"Well, I like weight through distance for it's simplicity, and then using idealizations of simple mechanical tools you can convert it in a consistent way.  Like a lever, you have one end moving a longer distance with less weight to move another end a shorter distance with more weight!  And to convert that to rotation, you can consider the circumference of a circle, which works out consistently if you are thinking in terms of large gears and small gears and which is more forceful.  I didn't actually know the math for exactly calculating it for a heavy wheel with significant weight in the middle and not just along the outer circumference... but it sounds like the right integral might solve that!  Like dividing up the shape the right way."

"And the Ba Sing Se academics don't exactly sound pleasant to deal with, but I assume they at least don't challenge each other to lethal duels over the pretext of the engineering standard they are using."

"They try to embarrass each other at fancy dinners, which is considered much the same level of escalation, from what I hear. But yes, no lethal duels. Mostly. So how do you use weight through distance as a measure of energy, can you give an example?"

“Well I mentioned levers and gears, it applies to pulley systems and other mechanisms as well.  To outline some actual practical machinery…”

He gives several examples. His examples include a water pump (which leads into a diversion about the pull of the earth and weight, but the math still balances out), a machine to wedge and crack rocks open (which ran into limits of the metal’s strength and thus proved impractical), and several setups for propulsion of Fire Nation ships.  The basic principle is ultimately an extension of the same math as levers, shorter and stronger motion or longer but weaker motion, and exchanging between the two.  He notes you can lose energy to friction (which, fascinatingly, generates heat) but otherwise the equations balance out.  His explanations are somewhat hampered by a lack of writing material, but he can accurately recite specific numbers from memory.  He ties off his summary by bringing it back to the abstract and the question of the energy of heat.

“And in principle, if you had some theoretical idea of what and how a perfect steam (or in your case air) engine operated, you would then have a conversion ratio of heat to physical energy.  And you could do the opposite to get another ratio if you could optimally maximize heat from friction…but to be honest I’m not sure those ratios would actually be the same.  Also, I’m not sure ‘perfect’ engines would have the same theoretical conversion ratios for steam vs. air vs. more exotic substances.  Do you know if air expands linearly with heat like steam (once it’s converted from liquid water that is) does?  Do you know of any exotic substances that expand non-linearly?”

He does have half a dozen lap-sized blackboards for sketching, in terms of writing materials. He can follow along and then try to estimate some analogous examples...

"Someone who is not me and has more time and materials should do careful experiments on heat produced in various mechanisms and how it relates to force. I'm not aware of the details of how various substances expand and contract. I do know in terms of air it's not exactly... Well, you can force air into a smaller space with more force and pressure. It's more like the product of pressure and volume varies with temperature. That's the whole way my engine works, generating pressure, and thus force, under changing volumes. I must admit I'm considerably more interested in practical application than theory. For that, commonly available materials are best- Air, water, steel or brass, leather, and so on. My poor Fire Ox can't pull this thing at very good speed all by himself, especially up slopes or on soft ground. I have to engage the engine often there."

"For metals, they expand or contract by a linear amount, but it is obviously a lot less than steam does.  You can build some pretty neat thermometers by using two types of metals coupled and wrapped in a spiraling shape, and some important projects actually use these for consistency even though 'real firebenders can gauge the heat themselves accurately.'" his voice is dripping with exaggerated sarcasm.

"But ah yes as for practical matters... do you have an estimate of this cart's weight?  With that, and the incline, as well as bit of trigonometry, we could calculate how much energy the engine is adding.  I don't know as much about calculating for soft ground, it doesn't come up with water based propulsion.  I would imagine you could treat it sort of like friction or maybe drag in water, but the way soft ground can stick and cling and freeze up motion is probably a critical difference."

"Huh. That does sound very neat. I wonder if you could, say, have it pull a string and close or open a valve automatically if things get too hot? There must be other ways to do that too, I have a spring valve so the steam vents instead of exploding if I push it too far... Oh yes, a lot of energy is lost in friction and pushing against soft ground. I had to re-design this thing with wider wheels several times, there's just a strict limit on practical ground pressure. How heavy it can be for every bit that touches the earth. I also have to gear it very far down, turning the wheel's fast speed to a much slower but more forceful motion. The wagon, empty, weighs about fifty five, sixty dan*, but when you add all my furnishings and current cargo and myself and you it's probably... Closer to a hundred and twenty?"

*A dan is "a bag of rice" and is about 50-60 kg

"For purposes of getting numbers to do calculations from, maybe we could just record numbers while on harder sections of road?  Do you know from memory if there are any decently long sections of the road which are hard and well-maintained and have at least a slight incline?"

"The roads immediately around Omashu would be best for that, in this region, but we are heading in the opposite direction..."

“Well that question is more of theoretical interest than immediate practical application, and I can write down everything I know for you to collect and analyze the data on your way back.  In the short term, what matters most is how much efficiency I can add to your use of fuel and/or how much fuel I can entirely replace.  I have several possible techniques in mind.  At a wildly optimistic guess, (extrapolating from manipulating candle flames purely via meditation and breathing) I could keep your furnace at least at partial heat with a tenth of the fuel it would normally take, and do that as long as I can meditate.  As for a more modest guess as to what I could do with constant meditation… maybe somewhere between a quarter as much to half as much fuel?  Usually with steam engine’s furnaces I’m directly bending fire at the furnace and I can do that for an hour, maybe two if I’m pacing myself carefully.  With your furnace’s size, directly using fire bending could entirely replace the fuel.  But it’s smaller size means doing that way would be irregular in heat (spiking with each movement in fire bending).  As a third technique (probably not effective but worth testing), I could directly transfer heat (either from my fire bending or from the furnace) to the air within your engine.  Without a flame inside your engine’s chamber, this technique is probably too inefficient to be worth using, but maybe getting the heat directly where it needs to go would make up for that?”

"We should design such experiments and write it all down even if I can't implement them right now... Directly putting heat into the engine chamber itself would be counter-productive unless you can do it in time with the pulses. It depends on the cycle of heat and cooling. Or did you mean firebending on the boiler? I don't even use the steam except as a heat store and an enormously oversized kettle, hah. So that might work. If you can boost a low coal flame to a higher one, that will suffice. It won't be a quick journey."