Question

Calculate the rate at which body heat is conducted through the clothing of a skier in a steady-state process, given the following data; the body surface is $1.8{{m}^{2}}$, and the clothing is $1.0cm$ thick, the skin surface temperature is ${{33}^{\circ }}C$and the outer surface of the clothing is at ${{1.0}^{\circ }}C$; the thermal conductivity of the clothing is $0.040\dfrac{W}{m.K.}$

Answer 1

When attempting questions based on heat transfer, keep in mind the thermal conductive properties of substances and the various factors affecting the rate of flow of heat like surface area, difference in temperatures, thickness of various materials et cetera.

Complete step-by-step answer:
For better understanding of the question, let us know more about conduction first. Conduction is the passing of heat energy between two objects that are in direct, physical contact. It is one of the three types of heat transfer, the other two being convection and radiation. Whenever two objects of different temperatures are in contact with each other, heat energy will pass between them.
This is because temperature is the average kinetic energy of the molecules in a substance. Hotter materials have molecules that are moving faster. So when a cold object and a hot object are touching, the fast moving hot molecules will hit the colder molecules, spreading the heat from the hot object into the cold object. This will keep happening until they reach the same temperature.
So, the rate of heat transfer to an object is equal to the thermal conductivity of the material the object is made from, multiplied by the surface area in contact, multiplied by the surface area in contact, multiplied by the difference temperature between the two objects and divided by the thickness of the material.
Assuming the thermal conductivity to be $k$ which will be $0.040\dfrac{W}{m.K.}$
The thickness of clothing to be $L$which is given to us in $cm$which we will convert to $m$and write as $1.0\times {{10}^{-2}}m$
The area to be $A$which is given to us as $1.8{{m}^{2}}$
The temperature of skin to be ${{T}_{H}}$which is given to us as ${{33}^{\circ }}C$
The outer temperature to be ${{T}_{C}}$which is given to us as ${{1.0}^{\circ }}C$
We need to find the rate of flow of heat which for easier calculation we will take as ${{P}_{cond}}$
Now putting all values into the equation we get ;
${{P}_{cond}}=\dfrac{kA({{T}_{H}}-{{T}_{C}})}{L}$
$\Rightarrow \dfrac{(0.040\dfrac{W}{m.K})(1.8{{m}^{2}})({{33}^{\circ }}C-{{1.0}^{\circ }}C)}{1.0\times {{10}^{-2}}m}$
$\therefore 2.3\times {{10}^{2}}J/s$
Hence the rate at which body heat is conducted through the clothing of a skier is $2.3\times {{10}^{2}}J/s$

Note: Heat naturally flows from a region of higher to a region of lower temperature. In fluids a hotter region has lower density than a colder region. Near the surface of the earth, where the gravitational acceleration points downward, buoyancy causes the hotter fluid to rise, setting up convection currents. The rate of heat flow depends on the heat capacity and mobility of the fluid, i.e. how quickly heat flows into or out of the fluid and how well the fluid circulates because of buoyancy.