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Asian Journal of Multidimensional Research (AJMR)
https://www.tarj.in
64
AJMR
Fig. 1.
Determination of the relationship of optical-geometric parameters of a paraboloid of
revolution
2
2
;
;
2
;
2
2
2
4
;
.
2
Y
Y
U
U
tgU
tg
y
f tg
f
x
f
U
Y
f x
x
f tg
(1)
𝑟
2
= 𝑓 ⋅ (1 + 𝑡𝑔
2 𝑢
2
) ⋅ 𝑡𝑔(𝛾
0
+ ∆𝛼)
(2)
𝑟
3
= 𝑓
(1+𝑡𝑔
2𝑢
2
)⋅𝑡𝑔(𝛾+∆𝛼)
1−𝑡𝑔
2𝑢
2
+2⋅𝑡𝑔
𝑢
2
𝑡𝑔(𝛾+∆𝛼)
(3)
𝑟
4
= 𝑓
(1+𝑡𝑔
2𝑢
2
)
2
𝑡𝑔(𝛾+∆𝛼)
1−𝑡𝑔
2𝑢
2
−2⋅𝑡𝑔
𝑢
2
𝑡𝑔(𝛾+∆𝛼)
(4)
0 ÷ 𝑟
0
, 𝑟
0
÷ 𝑟
2
, 𝑟
2
÷ 𝑟
3
, 𝑟
3
÷ 𝑟
4
.
(5)
где,
- видимый угловой размер солнца;
- среднеквадратическая ошибка поверхности;
r
0
, r
2
, r
3
, r
4
- характерные радиусы пятна рассеяния на фокальной плоскости концентратора.
Where,
is the apparent angular size of the sun;
is the root mean square error of the surface;
r
0
, r
2
, r
3
, r
4
are the characteristic radius of the scattering spot on the focal plane of the
concentrator.
As you know, the main characteristic of concentrators is their degree of concentration. This value
is equal to the ratio of the area of the MCS midship to the area of the focal spot [4-6]:
)
(
)
(
2
2
4
u
tg
u
tg
x
f
tg
x
f
r
y
С
(6)
)
(
)
(
2
2
4
u
tg
u
tg
tg
r
y
С
(7)
ISSN: 2278-4853 Vol 10, Issue 9, September, 2021 Impact Factor: SJIF 2021 = 7.699
Asian Journal of Multidimensional Research (AJMR)
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AJMR
2
0
2
)
(
)
(
u
tg
u
tg
tg
С
(8)
Knowing the degree of concentration C for a paraboloid of revolution and taking into account
0
0
0
)
(
1
)
(
)
(
tg
u
tg
tg
u
tg
u
tg
we find the optimal value of the degree of
concentration.
Зная степень концентрации С для параболоида вращения и учитывая находим
оптимальное значение степени концентрации. It is found from the expression C taking the
differential with respect to tg (u) and the resulting equation is taken equal to zero, that is:
Сd(tg(u))=0;
0
)
(
)
(
))
(
(
du
u
tg
u
tg
tg
u
tg
Сd
(9)
Solution:
du
u
tg
tg
tg
u
tg
u
tg
u
tg
Сd
))
(
1
(
)
(
1
)
(
))
(
(
2
0
0
(10)
We make the replacement tg (u) = x, then: and
0
1
2
x
откуда
0
1
2
1
1
2
2
2
2
1
)
1
(
1
2
2
1
2
1
)
1
(
1
0
2
2
2
0
2
2
0
2
2
0
2
0
3
2
0
3
2
0
2
2
0
2
0
3
2
0
2
0
0
tg
x
x
tg
x
xtg
x
tg
tg
x
x
tg
x
x
xtg
x
tg
tg
x
x
x
x
tg
x
dx
x
tg
tg
x
x
Сdx
0
2
0
2
,
1
1
tg
tg
x
finally
0
при
1
)
(
)
(
0
2
tg
tg
x
u
tg
опт
(11)
tg(u
opt
)=44
0
52’ Table 1 below shows the change in the tg (u
opt
) value for different values
.
TABLE 1
DEPENDENCE OF TAN (U
OPT
) ON VARIOUS VALUES
,
0
4’
8’
12’
16’
46’
106’
136’
450’
tg(u/2)
0,418
0,4125
0,412
2
0,4118 0,411
5
0,4091 0,405
2
0,402
7
0,298
9
U, 0
44,87
44,83
44,79
44,77
44,73 44,49
44,12
43,87
22,30
tg (u)
0,837
0,8250
0,824
2
0,8236 0,822
8
0,8182 0,810
4
0,805
4
0,597
8
U/2, 0
22,43
22,41
22,39
22,38
22,36 22,24
22,06
21,98
21,15
ISSN: 2278-4853 Vol 10, Issue 9, September, 2021 Impact Factor: SJIF 2021 = 7.699
Asian Journal of Multidimensional Research (AJMR)
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AJMR
Table 1 shows that the influence on the optimal value of U
opt
at small values of the root-mean-
square error
α is insignificant and is in the range U
opt
= 44 ÷ 450 (
α up to 1.5
0
). This
corresponds to the manufacture of even folding designs of space concentrators [7].
On the basis of the optimal aperture angle, first of all, the focal length for the selected diameter
(power) of the concentrator is found. For example, the MCS of a modular or autonomous version
of a concentrator with a diameter of D = 6.3 m, we find the focal length:
𝑓 =
𝐷
4⋅𝑡𝑔(
𝑢𝑜𝑝𝑡
2
)
(12)
When the value of manufacturing accuracy
α, for example, a mosaic concentrator, where with
the optimal element size - facets, which is calculated as follows. As it was said, the most typical
for power MCS is the size of the midsection, which is determined by the value
у
of the radius of
the round-symmetric concentrator, which characterizes the power. The power of the concentrator
characterizes the main requirement from the customer.
The next important characteristic is the optimal value of the aperture angle, as indicated above.
On the basis of this, the value of the focus is determined, as well as the optimal size of the
receiving surface of the receiver.
For example, D=6,3 м
𝑓 =
𝐷
4⋅𝑡𝑔(
𝑢𝑜𝑝𝑡
2
)
.
0
2
0
1
)
(
tg
tg
u
tg
opt
or expression
)
2
(
1
)
2
(
2
2
opt
opt
opt
u
tg
u
tg
u
tg
tg
tg
tg
tg
tg
U
tg
opt
2
2
2
1
1
2
1
2
Whence from the table of values
α = 0 ÷ 2
0
and u
opt
= 44
0
87 ÷ 43
0
87 the change in U
opt
is a
small value within one degree. If the root-mean-square error is taken equal to
α = 30’, then the
value of the optimal angle:
8899
,
0
)
tg(u
и
4091
,
0
2
opt
opt
u
tg
(13)
Then
for
the
concentrator
D=6,3
mfocal
length
u
f
opt
64
,
0
4091
,
0
85
,
3
2
tg
f
x
;
85
,
3
4091
,
0
4
3
,
6
2
2
(14)
We find the optimal size of the ray-receiving surface from the expression:
ISSN: 2278-4853 Vol 10, Issue 9, September, 2021 Impact Factor: SJIF 2021 = 7.699
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