| Once you have 4 or more current carrying conductors in a conduit, you must begin derating. Sleeving NM cable is not prohibited by code specifically, but only practical if you want to sleeve a cable or two a short distance. I would not try pulling 17 cables thru 30 feet of conduit. Not only do you need to size the conduit for the actual wires, you would also need to allow for the sheathing of each cable, and not fill the conduit more than 40 % with the cable. Without going thru all the calcs, you would probably need a 4-5 inch conduit.
Do exactly what Labman suggested, run single conductors thru the conduit. The most common used today is THHN insulation. The insulation is rated to carry much higher current, and helps a great deal with derating.
With the cables you mention, you have 37 current carrying conductors, which must be derated 40 %. Installing all in one conduit requires some calculations. I know I will get some comments on this, but #12 THHN is rated to carry 30 amps, derating by 40% will only allow 12 amps. So running 37 #12 in one conduit cannot be done if all,I assume , are connected to a 20 amp breaker.
There are two methods to "derate", one is to lower the size of the breaker, the other is to use larger wire. If all are #12 they as I said, probably all 20 amp circuits with 20 amp breakers.
One simple way of doing this is to use one 1-1/2 in EMT conduit and install 38 #10 wires. This size conduit can hold exactly 38 #10 THHN wires, and 37 #10 current carriying wires need to be derated 40 % of 40 amps that #10 THHN can carry. The 38th wire is the equipment ground that will take the place of each of the bare grounds in each cable, you do not need to run each ground in each cable.
To make it simple I would run the following:
1-1/2 in EMT conduit with 38 #10 THHN wires consisting of: 20-hots(colors) Labeled #1 thru 20
17-neutrals(white)
1-green
If you want to use any other type of conduit, you better check back because all the other types, such as flexible, or PVC, etc. has lesser amounts of #10 allowed.
There is a minimum size junction box allowed, the box must have at least 217 cu inch of volume, allowing for 2.5 CI for each #10 and 2.25 CI for each #12. A 6" x 6 " x 6" only has 216 cu in, so thats too small, the next size available is probably a 8" x 8" x6", has 384 CI of volume. A larger box can be used if you have the space, and would make splicing all the easier.
There is a lot of info here, review it and let me know if there are any other questions. | 
about the heat dissipation because none of us wants an electrical issue like that. I am going to use my old 100 amp breaker box as a pull/splice box thats 12 x 18 x 4 so that gives plenty of space for the connections to "breathe" . Using 2" conduit and pulling the 240 lines directly to the new box, I should be good without heat issues. 
Linear Mode
