An alloy rod of length and thermal diffusivity has both ends held at for all

PHYSICS · Subjective · 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

An alloy rod of length

L=0.60\,\mathrm{m}

and thermal diffusivity

D=1.10\times10^{-4}\,\mathrm{m^2\,s^{-1}}

has both ends held at

0^\circ\mathrm{C}

for all

t>0

. At

t=0

its temperature profile is the symmetric tent

T(x,0)=\begin{cases}\dfrac{2T_0}{L}x, & 0\le x\le L/2\\[4pt] \dfrac{2T_0}{L}(L-x), & L/2\le x\le L\end{cases}

with peak temperature

T_0=90^\circ\mathrm{C}

at the midpoint. The temperature obeys the diffusion equation

\partial T/\partial t=D\,\partial^2T/\partial x^2

. (a) Write the general solution satisfying the boundary conditions as

T(x,t)=\sum_nb_n\sin(n\pi x/L)e^{-t/\tau_n}

, giving

\tau_n

in terms of

L

D

and

n

. (b) By direct integration of the Fourier sine coefficient formula, show that

b_n=0

for every even

n

and that

b_n=\dfrac{8T_0}{n^2\pi^2}\sin(n\pi/2)

for odd

n

. Evaluate

b_1

and

b_3

. (c) Find the time

t_{5\%}

at which the fundamental mode's amplitude has fallen to

5\%

of its value at

t=0

, and check that the third mode is by then utterly negligible.

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