1. Consider a long, solid cylindrical rod with radius R and thermal diffusivity a = k/pcp....

 

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1. Consider a long, solid cylindrical rod with radius R and thermal diffusivity a = k/pcp. Initially, the rod is at a uniform temperature To. At time t = 0, the rod is exposed to an environment that causes the rod's surface temperature to be raised to and maintained at + . From the symmetry of the problem, it is clear that the temperature T will depend on the time t and the radial distance r from the rod axis, i.e. T = T(r, t) where r is the radial distance in the context of cylindrical coordinates. (a) Give the governing equation (a PDE), the initial condition, and the two boundary conditions for T(r, t). Using separation of variables, we write T(r, t) = A(r) B (t) (b) What is the ODE for B (t)? What is the solution to this ODE? Explain all steps. (c) What is the ODE for A(r)? (You do not need to find the solution to this ODE.) 1. Consider a long, solid cylindrical rod with radius R and thermal diffusivity a = k/pcp. Initially, the rod is at a uniform temperature To. At time t = 0, the rod is exposed to an environment that causes the rod's surface temperature to be raised to and maintained at + . From the symmetry of the problem, it is clear that the temperature T will depend on the time t and the radial distance r from the rod axis, i.e. T = T(r, t) where r is the radial distance in the context of cylindrical coordinates. (a) Give the governing equation (a PDE), the initial condition, and the two boundary conditions for T(r, t). Using separation of variables, we write T(r, t) = A(r) B (t) (b) What is the ODE for B (t)? What is the solution to this ODE? Explain all steps. (c) What is the ODE for A(r)? (You do not need to find the solution to this ODE.)

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