Question 12.7: Measurements of the spectral, directional emissivity of a me...
Measurements of the spectral, directional emissivity of a metallic surface at T = 2000 K and λ = 1.0 μm yield a directional distribution that may be approximated as follows:
Determine corresponding values of the spectral, normal emissivity; the spectral, hemispherical emissivity; the spectral intensity of radiation emitted in the normal direction; and the spectral emissive power.
Known: Directional distribution of ε_{λ,θ} at λ = 1 μm for a metallic surface at 2000 K.
Find:
1. Spectral, normal emissivity ε_{λ,n} and spectral, hemispherical emissivity ε_{λ}.
2. Spectral, normal intensity I_{λ,n} and spectral emissive power E_{λ}.
Analysis:
1. From the measurement of ε_{λ,θ} at λ = 1 μm, we see that
ε_{λ,n} = ε_{λ,θ}(1 μm, 0°) = 0.3
From Equation 12.42, the spectral, hemispherical emissivity is
ε_{λ}(λ, T) = 2\int_{0}^{\pi/2}{ε_{λ,θ}(λ, θ, T) \cos θ \sin θ dθ} (12.42)
ε_{λ}(1 μm) = 2\int_{0}^{\pi/2}{ε_{λ,θ} \cos θ \sin θ dθ}
or
ε_{λ}(1 μm) = 2\left[0.3\int_{0}^{\pi/3}{\cos θ \sin θ dθ} + 0.6\int_{\pi/3}^{4\pi/9}{\cos θ \sin θ dθ}\right]\\ ε_{λ}(1 μm) = 2\left[0.3 \frac{\sin^{2} θ}{2}\bigg|_{0}^{\pi/3} + 0.6 \frac{\sin^{2} θ}{2}\bigg|_{\pi/3}^{4\pi/9}\right]\\ = 2\left[\frac{0.3}{2}(0.75) + \frac{0.6}{2} (0.97 – 0.75)\right]\\ ε_{λ}(1 μm) = 0.36
2. From Equation 12.38 the spectral intensity of radiation emitted at λ = 1 μm in the normal direction is
ε_{λ,θ}(λ, θ, \phi, T) \equiv \frac{I_{λ,e}(λ, θ, \phi, T)}{I_{λ,b}(λ, T)} (12.38)
I_{λ,n}(1 μm, 0°, 2000 K) = ε_{λ,θ}(1 μm, 0°)I_{λ,b}(1 μm, 2000 K)
where ε_{λ,θ}(1 μm, 0°) = 0.3 and I_{λ,b}(1 μm, 2000 K) may be obtained from Table 12.2. For λT = 2000 μm · K, (I_{λ,b}/σT^{5}) = 0.493 × 10^{-4} (μm · K · sr)^{-1} and
| TABLE 12.2 Blackbody Radiation Functions | |||
| \pmb{\frac{I_{λ, b}(λ, T)}{I_{λ,b}(λ_{\max}, T)}} | \pmb{I_{λ, b}(λ, T)/σT^{5} (μm · K · sr)^{-1}} | \pmb{F_{(0 → λ)}} | λT (μm · K) |
| 0.000000 | 0.375034 × 10^{-27} | 0.000000 | 200 |
| 0.000000 | 0.490335 × 10^{-13} | 0.000000 | 400 |
| 0.000014 | 0.104046 × 10^{-8} | 0.000000 | 600 |
| 0.001372 | 0.991126 × 10^{-7} | 0.000016 | 800 |
| 0.016406 | 0.118505 × 10^{-5} | 0.000321 | 1,000 |
| 0.072534 | 0.523927 × 10^{-5} | 0.002134 | 1,200 |
| 0.186082 | 0.134411 × 10^{-4} | 0.007790 | 1,400 |
| 0.344904 | 0.249130 | 0.019718 | 1,600 |
| 0.519949 | 0.375568 | 0.039341 | 1,800 |
| 0.683123 | 0.493432 | 0.066728 | 2,000 |
| 0.816329 | 0.589649 × 10^{-4} | 0.100888 | 2,200 |
| 0.912155 | 0.658866 | 0.140256 | 2,400 |
| 0.970891 | 0.701292 | 0.183120 | 2,600 |
| 0.997123 | 0.720239 | 0.227897 | 2,800 |
| 1.000000 | 0.722318 × 10^{-4} | 0.250108 | 2,898 |
| 0.997143 | 0.720254 × 10^{-4} | 0.273232 | 3,000 |
| 0.977373 | 0.705974 | 0.318102 | 3,200 |
| 0.943551 | 0.681544 | 0.361735 | 3,400 |
| 0.900429 | 0.650396 | 0.403607 | 3,600 |
| 0.851737 | 0.615225 × 10^{-4} | 0.443382 | 3,800 |
| 0.800291 | 0.578064 | 0.480877 | 4,000 |
| 0.748139 | 0.540394 | 0.516014 | 4,200 |
| 0.696720 | 0.503253 | 0.548796 | 4,400 |
| 0.647004 | 0.467343 | 0.579280 | 4,600 |
| 0.599610 | 0.433109 | 0.607559 | 4,800 |
| 0.554898 | 0.400813 | 0.633747 | 5,000 |
| 0.513043 | 0.370580 × 10^{-4} | 0.658970 | 5,200 |
| 0.474092 | 0.342445 | 0.680360 | 5,400 |
| 0.438002 | 0.316376 | 0.701046 | 5,600 |
| 0.404671 | 0.292301 | 0.720158 | 5,800 |
| 0.373965 | 0.270121 | 0.737818 | 6,000 |
| 0.345724 | 0.249723 × 10^{-4} | 0.754140 | 6,200 |
| 0.319783 | 0.230985 | 0.769234 | 6,400 |
| 0.295973 | 0.213786 | 0.783199 | 6,600 |
| 0.274128 | 0.198008 | 0.796129 | 6,800 |
| 0.254090 | 0.183534 | 0.808109 | 7,000 |
| 0.235708 | 0.170256 × 10^{-4} | 0.819217 | 7,200 |
| 0.218842 | 0.158073 | 0.829527 | 7,400 |
| 0.203360 | 0.146891 | 0.839102 | 7,600 |
| 0.189143 | 0.136621 | 0.848005 | 7,800 |
| 0.176079 | 0.127185 | 0.856288 | 8,000 |
| 0.147819 | 0.106772 × 10^{-4} | 0.874608 | 8,500 |
| 0.124801 | 0.901463 × 10^{-5} | 0.890029 | 9,000 |
| 0.105956 | 0.765338 | 0.903085 | 9,500 |
| 0.090442 | 0.653279 × 10^{-5} | 0.914199 | 10,000 |
| 0.077600 | 0.560522 | 0.923710 | 10,500 |
| 0.066913 | 0.483321 | 0.931890 | 11,000 |
| 0.057970 | 0.418725 | 0.939959 | 11,500 |
| 0.050448 | 0.364394 × 10^{-5} | 0.945098 | 12,000 |
| 0.038689 | 0.279457 | 0.955139 | 13,000 |
| 0.030131 | 0.217641 | 0.962898 | 14,000 |
| 0.023794 | 0.171866 × 10^{-5} | 0.969981 | 15,000 |
| 0.019026 | 0.137429 | 0.973814 | 16,000 |
| 0.012574 | 0.908240 × 10^{-6} | 0.980860 | 18,000 |
| 0.008629 | 0.623310 | 0.985602 | 20,000 |
| 0.003828 | 0.276474 | 0.992215 | 25,000 |
| 0.001945 | 0.140469 × 10^{-6} | 0.995340 | 30,000 |
| 0.000656 | 0.473891 × 10^{-7} | 0.997967 | 40,000 |
| 0.000279 | 0.201605 | 0.998953 | 50,000 |
| 0.000058 | 0.418597 × 10^{-8} | 0.999713 | 75,000 |
| 0.000019 | 0.135752 | 0.999905 | 100,000 |
I_{λ,b} = 0.493 × 10^{-4} (μm · K · sr)^{-1} × 5.67 × 10^{-8} W/m^{2} · K^{4} (2000 K)^{5}\\ I_{λ,b} = 8.95 × 10^{4} W/m^{2} · μm · sr
Hence
I_{λ,n}(1 μm, 0°, 2000 K) = 0.3 × 8.95 × 10^{4} W/m^{2} · μm · sr\\ I_{λ,n}(1 μm, 0°, 2000 K) = 2.69 × 10^{4} W/m^{2} · μm · sr
From Equation 12.40 the spectral emissive power for λ = 1 μm and T = 2000 K is
ε_{λ}(λ, T) \equiv \frac{E_{λ}(λ, T)}{E_{λ,b}(λ, T)} (12.40)
E_{λ}(1 μm, 2000 K) = ε_{λ}(1 μm)E_{λ,b}(1 μm, 2000 K)
where
E_{λ,b}(1 μm, 2000 K) = \pi I_{λ,b}(1 μm, 2000 K)\\E_{λ,b}(1 μm, 2000 K) = \pi sr × 8.95 × 10^{4} W/m^{2} · μm · sr\\ = 2.81 × 10^{5} W/m^{2} · μm
Hence
E_{λ}(1 μm, 2000 K) = 0.36 × 2.81 × 10^{5} W/m^{2} · μm
or
E_{λ}(1 μm, 2000 K) = 1.01 × 10^{5} W/m^{2} · μm