Sympathy: Give me the chills!

Question

This question is inspired by our fortnighly challenge!

In the Kingkiller Chronicle, there is a type of magic called "sympathy." This magic works by magicians setting up an energetic "link" between objects. Energy is transferred from one object to another through the link. The magic user needs some source of energy (usually a fire, candle, or sometimes their own body's thermal energy, but not always thermal energy) and connects it to something to achieve some effect. For instance, the protagonist gives an individual a "hot foot" by forming a magical connection with the person's foot with something like a voodoo doll, and moving the voodoo doll's foot over a candle. Thermal energy went from the candle, to the doll, through the link (with energetic losses), and into the person's foot.

There are few points in the book where the protagonist uses his own body's thermal energy for magic effects. Since he is using his body's own heat energy, he suffers what is known as "Binder's Chills," or magically induced hypothermia. As an additional layer of "realism," the author included losses during this energy transfer.

Assuming a link with 2/5 energy efficiency (3/5 energy loss), what can an average mage, using sympathy, do with just their body's thermal energy without killing themselves via Binder's Chills? (See the summary below for specifics!)

I'm also aware of this related question and answers, but they do not go into specifics. I want to know how much energy (Joules are preferred) they can sap from their bodies without killing themselves, and what the maximum effects of the sympathy can be. We're looking for something like this question, but for this sympathy.

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A summary of what you ought to consider in your answer:

1. The mage you ought to consider has a normal body temperature. They are not already cold nor are they already hot.
2. This mage is of average weight and size for Europe (the protagonist is a redheaded, pale-skinned male. The UK-Wales data for men in the link is ideal.)
3. The binder's chills can induce severe hypothermia, but cannot drop the mage's body temperature below 29 degrees C or 82 degrees F.
4. Sympathy takes energy evenly from the body, and you may consider a individual's body to have constant temperature throughout. (This is a simplification, but one I find reasonable.)
5. This magical link transfers 2/5 of the energy put into it from the magician to the object. 3/5 of that energy is "lost" in maintaining that link. The energy is transferred through the link, for all practical purposes, instantly.
6. This thermal energy can be converted into other types of energy. Consider the conversion costs as part of the lost energy in #5.

For sweet over-achievement: describe extreme situations by changing the weight or body temperature (due to exercise, etc.) of the mage!

Answer 1

Well, "what can they do" is rather broad given your parameters. Since the energy is taken from everywhere on the casters body, but can be specifically applied to the victim (like the hotfoot example), the effect increases when you narrow the area. This quickly gets very dangerous.

1. The brain comprises ~2% of the body by weight, so any heat lost is multiplied by a factor of 20 (100%/2%=50*2/5=20), less if we're talking about smaller pieces of the brain. Drop your whole body one degree to instantly boil someone's frontal lobe.
2. The average human skin weighs 8-10 pounds (3.6-4.5 kg, ~5% of the body, factor of 20*2/5=8) and covers 22 square feet (2.04 square meters). The head in total is ~9% of that for a heating factor of 100%/9%*8=88. It would be trivial to set someone's head on fire or literally melt the skin off someone's face.
3. My browser history is starting to get disturbing, so I'm not even providing sources for these: cauterize the heart, or coagulate blood in the surrounding arteries, or boil the acid in their stomach.

The point is, doing these would require very little loss of energy from the perspective of the caster. Figuring out the energy cost in Joules is extremely complex and unnecessary for these cases. Unless the caster is on a very tight energy budget, they'd be losing more just from normal thermal dissipation at room temperature.

For some non-gory applications:

There are so many compensating systems in the human body that getting an accurate look at our energy budget is very difficult. We're going to do this as simply as possible. The average human produces ~100 watts of waste heat, comparable to an incandescent bulb. That means our wizard can easily power a lightbulb with this otherwise wasted heat at no real cost. This also equates to 2.4 kWh in one day, which is the same as saying 2.4 kJ.

It's estimated that the average BMR for an adult male is approximately 1500 kcal/day, though there is HUGE variances between people. One thermochemical calorie = ~4.184 J, so the total daily energy use of this hypothetical human is 1500*1000*4.184 = 6,276 kJ. Put in that context, the rate of waste heat is surprisingly low.

Maintaining body temp is only one of the vital tasks of the metabolism. Average body temp is 37C/98.6F and you've limited us to a minimum of 29C for a total available energy loss of 8C. For a simplistic BON, 1kC = an increase of 1C for 1kg. Our hypothetical person is ~71kg so a drop of 1C= 71kC = ~297kJ. Each loss of 1 degree C frees up ~300kJ*2/5=120kJ of energy for your magic.

For reference, this is almost eight times the energy capacity of a standard AA battery. Draining a car battery would be out of the question (2.6 MJ/120kJ/C = ~21C loss of temp, beyond your stated limits of 8C), but you could easily start a car.

Total energy budget = 8*120kJ=960kJ, or just under 1 MJ.

Apparently, that's just about the kinetic energy of a 1 tonne vehicle moving at 160 km/h. So I guess you could stop a car at your energy limit.

Today I learned: body fat metabolism has 200 times the energy density as an automotive lead-acid battery (35 vs .17 MJ/kg).

Answer 2

Burst: Energy

As Isaac already pointed out, we can get a pretty good approximation of a mage's heat capacity using just the heat capacity of water (since pretty much all of our tissues are mostly water). This gives us a maximum energy burst of:

$$ \begin{align} & 40\% \times 8~\text{°C} \times 4184~\text{J}/\text{kg}\cdot\text{K} \times 71~\text{kg} \\ &= 951~\text{kJ} \\ &= 264~\text{Wh} \\ &= 22.0~\text{Ah}~@~12~\text{V} \end{align} $$

• This is equivalent to around $315~\text{g}$ ($4900~\text{gr}$) of gunpowder (energy density about $3~\text{kJ}/\text{g}$), or over 100 M-80s .
• This is enough energy to fire almost 2000 .45 ACP rounds ($230~\text{gr}~@~830~\text{fps}=475~\text{J}$), or five 30 mm, armor-piercing, incendiary autocannon rounds ($14~\text{oz}~@~3250~\text{fps}=190~\text{kJ}$). Let's hope this doesn't violate conservation of momentum.
• This is enough energy to raise $29~\text{g}$ to low Earth orbit ($250~\text{mi}=410~\text{km}$ altitude, specific orbital energy $\approx 33~\text{kJ}/\text{g}$), or to raise $985~\text{g}$ up $100~\text{km}$ to the edge of space (enough to take my camera, $\approx 700~\text{g}$ with lens, with a $40\%$ mass margin).
• Assuming no violation of thermodynamics, this is enough energy to extract about $2.8~\text{kg}$ of gold ($92~\text{oz}$, around \$100,000 worth) from seawater (abundance around $10^{-11}$, entropy of mixing about $240~\text{J}/\text{mol}\cdot\text{K}$).
• Enough energy to charge my phone ($3300~\text{mAh}~@~3.7~\text{V}$) 21 times over.

Draw: Power

Although the burst energy is interesting, there's still a crucial factor missing: we don't know if "recharging" a mage's internal heat reservoir takes seconds or hours.

As a rough estimate, the amount of heat generated by the average $70~\text{kg}$ male is around $70-100~\text{W}$. During exercise this can rise to $200~\text{W}$. (Endurance cyclists generate a long-term average of around $185~\text{W}$, while world-class rowers can peak at $3/4~\text{HP}$ of mechanical energy during a $2000~\text{m}$ race, corresponding to a heat output of almost $1.9~\text{kW}$!)

Assuming that heat is generated at $150~\text{W}$ while recharging, and that heat loss is roughly proportional to temperature (with standard body temperature corresponding to $100~\text{W}$ of heat output), we get an equation like this:

$$ 300~\text{kJ}/\text{K}\times\dot{T} = 150~\text{W}-100~\text{W}\frac{T-25~\text{°C}}{37~\text{°C}-25~\text{°C}} \\ T=(43-14e^{-t/9.9~\text{h}})~\text{°C} \\ t_{T=37~\text{°C}} = 8~\text{hours}~20~\text{minutes} $$

That means that peak burst power represents the better part of a day's reserve, unless your mage takes the time to warm himself by the fire (which can deliver a significant amount of heat).

But hold on a second... what if you can augment your internal heat with environmental heat? Unlike for a person generating heat, for a person drawing heat from the surroundings the extremities will be warmest and their core will be coldest. We can approximate the torso as a uniform cylinder and use the steady-state heat equation in cylindrical coordinates:

$$ 0=k\nabla^2u+q=k(\frac{u'(r)}{r}+u''(r))+q\\ u(r)-u(0)=\frac{q}{4k}r^2 $$

Assuming that the maximum temperature your mage is willing to stand is around $120~\text{°F}$ ($49~\text{°C}$), the minimum core temperature is the same as before, a torso radius of $14~\text{cm}$, and a conductivity of about $0.5~\text{W}/\text{m}\cdot\text{K}$, the maximum energy draw can be about:

$$ \begin{align} q=\frac{4\times 20~\text{°C}\times 0.5~\text{W}/\text{m}\cdot\text{K}}{14~\text{cm}}&\approx 2~\text{W}/\text{L} \\ &\approx 2~\text{W}/\text{kg} \end{align} $$

Add on another $1.4~\text{W}/\text{kg}$ from the body's own heat generation, and account for the $40\%$ efficiency, and you get a total output of:

$$ 40\%\times(140~\text{W}+100~\text{W})\approx 100~\text{W} $$

That's not much. Even in this best-case scenario, you could exert more power mechanically. In a more realistic scenario (without absorbing environmental heat) the long-term average output would be in the tens of watts range.

Summary

Although the body contains a large amount of thermal energy, the body's limited ability to thermoregulate restricts us from drawing lots of power continuously. This form of sympathetic magic should be used more as a tool to precisely deliver small amounts of energy (see Isaac's answer for some good ideas) than to deliver huge amounts of energy (although it is capable of doing so at great cost to the mage).

Coming soon: Cheating Thermodynamics