Chapter 17, Problem 119RQ

To determine

The fraction of heat lost from each person by respiration.

Explanation of Solution

Given:

Thermal conductivity of the first person $\left(k_1\right)$ is $0.159\text{W}/\text{m}\cdot \text{K}$.

Thermal conductivity of the second person $\left(k_2\right)$ is $0.13\text{W}/\text{m}\cdot \text{K}$.

Heat transfer coefficient $\left(h\right)$ is $0.15\text{W}/{\text{m}}^2\cdot \text{K}$.

Diameter of the cylinder $\left(D\right)$ is 25 cm.

Length of the cylinder $\left(L\right)$ is 1.7 m.

Calculation:

Determine the surface area of the first person’s body.

  $\begin{array}{c}{A}_1=\frac{\pi }{2}DL\\ =\frac{\pi }{2}\left(0.25\text{ m}\right)\left(1.7\text{ m}\right)\\ =0.6675{\text{ m}}^2\end{array}$

Determine the surface area of the second person’s body.

  $\begin{array}{c}{A}_2=2A_1\\ =2\times 0.6675{\text{ m}}^2\\ =1.335{\text{ m}}^2\end{array}$

Determine the thermal resistance from leather.

  $\begin{array}{c}{R}_{\text{leather }}=\frac{L}{kA}\\ =\frac{0.001\text{ m}}{\left(0.159\text{W}/\text{m}\cdot °\text{C}\right)\left(0.6675{\text{ m}}^2\right)}\\ =0.00942°\text{C}/\text{W}\end{array}$

Determine the thermal resistance from convection.

  $\begin{array}{c}{R}_{\text{conv }}=\frac{1}{hA}\\ =\frac{1}{\left(15\text{W}/{\text{m}}^2\cdot °\text{C}\right)\left(0.6675{\text{ m}}^2\right)}\\ =0.09988°\text{C}/\text{W}\end{array}$

Determine the total thermal resistance from first person’s body.

  $\begin{array}{c}{R}_{\text{total }}=R_{\text{leather }}+R_{\text{conv }}\\ =0.00942°\text{C}/\text{W}+0.09988°\text{C}/\text{W}\\ =0.1093°\text{C}/\text{W}\end{array}$

Determine the heat transfer through clothes.

  $\begin{array}{c}{\dot{Q}}_{\text{clothes }}=\frac{T_1-T_{\infty 2}}{R_{\text{total }}}\\ =\frac{\left(32°\text{C}-30°\text{C}\right)}{0.1093°\text{C}/\text{W}}\\ =18.3\text{ W}\end{array}$

Determine heat transfer through skin.

  $\begin{array}{c}{\dot{Q}}_{\text{skin }}=\frac{T_1-T_{\infty 2}}{R_{\text{conv }}}\\ =\frac{\left(32°\text{C}-30°\text{C}\right)}{0.09988°\text{C}/\text{W}}\\ =20\text{ W}\end{array}$

Determine the total sensible heat transfer.

  $\begin{array}{c}{\dot{Q}}_{\text{sensible }}={\dot{Q}}_{\text{clothes }}+{\dot{Q}}_{\text{skin }}\\ =18.3\text{ W}+20\text{ W}\\ =38.3\text{ W}\end{array}$

Determine the fraction of heat lost by respiration.

  $\begin{array}{c}f\doteq \frac{{\dot{Q}}_{\text{respiration }}}{{\dot{Q}}_{\text{total }}}\\ =\frac{{\dot{Q}}_{\text{total }}-{\dot{Q}}_{\text{sensible }}}{{\dot{Q}}_{\text{total }}}\\ =\frac{60\text{ W}-38.3\text{ W}}{60\text{ W}}\\ =0.362\end{array}$

Thus, the fraction of heat lost by respiration is $\underset{\_}{0.362}$.

Determine the thermal resistance from leather.

  $\begin{array}{c}{R}_{\text{synthetic}}=\frac{L}{kA}\\ =\frac{0.001\text{ m}}{\left(0.13\text{W}/\text{m}\cdot °\text{C}\right)\left(1.335{\text{ m}}^2\right)}\\ =0.00576°\text{C}/\text{W}\end{array}$

Determine the thermal resistance from convection.

  $\begin{array}{c}{R}_{\text{conv }}=\frac{1}{hA}\\ =\frac{1}{\left(15\text{W}/{\text{m}}^2\cdot °\text{C}\right)\left(1.335{\text{ m}}^2\right)}\\ =0.04994°\text{C}/\text{W}\end{array}$

Determine the total thermal resistance.

  $\begin{array}{c}{R}_{\text{total }}=R_{\text{synthetic }}+R_{\text{conv }}\\ =0.00576°\text{C}/\text{W}+0.04994°\text{C}/\text{W}\\ =0.0557°\text{C}/\text{W}\end{array}$

Determine the total sensible heat transfer.

  $\begin{array}{c}{\dot{Q}}_{\text{sensible }}=\frac{T_1-T_{\infty 2}}{R_{\text{total }}}\\ =\frac{\left(32°\text{C}-30°\text{C}\right)}{0.0557°\text{C}/\text{W}}\\ =35.9\text{ W}\end{array}$

Determine the fraction of heat lost by respiration.

  $\begin{array}{c}f\doteq \frac{{\dot{Q}}_{\text{respiration }}}{{\dot{Q}}_{\text{total }}}\\ =\frac{{\dot{Q}}_{\text{total }}-{\dot{Q}}_{\text{sensible }}}{{\dot{Q}}_{\text{total }}}\\ =\frac{60\text{ W}-35.9\text{ W}}{60\text{ W}}\\ =0.402\end{array}$

Thus, the fraction of heat lost by respiration is $\underset{\_}{0.402}$.