Radiation fundamentals

Tomás's infrared survey of the boiler hall raised a puzzle in Lesson 1: surfaces at the same temperature looked very different on the camera's screen. The explanation lies in how surfaces emit, absorb and reflect thermal radiation, the subject of this lesson. Radiation dominates in Northgate's boiler furnaces, matters as much as convection for hot pipes and room heaters, sets the temperature of a sunlit roof and a clear-night windshield, and is the principle behind every infrared thermometer and camera. We start with the ideal emitter, the blackbody, and then describe how real surfaces depart from it.

The nature of thermal radiation

Thermal radiation is electromagnetic radiation emitted by matter because of its temperature. It occupies roughly the wavelength range from 0.1 to 100 μm, which includes part of the ultraviolet, all of the visible band from about 0.38 to 0.76 μm, and the infrared. Radiation travels at the speed of light, \(c = \lambda\nu\), and in a vacuum or in air it passes without loss. Unlike conduction and convection, which depend on temperature differences, emission depends on the absolute temperature of the emitter alone; net exchange between surfaces depends on the difference of their fourth powers.

Radiation from a surface goes out in all directions and at all wavelengths. To describe it completely we would need its spectral and directional distribution. For most engineering work it is enough to know the total emissive power \(E\) (W/m²), the rate of emission per unit area integrated over all wavelengths and directions, and the irradiation \(G\), the rate at which radiation arrives at a surface per unit area. Surfaces whose emission is the same in all directions are called diffuse, and most rough industrial surfaces are close to diffuse.

The blackbody

A blackbody is an ideal surface that absorbs all incident radiation, whatever its wavelength or direction, and for a given temperature and wavelength emits more than any other surface. Its emission is diffuse. A small hole in a large isothermal cavity is very nearly a blackbody, which is how calibration sources for infrared thermometers are built. Planck's law gives the blackbody's spectral emissive power:

At each temperature the spectrum rises to a peak and falls off. As temperature rises, emission increases at every wavelength and the peak shifts to shorter wavelengths, a result known as Wien's displacement law: \(\lambda_{\max}T = 2898\) μm·K. The sun, approximately a blackbody at 5800 K, peaks at 0.50 μm in the middle of the visible band; a flame at 1800 K peaks at 1.6 μm in the near infrared and looks orange because only the short-wavelength tail of its emission is visible; a steam main at 450 K peaks at 6.4 μm; the ground on a winter night peaks near 12 μm. Integrating Planck's law over all wavelengths gives the Stefan-Boltzmann law, \(E_b = \sigma T^4\).

Band emission

It is often necessary to know what fraction of a blackbody's emission falls in a band of wavelengths, for example the fraction of sunlight that is visible or the fraction of a pipe's emission that an infrared camera's detector can see. Because Planck's law depends on \(\lambda\) and \(T\) only through the product \(\lambda T\), the fraction emitted below a wavelength depends only on \(\lambda T\): \(F_{(0\to\lambda)} = \int_0^\lambda E_{\lambda,b}\,d\lambda/\sigma T^4\). This function is tabulated, and it can be computed from a rapidly converging series. The fraction in a band from \(\lambda_1\) to \(\lambda_2\) is then \(F_{(0\to\lambda_2)} - F_{(0\to\lambda_1)}\).

# Radiation fundamentals: blackbody emission, band fractions, surface properties and an infrared survey
import math
sigma, C1, C2, C3 = 5.670e-8, 3.742e8, 1.4388e4, 2897.8     # W/m2.K4; W.um4/m2; um.K; um.K

def planck(lam, T):                                           # spectral emissive power, W/m2.um
    return C1 / (lam**5 * (math.exp(C2 / (lam * T)) - 1))
def F0(lamT):                                                 # fraction of blackbody emission below lambda*T
    z = C2 / lamT
    return 15 / math.pi**4 * sum(math.exp(-n*z) / n * (z**3 + 3*z**2/n + 6*z/n**2 + 6/n**3) for n in range(1, 60))
def band(l1, l2, T): return F0(l2 * T) - F0(l1 * T)

for name, T in (("sun", 5800), ("furnace flame", 1800), ("steam main", 450), ("ground in winter", 250)):
    lmax = C3 / T
    print(f"{name:16s} T = {T:5d} K: E_b = {sigma*T**4/1000:9.3f} kW/m2, peak at {lmax:6.2f} um, "
          f"visible 0.38-0.76 um {band(0.38,0.76,T):6.1%}, beyond 3 um {1-F0(3*T):6.1%}")

# Check the band function against direct numerical integration of Planck's law for the sun
T, n = 5800.0, 4000
h = (0.76 - 0.38) / n
num = sum(planck(0.38 + (i + 0.5) * h, T) for i in range(n)) * h / (sigma * T**4)
print(f"visible fraction of sunlight by numerical integration: {num:.4f} (series: {band(0.38,0.76,T):.4f})")

# Equilibrium temperature of a sunlit plate, insulated behind, with only radiation exchange to a 270 K sky
G = 900.0
for name, a_s, eps in (("white paint", 0.20, 0.90), ("black paint", 0.95, 0.95), ("selective absorber", 0.95, 0.10)):
    Teq = (a_s * G / (eps * sigma) + 270.0**4) ** 0.25
    print(f"{name:18s} alpha_s/eps = {a_s/eps:5.2f}: no-convection equilibrium {Teq - 273.15:6.1f} C")

# Infrared survey of the steam main: camera set to emissivity 1.0 sees apparent temperatures
T_refl = 22 + 273.15
for surf, eps, T_app in (("painted pipe", 0.93, 168.0), ("bare stainless flange", 0.15, 55.0), ("aluminum jacket", 0.10, 23.5)):
    Ta = T_app + 273.15
    T_true = ((Ta**4 - (1 - eps) * T_refl**4) / eps) ** 0.25
    print(f"{surf:22s} eps = {eps:.2f}: camera shows {T_app:5.1f} C, surface is really about {T_true - 273.15:6.1f} C")
print(f"fraction of a 450 K surface's emission inside an 8-14 um camera band: {band(8, 14, 450):.1%}")

sun              T =  5800 K: E_b = 64164.532 kW/m2, peak at   0.50 um, visible 0.38-0.76 um  44.8%, beyond 3 um   2.1%
furnace flame    T =  1800 K: E_b =   595.214 kW/m2, peak at   1.61 um, visible 0.38-0.76 um   0.7%, beyond 3 um  32.0%
steam main       T =   450 K: E_b =     2.325 kW/m2, peak at   6.44 um, visible 0.38-0.76 um   0.0%, beyond 3 um  99.4%
ground in winter T =   250 K: E_b =     0.221 kW/m2, peak at  11.59 um, visible 0.38-0.76 um   0.0%, beyond 3 um 100.0%
visible fraction of sunlight by numerical integration: 0.4484 (series: 0.4484)
white paint        alpha_s/eps =  0.22: no-convection equilibrium   33.5 C
black paint        alpha_s/eps =  1.00: no-convection equilibrium  108.4 C
selective absorber alpha_s/eps =  9.50: no-convection equilibrium  355.4 C
painted pipe           eps = 0.93: camera shows 168.0 C, surface is really about  174.5 C
bare stainless flange  eps = 0.15: camera shows  55.0 C, surface is really about  157.2 C
aluminum jacket        eps = 0.10: camera shows  23.5 C, surface is really about   36.1 C
fraction of a 450 K surface's emission inside an 8-14 um camera band: 35.8%

Real surfaces: emissivity

Real surfaces emit less than a blackbody. The total emissivity is the ratio \(\varepsilon = E/E_b\), between 0 and 1. It depends on the material, the surface finish, the temperature and, in general, the wavelength and direction. Polished metals have low emissivities, about 0.02 to 0.1, because their free electrons reflect most radiation; oxidation, roughness and dirt raise them sharply. Non-metals, including paints of any color, brick, concrete, water, skin and most plastics, have high emissivities in the infrared, about 0.85 to 0.95. Color describes behavior in the visible band only and says almost nothing about infrared emissivity: white paint and black paint both emit strongly at room temperature.

Absorption, reflection and transmission

Radiation arriving at a surface is partly absorbed, partly reflected and, if the material is semitransparent, partly transmitted. The fractions are the absorptivity \(\alpha\), reflectivity \(\rho\) and transmissivity \(\tau\), and they sum to one: \(\alpha + \rho + \tau = 1\). For opaque surfaces, \(\tau = 0\). Glass is transparent between about 0.3 and 2.5 μm but nearly opaque at longer wavelengths, so it admits sunlight and blocks the long-wave radiation emitted by the interior, which is how a greenhouse, a sunroom or a parked car heats up.

Kirchhoff's law links absorption to emission. For a surface in thermal equilibrium with its surroundings, the spectral, directional emissivity equals the spectral, directional absorptivity. For a diffuse surface whose properties do not depend on wavelength, called a gray surface, it follows that total \(\varepsilon = \alpha\). Real surfaces are rarely gray over all wavelengths, but many are close to gray over the range in which they both emit and absorb, for example a pipe exchanging radiation with room-temperature walls. Treating them as gray is the basis of the exchange calculations in Lesson 11.

Solar absorptivity and selective surfaces

The gray approximation fails when the incoming and outgoing radiation lie in different bands, as with sunlight falling on a surface near room temperature. Such a surface absorbs in the solar band, characterized by its solar absorptivity \(\alpha_s\), and emits in the infrared, characterized by \(\varepsilon\). The ratio \(\alpha_s/\varepsilon\) controls its temperature in the sun. White paint has a low \(\alpha_s\), about 0.2, and a high \(\varepsilon\), about 0.9, so it stays cool; this is the basis of cool roofs. Black paint has both near 0.95. Selective absorbers for solar collectors combine a high \(\alpha_s\), about 0.95, with a low infrared \(\varepsilon\), about 0.1, so they absorb sunlight well and lose little by re-emission.

Infrared thermography

An infrared camera measures the radiation reaching its detector from each point of the scene, usually in a band such as 8 to 14 μm, and converts it to a temperature using an emissivity that the operator sets. If the true emissivity is lower than the setting, the surface emits less than the camera assumes and also reflects radiation from its surroundings. The camera then reads a temperature between the true surface temperature and the reflected temperature. A simple total-radiation correction, accurate enough to show the size of the error though not a substitute for the camera's own band-based software, is

In the United States, infrared thermographers are commonly trained and certified under the American Society for Nondestructive Testing practice for personnel qualification, with ASTM standards covering building envelope and electrical surveys. In US units, the blackbody emissive power at 450 K is about 740 Btu/h·ft², and the Stefan-Boltzmann constant is 0.1714 × 10⁻⁸ Btu/h·ft²·R⁴.