Question

The Stefan-Boltzmann law can be employed to estimate the rate of radiation of energy H from a surface, as in

H = Aeσ^4

where H is in watts, A = the surface area (m^2 ), e = the emissivity that characterizes the emitting properties of the surface (dimensionless), σ = a universal constant called the Stefan-Boltzmann constant (= 5.67 x 10^8 W m^2 K^-4 ), and T = absolute temperature (K).

Required:
Determine the error of H for a steel plate with A = 0.15 m^2 , e = 0.90, and T 5 650 ± 20.
b. Compare your results with the exact error. Repeat the computation but with T = 650 ± 40. Interpret your results.

Answer 1

Step-by-step explanation:

A.

H = Aeσ^4

Using the stefan Boltzmann law

When we differentiate

dH/dT = 4AeσT³

dH/dT = 4(0.15)(0.9)(5.67)(10^-8)(650)³

= 8.4085

Exact error = 8.4085x20

= 168.17

H(650) = 0.15(0.9)(5.67)(10^-8)(650)⁴

= 1366.376watts

B.

Verifying values

H(T+ΔT) = 0.15(0.9)(5.67)(10)^-8(670)⁴

= 1542.468

H(T+ΔT) = 0.15(0.9)(5.67)(10^-8)(630)⁴

= 1205.8104

Error = 1542.468-1205.8104/2

= 168.329

ΔT = 40

H(T+ΔT) = 0.15(0.9)(5.67)(10)^-8(690)⁴

= 1735.05

H(T-ΔT) = 0.15(0.9)(5.67)(10^-8)(610)⁴

= 1735.05-1059.83/2

= 675.22/2

= 337.61