Question

How can a node on an insulated boundary be treated as an interior node in the finite difference formulation of a plane wall? Explain.

Short answer

Answer: A node on an insulated boundary can be treated as an interior node by modifying the finite difference equation at the boundary. We first use a forward difference approximation to represent the insulated boundary condition. Following this, we replace the term in the second derivative approximation equation with the result from the forward difference approximation. This results in an equation with the same format as the equation for interior nodes, allowing us to treat the insulated boundary node as an interior node in the finite difference formulation.

Step-by-step solution

1. Recap of Finite Difference Method for Plane Wall Heat Conduction

The Finite Difference Method (FDM) is a numerical technique used to solve partial differential equations by discretizing the domain into discrete nodes. In the context of heat conduction in a plane wall, we use FDM to solve the 1-dimensional steady state heat conduction equation. The governing equation for 1D steady state heat conduction is: \(\frac{d^2T}{dx^2} = 0\) where T is the temperature and x is the spatial coordinate.

2. Discretization of the Governing Equation

To apply the finite difference formulation, we need to discretize the governing equation. For a node i, we can approximate the second derivative using the central difference approximation, which results in: \(\frac{T_{i-1} - 2T_i + T_{i+1}}{(\Delta x)^2} = 0\) where \(\Delta x\) is the distance between the nodes, and \(T_{i-1}\), \(T_i\), \(T_{i+1}\) are the temperatures of nodes i-1, i, and i+1, respectively.

3. Insulated Boundary Condition

An insulated boundary condition means that there is no heat flux across the boundary. Mathematically, this means the derivative of temperature with respect to x is zero at the boundary: \(\frac{dT}{dx} = 0\) at the insulated boundary.

4. Treatment of Insulated Boundary as an Interior Node

To treat a node on an insulated boundary as an interior node, we need to modify the finite difference equation at the boundary. We can use a forward difference approximation to represent the insulated boundary condition: \(\frac{T_{i+1} - T_i}{\Delta x} = 0\) Solving for \(T_{i+1}\), we get: \(T_{i+1} = T_i\) Now, we can replace \(T_{i+1}\) in the second derivative approximation equation with \(T_i\): \(\frac{T_{i-1} - 2T_i + T_i}{(\Delta x)^2} = 0\) which simplifies to: \(\frac{T_{i-1} - T_i}{(\Delta x)^2} = 0\) Now the equation for the boundary node has the same format as the equation for the interior nodes. This allows us to treat the insulated boundary node as an interior node in the finite difference formulation of a plane wall.