ESTIMATES OF CONCENTRATOR TEMPERATURES
A rough estimate was made of the various temperatures in the concentrator
using a two-plate model. An important part of the analysis involves
tracing the pathways of light reflected inside the concentrator
from the mirror, the grids and other surfaces. This was done very
crudely here and should be revisited at a later date. It is assumed
that the grids have half-round wires with the rounds facing upward.
The results are:
Mirror: The mirror will reach 244°C if it faces ceramic
behind it, 206°C if it is gold plated on the back and faces
a gold plated "hatbox" wall, and temperatures in the
range 170°C down to 120°C as better radiant surfaces
are provided for the back of the mirror and the front face of
the hatbox wall.
Grids: The grids will reach around 240°C.
Target: The target temperature depends on several factors. The 15 support rods for the mirror grid provide a significant solar input to the bottom of the target. In the conventional design, if the top of the target disc is gold plated, the target temperature is 292°C with round rods and 253°C with triangular support rods. In the alternate design with the ground grid covered by a gold plated disc for its central 6 cm, the target temperature is 234°C with round rods and 160°C with triangular support rods. The most effective way to decrease the target temperature is to texture its top surface by micromachining so that its absorptance = emittance ~ 0.5. In this case, with the conventional design, the target temperature drops to 128°C with round rods.
1. Some Simple Basics
A flat disc facing the sun on one side and facing empty space
on the other side will attain a temperature T, dependent on its
properties.
We have that
a S A = e s
(2 A) T4
| a/e | 1 | 2 | 3 | 4 | 5 |
| T(K) | 331 | 394 | 436 | 468 | 495 |
A flat disc facing the sun which has a back side that faces a
heat source at the same temperature as the disc will neither gain
nor lose heat from the back face. For this disc, the factor of
2 disappears from the above equations. Thus the appropriate expression
becomes
and the table becomes:
| a/e | 0.2 | 1 | 2 | 3 | 4 | 5 |
| T (K) | 263 | 394 | 468 | 519 | 557 | 589 |
For a disc which has a back side that faces a realm with lower
temperature than the disc, but not empty space, the temperature
will be intermediate between the two cases above (with and without
the factor of 2).
Note that it is the ratio of alpha/epsilon that is important.
The problem with gold is that alpha is around 0.2 and epsilon
Is around 0.05 for a ratio around 4.
If a gold surface in the sun can be textured to make alpha = epsilon
= Say 0.5, the thermal problem disappears despite the fact that
It actually increases the solar absorption!
2. Solar irradiance pathways
The concentrator is shown in Figures 1-3. Solar rays impinge upon the concentrator and meet in serial order:
If 1 sun of solar irradiance impinges on the ground grid, 0.07
is reflected into space, and 0.93 is transmitted. Similarly, 0.86
is transmitted by the proton rejection grid, and 0.80 is transmitted
by the accelerator grid. (See Figure 4). The 20% lost in these
three grids is 80% reflected back into space and does not appear
within the concentrator. The remaining 20% of the 20% is absorbed
by the wires and re-radiated outward.
Approximately 0.8 sun of solar irradiance impinges on the mirror
grid, of which ~0.05 sun is reflected upward and 0.74 sun is transmitted.
The 0.74 sun of downward solar irradiance impinges on the stepped
parabolic mirror where 20% is absorbed and 80% is reflected upward.
Thus 0.59 sun is reflected upward. Moving upward, at each successive
grid, 93% is transmitted. Thus the irradiance on the various grids,
both from incident solar irradiance moving downward, as well as
from upward irradiance from sunlight reflected from the stepped
mirror, impinging on the grids from below, are as given in the
following table:
| Grid | from below | from above | total |
| ground grid | 0.44 sun | 1.00 | 1.44 |
| proton rejection grid | 0.48 sun | 0.93 | 1.41 |
| accelerator grid | 0.51 sun | 0.86 | 1.37 |
| mirror grid | 0.55 sun | 0.80 | 1.35 |
| microstepped mirror | 0.59 sun | 0.74 | 1.33 |
3. Temperature of the Microstepped Mirror
The present plan for the mirror is that it mount directly on a
ceramic insulator on the inside face of the concentrator enclosure
wall ("hatbox"). I would like to see the mirror mounted
on stand-offs, facing a grounded metal inside face of the concentrator
enclosure wall and I will estimate the mirror temperatures, both
ways.
3.1 Backside of Mirror facing a grounded metal inside face
of the concentrator enclosure wall
The microstepped mirror receives a solar input of 0.74 sun. In
the downward direction, it "sees" the base of the concentrator
enclosure which I assume is ~ 80°C, and in the upward direction
it "sees" 4 grids which amounts to 26% of the view,
and empty space which amounts to 74% of the view. (Actually, it
sees a moderate solid angle of the side walls of the concentrator
enclosure, which I assume to be at 80°C, and since I have
neglected this, my calculation will be an overestimate of the
mirror temperature). I calculated the temperature of the mirror,
using a simplified thermal model. In this calculation, the mirror
is represented as a flat plate facing the sun with properties
of gold (absorptivity = 0.2 and emittance = 0.05) and an incident
irradiance of 0.74 sun. The backside of the mirror is in radiant
equilibrium with the concentrator enclosure wall facing it, and
we adjust the emittance of the backside of the concentrator enclosure
wall so that the temperature of the concentrator enclosure wall
comes out to ~ 80°C. The results of the calculation depend
upon assumptions made about the backside of the mirror and the
inside face of the concentrator enclosure. If we assume that all
surfaces are gold plated polished aluminum with an emittance of
0.05, the mirror attains a temperature of 206°C. However
it may be allowable to roughen the backside of the mirror and
the inside face of the concentrator enclosure wall, or otherwise
treat them (such as anodization) to increase their emittances.
For example, if we increase the emittances of these surfaces to
0.20, the temperature of the mirror drops to 153°C. For various
combinations of emittance of the backside of the mirror and the
inside face of the concentrator enclosure wall, the mirror temperature
is as given in the following table:
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In these calculations I have neglected the (almost vertical) steps
between horizontal faces on the mirror. These steps will radiate
to the side walls of the concentrator enclosure, and add to the
cooling of the mirror. Therefore the mirror will run cooler than
I have estimated due to this effect. This effect will depend upon
the ratio of step height to step width. If this ratio is small,
then the extra cooling from this effect will be small. On the
other hand, additional solar irradiance will fall on the mirror
from light reflected from the bottoms of the grids above it and
this will tend to increase the temperature of the mirror. This
was also neglected in this calculation.
One conclusion that can be drawn is that it is is very desirable
to treat the backside of the mirror and the inside face of the
concentrator enclosure wall to increase their emittances. It is
important to treat both surfaces, not just one.
3.2 Backside of Mirror Facing a Ceramic Insulator
If the backside of the mirror faces a ceramic insulator, I will
assume that no heat transfer occurs downward off the back of the
mirror. In this case, the mirror attains a temperature of 244°C.
This is the temperature of a gold plated disc in 0.74 sun which
can only radiate in one direction.
4. Temperatures of the Grids
The grids "see" empty space above, other grids above
and below, and the mirror below (I have neglected the moderate
solid angle to the side walls of the concentrator enclosure at
80°C which means that I overestimate the grid temperatures
due to this neglect). The cross section of the wires from which
the grids are made are of great importance. A round wire will
attain absolute temperatures a factor of (2/p)1/4
= 0.893 times that of a flat sheet because there is more area
to radiate than to absorb sunlight. I will assume all of the grids
are made from half-round wires, and that all the grids have the
rounds facing upward toward the sun.
If we make the assumption that a grid "sees" only space
above and the mirror below, a simple estimate can be made of the
grid temperature for any mirror temperature, assuming each grid
receives 1.4 suns. Here, I treat the grid as a flat disk with
an emittance = 0.05 x (p/2) = 0.79.
The results are shown in the following table:
The mirror temperature has only a weak effect on the grid temperatures.
5. Temperature of the Target
We consider here two possible designs:
(i) the conventional design where the three top grids are stretched
across the central region of the concentrator to a small diameter
ceramic tube (see Figures 3 and 4) so the top of the target sees
three grids above it; and
(ii) an alternate design where the ground grid is covered by thin
gold plated disc for its innermost 6 cm of diameter, thus shielding
the top of the target disc from incident sunlight from above,
and yet leaving a significant view of space to the top of the
target disc around this gold plated disc (see Figure 5).
The solar input to the target disc from above is either 0.8 sun
in the conventional design or nil in the alternate design.
The solar input to the target disc from below is the same in both
cases, but is difficult to estimate. On the bottom (where the
diamond film is) the disc sees the following:
(a) The mirror at 150 - 200 °C, depending on the design.
Although the mirror has horizontal facets, the concentrator is
pointed 4° ahead of the sun so rays strike the surfaces of
the mirror at an angle of 4±1° to the vertical. Therefore
a crescent shaped element of the mirror can reflect sunlight onto
the target. This crescent area has a maximum width of roughly
21 cm x 2 tan(4°) = 3 cm and is 6 cm long. The area of the
crescent is roughly 10 cm2. Since the area
of the target is 28.3 cm2, this adds about
(10/28.3)(0.74 sun) = 0.26 sun solar input to the target.
(b) The mirror grid. An irradiance of 0.80 sun falls downward
on the top of the mirror grid. Reflections from the mirror grid
will add another solar input to the target. This grid has an area
of 0.07 x 1660 cm2 = 116 cm2
which is more than 4 times the area of the target. This indicates
that a significant amount of solar irradiance relative to the
target size is reflected from the mirror grid. If this reflected
light is uniformly dispersed from round wires over the hemisphere,
then the fraction that falls on the target is just the solid angle
subtended by the target divided by 2p.
The target is about 19 cm away from the mirror grid, and the solid
angle subtended by the target is about 1.2% of the total hemisphere
above the mirror grid. Thus, the solar input from reflections
off the mirror grid add an input to the target of ~ 0.012 x 116/28.3
= 0.05 sun to the target.
(c) The 15 support rods for the mirror grid, each about 42 cm long by 0.1 cm in diameter, for a total area of 15 x 42 x 0.1 = 63 sq cm. If the support rods had flat tops, they would reflect light into a small region centered on the target disc because each rod has parabolic curvature. However, if the support rods have round tops, then only a region near the top of each rod will focus onto the target disc. The width of this region can be estimated from the fact that the angle subtended by the target disc at the support rods is 17°, whose tangent is 0.3. Since the slope of a circle is
it follows that the region of x of a circle which encompasses
a 17° range in slope is about 30% of the radius. Hence about
30% of the round surface at the top of the will concentrate light
onto the target. Thus the reflected irradiance onto the target
is (0.80 sun)(63/28.3)(0.3) = 0.5 sun. If the support rods had
triangular tops, this might be greatly reduced.
Thus the total solar input to the target from below is estimated
to be roughly 0.8 sun for round support rods and 0.3 sun for triangular
support rods.
5.1 Conventional Design
The temperature of the target is determined by the fact that on
top, it receives a solar input of 0.80 sun, and it "sees"
80% into space, and 20% grids, and on the bottom it receives 0.8
sun for round support rods and 0.3 sun for triangular support
rods, and it sees mainly the mirror. I will assume that the mirror
is at 170°C and I will assume that on top the target sees
100% space (I neglect the grids).
I did not know what to use as the absorptivity and emittance of
diamond so I guessed at 0.2 and 0.2, respectively.
Using a simple model we find that if the mirror temperature is
170°C, the target is 292°C for round support rods. For
triangular support rods, the target T drops to 253°C.
Now suppose we textured the top of the target disc facing the
sun so that it had absorptivity = emittance = 0.9. Then the temperature
of the target drops to 117°C for round rods. If absorptivity
= emittance = 0.5, the target temperature is 128°C for round
rods. These show that the entire problem is solved with a textured
top to the target disc.
5.2 Alternate Design
In the alternate design, there is no solar input to the target
from on top, but the view to space is partly restricted. I will
try to take this restricted view of space into account by assigning
a front surface emittance of 0.035 instead of 0.05. For round
support rods, the target temperature is 234°C. For triangular
support rods, the target T drops to 160°C.
Now suppose we textured the top of the target disc facing the
sun so that it had absorptivity = emittance = 0.5. The target
temperature is about 40°C for round rods.
Figure 1. Perspective view of concentrator
Figure 2. Side view of concentrator
Figure 3. Close-up of target support disc showing radial grid
support rods, ceramic tube, typical attachment ring, and segments
of the three horizontal grids.
Figure 4. Solar irradiance from above onto target
Figure 5. Alternate design with gold plated disc on ground grid.