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:
$$
\begin{aligned}
&r_{m}=0.05 \mathrm{~m}, t_{s f}=0.003 \mathrm{~m}, \dot{e}_{\mathrm{m}}=700
\mathrm{~W} / \mathrm{m}^{3}, \dot{p}=0.00051 / \mathrm{s}, \\
&T_{a}=37^{\circ} \mathrm{C}, T_{\text {co }}=T_{\text {surr }}=24^{\circ}
\mathrm{C}, \varepsilon=0.95 \\
&\rho_{b}=1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{b}=3600 \mathrm{~J} /
\mathrm{kg} \cdot \mathrm{K}, k_{m}=0.5 \mathrm{~W} / \mathrm{m} \cdot
\mathrm{K} \\
&k_{s f}=0.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{\text {conv }}=2
\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}, h_{\text {rad }}=5.9
\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}
\end{aligned}
$$
