|Thermodynamics and Propulsion|
With reference to Figure 18.1(b), a steady-state energy balance yields an equation for the heat flux, :
There is a change in heat flux with due to the presence of the heat sources. The equation for the temperature is
Equation (18.1) can be integrated once,
and again to give
where and are constants of integration. The boundary conditions imposed are . Substituting these into Equation (18.2) gives and . The temperature distribution is thus
Writing (18.3) in a normalized, non-dimensional fashion gives a form that exhibits in a more useful manner the way in which the different parameters enter the problem:
18.2. It is symmetric about the mid-plane at , with half the energy due to the sources exiting the slab on each side.
The heat flux at the side of the slab, , can be found by differentiating the temperature distribution and evaluating at :
This is half of the total heat generated within the slab. The magnitude of the heat flux is the same at , although the direction is opposite.
A related problem would be one in which there were heat flux (rather than temperature) boundary conditions at and , so that is not known. We again determine the maximum temperature. At and , the heat flux and temperature are continuous so
Referring to the temperature distribution of Equation (18.2) gives for the two terms in Equation (18.5),
Evaluating (18.6) at and allows determination of the two constants and . This is left as an exercise for the reader.
For an electric heated strip embedded between two layers, what would the temperature distribution be if the two side temperatures were not equal? (MP 18.1)