As a design project, about a million times more worthy than a cell phone.

Great thought but not efficient unless several floating "funnels" were lined up behind one another as well as being several meters in depth. Rough seas, floating but submerged plastic bits and the potential to be run over by transport ships are all considerations.
A better method would be to deploy manned or autonomous cargo ships to the area of greatest concentration(s) of floating debris. Using both tight nets and skimmers, great swaths of material could be harvested for recycling, sorted and then delivered to the appropriate port. A further advancement might be the ability to power the ships using converted plastic as fuel - provided that solar concentrators could work efficiently at sea.

the very low concentration of garbage per square mile make me question the feasibility of this proposal. How wide will the booms be, they will need to be tens of miles wide to even begin to make a dent. If made wide enough the booms become a navigation hazard or risk being destroyed by ships.

With plastics being recycled to make a variety of products - pens, clothing, 'wood', etc - it could potentially pay for itself. As stated above, it could be green to make some green.

I agree with Mirmillion, the concept is worth of the effort, however there are many holes. Most do not see that your system does not need to collect all plastic on the first pass, being powered by a self sustaining solar or other power supply it could run for years. The plastic on the bottom needs a different method of collection. The recycling is good if even just for fuel. I do see a vast amount of small organisms being collected with the plastic so a separation method is needed. I wish you good luck in your endeavor.

It doesn't have to be 100% effective on it's "first pass" (and since its stationary, that has to be analogous) or even through a storm and/or turbulence.
It will start gathering debris. That's what counts - and then that debris can be recycled or disposed of appropriately.
It's a great idea. I think the more appropriate worry is if a large "clump" approaches. . .

Nice concept for collecting floating plastic but not cost effective even if factoring in recycling value. The costs of collection devices and recovery of the plastic would far exceed recycle value if any. Better fix is preventing overboard dumping by segregation of plastic waste and compacting or shredding at the point of origin aboard ships and also to minimize all single use plastic packaging aboard ships. Passengers have enough idle time to segregate personal care containers for compacting or returning to shore.

If this works then they can try their hand at collecting all the radioactive elements that are being released from the three melted down cores at the Fukushima plant. That's been going on since the start. Before we know it it will be three years since that happened and it has been dumping radiation into the ocean since the start with virtually no intervention or reduction.
Ooops! Oh, we're not suppose to know about that. Never mind! I hope this plastic thing works out. Really everybody, reactor accidents are very rare an will never happen in the US, the NRC have got that covered. Our plants can't have a disastrous accident with their over sight. So far so good, right?

Biomimicry might be helpful here in scaling up already proven processes. What biological process is able to absorb food over great areas?

Government funding and maybe an xprize if they don't build one soon enough.
I also suggest making the feasibility study open source along with the design unless you really think you can do every ocean yourself.