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_sf 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 (ė_m) and perfusion rate (ṗ) are both constant throughout the muscle. The blood density and specific heat are p_b and c_b, respectively. The core body temperate (T_c) and the arterial blood temperature (T_a) are both assumed to be the same and constant. The muscle and the skin/fat layer thermal conductivities are k_m and k_sf, respectively. The skin has an emissivity of e and the forearm is subjected to an air environment with a temperature of T_∞, a convection heat transfer coefficient of hconv, and a radiation heat transfer coefficient of h_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 (T_i) 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 (T_i) and the maximum temperature in the forearm (T_max) for the following conditions:

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0.05 m, tf 0.003 m, em m T₁ = 37°C, T = Tsurr = 24°C, & = 0.95, 00 a = = Pb 1000 kg/m³, Cp Ksf = 0.3 W/m-K, hcony = 2 W/m².K, hrad = 5.9 W/m².K Skin/Fat = Muscle- I'm 700 W/m³, p = 0.0005 1/s, 3600 J/kg-K, Km 0.5 W/m-K,