🌬️Heat and Mass Transport Unit 4 – Radiation Heat Transfer

Key Concepts and Definitions

• Radiation heat transfer involves the transfer of energy through electromagnetic waves without requiring a medium
• Electromagnetic spectrum encompasses a wide range of wavelengths, including visible light, infrared, and ultraviolet radiation
• Blackbody an idealized surface that absorbs all incident radiation and emits the maximum amount of energy at a given temperature
• Emissivity ($\varepsilon$) a material property that quantifies the ability of a surface to emit radiation relative to a blackbody (ranges from 0 to 1)
  • Real surfaces have emissivities less than 1, while a blackbody has an emissivity of 1
• Absorptivity ($\alpha$) the fraction of incident radiation that a surface absorbs (ranges from 0 to 1)
  • A blackbody has an absorptivity of 1, absorbing all incident radiation
• Reflectivity ($\rho$) the fraction of incident radiation that a surface reflects (ranges from 0 to 1)
• Transmissivity ($\tau$) the fraction of incident radiation that a surface transmits (ranges from 0 to 1)
  • For opaque surfaces, transmissivity is zero

Fundamental Laws and Equations

• Stefan-Boltzmann law quantifies the total radiant heat energy emitted by a blackbody per unit area and time: $E_b = \sigma T^4$
  • $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m²·K⁴)
  • $T$ is the absolute temperature of the surface (in Kelvin)
• Planck's law describes the spectral distribution of blackbody radiation as a function of wavelength and temperature
• Wien's displacement law states that the wavelength of maximum emission from a blackbody is inversely proportional to its temperature: $\lambda_{\max} = \frac{2898 \mu m \cdot K}{T}$
• Kirchhoff's law of thermal radiation relates the emissivity and absorptivity of a surface at a given temperature and wavelength: $\varepsilon_{\lambda} = \alpha_{\lambda}$
• Net radiation heat transfer between two surfaces depends on their temperatures and radiative properties: $Q_{net} = \varepsilon \sigma A (T_1^4 - T_2^4)$
  • $\varepsilon$ is the emissivity of the surface
  • $A$ is the surface area
  • $T_1$ and $T_2$ are the absolute temperatures of the surfaces

Types of Radiation Heat Transfer

• Surface radiation occurs when electromagnetic waves are emitted, absorbed, or reflected by surfaces
  • Emission originates from the thermal energy of matter, with the rate depending on surface temperature and emissivity
• Gas radiation involves the emission and absorption of radiation by gases, such as carbon dioxide and water vapor
  • Gases can be transparent to certain wavelengths while absorbing or emitting others
• Participating media radiation considers the interaction of radiation with matter within a medium, such as in furnaces or combustion chambers
  • Scattering, absorption, and emission processes can occur within the medium
• Solar radiation the energy emitted by the sun, which can be harnessed for various applications (solar thermal collectors, photovoltaic cells)
  • Earth's atmosphere absorbs and scatters a portion of the incoming solar radiation
• Thermal radiation the electromagnetic radiation emitted by matter due to its temperature
  • All objects with a temperature above absolute zero emit thermal radiation

Properties of Radiating Surfaces

• Emissivity depends on factors such as material composition, surface finish, temperature, and wavelength
  • Polished metals generally have low emissivities, while rough and oxidized surfaces have higher emissivities
• Selective surfaces have emissivities that vary significantly with wavelength
  • Used in applications where specific wavelength ranges need to be absorbed or emitted (solar collectors, thermal insulation)
• Diffuse surfaces reflect radiation equally in all directions, following Lambert's cosine law
  • Matte surfaces exhibit diffuse behavior
• Specular surfaces reflect radiation in a mirror-like manner, with the angle of reflection equal to the angle of incidence
  • Polished surfaces and mirrors exhibit specular behavior
• Gray surfaces have emissivities that are independent of wavelength
  • Simplifies radiation heat transfer calculations
• Real surfaces often exhibit a combination of diffuse and specular characteristics and have emissivities that vary with wavelength and temperature

View Factors and Geometric Considerations

• View factor ($F_{ij}$) represents the fraction of radiation leaving surface $i$ that is intercepted by surface $j$
  • Also known as shape factor or configuration factor
• Reciprocity relation states that the product of the view factor and area for two surfaces is equal: $A_i F_{ij} = A_j F_{ji}$
• Summation rule the sum of all view factors from a surface $i$ to all other surfaces in an enclosure, including itself, is equal to unity: $\sum_{j=1}^{n} F_{ij} = 1$
• View factors depend on the size, shape, and orientation of the surfaces involved
  • Analytical expressions exist for simple geometries (parallel plates, perpendicular plates, concentric cylinders)
• Crossed-strings method a graphical technique for determining view factors between surfaces using crossed strings and uncrossed strings
• Contour integral method an analytical approach for calculating view factors using double integrals over the surfaces
• Hottel's crossed-string method an algebraic method for calculating view factors in enclosures with three or more surfaces

Radiation Heat Exchange Between Surfaces

• Net radiation heat exchange between two surfaces depends on their temperatures, emissivities, and view factor
  • $Q_{1\rightarrow2} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1-\varepsilon_1}{A_1 \varepsilon_1} + \frac{1}{A_1 F_{12}} + \frac{1-\varepsilon_2}{A_2 \varepsilon_2}}$
• Radiosity ($J$) the total radiation energy leaving a surface per unit area and time, including emitted and reflected radiation
  • $J_i = \varepsilon_i \sigma T_i^4 + (1-\varepsilon_i) \sum_{j=1}^{n} F_{ij} J_j$
• Irradiation ($G$) the total radiation energy incident upon a surface per unit area and time
  • $G_i = \sum_{j=1}^{n} F_{ij} J_j$
• Radiation network method models the radiation heat exchange between surfaces using a network of resistances and potentials
  • Surfaces are represented as nodes, and view factors and emissivities determine the resistances between nodes
• Enclosure analysis involves solving a system of equations for the radiosities or heat transfer rates in an enclosure with multiple surfaces
  • Matrix inversion or iterative methods can be used to solve the system of equations

Applications in Engineering and Industry

• Thermal insulation materials with low emissivities (reflective foils) can reduce radiation heat transfer and improve energy efficiency
• Solar thermal collectors harness solar radiation to heat fluids for various applications (water heating, space heating, power generation)
  • Selective surfaces with high absorptivity in the solar spectrum and low emissivity in the infrared spectrum enhance collector efficiency
• Radiative cooling systems exploit the emission of infrared radiation to the cold sky to achieve cooling without the need for external power
  • Used in passive cooling of buildings and electronic devices
• Furnaces and combustion chambers involve radiation heat transfer between high-temperature gases, flames, and surfaces
  • Proper design and material selection can optimize heat transfer and energy efficiency
• Spacecraft thermal control relies on radiation heat transfer to maintain acceptable temperature ranges in the harsh space environment
  • Radiators, insulation, and surface coatings are used to manage heat rejection and absorption
• Greenhouse effect the trapping of infrared radiation by atmospheric gases, leading to increased surface temperatures on Earth
  • Relevant to climate change and global warming studies

Problem-Solving Techniques and Examples

• Identify the mode(s) of radiation heat transfer involved (surface, gas, participating media)
• Determine the relevant surface properties (emissivity, absorptivity, reflectivity) and geometric factors (view factors, surface areas)
• Apply the appropriate equations or methods for the given scenario (Stefan-Boltzmann law, radiosity, radiation network)
  • Example: Calculate the net radiation heat transfer between two parallel plates with known temperatures and emissivities
• Simplify the problem by making reasonable assumptions (gray surfaces, diffuse surfaces, nonparticipating media)
  • Example: Assume a gas mixture is nonparticipating to focus on surface radiation heat transfer
• Use tabulated data, charts, or correlations to obtain view factors for common geometries
  • Example: Determine the view factor between two perpendicular rectangles using a chart or analytical expression
• Solve systems of equations for enclosures using matrix inversion or iterative methods
  • Example: Set up and solve the radiosity matrix for a three-surface enclosure
• Interpret the results and consider the implications for the specific application or system
  • Example: Evaluate the effectiveness of a radiative cooling system based on the calculated heat rejection rate