Question 12.111: KNOWN: Plate temperature and spectral and directional depend......

ANALYSIS: (a) For λ < λ_c  \text {and}  θ = 45^{\circ}, α_λ = α_1  \cos θ = 0.707 α_1. From Eq. (12.47) the total absorptivity is then

\alpha_{ S }=0.707 \alpha_1\left\{\frac{\int_0^{\lambda_{ c }} E _{\lambda, b }(\lambda, 5800  K ) d \lambda}{ E _{ b }}\right\}+\alpha_2\left\{\frac{\int_{\lambda_{ c }}^{\infty} E _{\lambda, b }(\lambda, 5800  K ) d \lambda}{ E _{ b }}\right\}

\alpha_{ S }=0.707 \alpha_1 F _{\left(0 \rightarrow \lambda_{ c }\right)}+\alpha_2\left[1- F _{\left(0 \rightarrow \lambda_{ c }\right)}\right]

For the prescribed value of \lambda_{ c }, \lambda_{ c } T =11,600  \mu m \cdot K and, from Table 12.1, F _{(0 \rightarrow \lambda c )}=0.941 . Hence,

\alpha_{ S }=0.707 \times 0.93 \times 0.941+0.25(1-0.941)=0.619+0.015=0.634

(b) With \varepsilon_{\lambda, \theta}=\alpha_{\lambda, 0}, Eq (12.36) may be used to obtain \varepsilon_\lambda \text { for } \lambda<\lambda_c \text {. }

\varepsilon_\lambda(\lambda, T )=2 \alpha_1 \int_0^{\pi / 2} \cos ^2 \theta \sin \theta d \theta=-\left.2 \alpha_1 \frac{\cos ^3 \theta}{3}\right|_0 ^{\pi / 2}=\frac{2}{3} \alpha_1

From Eq. (12.38),

\varepsilon=0.667 \alpha_1 \frac{\int_0^{\lambda_{ c }} E _{\lambda, b }\left(\lambda, T _{ p }\right) d \lambda}{ E _{ b }}+\alpha_2 \frac{\int_{\lambda_{ c }}^{\infty} E _{\lambda, b }\left(\lambda, T _{ p }\right) d \lambda}{ E _{ b }}

\varepsilon=0.667 \alpha_1 F _{\left(0 \rightarrow \lambda_{ c }\right)}+\alpha_2\left[1- F _{\left(0-\lambda_{ c }\right)}\right]

For \lambda_{ c }=2  \mu m \text { and } T _{ p }=333  K , \lambda_{ c } T =666  \mu m \cdot K and, from Table 12.1, F _{(0-\lambda c )}=0 . Hence,

\varepsilon=\alpha_2=0.25

(c)      q _{\text {net }}^{\prime \prime}=\alpha_{ S } q _{ S }^{\prime \prime}-\varepsilon \sigma T _{ p }^4=634  W / m ^2-0.25 \times 5.67 \times 10^{-8}  W / m ^2 \cdot K ^4(333  K )^4

q _{\text {net }}^{\prime \prime}=460  W / m ^2

(d) Using the foregoing model with the Radiation/Band Emission Factor option of IHT, the following results were obtained for α_S~\text{and}~ε. The absorptivity increases with increasing λ_c, as more of the incident solar radiation falls within the region of α_1 > α_2. Note, however, the limit at λ ≈ 3 µm, beyond which there is little change in α_S. The emissivity also increases with increasing λ_c, as more of the emitted radiation is at wavelengths for which ε_1 = α_1 > ε_2 = α_2. However, the surface temperature is low, and even for λ_c = 5 µm, there is little emission at λ < λ_c. Hence, ε only increases from 0.25 to 0.26 as λ_c increases from 0.7 to 5.0 µm.

The net heat flux increases from 276 W/m² at λ_c = 2 µm to a maximum of 477 W/m² at λ_c = 4.2 µm and then decreases to 474 W/m² at λ_c = 5 µm. The existence of a maximum is due to the upper limit on the value of α_S and the increase in ε~\text{with}~λ_c.

COMMENTS: Spectrally and directionally selective coatings may be used to enhance the performance of solar collectors.