A body at 1000°C in black surroundings at 500°C has an emissivity of

1.	A body at 1000°C in black surroundings at 500°C has an emissivity of 0.42 at 1000°C and an emissivity of 0.72 at 500°C. Calculate the rate of heat loss by radiation per m2. (i) When the body is assumed to be grey with ε = 0.42. (ii) When the body is not grey. Assume that the absorptivity is independent of the surface temperature.
Answer» (i) When the body is grey with ε = 0.42 : T₁= 1000 + 273 = 1273 K T₂ = 500 + 273 = 773 K ε at 1000°C = 0.42 ε at 500°C = 0.72 σ = 5.67 × 10^–8 Heat loss per m² by radiation, q = εσ (T₁⁴ - T₂⁴) = 0.42 × 5.67 × 10^–8 [(1273)⁴ – 773)⁴] = 54893 W i.e., Heat loss per m² by radiation = 54.893 kW. (ii) When the body is not grey : Absorptivity when source is at 500°C = Emissivity when body is at 500°C i.e., absorptivity, α = 0.72 Then, energy emitted = εσ T₁⁴ = 0.42 × 5.67 × 10^–8 × (1273)⁴ Energy absorbed = ασT₂⁴ = 0.72 × 5.67 × 10^–8 × (773)⁴ i.e., q = Energy emitted – Energy absorbed = 0.42 × 5.67 × 10^–8 × (1238)⁴ – 0.72 × 5.67 × 10^–8 × (773)⁴ = 62538 – 14576 = 47962 W i.e., Heat loss per m² by radiation = 47.962 kW.

A body at 1000°C in black surroundings at 500°C has an emissivity of 0.42 at 1000°C and an emissivity of 0.72 at 500°C. Calculate the rate of heat loss by radiation per m2. (i) When the body is assumed to be grey with ε = 0.42. (ii) When the body is not grey. Assume that the absorptivity is independent of the surface temperature.

Answer»

(i) When the body is grey with ε = 0.42 :

T₁= 1000 + 273 = 1273 K

T₂ = 500 + 273 = 773 K

ε at 1000°C = 0.42

ε at 500°C = 0.72

σ = 5.67 × 10^–8

Heat loss per m² by radiation,

q = εσ (T₁⁴ - T₂⁴)

= 0.42 × 5.67 × 10^–8 [(1273)⁴ – 773)⁴] = 54893 W i.e.,

Heat loss per m² by radiation = 54.893 kW.

(ii) When the body is not grey :

Absorptivity when source is at 500°C = Emissivity when body is at 500°C i.e., absorptivity, α = 0.72

Then, energy emitted = εσ T₁⁴ = 0.42 × 5.67 × 10^–8 × (1273)⁴

Energy absorbed = ασT₂⁴ = 0.72 × 5.67 × 10^–8 × (773)⁴ i.e.,

q = Energy emitted – Energy absorbed

= 0.42 × 5.67 × 10^–8 × (1238)⁴ – 0.72 × 5.67 × 10^–8 × (773)⁴

= 62538 – 14576 = 47962 W

i.e., Heat loss per m² by radiation = 47.962 kW.