Question

A batch of 2 -cm-thick stainless steel plates $\left(k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8000 \mathrm{~kg} / \mathrm{m}^{3}\right.$, and \(\left.c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) are conveyed through a furnace to be heat treated. The plates enter the furnace at \(18^{\circ} \mathrm{C}\), and they travel a distance of \(3 \mathrm{~m}\) inside the furnace. The air temperature in the furnace is maintained at $950^{\circ} \mathrm{C}\( with a convection heat transfer coefficient of \)150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Using appropriate software, determine how the velocity of the plates affects the temperature of the plates at the end of the heat treatment. Let the velocity of the plates vary from 5 to $60 \mathrm{~mm} / \mathrm{s}$, and plot the temperature of the plates at the furnace exit as a function of the velocity.

Short answer

Answer: To determine the effect of the velocity of stainless steel plates on their temperature after heat treatment, we can calculate the final temperature of the plates for different velocities by considering the thermal resistance, heat transferred, and time spent in the furnace. After calculating the temperatures for various velocities, we can plot the results to visualize the relationship between the velocity and temperature of the plates. This will help us understand how the temperature of the plates changes with different velocities during the heat treatment process.

Step-by-step solution

Calculate the thermal resistance

The thermal resistance can be calculated using the convection heat transfer coefficient (h) and the surface area (A) of the plates. We know the thickness of the plates, and the heat transfer coefficient value is given. The area of the plates is not directly given, you will find it in terms of length as \(A = L \cdot width\), where L is the distance the plates travel in the furnace. \(h = 150 \frac{W}{m^{2} \cdot K}\) \(L = 3 m\) Use the following formula to find out the thermal resistance: \(R = \frac{1}{hA}\)

Calculate the temperature difference

The temperature difference between the furnace and the plates depends on the heat transferred through the plates over their area and the time they spend in the furnace. Since the velocity is changing from 5 to 60 mm/s, we will calculate the time taken by the plates to travel in the furnace using the following formula: \(t = \frac{L}{v}\) The heat transferred through the plates can be found using the formula: \(Q = \Delta T \cdot A \cdot h \cdot t\) The final temperature of the plates can then be calculated as: \(T_{final} = T_{initial} + \Delta T\)

Calculate the temperature of the plates for different velocities

Now we will calculate the final temperature of the plates for different velocities varying from 5 mm/s to 60 mm/s (with an increment of 5 mm/s). Repeat the calculations from steps 1 and 2 for each velocity value.

Plot the temperature of the plates as a function of velocity

Finally, plot the temperature of the plates at the furnace exit (y-axis) against the velocity of the plates (x-axis). This plot will show how the temperature of the plates changes with different velocities.