A large slab of 3 cm-thick Teflon at 20^∘ C is suddenly exposed to a radiative heat flux of 350

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A large slab of 3 cm-thick Teflon at 20^∘ C is suddenly...

A large slab of $3 \mathrm{~cm}$-thick Teflon at $20^{\circ} \mathrm{C}$ is suddenly exposed to a radiative heat flux of $3500 \mathrm{~W} / \mathrm{m}^{2}$ from a high temperature source on one side, and is insulated on the other. Reradiation and convection from the exposed side can be assumed to be negligible. Using the implicit method, give the finite-difference approximations for the $1 \mathrm{~cm}$ mesh shown, and perform five iterations at the first time step using Gauss-Seidel iteration. Also check the temperatures at the first time step by direct solution of the difference equations. Use Fo $=0.4$.

Key Concepts

Implicit Finite Difference Method

This numerical technique involves approximating the heat conduction equation by discretizing both time and space, where the temperature at a new time step is determined by a system of algebraic equations that incorporate unknown future temperatures. It is unconditionally stable, making it well?suited for problems with rapid transients or stiff systems, and requires solving a system of simultaneous equations at each time step.

Gauss-Seidel Iteration

This is an iterative method for solving systems of linear equations derived from discretizing partial differential equations. In this technique, the solution is updated sequentially for each node using the most recent available values, which can make convergence faster compared to methods using only previous time step data. It is particularly useful in implicit schemes where direct methods may be computationally expensive.

Mesh Discretization

Mesh discretization involves dividing the spatial domain into finite segments or nodes, creating a grid over which the differential equations are approximated by algebraic equations. This approach transforms continuous problems into discrete ones, allowing for numerical solution. The choice of mesh size affects both the accuracy and computational cost of the simulation.

Boundary Conditions

Boundary conditions provide the necessary additional information to properly solve a differential equation over a finite domain. In heat conduction, these can include specified heat fluxes, convective losses, or insulated surfaces. Correctly applying these conditions ensures that the numerical method accurately models the physical scenario at the edges of the domain.

Fourier Number

The Fourier Number is a dimensionless quantity that represents the ratio of heat conduction to thermal energy storage. Defined as ?t/L², where ? is thermal diffusivity, t is time, and L is a characteristic length, it is used to characterize the transient behavior of heat conduction problems. In numerical methods, it often appears as an important parameter that influences the stability and convergence of the solution.

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