Question

A long metal bar \(\left(c_{p}=450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=\right.\) \(7900 \mathrm{~kg} / \mathrm{m}^{3}\) ) is being conveyed through a water bath to be quenched. The metal bar has a cross section of \(30 \mathrm{~mm} \times 15 \mathrm{~mm}\), and it enters the water bath at \(700^{\circ} \mathrm{C}\). During the quenching process, \(500 \mathrm{~kW}\) of heat is released from the bar in the water bath. In order to prevent thermal burn on people handling the metal bar, 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 \(1 \mathrm{~m}\) from the bar to monitor the exit temperature. The radiometer receives radiation from a target area of \(1 \mathrm{~cm}^{2}\) of the bar surface. Irradiation signal detected by the radiometer is used to control the speed of the bar being conveyed through the water bath so that the exit temperature is safe for handing. If the radiometer detects an irradiation of \(0.015 \mathrm{~W} / \mathrm{m}^{2}\), determine the speed of the bar bar can be approximated as a blackbody.

Short answer

Based on the provided information and calculations, the metal bar should be conveyed at a speed of 0.206 m/s to ensure it reaches a safe handling temperature below 45°C when it exits the water bath.

Step-by-step solution

Calculate the corresponding temperature using blackbody radiation

We will use the Stefan-Boltzmann Law to determine the temperature of the metal bar corresponding to the given irradiation. The Stefan-Boltzmann Law states that \(E = \sigma T^4\) where E is the emissive power, \(\sigma\) is the Stefan-Boltzmann constant (5.67 x 10^{-8} W/m²K⁴), and T is the temperature in K. We can rearange this formula to solve for T so that we can find the temperature corresponding to the given irradiation: \(T = \sqrt[4]{\frac{E}{\sigma}}\) Given irradiation is \(0.015 \mathrm{~W} / \mathrm{m}^{2}\) From that we can find the emissive power per unit area E as \(0.015 \mathrm{~W}/1 \mathrm{~cm}^{2} = 0.015 \times 10^{4} \mathrm{~W}/\mathrm{m}^{2}\) Now we calculate the corresponding temperature using the Stefan-Boltzmann Law: \(T = \sqrt[4]{\frac{0.015\times10^4}{5.67 \times 10^{-8}}} = 317.45 \mathrm{~K}\)

Calculate the heat transfer from the metal bar surface

We know that during the quenching process, 500 kW of heat is released from the bar. The cross-sectional area of the bar is 30 mm x 15 mm = \(450 \mathrm{~mm}^{2}\) or \(450 \times 10^{-6} \mathrm{~m}^{2}\). The total surface area of the bar taking part in the heat transfer is thus: \(A = 2\times\)(30 x 15) \(mm^2 = 900 \times 10^{-6} \mathrm{~m}^{2}\) The heat transfer per unit area is: \(q = \frac{500}{900 \times 10^{-6}} = 555555.56 \mathrm{~W/m}^{2}\)

Calculate the time needed to cool the metal bar to the safe temperature

To calculate the time needed to cool the metal bar from 700°C to the safe temperature (45°C), we use the heat transfer formula: \(Q = mc_p\Delta T\) Rearrange for the time, and use the heat transfer per unit area: \(t = \frac{mc_p\Delta T}{qA}\) The mass of the metal bar per meter \(m = Volume \times \rho = (450 \times 10^{-6}) \times 7900 = 3.555 \mathrm{~kg/m}\) The temperature difference is: \(\Delta T = T_{initial} - T_{final} = 700 - (317.45-273.15) = 245.7 \mathrm{~K}\) Substituting all the values: \(t = \frac{3.555 \times 450 \times 245.7}{555555.56 \times 900 \times 10^{-6}} = 4.842 \mathrm{~s}\)

Calculate the speed of the metal bar

The final step is to calculate the speed of the metal bar, knowing the distance traveled (1 meter) and the time needed to cool down: \(v = \frac{d}{t} = \frac{1}{4.842} = 0.206 \mathrm{~m/s}\) So, the metal bar should be conveyed at a speed of 0.206 m/s to ensure it reaches a safe handling temperature below 45°C when it exits the water bath.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a crucial principle in understanding thermal radiation, particularly when dealing with objects that can be approximated as blackbodies. In essence, this law allows us to calculate the power radiated from an object in the form of thermal radiation. According to this law, the total energy radiated per unit surface area of a blackbody is directly proportional to the fourth power of the blackbody's temperature. The mathematical representation is given by:
\[\begin{equation}E = \text{\(\sigma T^4\)}\end{equation}\]
where \text{\(E\)} is the emissive power per unit area, \text{\(\sigma\)} is the Stefan-Boltzmann constant, and \text{\(T\)} is the absolute temperature in kelvins. The Stefan-Boltzmann constant, \text{\(\sigma\)}, has a value of approximately \text{\(5.67 x 10^{-8} W/m^2K^4\)}.
In the context of the exercise, the Stefan-Boltzmann Law was employed to back-calculate the temperature of the metal bar given the emissivity detected by the radiometer, a vital step in ensuring the safe handling of the metal upon exiting the water bath during the quenching process.

Thermal Radiation

Thermal radiation is a form of heat transfer that occurs through the emission of electromagnetic waves. All objects, depending on their temperature, emit radiation across a spectrum of wavelengths. However, an idealized object known as a 'blackbody' is a perfect emitter and absorber of thermal radiation.
Thermal radiation is significant in the quenching process, as the metal bar radiates energy to its cooler surroundings. This transfer of energy, detectable as irradiation by the radiometer, is quantifiable and allows us to monitor the temperature of the object without making direct contact. Understanding the principles of thermal radiation is essential for interpreting the radiometer readings and controlling the quenching process to avoid burns or overheating.

Cooling Rate Calculation

The cooling rate is instrumental when determining the time required for an object to reach a safe temperature, as seen in the quenching of a metal bar. To calculate the cooling rate, we employ an understanding of heat transfer principles and thermodynamics. In a practical application, given the mass of the object, its specific heat capacity, and the initial and final temperatures, we can estimate the heat absorbed or released during the cooling process using the equation:
\[\begin{equation}Q = mc_p\text{\(\Delta T\)}\end{equation}\]
where \text{\(Q\)} is the heat transfer, \text{\(m\)} is the mass of the object, \text{\(c_p\)} is the specific heat capacity, and \text{\(\Delta T\)} is the change in temperature. By relating this heat transfer to the rate of energy loss per unit area, we can determine the time it takes to cool the object to the desired temperature. In the given exercise, this calculation was essential to establish the required speed of the metal bar through the water bath to ensure it exits at a safe handling temperature.

Conduction Heat Transfer

Conduction heat transfer is the process by which heat is transferred within a body or between two bodies in direct contact, due to a temperature gradient. In solids, particularly metals like the one in the exercise, conduction occurs as the high-energy particles within the material collide with lower energy particles, transferring energy in the form of heat. The conduction heat transfer rate can be described by Fourier's law, which states:
\[\begin{equation}q = -k\text{\(\frac{\partial T}{\partial x}\)}\end{equation}\]
where \text{\(q\)} is the heat transfer per unit area, \text{\(k\)} is the thermal conductivity of the material, and \text{\(\frac{\partial T}{\partial x}\)} represents the temperature gradient within the material. Although the primary mode of heat transfer in quenching is through convection with the water and radiation to the surroundings, conduction through the bar's material dictates how quickly heat is distributed within the bar. Understanding conduction is essential for predicting temperature variations within the bar and ensuring uniform cooling.