An average person generates heat at a rate of 84 W while resting. Assuming one-quarter of this heat is lost from the head and disregarding radiation, determine the average surface temperature of the head when it is not covered and is subjected to winds at 10°C and 29 km/h. The head can be approximated as a 0.2-m-diameter sphere. Assume a surface temperature of 15ºC for evaluation of μs. The properties of air at 1 atm pressure and the free stream temperature of 10°C are k

Average surface temperature = 11.4 C

Explanation:

Temperature of the surface = Incoming Temperature + (Heat generated by the Average person's head / Coefficient of heat transfer x Surface Area of head)

Area = πD² = π (0.2) ² = 0.1256 m²

h = (Thermal Conductivity/Diameter of Surface) (Nusselt Number)

Nu = 2 + [0.4Re^0.5 + 0.06Re^2/3] x Pr^0.4 x [μ∞/μs]^1/4

where Re is the Reynold Number, Pr = 0.7336 is the Prandtl Number, μ∞ = 1.778 x 10^-5 is the dynamic viscosity and μs = 1.802 x 10^-5 is the Dynamic Viscosity at 15 C

Re = VD/v where V is the velocity of wind, D is the diameter of the sphere and v is the kinematic viscosity

Re = (8.056 x 0.2) / (1.426 x 10^-5) = 1.13 x 10^6

Substituting value of Re into Nu equation

Nu = 2 + [0.4 (1.13 x 10^6) ^0.5 + 0.06 (1.13 x 10^6) 2/3] (0.7336 ^0.4) { (1.778 x 10^-5) / (1.802 x 10^-5) ^1/4}

Nu = 2 + [1076.1] (0.8835) (0.9967) = 949.5

Putting Nu values in h

h = (0.02439/0.2) (949.5) = 115.8 W/m² C

putting h value in Ts equation

Ts = 10 + [ (84/4) / (115.8 x 0.1256) ] = 11.4 C