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BACKGROUND

The pyromancers described here have found some very tall cliffs to jump off of, and they'd like to perform some experiments with gliders. They believe, or at least hope, that being able to magically heat substances at a distance might allow them to create personal thermal updrafts by heating the air directly beneath their gliders. But even if this is the case their ability to create heat is limited. To heat the air they have to burn kilocalories, and they can only produce so much wattage at a time. That's why before jumping off any tall cliffs they'd like to know how much thermal lift force they can generate per kilocalorie spent.

Going from there, I ... uh, these pyromancers, don't yet know a whole lot about the mechanics of flight, so if they can generate adequate thermal lift they'd also like to know if there are particularly energy efficient ways to use it, and if any particular kind of glider structure is best suited to their talents and limitations.


EDIT: Not to distract from the main question of lift, but I should mention that as this is currently a fairly low-tech world they don't have aluminum alloys or similar fancy materials to construct the glider. Pyromancy does allow for the cutting, welding and forging of materials very exactly however.


Heat transfer is pretty straightforward, as is converting the resulting joules to kcal. Air typically starts at 20 C° with specific heat of about 1, J/g °C, though that can vary. Assume that pyromancy is 100% efficient for simplicity, but keep in mind that probably won't be true for the final system. At that efficiency, a well-trained mage should be able to keep up 1000 W over an hour or so.

$ Q=mcΔT $

What's tricky (so far) is knowing how much heat generates approximately how much lift force, and how to most efficiently spread the energy under the glider, and at what range they should begin heating air; a range of within 30ft would be preferred for other aspects of the magic system, but that isn't yet finalized.


THE QUESTION

How much thermal lift can my pyromancers generate per unit of energy spent?


EDIT

In response to @A.C.A.C.'s comment, and in light of the recognition that this may very well be impossible. If that holds true, I'd like to add that I would accept answers that give any of the following, in order of preference.

1) Answers that show how much thermal lift pyromancy could generate, even if it's inadequate for powering gliders.

2) Answers showing in detail how the above pyromancy could supplement gliding rather outright powering it.

3) Answers showing how this could work with a higher maximum wattage.

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    $\begingroup$ Also remember that a glider (plane with long thin wings and no engine) works by moving forward through the air. Are you talking about creating a hot spot so the gliders can enter and ride the stationary thermal? $\endgroup$ – ShadoCat Jan 16 '18 at 18:49
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    $\begingroup$ If you are thinking of something similar to an ultralight (a kite that you ride), they would have to continuously project the heat source ahead of and below the the glider's path. $\endgroup$ – ShadoCat Jan 16 '18 at 18:50
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    $\begingroup$ You get lift from air movement. The more air moving the more lift. The farther below your flight path you begin the heating, the more air you will be moving and the more lift you will get. "Thermals" are columns of air that are rising. The bigger the column and the faster it is rising the better. The column is simply a wind heading up. $\endgroup$ – ShadoCat Jan 16 '18 at 19:05
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    $\begingroup$ This is a well-written question, but fundamentally what you're asking is how much thermal energy it takes to produce a column of air ascending at the exact speed to oppose the descent of the glider, which is a physics rather than worldbuilding question. You might consider asking on physics.se. $\endgroup$ – Catgut Jan 16 '18 at 19:32
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    $\begingroup$ Doing some back of the napkin math, Wikipedia says the Wright brother plane used a 12HP engine. Which is about 10000W. So your engine design with these powers would need to be about 10x more efficient at generating lift than that engine, give or take. Thermodynamics doesn't really deal with how you generate the lift but I think 1000W won't get you in the air no matter how well you design your mechanism to do it. Just to put into perspective, hot air balloon burners are 2-3 Megawatt of power. $\endgroup$ – A. C. A. C. Jan 16 '18 at 19:53
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Gliders are dumb, use balloons

If you heat up the air, that is going to dissipate quickly due to convection and winds and whatever. The first people trying to get airborne with hot air didn't use gliders. They used balloons. As an added bonus, if you are using primitive technologies, people went up in balloons in the 1700s, so the technology level is about as low as you are going to get.

Heat loss in a hot air baloon

Conveniently, Adriana Llado-Garcin, who I never met but now I love, did her Master's thesis on a Heat Transfer Model for Hot Air Balloons.

I started doing some fancy math, but then realized a few things. First: 1000 W is nothing. The burner studied in the paper was 3.2 MW.

Second, 1000 W is really nothing. A hot air balloon presents a surface area of about 400 m$^2$ to the sun, which on a sunny day will warm it with 1000 W/m$^2$. A dark colored balloon will absorb 90% of this radiation. Therefore, heat input to a large-ish hot air balloon from the sun on a sunny day is about 360 kW. So putting your balloon on a field on a sunny day is about the same heating as 360 of your pyromancers.

Conclusions

I didn't end up doing any math out, but the discrepancies in power numbers impressed on me that this scheme will not work. If you have a better chance of firing a hot air balloon by leaving it in the sun, and you certainly won't lift off by leaving it in the sun, then your pyromancers can't reasonably launch a balloon.

And if they can't reasonably launch a balloon, what chance do they have of keeping their glider airborne? I'm calling this myth busted. A 1000W pyromancer cannot generate any significant lift force. Better tell those pyromancers to step away from the cliff.

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    $\begingroup$ But stepping away from the cliff is exactly what they want to do... $\endgroup$ – Frostfyre Jan 17 '18 at 14:01

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