Yes, it is an S trap. If you look at it sideways, you see the shape of the letter 's'.
The problem with that is that the trap will siphon dry. There is not entry for the vent air . A p trap...shaped like the letter P on its side....as soon as the sink is drained, the vent air enters at a level above the water level in the U part of the trap. This level is called the weir of the trap, and the siphon is broken.
Just bought this house and want to add a disposer under the double kitchen sink. The "flipper" we're buying from moved the sink from across the kitchen and the new one isn't tied into the vent stack. Our home inspector said the nearest vent is seven or eight feet away...inaccessible and too far from the fixture to satisfy code anyway. He suggested we have a air admittance valve put in when we have the disposer installed. The under-the-sink drain plumbing doesn't seem to be well done and needs a makeover when a disposer's installed. I think that's an S trap they created in there.
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If you did use an AAV, it would be as high as possible under the counter where you could still unscrew it to replace. There a lots of diagrams of other projects where this has been done to go by.
Yes, that does look like an s-trap. Even if it had a loop vent (which would pass code most places), it's still not installed properly.
The disposal will have a DW inlet port, you can use that. You need a baffle T on the part connecting the two sinks to prevent the outlet of one from flowing to the other sink. The disposal acts like a pump, and without that, you'll have problems.
There is no reason to think about using an AAV.
Cut off all that junk you have in there and install a P-trap.
It looks to me there is room room to do this, despite that it will take away from the storage space in the cabinet.
The proper thing to do it open the wall and raise the level of the tee in the wall.
You connected the drain system the "hard" way by using glued sch. 40 PVC.
You also made it difficult to revise it because the tee at the wall may not have enough pipe available to cut the tee off and install a different fitting.
Your real problem is that you believe the guys at Home Depot when they tell you all you need is " a few fittings and our advice".
As an aside, plastic threads do NOT like metal slip nuts.
Once the existing drain piping is out of the way it would be a simple matter of cutting out an access panel in the back of the cabinet and then removing a 1x2 piece of wallboard to cut out and replace the tee. If it's an old house, it might not even have a vent and the riser may be accessible from below.
You might want to check your local codes, as some places require a separate P-trap for each sink basin.
If that were allowed, that is how it would have been installed originally. You cannot do it now, either.
A tee cannot be used on its back as a vent connection.
The tee must be horizontal and the waste connected to waste stack with a wye.
It can be done but you have to revent the L/T properly. As per IPC code.
This is awkward, but...
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