I am sure you would not need an entire tub of Liquid Nitrogen. The cold temperatures would probably just crack your hardware if you had it un-contained. (ie. A tub full of liquid nitrogen constantly pouring into it.)
I am sure if you could just relay the liquid around the casing via tubes you would be able to maintain a frosty temperature so your PC would never over-heat.
I am also afraid that if I put Dry Ice into my PC cooling set up it would not be such a good day. The Dry Ice as cold as it seems doesn't melt, it evaporates into a smoggish white gas. And I don't want my PC to be leaking freezing gas.