Question

Long aluminum wires of diameter \(5 \mathrm{mm}(\rho=2702\) \(\left.\mathrm{kg} / \mathrm{m}^{3} \text { and } c_{p}=0.896 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) are extruded at a temperature of \(350^{\circ} \mathrm{C}\) and are cooled to \(50^{\circ} \mathrm{C}\) in atmospheric air at \(25^{\circ} \mathrm{C}\) If the wire is extruded at a velocity of \(8 \mathrm{m} / \mathrm{min}\), determine the rate of heat transfer from the wire to the extrusion room.

Short answer

Question: Calculate the rate of heat transfer from a long aluminum wire being cooled from \(350^{\circ}\mathrm{C}\) to \(50^{\circ}\mathrm{C}\) in atmospheric air at a constant extrusion speed. The properties of aluminum are as follows: density (\(\rho\)) = 2702 kg/m³, specific heat capacity (\(c_p\)) = 0.896 kJ/kg·°C, diameter of the wire = 5 mm, and extrusion speed = 8 m/min. Answer: The rate of heat transfer from the aluminum wire is approximately \(95,424\,\mathrm{W}\).

Step-by-step solution

Calculate the mass flow rate of aluminum

We first need to calculate the mass flow rate (m_dot) of the aluminum wire, which is the mass of aluminum extruded per unit time. We can do this using the following formula: \(m_{\text{dot}} = \rho \cdot A \cdot v\), where \(\rho\) is the aluminum density, \(A\) is the area of the cross-section of the wire, and \(v\) is the extrusion speed. Given the diameter of the wire as \(5\mathrm{mm}\), we can calculate the area of the wire as follows: \(A = \dfrac{\pi d^2}{4}\), where \(d\) is the diameter of the wire. \(d = 5\mathrm{mm} = 0.005\mathrm{m}\) \(A = \dfrac{\pi (0.005)^2}{4} = \dfrac{\pi}{400} \mathrm{m}^2\) Now we convert the extrusion speed from m/min to m/s: \(v = 8 \mathrm{m/min} = \dfrac{8}{60} \mathrm{m/s} = \dfrac{2}{15}\mathrm{m/s}\) We can now find the mass flow rate of aluminum: \(m_{\text{dot}} = \rho \cdot A \cdot v = 2702 \cdot \dfrac{\pi}{400} \cdot \dfrac{2}{15}\mathrm{kg/s}\approx 0.355\mathrm{kg/s}\)

Calculate the temperature difference

Now we need to determine the difference in temperature between the initial and final states of the aluminum wire: \(\Delta T = T_{\text{initial}} - T_{\text{final}} = 350^{\circ}\mathrm{C} - 50^{\circ}\mathrm{C} = 300^{\circ}\mathrm{C}\)

Determine the heat capacity of aluminum

We are given the specific heat capacity of aluminum, \(c_p = 0.896\,\mathrm{kJ/kg}\cdot^{\circ}\mathrm{C}\), which is the amount of heat energy required to raise the temperature of one kilogram of aluminum by one degree Celsius.

Calculate the rate of heat transfer

We can now calculate the rate of heat transfer (Q_dot) using the formula: \(Q_{\text{dot}} = m_{\text{dot}} \cdot c_p \cdot \Delta T\) First, we have to convert the specific heat capacity from kJ to J: \(c_p = 0.896\,\mathrm{kJ/kg}\cdot^{\circ}\mathrm{C} = 896\,\mathrm{J/kg}\cdot^{\circ}\mathrm{C}\) Now we can calculate the rate of heat transfer: \(Q_{\text{dot}} = m_{\text{dot}} \cdot c_p \cdot \Delta T = 0.355\mathrm{kg/s} \cdot 896\,\mathrm{J/kg}\cdot^{\circ}\mathrm{C} \cdot 300^{\circ}\mathrm{C} \approx 95,424\,\mathrm{W}\) So, the rate of heat transfer from the aluminum wire to the extrusion room is approximately \(95,424\,\mathrm{W}\).

Key concepts

Mass Flow Rate

Understanding the concept of mass flow rate is crucial when dealing with heat transfer in various engineering applications, including manufacturing processes like extrusion. The mass flow rate, often denoted as \(\dot{m}\), is a measure of the amount of mass passing through a given surface per unit time. Think of it like the flow of cars on a highway; the mass flow rate would be analogous to the number of cars passing a toll booth every hour.

When calculating the mass flow rate, the formula \(\dot{m} = \rho \cdot A \cdot v\) is used, where \(\rho\) is the material density, \(A\) is the cross-sectional area through which the material moves, and \(v\) is the velocity at which it is moving. In our exercise, we used the mass flow rate to determine how much aluminum is continuously being cooled as it is extruded. This value is fundamental because it directly influences the total amount of heat exchanged between the aluminum wire and its surroundings.

Specific Heat Capacity

Specific heat capacity, often abbreviated as \(c_p\), is an intrinsic property of a material that indicates how much heat energy is required to raise the temperature of one kilogram of the material by one degree Celsius (or Kelvin). The higher the specific heat capacity, the more energy it can hold, making it something like the thermal 'storage capacity' of the substance.

In our example involving aluminum, the specific heat capacity \(c_p = 0.896 \mathrm{kJ/kg}\cdot^{\circ}\mathrm{C}\) means that to increase the temperature of one kilogram of aluminum by one degree Celsius, 0.896 kilojoules of heat energy are required. This property is vital for our heat transfer calculations because it allows us to quantify the energy associated with temperature changes in the extrusion process.

Temperature Difference

The temperature difference, denoted as \(\Delta T\), is simply the change in temperature that a substance undergoes. It plays a pivotal role in heat transfer calculations because the heat exchanged between two bodies or within a system is directly proportional to the difference in temperature between the initial and final states of the substance.

In the case of the aluminum extrusion process, we identified the temperature difference by subtracting the final cooling temperature from the initial extrusion temperature, resulting in \(\Delta T = 300^{\circ}\mathrm{C}\). This large temperature difference is responsible for driving the heat transfer, as heat moves from a hotter object (the wire) to a colder environment (the air). The greater the temperature difference, the greater the potential for heat transfer, following the basic principle that heat flows from warmer to cooler regions.

Extrusion Process Thermodynamics

The extrusion process thermodynamics involves complex heat transfer phenomena that must be managed to ensure product quality and process efficiency. Thermodynamics in the context of extrusion refers to the study of energy conversion, particularly the transfer of heat energy, during the shaping of materials. In extrusion, a material is forced through a die under controlled temperature conditions to create objects of a fixed cross-sectional profile.

During this process, the material often undergoes significant temperature changes, which can impact its physical properties and the energy required to shape it. Efficient cooling is necessary to manage these temperature changes and can be achieved through convection, conduction, or radiation. Calculating the rate of heat transfer, as we've done in our exercise, provides insights into the cooling requirements and helps maintain the material properties, ensuring the extrusion process is energy-efficient and produces high-quality components.