Write a computer program (or use a spreadsheet) to implement...

Write a computer program (or use a spreadsheet) to implement the solution procedure of Exercise $3-134$. Allow for an arbitrary mesh size and mesh Fourier number. Explore the effect of mesh size on the temperature profile at $t=1000 \mathrm{~s}$. Also compare your results with the exact solution from COND2.

Key Concepts

Finite Difference Methods

Finite difference methods involve approximating derivatives by differences between values at discrete grid points in space and time. This technique transforms differential equations into a set of algebraic equations that can be solved numerically, making it possible to approximate solutions to problems in heat transfer and other areas of physics and engineering.

Mesh Size

Mesh size refers to the distance between adjacent grid points in the numerical domain. The choice of mesh size impacts the accuracy and computational effort of a simulation; finer meshes typically provide more accurate approximations of the true solution but at the expense of increased computational cost.

Fourier Number

The Fourier number is a dimensionless parameter defined as the ratio of heat conduction rate to heat storage rate. It is commonly used in transient heat conduction analyses to characterize the balance between diffusive transport and energy storage, and it appears in the discretization schemes governing the temporal evolution of temperature.

Numerical Stability and Convergence

Numerical stability and convergence are critical concepts in the design and analysis of computational algorithms. Stability ensures that numerical errors do not amplify uncontrollably during simulation, while convergence guarantees that the numerical solution approaches the correct result as the mesh is refined or as time steps are reduced.

Verification with Analytical Solutions

Verification with analytical solutions, or exact solutions, serves as a benchmark for assessing the accuracy of numerical methods. Comparing numerical results with known analytical results helps identify errors in implementation and confirm that the numerical method is correctly capturing the underlying physical behavior.

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