I don't have part numbers and forum members don't have access to Dell's spare parts database.  The only way to find part numbers here is if somebody has seen a thread Dell-Chris M answered with part numbers, or another thread with part #'s, which I couldn't find for the T5810. The good news is I couldn't find any reason why the 825w PSU you want wouldn't work.  I see eBay and others sell them for your model. More info:  [URL] 
I have a 5810 desktop and it comes with a 685w PSU, can i buy this to replace it? My computer is working fine, just thought i should upgrade to have more watts considering i added more stuff to my computer. I just want to know would it work/fit? Dell Precision T5810 T7810 T7910 825W Power Supply 0K61PK?  
Dell precision 5810 Power Supply question