Though I agree with all the comments posted 'til now (especially jadnashua's who realizes you can't solve the problem by only looking "up" at the problem), I also think the answer may not be a beam...
You surely will want an engineer's input regardless, but it could be significantly easier... that is, if I understand the problem without pictures, or sketches.
My assumptions from what you have posted:
- The wall you are removing is parallel to the ridge
- The rooms have vaulted ceilings on both sides of the wall
- This wall's top plate is directly supporting the rafters
- The wall you want to remove does not go the entire length of the ridge
OK, so assuming those are all true...
You do not need to support the rafters at the ridge... as long as you have
something keeping the rafters from spreading. Basically, the roof load will be balanced quite nicely with the rafters in opposition, but without something holding the walls together they will push the walls apart. A typical house has ceiling joists to hold the walls from spreading, but a vaulted ceiling has no ceiling joists. The wall you want to remove isn't really keeping the walls from spreading (presumably other parts of the house are doing that), but it is keeping the ridge from sagging (and then pushing the walls out).
An engineer is likely going only going to suggest one solution: add rafter ties (or ceiling joists). A beam also would work, but will mean more work unless adding extra posts, hardware, and footings to support the beams doesn't involve opening up several extra finished walls and floors. (Need to worry about everything from squash-blocks in the floor diaphragm(s) to the soil properties below the foundation)
You will have to tell him/her where you want them... You can find prescriptive codes for rafter ties at I believe either 1/3 or 2/3 the rafter span (your "rafter span" is 12' 6" BTW), but I would still have an engineer do it, stamp it, and give you a fastener spec too (i.e. not just a few nails wherever you can squeeze a nail gun in). The lower down the rafter tie, the lower the force required to hold it together. If you want it quite high (i.e. within 12" of the peak), you will end up needing to have metal plates fabricated to sister on the joists.
You can also choose to add them lower down and either drop the ceiling or have the ties exposed.
There's a fine homebuilding or JLC from ~2007 sometime that talked about moving/removing collar ties and the tremendous forces he ended up dealing with.
Anyway, talk to an engineer/architect and have them design the best solution for your desired result. DON'T rely on the lumberyard to spec it. They will get the "correct" beam, but they won't take the whole picture into account and, generally, won't worry about taking a distributed load and making it a point load and cracking your foundation or your gable end wall blowing out during a big snowstorm.
BTW, it's pretty easy to size that beam if you know the designed roof load and the span (22'). It's not a secret for engineer's only (though it's outside of prescriptive codes, so in most parts of the country an engineer would have to spec it). For the reasons others have mentioned the answer to the specific question of what size beam is ignoring the rest of the problem. Specifically, ~14,000 lbs of a problem! (with a 40/10 roof load)
As a contractor (carpenter), I talk with clients about this kind of thing a lot. They usually are interested in getting their envisioned result only until the actual costs start adding up, then we work together to get a good result at a reasonable cost. If everything is torn up, then the beam is simplest, if they just want to play with cabinets and flooring and "open up that wall while you're at it" I'd probably float the idea of putting a flat top on the vaulted ceiling unless cost is not a priority (haven't seen too many of those jobs lately). Of course they usually ask me to price out their original option, which I'm happy to do after they pay for the engineering to tell me what they're asking me to build!