Question

A radiometer is employed to monitor the temperature of manufactured parts \(\left(A_{1}=10 \mathrm{~cm}^{2}\right)\) on a conveyor. The radiometer is placed at a distance of \(1 \mathrm{~m}\) from and normal to the manufactured parts. When a part moves to the position normal to the radiometer, the sensor measures the radiation emitted from the part. In order to prevent thermal burn on people handling the manufactured parts at the end of the conveyor, the temperature of the parts should be below \(45^{\circ} \mathrm{C}\). An array of spray heads is programmed to discharge mist to cool the parts when the radiometer detects a temperature of \(45^{\circ} \mathrm{C}\) or higher on a part. If the manufactured parts can be approximated as blackbody, determine the irradiation on the radiometer that should trigger the spray heads to release cooling mist when the temperature is not below \(45^{\circ} \mathrm{C}\).

Short answer

Answer: The irradiation on the radiometer that should trigger the spray heads is approximately 0.348 W/m².

Step-by-step solution

Write down the known values

In order to solve the problem, let's first list down the given values: - Area of the manufactured parts: \(A_{1} = 10 \ \text{cm}^2\) - Distance between the radiometer and the parts: \(d = 1 \ \text{m}\) - Temperature threshold: \(T_{threshold} = 45^{\circ} \mathrm{C}\)

Convert the temperature to Kelvin

Since we will be working with the Stephan-Boltzmann Law, it is necessary to convert the temperature threshold to Kelvin. To do this, add 273.15 to the Celsius temperature: \(T_{threshold (K)} = T_{threshold} + 273.15 = 45 + 273.15 = 318.15 \ \mathrm{K}\)

Apply the Stephan-Boltzmann Law

The power emitted by a blackbody is given by the Stephan-Boltzmann Law: \(P_{emitted} = A \sigma T^4\) Where: - \(P_{emitted}\) is the power emitted by the blackbody, - \(A\) is the area of the blackbody (in this case, the area of the manufactured parts), - \(\sigma\) is the Stefan-Boltzmann constant: \(5.67 \times 10^{-8} \mathrm{W}\cdot\mathrm{m}^{-2}\cdot\mathrm{K}^{-4}\) - \(T\) is the temperature of the blackbody in Kelvin. Since we are interested in the irradiation detected by the radiometer, we need to calculate the power per unit area that reaches the radiometer at a distance \(d\). To do this, divide the emitted power by the surface area of a sphere with radius \(d\): \(I = \frac{P_{emitted}}{4 \pi d^2}\) Combining these two equations, we get: \(I = \frac{A \sigma T^4}{4 \pi d^2}\)

Calculate the irradiation

Now, we can plug in the known values into the equation and solve for the irradiation on the radiometer, \(I\): - Convert the area of the manufactured parts to square meters: \(A_{1} = 0.0010 \ \text{m}^2\) - Distance between radiometer and parts: \(d = 1 \ \text{m}\) - Temperature threshold in Kelvin: \(T_{threshold (K)} = 318.15 \ \mathrm{K}\) \(I = \frac{0.0010 \mathrm{m}^2 \times 5.67 \times 10^{-8} \mathrm{W}\cdot\mathrm{m}^{-2}\cdot\mathrm{K}^{-4} \times (318.15 \ \mathrm{K})^4}{4 \pi \times (1 \ \text{m})^2}\) \(I \approx 0.348 \ \mathrm{W/m^2}\) Thus, the irradiation on the radiometer that should trigger the spray heads to release cooling mist when the temperature is not below \(45^{\circ} \mathrm{C}\) is approximately \(0.348 \ \mathrm{W/m^2}\).

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is crucial in determining how much radiation is emitted by an object based on its temperature. This law is particularly useful in the context of radiometers monitoring temperatures, where direct measurement of the heat can be challenging. The formula involved is: \[ P_{emitted} = A \sigma T^4 \]
Here:

• \( P_{emitted} \) represents the power emitted by an object.
• \( A \) stands for the object's area through which the heat is being emitted.
• \( \sigma \) is the Stefan-Boltzmann constant, which holds a value of \( 5.67 \times 10^{-8} \mathrm{W}\cdot\mathrm{m}^{-2}\cdot\mathrm{K}^{-4} \).
• \( T \) is the temperature of the emitter in Kelvin.

Through this equation, we assess the radiation emitted as a function of temperature raised to the fourth power. Therefore, small changes in temperature can lead to large differences in the emitted power. In practical applications, this principle helps set thresholds for cooling systems to maintain safe temperatures in manufactured components.

Blackbody radiation

Blackbody radiation describes the phenomenon where an object absorbs all incoming radiation and re-emits energy in a characteristic spectrum dependent only on its temperature. This concept is vital when assuming that manufactured parts behave like ideal blackbodies in the context of a radiometer monitoring system.

• A blackbody does not reflect or transmit any radiation.
• It is a perfect emitter, meaning all energy it absorbs is re-emitted.
• The radiation emitted by a blackbody spans a range of wavelengths, defined by its temperature.

When applying this idealization to manufactured components, it simplifies the calculation of the emitted radiation. The simplified assumption using blackbody radiation allows us to apply the Stefan-Boltzmann Law effectively. Here, it ensures that the calculations remain straightforward when assessing whether the temperature exceeds certain thresholds where cooling becomes necessary.

Cooling systems

Cooling systems are essential in processes where maintaining specific temperature limits is crucial to safety and efficiency, like in manufacturing settings. In this scenario, spray heads are employed to release a mist that cools the parts if their temperature becomes dangerously high.

• These systems are automatically triggered based on temperature readings from the radiometer.
• Predefined temperature thresholds, such as \( 45^{\circ} \mathrm{C} \), are used to initiate cooling.
• Mist cooling systems are particularly effective due to the high heat absorption capacity of water and rapid evaporation.

This automated response helps prevent thermal hazards and ensures that manufactured parts remain within safe handling temperatures. The integration of monitoring and cooling systems constitutes an effective feedback loop, optimizing operational safety and product quality by preventing overheating.