The Vacuum Airship

,

Archimedes’ Principle states that the (upthrust) force on an object that is displacing a fluid is equal to the weight of the fluid displaced. For example, a cube with sides of one metre, fully submerged in water, will experience an upward force of 9810 newtons, as this is the weight of one cubic metre of water. The material that the object is made of has no effect on this force, so if the object weighs more than 9810 newtons it will sink, and if it weighs less than 9810 newtons it will float.

The upthrust force on an object therefore depends only on the relative densities of the object and the fluid it is displacing. A bigger difference means a bigger force.

Hydrogen is the least dense gas, at 0.0898 kilograms per metre cubed, but hydrogen is rarely used in airships as it is highly flammable and therefore dangerous. Hydrogen was used in the Hindenburg because  helium was difficult to produce and the United States, the only country with significant reserves, had banned its export. Helium has a density of 0.179 kg/m3 and therefore produces only 93% of the lift of hydrogen, but it is far, far safer.

vaccum-airship

Ideally, to create the maximum upthrust force, we would want our balloon or airship’s envelope to be filled with something with the lowest possible density. The lowest possible density would be a vacuum, the total absence of anything, which would create a lifting force of 12.7 newtons per cubic metre (as opposed to 11.0 N/m3 for helium).

The problem with using a vacuum to lift an airship, is how to contain the vacuum. If a difference in pressure exists between two regions, then a difference in force exists between those two regions. In the case of a vacuum airship being used on Earth’s surface, that force would be 101325 newtons per square metre, the equivalent of more than ten tonnes pushing down on every square metre. No material on Earth is strong enough to withstand this force without being so heavy as to negate the point of the vacuum lifting effect in the first place. In order to still have lifting capability and withstand the stresses involved, we can calculate the minimum required ratio of Young’s modulus to density, and this yields a figure of around 450000 Pa/(kg/m3)2. Unfortunately, even the strongest materials, like diamond, have ratios that are only one-fifth of this, so it looks like we won’t be creating vacuum airships any time soon.

What Does “Fail-Safe” Mean?

, ,

The term “fail-safe” is often-used to refer to an object or a device, but it more properly refers to a condition. In this post I hope to explain what “fail-safe” actually actually means, with reference to how nuclear power station stay safe.

To “fail safe” means that in the event of a failure, the failure causes the device to fail in such a way that the device is rendered safe. In terms of deaths per gigawatt year nuclear power comes second only to hydroelectric power in terms of safety (Source: ExternE Externalities of Energy Project, European Commission). This is due to the incredible emphasis that is put on safety in nuclear power stations, and is a testament to nuclear power stations’ defence in depth concept.

Control Rods

One of the key parts of a nuclear reactor is the control rod assembly. When fission occurs in a fuel rod, neutrons are released and these neutrons go on to cause further fissions. The purpose of control rods is to “soak up” excess neutrons and prevent further fissions. Control rods are made of materials such as boron, cadmium and hafnium that have a large capture cross section, meaning that they have a high probability of capturing and absorbing neutrons.

crocus-control-rods

The control rod assembly for the CROCUS research reactor.

If the control rods are raised out of the reactor the excess neutrons are not absorbed and further fission occurs and the reactor releases more thermal energy. If the control rods are lowered into the reactor the neutrons are absorbed, fission does not occur and the amount of thermal energy released is decreased.

Control rods fail safe by being held up by electromagnets. In the event of a power failure the electromagnets are no longer powered and thus the control rods will fall into the reactor, shutting it down. Whilst we cannot be sure that the power supply to the reactor will not fail, we can be sure that gravity won’t fail. If the control rods weren’t held up by electromagnets then we’d run the risk of a fail dangerous situation, with the control rods raised up out of the reactor and no way for them to be reinserted to shut down the reactor.

Moderator and Coolant

The neutrons released in each fission process are travelling too fast to cause further fissions. (Imagine trying to putt a golf ball – hit it too hard and it will just skip over the hole.) The job of the moderator is to slow these neutrons down so that they are travelling at the correct speed to continue the chain reaction process.

The moderators used in nuclear reactors vary between different designs, but graphite and light- and heavy-water are common.

The job of the coolant in the reactor is to take thermal energy away from the nuclear fuel and transfer it (via a heat exchanger) to a steam generator that then drives a turbine and generates electricity. If coolant leaks from a reactor whilst the nuclear fission process continues this leads to thermal energy not being removed from the fuel, and the fuel heating up to the point at which is gets so hot that it melts – a meltdown.

In some reactors (e.g. PWRs, BWRs, SCWRs) the coolant is the moderator, and the reactor will fail safe in the event of a coolant leak because a coolant leak is a moderator leak and the reactor cannot continue the fission process without a moderator. Other reactor designs, that do not use a combined moderator-coolant, have different safety features in place to cope with a coolant leak.

Citrus

,

Oranges and lemons are the most commonly consumed citrus fruits, but the citrus family is a lot more complicated than that. There are four fruits from which most of the rest of the citrus family originate: the mandarin, the pomelo, the citron and the papeda.

mandarin

Mandarin

The mandarin (Citrus reticulata) is a small citrus fruit, and the only citrus that produces sweet fruit. Both the sweet orange (Citrus × sinensis, what you and I would simply call “an orange”) and the sour or marmalade orange (Citrus × aurantium) are crosses between the mandarin and the pomelo (Citrus maxima), and thus it would be more correct to refer to oranges as large mandarins rather than mandarins as small oranges.

The satsuma (Citrus unshiu), tangerine (Citrus tangerina) and clementine (Citrus × clementina) are all cultivars of the mandarin, and if the tangerine is crossed with the pomelo this yields the tangelo (Citrus × tangelo).

The blood orange (Citrus × sinensis) is a cultivar of the sweet orange, and the tangor (Citrus reticulata × Citrus sinensis) is a cross between the sweet orange and the mandarin.

pomelo

Pomelo

The pomelo is a large, thick-skinned citrus fruit. As explained above, the sweet orange is a cross between the pomelo and the mandarin, and if the sweet orange is crossed with a pomelo this yields the grapefruit (Citrus × paradisi). If the pomelo is crossed with the citron (Citrus medica) this yields the lemon (Citrus × limon)).

citron

Citron

The citron has a similarly thick skin to the pomelo. It is crossed with the pomelo to yield the lemon, and the lemon again with the pomelo and mandarin to create the rangpur (Citrus × limonia). It is also crossed with the papeda (Citrus subg. papeda) to yield the lime (Citrus × latifolia and others). (The key lime (Citrus aurantiifolia) is a separate species, like the kumquat (Citrus japonica), that does not originate from the four “fathers” of the citrus family.)

A further cross between the lemon and the citron yields the limetta (Citrus limetta), which when crossed with the marmalade orange yields the bergamot orange (Citrus bergamia).

papeda

Papeda

The papeda is a subgenus of citrus that contains a number of fruits. As mentioned above, a cross between the papeda and the citron yields the lime, and the yuzu (Citrus ichangensis × Citrus reticulata) is the result of a cross between the papeda and the mandarin.

Disclaimer: I did try to sum up this post in a diagram, but I gave up in the end.

Integer Sequences

,

Number can be broken up into many groups. Some of these groups have specific uses (for example, prime numbers are very important in cryptography) and some are just interesting for existing in the first place.

The natural numbers are what you might call “counting numbers”: 1, 2, 3, 4, … . Whether or not zero is included in the natural numbers is a matter of some discussion, and there doesn’t seem to be a consensus either way. The natural numbers does not include the negative integers, as it is not possible to have “minus one apples” or “minus two cars”.

The rational numbers are those that can be expressed as a simple fraction: \frac{p}{q}. Because $latex q$ can equal one, the rational numbers necessarily include the natural numbers, but also every possible other fraction: \frac{1}{2}, \frac{3}{4}, \frac{27}{31} and so on. The irrational numbers are those that cannot be expressed as fractions: e, \pi, \sqrt{2}, … and repeat indefinitely after their decimal points.

The square numbers are those that are the square of an integer: 1, 4, 9, 16, … . The cube numbers are those that are the cube of an integer: 1, 8, 27, 64 … . This process continues with x^4, x^5, and so on.

The prime numbers are those that are divisible only by one and themselves: 2, 3, 5, 7, … . The Mersenne primes are those prime numbers that are expressible as 2^n - 1, one less than a power of two: 3, 7, 31, 127, … . Sphenic numbers are the product of three primes: 30, 42, 66, 70, … . The semiprimes are natural numbers that can be expressed as the product of two prime numbers: 4, 6, 9, 10, … . There are also almost primes which are then the product of three primes, four primes, and so on. The composite numbers are those numbers that have a divisor other than one and itself, thus they are the set of numbers that are not prime.

Perfect numbers are numbers that are the sum of their divisors: 6, 28, 496, 8128, … . 6 is  a perfect number because the factors of 6 are 1, 2 and 3, and the sum of 1, 2 and 3 is 6, and so on. A number is semiperfect if it is equal to the sum of some of its divisors: 6, 12, 18, 20, … , whereas an untouchable number is a number that cannot be expressed as the sum of the divisors of any number: 2, 5, 52, 88, … . That is, there is no number whose divisors sum to 2, or to 5, or to 52, etc.

Abundant numbers are numbers whose divisors sum to a total greater than itself: 12, 18, 20, 24, … . For example, 12 is abundant because the divisors of 12 are 1, 2, 3, 4 and 6 which sum to 16 (i.e. the abundance of 12 is 4 and its abundancy is 12/4 or 3). Friendly numbers are pairs of numbers with the same abundancy: for example the abundancy of both 30 and 140 is 12/5 and therefore 30 and 140 form a friendly pair. Amicable numbers are pairs of numbers where the sum of the divisors of one number is equal to the other and vice versa. For example, 220 and 284 are amicable because the divisors of 220 add up to 284 and the divisors of 284 add up to 220. This concept can be expanded to the sociable numbers where each is part of a “loop” that arrives back at the first number (i.e. amicable numbers are sociable numbers with a period of two).  At the moment there are many known sequences of four amicable numbers, but far fewer with periods longer than this.

Weird numbers are those that are abundant but not semiperfect: 70, 836, 4030, 5830, … . The sum of their divisors is greater than the number itself, but no subset of their divisors add up to the number itself.

There are many, many other groups of numbers. Wikipedia is a good place to start looking for them.

Quotation Marks and Guillemets

In many languages, quotation marks are used to indicate that the source of a piece of text is not the author, but that the author is directly quoting someone else.

“The cat sat on the mat,” said the man.

Quotation marks can be double or single, and the nesting of double and single quotation marks is used to indicate when the person being quoted is themselves quoting someone else.

“Can you believe she said ‘The cat sat on the mat’?” asked the man.

In many languages, quotation marks are not used like this. German uses the same marks, but in different positions.

Sagte der Mann: „die Katze saß auf der Matte”.

In many languages, marks called guillemets are used instead.

« Le chat était assis sur le tapis, » dit l’homme.

In French (except Swiss French) spaces are placed between the marks and the speech, but in most languages that use guillemets (Arabic, Greek, Italian, Portuguese, Russian, etc.) there are no spaces, as with quotation marks. Some languages occasionally use guillemets pointing in the other direction (e.g. Danish):

»Katten sad på måtten,« sagde manden.

In some languages, (e.g. Polish) guillemets are used within quotation marks to indicate a quotation of a quotation, as with nested quotation marks as explained above.

„Czy uwa?asz, ?e powiedzia?a «kot siedzia? na macie»?”.

I’d be interested to hear from any MrReid.org readers on how quotations are indicated in their languages.

Apologies to any native speakers of the languages above. Blame Google Translate.