A final consideration for our thermal model is the possibility of the existence of temperature differences between the targets and base, due to high contact resistances between these bodies. This can be studied by using a separate lumped model for each heated body. In order to estimate the impact of contact resistance, we will ignore conduction losses to the chamber. These assumptions lead to the following equations:
and
where the subscript represents the target and the subscript
represents the base. Also,
is the contact area between the
target and base and
is the heat transfer coefficient between the
target and base (equivalent to the thermal conductivity of the
adhesive used to attach the target to the base divided by the adhesive
thickness). We can simplify this to the steady state problem, assume
that the chamber temperature is much less than the target or base
temperatures, assume that the target and base heat fluxes are
identical, and assume that the radiating and heated areas are the
same, giving:
and
For given area, emissivities, and values for , these equations
can be solved (numerically) for the target and base temperatures. It
is instructive, though, to consider some special cases. First, if the
emissivities of the two materials are equal, then the solution of the
above system is
. In this case, contact resistance is
irrelevant. The worst case, then, is a situation where the emissivity
of one body is 1.0, and that of the other body is zero. For
instance, if
and
, then we obtain:
and
The first of these equations gives:
This, then, is the largest temperature difference that can be supported in a PSII target/base combination. Unfortunately, this ratio can be quite large, so we must look at more realistic cases.
To further study the impact of contact resistance, we will numerically solve Equations 36 and 37 for the case where the emissivity of one body is small and the ratio of the area of that body to the contact are is large. In other words, this is a case where one body is poorly cooled. The results of this study are shown in Figure 7. In this figure, the base emissivity is taken to be 1.0, the base area is taken to be the same as the contact area, and the target area is taken to be 5 times that of the contact area. It can bee seen in this figure that for reasonable contact resistances and surface areas, the temperature difference between the target and base will generally be quite small as long as the difference in the target and base emissivities is not excessive. On the other hand, this figure also indicates that one can force the target and base temperatures to differ by significant amounts by obtaining small contact heat transfer coefficients or by using a small target emissivity and a large base emissivity.
