91影视

The generated energy by metabolic reactions,\({P_{meta}} = 280\;{\rm{W}}\)

Heat delivered to the skin by convection,\({P_{con}} = k'{A_{skin}}\left( {{T_{air}} - {T_{skin}}} \right)\)

Exposed skin area,\({A_{skin}} = 1.5\;{{\rm{m}}^2}\)

The value of \(k' = 54\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{h}} \cdot {\rm{C}}^\circ \cdot {{\rm{m}}^2}}}} \right.\\} {{\rm{h}} \cdot {\rm{C}}^\circ \cdot {{\rm{m}}^2}}}\)

Air temperature,\({T_{air}} = 47^\circ {\rm{C}}\)

Skin temperature,\({T_{skin}} = 36^\circ {\rm{C}}\)

Skin absorbs the radiated energy,\({I_{sun}} = 1400\;{{\rm{W}} \mathord{\left/{\vphantom {{\rm{W}} {{{\rm{m}}2}}}} \right.\\} {{{\rm{m}}2}}}\)

Emissivity of the skin,\(e = 1\)

Environment temperature,\(T = 47^\circ {\rm{C}}\)

The expression to calculate the heat delivered to the skin by the convection is given as follows.

\({P_{con}} = k'{A_{skin}}\left( {{T_{air}} - {T_{skin}}} \right)\) 鈥︹�� (i)

The expression to calculate the heat transferred to the skin by the sun is given as follows.

\({P_{sun}} = {I_{sun}} \times {A_{skin}}\) 鈥︹�� (ii)

The expression to calculate the heat transferred to the skin by the radiation is given as follows.

\({P_{rad}} = {A_{skin}}e\sigma \left( {{T^4} - T_s^4} \right)\) 鈥︹�� (iii)

Here,\(\sigma \)is the Stefan Boltzmann constant.

The expression to calculate the net rate at which person鈥檚 skin is heated is given as follows.

\(P = {P_{meta}} + {P_{con}} + {P_{sun}} + {P_{rad}}\) 鈥︹�� (iv)

The value of Stefan Boltzmann constant,\(\sigma = 5.67 \times {10^{ - 8}}\;{{\rm{W}} \mathord{\left/{\vphantom {{\rm{W}} {{{\rm{m}}^2}{{\rm{K}}^4}}}} \right.\\} {{{\rm{m}}^2}{{\rm{K}}^4}}}\)

Calculate the heat delivered to the skin by the convection.

Substitute \(54\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{h}} \cdot {\rm{C}}^\circ \cdot {{\rm{m}}^2}}}} \right.\\} {{\rm{h}} \cdot {\rm{C}}^\circ \cdot {{\rm{m}}^2}}}\) for \(k'\), \(1.5\;{{\rm{m}}^2}\) for \({A_{skin}}\),\(47^\circ {\rm{C}}\) for \({T_{air}}\) and \(36^\circ {\rm{C}}\) for \({T_{skin}}\) into equation (i).

\(\begin{array}{l}{P_{con}} = \left( {54\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{h}} \cdot ^\circ {\rm{C}} \cdot {{\rm{m}}^2}}}} \right.\\} {{\rm{h}} \cdot ^\circ {\rm{C}} \cdot {{\rm{m}}^2}}}} \right)\left( {1.5\;{{\rm{m}}^2}} \right)\left( {47^\circ {\rm{C - 36}}^\circ {\rm{C}}} \right)\\{P_{con}} = \left( {54\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{h}} \cdot ^\circ {\rm{C}} \cdot {{\rm{m}}^2}}}} \right.\\} {{\rm{h}} \cdot ^\circ {\rm{C}} \cdot {{\rm{m}}^2}}}} \right)\left( {\frac{{1\;{\rm{h}}}}{{3600\;{\rm{s}}}}} \right)\left( {1.5\;{{\rm{m}}^2}} \right)\left( {11^\circ {\rm{C}}} \right)\\{P_{con}} = \frac{{54 \times 1.5 \times 11}}{{3600}}\;{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {\rm{s}}}} \right.\\} {\rm{s}}}\\{P_{con}} = 0.247\;{\rm{W}}\end{array}\)

Calculate the heat transferred to the skin by the sun.

Substitute \(1400\;{{\rm{W}} \mathord{\left/{\vphantom {{\rm{W}} {{{\rm{m}}^2}}}} \right.\\} {{{\rm{m}}^2}}}\) for \({I_{sun}}\), and \(1.5\;{{\rm{m}}^2}\) for \({A_{skin}}\)into equation (ii).

\(\begin{array}{l}{P_{sun}} = \left( {1400\;{{\rm{W}} \mathord{\left/{\vphantom {{\rm{W}} {{{\rm{m}}^2}}}} \right.\\} {{{\rm{m}}^2}}}} \right)\left( {1.5\;{{\rm{m}}^2}} \right)\\{P_{{\mathop{\rm sun}\nolimits} }} = 1400 \times 1.5\;{\rm{W}}\\{{\rm{P}}_{sun}} = 2100\;{\rm{W}}\end{array}\)

Calculate the heat transferred to the skin by the radiation.

Substitute \(1.5\;{{\rm{m}}^2}\) for \({A_{skin}}\),\(1\) for \(e\), \(5.67 \times {10^{ - 8}}\;{{\rm{W}} \mathord{\left/ {\vphantom {{\rm{W}} {{{\rm{m}}^2}{{\rm{K}}^4}}}} \right. } {{{\rm{m}}^2}{{\rm{K}}^4}}}\) for \(\sigma \), \(320\;{\rm{K}}\) for \(T\) and \(309\;{\rm{K}}\) for \({T_{skin}}\) into equation (iii).

\(\begin{array}{l}{P_{rad}} = \left( {1.5\;{{\rm{m}}2}} \right)\left( {5.67 \times {{10}{ - 8}}\;{{\rm{W}} \mathord{\left/{\vphantom {{\rm{W}}{{{\rm{m}}2}{{\rm{K}}4}}}} \right.\\} {{{\rm{m}}2}{{\rm{K}}4}}}} \right)\left( 1 \right)\left( {{{\left( {320\;{\rm{K}}} \right)}4} - {{\left( {309\;{\rm{K}}} \right)}4}} \right)\\{P_{rad}} = \left( {1.5\;{{\rm{m}}2}} \right)\left( {5.67 \times {{10}{ - 8}}\;{{\rm{W}} \mathord{\left/{\vphantom {{\rm{W}} {{{\rm{m}}2}{{\rm{K}}4}}}} \right.\\} {{{\rm{m}}2}{{\rm{K}}4}}}} \right)\left( {1.369 \times {{10}9}\;{{\rm{K}}{\rm{4}}}} \right)\\{P_{rad}} = 1.5 \times 5.67 \times {10{ - 8}} \times 1.369 \times {109}\;{\rm{W}}\\{P_{rad}} = 116.43\;{\rm{W}}\end{array}\)

Calculate the net rate at which person鈥檚 skin is heated.

Substitute \(280\;{\rm{W}}\) for \({P_{meta}}\),\(0.247\;{\rm{W}}\) for \({P_{con}}\),\(2100\;{\rm{W}}\) for \({P_{sun}}\) and \(116.43\;{\rm{W}}\) for \({P_{rad}}\) into equation (iv).

\(\begin{array}{l}P = 280\;{\rm{W}} + 0.247\;{\rm{W}} + 2100\;{\rm{W}} + 116.43\;{\rm{W}}\\P = \left( {280 + 0.247 + 2100 + 116.43} \right)\;{\rm{W}}\\P = 2496.67\;{\rm{W}}\\P = 2.5 \times {10^3}\;{\rm{W}}\end{array}\)

Hence the net rate at which person鈥檚 skin is heated is \(2.5 \times {10^3}\;{\rm{W}}\).