Question

Consider transient heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(80^{\circ} \mathrm{C}\) while the right surface (node 6) is subjected to a solar heat flux of \(600 \mathrm{~W} / \mathrm{m}^{2}\). The wall is initially at a uniform temperature of \(50^{\circ} \mathrm{C}\). Express the explicit finite difference formulation of the boundary nodes 0 and 6 for the case of no heat generation. Also, obtain the finite difference formulation for the total amount of heat transfer at the left boundary during the first three time steps.

Short answer

Question: Find the explicit finite difference formulations for the boundary nodes 0 (left surface) and 6 (right surface) of a plane wall during transient heat conduction with no heat generation. Also, calculate the total amount of heat transfer at the left boundary during the first three time steps using the finite difference formulation. Answer: The explicit finite difference formulations for the boundary nodes are: For left boundary (node 0): \( T^{n+1}_0 = 80^{\circ} \mathrm{C}\) For right boundary (node 6): \( T^{n+1}_6 = T^n_6 + \frac{\alpha \Delta t}{\Delta x^2} \left( T^n_6 - \frac{600 \mathrm{~W} / \mathrm{m}^{2} \cdot \Delta x}{k} - 2T^n_6 + T^n_5 \right)\) The total amount of heat transfer at the left boundary during the first three time steps can be calculated using: 1. Calculate the temperature at nodes 1 to 5 during the first three time steps using the explicit finite difference method. 2. Compute the heat flux \(Q^n = -k \frac{T^n_1 - T^n_0}{\Delta x}\) for each time step. 3. Find the total amount of heat transfer during these time steps. Note that the physical properties of the material (thermal diffusivity \(\alpha\) and thermal conductivity \(k\)) are needed for these calculations.

Step-by-step solution

1. Governing equation for transient heat conduction

The governing equation for transient heat conduction in a plane wall without any heat generation is given by the unsteady state of Fourier's Law of Heat Conduction: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\) where \(t\) is time, \(T\) is temperature, \(x\) is the position along the wall, and \(\alpha\) is the thermal diffusivity.

2. Implementing the explicit finite difference method

We'll discretize the spatial and temporal domain using the explicit finite difference method. The general finite difference approximation for this problem can be written as: \(\frac{T^{n+1}_{i}-T^n_{i}}{\Delta t} = \alpha \frac{T^n_{i+1}-2T^n_{i}+T^n_{i-1}}{\Delta x^2}\) Rearranging the terms, we have: \( T^{n+1}_{i} = T^n_{i} + \frac{\alpha \Delta t}{\Delta x^2} \left( T^n_{i+1} - 2T^n_{i} + T^n_{i-1} \right)\) where \(n\) is the time step index, and \(i\) is the node index.

3. Apply boundary and initial conditions

In this problem, we have the following boundary and initial conditions: 1. \(T^0_i=50^{\circ} \mathrm{C}\) for all nodes \(i\) (initial condition) 2. \(T^n_0=80^{\circ} \mathrm{C}\) for all time steps \(n \ge 1\) (left boundary) 3. \(-k\frac{T^n_7-T^n_6}{\Delta x}=600 \mathrm{~W} / \mathrm{m}^{2}\) for all time steps \(n \ge 1\) (right boundary) where \(k\) is the thermal conductivity of the material.

4. Finite difference formulations for boundary nodes 0 and 6

Using the boundary conditions given, we can find the explicit finite difference formulations for the boundary nodes. For left boundary (node 0): \( T^{n+1}_0 = 80^{\circ} \mathrm{C}\) For right boundary (node 6): \( T^{n+1}_6 = T^n_6 + \frac{\alpha \Delta t}{\Delta x^2} \left( T^n_7 - 2T^n_6 + T^n_5 \right)\) However, we need to express \(T^n_7\) in terms of the given boundary condition. From the boundary condition of node 6: \( T^n_7 = T^n_6 - \frac{600 \mathrm{~W} / \mathrm{m}^{2} \cdot \Delta x}{k}\) Substituting this expression into the finite difference formulation for node 6, we get: \( T^{n+1}_6 = T^n_6 + \frac{\alpha \Delta t}{\Delta x^2} \left( T^n_6 - \frac{600 \mathrm{~W} / \mathrm{m}^{2} \cdot \Delta x}{k} - 2T^n_6 + T^n_5 \right)\)

5. Calculate the total amount of heat transfer at the left boundary during the first three time steps

The total amount of heat transfer through the left boundary, during the first three time steps, can be calculated using the finite difference formulation. Let \(Q^n\) represent the heat flux at the left boundary node 0 during time step \(n\). From Fourier's law, we have: \( Q^n = -k \frac{T^n_1 - T^n_0}{\Delta x}\) Now, we need to calculate \(Q^1\), \(Q^2\), and \(Q^3\). For this, we will first find temperatures at nodes 1 to 5 during the first three time steps using the explicit finite difference method as derived earlier and then compute the heat flux for each time step. Lastly, we will find the total amount of heat transfer during these time steps. It is important to note that you will need the physical properties of the material (thermal diffusivity \(\alpha\) and thermal conductivity \(k\)) to calculate the temperature distribution and the amount of heat transfer at the left boundary.