Question

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.

Short answer

Answer: The warning alarm should be set to detect an irradiation of \(1.048 \times 10^{-4} \mathrm{Wm^{-2}}\) or higher.

Step-by-step solution

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 \(T_{\mathrm{C}} = 45^{\circ} \mathrm{C}\): \(T_{K} = 45 + 273.15 = 318.15 \mathrm{K}\)

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 \(45^{\circ} \mathrm{C}\). \(P = 10^{-4} \times 1 \times (5.67\times10^{-8}) \times (318.15)^4\) \(P = 8.249 \times 10^{-5} \mathrm{W}\)

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 \(0.5 \mathrm{m}\). \(E = \dfrac{P}{4\pi (distance)^2}\) \(E = \dfrac{8.249 \times 10^{-5}\mathrm{W}}{4\pi (0.5)^2 \mathrm{m^2}}\) \(E = 1.048 \times 10^{-4} \mathrm{Wm^{-2}}\)

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.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in radiation heat transfer, which describes how much energy a perfect blackbody emits based on its temperature. It asserts that the power radiated per unit area of a blackbody is proportional to the fourth power of its absolute temperature. This relationship can be mathematically defined as:
\( P = Ae \sigma T^4 \)
where:

• \( P \) is the power radiated per unit area (W/m²),
• \( A \) is the surface area (m²),
• \( e \) is the emissivity of the material (for a blackbody, \( e = 1 \)),
• \( \sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \mathrm{Wm^{-2}K^{-4}}) \),
• \( T \) is the temperature in Kelvin (K).

The law is extremely useful for determining the energy emission for blackbody surfaces, such as the metal sheet described in the exercise.
In our problem, the Stefan-Boltzmann Law helps determine if the heat emitted from the sheet's surface at a particular temperature would trigger the radiometer, and hence trigger the thermal safety alarm. This powerful formula allows engineers and scientists to calculate the energy emissions required to heat management in many practical applications.

radiation heat transfer

Radiation heat transfer is one of the primary means by which energy can be transferred from one body to another without the need for a medium—unlike conduction or convection. It occurs through the emission of electromagnetic waves, primarily in the form of infrared radiation for thermal processes.
In the context of the given exercise, as the metal sheet exits the water bath, its temperature needs monitoring to ensure it has cooled sufficiently and is safe to handle.
Radiation heat transfer is modeled using Stefan-Boltzmann Law, as it determines the power being radiated based on the body's temperature. When the sheet's surface emits radiation due to its temperature, it carries energy that spreads out and can be detected by the radiometer, a device designed to measure the intensity of radiant energy.

• Key factors affecting radiation heat transfer include the temperature of the body, surface area, the distance over which the radiation travels, and the characteristics of the surface.
• In many practical applications, like in industrial settings, understanding radiation heat transfer is critical for ensuring processes remain safe and efficient.

Utilizing this principle allows precise temperature assessments, alerting operators if the sheet remains too hot.

thermal safety

Thermal safety is vital in environments where individuals handle heated objects or work with high-temperature processes. It ensures that an object's temperature remains within safe limits, preventing injuries like burns.
In the exercise, the goal is to avoid thermal burns by ensuring that the sheet's temperature remains below \(45^{\circ} \mathrm{C}\). This is accomplished using the radiometer, which monitors the radiation emitted from the sheet. The alarm is programmed to sound if the emitted radiation indicates an unsafe temperature.

• The calculation of irradiation using the Stefan-Boltzmann Law ensures that accurate monitoring can take place.
• Implementing vigilant temperature measurement systems, like radiometers, enhances safety outcomes.

By setting a warning alarm threshold, workers are alerted promptly to ensure they do not handle or approach potentially dangerous scenarios.

blackbody radiation

Blackbody radiation refers to the idealized emission of electromagnetic radiation from an object that perfectly absorbs all incoming radiation without reflecting any.
This concept is crucial in theoretical and practical heat transfer studies because it provides a benchmark for comparison. Blackbodies emit radiation most efficiently, based on their temperature, and are described by Planck's Law as well. In practical terms, most surfaces are not perfect blackbodies but can be approximated as such for calculations, especially metals in radiation studies.
In the problem scenario, the long metal sheet was treated as a blackbody to simplify calculations. By doing so, engineers could efficiently use the Stefan-Boltzmann Law to calculate the radiation emitted from the sheet's surface.

• Blackbody radiation studies aid in designing sensors and devices such as radiometers to accurately measure emitted infrared radiation.
• Understanding this concept allows for better design considerations in terms of material selection and safety plan formulation when designing systems that involve high temperatures.

These theoretical insights provide a framework for more accurate predictions in industrial applications and in safety equipment design.