We are interested in steady state heat transfer analysis from a human forearm
subjected to certain environmental conditions. For this purpose consider the
forearm to be made up of muscle with thickness \(r_{m}\) with a skin/fat layer
of thickness \(t_{s f}\) over it, as shown in the Figure P3-138. For simplicity
approximate the forearm as a one-dimensional cylinder and ignore the presence
of bones. The metabolic heat generation rate \(\left(\dot{e}_{m}\right)\) and
perfusion rate \((\dot{p})\) are both constant throughout the muscle. The blood
density and specific heat are \(\rho_{b}\) and \(c_{b}\), respectively. The core
body temperate \(\left(T_{c}\right)\) and the arterial blood temperature
\(\left(T_{a}\right)\) are both assumed to be the same and constant. The muscle
and the skin/fat layer thermal conductivities are \(k_{m}\) and \(k_{s f}\),
respectively. The skin has an emissivity of \(\varepsilon\) and the forearm is
subjected to an air environment with a temperature of \(T_{\infty}\), a
convection heat transfer coefficient of \(h_{\text {conv }}\), and a radiation
heat transfer coefficient of \(h_{\mathrm{rad}}\). Assuming blood properties and
thermal conductivities are all constant, \((a)\) write the bioheat transfer
equation in radial coordinates. The boundary conditions for the forearm are
specified constant temperature at the outer surface of the muscle
\(\left(T_{i}\right)\) and temperature symmetry at the centerline of the
forearm. \((b)\) Solve the differential equation and apply the boundary
conditions to develop an expression for the temperature distribution in the
forearm. (c) Determine the temperature at the outer surface of the muscle
\(\left(T_{i}\right)\) and the maximum temperature in the forearm \(\left(T_{\max
}\right)\) for the following conditions:
$$
\begin{aligned}
&r_{m}=0.05 \mathrm{~m}, t_{s f}=0.003 \mathrm{~m}, \dot{e}_{m}=700
\mathrm{~W} / \mathrm{m}^{3}, \dot{p}=0.00051 / \mathrm{s} \\
&T_{a}=37^{\circ} \mathrm{C}, T_{\text {co }}=T_{\text {surr }}=24^{\circ}
\mathrm{C}, \varepsilon=0.95 \\
&\rho_{b}=1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{b}=3600 \mathrm{~J} /
\mathrm{kg} \cdot \mathrm{K}, k_{m}=0.5 \mathrm{~W} / \mathrm{m} \cdot
\mathrm{K} \\
&k_{s f}=0.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{\text {conv }}=2
\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}, h_{\mathrm{rad}}=5.9
\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}
\end{aligned}
$$
