Well I use a immersian chiller and when it is cool I use my sanitized auto shipon and rack it....before that, EIther I just dumped everything in, without straining, just pour it in the bucket or in the funnel....Or I would use a big strainer that fit in the funnel for a carboy, or a sanitized 5 gallon nylon paint strainer bag in the bucket...
Many people don't bother straining at all..sometimes I don't bother....It really doesn't matter...anything will settle.
In other words, there is no wrong way to do it, or better way, or way that will make the best beer...they all work...the choice is what will work the best for you.
What I do with my IC, is chill the wort, then I lean the bottom of my autosiphon about two coils up from the bottom on the metal of the siphon. That rests it above most of the break material and trub, then I rack it to the fermenter until I'm down to that and carefully lower the siphon down into the gunk, just trying to get as much of the wort as possible without letting in the hops and break matter.
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