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Fruitbraker
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Homework Statement
In order to stay warm, divers often wear some sort of thermal protection, like a "wetsuit". Often this is a neoprene "foamed" material, which traps gas bubbles as the insulating material. For this problem, assume:
- the thermal conductivity is that of air (κ = 0.03 W/m-K)
- the suit thickness is d = 3.5 mm
- the area of the suit is A ~2 m2
- the diver's initial body temperature is Td,i = 37°C (98.6°F)
- the water temperature is Tw = 2°C
- the diver "weighs" m = 60 kg
- the specific heat of the diver is cd = 3480 J/kg-K (this is slightly less than the specific heat of water 4184 J/kg-K due to the presence of protein, fat, and minerals)
- the diver will start to experience loss of motor skills due to hypothermia when his core temperature cools to below Td,f = 35°C (95°F).
(Note: Throughout this problem we are also implicitly assuming that the diver is at a uniform temperature, which obviously is an over-simplification [since our bodies are evolutionarily engineered to maintain a stable core temperature, even if we have cold limbs...].)
Thermal Resistance of wet suit Rth: 0.058333 K/Watt
Heat Capacity of the diver: 208800 J/K
τ: (time constant for differential equation): 12179.9
Estimate how long (in minutes) the diver can stay in the water (before feeling the effects of hypothermia).
Homework Equations
The solution to a differential equation is as follows:
TA(t) = TB + (TA0 - Tb)e^-t/τ
The Attempt at a Solution
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Solving everything symbolically yields
τln( (TA - TB)/(TA0 - TB)) = -t
TA0 = 37°C
TA = 35°C
TB = 2°C
τ = 12179.9
∴t = 716.673s -> 11.94min
Apparently, the answer is wrong. I even had a friend check my work and he said my work is ok.
What's going on?
Thanks!