As we progress further in our knowledge of transport, we will actually calculate the full temperature fields within heat transfer problems.

An alternative way to determine the heat flow is to use the
temperature profile in conjuction with Fourier's Law; *this is
particularly useful (in fact, necessary) for problems where
the heat flow varies with position!*

Suppose a given problem yields a temperature profile of the following:

$\displaystyle{T =T_2 + \beta(1-r^2) + \gamma \ln(r)}$

We could plug this into Fourier's Law in cylindrical (or spherical) corrdinates:

$\displaystyle{\frac{q}{A} = -k \frac{dT}{dr}}$

where dt/dr is given as

$\displaystyle{\frac{dT}{dr} = -2\beta r + \frac{\gamma}{r}}$

Combining these equations yields a flux of

$\displaystyle{\frac{q}{A} = k\left (2\beta r-\frac{\gamma}{r} \right )}$

or (in cyliundrical coordinates), a flow of

$\displaystyle{q = 2\pi k L \left (2\beta r^2-\gamma \right )}$

where q is clearly now a function of position.

Determine the heat flow through these solids from their temperature profiles