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 subjected to specified heat flux, express the stability criterion for this problem in its simplest form.

Short answer

Define the stability criterion for a transient one-dimensional heat conduction problem in a plane wall with specified heat flux on both sides using the explicit method. The stability criterion for this problem, using the explicit method, is given by the maximum allowable time step, and it can be expressed in its simplest form as: Δt ≤ (Δx²) / (2α) where Δt is the time step, Δx is the spatial step, and α is the thermal diffusivity of the material.

Step-by-step solution

Understand the explicit method for solving heat conduction problems

In the explicit method, the temperature at a point (i, j) in the grid depends on the temperatures at points (i-1, j), (i+1, j), and (i, j-1) at the previous time step. The general equation for the explicit method is given by: T_i^{n+1} = T_i^n + \Delta t \cdot \frac{\alpha}{\Delta x^2} (T_{i+1}^n - 2T_i^n + T_{i-1}^n) where T_i^n is the temperature at a spatial grid point i and time step n, \Delta t is the time step, \Delta x is the spatial step, and \alpha is the thermal diffusivity of the material.

Find the stability criterion

The stability criterion for the explicit method is given by: Fo \leq \frac{1}{2} where Fo is the Fourier number, defined as: Fo = \frac{\alpha \Delta t}{\Delta x^2} Rearranging to solve for the maximum allowable time step, the stability criterion becomes: \Delta t \leq \frac{\Delta x^2}{2\alpha}

Simplify the stability criterion

In this case, there are specified heat fluxes on both sides of the plane wall, which implies a constant rate of heat flow. However, the stability criterion derived in Step 2 is a general one and does not account for this specific boundary condition. As both sides are subjected to specified heat flux, the behavior of temperatures within the wall remains the same and the stability criterion does not change. Thus, the simplest form of the stability criterion for this problem is: \Delta t \leq \frac{\Delta x^2}{2\alpha}