Question

In: Mechanical Engineering

In class, we learned that ”most” of the sun’s radiation happens in the band 0.2 ?...

In class, we learned that ”most” of the sun’s radiation happens in the band 0.2 ? ? ? 3µm, and Stephen asked what ”most” means. What proportion of solar radiation (for a black body sun) falls in the above wavelength range. Similarly, what proportion of the Earth’s radiation (black body Earth at 290K) occurs in the wavelength band 4 ? ? ? 40µm? What wavelength range contains the middle 95% of the sun’s radiation?

Expert Solution

1. Given that T = 5778 K, (for a blackbody sun)

?₁T = 0.2 * 5778 = 1156 ?mK and ?₂T = 3 * 5778 = 17334 ?mK

the blackbody radiation functions are determined from Table 11-12 , (Introduction to Thermodynamics and Heat Transfer, Yunus A. Cengel)

f _?1 = 0.00173514 f _?2 = 0.97851368

Then the fraction of radiation emitted between these two wavelengths becomes

f _?2 - f _?1 = 0. 97851368– 0. 00173514= 0.9768 ? 97.68%

2. Given that T = 290 K, (for a blackbody earth)

?₁T = 4 * 290 = 1160 ?mK and ?₂T = 40 * 290 = 11600 ?mK

the blackbody radiation functions are determined from Table 11-12 , (Introduction to Thermodynamics and Heat Transfer, Yunus A. Cengel)

f _?1 = 0.001714 f _?2 = 0.9409868

Then the fraction of radiation emitted between these two wavelengths becomes

f _?2 - f _?1 = 0. 9409868– 0. 001714= 0.9393 ? 93.93%

3. Let the range for this 95% radiation be 0 ?m to ? ?m

thus, f _?2 - f _?1 = 0. 95 i.e., f _?2 – 0 = 0.95 i.e., f _?2 = 0.95

from the table 11-2, for 0.95

?₂T = 12488 ?mK

?₂*5778 = 12488

?₂ = 2.16 ?m

thus, 95% of the radiation has a range of 2.16 ?m

samet mamat answered 2 years ago

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