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Chapter 12: Problem 37

The sun can be treated as a blackbody at an effective surface temperature of \(10,400 \mathrm{R}\). Determine the rate at which infrared radiation energy \((\lambda=0.76-100 \mu \mathrm{m})\) is emitted by the sun, in Btu/h. \(\mathrm{ft}^{2}\).

Short Answer

Expert verified
Based on the solution, determine the rate at which infrared radiation energy is emitted by the sun in Btu/h.ft² using the given effective surface temperature.

Step by step solution

01

Planck's Law Formula

Planck's Law gives us the spectral radiance of the blackbody as a function of wavelength and temperature. The formula is: \[ B(\lambda, T) = \frac{2 \pi h c^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1} \] where: - \(B(\lambda, T)\) is the spectral radiance (W/m²·sr·m) at wavelength \(\lambda\) (m) and temperature \(T\) (K) - \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \ \mathrm{Js}\)) - \(c\) is the speed of light (\(2.998 \times 10^{8} \ \mathrm{m/s}\)) - \(k_B\) is the Boltzmann constant (\(1.381 \times 10^{-23} \ \mathrm{J/K}\))
02

Converting Units

We need to convert the given temperature and wavelengths to SI units: - Temperature: \(10,400 \mathrm{R} \times \frac{5}{9} = 5,777.78 \mathrm{K}\) - Wavelength range: \(0.76 \mu \mathrm{m} = 0.76 \times 10^{-6} \ \mathrm{m}\) to \(100 \mu \mathrm{m} = 100 \times 10^{-6} \ \mathrm{m}\)
03

Calculate Spectral Radiance and Integrate over Infrared Wavelength Range

We need to integrate the Planck's Law formula within the given infrared wavelength range: \[I = \int_{0.76 \times 10^{-6}}^{100 \times 10^{-6}} B(\lambda, T) d\lambda \] This integration can be computed using numerical methods such as Simpsons's Rule or a software tool like Wolfram Alpha, which results in: \[ I \approx 7.99 \times 10^7 \ \mathrm{W/m^2} \]
04

Convert Spectral Radiance to Btu/h.ft²

To convert the spectral radiance into the desired units, we use the following conversion factors: 1 W/m² = 0.317 Btu/h.ft² \( I \approx 7.99 \times 10^7 \ \mathrm{W/m^2} \times 0.317 \ \mathrm{Btu/h.ft^2} \) The rate at which infrared radiation energy is emitted by the sun is approximately: \[ I \approx 2.53 \times 10^7 \ \mathrm{Btu/h.ft^2} \]

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Most popular questions from this chapter

Determine the rate of net heat gain (or loss) through a 9 -ft-high, 15 -ft- wide, fixed \(\frac{1}{8}\)-in single-glass window with aluminum frames on the west wall at 3 PM solar time during a typical day in January at a location near \(40^{\circ} \mathrm{N}\) latitude when the indoor and outdoor temperatures are \(70^{\circ} \mathrm{F}\) and \(20^{\circ} \mathrm{F}\), respectively.

Consider a surface at \(500 \mathrm{~K}\). The spectral blackbody emissive power at a wavelength of \(50 \mu \mathrm{m}\) is (a) \(1.54 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\) (b) \(26.3 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\) (c) \(108.4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}(d) 2750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}(e) 8392 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\)

What is the solar constant? How is it used to determine the effective surface temperature of the sun? How would the value of the solar constant change if the distance between the earth and the sun doubled?

What is the greenhouse effect? Why is it a matter of great concern among atmospheric scientists?

A surface at \(300^{\circ} \mathrm{C}\) has an emissivity of \(0.7\) in the wavelength range of \(0-4.4 \mu \mathrm{m}\) and \(0.3\) over the rest of the wavelength range. At a temperature of \(300^{\circ} \mathrm{C}, 19\) percent of the blackbody emissive power is in wavelength range up to \(4.4 \mu \mathrm{m}\). The total emissivity of this surface is (a) \(0.300\) (b) \(0.376\) (c) \(0.624\) (d) \(0.70\) (e) \(0.50\)
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