Closing 16 valve - by hand!

These beer valve are very loose by design. If you apply your counter pressure to the bowl before filling - and rotate the bowl - each valve will get closed and stay closed as the pressure builds.
Once you have pressure in the bowl and all the valves are closed - you can then add the beer.

http://www.youtube.com/watch?v=1HIVEHJ_jTQ

A suggestion from ProBrewer:

I have experience with many fillers but not that particular one. It sounds like you have too much pressure and you are forcing open the valves on the filler. The valves should always remain sealed. If I could see a better picture of your filler and your tank I may be able to help you more. Is there pressurized lift cylinders that are used to lift the bottle up to the seals? If so what is the pressure of that and what is the pressure on your beer tank? You don’t want more pressure on your beer tank than you have on your lift cylinders.

__________________
Cliff Panneton