The Stefan-Boltzmann Law is a key concept in heat transfer, describing how objects emit thermal radiation. It shows that the total heat power emitted by a surface is directly related to its temperature, raised to the fourth power. This relationship is crucial for understanding radiative heat transfer.

Real-world surfaces aren't perfect emitters, so we use emissivity to describe how well they radiate compared to an ideal blackbody. This concept helps us apply the Stefan-Boltzmann Law to practical situations, from everyday objects to complex engineering problems.

Stefan-Boltzmann Law

Mathematical Representation and Physical Meaning

• The Stefan-Boltzmann Law states that the total radiant heat power emitted from a surface is proportional to the fourth power of its absolute temperature
• The mathematical representation of the Stefan-Boltzmann Law is given by the equation: $Q = σ * A * T^4$
  • $Q$ is the heat transfer rate (W)
  • $σ$ is the Stefan-Boltzmann constant ($5.67 × 10^{-8} W/m^2⋅K^4$)
  • $A$ is the surface area ($m^2$)
  • $T$ is the absolute temperature (K)
• The Stefan-Boltzmann constant, $σ$, is a fundamental physical constant that relates the total radiant heat power to the absolute temperature of a surface
• The Stefan-Boltzmann Law applies to ideal blackbodies, which are perfect emitters and absorbers of thermal radiation

Emissivity and Real Surfaces

• Real surfaces are characterized by their emissivity, which is the ratio of the radiation emitted by the surface to that emitted by a blackbody at the same temperature
• Emissivity values range from 0 to 1, with a perfect blackbody having an emissivity of 1
• Examples of surfaces with high emissivity (close to 1) include rough, oxidized, or non-metallic surfaces such as asphalt, brick, and human skin
• Examples of surfaces with low emissivity (close to 0) include highly polished metals such as polished aluminum and gold

Blackbody Radiation

Characteristics of Blackbody Radiation

• A blackbody is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence
• Blackbody radiation is the thermal electromagnetic radiation emitted by a blackbody in thermodynamic equilibrium with its environment
• The radiation emitted by a blackbody depends only on the object's temperature and follows the Stefan-Boltzmann Law
• Examples of near-ideal blackbodies include a cavity with a small hole (cavity radiator) and the cosmic microwave background radiation

Relationship to Planck's Law and the Stefan-Boltzmann Law

• The spectral radiance of a blackbody is described by Planck's Law, which gives the intensity of radiation emitted by a blackbody as a function of wavelength for a given temperature
• The Stefan-Boltzmann Law is derived by integrating Planck's Law over all wavelengths, yielding the total radiant heat power emitted by a blackbody per unit area
• Real surfaces are not perfect blackbodies, but their radiation characteristics can be described using the concept of emissivity, which relates their emission to that of a blackbody at the same temperature
• Examples of the application of Planck's Law include the color of stars (related to their surface temperature) and the peak wavelength of the sun's emission spectrum

Thermal Radiation Heat Transfer

Calculating Heat Transfer Rate between Surfaces

• To calculate the net heat transfer rate between two surfaces using the Stefan-Boltzmann Law, consider the difference in the radiant heat power emitted by each surface
• The net heat transfer rate from surface 1 to surface 2 is given by: $Q_{net} = σ * A * (T_1^4 - T_2^4)$
  • $T_1$ and $T_2$ are the absolute temperatures of surfaces 1 and 2, respectively
• When dealing with real surfaces, the emissivity ($ε$) of each surface must be considered
  • The modified equation becomes: $Q_{net} = σ * A * ε (T_1^4 - T_2^4)$, assuming both surfaces have the same emissivity
• For heat transfer between a surface and its surroundings, the surroundings are often treated as a blackbody at a specific temperature
  • The net heat transfer rate is given by: $Q_{net} = σ * A * ε (T^4 - T_{surr}^4)$, where $T_{surr}$ is the absolute temperature of the surroundings

View Factors in Radiative Heat Transfer

• When multiple surfaces are involved in radiative heat transfer, the view factors between the surfaces must be considered
• The view factor ($F_{ij}$) is the fraction of radiation leaving surface $i$ that is intercepted by surface $j$
• View factors depend on the geometry and orientation of the surfaces involved
• Examples of view factor calculations include the heat transfer between parallel plates, concentric cylinders, and a small surface enclosed by a larger surface

Emissivity and Absorptivity

Factors Influencing Emissivity

• Emissivity ($ε$) is a material property that represents the ratio of the radiation emitted by a surface to that emitted by a blackbody at the same temperature
• Emissivity depends on the surface material, finish, and temperature
  • Highly polished metals generally have low emissivities, while rough, oxidized, or non-metallic surfaces have higher emissivities
  • Examples of low-emissivity surfaces include polished aluminum ($ε ≈ 0.05$) and polished gold ($ε ≈ 0.03$)
  • Examples of high-emissivity surfaces include asphalt ($ε ≈ 0.90$), red brick ($ε ≈ 0.93$), and human skin ($ε ≈ 0.98$)
• Emissivity can vary with the wavelength of the emitted radiation and the temperature of the surface
  • For example, the emissivity of a material may be different in the visible spectrum compared to the infrared spectrum

Absorptivity and Kirchhoff's Law

• Absorptivity ($α$) is the fraction of incident radiation absorbed by a surface
• According to Kirchhoff's Law of thermal radiation, the emissivity and absorptivity of a surface are equal at thermal equilibrium for a given wavelength and direction
• The color of a surface affects its absorptivity and emissivity in the visible spectrum
  • Dark-colored surfaces generally have higher absorptivities and emissivities than light-colored surfaces
  • For example, a black surface absorbs more visible light than a white surface
• Surface roughness can increase both the absorptivity and emissivity of a surface by increasing the surface area and promoting multiple reflections of incident radiation
• The angle of incidence of the radiation on the surface can affect the absorptivity, with normal incidence generally resulting in higher absorptivity than oblique incidence