A copper rod of length has thermal diffusivity

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

A copper rod of length L=1.2 mL = 1.2\,\mathrm{m} has thermal diffusivity D=1.11×10−4 m2 s−1D = 1.11\times10^{-4}\,\mathrm{m^2\,s^{-1}}. Both ends are held at 0∘C{0}{}^\circ\mathrm{C} for all t≥0t \ge 0. Lateral heat loss is ignored, so the rod obeys ∂u/∂t=D ∂2u/∂x2\partial u/\partial t = D\,\partial^2u/\partial x^2. At t=0t=0 a heater has raised the middle third of the rod, from x=L/3x=L/3 to x=2L/3x=2L/3, to a uniform T0=80∘CT_0 = {80}{}^\circ\mathrm{C}, while the rest of the rod is at 0∘C{0}{}^\circ\mathrm{C}. ((a)) Using the half-range sine expansion of u(x,0)u(x,0), show that only odd nn contribute and find b1b_1, b3b_3 and b5b_5, the coefficients of sin⁡(πx/L)\sin(\pi x/L), sin⁡(3πx/L)\sin(3\pi x/L), sin⁡(5πx/L)\sin(5\pi x/L). ((b)) Write the full separated solution u(x,t)u(x,t) as a sum over odd nn, stating the decay time constant τn\tau_n of each mode. ((c)) Evaluate τ1\tau_1, τ3\tau_3 and τ5\tau_5 numerically. ((d)) Using only the three terms found in (a), estimate the temperature at the centre of the rod, x=L/2x=L/2, at t=200 st=200\,\mathrm{s}.

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