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Purdue ME 606: Solar Constant and Emissivity (Lecture 4)

Table of Contents

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The Solar Constant

The solar constant is the heat flux from the Sun.

Known:

T=5762K; Radius of Sun, RS = 6.96x10 − 8m; SES = 1.496x1011m;

q_{solar} = E_b (T_s ) \cdot A_s = \sigma T_s^4 \cdot 4\pi R_s^2

q''_{ES} = \frac{{q_{sol} }}{{A_{ES} }} = \frac{{\sigma T_s^4 \cdot 4\pi R_s^2 }}{{4\pi S_{ES}^2 }} = 1353\frac{W}{{m^2 }} (solar constant)

The following figure shows the spectral irradiance [1]

Emissivity

\varepsilon _\lambda = \frac{{E_\lambda }}{{E_{\lambda b} }}


\varepsilon = \frac{{\int_0^\infty {E_\lambda } d\lambda }}{{\int_0^\infty {E_{b\lambda } } d\lambda }} = \frac{{\int_0^\infty {\varepsilon _\lambda E_{b\lambda } } d\lambda }}{{\int_0^\infty {E_{b\lambda } } d\lambda }} = \frac{E}{{E_b }}

Metals: good reflectors due to free electrons —> usually poor emitter

Gray Surface

No wave length of \varepsilon _\lambda ,\rho _\lambda ,\tau _\lambda etc.

\varepsilon \simeq 0.05 \sim 0.15 for metal

\varepsilon \simeq 0.7 \sim 1.0 for non-metal

Directional dependence of emissivity

Directional emissivities

“Spectral” emissivity: \varepsilon _\lambda (\theta ,\phi ) = \frac{{I_\lambda (\theta ,\phi )}}{{I_{b\lambda } }}

“Total” emissivity: \varepsilon (\theta ,\phi ) = \frac{{I(\theta ,\phi )}}{{I_b }}

For “Isotropic” surface, no Φ dependence

Polar plot:

Consider emission (not reflection) from radiating spheres,

Diffuse: uniformly bright

Non-metal: darker on edges

Metal: darker in the middle and extreme edges


Relate hemispherical (\varepsilon ,\varepsilon _\lambda ) to normal (θ = 0)

\varepsilon _n ,\varepsilon _{n\lambda } relatively easy to measure, and often be tabulated


Empirical relations:

\frac{{\varepsilon (hemi)}}{{\varepsilon _n }} \simeq 0.96 for non-metal

\frac{{\varepsilon (hemi)}}{{\varepsilon _n }} \simeq 1.2 for polished metal

Another empirical curve:

Emissivity Summary

Emission from a surface depends on λ,θ,Φ,T


1. Dependence on Φ is weak, but dependence on λ,θ,T can be strong


2. For point to point or surface to surface exchange, we often account for λ and/or θ dependence.

Spectral directional emissivity \varepsilon _\lambda (\theta ) = \frac{{I_\lambda (\theta )}}{{I_{b\lambda } }}


3. For emission to hemisphere above a surface,

Spectral hemispherical emissivity \varepsilon _\lambda = \frac{{E_\lambda }}{{E_{b\lambda } }} = \frac{1}{\pi }\int {\varepsilon _\lambda (\theta )\cos \theta } d\omega


4. Total hemispherical emissivity \varepsilon = \frac{E}{{E_b }} = \frac{1}{{E_b }}\int_0^\infty {\varepsilon _\lambda E_{b\lambda } d\lambda }


5. For gray, diffuse surface, \varepsilon = \varepsilon _\lambda = \varepsilon _\lambda (\theta ) = const.

Last modified on 15 Oct, 2008