Thursday, August 01, 2013

Physics Under The Dome

I got hooked on the new CBS TV serial, Under the Dome (Monday Nights). The suspense led to the Stephen King novel on which it is based. In the afterword, King makes a big point of noting his medical advisors, who ensured, I suppose, that he got the gruesome details right on the countless catstrophes he inflicted on his characters. But we won't worry about that here, or about the literary merits of the novel as a whole - what we care about is the physics.
Not the physics of the Dome - we don't learn anything about that - the physics of life under the dome.
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SPOILER ALERT (FOR NOVEL)
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The climactic catastrophe of the book is based on a monstrous explosion producing a giant fireball that spreads over most of the dome with terrible carnage, leaving the few survivors with no oxygen to breathe. Do the numbers add up?
In the book, the town of Chester's Mills is suddenly enclosed in a dome, impenetrable to anything they can throw at it, but transmitting sound, electromagnetic radiation, and somewhat permeable to air and water. The area enclosed is perhaps 10 or 20 square kilometers - call it 10^7 m^2. The maximum height is about 15 km, so the enclosed volume is something around 10^11 m^3 - 100 billion cubic meters.
The explosion was produced mostly by 10,000 gallons of propane - the fuel supply of an enormous Meth lab.
How much heat would that evolve? Complete combustion of one gallon of propane produces about 100 million Joules of heat, so 10,000 gallons produces something like a trillion Joules. For comparison, on a sunny October day at noon in Maine (the setting)the ground might receive 400 J/sec per sq meter, or 4 billion J/s for the whole area enclosed by the Dome. The propane would thus release the equivalent of about eight minutes sunlight for the whole Dome. That's a bunch, but not a Dome wide firestorm type bunch.

When I was a student, a guy had a fight with his wife and ran his car into the middle of a gasoline tanker truck not far from where I lived. The fireball melted the pavement and set fire to a couple of nearby roofs, but the tanker driver escaped by kicking out his front window and running like hell. Those trucks carry five or ten thousand gallons of gasoline, and gasoline has an energy content that is similar to (but slightly lower than) propane.

How about the notion that the fire would use up all the oxygen? Propane has 3 carbons and 8 hydrogens, so complete combustion consumes 10 Oxygen atoms (5 O2 molecules), producing 3 CO2 + 4 H2O. Hence each molecule of propane (molecular weight 44) gobbles up Oxygen atoms of total molecular weight 160, and so a gallon of propane, at a bit less than two kilograms, would consume rather less than 8 kilograms of Oxygen, or 80,000 kilos of Oxygen all told. Sea level air has about 0.2 kg of Oxygen/m^3, so the whole enterprise consumes the Oxygen in 400,000 m^3 of air - a pittance compared to the 100 billion m^3 in the Dome - even if you allow for the fact that the upper levels contain much smaller masses of air.

What about the 78,000 or so kilos of CO2 produced? It's a bunch, but not too bad compared to 50 billion or so kg of air in the dome - only about 2 ppm - not even a major local warming threat.
The complete combustion assumed above is highly unlikely. In the scenario of the book, even if most or all of the propane vessels were breached in the initial or subsequent explosions, the fuel-air mixtures would mostly lack enough oxygen for combustion.

Propane tanks are tough. Mythbusters took them on in this episode. Explosions at the Blue Rhino plant in Tavares, Florida destroyed many of the more than 50,000 20 gallon tanks stored there, but the 3 thirty thousand gallon tanks survived. This video captures far larger propane explosions in Korea, but even they are much too small to produce the kind of widespread catastrophe King describes.