For simplicity, approximate the forearm as a one-dimensional cylinder and ignore the presence of bones. The metabolic heat generation rate $\left(\dot{e}_{\mathrm{m}}\right)$ 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 $T_{\infty 0}$, a convection heat transfer coefficient of $h_{\text {com }}$, and a radiation heat transfer coefficient of $h_{\text {rad }}$. 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:
$$
\begin{aligned}
&r_{m}=0.05 \mathrm{~m}, t_{s f}=0.003 \mathrm{~m}, \dot{e}_{\mathrm{m}}=700 \mathrm{~W} / \mathrm{m}^{3}, \dot{p}=0.00051 / \mathrm{s}, \\
&T_{a}=37^{\circ} \mathrm{C}, T_{\text {co }}=T_{\text {surr }}=24^{\circ} \mathrm{C}, \varepsilon=0.95 \\
&\rho_{b}=1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{b}=3600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k_{m}=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} \\
&k_{s f}=0.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{\text {conv }}=2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}, h_{\text {rad }}=5.9 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}
\end{aligned}
$$