Thermodynamics in Game of Thrones

Dragonfire vs. Ice

(Image credit: hdwallpapers)

CAVEATS

1. I have not read George R. R. Martin’s Game of Thrones —this is purely reacting to the HBO series depiction.
2. Though I work with many, I am not a mechanical engineer myself, nor have I personally done any technical work in thermodynamics.
3. Yes, Game of Thrones is fiction — in fact, it’s fantasy and its world has magic that is clearly beyond anything in our world. So it can be taken as a given whatever is depicted is “possible” — I am not disputing that point. I wanted to consider consistency in the scale of what is depicted — that something applied once would apply the same other times as well.
4. I’ve seen some interpretations that the force involved is not heat (melting the wall) but something else (acoustic force?) “breaking” the wall. But it looks like fire, sounds like fire, behaves like fire — so I call it fire.

Here is the scene in question:

This scene is a huge moment in the series, as that wall has been holding back the bad nasties for a long time, and now that it has a hole in it, an army of them will sweep into the kingdoms of the south (a.k.a. “Winter is Coming”).

What nagged me about this scene is the energy transfer involved. Again — magic, fine — I won’t dispute the dragon’s ability to launch the energy we calculate — but that’s how much it should also be able to cough up on a battlefield, and whatever radiant effects of heat pops out of this would be experienced by the dragon’s rider (The Night King) and really all the other animate things in the area.

THE WALL

Let’s start with the wall itself. Conveniently, Time Magazine did a feature on the topic: http://time.com/4912209/game-of-thrones-wall-history-magic/

• Dimensions: 300 miles long, 700 feet high, averaging 300 feet thick
• Composition: Ice and magic (runes?), some areas built into mountains
• Shape: The wall is tapered — wider at the bottom, narrower at the top. (Apparently there is a reference to a dozen horsemen can ride abreast atop the wall.)

Just to avoid dickering on specifics, I’ll pile on a bunch of factors ignored in this that would make this even more extreme:

1. Massive amounts of super-heated steam being created and its effects on all around (and also on cooling the energy delivery).
2. Runoff-refreezing/sealing countered by any kinetic or acoustic assistance the dragonbreath might deliver (but also we ignore those and focus on thermal aspects).
3. [60 feet thick] Assume wall does not even taper out, but is merely the 12 horsemen wide for the melted area (and assume tightly-packed riders — so only 5 feet wide x 12 = 60 feet — far from the average of 300).
4. [100 feet wide] Assume the impacted melt-area half the wingspan of the dragon, and the dragon no bigger than a 747 (so ~100 feet across)
5. [350 feet high] (There seems to be kind of a collapse rather than a melt at the end — I guess it becomes enough a breach that it forms a hill over which the bad nasties just hike — but you have to admit these are already some really monster reductions to the mass we will calculate).

60′ x 100′ x 350′ = 1,750,000 cubic feet of ice

1 cubic foot of ice = 26 kilograms of ice (so a total of 45.5 million kilograms of ice)

MELTING ICE

It takes 333 kilojoules to melt 1 kilogram of ice.

I don’t really understand the bursting vs. melting in the scene, but hopefully given the mega-reducing assumptions above we can agree we’re talking about melting at least this amount.

45,500,000 kg ice x 333 kJ to melt = 15,152,000,000,000 joules (or 15.2 terajoules — or TJ).

To chart the region of energy we have just reached:

• The atom bomb at Hiroshima was 63 TJ (so the dragon produced roughly 1/4 that energy)
• All of (2014) Australia consumes that amount of energy in just under a week.

The rate of transfer of this energy is the real centerpiece, however. The footage taken literally, not assuming cut-aways would have this energy delivered in roughly 2 minutes. I will be wildly generous and say that it was perhaps more like 10 minutes (people were still hurriedly running and getting out of the way, but there could have been confusion and the like).

(15.2 TJ / 10 minutes) / (60 seconds / minute) = 25.3 gigajoules per second

1 watt = 1 joule per second (so the dragonbreath delivered 25.3 gigawatts of power)

The largest operating nuclear power plant in the world (Bruce, in Ontario) generates 6.4 gigawatts — so if the dragon was directly connected to nuclear power plants to do this, it would take the full output of 4 of the largest operating nuclear power plants to feed that rate.

CONVECTION AND RADIANT HEAT

I need some help here on this one, so as a proxy, let’s assume there is a fairly major thermal flow of air created by this energy transfer. Ignoring the potential wind and weather effects (substantial) simply considering that air separating the dragon’s mouth and the wall must be heated, and that Night King thing is on its back must contend with it (assume the dragon is heat-immune and the bad nasty army too far away).

The 15.2 TJ assumes perfect transfer through air, but clearly the air is heating up and there is loss in the process. We’re so far beyond a “fire” here that that blue flame stuff is really a plasma. To make a guess at the temperature rise, I’ll confine as follows:

Volume of a 747 passenger area ~31,000 cubic feet — assuming the flame is roughly that big:

25.3 GJ / (specific heat of air ~ 1.006 kJ/kgC) / (density of air ~ 0.036 kg/cubic foot) / (31,000 cubic feet) x (1,000,000 giga / kilo) = raise the temperature 22,535 degrees Celsius

Interestingly — this is about the temperature of a blue star!!

(But the air itself is not confined — lots and lots of convection here, surely, so let’s just take the temperature up to the melting point of iron (1538C)

25.3 GJ / (specific heat of air ~ 1.006 kJ/kgC) / (density of air ~ 0.036 kg/cubic foot) / 1538C x (1,000,000 giga / kilo) = 454,217 cubic feet

That’s a radius of 330 feet of iron-melting hot.

(ASIDE: my combustion-minded colleagues note there is also an “impinged flame” here — think of the forces involved in rocket launch or jet fighter afterburners. Topic for another rant…)

SO WHAT IS MY POINT?

Consistency. And if the Night King making the dragon undead made its red flame blue and that much an amplification of power, then game over. But assuming the flame is relatively on par with the flame when it was alive (and fully functional / no leaks in the seals), then these flames should have been way WAY more devastating — blasting the ground surface into glass and killing the entire battlefield (both sides) with super-heated air.

So — if I buy the dragon melted the wall. How did the Night King on its back survive? Why is Daenerys holding back on this tactical nuclear-weapon class power in the Lannister battle?

Something doesn’t add up! (Maybe my math — check my work?)

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