6.4 Radiation

Thermal Radiation Fundamentals

Radiation is the one mode of heat transfer that doesn't need a medium. Unlike conduction and convection, radiative energy travels as electromagnetic waves and can move through a vacuum. This is how the Sun heats the Earth across empty space, and it becomes the dominant heat transfer mechanism at high temperatures (think furnaces, combustion chambers, and re-entry vehicles).

This section covers the physics behind thermal radiation, the key laws that govern it, and how to calculate radiative heat exchange between surfaces.

Blackbody Radiation and Planck's Law

Thermal radiation is electromagnetic radiation emitted by all matter due to its temperature. Every object above absolute zero emits some thermal radiation.

A blackbody is an idealized surface that absorbs all incident radiation and emits the maximum possible radiation at every wavelength for a given temperature. No real surface does this perfectly, but it sets the theoretical upper limit.

The spectral distribution of blackbody radiation shifts with temperature. Planck's law describes the spectral emissive power of a blackbody as a function of wavelength $\lambda$ and absolute temperature $T$:

$E_{\lambda,b} = \frac{C_1}{\lambda^5 \left[\exp\left(\frac{C_2}{\lambda T}\right) - 1\right]}$

where $C_1$ and $C_2$ are radiation constants. You don't usually need to evaluate this integral directly in an intro course, but you should understand what it tells you:

• At higher temperatures, the peak of the emission spectrum shifts to shorter wavelengths (this is Wien's displacement law: $\lambda_{\max} T = 2898 \; \mu\text{m·K}$)
• The total energy emitted increases dramatically with temperature (proportional to $T^4$)

Emissivity and Surface Properties

Real surfaces don't emit as much radiation as a blackbody. Emissivity ($\varepsilon$) quantifies how close a real surface comes to blackbody behavior. It's defined as:

$\varepsilon = \frac{\text{actual radiation emitted}}{\text{blackbody radiation at the same temperature}}$

Emissivity ranges from 0 to 1, where $\varepsilon = 1$ is a perfect blackbody.

What affects emissivity:

• Material type: Metals tend to have low emissivity (polished aluminum ≈ 0.04), while non-metals tend to have high emissivity (brick ≈ 0.93, human skin ≈ 0.95)
• Surface finish: A polished copper surface has much lower emissivity than an oxidized one
• Wavelength and temperature: Emissivity can vary across the spectrum and change with temperature

For many engineering calculations, surfaces are treated as gray bodies, meaning their emissivity is assumed constant across all wavelengths. This simplifies the math considerably.

Radiative Heat Transfer Calculations

Stefan-Boltzmann Law

The Stefan-Boltzmann law gives the total emissive power of a blackbody by integrating Planck's law over all wavelengths:

$E_b = \sigma T^4$

where:

• $E_b$ = total emissive power (W/m²)
• $\sigma$ = Stefan-Boltzmann constant = $5.67 \times 10^{-8}$ W/(m²·K⁴)
• $T$ = absolute temperature (K)

For a real surface with emissivity $\varepsilon$:

$E = \varepsilon \sigma T^4$

Notice the $T^4$ dependence. This means radiation becomes extremely important at high temperatures. Doubling the absolute temperature increases emitted energy by a factor of 16.

Radiative Heat Exchange Between Surfaces

The net radiative heat transfer between two surfaces depends on their temperatures, emissivities, areas, and how much they "see" each other (the view factor, covered below).

For two blackbody surfaces:

$Q = \sigma A_1 F_{12} (T_1^4 - T_2^4)$

where:

• $A_1$ = area of surface 1
• $F_{12}$ = view factor from surface 1 to surface 2
• $T_1, T_2$ = absolute temperatures of the two surfaces

For two gray surfaces in a simple two-surface enclosure, the calculation gets more involved because you need to account for the resistance to radiation at each surface and between them. A common simplified form for two large parallel gray plates is:

$Q = \frac{\sigma A (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1}$

The denominator acts like a "resistance" that reduces heat transfer below the blackbody case. The lower either emissivity, the less heat is exchanged.

These calculations matter for designing furnaces, boilers, combustion chambers, and any system where surfaces exchange heat at high temperatures.

View Factors in Radiative Heat Transfer

Definition and Reciprocity Relation

A view factor $F_{12}$ (also called a shape factor or configuration factor) is the fraction of radiation leaving surface 1 that directly strikes surface 2. It's purely geometric and depends only on the size, shape, and relative orientation of the two surfaces.

Key properties of view factors:

• View factors range from 0 to 1
• A view factor of 0 means surface 1 can't "see" surface 2 at all
• A view factor of 1 means all radiation leaving surface 1 hits surface 2

The reciprocity relation connects the view factors between two surfaces:

$A_1 F_{12} = A_2 F_{21}$

This is very useful. If you know $F_{12}$ and both areas, you can immediately find $F_{21}$ without any additional geometric calculation.

The summation rule states that for an enclosure of $N$ surfaces, the view factors from any surface $i$ must sum to 1:

$\sum_{j=1}^{N} F_{ij} = 1$

This accounts for all the radiation leaving surface $i$. For a flat or convex surface, $F_{ii} = 0$ (it can't see itself). A concave surface can have $F_{ii} > 0$.

Calculating View Factors

For common geometries, view factors are tabulated in heat transfer references. You'll typically look them up rather than derive them. Some examples:

• Two large parallel plates directly facing each other: $F_{12} \approx 1$ (nearly all radiation from one hits the other)
• Small surface to a much larger enclosing surface: $F_{12} = 1$ (all radiation from the small surface reaches the enclosure)
• Perpendicular plates sharing a common edge: View factor depends on the aspect ratios; charts and formulas are available

For complex geometries, view factors can be determined using:

• Tabulated charts and analytical formulas (most common in coursework)
• Numerical integration methods
• Monte Carlo ray-tracing simulations (used in practice for complicated 3D geometries)

Accurate view factors are critical for predicting radiative heat transfer in applications like spacecraft thermal control, solar energy systems, and industrial furnace design.

Surface Properties and Radiative Exchange

Kirchhoff's Law and Surface Properties

Three properties describe how a surface interacts with incident radiation:

• Absorptivity ($\alpha$): fraction of incident radiation absorbed
• Reflectivity ($\rho$): fraction of incident radiation reflected
• Transmissivity ($\tau$): fraction of incident radiation transmitted through the material

For an opaque surface ($\tau = 0$, which covers most solids in engineering):

$\alpha + \rho = 1$

Kirchhoff's law states that at thermal equilibrium, for a given temperature and wavelength:

$\varepsilon_\lambda = \alpha_\lambda$

A surface that's a good emitter at a particular wavelength is equally good at absorbing that wavelength. For gray surfaces (where properties don't vary with wavelength), this simplifies to $\varepsilon = \alpha$, which means:

$\rho = 1 - \varepsilon$

This is why shiny, low-emissivity surfaces (like polished aluminum) are also highly reflective and poor absorbers.

Selective Surfaces and Applications

Selective surfaces are engineered to have different radiative properties at different wavelengths. This is possible because solar radiation and thermal radiation from room-temperature objects occupy different parts of the electromagnetic spectrum.

Solar collectors use surfaces with:

• High absorptivity in the visible and near-infrared range (0.3–2.5 μm), where most solar energy arrives
• Low emissivity in the mid- and far-infrared range (>2.5 μm), where the collector would otherwise lose heat by re-radiation

This combination maximizes energy capture while minimizing thermal losses.

Low-emissivity (low-e) coatings on windows and reflective foil insulation work the opposite way for thermal management. They reflect infrared radiation, reducing radiative heat transfer through building envelopes or around hot pipes.

Other applications include:

• Thermophotovoltaic systems, where a high-temperature emitter with a tailored emission spectrum drives photovoltaic cells to generate electricity
• Spacecraft thermal control, where surfaces are coated to balance solar absorption and thermal emission in the vacuum of space

Understanding how to manipulate surface properties gives engineers a powerful tool for controlling radiative heat transfer across a wide range of applications.