Problem $5($from Module $2$ Chapter $3)$: THE answer is at the bottom

Consider a region of muscle with a skin$/$fat layer over it$.$ At a depth of $L_m=30mm$ into the muscle, the temperature can be assumed to be at the core body temperature of $T_c=37C.$ The muscle thermal conductivity is $k_m=0\mathrm{.}5\frac{W}{m}*K.$ The perfusion rate is $p^{}=0\mathrm{.}0005s^{-1}$; the blood density and specific heat are ${}_b=1000k\frac{g}{m^3}$ and $c_b=3600\frac{J}{k}g*K,$ respectively, and the arterial blood temperature $T_a$ is the same as the core body temperature. The thickness, emissivity, and the thermal conductivity of the skin$/$fat layer are as given in the picture.

Consider the conditions for an air environment, where the air and surroundings temperatures are both $15C.$ Humans respond to cold by shivering, which increases the metabolic heat generation rate. What would the metabolic heat generation rate $($per unit volume$)$ have to be to maintain a comfortable skin temperature of $33C$ under these conditions?

Hint: $1)$ find the heat loss rate $q^{}$ from the skin surface to the environment based on the given conditions regarding convection and radiation; $2)$ find $T_i$ based on $q^{}$ and the resistant in the skin$/$fat layer; and $3)$ find $e_m^{}$ based on the equation for heat transfer rate in the muscle layer. The equation is available in the lecture video. You may use the equation directly.

Answer$($s$)$: $4620\frac{W}{m^3}$