Why was this the opportunity you chose to pursue after private equity?
I know you are planning to go back to business school and maybe you needed something to fill the time. However, there would have been other more "enriching" experiences you could have volunteered for or you could have chosen to engage that would have been much more consistent with the story you want to tell an admissions committee.
Unfortunately, this whole 10-month experience is going to be a big question mark in the eyes of the admissions committee. They will think along the lines of the question I posed to you above - "why on Earth did he choose this opportunity?" In my opinion, it won't show a lot of initiative to the admissions committee. In fact, it shows the opposite. They may question your judgement.
The only potential spin you could put on this, and it would definitely be a big spin, would be that it was some sort of operations play. After working in private equity, you wanted to get a Hands-On feel for operations ... But even this wouldn't be very realistic. It's too far of a stretch IMO.
Which would have been better, as far as weaving a consistent story that shows progression to the adcom, would have been if you worked in ops for a startup in the industry or sector where you truly have an interest.