Middle support for a 9 Foot Farmhouse Table?

Hi folks! I'm getting ready to build my first carpentry project, a 9-foot L-table based on this plan:
http://ana-white.com/2010/02/plans-narrow-farmhouse-table-beginner.html

I'm making a few modification to make table bigger than the plan, so I'm concerned about the need for a middle-support, since the table will be used as a working desk (computer, sewing, propping legs).

Here are the list of modifications:
1. Use 2 x 2 for the legs
2. Use 2 x 2 for end stretchers and move it down to the bottom of the legs. (a little more Parson-like)
3. Use 2 x 6 for aprons.
3. The depth will be about 28 inches (3 planks of 1x10's, rounded), just enough depth for the keyboard/monitor.
4. The width will be 9 feet.
5. Within the 9-foot width, there will be 32-inches-wide unit at one end of the table, making an L-shape. So the maximum depth of the desk is 28" + 32" = 60" at the bottom of the "L"

About the 32-inch unit:
It will be screwed to the 9-foot unit, apron to apron. It will have only 2 legs + stretcher at the other end. So the end that gets attached to the 9-foot unit doesn't have legs.

My question is, what with the table being very long, and the apron has an added burden of supporting the 32" unit, do I need to insert additional leg (or legs) in the middle of the 9-foot table? Would the aprons alone be strong enough?

Thank you all in advance for your input!
Rita