Minimum emissivity limits given in the literature for high temperature selective surfaces may be too pessimistic. To reassess these limits, hypothetical bulk absorbing semiconductor–metal (SM) and semiconductor-insulator-metal (SIM) coatings were modeled using plausible high temperature semiconductor optical constants. Careful attention was paid to the positioning of the exponential region of the semiconductor absorption edge. Values of α/∊ near those of an ideal selective surface on copper were obtained with SIM surfaces, which use a dielectric refractive-index mismatch layer to reduce emittance. It is suggested that an ideal selective surface on copper (or silver) be regarded as the approachable limiting performances case for most applications. If favorable but extreme values of refractive index can be utilized, the ideal α/∊ may even be exceeded using interference effects to limit copper emissivity to below vacuum values.
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Values of Semiconductor n and k Used in Selective Surface Optical Calculations as a Function of Wavelengtha
λ(μm)
n(high)
n(low)
k(hω0 = 1.60 eV)
k(hω0 = 1.24 eV)
0.3
8.00
3.0
2.39
2.39
0.35
7.71
2.89
2.79
2.79
0.4
7.50
2.81
3.18
3.18
0.45
7.33
2.75
3.58
3.58
0.5
7.20
2.70
3.98
3.98
0.55
7.09
2.66
4.35
4.38
0.6
7.00
2.63
4.66
4.77
0.65
6.92
2.60
4.76
5.15
0.7
6.86
2.57
4.42
5.49
0.75
6.80
2.55
3.57
5.76
0.8
6.75
2.53
2.50
5.88
0.85
6.71
2.52
1.60
5.77
0.9
6.67
2.50
1.00
5.39
0.95
6.63
2.49
0.625
4.75
1.0
6.60
2.48
0.403
3.98
1.1
6.55
2.46
0.184
2.52
1.2
6.50
2.44
0.095
1.52
1.3
6.46
2.42
0.055
0.940
1.4
6.43
2.41
0.034
0.608
1.5
6.40
2.40
0.023
0.414
1.6
6.38
2.39
0.016
0.252
1.7
6.35
2.38
0.012
0.218
1.8
6.33
2.38
0.009
0.168
1.9
6.31
2.37
0.007
0.133
2.0
6.29
2.36
0.006
0.118
2.5
6.25
2.34
0.003
0.051
3.0
6.21
2.33
0.002
0.031
4.0
6.15
2.31
0.001
0.018
∞
6.00
2.25
0.000
0.000
Equation (2) is used to calculate k with the values of ω0 shown, using k = 4πd/λ.
Table II
α/∊ of SM Surface for Different Temperatures and Semiconductor Layer Thicknesses (ds)a
Ee = 0.1242 eV
ℏω0(eV)
n(ω)
ds(μm)
T(°C)
αH
∊H
αH/∊H
F
1.60
High
2
300
0.86
0.074
12
0.24
High
2
500
0.86
0.083
10
0.26
High
2
700
0.87
0.105
8.3
0.29
Low
2
300
0.87
0.039
22
0.43
Low
2
500
0.87
0.052
17
0.45
Low
2
700
0.87
0.073
12
0.41
High
3
300
0.89
0.075
12
0.24
High
3
500
0.89
0.097
9.2
0.24
High
3
700
0.89
0.128
7.0
0.24
Low
3
300
0.88
0.040
22
0.43
Low
3
500
0.89
0.054
16
0.42
Low
3
700
0.89
0.079
11
0.38
1.24
High
2
300
0.91
0.129
7.1
0.14
High
2
500
0.91
0.189
4.8
0.13
High
2
700
0.91
0.266
3.4
0.12
Low
2
300
0.91
0.093
9.8
0.19
Low
2
500
0.91
0.151
6.0
0.16
Low
2
700
0.91
0.230
4.0
0.14
High
3
300
0.93
0.136
6.6
0.13
High
3
500
0.93
0.204
4.6
0.12
High
3
700
0.93
0.288
3.3
0.11
Low
3
300
0.93
0.116
8.0
0.16
Low
3
500
0.93
0.185
5.0
0.13
Low
3
700
0.93
0.272
3.4
0.12
Both high and low ranges of refractive index and two values of ℏω0 are represented (see Table I). The value ℏω0 = 1.60 eV clearly gives better results at high temperatures for dd of both 2 and 3 μm. Edge position is more important than thickness for this type of surface. F is the ratio of calculated α/∊ compared with that in Fig. 5.
Table III
λ/∊ of SIM Surface with High ℏω0 and High n(ω)
ds(μm)
ℏ0 = 1.604 eV
Ee = 0.1242 eV
Dielectric n = 1.5
n(ω) = high
F
dd(μm)
T(°C)
αH
∊H
αH/∊H
2
0.5
300
0.86
0.017
51
1.00
0.5
500
0.86
0.022
39
1.03
0.5
700
0.86
0.036
24
0.83
1.0
300
0.86
0.014
61
1.20
1.0
500
0.86
0.024
33
0.87
1.0
700
0.86
0.043
20
0.69
1.5
300
0.86
0.026
33
0.65
1.5
500
0.86
0.041
21
0.55
1.5
700
0.86
0.061
14
0.48
2.0
300
0.85
0.032
27
0.53
2.0
500
0.86
0.041
21
0.55
2.0
700
0.86
0.055
15
0.52
3
0.5
300
0.88
0.025
35
0.69
0.5
500
0.88
0.031
28
0.74
0.5
700
0.88
0.049
18
0.62
1.0
300
0.88
0.020
44
0.86
1.0
500
0.88
0.034
26
0.68
1.0
700
0.88
0.058
15
0.52
1.5
300
0.88
0.023
38
0.75
1.5
500
0.88
0.038
23
0.61
1.5
700
0.88
0.062
14
0.48
2.0
300
0.88
0.036
24
0.47
2.0
500
0.88
0.048
18
0.47
2.0
700
0.88
0.068
13
0.45
Performance is dramatically enhanced, and α/∊ values greater than those in Fig. 5 may be obtained with high values of n(ω) due to emittance being restricted by interference. Dielectric layer thickness is given by dd.
Calculation of Net Energy Gain Using Surfaces in Tables III–VI on an Absorber of a Perfect Concentrating Systema
Average insolation = 350 W/m2
T(°C)
αH
Pin(W/m2)
∊H
X
Pout(W/m2)
Pnet(W/m2)
500
0.86
301
0.022
1
435
0
5
87
214
25
17
284
100
4
297
0.88
308
0.031
1
613
0
5
123
185
25
25
283
100
6
302
0.91
319
0.109
1
2157
0
5
431
0
25
86
233
100
22
297
700
0.86
301
0.036
1
1830
0
5
366
0
25
73
228
100
18
283
0.88
308
0.049
1
2490
0
5
498
0
25
100
208
100
25
283
0.91
319
0.180
1
9148
0
5
1830
0
25
366
0
100
91
228
Calculations for real systems would include optical losses and a sophisticated model of solar insolation, but the input power Pin chosen is not inconsistent with more elaborate models.
Tables (7)
Table I
Values of Semiconductor n and k Used in Selective Surface Optical Calculations as a Function of Wavelengtha
λ(μm)
n(high)
n(low)
k(hω0 = 1.60 eV)
k(hω0 = 1.24 eV)
0.3
8.00
3.0
2.39
2.39
0.35
7.71
2.89
2.79
2.79
0.4
7.50
2.81
3.18
3.18
0.45
7.33
2.75
3.58
3.58
0.5
7.20
2.70
3.98
3.98
0.55
7.09
2.66
4.35
4.38
0.6
7.00
2.63
4.66
4.77
0.65
6.92
2.60
4.76
5.15
0.7
6.86
2.57
4.42
5.49
0.75
6.80
2.55
3.57
5.76
0.8
6.75
2.53
2.50
5.88
0.85
6.71
2.52
1.60
5.77
0.9
6.67
2.50
1.00
5.39
0.95
6.63
2.49
0.625
4.75
1.0
6.60
2.48
0.403
3.98
1.1
6.55
2.46
0.184
2.52
1.2
6.50
2.44
0.095
1.52
1.3
6.46
2.42
0.055
0.940
1.4
6.43
2.41
0.034
0.608
1.5
6.40
2.40
0.023
0.414
1.6
6.38
2.39
0.016
0.252
1.7
6.35
2.38
0.012
0.218
1.8
6.33
2.38
0.009
0.168
1.9
6.31
2.37
0.007
0.133
2.0
6.29
2.36
0.006
0.118
2.5
6.25
2.34
0.003
0.051
3.0
6.21
2.33
0.002
0.031
4.0
6.15
2.31
0.001
0.018
∞
6.00
2.25
0.000
0.000
Equation (2) is used to calculate k with the values of ω0 shown, using k = 4πd/λ.
Table II
α/∊ of SM Surface for Different Temperatures and Semiconductor Layer Thicknesses (ds)a
Ee = 0.1242 eV
ℏω0(eV)
n(ω)
ds(μm)
T(°C)
αH
∊H
αH/∊H
F
1.60
High
2
300
0.86
0.074
12
0.24
High
2
500
0.86
0.083
10
0.26
High
2
700
0.87
0.105
8.3
0.29
Low
2
300
0.87
0.039
22
0.43
Low
2
500
0.87
0.052
17
0.45
Low
2
700
0.87
0.073
12
0.41
High
3
300
0.89
0.075
12
0.24
High
3
500
0.89
0.097
9.2
0.24
High
3
700
0.89
0.128
7.0
0.24
Low
3
300
0.88
0.040
22
0.43
Low
3
500
0.89
0.054
16
0.42
Low
3
700
0.89
0.079
11
0.38
1.24
High
2
300
0.91
0.129
7.1
0.14
High
2
500
0.91
0.189
4.8
0.13
High
2
700
0.91
0.266
3.4
0.12
Low
2
300
0.91
0.093
9.8
0.19
Low
2
500
0.91
0.151
6.0
0.16
Low
2
700
0.91
0.230
4.0
0.14
High
3
300
0.93
0.136
6.6
0.13
High
3
500
0.93
0.204
4.6
0.12
High
3
700
0.93
0.288
3.3
0.11
Low
3
300
0.93
0.116
8.0
0.16
Low
3
500
0.93
0.185
5.0
0.13
Low
3
700
0.93
0.272
3.4
0.12
Both high and low ranges of refractive index and two values of ℏω0 are represented (see Table I). The value ℏω0 = 1.60 eV clearly gives better results at high temperatures for dd of both 2 and 3 μm. Edge position is more important than thickness for this type of surface. F is the ratio of calculated α/∊ compared with that in Fig. 5.
Table III
λ/∊ of SIM Surface with High ℏω0 and High n(ω)
ds(μm)
ℏ0 = 1.604 eV
Ee = 0.1242 eV
Dielectric n = 1.5
n(ω) = high
F
dd(μm)
T(°C)
αH
∊H
αH/∊H
2
0.5
300
0.86
0.017
51
1.00
0.5
500
0.86
0.022
39
1.03
0.5
700
0.86
0.036
24
0.83
1.0
300
0.86
0.014
61
1.20
1.0
500
0.86
0.024
33
0.87
1.0
700
0.86
0.043
20
0.69
1.5
300
0.86
0.026
33
0.65
1.5
500
0.86
0.041
21
0.55
1.5
700
0.86
0.061
14
0.48
2.0
300
0.85
0.032
27
0.53
2.0
500
0.86
0.041
21
0.55
2.0
700
0.86
0.055
15
0.52
3
0.5
300
0.88
0.025
35
0.69
0.5
500
0.88
0.031
28
0.74
0.5
700
0.88
0.049
18
0.62
1.0
300
0.88
0.020
44
0.86
1.0
500
0.88
0.034
26
0.68
1.0
700
0.88
0.058
15
0.52
1.5
300
0.88
0.023
38
0.75
1.5
500
0.88
0.038
23
0.61
1.5
700
0.88
0.062
14
0.48
2.0
300
0.88
0.036
24
0.47
2.0
500
0.88
0.048
18
0.47
2.0
700
0.88
0.068
13
0.45
Performance is dramatically enhanced, and α/∊ values greater than those in Fig. 5 may be obtained with high values of n(ω) due to emittance being restricted by interference. Dielectric layer thickness is given by dd.
Calculation of Net Energy Gain Using Surfaces in Tables III–VI on an Absorber of a Perfect Concentrating Systema
Average insolation = 350 W/m2
T(°C)
αH
Pin(W/m2)
∊H
X
Pout(W/m2)
Pnet(W/m2)
500
0.86
301
0.022
1
435
0
5
87
214
25
17
284
100
4
297
0.88
308
0.031
1
613
0
5
123
185
25
25
283
100
6
302
0.91
319
0.109
1
2157
0
5
431
0
25
86
233
100
22
297
700
0.86
301
0.036
1
1830
0
5
366
0
25
73
228
100
18
283
0.88
308
0.049
1
2490
0
5
498
0
25
100
208
100
25
283
0.91
319
0.180
1
9148
0
5
1830
0
25
366
0
100
91
228
Calculations for real systems would include optical losses and a sophisticated model of solar insolation, but the input power Pin chosen is not inconsistent with more elaborate models.