You will need two chases/plenum in your DC closet. The cross-section of a stub wall sheathed with dry wall will not have enough volume. As McRabbet mentions in his post, you will need to tape and bed the entire interior of your closet to get the right effect. I learned the hard way that one chase isn't enough. Locate one chase with an entry point near the floor on the inside and one chase near the ceiling on the inside. Put your outlet grilles opposite on the outside wall. My closet is sheetrocked, taped and bedded both on the inside and outside with R-13 insulation, including the ceiling. I have a double path chase for both of my chases: built as a add on in the interior of the closet is a 2x4 stud wall sealed with 1/4 plywood, one with an entry on the top, the other (opposite wall in the closet) near the floor. Air goes up the chase, thru an opening in the sheet rock, down between the studs encased by the exterior sheetrock and out the grille located opposite the intake. So basically the exhaust air travels 16+ feet from intake to exhaust. The turbulence of air leaving the filters and trying to get out of the closet via a high/low exit cools by closet just fine.
As I've posted before, I have two solid wood exterior doors, fully weather stripped, with a frame/threshold in my closet. I figured out the lack of sufficient air volume exiting the closet when I could not push the doors closed with my 320 pound body. Air can't get to the cyclone if the air can't get out of the filters. Don't make the same mistake I did. Retrofitting fully taped, bedded sheetrock is a nasty business.
Good Luck!