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I'm trying to calculate heat loss when a human body is simply suspended in $20 \sideset{^{\circ}}{}{\mathrm{C}}$ water, but I keep getting absurdly high results. It should be a simple application of this equation$$ H = \frac{K \cdot A \cdot \Delta T}{\Delta x} \,,$$where:

  • $H$ is heat loss,

  • $K$ is the thermal conductivity of skin ($0.3\frac{\mathrm{W}}{\mathrm{m} \cdot \mathrm{K}}$, according to this page),

  • $A$ is the surface area of a human ($1.7 \, {\mathrm{m}}^{2}$, according to this page),

  • $\Delta T$ is the temperature difference $\left(37 - 20 = 17\right) ,$ and

  • $\Delta x$ is the thickness of the skin ($\sim 1 \, \mathrm{mm}$, according to this site).

Plugging those numbers in, we get$$ H ~=~ \frac{K \cdot A \cdot \Delta T}{\Delta x} ~ \approx ~ \frac{0.3 \cdot 1.7 \cdot 17}{0.001} ~ \approx ~ 8670 \, \mathrm{W} \,.$$ Now, that seemed fine until I realized that $8500 \, \mathrm{W}$ is a lot of energy – essentially 2 food calories per second. That seems highly inaccurate and implies that you're incapable of surviving more than 20 minutes in $20 \sideset{^{\circ}}{}{\mathrm{C}}$ water before using up an entire day's worth of calories.

Is there an error in my calculations somewhere, or an error in the numbers I'm using?

I also found this source which suggests that the thermal conductivity of water is around $83\frac{\mathrm{W}}{{\mathrm{m}}^2 \cdot \mathrm{K}},$ which is both a very different number and has different units.

I don't think that this discrepancy is due to a protective layer of warmer water forming near the body, because swimmers also survive longer than half an hour and they're constantly disrupting any warm boundary layer that may form.

This question is similar, but I can't figure out how to reverse-engineer it. Perhaps someone else can explain the logic there more clearly?

Nat
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Dubukay
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  • The thickness is the suspicious part of this to me- even if you had no skin you wouldn't expect to reach thermal equilibrium instantly, which is what your equation implies. – Chris Nov 30 '17 at 07:19
  • The skin is not 37 degrees. –  Nov 30 '17 at 08:08
  • Hi and welcome to physics.SE! Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. – ACuriousMind Nov 30 '17 at 08:31
  • @Pieter I’m working with the simplifying assumption that heat is moving directly from the internal body (~37 degrees C) through the skin into the water. The skin will therefore be a gradient between 37 and 20 – Dubukay Nov 30 '17 at 15:53
  • @ACuriousMind Thanks for the welcome! How would you recommend broadening this question? I tried to frame it as a “this equation doesn’t seem to be working, what’s up” kind of conceptual question with the one worked-out example to give some concrete details and prove that this isn’t coming from a problem set. – Dubukay Nov 30 '17 at 15:55
  • @Chris I agree! Do you think that the skin should be much thicker, like 1 cm instead? I’m finding it hard to justify that arbitrary thickening... – Dubukay Nov 30 '17 at 15:57
  • That "thermal conductivity" that you cite is undoubtedly (judging from the units) a typical heat transfer coefficient, rather than a thermal conductivity. The thermal conductivity from your first table is about 0.6, which is similar to the value given in other references. I think you are underestimating the resistance to heat flow in the water boundary layer. I would model this by considering the body as a cylinder with water flowing across at a nominal rate of about 10 cm/sec. I would use the correlation in the literature for flow past a cylinder to get the heat transfer coefficient. – Chet Miller Nov 30 '17 at 20:26
  • From the values entered here, you assume the body temperature to be 37 degrees C 1mm from the skin surface over the entire body. I think this estimate is too rough. Places where the body temperature indeed is 37 degrees C 1mm from the skin, such as blood veins, do not constitute the entire body area. Large portions of your body have a much deeper tissue before you reach 37 degree C; most of e.g. hands and feet don't reach this temperature throughout their thickness. – Steeven Aug 22 '18 at 08:56

1 Answers1

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It is rash to assume that past the 1mm chosen thickness of skin, there is zero thermal resistance to the heat capacitance in the blood stream. By reverse calculation, an effective skin thermal thickness might be 10mm, then the results look believable.

James Fricker
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