## Sunday, January 31, 2010

### Eureka! Smells Like Burnt Ship in Here

It is commonly believed that Archimedes devised a scheme using only sunlight and mirrors to burn Roman ships that were attacking Syracuse. The mirrors act as a parabolic reflector that focuses heat on the ship until the wood ignites. A group of MIT students and MythBusters tested whether this “Archimedes Heat Ray” was feasible and determined that it would have been far easier to burn the ships with more conventional weapons. As any 10 year-old boy with a magnifying glass and an ant hill can tell you, the Sun produces more than enough energy to set things ablaze under the right conditions. That said, perhaps the MIT-Mythbusters team could have gone after a more contemporary problem. How many mirrors would you need to melt an aircraft carrier in one second?

In order to melt an aircraft carrier, you first need to raise the temperature to its melting point. Aircraft carriers are probably made of steel or some other material with a similar melting point. The melting point of steel is different for different alloys, but according to Google it's generally in the vicinity of 1300°C. Since room temperature is about 25°C, this means we must first raise the temperature by 1275°C = 1275 K.

To raise the temperature, we must add heat to the aircraft carrier. The heat capacity (i.e. the amount of heat energy needed to raise one kilogram of a substance by 1°C) is 460 J/kg·K. An aircraft carrier weighs about 2.0x108 kg. Using this info, we can compute how much heat we need to raise the steel in the carrier to its melting temperature,

heat needed = (heat capacity) · (temperature change) · (mass)
= (460 J/kg·K) · (1275 K) · (2.0x108 kg)
=1.1x1014 J

OK, so we’re done, right? Not necessarily. So far we’ve only raised the temperature to the melting point of steel. At this point, it’s just really hot solid metal. In principle, we still have to add more heat to change the solid metal into molten metal. The amount of heat we need to induce the phase change is called the latent heat. To calculate the latent heat, we need the heat capacity (i.e. the amount of heat needed to transform one kilogram of a substance from one phase to another), which for steel is about 25.5 J/kg. We can then compute the amount of energy needed to make the phase change,

latent heat = (heat capacity) · (mass)
= (25.5 J/kg) · (2.0x108 kg)
= 5.1x109 J

This is a large amount of energy by most standards, but, as it turns out, it’s very small compared to how much we need to raise the temperature in the first place, so we can neglect it. In total, we’ll need about 1.1x1014 J of energy to melt the ship. All of this energy has to come from sunlight, which hits the Earth with 1360 W/m2 (this is called a solar flux.) From this we can use dimensional analysis to compute the total area of mirrors we would need to melt the air craft carrier in one second,

area = (energy needed) / [ (energy flux) · (time) ]
= (1.1x1014 J) / [ (1360 W/m2) · (1.0 s) ]
8.1x1010 m2

This means our mirrors, when laid down, would take up more than half of New York state.