Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
of the fluid temperature profile in Fig. 2 is due to a reduced
mass flow rate.
B. Minimal thermal resistance – minimization of J
2
The same case as in A was considered for optimizing the
thermal resistance, using objective function J . When axial
2
conduction was modeled, the following parameters were
used: heat conductivity ratio between solid and fluid
k
s
= 241.4 , ratio of total height to channel height h
t
= 1. 5 and
ratio of length to channel height L = 16. 7 . Fig. 3 shows the
results for three different optimization settings. In the first
setting, a uniform channel is optimized to allow comparison.
The second and third setting concern the optimization of a
non-uniform channel using respectively a model without and
with axial conduction. The top of the figure shows the
optimal width distribution in each of the three cases; the
bottom shows the accompanying wall temperature profiles.
Unexpectedly, the optimal non-uniform channel appears to
have two distinct parts. The first part of the channel has a
uniform width and thus an increasing wall temperature. The
second part is similar to the profile observed in subsection A:
the width is decreasing to keep the wall temperature at a
uniform level. Again the decrease in width towards the end
of the channel improves the local convective heat transfer,
thereby preventing the maximal wall temperature from
getting too high. At first sight, it seems suboptimal that this
trend does not fully continue towards the beginning of the
channel. Not all potential of pushing the wall temperature
against the maximum is being used. However would this
trend continue, the inlet width of the channel would become
very broad. This would make α max
and thus the total width
very large, thereby increasing the amount of heat that each
single channel of the cooler has to get rid of. Equation (6)
incorporates this effect.
The first part of the resulting profile therefore stems from a
trade-off between enlarging α to allow more mass flow and
reducing α to reduce the heat load of a single channel. It is
logical that this results in a part with uniform width. Indeed,
only the maximal width has an influence on the channel
density. This is similar to what happens in the second part.
Here, the trade-off is between a large α to increase mass flow
and a low α to increase the convective heat transfer.
In the last optimization setting, the effect of axial
conduction is investigated. This introduced only a minor
difference in the channel shape. At the breakpoint between
the two lines, a truncated peak is added to compensate for the
flattening of the wall temperature profile due to the axial
conduction.
Table I presents a comparison between the three obtained
shapes in terms of their thermal resistance. The table also
shows the inlet and outlet widths of the optimized channel in
each of the cases. It is obvious that using the most accurate
model including axial conduction and the largest number of
degrees of freedom (for a non-uniform channel) gives the best
result. We see however that the improvement compared to
the model without axial conduction is negligible. Therefore,
taking into account practical considerations such as
computational effort etc., leads to the conclusion that the
optimization with a model without axial conduction will lead
to sufficiently accurate results for technical implementation.
C. Parameter sensitivity analysis
The main model parameters determining the solution are
the dimensionless quantities w and χ . The influence of both
f
parameters on the optimal result was investigated by scanning
the parameter field, for both objectives J and
1
J .
2
One can see from Fig. 1 that w has an influence on the
f
density of the channels only. This characterizes the amount
of heat to be cooled by a single channel. An increase in w
f
will result in a larger total width, thereby reducing the density
of the channels and a corresponding increase in heat transfer
to each individual channel.
For objective J 1
, we observe that
w does not have any
f
, at least when the inlet
effect on the width distribution α (x)
width is wide enough to allow a uniform wall temperature
profile. It can be easily seen from (6) that w f
does not
influence the temperature gradient if this is already zero.
There is off course an effect on the actual level of the wall
temperature, which increases when w increases.
f
Fig. 3: Optimal channel shape and wall temperature profiles w.r.t.
objective J for 3 different optimization settings.
2
TABLE I
COMPARISON OF 3 SHAPES OPTIMIZED FOR THERMAL RESISTANCE
Optimization setting α0
α
e θ max (10 -3 )
s
∆ (10 -3 ) / %
Uniform channel 0.1253 0.1253 4.806 - / -
No axial conduction 0.1313 0.0979 4.511 0.295 / 6.1 %
Axial conduction 0.1319 0.0980 4.508 0.298 / 6.2 %
©EDA Publishing/THERMINIC 2009 160
ISBN: 978-2-35500-010-2