A laterally insulated bar of length L = 10cm, density rho = 10.6 gm/cm^3, therma

Question

A laterally insulated bar of length L = 10cm, density rho = 10.6 gm/cm^3, thermal conductivity K = 1.04 cal/(cm s degreeC), and specific heat sigma = 0.056 cal/(gm degreeC) (this corresponds to silver, a good heat conductor) has initial temperature f(x) (degreeC) and is kept at 0 degreeC at the ends x = 0 cm and x = 10 cm. Find the temperature u(x, t) at later times if f(x) = { F middot x (degreeC), 0 less than x less than 5 (cm), 0degreeC otherwise, where F = 0.2degreeC/cm. Note: obtain the complete infinite series solution in a fully symbolic form first and only then substitute the given numerical values keeping track of the units. Make sure that your answer is dimensionally consistent. Finally, use your symbolic solution and the given values to write a computer code (in Mathematica or Matlab) and plot the solution to this problem involving n = 500 terms for time t = 0s and t = 1s.

Explanation / Answer

temperature , u(x,t) =    E/KA

                                   = (0.056/10 * 10.6 * 1.04) * x * t   + (0.056/10 * 10.6 * 1.04)2 * x2 * t2

                                                   =   5.08 * 10-4 * x * t   + 2.58 * 10-7 * x2 * t2              for (0<x<5) cm

                                    =    2.54 * 10-4 * x * t   + 1.29 * 10-7 * x2 * t2              for otherwise

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