Question

Radiation from the head is a major source of heat loss from the human body. Model a head as a

20-cm-diameter, 20-cm-tall cylinder with a flat top.

If the body's surface temperature is 36 ∘C , what is the net rate of heat loss on a chilly 5 ∘C day? All skin, regardless of color, is effectively black in the infrared where the radiation occurs, so use an emissivity of 0.95.

Express your answer to two significant figures and include the appropriate units.

Answer 1

Use Stefan‒Boltzmann law in the form ---------------- P = εσA [ T₂ ⁴ ‒ T₁ ⁴ ]

where ε = 0.95 emissivity σ = 5.6703 × 10 ⁻⁸ J s ⁻¹ m ⁻² K ⁻⁴

T₁ = 5 °C + 273.15 = 278.15 K
T₂ = 36 °C + 273.15 = 309.15 K
T₂ ⁴ ‒ T₁ ⁴ = 309.15 ⁴ ‒ 278.15 ⁴ = 3148618368.21 K⁴
For the total surface area A = πR ² + 2π R H = ¼πD ² + 2π [ ½D ] H
D = 20 cm = 0.20 m H = 20 cm = 0.20 m
A = ¼π [ 0.20 m ] ² + 2π [ ½(0.20 m ) ] [ 0.20 m ]
= ( ⁵/₄ ) π [ 0.20 m ] ² = 0.1571 m ²
Then, as given in the expression above
P = ( 0.95 ) ( 5.6703 × 10 ⁻⁸ J s ⁻¹ m ⁻² K ⁻⁴ ) ( 0.1571 m ² ) ( 3.14861836821 x 10⁹ K ⁴ )
= 26.6456213389 J s ⁻¹ = 26.6456213389 W … is the power emitted by the body

P = ΔE / Δt = ΔQ / Δt = 26.6456213389 W