Question

Quench hardening is a mechanical process in which the ferrous metals or alloys are first heated and then quickly cooled down to improve their physical properties and avoid phase transformation. Consider a $40-\mathrm{cm} \times 20-\mathrm{cm}\( block of copper alloy \)\left(k=120 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=3.91 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ being heated uniformly until it reaches a temperature of \(800^{\circ} \mathrm{C}\). It is then suddenly immersed into the water bath maintained at \(15^{\circ} \mathrm{C}\) with $h=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ for quenching process. However, the upper surface of the metal is not submerged in the water and is exposed to air at $15^{\circ} \mathrm{C}\( with a convective heat transfer coefficient of \)10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Using an explicit finite difference formulation, calculate the temperature distribution in the copper alloy block after 10 min have elapsed using \(\Delta t=10 \mathrm{~s}\) and a uniform mesh size of \(\Delta x=\Delta y=10 \mathrm{~cm}\).

Short answer

*Answer:* The number of nodes in the x direction, Nx, is 5, while the number of nodes in the y direction, Ny, is 3.

Step-by-step solution

List the given information and calculate the number of nodes

We are given: - Block dimensions: \(40\,\text{cm}\times20\,\text{cm}\) - Copper alloy properties: \(k=120\,\frac{\mathrm{W}}{\mathrm{m}\cdot\mathrm{K}}\), \(\alpha = 3.91 \times 10^{-6} \frac{\mathrm{m}^2}{\mathrm{s}}\) - Initial temperature: \(800^{\circ}\mathrm{C}\) - Water bath temperature and convective heat transfer coefficient: \(15^{\circ}\mathrm{C}\), \(h_w =100\,\frac{\mathrm{W}}{\mathrm{m}^2\cdot\mathrm{K}}\) - Air temperature and convective heat transfer coefficient: \(15^{\circ}\mathrm{C}\), \(h_a =10\,\frac{\mathrm{W}}{\mathrm{m}^2\cdot\mathrm{K}}\) - Time step and mesh size: \(\Delta t=10\,\mathrm{s}\), \(\Delta x=\Delta y=10\,\mathrm{cm}\) We first need to find the number of nodes in the x and y directions: $$N_x = \frac{40\,\text{cm}}{10\,\text{cm}} + 1 = 5$$ $$N_y = \frac{20\,\text{cm}}{10\,\text{cm}} + 1 = 3$$

Set up the initial temperature distribution

The initial temperature of the block is \(800^{\circ}\mathrm{C}\). Represent this temperature at each node in a temperature matrix of size \(N_x\times N_y\). We will update this matrix during the quenching process.

Implement the explicit finite difference method

We can apply the following explicit finite difference formula for the heat conduction problem: $$T_{i,j}^{n+1} = T_{i,j}^n + \alpha\Delta t\left(\frac{T_{i+1,j}^n - 2T_{i,j}^n + T_{i-1,j}^n}{\Delta x^2} + \frac{T_{i,j+1}^n - 2T_{i,j}^n + T_{i,j-1}^n}{\Delta y^2}\right)$$ We will implement this formula in a for loop to iteratively update the temperature matrix for each time step and for each node.

Account for the convective boundary conditions

Since the side nodes and bottom nodes are submerged in the water, we need to consider the convective heat transfer due to the water. We can implement this boundary condition using the following formula: At side and bottom nodes only: $$T_{i,j}^{n+1} = \frac{h_w\Delta t}{k}\left(T_{\text{water}} - T_{i,j}^{n}\right) + T_{i,j}^n$$ For the top nodes, we need to consider the convective heat transfer due to the air. We can implement this boundary condition using the following formula: At top nodes only: $$T_{i,j}^{n+1} = \frac{h_a\Delta t}{k}\left(T_{\text{air}} - T_{i,j}^{n}\right) + T_{i,j}^n$$

Simulate the quenching process for 10 minutes

Now that we have the explicit finite difference method and the convective boundary conditions implemented, we can simulate the quenching process for 10 minutes using the given time step of 10 seconds: Number of time steps in 10 min: $$\frac{10 \times 60\,\text{s}}{10\,\text{s}} = 60$$ Iterating over each time step and updating the temperature matrix, we can obtain the final temperature distribution of the copper alloy block after 10 minutes of quench hardening.

Report the final temperature distribution

After simulating the quenching process, we can report the final temperature distribution in the copper alloy block after 10 minutes have elapsed by presenting the updated temperature matrix.