Go to the Projects Module > List View.
Leave the Title box blank (1) but select the Department you want to clean up the projects for (2). Click the Apply and Refresh button (3).
Sort the results by the Last Modified column. (Sort it so the black triangle indicator is pointing up. this puts the older projects at the top of the list)
Select then R-Click (Control Click) on a project. Select Archive from the Pop-up menu. You can select multiple Projects at one time to archive.
*Note The more projects you have selected the longer the archiving process will take. 10 projects at a time is a good amount.