Just what the title states; there's a good deal of noise made about transport, and storage of spent nuclear fuel. Why all the hullabaloo when the fuel is all spent?
[Physics] Why is storage of spent nuclear fuel dangerous
nuclear-physics
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A lot can be, and has been, written on the subject, but I'll give you the short and sweet version.
Does nuclear chain reaction start in fuel pellets even before they being installed in reactor? -- No
There are several reasons why this is so.
The number of spontaneous fissions of $^{235}$U is minimal. The branching ratio for that mode of decay is $7 \cdot 10^{-9} \%$, which means that for every billion $^{235}$U atoms that decay, only $7$ of them do so by spontaneous fission. This does not produce enough neutrons to start a chain reaction.
The neutrons released from fission have too much energy to induce many more reactions. The probability of an atomic event is characterized by the associated cross-section. The cross-section for the relevent fissions of $^{235}$U at the fission spectrum average is 1.235 barns. This is not zero, but it isn't very large; compare this to neutrons in the 0.025 eV range where the cross-section is 584 barns.
Fresh fuel rods are not typically enriched very much. The exact enrichment varies depending on a variety of factors, but fresh fuel is typically on the order of 2-5% $^{235}$U; most of the rest of the fuel is $^{238}$U which is significantly less likely to fission due to neutrons in the average fission spectrum.
As to your confusion, yes, the fuel is sufficiently enriched to sustain a chain reaction; that is what it is designed for. It is designed, however, to be inside a reactor when that happens. Inside a reactor, there are other things that start and sustain the chain reaction. The primary of these is a moderator.
A moderator is a substance that slows the neutrons down from the fission energy of around 2 MeV to the average temperature of the moderator, around an eV or so. In all commercial reactors in the United States, this moderator is plain old water. Some reactors, though, use heavy water and others use graphite. Either way, the function is critical for (most) nuclear reactors. There are such things as fast reactors, but I'll let you research that on your own.
Also, for fuel pellets and rods are radioactive, how do we transport them? -- Very carefully.
Fresh fuel rods, as explained above, are not dangerously radioactive and can be handled without a great deal of extra caution. Fresh fuel is made into pellets that go into rods; the rods are assembled into assemblies which are transported to the power plant in transportation casks. These are often times not much more than a wooden box with packaging material. They can be loaded on the back of a semi or onto a train and shipped to the power plant.
Spent fuel rods are typically moved by machine and only then a very short distance into cooling pools. These pools provide sheilding while allowing the removal of excess heat generated by fission products. After many years in a cooling pond, many nuclear power companies have started to move spent fuel to dry cask storage containers. These containers provide the same benefits of the cooling ponds, but require less maintenance.
You are missing the concept of critical mass. In a small amount of uranium, like a single pellet, some fraction of the atoms will decay spontaneously every second. When enough of this is put in proximity, then the emissions of some of the decaying atoms kick other atoms just right so that they fission too. As a result, more atoms fission than you would predict from just the probability of a single atom doing so.
The main difference between a small pile of uranium and a large one is that in a large one you get a chain reaction. Power plant reactors rely on this chain reaction mechanism to get the large amounts of output power. The control rods control how much the emissions of fissioning atoms can hit other atoms, thereby controlling the overall reaction rate.
In reality, this description is over simplified. There can also be moderators envolved that sortof convert some of the fission results into stuff that can kick other atoms to fission when without the moderator they would not. However, that is a aside to this question.
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Mart's answer gets to some of the problem (e.g. ensuring that storage remains stable for the period of decay) but I believe the other answers here are a bit off base.
Nuclear fuel is determined to be "spent" when the nuclear engineers determine it is no longer economically warranted to continue using it. On a physical sense, the fuel could certainly be used for much longer. Besides, it depends on what kind of reactor you are talking about as to how long the fuel is used and what the composition of the spent fuel is. The plutonium that is such a dangerous part of most used fuel is actually a major contributor to the energy output of a CANDU (Canadian heavy water) reactor toward the end of the useful fuel life.
As for composition of waste, the majority is composed of rather inert U238 (~91%) that did not transmute through neutron capture or fission. This material is part of the original fuel composition and is not harmful. A small percentage (~1%) consists of the remaining U235 that did not transmute or fission. About the same amount is plutonium that results from neutron capture by U238 and subsequent decay. Depending on reactor operation, about 4% is daughter products and the rest is actinides and activation products.
The half-lives of these isotopes varies significantly but it is a convenient fact that the more radioactive a material is, the shorter its halflife and consequently the shorter time before it is "safe." There are many graphs (e.g. see second graph) out there of decay times for spent fuel, but as I said above, the exact time for decay depends on a lot of things like the original composition of the fuel, what kind of reactor was used, and the final processing of the used fuel.
Several options for processing spent fuel exist including recycling it to retrieve the useable uranium and plutonium. Doing so reduces the waste volume considerably but also necessitates the development of separations technologies and the handling of concentrated waste. It is also possible to place the waste in special reactors that are dedicated to "burning" the waste with high neutron flux; even still, there will always be some waste. Waste that is slated for disposal is often vitrified, that is, it is mixed with borated glass. These glass logs are put into steel containers and then stored in concrete.
Whatever is done with the waste, we must be confident in the stability of the storage for at least several hundred years (though thousands of years in some cases). A great deal of research continues on this subject. That being said, the absolute amount of waste is quite small. I've read various numbers, but for order of magnitude we are talking about one football field 20 feet deep of waste for all of the nuclear reactors in the United States for the last 60 years. That is a lot of bad stuff, but in comparison, it seems quite manageable. If we recycled the fuel, that would reduce to about 6 inches of waste spread over one foodball field. Coal plants, in general produce on the order of 10,000 times more waste by volume which contains more radioactive material in absolute terms than nuclear plant waste.