Question

Long cylindrical steel rods \(\left(\rho=7833 \mathrm{kg} / \mathrm{m}^{3}\) and \right. \(\left.c_{p}=0.465 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) of \(8-\mathrm{cm}\) diameter are heat-treated by drawing them at a velocity of \(2 \mathrm{m} / \mathrm{min}\) through an oven maintained at \(900^{\circ} \mathrm{C}\). If the rods enter the oven at \(30^{\circ} \mathrm{C}\) and leave at a mean temperature of \(700^{\circ} \mathrm{C},\) determine the rate of heat transfer to the rods in the oven.

Short answer

Answer: The rate of heat transfer to the steel rods in the oven is approximately 4,256.82 kJ/s.

Step-by-step solution

Calculate the cross-sectional area of the rod

To find the cross-sectional area (\(A\)) of the rod, we can use the formula for the area of a circle, which is \(A = \pi r^2\), where \(r\) is the radius of the rod. Given the diameter of the rod is 8 cm, we can find the radius: \(r = \frac{d}{2} = \frac{8\,\text{cm}}{2}=4\,\text{cm}\) Now convert the radius to meters: \(r = 4\,\text{cm} \cdot\frac{1\,\text{m}}{100\,\text{cm}}=0.04\,\text{m}\) Now we can calculate the cross-sectional area: \(A=\pi r^2=\pi (0.04\,\text{m})^2=0.005027\,\text{m}^2\)

Determine the mass flow rate of the rod

To calculate the mass flow rate (\(\dot{m}\)), we will use the relation: \(\dot{m}=\rho \cdot V \cdot A\), where \(V\) is the velocity of the rods through the oven. First, convert the velocity from meters per minute to meters per second: \(V=2\frac{\text{m}}{\text{min}}\cdot \frac{1\,\text{min}}{60\,\text{s}}=\frac{1}{30}\,\text{m/s}\) Now calculate the mass flow rate: \(\dot{m}=\rho \cdot V \cdot A=7833\frac{\text{kg}}{\text{m}^3} \cdot \frac{1}{30}\frac{\text{m}}{\text{s}} \cdot 0.005027\,\text{m}^2= 13.086\frac{\text{kg}}{\text{s}}\)

Calculate the change in temperature of the rod

The change in temperature (\(\Delta T\)) can be calculated by finding the difference between the final temperature and the initial temperature of the rod: \(\Delta T = T_\text{final} - T_\text{initial} = 700^{\circ}\text{C} - 30^{\circ}\text{C} = 670^{\circ}\text{C}\)

Determine the heat transfer rate

Now we calculate the rate of heat transfer (\(\dot{Q}\)) using the specific heat equation: \(\dot{Q}=\dot{m} \cdot c_p \cdot \Delta T\) Substituting the known values: \(\dot{Q}=13.086 \frac{\text{kg}}{\text{s}}\cdot 0.465\frac{\text{kJ}}{\text{kg}\cdot^{\circ}\text{C}}\cdot 670^{\circ}\text{C}=4,256.82\frac{\text{kJ}}{\text{s}}\) Thus, the rate of heat transfer to the rods in the oven is approximately \(4,256.82\) kJ/s.

Key concepts

Mass Flow Rate

Understanding the concept of mass flow rate is essential for a variety of engineering applications, including the heat-treatment process of cylindrical steel rods as described in our exercise. The mass flow rate, denoted as \(\dot{m}\), represents the quantity of mass passing through a given surface per unit time. It's a measure of the 'mass throughput' or 'mass flux'.

To calculate the mass flow rate, one would typically employ the formula \(\dot{m}=\rho \cdot V \cdot A\), where:\

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• \(\rho\) is the density of the material (in \(\text{kg}/\text{m}^3\)).
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• \(V\) is the velocity at which the material moves through a cross-section (in \(\text{m}/\text{s}\)).
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• \(A\) is the cross-sectional area through which the material flows (in \(\text{m}^2\)).
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\In the context of our exercise, the steel rods have a density of \(7833 \text{kg}/\text{m}^3\) and are moving at a velocity of \(2 \text{m}/\text{min}\), which we convert to \(\text{m}/\text{s}\) for consistency with the SI unit system. The mass flow rate is imperative for calculating the total heat transfer, as it indicates how much steel is being heated per second, directly affecting the thermal energy absorption rate.

Specific Heat Capacity

The specific heat capacity, usually symbolized as \(c_p\), is another vital concept in thermodynamics. It is the amount of energy required to raise the temperature of one kilogram of a substance by one degree Celsius (or Kelvin). Specific heat capacity is a property intrinsic to the material and significant in understanding how different materials respond to heat.\
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In our exercise, the specific heat capacity of steel is given as \(0.465 \text{kJ}/\text{kg}\cdot^\circ\text{C}\). This value indicates how much heat energy the steel rods need to absorb to increase their temperature, which is crucial in determining the total energy required to attain the desired heat-treatment temperature.\
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By relating this to the mass flow rate, we can infer that a certain mass of steel flowing through the oven every second will require a specific amount of heat energy (derived from the product of mass flow rate, specific heat capacity, and temperature change) to reach from the initial temperature of \(30^\circ\text{C}\) to the final temperature of \(700^\circ\text{C}\). This relationship is the cornerstone of the process in calculating the heat transfer rate and designing efficient heating systems.

Thermodynamics

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy. In the realm of thermodynamics, we look at how heat energy transfers from one system to another and how this energy can be transformed into different types of work. The exercise we are analyzing involves the application of thermodynamic principles to calculate the rate of heat transfer.\
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Key concepts such as temperature change (\(\Delta T\)), heat transfer rate (\(\dot{Q}\)), and heat capacity play pivotal roles in the analysis of the energy exchange within heating systems like our steel rod oven. In this scenario, thermodynamics informs how heat from the oven is absorbed by the steel rods and how the rods' temperature increases as a result.\
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The final calculation of the heat transfer rate to the rods requires a combination of the mass flow rate, specific heat capacity, and the temperature differential, showcasing a practical application of thermodynamics in an industrial process. Such calculations are essential in designing and optimizing heating systems to ensure the efficient use of energy and the required outcome of the heat-treatment process.