Question

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?

Short answer

Answer: No, there is no risk of fire hazard in this case, as the calculated surface temperature of the engine is 100.52°C, which is below the threshold of 180°C.

Step-by-step solution

Compute the irradiation from the engine surface

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

Compute the surface temperature

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

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: \(T_{Celsius} = T_{Kelvin} - 273.15 = 373.67 K - 273.15 = 100.52°C\)

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.

Key concepts

Blackbody Radiation

Blackbody radiation is a type of thermal radiation emitted by an object due to its temperature. The concept of a blackbody refers to an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Blackbodies are perfect emitters of radiation, meaning they have a characteristic spectrum of emitted radiation that depends solely on their temperature.

In practical terms, blackbody radiation is significant for understanding how objects emit heat. For example, the engine in the exercise is treated as a blackbody. This approximation helps in accurately predicting how much heat the engine's surface will radiate based on its temperature. The emitted radiation helps us determine the temperature of the engine's surface – a critical factor when assessing fire hazards in the context of oil leakage.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in thermal physics that describes the power radiated from a blackbody in terms of its temperature. It is given by the equation: \[ I = \sigma T^4 \] Here, \( I \) is the total energy radiated per unit surface area, \( \sigma \) is the Stefan-Boltzmann constant \( (5.670 \times 10^{-8} \frac{W}{m^2 K^4}) \), and \( T \) is the absolute temperature in Kelvin.

In the original exercise, this law is used to calculate the temperature of the engine's surface based on the radiation measurement detected. By rearranging the formula, you can solve for \( T \), the temperature of the surface:
\[ T = \sqrt[4]{\frac{I}{\sigma}} \]
This calculation is essential for assessing whether the surface temperature is safe and below the oil's autoignition temperature, ensuring fire hazard prevention.

Fire Hazard Prevention

Preventing fire hazards is crucial, especially in engineering applications like engines that operate under high temperatures. The primary goal is to keep the surface temperature of potentially flammable surroundings, such as oil, below its autoignition temperature. This is the lowest temperature at which it can spontaneously ignite without an external flame or spark.

In the exercise scenario, the oil must not reach 180°C in contact with the heated engine surface. Regular monitoring using radiation measurement techniques, as depicted in the exercise, ensures that the surface temperature stays safe, thereby preventing potential fire hazards. Understanding and controlling heat transfer in this manner is a practical application of blackbody and Stefan-Boltzmann principles.

Radiation Measurement

Radiation measurement is a powerful technique used to gauge the temperature of objects remotely by examining the radiation they emit. This process is integral to applications where direct temperature measurement is impractical or unsafe. The radiometer, a device for measuring radiation, plays a vital role here.

In the original exercise, a radiometer is used to monitor the surface temperature of the engine from a safe distance of 1 meter. The measurement of irradiation received helps in calculating the actual surface temperature using laws of thermal radiation, such as the Stefan-Boltzmann law. By knowing the local temperature, engineers can ensure that the machinery operates safely and is within controlled thermal limits to prevent overheating and fire hazards.