Two lagged bars

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 The classical partial differential equations of physics and their boundary conditions

Problem

Two lagged bars, each L=0.30 mL = 0.30\,\mathrm{m} long, start at 20 ∘C20\,{}^\circ\mathrm{C} throughout. One is aluminium, with diffusivity DAl=9.7×10−5 m2 s−1D_{\mathrm{Al}} = 9.7\times10^{-5}\,\mathrm{m^2\,s^{-1}}, and one is stainless steel, with DSt=4.0×10−6 m2 s−1D_{\mathrm{St}} = 4.0\times10^{-6}\,\mathrm{m^2\,s^{-1}}. From t=0t=0 the end x=0x=0 of each bar is held at 60 ∘C60\,{}^\circ\mathrm{C} and the end x=Lx=L at 20 ∘C20\,{}^\circ\mathrm{C}. ((a)) Find the steady temperature profile T(x)T(x) that each bar approaches, and explain why it is the same for both. ((b)) Evaluate the time scale L2/DL^2/D for each bar, and explain why the material does matter here.

Browse the problem library