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Chapter 12: Problem 56

A small circular plate has a diameter of \(2 \mathrm{~cm}\) and can be approximated as a blackbody. To determine the radiation from the plate, a radiometer is placed normal to the direction of viewing from the plate at a distance of \(50 \mathrm{~cm}\). If the radiometer measured an irradiation of \(85 \mathrm{~W} / \mathrm{m}^{2}\) from the plate, determine the temperature of the plate.

Short Answer

Expert verified
Answer: The temperature of the plate is approximately 1365.53 K.

Step by step solution

01

Recall Stefan-Boltzmann Law

Stefan-Boltzmann Law states that the radiated power from a blackbody is proportional to the fourth power of its temperature (\(T\)). Mathematically, it can be written as: \(P = \epsilon \sigma A T^{4}\) \(where \text{~}\) \(P\) - radiated power, \(\epsilon\) - emissivity, \(\sigma\) - Stefan-Boltzmann constant \(= 5.670 \times 10^{-8} \mathrm{W} / \mathrm{m}^{2} \mathrm{K}^{4}\), \(A\) - surface area, \(T\) - temperature. Here, we assume the plate is a perfect blackbody, so the emissivity (\(\epsilon\)) is equal to 1.
02

Find the surface area of the circular plate

The surface area of the circular plate can be found using the formula for the area of a circle: \(A = \pi r^2\) where \(r\) is the radius of the plate. Given that diameter of the circular plate is \(2\mathrm{~cm}\), its radius is: \(r= \frac{2}{2} = 1\mathrm{~cm} = 0.01\mathrm{~m}\) (converted to meters) Now, compute the surface area: \(A = \pi (0.01)^{2} = 3.14 \times 10^{-4}\mathrm{~m}^{2}\)
03

Find the radiated power of the plate

The irradiation measured by the radiometer is given as \(85\mathrm{~W} / \mathrm{m}^{2}\). To find the radiated power of the plate, we must consider the area occupied by the radiation at a distance of \(50\mathrm{~cm}\) \((\) or \(0.5\mathrm{~m})\) from the plate: \(A_{irradiation} = 4 \pi (0.5)^{2}\) Now compute the radiated power: \(P = (\text{Irradiation})(A_{irradiation})\) \(P = (85\mathrm{~W} / \mathrm{m}^{2})(4 \pi (0.5)^{2})\) \(P = 85 \times \pi \times 0.5^{2} = 33.5175\mathrm{~W}\)
04

Determine the temperature of the plate

Now, using the Stefan-Boltzmann Law, we can find the temperature \(T\): \(P = \epsilon \sigma A T^{4}\) That means: \(33.5175 = (1)(5.670 \times 10^{-8})(3.14 \times 10^{-4}) T^{4}\) We can now solve for \(T\): \(T^{4} = \frac{33.5175}{(5.670 \times 10^{-8})(3.14 \times 10^{-4})}\) \(T^{4} = 18866357.75\) \(T = \sqrt[4]{18866357.75} = 1365.53\mathrm{~K}\) So, the temperature of the plate is approximately \(1365.53\mathrm{~K}\).

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Most popular questions from this chapter

A long metal sheet that can be approximated as a blackbody is being conveyed through a water bath to be cooled. In order to prevent thermal burn on people handling the sheet, it must exit the water bath at a temperature below \(45^{\circ} \mathrm{C}\). A radiometer is placed normal to and at a distance of \(0.5 \mathrm{~m}\) from the sheet surface to monitor the exit temperature. The radiometer receives radiation from a target area of \(1 \mathrm{~cm}^{2}\) of the metal sheet surface. When the radiometer detects that the metal sheet temperature is not below \(45^{\circ} \mathrm{C}\), an alarm will go off to warn that the sheet is not safe to touch. Determine the irradiation on the radiometer that the warning alarm should be triggered.

A horizontal opaque flat plate is well insulated on the edges and the lower surface. The top surface has an area of \(5 \mathrm{~m}^{2}\), and it experiences uniform irradiation at a rate of \(5000 \mathrm{~W}\). The plate absorbs \(4000 \mathrm{~W}\) of the irradiation, and the surface is losing heat at a rate of \(500 \mathrm{~W}\) by convection. If the plate maintains a uniform temperature of \(350 \mathrm{~K}\), determine the absorptivity, reflectivity, and emissivity of the plate.

Irradiation on a semi-transparent medium is at a rate of \(520 \mathrm{~W} / \mathrm{m}^{2}\). If \(160 \mathrm{~W} / \mathrm{m}^{2}\) of the irradiation is reflected from the medium and \(130 \mathrm{~W} / \mathrm{m}^{2}\) is transmitted through the medium, determine the medium's absorptivity, reflectivity, transmissivity, and emissivity.

A small surface of area \(A=1 \mathrm{~cm}^{2}\) is subjected to incident radiation of constant intensity \(I_{i}=2.2 \times 10^{4} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{sr}\) over the entire hemisphere. Determine the rate at which radiation energy is incident on the surface through \((a) 0 \leq \theta\) \(\leq 45^{\circ}\) and (b) \(45 \leq \theta \leq 90^{\circ}\), where \(\theta\) is the angle a radiation beam makes with the normal of the surface.

A 5-in-diameter spherical ball is known to emit radiation at a rate of \(550 \mathrm{Btu} / \mathrm{h}\) when its surface temperature is \(950 \mathrm{R}\). Determine the average emissivity of the ball at this temperature.
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