal. The steel produced was rather hard as a result of all the nitrogen introduced into it from the air blast. Today we use the “basic oxygen process” where a lance blows a jet of pure oxygen into a cauldron of molten blast furnace iron. It’s quite spectacular.
On re-reading the OP, I think I’ve misunderstood the question. A “blast furnace” is a specific kind of furnace for reducting iron oxide using carbon. Preheated air is blown in from the bottom, but the furnace contains both iron oxide and coke (carbon). It is combination of heat and carbon that reduces the iron oxide - in effect the carbon strips the the oxygen from the iron to produce molten iron and carbon dioxide.
Simply heating iron oxide or rust to a high temperature and and blowing air or oxygen through it will not reduce the iron oxide to iron.
Right…I knew there was fuel in there somewhere.
A blast furnace has different zones, reducing zone, heating zone, etc.
The actual reduction of Iron Oxide is accomplished by CO (Carbon Monoxide)
If you want to reduce the rust to iron without using C, you may do so by using H2 (Hydrogen) - Heat the rust up and pass H2 through it (thats called Sponge Iron - BTW).
Bessemer Convertor is used to remove the C in the Iron obtained from the Blast Furnace. Oxygen is purged at supersonic speeds into the molten metal (the temperature of the metal increases in this process) using a lance made of tungsten or copper cooled with water. Alloys are added to give steel desired properties during this process.