Question

Consider steady heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(40^{\circ} \mathrm{C}\) while the right surface (node 8 ) is subjected to a heat flux of \(3000 \mathrm{~W} / \mathrm{m}^{2}\). Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heat generation. Also obtain the finite difference formulation for the rate of heat transfer at the left boundary.

Short answer

Question: Express the finite difference formulation for the boundary nodes 0 and 8 and find the finite difference formulation for the rate of heat transfer at node 0 in a steady heat conduction problem with no heat generation inside the wall. Given that the temperature at node 0 is 40ºC and the heat flux at node 8 is 3000 W/m². Answer: The finite difference formulations for the boundary nodes are: 1. For node 0: \(\frac{\partial T}{\partial x} = \frac{T_1 - 40}{\Delta x}\) 2. For node 8: \(3000 = -k \frac{T_8 - T_7}{\Delta x}\) The finite difference formulation for the rate of heat transfer at node 0 is: \(q_0 = -k \frac{T_1 - 40}{\Delta x}\).

Step-by-step solution

1. Discretize the heat equation.

Start by discretizing the one-dimensional heat equation without heat generation using central differencing. We can write the second derivative as follows: \(\frac{\partial^2 T}{\partial x^2} = \frac{T_{i-1} - 2T_i + T_{i+1}}{\Delta x^2}\), where \(T_i\) is the temperature at node i and \(\Delta x\) is the distance between two adjacent nodes.

2. Apply boundary conditions at node 0.

At node 0, the temperature is fixed at \(40^{\circ} C\). Since we are using central differencing, we need to use a backward difference at node 0 for the finite difference formulation: \(\frac{\partial T}{\partial x} = \frac{T_1 - T_0}{\Delta x}\), where \(\Delta x\) is the distance between nodes 0 and 1.

3. Apply boundary conditions at node 8.

At node 8, the heat flux is fixed at \(3000 \, W/m^2\). The heat flux can be expressed in terms of temperature gradient, using Fourier's law of heat conduction: \(q = -k \frac{\partial T}{\partial x}\), where k is the thermal conductivity of the material. Since we are using central differencing, we need to use a forward difference at node 8 for the finite difference formulation: \(\frac{\partial T}{\partial x} = \frac{T_8 - T_7}{\Delta x}\), where \(\Delta x\) is the distance between nodes 7 and 8. Now, substitute the given heat flux into Fourier's law: \(3000 = -k \frac{T_8 - T_7}{\Delta x}\).

4. Obtain the finite difference formulation for the rate of heat transfer at node 0.

To find the rate of heat transfer at the left boundary (node 0), we can use Fourier's law of heat conduction and the previously found temperature gradient at node 0. The rate of heat transfer \(q_0\) can be expressed as follows: \(q_0 = -k \frac{T_1 - T_0}{\Delta x}\), where k is the thermal conductivity of the material and \(T_0\) is the temperature at node 0 (\(40^{\circ} C\)). So, the finite difference formulations we have obtained are: 1. For node 0: \(\frac{\partial T}{\partial x} = \frac{T_1 - 40}{\Delta x}\) 2. For node 8: \(3000 = -k \frac{T_8 - T_7}{\Delta x}\) 3. For the rate of heat transfer at node 0: \(q_0 = -k \frac{T_1 - 40}{\Delta x}\)