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.

Step by step solution

01

Convert the temperature to Kelvin

We need to work in Kelvin, so convert the temperature from Celsius to Kelvin. \(T_{K} = T_{\mathrm{C}} + 273.15\) For \(T_{\mathrm{C}} = 45^{\circ} \mathrm{C}\): \(T_{K} = 45 + 273.15 = 318.15 \mathrm{K}\)

02

Use Stefan-Boltzmann Law

The Stefan-Boltzmann Law relates the radiant energy emitted by a blackbody to its temperature, which is given by the formula: \(P = Ae \sigma T^4\) where - \(P\) is the radiated power, - \(A\) is the area of the emitter, - \(e\) is the emissivity (for a blackbody, \(e=1\)), - \(\sigma\) is the Stefan-Boltzmann constant (\(5.67\times10^{-8} \mathrm{Wm^{-2}K^{-4}}\)), - \(T\) is the temperature in Kelvin. For our problem, \(A = 1 \mathrm{cm^2} = 10^{-4} \mathrm{m^2}\), \(T = 318.15 \mathrm{K}\), and \(e=1\). Let's find the radiated power at \(45^{\circ} \mathrm{C}\). \(P = 10^{-4} \times 1 \times (5.67\times10^{-8}) \times (318.15)^4\) \(P = 8.249 \times 10^{-5} \mathrm{W}\)

03

Calculate the irradiation on the radiometer

Now we need to find the irradiation on the radiometer, which is the radiant energy received per unit area. The irradiation can be calculated by dividing the radiated power by the area of the radiometer's target over which it is spread. The distance between the sheet and radiometer is given as \(0.5 \mathrm{m}\). \(E = \dfrac{P}{4\pi (distance)^2}\) \(E = \dfrac{8.249 \times 10^{-5}\mathrm{W}}{4\pi (0.5)^2 \mathrm{m^2}}\) \(E = 1.048 \times 10^{-4} \mathrm{Wm^{-2}}\)

04

Set the warning alarm threshold

The irradiation which the warning alarm should be triggered at is: \(1.048 \times 10^{-4} \mathrm{Wm^{-2}}\) So, the warning alarm should be set to detect an irradiation of \(1.048 \times 10^{-4} \mathrm{Wm^{-2}}\) or higher to indicate that the metal sheet is not safe to touch.

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