Breath that lifts

When I was reading Moonfire, one statement really jumped out at me and made me think.

“What a vehicle was the spaceship! A planet-traveler massive as a destroyer, delicate as a silver arrow. At the moment it lifted off from the earth it would be burning as much oxygen as is consumed by half a billion people taking their breath – that was twice, no, more than twice the population of America. What a deep breath must then have been concentrated into the liquid oxygen they were passing into its tanks right now, a liquid oxygen cooled to 297 degrees below zero and thereby turning air to cloud at every hint of contact with the pipes which were in turn contained within other pipes two feet think to insulate the fuel.”

I had never thought about comparing the volume of oxygen used by the Saturn V engines to the breath of millions of people. So, I wanted to look into this in further detail.

F-1 engine from Saturn V (S-IC first stage)
F-1 engine from Saturn V (S-IC first stage)

The human lung has an average total volume of 4 to 6 litres at sea level. That’s the total volume of both lungs combined, including residual volume that cannot be exhaled. Of that volume, an average breath (tidal volume) is 500ml, or about 10% of the total capacity.

Oxygen is about 21% of the air at sea level. Meaning that an average breath contains 105ml of oxygen. Humans breathe an average of 20 times per minute (10-20 resting, I’ll use the higher number as a guesstimate for a daytime office worker). That is a per minute average of 2.1 litres of oxygen.

The first stage of the Saturn V rocket, the S-IC stage, burns liquid oxygen (LOX) and RP-1 (refined kerosene) rocket fuel for the first 160 seconds of flight, and pushes the 3,039 ton rocket from the launch pad to a height of 56km and a speed of 8,530km/h. The thrust is generated from five F-1 engines. Each F-1 engine consumes 93,920 litres of liquid oxygen per minute. The total capacity of the LOX tank in the S-IC stage is 1,204,000 litres (which can also be derived by multiplying 93,920 litres/minute by five F-1 engines by roughly 160 seconds of flight).

Liquid oxygen has an expansion ratio of 860.6:1, meaning that gaseous oxygen at sea level has about 861 times larger volume than the liquid phase.

Translating 93,920 litres per minute of LOX into the gaseous form means that that a single F-1 engine uses the equivalent of 80,827,552 litres per minute of gaseous oxygen (at sea level). The entire S-IC stage would therefore consume 404,137,760 litres per minute of gaseous oxygen (sometimes abbreviated as GOX) equivalent. A human uses 2.1 litres per minute, so the S-IC stage is consuming LOX at the same rate as 192,446,552 adults and one baby. If the lower rate of 10 breaths per minute were used, it would increase to nearly 400 million people.

Brazil has the fifth largest population in the world, 190,732,694 (August 1, 2010, estimated). So the S-IC stage LOX consumption rate is equivalent to the entire population of Brazil breathing. The world’s population in 1969 was 3.6 billion. 192 million people would be more than 1 in every 20 people on the planet.

The island of Manhattan in New York city has a population density of 71,201 people per square kilometre. At the same population density, the S-IC would consume the same amount of oxygen as a city like Manhattan 62,057 square kilometres in size – a little less than the area of the island of Sri Lanka. That’s a city with a radius of 140.5 kilometres (281 kilometres in diameter).

Another interesting statistic is the weight. The F-1 engine burns LOX and RP-1 at the rate of 1,789 kilograms and 788 kilograms per second, respectively.

BMW powered Saturn V
BMW powered Saturn V

My (red) 2003 BMW 330Ci is 1,490 kilograms, unloaded. The weight of the S-IC stage decreases at a rate that is nearly equivalent to 8.6 BMWs per second. Imagine that as a stream of BMWs blowing from the bottom of the Saturn V at 2,989 metres per second to lift the massive 3,039 ton rocket out of the Earths gravity well. The high speed is required. Newton’s Third Law (for every action, there is an equal and opposite reaction) means that to push a much heavier weight up, a smaller weight needs to be pushed down at a much greater speed, proportional to the difference in the weights.

Or maybe I’m just getting too silly now.

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