Question

Find the net rate of heat transfer by radiation from a skier standing in the shade, given the following. She is completely clothed in white (head to foot, including a ski mask), the clothes have an emissivity of 0.200 and a surface temperature of 10.0ºC , the surroundings are at −15.0ºC , and her surface area is\({\bf{1}}{\bf{.60}}\;{{\bf{m}}^{\bf{2}}}\).

Short answer

The net rate of heat transfer from skier by radiation is \({\bf{36}}\;{\bf{J/s}}\).

Step-by-step solution

Determination of formula for net rate of heat transfer by radiation from skier

Given Data:

The surface area of skier is\(A = 1.60\;{{\rm{m}}^2}\)

The temperature of skier is\({T_1} = 10^\circ {\rm{C}} = 283\;{\rm{K}}\)

The temperature of surrounding is\({T_2} = - 15^\circ {\rm{C}} = 258\;{\rm{K}}\)

The emissivity of skier clothes is\(\varepsilon = 0.200\)

The net rate of heat transfer is calculated by using the Stefan’s law. This law gives the heat transfer without any medium and difference in temperature between surrounding and skier.

The net rate of heat transfer by radiation from skier is given as:

\(Q = \varepsilon \sigma A\left( {T_1^4 - T_2^4} \right)\)

Here, \(\sigma \) is Stefan’s constant and its value is \(5.67 \times {10^{ - 8}}\;{\rm{W}} \cdot {{\rm{m}}^{ - 2}} \cdot {{\rm{K}}^4}\)

Determination of net rate of heat transfer by radiation from skier

Substitute all the values in the above equation.

\(\begin{aligned}{}Q &= \left( {0.200} \right)\left( {5.67 \times {{10}^{ - 8}}\;{\rm{W}} \cdot {{\rm{m}}^{ - 2}} \cdot {{\rm{K}}^4}} \right)\left( {1.60\;{{\rm{m}}^2}} \right)\left[ {{{\left( {283\;{\rm{K}}} \right)}^4} - {{\left( {258\;{\rm{K}}} \right)}^4}} \right]\\Q &= 36\;{\rm{W}}\\Q &= \left( {36\;{\rm{W}}} \right)\left( {\frac{{1\;{\rm{J}}/{\rm{s}}}}{{1\;{\rm{W}}}}} \right)\\Q &= 36\;{\rm{J}}/{\rm{s}}\end{aligned}\)

Therefore, the net rate of heat transfer from skier by radiation is \(36\;{\rm{J}}/{\rm{s}}\).