Static and quasi-static concentrators present interesting characteristics for obtaining photovoltaic solar energy. In this work we study the characteristics of the crossed compound parabolic concentrator, formed by the intersection of two cylindrical compound parabolic concentrators (CPC). Bifacial cells are used in this concentrator as a requirement for obtaining higher concentrations. Static and quasi-static concentrators see the sun as an extended source, so a simplified source model of radiance for the sky of Madrid is used. The figures of merit of a lossless concentrator are studied and the most important parameters influencing its optical behavior are discussed. We conclude that these concentrators obtain results that lead to a decrease in the cost of photovoltaic energy.
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Geometrical Characteristics for the Static Crossed Concentrators with ϕq = 30° and ϕp = 62.778°
Case
L (cm)
h (cm)
W (cm)
fp (%)
fq (%)
Cg
1a
L
0.75L
L/6
54
56
7.29
2a
L
0.75L
L/8
54
41.6
7.81
3a
L
0.75L
L/10
54
27
8.13
1b
L
L
L/6
39
42
8.17
2b
L
L
L/8
39
22.2
8.61
3b
L
L
L/10
39
2.7
8.75
1c
L
1.25L
L/6
23
29
8.8
2c
L
1.25L
L/8
23
4
9.1
3c
L
1.25L
L/10
23
Not possible
Table II
Geometrical Characteristics for Quasi-Static Concentrators with Two Seasonal Adjustments for ϕq = 15° and ϕp = 60°
Case
L (cm)
h (cm)
W (cm)
fp (%)
fq (%)
Cg
1a
L
0.75L
L/6
56.6
87.2
9.03
2a
L
0.75L
L/8
56.6
83
10.16
3a
L
0.75L
L/10
56.6
78.7
11.1
1b
L
L
L/6
42.1
83
10.7
2b
L
L
L/8
42.1
77.3
11.98
3b
L
L
L/10
42.1
71.6
13.01
1c
L
1.25L
L/6
27.7
78.7
12.07
2c
L
1.25L
L/8
27.7
71.6
13.43
3c
L
1.25L
L/10
27.7
64.5
14.49
Table III
Figures of Merit for the Static Crossed Concentrator (ϕq = 30°, ϕp = 62.778°, and n = 1.425) a
Case
Dimension
Cg
Ib
Id
Cob
Cod
g ≃ γd
γb
γ31
I31
Co31
1a
h = 0.75L, W = L/6
7.21
0.589
0.383
4.293
2.79
0.687
0.758
0.778
0.522
3.805
2a
h = 0.75L, W = L/8
7.81
0.732
0.437
5.714
3.407
0.839
0.827
0.834
0.636
4.966
3a
h = 0.75L, W = L/10
8.13
0.849
0.482
6.9
3.92
0.965
0.868
0.865
0.73
5.934
1b
h = L, W = L/6
8.17
0.52
0.322
4.249
2.631
0.648
0.796
0.808
0.456
3.724
2b
h = L, W = L/8
8.61
0.721
0.41
6.21
3.529
0.869
0.867
0.864
0.62
5.34
3b
h = L, W = L/10
8.75
0.831
0.461
7.266
4.034
0.993
0.888
0.881
0.711
6.218
1c
h = 1.25L, W = L/6
8.8
0.446
0.266
3.915
2.334
0.575
0.826
0.833
0.388
3.403
2c
h = 1.25L, W = L/8
9
0.703
0.394
6.331
3.543
0.872
0.881
0.875
0.603
5.427
3c
h = 1.25L, W = L/10
Not possible
The subscripts b, d, and 31 belong to the solar beam source diffuse radiation and the three-level model of global radiation, respectively.
Table IV
Figures of Merit of the Quasi-Static Crossed Concentrator, with Two Adjustments a Year (ϕq = 15° and ϕp = 60°), for the Heights Considered for Static Crossed Concentrators
Case
Dimension
Cg
Ib
Id
Cob
Cod
g ≃ γd
γb
γ31
I31
Co31
1a
h = 0.75L, W = L/6
9.03
0.573
0.309
5.171
2.794
0.688
0.456
0.511
0 487
4.4
2a
h = 0.75L, W = L/8
10.16
0.725
0.339
7.368
3.449
0.849
0.526
0.573
0.6
6.09
3a
h = 0.75L, W = L/10
11.1
0.825
0.352
9.158
3.907
0.962
0.577
0.619
0.672
7.455
1b
h = L, W = L/6
10.7
0.502
0.245
5.368
2.625
0.646
0.504
0.554
0.418
4.47
2b
h = L, W = L/8
11.98
0.698
0.294
8.363
3.517
0.866
0.586
0.627
0.567
6.79
3b
h = L, W = L/10
13.01
0.809
0.31
10.525
4.026
0.991
0.644
0.679
0.647
8.41
1c
h = 1.25L, W = L/6
12.07
0.435
0.196
5.248
2.366
0.583
0.547
0.590
0.357
4 31
2c
h = 1.25L, W = L/8
13.43
0.682
0.264
9.167
3.549
0.874
0.636
0.672
0.547
7.34
3c
h = 1.25L, W = L/10
14.49
0.792
0.278
11.476
4.033
0.993
0.702
0.73
0.625
9.06
Table V
Figures of Merit of the Quasi-Static Crossed Concentrator, with Two Adjustments a Year (ϕq = 20° and ϕp = 60°), for the Same Heights Considered in the Previous Cases
Case
Dimension
Cg
Ib
Id
Cob
Cod
g ≃ γd
γb
γ31
I31
Co31
1a
h = 0.75L, W = L/6
8.45
0.6
0.348
5.072
2.94
0.724
0.425
0.483
0.519
4 381
2a
h = 0.75L, W = L/8
9.38
0.742
0.37
6.954
3.468
0.854
0.494
0.545
0.621
5.824
3a
h = 0.75L, W = L/10
10.11
0.857
0.387
8.635
3.913
0.963
0.544
0.59
0.703
7.104
1b
h = L, W = L/6
9.88
0.505
0.264
4.989
2.61
0.643
0.471
0.524
0.427
4.218
2b
h = L, W = L/8
10.86
0.714
0.323
7.76
3.506
0.863
0.545
0.591
0.587
6.381
3b
h = L, W = L/10
11.59
0.84
0.348
9.739
4.028
0.992
0.595
0.636
0.681
7 887
1c
h = 1.25L, W = L/6
10.99
0.442
0.216
4.853
2.37
0.583
0.506
0.555
0.368
4 048
2c
h = 1.25L, W = L/8
11.97
0.7
0.296
8.381
3.54
0.872
0.584
0.625
0.569
6 811
3c
h = 1.25L, W = L/10
12.63
0.823
0.32
10.356
4.042
0.995
0.634
0.67
0.66
8.334
Tables (5)
Table I
Geometrical Characteristics for the Static Crossed Concentrators with ϕq = 30° and ϕp = 62.778°
Case
L (cm)
h (cm)
W (cm)
fp (%)
fq (%)
Cg
1a
L
0.75L
L/6
54
56
7.29
2a
L
0.75L
L/8
54
41.6
7.81
3a
L
0.75L
L/10
54
27
8.13
1b
L
L
L/6
39
42
8.17
2b
L
L
L/8
39
22.2
8.61
3b
L
L
L/10
39
2.7
8.75
1c
L
1.25L
L/6
23
29
8.8
2c
L
1.25L
L/8
23
4
9.1
3c
L
1.25L
L/10
23
Not possible
Table II
Geometrical Characteristics for Quasi-Static Concentrators with Two Seasonal Adjustments for ϕq = 15° and ϕp = 60°
Case
L (cm)
h (cm)
W (cm)
fp (%)
fq (%)
Cg
1a
L
0.75L
L/6
56.6
87.2
9.03
2a
L
0.75L
L/8
56.6
83
10.16
3a
L
0.75L
L/10
56.6
78.7
11.1
1b
L
L
L/6
42.1
83
10.7
2b
L
L
L/8
42.1
77.3
11.98
3b
L
L
L/10
42.1
71.6
13.01
1c
L
1.25L
L/6
27.7
78.7
12.07
2c
L
1.25L
L/8
27.7
71.6
13.43
3c
L
1.25L
L/10
27.7
64.5
14.49
Table III
Figures of Merit for the Static Crossed Concentrator (ϕq = 30°, ϕp = 62.778°, and n = 1.425) a
Case
Dimension
Cg
Ib
Id
Cob
Cod
g ≃ γd
γb
γ31
I31
Co31
1a
h = 0.75L, W = L/6
7.21
0.589
0.383
4.293
2.79
0.687
0.758
0.778
0.522
3.805
2a
h = 0.75L, W = L/8
7.81
0.732
0.437
5.714
3.407
0.839
0.827
0.834
0.636
4.966
3a
h = 0.75L, W = L/10
8.13
0.849
0.482
6.9
3.92
0.965
0.868
0.865
0.73
5.934
1b
h = L, W = L/6
8.17
0.52
0.322
4.249
2.631
0.648
0.796
0.808
0.456
3.724
2b
h = L, W = L/8
8.61
0.721
0.41
6.21
3.529
0.869
0.867
0.864
0.62
5.34
3b
h = L, W = L/10
8.75
0.831
0.461
7.266
4.034
0.993
0.888
0.881
0.711
6.218
1c
h = 1.25L, W = L/6
8.8
0.446
0.266
3.915
2.334
0.575
0.826
0.833
0.388
3.403
2c
h = 1.25L, W = L/8
9
0.703
0.394
6.331
3.543
0.872
0.881
0.875
0.603
5.427
3c
h = 1.25L, W = L/10
Not possible
The subscripts b, d, and 31 belong to the solar beam source diffuse radiation and the three-level model of global radiation, respectively.
Table IV
Figures of Merit of the Quasi-Static Crossed Concentrator, with Two Adjustments a Year (ϕq = 15° and ϕp = 60°), for the Heights Considered for Static Crossed Concentrators
Case
Dimension
Cg
Ib
Id
Cob
Cod
g ≃ γd
γb
γ31
I31
Co31
1a
h = 0.75L, W = L/6
9.03
0.573
0.309
5.171
2.794
0.688
0.456
0.511
0 487
4.4
2a
h = 0.75L, W = L/8
10.16
0.725
0.339
7.368
3.449
0.849
0.526
0.573
0.6
6.09
3a
h = 0.75L, W = L/10
11.1
0.825
0.352
9.158
3.907
0.962
0.577
0.619
0.672
7.455
1b
h = L, W = L/6
10.7
0.502
0.245
5.368
2.625
0.646
0.504
0.554
0.418
4.47
2b
h = L, W = L/8
11.98
0.698
0.294
8.363
3.517
0.866
0.586
0.627
0.567
6.79
3b
h = L, W = L/10
13.01
0.809
0.31
10.525
4.026
0.991
0.644
0.679
0.647
8.41
1c
h = 1.25L, W = L/6
12.07
0.435
0.196
5.248
2.366
0.583
0.547
0.590
0.357
4 31
2c
h = 1.25L, W = L/8
13.43
0.682
0.264
9.167
3.549
0.874
0.636
0.672
0.547
7.34
3c
h = 1.25L, W = L/10
14.49
0.792
0.278
11.476
4.033
0.993
0.702
0.73
0.625
9.06
Table V
Figures of Merit of the Quasi-Static Crossed Concentrator, with Two Adjustments a Year (ϕq = 20° and ϕp = 60°), for the Same Heights Considered in the Previous Cases
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