Because the PSII process features very large heat fluxes which are
applied to the target and base plate over a very short period of time,
a specific concern is the peak target surface temperature at the end
of the (typically 10 s) pulse. Because of the short duration of
this pulse, this situation can be modeled as a semi-infinite,
1-dimensional solid with a uniform surface heat flux. The governing
equation then becomes:
where is the distance from the target surface. The boundary condition for this calculation is
where is the applied surface heating, which is assumed to be
known,
is the heat flux lost as radiation, and
is the
heat flux lost due to conduction.
Due to the radiation term, this equation is nonlinear and no analytical solution exists except for special cases. We can simplify this equation by eliminating the conduction and radiation terms, thus allowing an analytical solution. This will provide an upper bound for the surface temperature achieved during a single pulse. The boundary condition then becomes
The solution for this problem is [4]
The surface temperature is obtained by setting , giving:
This result is plotted in Figure 3, which shows the
surface temperature as a function of the applied heat flux for three
different materials: 304 stainless steel, Ti-6Al-4V, and silicon.
These results were achieved using the material properties shown in
Table 1 and assuming a 20 s pulse width. Note that
the material properties used are room-temperature properties. They
will vary somewhat with temperature, but this has not been accounted
for in this study.
Typical heat fluxes in the PSII chamber at the University of Wisconsin
- Madison are on the order of 10 MW/m. As seen in
Figure 3, this would produce temperature rises of
less than 12
C for each of the materials shown.
To estimate the effect of radiation in this analysis, we can compare
the radiative heat flux to the applied heat flux. For a target
temperature of 600 C (873 K) and an emissivity of 1.0, the heat flux
due to radiation to a body at 0 K would be 0.03 . This is
much less than the heat flux during a typical PSII pulse.
Furthermore, the effect of radiation would be even smaller for more
realistic conditions (lower target temperature, lower emissivity,
larger sink temperature). Hence, radiation can be ignored on these
time scales (tens of microseconds), and Figure 3 is
an accurate representation of the target surface temperatures to be
expected during a PSII pulse.