In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas gives both an air temperature and the wind chill temperature. The air temperature is measured with a thermometer that is not affected by the wind. However, the rate of heat loss from the skier increases with wind velocity, and the wind-chill temperature is the temperature that would result in the same rate of heat loss in still air as occurs at the measured air temperature with the existing wind. Suppose that the inner temperature of a 3-mm-thick layer of skin with a thermal conductivity of 0.35 W/(m K) is 35°C and the ambient air temperature is –20°C. Under calm ambient conditions the heat transfer coefficient at the outer skin surface is about 20 W/(m2 K) but in a 40 mph wind it increases to

Question

75 W/(m2 K). (a) If frostbite can occur when the skin temperature drops to about 10°C, would you advise the skier to wear a face mask? (b) What is the skin temperature drop due to wind?



GIVEN

• Skier’s skin exposed to cold air

• Skin thickness (L) = 3 mm = 0.003 m

• Inner surface temperature of skin (Tsi) = 35°C

• Thermal conductivity of skin (k) = 0.35 W/(m K)

• Ambient air temperature (T?) = –20°C

• Convective heat transfer coefficients

? Still air (hc0) = 20 W/(m2 K)

? 40 mph air (hc40) = 75 W/(m2 K)

• Frostbite occurs at an outer skin surface temperature (Tso) = 10°C FIND

(a) Will frostbite occur under still or 40 mph wind conditions? (b) Skin temperature drop due to wind chill.

ASSUMPTIONS

• Steady state conditions prevail

• One dimensional conduction occurs through the skin

• Radiative loss (or gain from sunshine) is negligible

SKETCH

Answer 1

The thermal circuit for this system is shown below



(a) The rate of heat transfer is given by



The outer surface temperature of the skin in still air can be calculated by examining the conduction

through the skin layer



Solving for the outer skin surface temperature



The rate of heat transfer by conduction through the skin must be equal to the total rate of heat transfer,

therefore



Solving this for still air



For a 40 mph wind



Therefore, frostbite may occur under the windy conditions.

(b) Comparing the above results we see that the skin temperature drop due to the wind chill was

15°C.