Solution to Homework Problem 5a

(3.1)

If Qgen is a constant then we can integrate twice to get T(z).

(3.1a)

The boundary conditions are given by equation (3.9)

(3.9)
(3.1b)
           At z = 0            T0 = c2            so that
(3.1c)
At z = L
(3.1d)
Thus
(3.1e)

The T vs z plot can be made dimensionless. Select
(3.1f)
(3.1g)
and insert these variables into equation (3.1e)

(3.1h)
(3.1i)
(3.1j)

We see that there is one parameter in the solution which must represent the ratio of the heat generation to conduction terms of the ODE Equation (3.1). The parameter, P, is
(3.1k)

Since Qgen is the heat generation per unit volume, Qgen Ac L is the heat generation in the rod and k Ac(TL - T0) / L represents the heat conduction in the rod.

Now since Qgen is a constant and
(3.1l)

Thus
(3.1m)
(3.1n)
and
(3.1o)

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