This is 2 questions - 1) how to make a chute that deals w/ miasma, and 2) how to designate that for miasma-causing stuff.
Part #1, blocking miasma, is easy, but it depends on how you build the 2nd, so first, ... how to designate it.
There are 2 ways to do this, and I actually recommend the non-dump approach, but ymmv
When you designate a dump area, it can be just 1-tile big, and endless amounts of stuff can be put there - 1,000 stones plus misc. rotting vermin on 1 tile, no problem. Handy for all kinds of purposes. The problem is when you have several of these scattered across your fortress - a dwarf who accepts the [j]ob to "dump item" may drop that item in one nearby, or the furthest away - and that's a loooong slooooow haul if he's carrying stones. Similarly, if you're trying to send bones and skulls close to a workshop, it's pretty random where they'll end up if you have more than 1 Dump Zone designated. So, if you can, it's best to have only 1 "dump zone" active at any one time, for one given job.
Meanwhile "refuse" can be stored in a refuse stockpile by any dwarf w/ "Refuse Hauling" enabled. You can have many different stockpiles customized to accept different kinds of refuse (from butcherable meat to vermin to body parts, etc.) - BUT a stockpile only accepts 1 item/square, so no "quantum dumping".
(Note that you can chain refuse stockpiles, from local small ones to feed a larger one off somewhere else. And if you aren't using a Dump Zone for anything else atm, you can remove that stockpile temporarily* and Dump all the refuse into a single miasma-locked dumpsite tile - see next.)
(* Vanilla DF saves the last custom stockpile {s}ettings you designated or {q}ueried, so that might work if you're careful. DFHack allows you to Save custom stockpile profiles with significant names and Load them later onto any new Stockpile - handy for this.)If you have a large(ish) stockpile of refuse, yes - that room will be filled with stenchy, bad-thougth-causing miasma. Which is why you only designate refuse-haulers who have happy thoughts in general. and never have Urist McBorderlineDepressive to haul refuse. Or just use dumpzones and air-locked chutes (see next), and sort it all out later.
So you have to first decide which approach you'll use - multiple Dumpzones (and hope for the best) OR multiple Refuse Stockpiles, and accept the larger footprint they'll need and associated miasma in that room.
Now, either way, stopping miasma is easy - miasma doesn't spread diagonally, so dig a diagonal access and you're good.
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The refuse goes into the inner open area, and can be left there to rot safely. 1 tile is all you need for a Dump zone, or make that area larger for a full stockpile. (You can also simply put 2 doors into the area, and let those act as an "airlock" - either/or.)
If you use the Dumpzone and channel out a chute, yes - if the dumpzone covers both a hole in the ground and a tile next to it, everything goes down the hole. What's at the bottom of the hole is up to you - a diagonal access for craftsshops, magma, an atom smasher - ymmv.
But if you don't dig that hole in a diagonal, yes again - it has to be deeeep to avoid miasma wafting up.
... Same w/ lye bags -> soap...
Lye is stored in buckets and barrels, not bags. I usually embark with 50 sand bags and make glass things to free some up.
Hmmm, true, no idea how I jumped track there. Except the OP was talking about food stockpiles, and lye is stored in food stockpiles*, so - dunno. <shrugs>
Still, could have said
'..."stone bags" -> stone blocks + empty bag...', and it would have illustrated the concept, which was the important part, even if based on gibberish.
@ all - But while talking about lye and stockpiles, NOTE - First, lye is, by default, located in Food Stockpiles - it's one of the last categories listed, so often gets overlooked. Uncheck this to get lye near your soap-maker. But more important -
there's a bug that puts buckets of lye into barrels -
with the bucket - making both unusable! So, if you have "lye" designated in any stockpile, have ZERO BARRELS in that stockpile and avoid that particular frustration. Just so you know.