How can i conjure a fire elemental and trap it in a forge for eternal use

Are you asking as a player or GM?

As a player I would say ask your GM. Everybody will have a different idea as to how this can be achieved.

As a GM I'd say make up whatever you feel is appropriate to the situation. Keep in mind that this would end up as a form of slavery or kidnapping. Fun stuff you can have the elementals allies try to rescue it. 

The work could be a trade off.

Maybe the elemental could use a favor. Help him free a buddy from some other person that has enslaved him into firing a forge.

Planar Binding would be the best hard RAW method to attempt this, although you can't get "eternal" that way; year and a day max with a 9th level cast.

Cast Magic Circle at 3rd level using the inverted option.

Cast Conjure Elemental to summon the elemental into the magic circle.

Immediately begin casting Planar Binding at your highest level. If the DM is lenient about timing then your elemental is bound - you can then wait out or dispel the magic circle. However, if the DM is an absolute stickler they can say that since CE only lasts 1 hour and PB takes 1 hour to cast and you would have had to wait 6 seconds between conjuring the elemental and starting the casting of PB, they can say the elemental disappears 6 seconds (1 turn) before you complete PB. Although it is worth noting that the MC should be preventing the elemental leaving as a result of the CE spell ending but there will be those that argue against that. Also, even the devs like JC state they'd allow this to work - given this is the point of the spell. 

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Although there are, of course, ways to mitigate the issue by capturing an elemental from its native plane and bringing it back with you (a rather difficult task). Or by making the PB be an action to cast or come into effect, such as through magic items like Ring of Spell Storing or a Spell Gem - or by using spells like Glyph of Warding or Wish.

Quote from Wysperra >>

As a GM I'd say make up whatever you feel is appropriate to the situation. Keep in mind that this would end up as a form of slavery or kidnapping. Fun stuff you can have the elementals allies try to rescue it. 

I once had an evil scientist use trapped steam mephits in jars to heat his experiments. The PCs freed them and gained some very useful allies from the encounter.

Well, in the novels the way to trap a fire elemental the size of a volcano is to bind smaller water elementals around it …which effectively traps the fire elemental (and in this case it’s a whopping primordial -which makes for the most powerful forge in the world). 

pure homebrew but I’d say they’d need to pay a wizard to cast conjure minor elemental (or conjure elemental) for a year to make them permanent….and then cast planar ally to get a nice big fat fire elemental -that would then be trapped.

…something like that anyway, but all homebrew

There’s the red corundum elemental gem. RAW it doesn’t contain an elemental, but can be used to summon one. Since homebrew is likely the answer, maybe someone figured a way to actually contain it, but it gives off crazy amount of heat. 

If you speak primordial/ignan then there is the possibility of negotiating a period of service. Forever is out but a century at a time might not be unreasonable.

Granted, the lore on what exactly a discrete elemental represents and its ability to relate with humanoids is sketchy, so it's up to the DM if your basic Fire Elemental is cognizant of and/or amicable to the concept of service for payment.