# Summary

At any given temperature, real materials emit less energy than that of a black body. The effectiveness of a material at emitting energy is represented by a radiative property called emissivity, which is the ratio of the actual energy emitted by the material to that of a black body at the same temperature. This article will provide an overview of the methods available for calculating the spectral, spectral-directional, hemispherical and total hemispherical emissivity for metals.

# Definitions

: | Imaginary component of complex index of refraction (dimensionless) | |

: | Refractive index (dimensionless) | |

: | Surface emittance (dimensionless) | |

: | Polar angle (radians) | |

: | Wavelength | |

: | Reflectivity of a material (dimensionless) | |

: | DC conductivity of a material | |

: | Indicates a parameter is directional and corresponds to the surface normal | |

: | Indicates a parameter is directional and corresponds to the direction parallel to a surface |

# Introduction

Radiative properties are strong functions of a material’s electrical conductivity and therefore calculation methods for properties such as emissivity are typically generalised for either conductors or non-conductors.

Conductors and more specifically metals, behave similarly to non-metals for short wavelengths but tend to have lower emittances (shown below) with more regular spectral dependence at longer wavelengths such as the infrared part of the spectrum.

The emissivity of conductors is relatively well described by electromagnetic theory, particularly at moderate temperatures. Therefore for most applications the calculation methods described in the subsequent sections will be adequate.

# Overview of Theories for the Calculation of Emissivity

Several theories are referenced when describing the calculation methods of each emissivity variant. While a detailed description of the theory is beyond the scope of this article, a brief overview of the key theories is provided below for context. It should be noted that these theories describe either the movement of electrons within a material which is key to understanding the a materials response to radiation or the behaviour of radiation as it intersects media.

**Maxwell’s equations** - A set of partial differential equations that describe how electric charges and currents can create electric and magnetic fields. The Maxwell equations form the basis of classical electromagnetic theory and demonstrate how electromagnetic radiation propagates at the speed of light.

**Fresnel’s equations** - Describes the behaviour of electromagnetic radiation when moving between media of differing refractive indices. Fresnel’s equations allow the reflection of radiation to be predicted and when coupled with Kirkhoff’s law allow the determination of the spectral-directional emissivity of an opaque medium in a vacuum.

**Lorentz Model** - Describes the absorption of radiation in dielectric materials (electrical insulator that become polarized when placed in an electric field). Lorentz treated electrons as simple harmonic oscillators and postulated that electromagnetic fields such as radiation are the driving force for the oscillation.

**Drude Model** - Describes the movement of electrons in conductors and provides methods for the approximation of a metal’s complex refractive index and absorptivity which can be used to calculate emissivity. At longer wavelengths (generally greater than
) the Drude model can be simplified to the Hagen-Rubens relation.

**Hagen-Ruben relation** - Is a simplification of the Drude model and describes the relationship between the coefficient of reflection and the conductivity of a material.

# General Emissivity Calculation

## Spectral Emissivity

For wavelengths in the infrared region the spectral emittance of a metal in the direction of the normal may be approximated using the Hagen-Rubens relation as follows:

Here is the refractive index, is the dc conductivity of the material in and is the wavelength of the radiation in .

Experimental comparison has indicated that that the Hagen-Rubens relationship provides a good approximation of the radiative properties of (not quite optically smooth) polished metals, however for optically smooth metal surfaces such as vapour-deposited aluminium on glass the more sophisticated Drude theory is recommended.

## Spectral-Directional Emissivity (Real Surface)

The spectral-directional emissivity for optically smooth surfaces can be approximated by first calculating the spectral-directional reflectance using the Fresnel relations. For wavelengths in the infrared region of the spectrum, the real and imaginary components of the complex refractive index are large for metals and allowing the Fresnel’s relations can be simplified as follows:

Here and are the spectral reflectance of electromagnetic radiation that has been polarised in the parallel and perpendicular planes respectively. As radiative heat transfer is often due to exposure of many randomly orientated waves the spectral-directional emittance and reflectance can be evaluated from these two valves as follows:

While at a glance it might not be immediately clear how these relationships have a spectral dependence, remember that the refractive index is a function of wavelength.

An application of these equations to determine the spectral-directional reflectance (and therefore emissivity) of room temperature Titanium at a wavelength of is shown below.

Here the refractive index, and the extinction coefficient .

As seen in in the figure above the reflectance is large for near normal incidence but has a sharp drop at incident angles of . The drop in reflectance has a corresponding increase in emissivity which is consistent with the general conductor hemispherical emissivity presented earlier.

## Spectral Hemispherical Emissivity (Diffuse Surface)

The hemispherical emittance of metals can be obtained by integrating the equations for and over all directions . This is most easily done using numerical integration (noting that and are not ratios and can have values greater than 1).

For example numerically integrating and from the figure above using the trapezoid rule gives us the following result:

## Total Hemispherical Emissivity (Diffuse, Gray Surface)

The total hemispherical emissivity for metals can be obtained by integrating the Hagen-Rubens relation for normal spectral emissivity ( ) over all wavelengths then multiplying the result by the ratio of hemispherical emittance to normal emittance as calculated using the spectral-directional emissivity formulas. This gives the approximation for total hemispherical emittance of metals as:

As this equation is derived from the Hagen-Rubens relation it is subject to the same limitations and therefore best applied to polished metals where peak blackbody emission is at longer wavelengths.

# Further Reading

- Heat Transfer Handbook
- Heat Transfer A Practical Approach
- Chemical Engineering Volume 1, Sixth Edition: Fluid Flow, Heat Transfer and Mass Transfer
- Radiative Heat Transfer, Third Edition

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