Consider one-dimensional unsteady heat conduction in a rod with constant interna

Question

Consider one-dimensional unsteady heat conduction in a rod with constant internal thermal generation (q) and thermal conductivity (k), as sketched in the figure below: Boundary conditions: The temperature is fixed at zero at the left end (T_l = 0) and at one at the right end (T_r = 1). Initial condition: Initially, the whole rod temperature is at zero (T_initial = 0). The thermal conductivity k = 1 and the internal thermal generation q = 10. The density of the rod (rho) is 250. Its specific heat (c) is 750. The overall length of the rod is L = 1. Approximate the solution to this unsteady heat conduction problem using the finite volume method as discussed in class. Uniform grid spacing (delta x = constant) can be used. Find: Use Crank-Nicolson Method to obtain the temperature distribution along the rod at different times and plot the center temperature versus time. (0 lessthanorequalto t lessthanorequalto 150.000) Plot the temperature distribution along the rod at t = 75.000 and t = 150,000.

Explanation / Answer

In some transient problems , the internl tempertaure gradients in the body may be quite small and insignnificant . Yet the tempertaure at a given location. or the average temperature. of theobjectl,may be changing quite rappidly with time. we can note that such could be the case for large thermal diffusivity alpha.

A more meaningful approach is to consider the general problem of transient cooling of an object, such as the hollow cylinder . For very lare ri , the heat transfer rate by conduction through the cylinder wall is approximately.

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