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Just what the title states. What is the reason that a Nuclear reactor has a characteristic dome/bell shape?

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My apologies. I did mean the structure that is visible in photographs (never seen a reactor 101) –  Everyone Oct 18 '11 at 20:15
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Same reason scuba tanks are. –  Mike Dunlavey Oct 18 '11 at 20:29
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http://en.wikipedia.org/wiki/Containment_building

By "nuclear reactor" I take it you mean the containment building that is visible looking at a nuclear reactor site. The pressure vessel which houses the nuclear reactor itself is not quite bell shape, and the containment building itself isn't always bell shape either. Here is an image from that article. You can browse through Wikipedia for actual pictures of examples.

Containment building

I typically refer to these containment buildings being "can" shape or "sphere" shape. The best way to explain the reason for these two extreme shapes is to consider the two extremes from a design perspective. That is:

  1. Limiting factor is the internal steam pressure in an accident
  2. Limiting factor is material strength to hold up the containment itself

If #1 is the case, the objective is to keep the containment from leaking or blowing apart, and you will make the containment out of steel. Steel has good tensile strength and you will also build it spherical. This should probably go without saying, but the strongest structure to hold a pressure inside is a sphere. But that's not always the restricting engineering limit. Consider the case that a massive massive volume is needed. Large volume, lower maximum pressure. In that case, the ideal material selection will change, and you will objectively use a material that is cheaper and still has good specific compressive strength. Trying to build a large volume, lower maximum pressure, containment will lead to a more "can" like shape, where you basically build a cylinder. This lets you build tall, keeping the forces compressive, and still not sacrifice too much in terms of strength against an internal pressure. Of course, you still need a roof.

Generally, containment buildings exhibit a compromise between these two cases, which is what I was intending to demonstrate in that sketch. Really, it's incorrect to say they're always dome/bell shape. The shape varies, and varies strongly country to country, and generation to generation, reflecting different engineering design choices.

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Martin is correct that the shape is the solution to an optimization problem of pressure vs. material and construction costs. In order to avoid confusion, however, it should be pointed out that reactors are not the things you are talking about; the domes you see are containment buildings.

The reactors are located inside the containment buildings inside steel reactor vessels. The reactor vessel is, under normal conditions, isolated from the atmosphere of the containment vessel. In an accident scenario, however, the containment building is designed to contain the contents of the reactor to prevent environmental damage.

The buildings are designed to withstand the very large pressures resulting from water evaporating due to core heat. In the case of fukushima, some "containment" [1] buildings were destroyed when the zirconium alloy fuel rod cladding combined with the oxigen in the cooling water and formed explosive hydrogen gas.

[1] See comments below.

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The terminology "some containment vessels were destroyed" is problematic. The word vessel fairly specifically refers to the wrong thing. However much of a "containment" that building was is disputable in the first place, and in one extreme I hear people call it a "weather building". Officially, it's called the reactor building and I believe that the area where the crane is, the area that blew up, isn't even airtight. Even if it was airtight, that wouldn't mean it was a containment. A containment can hold pressure inside. The question is about PWRs I think. –  AlanSE Oct 18 '11 at 14:16
    
Good points, I didn't follow the Fukushima situation closely and I took that wording from a news article I Googled quickly. PWRs and BWRs both typically have containment buildings. –  AdamRedwine Oct 18 '11 at 14:25
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The same reason the bottom of Coke Cola cans are - it's the best shape to contain internal pressure withe the least material

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If you're referring to the typical "coke can optimization problem", that's a problem of minimizing the surface area to volume ratio, and the fact that it's a cylinder is given a priori. –  AlanSE Oct 18 '11 at 14:19
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The containments of PWRs and BWRs are indeed different:

PWRs have the containment buildings you are talking here about. That's because in PWRs, the containment contains much more than in a BWR and is much larger. In a PWR, the reactor building is itself the containment vessel (hence the term containment building) and contains not only the reactor vessel but also the entire primary loop of the plant including the reactor coolant pumps and the steam generators. In some PWRs the containment building is actually made of two separate structures. An inner steel containment vessel which serves the role of containing the high pressures of an accident, and an outer concrete shield building which protects against outside threats (such as the impact of a jet liner.) The concrete wall is located directly outside the steel but structurally is separate.

In a BWR, the containment vessel is much smaller and surrounds only the reactor pressure vessel and surrounding pipes. It consists of the drywell which is the steel/concrete structure that surrounds the reactor vessel and the wetwell (also known as the suppression pool or torus) which is a pool of water that is used to control reactor pressure in an emergency. Together they are known as primary containment. Surrounding primary containment is the outer concrete reactor building which is usually square or rectangular in shape, this area is sometimes called secondary containment. It is kept at a negative pressure so that any radioactive materials can be quickly removed if necessary. The outer reactor building also serves the shield building role protecting the reactor from outside threats. At Fukushima the explosions occurred on the refuel floor within secondary containment. While this was certainly a problem, the primary containment vessels remained, overall, intact although there has been leakage from it. In the Fukushima reactor buildings, the construction was far lighter in the area surrounding the refuel floor, while the lower part of the building was much heavier construction which is why that top area was blown away but the lower portion is still somewhat intact.

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