# Heat Conduction - How long does it take?

1. Oct 23, 2016

### Fruitbraker

1. The problem statement, all variables and given/known data
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...].)
Values I have calculate so far:
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).

2. Relevant equations

The solution to a differential equation is as follows:

TA(t) = TB + (TA0 - Tb)e^-t/τ

3. The attempt at a solution

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!!

2. Oct 23, 2016

### Staff: Mentor

Simple cross-check: Initial power is 600 W, which won't change much during the process. At 209 kJ/K, a change by 2 K is 418 kJ, and 418kJ/600W = 700 seconds. A bit more because the temperature difference goes down slightly. Your answer is right.

An actual diver won't die after 12 minutes, of course - their body will produce heat, the skin temperature will get lower, limiting heat loss, and so on.