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BACKGROUND

I'd like to create a world that includes pyrokinesis, the ability to heat and even combust materials at a distance. This is a low-magic setting, so what I have in mind is not very flashy. I'd like my mages to be able to heat good thermal conductors, spark combustibles, and that's it. For anything flashier it's BYOB (Bring Your Own Blackpowder). The mechanism they use to conjure heat is innate and handwavy, to be left ambiguous. The pyrokinetics believe they get their power from the sun.


I want to be carefully consistent about measuring what my pyrokinetics can and can't do. Ideally, I'd like to know that a trained pyrokinetic can create X amount of heat inside a body of material over whatever time and distance, and then plug that into a formula and see how hot the material gets, whether it catches fire, etc.

Strangely, I can't find any plug and play methods of efficiently estimating spontaneous combustion, so I've had to turn to thermodynamics in general, about which I know only a very little. My best bet seemed to be the formula for heat transfer but I quickly realized that (naturally enough) it assumes heat is being transferred from one object to another through a two-dimensional surface, rather than being produced within a three dimensional object. This seems like it would skew my results. I also read several articles on thermal radiation, but while that seemed promising I didn't understand it well enough to apply it. (There may well be obvious solution I'm looking right past.)


QUESTION

  • What's the best way to calibrate and estimate the abilities of a pyrokinetic, along the lines described?
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    $\begingroup$ Have your firebending forms face off in agni kai? $\endgroup$ – pojo-guy Jan 3 '18 at 22:28
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    $\begingroup$ @pojo-guy: I would, but my world is intended to be pretty low-magic. Everything would change if the fire nation attacked. $\endgroup$ – Era Jan 3 '18 at 22:29
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    $\begingroup$ you may want to read "the name of the wind" and its sequels by patrick rothfuss, to heat something you would be using your own body heat as a source so you cannot generate much heat before you risk putting yourself in hypothermia. $\endgroup$ – John Jan 3 '18 at 22:45
  • $\begingroup$ @John not necessarily. If there's a specially developed 'magic' organ that, through magical means turns chemical energy into pyrokinesis, then all you risk is exhaustion. $\endgroup$ – Jakob Lovern Jan 3 '18 at 22:46
  • $\begingroup$ what is the technology level of the world? Do they have thermometers? (Another way to ask this question is, "what level of precision do your characters, not you, need?") $\endgroup$ – JBH Jan 3 '18 at 23:55

11 Answers 11

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There are only two numbers of interest, energy and power.

  • Energy is how much heat can the magician produce; it is measured in joules, or small calories (1 cal = 4.2 J), or kilocalories (aka food calories, 1 kcal = 1 Cal = 4.2 kJ). A human consumes (that is, dissipates) about 2000 kcal (8.4 MJ) per day at rest, more if he is male, less if she is female; a man employed at heavy manual labor uses some 4500 to 5000 kcal (19 to 21 MJ) per day.

  • Power is how much heat can the mage produce per second; it is measured in watts; 1 watt is 1 joule per second. At rest, a human dissipates about 100 W; a trained athlete can sustain about 250 to 300 W of mechanical effort, using up some 750 to 1000 W of stored energy (muscles have an efficiency of about 30% at converting the stored chemical energy into mechanical energy).

So if the magicians have a metabolism similar to regular humans, the only magical property being their ability to project heat, one could assume that a trained magician can project a total of about 750 to 3000 kcal of heat per day, with a power of about 250 to 1000 W, depending on how efficient is the mechanism which projects the heat; specifically, 750 kcal / 250 W at 25% efficiency, up to 3000 kcal / 1000 W at 100% efficiency.

If we take the higher numbers, 3000 kcal (12.6 MJ) is enough heat to boil off completely about 4.8 kg (1.2 U.S. gallons) of water; supposing the magician is able to transfer a power of 1000 W, completely transforming that amount of water into vapor would take about 3 hours 30 minutes. (Assuming standard pressure and water starting at 20 degrees Celsius.)


To boil off 1 gram of water:

  • one first needs to warm it up from 20° C (assumed) to 100° C; this uses (100 − 20) * 4.2 = 336 J;
  • then one needs to supply the heat of vaporization, 2.26 kJ.

So overall one needs 2.6 kJ to boil off completely 1 gram of water; dividing 2.6 kJ/gram into 12,600 kJ of available heat we get 4,859 grams.

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    $\begingroup$ Wikipedia has a couple of useful pages for getting a feel for these numbers: Orders of magnitude (power) and Orders of magnitude (energy) $\endgroup$ – Mark Jan 4 '18 at 1:37
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    $\begingroup$ It is my opinion that you should drop all the kcal numbers and leave everything in J. Since a W is a J/s, this will make the numbers more tractable, I think. $\endgroup$ – kingledion Jan 4 '18 at 5:38
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    $\begingroup$ Athletes can consume upward of 10,000 kcal/day, so 3,000 kcal/day seems a low estimate. $\endgroup$ – Matthieu M. Jan 4 '18 at 8:06
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    $\begingroup$ The one thing missing from this is the question of how the power is reduced due to distance from the target. I see it as an additional parameter, orthogonal to the power that can be brought to bear on a target in contact with the mage. If there's an inverse-square dissipation of that power, you can probably forget "Battle Mage" as a job description. $\endgroup$ – Monty Harder Jan 4 '18 at 19:36
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    $\begingroup$ Note that boiling water is very hard; water has a huge heat capacity and transferring from liquid to gas has a huge cost. Lighting paper on fire, making iron hot to the touch -- all extremely easy compared to boiling water, in terms of power and energy. If you build your pyros around "boiling water", and you don't realize how hard that is, they'll be able to do surprising things on things that aren't water. $\endgroup$ – Yakk Jan 4 '18 at 20:13
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In addition to AlexP great answer, if you want to measure heat, then you are referring to a calorimeter, my favorite being the bomb calorimeter.

In a real world calorimeter, heat is generated inside a container, e.g. via combustion, or as a product of a chemical reaction. The exchange part happens entirely within the instrument and it is needed to measure the heat.

To simplify the process: the pyro-agent heats a target metal ball inside the inner container of the calorimeter. The heating occurs from a fixed distance and for an agreed amount of time.

The outer shell of the calorimeter is made of refractory material. Inside this shell there is water, and another container, surrounded by a thin metal pipe, acting as a heat exchanger. Inside the inner container the metal ball is held in place by rods of refractory material, such that the only heat exchange is with the air around the ball. As the air heats up, it expands and warms up the water around the inner container via the heat exchanger. The change of water temperature is measured on a roll of paper by a pen connected to a thin rod of metal submersed in water and coming out from a tiny tiny hole at the top of the instrument. The difference between the temperature prior to heating and the maximum temperature reached by the water is the $\Delta T$ you have been looking at.

The relevant formula is

$\Delta \text{Energy} = C * \Delta T$

where $C$ is the capacity of the calorimeter, as explained here.

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Raise the average temperature of the grey matter between an enemy's ears by 0.7 degrees Celsius and their combat effectiveness falls to zero. They will be lucky to survive without brain damage.

Using @AlexP's math, that is 1300 grams (of mostly water) raised 0.7 degrees equals 910 Joules (roughly a KiloJoule), to take down any opponent at any reasonable distance.

In a pre-firearms world (primitive thermometers), beings, with such an ability, would be overwhelmingly powerful. They would literally possess the power to kill with a thought.

With that in mind, I suggest the following categorization...

god - a pyrokinetic who can only kill one enemy warrior at a time

God - a pyrokinetic who can kill an enemy squad at once

Oh God! - a pyrokinetic who can kill whole armies with an angry glance

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  • $\begingroup$ I wondered when someone would spot that. Thankfully, with a little tinkering it'll be a feature and not a bug. $\endgroup$ – Era Jan 4 '18 at 2:32
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    $\begingroup$ Reminds me of a game whose name escapes me at the moment. It had four classes of weapon. Small pointy thing, pointy thing, large pointy thing, and don't point that thing at my planet. $\endgroup$ – pojo-guy Jan 4 '18 at 3:20
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    $\begingroup$ @pojo-guy Evil Overlord List #150: I will provide funding and research to develop tactical and strategic weapons covering a full range of needs so my choices are not limited to "hand to hand combat with swords" and "blow up the planet". $\endgroup$ – Perkins Jan 4 '18 at 23:21
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    $\begingroup$ How can pyrokinetics aim? If they can heat arbitrary point regardless of what is or isn't in the way, that's a terrifying weapon. If matter on the way affects losses, water is a good protection, and they may be limited to heating the skin. They can still probably blind someone facing them unprotected, though. Goggles could protect, but pre-modern goggles (possibly water-filled) would probably loose their transparency fast. $\endgroup$ – Eth Jan 5 '18 at 17:55
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    $\begingroup$ Magic often has restrictions such as "cannot be used on living things." That said, though, I once read a story where someone's magic tutor casually mentioned a historical mage who specialized in things like setting the hem of enemy mages on fire at just the wrong time. $\endgroup$ – arp Jan 9 '18 at 15:33
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To set the mood for this answer, we should all remember how this works. Clear your mind, and will the paper to burst into flames.

Now, let's start with two assumptions:

  • Your world doesn't have access to thermometers, much less any more precise technologies.

  • You're looking to create a "hierarchy within a guild," meaning a means of comparing one practitioner to another ... not necessarily a way to say "that guy can produce 2,000 joules/second!"1

Under these assumptions your easiest solution is for your well-developed, respected, and ancient guild to have selected a half-dozen materials with specific quantities. Let's say (somewhat arbitrarily)...

  • a 1/2 cubic ft block of aspen wood (how long to burn to ash),
  • a gallon of water (how long to bring to boil),
  • an ounce ingot of copper (how long to melt),
  • a 1# ingot of steel (how long to melt),
  • A 5# block of glass (how long to melt),
  • a 10# block of obsidian (how long to melt).

The effort, because there are sub-achievements (like cracking the glass, warming the obsidian, etc.) would provide a basic gradation of skill and power. An initiate might be able to get the wood to burn, the water to steam, and the copper warm. An archmaster might be able to vaporize everything in under an hour.

This method is imprecise, especially since weather is a factor. The job is simpler (if only fractionally) when done on a hot summer day than the dead of winter. But, it would create a viable pecking order.2

So, let's change our assumptions:

  • Your world has access to thermometers and stopwatches.

  • Practitioners carry around government-mandated identity cards that list the statistical capabilities of the practitioner somewhat like a passport.

Under these assumptions your solution isn't much different, but demands fewer materials. Probably just one. Let's say the goal is to see how much the practitioner can raise the temperature of a sphere of Inconel with a radius of 0.5m having a mass of approximately 4.3kg in 20 seconds. What we're looking for is the measure of ℃/s.

This measure (℃/s) has, of course, become known as the Naidoo after the creator of the measure for pyrokinetic purposes, practitioner Iminathi Naidoo. The Naidoo reflects the potential of the practitioner to heat or combust anything. The ability to raise the Inconel sphere 20℃ in 20 seconds means your Naidoo rating is 1. But, what does this mean?

I'm glad you asked! The "Naidoo Combustibility Index" or NCI (the measure of how many Naidoo are required to "spontaneously combust" or bring a material to flame in one second or less) was developed and referenced by practitioner Naidoo in the NCI Reference Guide.

Look at it this way. The thermal conductivity of Inconel is (basically, there are a lot of kinds of Inconel) 73 kW/m℃. That of copy paper is 0.33 kW/m℃. This means you need 221 times the Naidoo rating to heat paper as fast as you heated Inconel.3

But, the NCI is a measure of spontaneous combustion. Paper's autoignition temperature is between 218℃ and 246℃ (call it 230℃ for fun). With a room temperature of 25℃, we need to lift the temperature 205℃ in one second or less. Thus, the NCI rating of copy paper is 221 * 205 = 45,305 (let's call it 45,000 Naidoo or 45kN4).

Now, if you can't spontaneously combust a piece of paper, then you have no business joining the Pyrokinesis User's Guild (PUG), minimum of 45kN to join.5


1Not to be confused with Watts, right?

2"You mean you couldn't melt the copper? Hahahahahaha!"

3Oh, I'm simplifying this process something awful, but I'm having fun doing it.

4Anyone thinking that N means "newtons" is being way too serious about this....

5And I am having way too much fun putting the "fiction" back into "science fiction."

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  • $\begingroup$ In addition to the suggestions here, you could also add a numbering or lettering system to allow people and yourself to keep track of the power and extent. You could apply the classic S, A,B,C lettering system and apply it to things like the intensity of heat, range and duration. So someone with SSS would be the best in all 3 of these, while sometime with SSF could create a very intense flame over a large distance but only for a fraction of a second. $\endgroup$ – Shadowzee Jan 4 '18 at 4:42
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    $\begingroup$ This sounds like a great approach(+1), but also paints a world with naively nice metrology. For a realistic world, the value of one traditional Naidoo will vary by up to 20% between cities in the same country and a bit more in other countries, except those who use a differently named unit based on s/°C and the one which is simply doing everything in aspens (from the cubic foot of aspen wood), and there will be three different officially recognized Naidoos, each left over from a different attempt at standardization. $\endgroup$ – rumtscho Jan 4 '18 at 20:16
  • $\begingroup$ @rumtscho, I about died laughing! This, of course, is also affected by the Kelley, a competing unit of measure that uses °K/s rather than °C/s. $\endgroup$ – JBH Jan 4 '18 at 20:28
  • $\begingroup$ There is a thriving market in adulterated aspen blocks that help marginally-qualified pyros pass the entrance exams. $\endgroup$ – arp Jan 9 '18 at 15:38
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I've definitely been in your shoes. It turns out the question you're trying to answer is an inexorably difficult one, but there is a couple ways we can approach it with simple formulas..

Why is it difficult? As you already mentioned, heat transfer is complicated. It involves conduction, forced and free convection and thermal radiation all applying in 3 dimensions with complex surfaces and an enormous range of non-constant material properties. It's a mess except in the most simple of cases. Here let's look at 2 simple cases: heating something fast enough that it doesn't have time to lose heat to the surroundings, and splatting heat just onto the surface of a thing like a laser would.

Here if your mages are trying to ignite something, they're trying to focus enough heat into a small enough space. Only one point needs to become hot enough to ignite (250C-ish) and the rest will burn from there. Despite having much less energy, a tiny spark is better at igniting things than an oven. The necessary temperature to ignite something is called the autoignition temperature, and can often be found through google searches.

So your mages should be able to apply a certain amount of heating power into a certain volume (smaller = better) over a given length of time. Let's look at some first order equations to help shape our understanding. Here let's assume they're applying heat in a cylinder shape with a diameter+length we determine because it's a nice shape for the focus of a magic heat-beam and convenient mathematically. Also, your mage would need to hold their heat-beam steady for the duration of the time, or the heat would spread out too much.

Heating the Inside
Time to Ignition = (Autoignition Temp - Room Temp) * (Specific Heat * Density * (pi/4) * diameter^2 * length) / Power(magic)
units conventionally in C, cm, g, J and W. Let's look at igniting paper:
Time to Ignition = (230C - 30 C) * (1.34 J/gK * 0.8 g/cm^3 * (3.14/4) * 1cm^2 * 2cm) / 500W = 1.3s

The actual length of time is easily tuneable by making the mages more powerful, or focusing their beams tighter or broader. To me, 1.3s sounds like a nice amount of time to make a character wonder if it's working or not. As a reference, 500W can heat a cup of coffee to boiling in a few minutes. Microwaves are often 1000W. Also, this formula completely ignores heat conducting away, which is fair enough for these power levels and the accuracy we're aiming for.

An alternative, and the last stop on the trip to way too much information, is heating the material at its surface rather than in the body. This approach would require less power but be more complicated.

Heating the Surface
Time to Ignition = ((Autoignition Temp - Room Temp) * Thermal Conductivity * diameter^2 / Power(magic) )^2 * (pi/4)^3 / Thermal Diffusivity
units this time traditionally in W, m, K, s. Example for wood:
Time to Ignition = ((230C - 30C) * 1.26 W/mK * .01^2 m^2 / 5W)^2 * (3.14/4)^3 / (1.1*10^-5m^2/s) = 1.1s

Notice in this analysis the pyromancer is 100 times less powerful than before (Now 5W), but still able to heat things enough to burn, like a high power laser pointer. That's because we're not heating a large volume, just the surface. This formula is much more prone to inaccuracy though, as it doesn't account for convection or multidimensional heat flow. I'd probably multiply it by 5 or 10 in practice, and don't trust any times over a couple seconds from this equation.

I hope that helps. Best of luck with your world!

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What you are looking for is a bomb calorimeter. Instead of providing a known ignition source (spark) to an unknown material sample, you provide a material samples of known combustibility, and have your fire bender ignite the contents of the bomb. Measure the temperature change compared to the temperature change for the known ignition source.

There is likely to be more than raw ignition power involved in measuring the total ability of your mages. Distance, precision, sustainability over time, spark rise time, etc are all factors that will come into play for determining where each Mahe would be most effective.

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Pre thermometer world:

One simple test would be the time it takes to make a small bowl of water start to steam. This is dependent on temperature and humidity. For timing either a count the swings of a pendulum, or use a water clock.

Another test is to ignite a ball of dandelion fluff. This requires less power, but greater focus.

Such a power may be expressed in different ways.

Consider: If I can deliver a kW to a cup of water that is very different from being able to deliver the same power to a cubic millimetre. The latter has a power density 250 thousand times as great.

On the other hand, suppose that you had a practicianer that could move gigajoules from 1 cubic kilometre of air to another. The power density is low. The power level is high. This lad would be rejected by the guild because he couldn't make a cup of water even warm, but he can herd clouds, and make rain -- maybe.

Consider how long they can work. One mage may be able to boil that cup of water in to steam. Another may be able to run a slow cooker all day. Same energy, different power rate.

Consider one practicianer who has to see his target, and can only heat while looking at it, another who can heat anything he has seen or knows well, a third who can work by holding an artefact that was once in intimate contact with the target (law of contagion) Some might be able to see something and move their focus inward -- the brain cooker mentioned in another answer. (Personally I think that cooking their eyeballs would be both easier and more effective.)

A person who had a small cloud herding gift, but had great stamina could power his own hot air balloon.

Someone who had great focus and power would make an awesome black smith's assistant.

Someone who didn't need to see his target may be able to cauterize internal bleeding. (Requires some form of clairvoyance or seeing around corners.)

You could have fun with different orders that have regimes for training different aspects of this skill.

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You would also need to consider the range of the ability in addition to the other answers (ie. energy and power).

A pyrokinetic that can set something on fire from a range of only 1 meter is far less dangerous then someone who can do the same thing at a range of 100 meters or even more.

Most likely the ability becomes weaker over a longer distance so you would need two tests:

  1. Performing a simple task everyone can do, say ignite a match, from a set distance. Keep repeating the test at increasing distances until the participant can no longer ignite the match.

  2. Measure the person's energy and power as per the other answers but at a set reference distance.

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Hmm, it might be useful to imagine magic as some sort of catalytic effect; rather than dumping energy into things to make them burn, you just reduce the normal activation energy for oxidation of combustibles they carry on them (or on their enemies). That might make it a lot more feasible, or else your pyromancers will be bingeing at the spaghetti warehouse before every battle.

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What if instead of transferring heat over a distance, which in itself would mean that energy losses would occur, you could instead add kinetic energy (I mean, they are called pyroKINETICs) to the particles of the matter that you're trying to burn?

It's more handwavy to explain how the energy got there, but it is still believable as long as it's magic. (for instance, since you said the pyrokinetics believe their power comes from the sun, you could visualize this effect by having a magical ray of sunlight cast down on the object, even in closed spaces, that would begin to accelerate the movement of particles in that spot.)

As to how to measure it, it would be simply how fast you'd reach your intended heat level: have something burst into flames, boil, melt etc.

I believe this would fit with your idea of pyrokinetics since it would be impossible to create flashy floating fireballs, but would be very believable to boil the air around a target or burn the target's skin, perhaps even internal organs if your handwaving would allow that.

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Great answers on measuring energy and power of a fire blast. What about a different scale, inspired by "Worm":

How dangerous?

Rank your espers in the way how fast, how bad, and how many people are needed to subdue their powers. It is an indirect scale, generating one small superhot fire ball and generating a slightly larger but much cooler fire ball might differ a lot in terms of energy, but little in terms of damage (if both are hot enough they both are fourth degree burns anyway).

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