Would silica gel inside my bike frame be effective in preventing corrosion?
No, it wouldn't work as long as the frame is not hermetically sealed to keep water from entering (I don't know why manufacturers don't do that).

The silica gel can only absorb a tiny quantity of water, a drop or so per packet, then it becomes ineffective. It is only effective to absorb water vapor.

You can bake it in the oven to regenerate it. 
If you want to play with it, there is a version with a water-sensitive dye (blue -> pink, now illegal because of cobalt content) and orange to a bit less orange. If left open, it will change color quickly.