Consider the one-dimensional rod with the lateral surface flux specified by w(x,

Question

Consider the one-dimensional rod with the lateral surface flux specified by w(x, t) per unit area per unit time. (a).Derive the PDE for the temperature in the rod: (b) if w is proportional to the difference between the surrounding temperature and the rod temperature, with a constant proportionality, determine the sign of the constant and write down the PDE for this case: (c) lf a perfect insulation condition is applied at the ends, and we look for solutions that are uniform in x, what differential equation does it reduce to? (d).What do you expect the equilibrium temperature distribution to be? Explain you answer.

Explanation / Answer

Thermal energy density e(x,t) = the amount of thermal energy per unit volume.
Heat ux (x,t) = the amount of thermal energy owing across boundaries per unit surface area per unit time.
Temperature u(x,t).
Specic heat c = the heat energy that must be supplied to a unit mass of a substance to raise its temperature one unit.
Mass density (x) = mass per unit volume.
Fourier’s Law: the heat ux is proportional to the temperature gradient

      = K0u   

Conservation of heat energy: Rate of change of heat energy in time = Heat energy owing across boundaries per unit time + Heat energy generated insider per unit time

heat energy = e(x,t)Ax.
Heat energy owing across boundaries per unit time = (x,t)A (x + x,t)A.
Heat energy owing out of the lateral sides per unit time = w(x,t)Px = [u(x,t) (x,t)]h(x)Px, w

here h(x) is a proportionality.
Then

t [e(x,t)A(x)x] = (x,t)A(x) (x + x,t)A(x + x) [u(x,t) (x,t)]h(x)Px.

Dividing it by Ax and letting x go to zero give
e t = x P/A[u(x,t) (x,t)]h(x)

Heat energy per unit mass = c(x)u(x,t)Ax.

So
e(x,t)Ax = c(x)u(x,t)Ax,
and then
e(x,t) = c(x)u(x,t). I

t then follows from Fourier’s law that
cu/ t
= /x( µK0u/x )P/A[u(x,t) (x,t)]h(x)

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