Heat dissipated from an engine in operation can cause hot spots on its surface. If the outer surface of an engine is situated in a place where oil leakage is possible, then when the leaked oil comes in contact with hot spots above the oil's autoignition temperature, it can ignite spontaneously. Consider an engine outer surface that can be approximated as a blackbody. To prevent fire hazard in the event of oil leak on the engine surface, the surface temperature of the engine should be kept below \(180^{\circ} \mathrm{C}\). A radiometer is placed normal to and at a distance of \(1 \mathrm{~m}\) from the engine surface to monitor the surface temperature. The radiometer receives radiation from a target area of \(1 \mathrm{~cm}^{2}\) of the engine surface. If the radiometer detects an irradiation of \(0.1 \mathrm{~W} / \mathrm{m}^{2}\), would there be any risk of fire hazard?

Step by step solution

01

Compute the irradiation from the engine surface

Using the inverse square law, we can calculate the irradiation emitted by the engine's surface: \(I = \frac{I_R \cdot D^2}{A} = \frac{0.1 \frac{W}{m^2} \cdot (1 m)^2}{1 \times 10^{-4} m^2} = 1000 \frac{W}{m^2}\)

02

Compute the surface temperature

Using the Stefan-Boltzmann law, we can now find the temperature of the engine's surface: \(I = \sigma T^4 \Rightarrow T = \sqrt[4]{\frac{I}{\sigma}} = \sqrt[4]{\frac{1000 \frac{W}{m^2}}{5.670 \times 10^{-8} \frac{W}{m^2 \cdot K^4}}} = 373.67 K\)

03

Convert temperature to Celsius

To compare the calculated temperature with the given threshold of 180°C, we need to convert the temperature from Kelvin to Celsius: \(T_{Celsius} = T_{Kelvin} - 273.15 = 373.67 K - 273.15 = 100.52°C\)

04

Determine if there is a risk of fire hazard

The calculated surface temperature of the engine is 100.52°C, which is below the threshold of 180°C. Therefore, there wouldn't be any risk of fire hazard in this case.

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