Question

Consider transient one-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall are at specified temperatures, express the stability criterion for this problem in its simplest form.

Short answer

Answer: The stability criterion for one-dimensional transient heat conduction in a plane wall using the explicit method is \(S \leq 0.5\), where \(S\) is defined as \(S = \frac{\alpha \Delta t}{(\Delta x)^2}\) with \(\alpha\) representing the thermal diffusivity of the material, \(\Delta t\) representing the time step, and \(\Delta x\) representing the grid spacing. It is important to ensure this criterion is met to avoid numerical instability.

Step-by-step solution

Understand the explicit method

The explicit method is a numerical technique used for solving partial differential equations (PDEs), in this case, the heat conduction equation. The explicit method uses a finite difference approximation of derivatives to advance the solution in time.

The heat conduction equation

One-dimensional transient heat conduction in a plane wall can be described by the following PDE: \[ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \] Where \(T\) is the temperature, \(t\) is time, \(x\) is the spatial coordinate, and \(\alpha\) is the thermal diffusivity of the material.

Discretizing the heat conduction equation

We will now discretize the heat conduction equation for the plane wall using the explicit method, which involves replacing partial derivatives with finite difference approximations. For this exercise, we can use forward difference in time and central difference in space. This will transform the PDE into the following equation: \[ \frac{T^{n+1}_i - T^n_i}{\Delta t} = \alpha \frac{T^n_{i+1} - 2T^n_i + T^n_{i-1}}{(\Delta x)^2} \]

Rewrite the equation for stability criteria

To determine stability, we need to rewrite the equation: \[ T^{n+1}_i = T^n_i + S\left(T^n_{i+1} - 2T^n_i + T^n_{i-1}\right) \] Where \(S = \frac{\alpha \Delta t}{(\Delta x)^2}\) is the stability criteria.

Apply the stability criterion

The explicit method is stable if the stability criterion, \(S\), is less than or equal to 0.5, which can be formally expressed as: \[ S \leq 0.5 \] This is the simplest form of the stability criterion for this problem. It is essential to ensure this criterion is met while choosing the time step and grid spacing to avoid numerical instability.