Grey Body Radiation .pptx

Detailed explanation of Grey Body Radiation

1. Grey Body Radiation By Narendra Prasath N J IBT-B

2. agenda Introduction 3 Definition of a Grey Body 4 Spectral Emission Characteristics 5 Relation Between Emissivity and Absorptivity 6 Engineering Significance & Limitations 11

3. Introduction In radiation heat transfer, real surfaces are rarely perfect absorbers or emitters of thermal energy. While an ideal black body absorbs and emits radiation at the maximum theoretical limit for any given temperature, real materials emit less energy. Such real surfaces are often idealized as grey bodies — an important simplification in thermal radiation analysis that allows engineers to accurately model complex radiative behavior without resorting to full spectral calculations. The grey body assumption provides a bridge between ideal theoretical radiation and practical engineering applications . 3

4. Definition of a Grey Body A grey body is defined as a real surface whose emissivity (ε) is independent of wavelength over the spectral range of interest. Mathematically, ε_λ = constant with wavelength ⇒ ε_λ = ε Thus, the emissive power of a grey body at temperature T is given by: E = ε E_b = ε σ T⁴ where: E = total emissive power of the grey body (W/m²) E_b = blackbody emissive power at the same temperature ε = emissivity of the surface (0 < ε < 1) σ = Stefan–Boltzmann constant (5.67 × 10⁻⁸ W/m²·K⁴) Hence, a grey body emits only a fraction (ε) of the radiation emitted by a black body at the same temperature. 4

5. Spectral Emission Characteristics For an ideal black body, the spectral distribution of emitted energy at each wavelength (λ) follows Planck’s Law: E_bλ = (2πhc²/λ⁵) * [1 / (e^( hc / λkT ) - 1)] For a grey body: E_λ = ε E_bλ This means: - The shape of the spectral emission curve ( Eλ vs λ) is identical to that of a black body at the same temperature. - The magnitude is uniformly reduced by the factor ε. Thus, the grey body behaves as a scaled version of a black body, radiating at a reduced intensity but preserving the same spectral pattern. 5

6. Physical Interpretation The assumption of wavelength-independent emissivity implies that the material’s surface emits and absorbs the same proportion of radiation at all wavelengths. In physical terms: - The grey body radiates energy less efficiently than a black body but with the same spectral distribution. - The surface emits and absorbs a constant fraction (ε) of the incident radiation across all wavelengths. Hence, the grey body model is a simplified representation of a real surface that eliminates the complexity of spectral variations while maintaining sufficient physical realism for engineering calculations. 6

7. Emissivity (ε) Emissivity is a dimensionless property (0 ≤ ε ≤ 1) that indicates how closely a surface approaches blackbody behavior. - ε = 1 → Perfect black body - ε = 0 → Perfect reflector (no emission) For most engineering materials, ε remains nearly constant over the thermal infrared range (2–15 μm ), which justifies treating them as grey bodies. Typical emissivity values: Human skin – 0.95 Oxidized steel – 0.70 Black paint – 0.97 Polished aluminum – 0.05–0.10 7

8. Relation Between Emissivity and Absorptivity According to Kirchhoff’s Law of Thermal Radiation, at thermal equilibrium: ε_λ = α_λ For a grey body (wavelength-independent properties): ε = α That means a grey body absorbs and emits the same proportion of energy. If a surface emits 70% of blackbody radiation (ε = 0.7), it also absorbs 70% of the incident radiation. This relationship is fundamental in radiation exchange calculations between surfaces. 8

9. The total emissive power of a grey body is derived from integrating its spectral emissive power over all wavelengths: E = ∫₀^∞ E_λ dλ = ε ∫₀^∞ E_bλ dλ = ε σ T⁴ This is one of the most important equations in radiative heat transfer and forms the basis of the grey surface model. When a grey surface at temperature Ts exchanges radiation with its surroundings at temperature Tsur , the net radiative heat flux is given by: q_net = ε σ (Ts⁴ - Tsur ⁴) This expression is widely applied in: - Furnace wall heat losses - Thermal design of heat exchangers - Cooling of electronic and mechanical systems - Radiative heat exchange in enclosures 9 Total Emissive Power of a Grey Body and Net Radiative Heat Exchange

10. Çengel and Ghajar highlight that the grey body model provides an excellent engineering approximation for many materials, particularly those used in thermal systems. It simplifies radiation analysis by: - Treating emissivity as a single constant value instead of a function of wavelength. - Making the radiation network analogy (radiosity–irradiation method) feasible. - Enabling analytical or numerical solutions in multi-surface enclosures. Because most solid materials exhibit nearly constant emissivity in the infrared range, the grey body model is sufficiently accurate for a wide range of engineering applications. 10 Engineering Significance

11. The grey body assumption fails for materials that have strong spectral selectivity, such as: - Polished metals (wavelength-dependent reflectivity) - Gaseous media (CO₂, H₂O vapor, selective emitters) - Solar absorber coatings (designed for selective emission) In such cases, a non-grey analysis is required, considering emissivity variation with wavelength or using band models and emissivity functions. Property | Grey Body Emissivity (ε) | Constant, less than 1 Emission Spectrum | Same shape as black body, scaled down Absorptivity | Equal to ε Total Emissive Power | E = ε σ T⁴ Net Heat Flux | q = ε σ (Ts⁴ - Tsur ⁴) Engineering Use | Realistic approximation for solids and surfaces 11 Limitations and summary

12. thank you…😊 Reference Çengel , Y. A., & Ghajar , A. J. (2015). Heat and Mass Transfer: Fundamentals and Applications (5th ed.). New York: McGraw-Hill Education. (Chapter 13 — Radiation Heat Transfer, Sections: Blackbody Radiation, Emissivity, and Gray Surfaces.)