Question

Engine valves \(\left(c_{p}=440 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\) and \(\left.\rho=7840 \mathrm{~kg} / \mathrm{m}^{3}\right)\) are to be heated from \(40^{\circ} \mathrm{C}\) to \(800^{\circ} \mathrm{C}\) in \(5 \mathrm{~min}\) in the heat treatment section of a valve manufacturing facility. The valves have a cylindrical stem with a diameter of \(8 \mathrm{~mm}\) and a length of \(10 \mathrm{~cm}\). The valve head and the stem may be assumed to be of equal surface area, with a total mass of \(0.0788 \mathrm{~kg}\). For a single valve, determine ( \(a\) ) the amount of heat transfer, \((b)\) the average rate of heat transfer, \((c)\) the average heat flux, and \((d)\) the number of valves that can be heat treated per day if the heating section can hold 25 valves and it is used 10 h per day.

Short answer

Answer: The average rate of heat transfer is 40.22 W, the average heat flux is 4,022 W/m², and 3000 valves can be heat treated in one day.

Step-by-step solution

Calculate the mass of the cylindrical stem and the valve head

Given that the total mass of the stem and the valve head is 0.0788 kg. Divide it by 2 to get the mass of each part. Mass of each part: \( m_{1} = m_{2} = \frac{0.0788}{2} \) \(m_{1} = m_{2} = 0.0394 \,\textup{kg}\)

Calculate the temperature difference

We are given the initial and final temperature. Initial temperature, \( T_{1} = 40^{\circ} \mathrm{C}\) Final temperature, \( T_{2} = 800^{\circ} \mathrm{C}\) Temperature difference, \(\Delta T = T_{2} - T_{1}\) <=> \(\Delta T = 760 \,\text{k}\)

Calculate the amount of heat transfer (a)

Use the formula for heat transfer, \( Q = m\, c_{p}\, \Delta T \) \( Q = (m_{1} + m_{2}) c_{p} \Delta T \) \( Q = (0.0394 + 0.0394)(440)(760)\) \( Q = 12,064.56 \,\textup{J}\)

Calculate the average rate of heat transfer (b)

We know that it takes 5 minutes to heat the valve. Thus, Heat treatment time, \(t = 5\,\textup{min} = 300\,\textup{s}\) Average rate of heat transfer is given by: \(P = \cfrac{Q}{t}\) \(P = \cfrac{12,064.56}{300}\) \(P = 40.22\,\textup{W}\)

Calculate the average heat flux (c)

The average heat flux is defined as the rate of heat transfer per unit area. First, calculate the surface area of the stem and the valve head. \(\textit{Surface area (stem)} = 2\pi r L = 2\pi \cfrac{8\times 10^{-3}}{2}(0.1)= 0.005\,\textup{m}^{2}\) Assuming the surface area of the head is equal to the stem, the total surface area is 0.01 m². Average heat flux, \(q'' = \cfrac{P}{\textit{Surface area}}\) \(q'' = \cfrac{40.22 \,\textup{W}}{0.01\,\textup{m}^{2}}\) \(q'' = 4,022\,\textup{W/m}^{2}\)

Calculate the number of valves that can be treated per day (d)

The heating section can hold 25 valves and is used for 10 hours per day. So, the total daily working time is \( 10 \,\textup{h} = 36,000\,\textup{s}\). Each valve is treated for 5 minutes or 300 seconds. The number of times the heat treatment section can process a batch of valves in a day is \(\cfrac{36,000}{300} = 120\) The total number of valves that can be treated per day is: \(\textit{Number of valves per day} = 25 \times 120\) \(\textit{Number of valves per day} = 3000\) Thus, 3000 valves can be heat treated per day.

Key concepts

Heat Flux

Heat flux is a measure of the rate of heat energy transfer through a surface. It is usually expressed in watts per square meter (\(\text{W/m}^2\)). In the case of heating engine valves, the average heat flux provides insight into how efficiently and quickly heat is being transferred across the surface area of the valve's components.
To calculate the average heat flux, you first need to know the average rate of heat transfer (in watts) and the total surface area through which this heat is being transferred. For instance, if the engine valve has a total surface area of 0.01 m², and the average rate of heat transfer is 40.22 W, the average heat flux can be determined by dividing the power by the area:
\[q'' = \frac{40.22 \,\text{W}}{0.01 \,\text{m}^2} = 4,022 \,\text{W/m}^2\]Understanding heat flux is crucial for optimizing heating processes in industrial applications, ensuring that components like engine valves reach the required temperature efficiently, without overuse of energy or time.

Heat Treatment

Heat treatment is a controlled process used in manufacturing to alter the physical, and sometimes chemical, properties of a material. It is especially significant in industries like automotive where it is used to strengthen engine components, including valves.
In heat treatment, materials like metals are subjected to cycles of heating and cooling. The main objectives include:

• Increasing the hardness
• Improving the strength
• Enhancing wear resistance
• Relieving internal stresses

In the exercise, the engine valves are heat treated by heating them from 40°C to 800°C. This final temperature is selected based on the material's specific properties and desired performance characteristics. Efficient heat treatment ensures that each part reaches a uniform temperature, minimizing the risks of deformation or structural weakness.
During the process, calculating the amount of heat required, as well as controlling the timing, ensures that the engine valves undergo consistent treatment without overuse of resources or energy. The treatment duration along with the capacity of the heating section dictates how many valves can be processed within a specific period, aiding in maintaining an efficient production schedule.

Engine Valve

Engine valves are crucial components in an engine, typically made from durable materials to withstand high pressures and temperatures. They play a key role in controlling the intake and exhaust of gases in the engine, which in turn, affects the engine's performance and efficiency.
For proper functioning, engine valves need specific treatments, like heat treatment, to endure the demanding conditions. This involves improving their mechanical properties such as toughness and thermal resistance.
The structure of an engine valve includes a cylindrical stem and a head, both contributing to the valve's mass and surface area. In the discussed exercise, the valve was modeled based on these parameters for efficient heat treatment analysis. Knowing the material properties, such as specific heat capacity (\(c_p = 440 \,\text{J}/\text{kg} \cdot \text{K}\)) and density (\(\rho = 7840 \,\text{kg}/\text{m}^3\)), allows for precise calculations of the heat required to achieve the desired thermal state.
Properly treated engine valves contribute to a well-functioning engine, offering advantages such as reduced fuel consumption, lower emissions, and increased durability, making it a significant focus in engine manufacturing.