I'm trying to devise a plot point where a great sorcerer (who's also a bit of a renaissance man, alchemist, inventor, politician, etc.) who slays a dragon and use its corpse to carburize iron ore inside a volcanic mountain into steel (or tungsten into tungsten carbide) that locals might call it "dragon steel". Does this make sense? If not, what needs to be changed to make sense, or is there a way to create a similar effect?
3 Answers
You need to pyrolyse your dragon to remove impurities, particularly the phosphorus and sulfur as they have the most detrimental effects, that would damage the quality of the result. You could conceivably then use it as fuel in a Tatara style furnace to produce Tamahagane the high quality "jewel steel" of ancient, and modern, Japan. Tamahagane is an outlier in the world of ancient metallurgy in that it is produced in a one step process that goes from pure, high quality, ore and fuel directly to steel without carburising bloomery or decarburising cast iron.
If you want to use a volcano though I would suggest that blister and shear is the process for you. Low carbon reasonably pure bloomery iron is beaten into fairly thin bars that are packed in a carbon donor, usually charcoal or coke but in this case cooked dragon, in a stone box. The box is then brought to high temperatures, usually 600-800°C, and held there for days, or even weeks, on end while the metal absorbs carbon. This process used to be ruinously expensive because of the fuel consumption required to maintain the oven at working heat for such long periods, the volcano will be much cheaper. The name blister steel comes from the blistered appearance the bars take on where the absorbed carbon distorts their structure. Shear steel is the result of forge welding the bars together to more evenly distribute the carbon they had absorbed.

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Tamahagane is far from a 1 step process, and it does include carburizing in the refinement process (normally done using pyrolyzed rice grass). http://www.ksky.ne.jp/~sumie99/forging.html – Nosajimiki Jan 04 '22 at 15:52
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@Nosajimiki Only for the parts of the bloom, sometimes all of it, that are lower quality. – Ash Jan 05 '22 at 02:45
It's Magic
Assuming your "great sorcerer" is in fact a sorcerer and your "dragon" is in fact a dragon, then it makes perfect since. Dragons are considered in many mythoses to be the most magical of all creatures or even demigods. And a sorcerer is of course a person who specializes in the manipulation of magic, not science.
There is not a scientific reason to do this, but there is precedent
In many magic systems there is the concept of the reagent: an ingredient used in casting spells for the magical properties that it contributes, and in these systems reagents that are derived from dragons are often considered among the most potent.
In fact, some cultures historically did add un-pyrolyzed biological components to their steel specifically because they believed it imparted some mystical properties into their blades. Some Viking cultures for example would add a raven feather to their steel to impart the favor of Oden on it.
So sure, it could be argued that mixing dragon parts into your steel would weaken it because of the sulfur impurities... but so what? A sword made from sub-standard steel, but is also held together by primordial magic powers beyond our human comprehension is still going to be a large step above any steel made by non-magical means.
But maybe it's not real magic...
What if your great sorcerer is NOT in fact a great sorcerer, and your slain dragon is just some dinosaur fossil he found? Maybe your sorcerer is just an exceptional blacksmith with a flare for the dramatic.
Before modern science no body really understood WHY one steel was better than another, they only understood if you do the right steps you get a better result which sometimes creates superstitious extra steps. Going back to the Vikings, the metallurgy techniques invented in Northern Europe during the early medieval period were so much better than other European techniques at the time that Viking swords were often considered magical. So while that raven feather may have been an unnecessary step, the care they took in preparing their ores and the multi-alloyed pattern welding techniques they used did make their swords much better than those made by other smiths... so as far as anyone at the time was concerned, the raven feather worked.
Likewise, if your smith throws a little bit of fossilized dinosaur into his furnace, it will just melt out with all the other silicates, no harm, no foul. But as long as he does enough other things right with the carbon levels and the purification, and the tempering, etc. then his Dragon Steel blades could still be the best blades in the region.

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This is the "it's magic don't worry about the science" answer, I like and loath this answer but it's always an important point to make +1. – Ash Jan 04 '22 at 05:14
Biomass is not made of carbon only: it contains also, among others, nitrogen, hydrogen, sulfur, phosphorus.
Most of them are normally undesired elements into steel (one of the theories on the sinking of the Titanic claims its steel was very brittle due to the sulfur content), therefore using biomass to dope steel will work, but not in a way that makes the steel particularly attractive property-wise.

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1Most usually you first convert the biomass into charcoal... After all, charcoal made from biomass actually was the source of carbon used in steelmaking for about 2,000 years. – AlexP Jan 03 '22 at 10:14
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@AlexP and when that carbon happens to be rich in sulfur one gets poor steel, until controlling the process is learned – L.Dutch Jan 03 '22 at 10:26
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Doesn't the charcoalification process get rid of most of the sulfur automatically? – AlexP Jan 03 '22 at 10:48
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@AlexP, no. Coal from Sulcis mines, in Sardinia, is an example of high sulfur carbon. – L.Dutch Jan 03 '22 at 10:57
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3Charcoal is not mined coal. Making charcoal involves partial combustion of biomass; I would think that any sulfur will be lost as sulfur dioxide gas. – AlexP Jan 03 '22 at 14:03
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1@AlexP The technical name for the "charcoalification process" is "pyrolytic carburisation" by the way, usually it's just "charcoal making", and yes the sulfur does burn off as a gas, (due to the low oxygen conditions more often disulfur monoxide than the dioxide though), along with the nitrogen, hydrogen, oxygen, and phosphorus. – Ash Jan 04 '22 at 02:24
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1@Ash: Charcoalification was intended to be humorous. I can see it failed. – AlexP Jan 04 '22 at 02:25
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@AlexP Sorry usually does with me, I'm that guy who takes things literally regardless, especially in written media. – Ash Jan 04 '22 at 02:27
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1Nor is there any reason to prefer dragon charcoal over far easier to acquire tree charcoal. – John Jan 04 '22 at 03:19
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@John That part is easy: Why hire entire villages to carry a forest into the volcano when you can just piss off one dragon and have the charcoal move itself? – Fantasy Science Jan 04 '22 at 04:48
Can the dragon be turned to charcoal by a massive fireball and directed to crash over iron/tungsten and future smiths remove the slag to create steel/tungsten carbide?
– Fantasy Science Jan 03 '22 at 16:45