physics......>>>>>>?
The storage of radioactive waste is a massive problem in the nuclear industry. One problem that scientists must take into account is volumetric expansion. Consider a 50.0-gallon metal canister of diameter 22.0 inches filled with a radioactive waste possessing a coefficient of volume expansion of 9.60 × 10-4 (°C)–1. This canister will be filled at room temperature (23.0°C) and lowered deep into the Earth (where temperatures can reach 155°C) for storage. If the coefficient of volume expansion of the metal is 4.10 × 10-5 (°C)–1, how much waste will spill over the top if the lid becomes loose?
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Okay, Tasha, since you asked me and since no one has answered this question yet, I'll try, but let me warn you that I'm not at all sure I'm right. Let's do the easy part first. If you assume they filled the tank all the way to the top as the question implies and which is dumb because you know it'll expand, then we originally have 50 gallons of gunk. If the coefficient of expansion for the gunk is 9.6 x 10^-4 per degree Celsius, and the temperature changes from 23 to 155 degrees, we have a 132 degree temperature change and the expansion of the gunk will be 50 gal x 9.6 x 10^-4 x 132 = 6.34 gallons.
Here's the hard part and the part I'm not sure of. The coefficient of expansion of the metal is 4.1x10^-5
so for a 132 degree temperature change the metal will expand 132 x 4.1 x 10^-5 = 5.4 x 10^-3 x the volume (I think). The thing is, the metal will expand in all directions and I'm not sure if the expansion will directly increase the volume to the same degree or if you would have to calculate the change in height and diameter of the canister in order to determine the increase in volume. I'm guessing the former is the correct assumption. If it is, then the increase in the volume of the tank will be5.4 x 10^-3 x 50gallons = 0.27 gallons. Okay IF that's the case then the difference in the volume of the gunk (which is now 50 + 6.34 ) 56.34 gallons and the tank which is (50 +0.27) 50.27 gallons is the amount of gunk which will overflow. That's 56.34 - 50.27 gallons = 6.07 gallons. I make enough mistakes (usually through carelessness) when I know what I'm doing, which is why I never answer questions I'm not sure of, but you asked me for help and I can't let you down. I hope I'm right. I'd really appreciate it if you'd email me when you get the correct answer and tell me if I'm right or wrong.