Question

A solid cylinder of radius r₁ = 2.2 cm, length h₁ = 5.9 cm, emissivity 0.91, and temperature 38°C is suspended in an environment of temperature 70°C. (a) What is the cylinder's net thermal radiation transfer rate P₁? (b) If the cylinder is stretched until its radius is r₂ = 0.55 cm, its net thermal radiation transfer rate becomes P₂. What is the ratio P₂/P₁?

Answer 1

a)

r₁ = radius = 2.2 cm = 0.022 m

h₁ = height = 5.9 cm = 0.059 m

A₁ = Area of cylinder = 2 $\pi$ r₁ (r₁ + h₁) = (2 x 3.14 x 0.022) (0.022 + 0.059) = 0.011 m²

T = temp. of surrounding = 70 C = 70 + 273 K = 343 K

T_o = cylinder's temp. = 38 C = 38 + 273 = 311 K

P₁ = $\epsilon$ $\sigma$ A₁ (T⁴ - T_o⁴)

P₁ = (0.91) (5.67 x 10^-8) (0.011) (343⁴ - 311⁴)

P₁ = 2.55 Watt

b)

initial Volume of cylinder = final volume of cylinder after being stretched

$\pi$ r₁² h₁ = $\pi$ r₂² h₂

(0.022)² (0.059) = (0.0055)² h₂

h₂ = 0.944 m

A₂ = Area of cylinder = 2 $\pi$ r₂ (r₂ + h₂) = (2 x 3.14 x 0.0055) (0.0055 + 0.944) = 0.033 m²

P₂ = $\epsilon$ $\sigma$ A₂ (T⁴ - T_o⁴)

P₂ = (0.91) (5.67 x 10^-8) (0.033) (343⁴ - 311⁴)

P₂ = 7.64 Watt

Ratio : P₂/P₁ = 7.64/2.55 = 3