Radiation Basics: Making Sense of Emissivity & Absorptivity

Introduction

This is the second installment in a series of articles that aims to explore a range of practical topics on radiation that are relevant to those of us focused on electronics cooling and thermal design.

The last article [3] provided guidance regarding when radiation matters, with examples being natural convection environments, space applications, solar collectors, and many more.

Let’s start with equation 1, which quantifies the net radiative heat rate between a grey body and its surroundings. In this equation Q is the heat rate in Watts, ε is the emissivity of the surface, F is the view factor between the surface and its surroundings, A is the surface area, σ is the Stefan-Boltzmann constant (5.67e-8 W/m²-K⁴), T is the surface temperature of the body in K, and T_∞ is the temperature of the surroundings in K.

This is one of the most frequently used radiation equations by the practicing thermal engineer. To use it correctly, one must understand each of the factors. In this article we will deep dive into emissivity, ε.

While you have probably heard of several different types of emissivity, do you know which one is assumed in the above equation? Do you recall what is meant by the grey body assumption?

The various emissivity definitions and the radiation assumptions have confused me in the past, so this is an attempt to demystify emissivity and absorptivity for the reader.

Emissivity

Now that we have identified a common equation where emissivity is used, let’s step back to a few fundamental definitions.

The total emissive power, E, (W/m²) is the rate at which radiation is emitted from a surface when integrated over all wavelengths and directions and is defined in equation 2.

The theoretical maximum emissive power that can be emitted will occur from a blackbody (E_b) as a function of its temperature. As illustrated in Figure 1, a blackbody is a perfect absorber and emitter.

 

The emissivity in equation 2, ε, is often referred to as the total hemispherical emissivity, or the emissivity integrated over all angles and wavelengths. Emissivity is essentially the ratio of the emissive power of a real body, compared to that of a blackbody. Thus, emissivity always falls between 0 and 1.

While the total hemispherical emissivity is convenient for quick engineering calculations, it will not be appropriate for all situations. More generally, emissivity can be a function of temperature, wavelength, and direction.

Most engineering surfaces can be reasonably assumed to behave as diffuse surfaces, in which the emissivity is independent of direction (θ,ϕ). For simplicity, we will assume the radiation surfaces are diffuse in the remainder of this article. The reader should always verify that this assumption holds for their situation, specifically with specular surfaces or with significant radiative transfer occurring at highly non-normal angles. For a diffuse surface, the total spectral emissivity (ε_λ) is defined per equation 4, where ε_λ is the emissive power of a real surface at a specific wavelength, and E_b,_λ is the emissive power of a blackbody at the same wavelength.

One can always recover the total hemispherical emissivity (ε) from the total spectral emissivity (ε_λ) by integrating it across all wavelengths, as illustrated in equation 5. This calculation will be demonstrated shortly.

In this short section we have discussed various forms of emissivity, from the most general definition that is a function of many variables (ε_λ,θ(T,λ,θ,ϕ), to a single integrated value (ε). It is always important to identify the type of emissivity required to appropriately represent your situation. Care must also be taken to ensure you understand the meaning of emissivity values found in the literature.

Absorptivity

When dealing with radiation between interacting bodies, our discussion of emissivity must be extended to include absorptivity, as described in equation 1.

The light arriving at any surface must either be transmitted, reflected, or absorbed. This is mathematically described by equation 6, where α is the absorptivity, ρ is the reflectivity, and τ is the transmissivity.

Many engineering surfaces can be considered opaque, in which there is zero transmission through the body (τ=0), and the radiation exchange can be assumed to be limited to the surface. This results in equation 7.

While this is appropriate for the painted surfaces that will be considered in this article, it is not appropriate for all engineering applications. Examples include transparent materials such as water, glass, or some plastics. It must also be remembered that radiation occurs outside of the visible spectrum, so this may not hold for some visibly opaque materials (e.g. opaque IR transparent plastics).

Now, how do we relate absorptivity to emissivity? This is where Kirchoff’s Law comes in (equation 8). Kirchoff’s Law states that the total hemispherical emissivity (ε) and total hemispherical absorptivity (α) are equal for surfaces in an isothermal enclosure at thermal equilibrium (like the body in the Figure 1 enclosure). Said in a different way, the body is irradiated by a blackbody at the same temperature as the body itself.

This relation also holds for the spectral properties of each surface in the enclosure, per equation 9. Equation 9 holds so long as either the irradiation and/or the surface is diffuse.

The above relation greatly simplifies the calculation of radiant energy exchange between surfaces, enabling equation 1 to be used. Equation 1 holds so long as either of the following criteria are met.

• The surface is only irradiated by a blackbody that is maintained at the same temperature as the surface in question.
• The surface is grey, meaning that αλ and ελ are constant over the range of wavelengths relevant to the problem.

Solar Absorptivity

While Kirchoff’s Law holds for many engineering applications, solar heating is one scenario that frequently breaks this assumption. In typical terrestrial applications, radiation is emitted from a stationary body at a temperature on the order of ~300K. While the terrestrial surroundings irradiating the body may also be on the order of ~300K, the sun is a different story. The sun acts as a blackbody irradiation source at ~5800K. In these scenarios, emission to the surroundings may occur at wavelengths far removed from that of solar irradiation. This is illustrated in Figure 2, which shows the Plank distribution, normalized by the maximum spectral emissive power at the blackbody temperatures of the surroundings and the sun.

 

Given the large spectral separation in irradiation sources in terrestrial solar heating applications, it is typically necessary to modify equation 1 into equation 10. The additional term accounts for the solar flux (q_s) that is typically on the order of ~1000 W/m², the illuminated area (A), and the solar absorptivity (α_s).

As you can see, the emissivity is no longer sufficient to solve the problem. The solar absorptivity is now also required (α_s). The solar absorptivity can be measured via processes similar to that of emissivity, only integrated and weighted for the solar spectrum (0.3um to 2.5um). The reader is referred to reference [2] for detailed solar absorptivity measurement and calculation techniques.

Examples in the Real World

Let’s characterize the emissivity and solar absorptivity of some real-world surfaces to put the principles that we have talked about into practice. We will consider the generic black and a generic white painted surfaces illustrated in Figure 3.

 

There are many ways to measure the emissivity of a surface, with the appropriate measurement technique being a function of your needs (e.g. total hemispherical emissivity, total spectral emissivity, etc.).

Since the spectral dependence is of interest in this example, the total spectral reflectivity of the samples is measured in a manner like the approach described in reference [2]. This measurement leverages a spectrophotometer and integrating sphere to capture the ultraviolet to near-infrared (0.3-2.5um), as well as a FTIR and integrating sphere to capture the mid-infrared (2.5um-20um). The use of an integrating sphere, as illustrated in Figure 4, integrates the measurement across all angles.

 

The output of the measurement is the spectral reflectivity illustrated in Figure 5.

 

The spectral reflectivity can easily be converted to the spectral emissivity (or absorptivity) using equation 7. The resulting spectral emissivity is illustrated in Figure 6.

Two distinct trends are observed in the two paint samples.

• The black paint exhibits a high emissivity across the entire wavelength spectrum. The nearly constant emissivity value suggests that it will satisfy the grey body assumption over a wide range of temperatures.
• The white paint exhibits spectral variation. This is expected as the white color reflects in the visible spectrum. Interestingly, the emissivity at higher wavelengths is high. This suggests that separate emissivity and solar absorptivity values may be required to accurately represent the surface. The rapid changes with wavelength also suggest that the grey body assumption may break down at some temperature.

While the total spectral emissivity enables us to understand the surface behavior such that we make the right assumptions, it is typically convenient to work with total hemispherical emissivity in first-order calculations. Luckily, equation 5 provides us a means to convert between the two.

In practice, the total hemispherical emissivity is typically calculated using the F-Tables found in standard heat transfer textbooks, such as [1]. Figure 7 provides a visual representation of the F-Table.

We will now demonstrate the calculation for the white paint surface that we characterized above. To clearly demonstrate the process, let’s start with the approximate spectral variation shown by the dashed green line in Figure 8.

 

The spectrum is simplified into three zones that are summed in equation 11 to obtain an approximation of the total hemispherical emissivity.

Let’s start by calculating the total hemispherical emissivity for a surface at 300K. The F-values taken from Figure 7 are as follows.

Plugging the emissivity and F-value for each zone into equation 11 yields an approximate total hemispherical emissivity of 0.87. It is interesting to note that the low emissivity zone at low wavelengths has no contribution. This makes sense as the emissive power at 300K is in the mid-infrared (see Figure 2).

Let’s repeat this calculation for a temperature of 5800K. While there are more accurate methods to calculate the solar absorptivity [2], this provides a quick approximation for illustrative purposes. The F-values taken from Figure 7 are as follows.

Plugging the emissivity and F-value for each zone into equation 11 yields an approximate solar absorptivity of 0.29. In this example the main contribution is from the lower wavelengths, while the mid-infrared has negligible contribution. Again, this makes sense as the peak emissive power at 5800K is in the visible spectrum (see Figure 2).

While the simplified spectral emissivity of Figure 8 was useful for demonstrating the calculation process, an accurate calculation using the entire spectral data sets is provided for both the black and white paints in Table 1.

	Solar Absorptivity, αs	Total Hemispherical Emissivity, ε	αs / ε
White Paint1	0.19	0.91	0.21
Black Paint	0.94	0.89	1.06

Table 1: Emissivity and solar absorptivity

¹ Deviations from the approximated spectrum based on Figure 8 are expected and suggest the original approximation was too aggressive

As expected from the spectral distribution, the total hemispherical emissivity and the solar absorptivity of the black paint are similar. In contrast, the white paint’s solar absorptivity is much smaller than its total hemispherical emissivity. Surfaces with low ratios of solar absorptivity to total hemispherical emissivity (α_s/ε) are useful for electronics cooling applications as they remain effective at re-radiating to the environment, while also minimizing solar heat absorbed.

Concluding Remarks

Emissivity is an important factor in radiative heat transfer. Although it may appear simple at first glance, care should be taken to ensure that it is handled appropriately in thermal design. This article serves as a high-level checklist to get you started.

• Determine which type of emissivity value is required for your application (e.g. total hemispherical, total spectral, etc.)
• Ensure that you know the type of emissivity values that you use from literature
• Review the underlying assumptions for appropriateness in your radiation calculations (e.g. diffuse surface, grey body, opaque, blackbody, etc.)
• Watch out for scenarios where emissivity and absorptivity cannot be assumed equal, such as with solar heating

As this article only captures the tip of the iceberg, it is recommended that the interested reader refer to their favorite textbook for a deep dive into areas not sufficiently captured here.

References

[1] Frank Incropera and David DeWitt, Fundamentals of Heat and Mass Transfer, 4th Edition, Wiley (1996)

[2] ASTM E903-20, Standard Test Method for Solar Absorptance, Reflectance, and Transmittance of Materials Using Integrating Spheres, ASTM International

[3] Alex Ockfen, Radiation Basics: When Does It Matter, Electronics Cooling Magazine, September 2024