Formula Sheet: Radiation

Fundamental Radiation Concepts

Electromagnetic Radiation Spectrum

• Thermal radiation: Emission of electromagnetic waves from a surface due to its temperature
• Wavelength range: Thermal radiation occurs primarily in the range of 0.1 to 100 μm
• Speed of light: \(c = 2.998 \times 10^8\) m/s (in vacuum)

Wavelength-Frequency Relationship

\[ \lambda = \frac{c}{\nu} \] Where:

• \(\lambda\) = wavelength (m or μm)
• \(c\) = speed of light = \(2.998 \times 10^8\) m/s
• \(\nu\) = frequency (Hz or s^-1)

Photon Energy

\[ E = h\nu = \frac{hc}{\lambda} \] Where:

• \(E\) = photon energy (J)
• \(h\) = Planck's constant = \(6.626 \times 10^{-34}\) J·s
• \(\nu\) = frequency (Hz)
• \(c\) = speed of light (m/s)
• \(\lambda\) = wavelength (m)

Blackbody Radiation

Blackbody Definition

• A blackbody is an ideal surface that absorbs all incident radiation regardless of wavelength and direction
• A blackbody also emits the maximum possible radiation at any given temperature
• No real surface is a perfect blackbody, but it serves as an idealization

Planck's Law

\[ E_{b\lambda}(T) = \frac{C_1}{\lambda^5 \left[e^{C_2/\lambda T} - 1\right]} \] Where:

• \(E_{b\lambda}\) = spectral (monochromatic) emissive power of a blackbody (W/m²·μm or W/m³)
• \(T\) = absolute temperature (K)
• \(\lambda\) = wavelength (μm or m)
• \(C_1\) = first radiation constant = \(3.742 \times 10^8\) W·μm⁴/m² = \(2\pi hc^2\)
• \(C_2\) = second radiation constant = \(1.439 \times 10^4\) μm·K = \(hc/k\)
• \(k\) = Boltzmann constant = \(1.381 \times 10^{-23}\) J/K

Wien's Displacement Law

\[ \lambda_{max} T = C_3 \] Where:

• \(\lambda_{max}\) = wavelength at which maximum spectral emissive power occurs (μm)
• \(T\) = absolute temperature (K)
• \(C_3\) = Wien's displacement constant = 2898 μm·K

Note: As temperature increases, the peak wavelength shifts to shorter wavelengths (higher frequencies).

Stefan-Boltzmann Law

\[ E_b = \sigma T^4 \] Where:

• \(E_b\) = total (integrated over all wavelengths) emissive power of a blackbody (W/m²)
• \(\sigma\) = Stefan-Boltzmann constant = \(5.670 \times 10^{-8}\) W/(m²·K⁴)
• \(T\) = absolute temperature (K)

Alternative form: \[ E_b = \int_0^\infty E_{b\lambda} \, d\lambda = \sigma T^4 \]

Blackbody Radiation Intensity

\[ I_b = \frac{E_b}{\pi} = \frac{\sigma T^4}{\pi} \] Where:

• \(I_b\) = blackbody intensity (W/m²·sr)
• \(E_b\) = blackbody emissive power (W/m²)

Note: The factor π arises because blackbody emission is diffuse (Lambertian).

Real Surface Radiation Properties

Emissivity

Total hemispherical emissivity: \[ \varepsilon = \frac{E}{E_b} = \frac{E}{\sigma T^4} \] Where:

• \(\varepsilon\) = total hemispherical emissivity (dimensionless, 0 ≤ ε ≤ 1)
• \(E\) = total emissive power of the real surface (W/m²)
• \(E_b\) = blackbody emissive power at the same temperature (W/m²)

Spectral emissivity: \[ \varepsilon_\lambda = \frac{E_\lambda}{E_{b\lambda}} \] Where:

• \(\varepsilon_\lambda\) = spectral (monochromatic) emissivity
• \(E_\lambda\) = spectral emissive power of the real surface
• \(E_{b\lambda}\) = spectral emissive power of a blackbody

Absorptivity

Total hemispherical absorptivity: \[ \alpha = \frac{G_{abs}}{G} \] Where:

• \(\alpha\) = total hemispherical absorptivity (dimensionless, 0 ≤ α ≤ 1)
• \(G_{abs}\) = absorbed radiation (W/m²)
• \(G\) = incident radiation (irradiation) (W/m²)

Spectral absorptivity: \[ \alpha_\lambda = \frac{G_{\lambda,abs}}{G_\lambda} \]

Reflectivity

\[ \rho = \frac{G_{ref}}{G} \] Where:

• \(\rho\) = total hemispherical reflectivity (dimensionless, 0 ≤ ρ ≤ 1)
• \(G_{ref}\) = reflected radiation (W/m²)
• \(G\) = incident radiation (W/m²)

Transmissivity

\[ \tau = \frac{G_{tr}}{G} \] Where:

• \(\tau\) = total hemispherical transmissivity (dimensionless, 0 ≤ τ ≤ 1)
• \(G_{tr}\) = transmitted radiation (W/m²)
• \(G\) = incident radiation (W/m²)

Conservation of Incident Radiation

\[ \alpha + \rho + \tau = 1 \] For opaque surfaces (τ = 0): \[ \alpha + \rho = 1 \]

Kirchhoff's Law

For surfaces in thermal equilibrium: \[ \varepsilon = \alpha \] Spectral form: \[ \varepsilon_\lambda = \alpha_\lambda \] Important conditions:

• Valid when the surface temperature equals the temperature of the incident radiation source
• For diffuse, gray surfaces at different temperatures, often approximated as ε = α

Gray Surface Assumption

• A gray surface has spectral properties independent of wavelength: \(\varepsilon_\lambda = \varepsilon = \text{constant}\)
• Similarly: \(\alpha_\lambda = \alpha = \text{constant}\)
• Simplifies radiation calculations significantly

Diffuse Surface Assumption

• A diffuse surface has radiation properties independent of direction
• Emitted and reflected radiation intensity is uniform in all directions
• Follows Lambert's cosine law

Radiation Heat Transfer Between Surfaces

Radiation Emitted by a Real Surface

\[ q_{emit} = \varepsilon A \sigma T^4 \] Where:

• \(q_{emit}\) = total radiation emitted by the surface (W)
• \(\varepsilon\) = emissivity of the surface
• \(A\) = surface area (m²)
• \(\sigma\) = Stefan-Boltzmann constant = \(5.670 \times 10^{-8}\) W/(m²·K⁴)
• \(T\) = absolute temperature of the surface (K)

Net Radiation Heat Transfer

For a single surface exchanging radiation with surroundings: \[ q_{net} = \varepsilon A \sigma (T^4 - T_{surr}^4) \] Where:

• \(q_{net}\) = net radiation heat transfer from the surface (W)
• \(T\) = surface temperature (K)
• \(T_{surr}\) = temperature of surroundings (K)

Assumption: Surroundings act as a blackbody and completely enclose the surface.

Radiation Between Two Infinite Parallel Plates

\[ q = \frac{\sigma A (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1} \] Where:

• \(q\) = heat transfer rate (W)
• \(A\) = area of either plate (m²)
• \(T_1, T_2\) = absolute temperatures of plates 1 and 2 (K)
• \(\varepsilon_1, \varepsilon_2\) = emissivities of plates 1 and 2

Radiation Between Two Concentric Cylinders or Spheres

For concentric cylinders (per unit length): \[ q = \frac{\sigma (2\pi r_1) (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{r_1}{r_2}\left(\frac{1}{\varepsilon_2} - 1\right)} \] For concentric spheres: \[ q = \frac{\sigma (4\pi r_1^2) (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{r_1^2}{r_2^2}\left(\frac{1}{\varepsilon_2} - 1\right)} \] Where:

• \(r_1\) = radius of inner surface (m)
• \(r_2\) = radius of outer surface (m)
• \(T_1, T_2\) = absolute temperatures of inner and outer surfaces (K)
• \(\varepsilon_1, \varepsilon_2\) = emissivities of inner and outer surfaces

Special case: If inner surface is small compared to outer surface (\(r_1 \ll r_2\)): \[ q \approx \varepsilon_1 A_1 \sigma (T_1^4 - T_2^4) \]

View Factors (Configuration Factors)

View Factor Definition

• The view factor \(F_{ij}\) is the fraction of radiation leaving surface i that is directly intercepted by surface j
• Also called shape factor or configuration factor
• Purely geometric quantity; independent of surface properties and temperatures

View Factor Properties

Reciprocity relation: \[ A_i F_{ij} = A_j F_{ji} \] Where:

• \(A_i, A_j\) = areas of surfaces i and j (m²)
• \(F_{ij}\) = view factor from surface i to surface j
• \(F_{ji}\) = view factor from surface j to surface i

Summation rule (enclosure): \[ \sum_{j=1}^{N} F_{ij} = 1 \]

• For an enclosure with N surfaces, all radiation leaving surface i must be intercepted by the N surfaces (including itself)

For flat or convex surfaces: \[ F_{ii} = 0 \]

• A flat or convex surface cannot see itself
• For concave surfaces, \(F_{ii} > 0\)

Common View Factor Relations

Two parallel, identical rectangles or disks:

• Use charts or tables from NCEES PE Reference Handbook or standard references

Perpendicular rectangles with common edge:

• Use charts or tables from standard references

Sphere inside a cube or cylinder:

• Use reciprocity and summation rules with known geometric relations

View Factor Algebra

Subdivision of surfaces: \[ F_{i(j,k)} = F_{ij} + F_{ik} \] Where:

• Surface composed of j and k can be analyzed using subdivision

Example application: \[ A_i F_{i(j+k)} = A_i F_{ij} + A_i F_{ik} \]

Radiation Heat Transfer in Enclosures

Radiosity

\[ J = \varepsilon E_b + \rho G \] Where:

• \(J\) = radiosity, total radiation leaving a surface per unit area (W/m²)
• \(\varepsilon\) = emissivity
• \(E_b = \sigma T^4\) = blackbody emissive power (W/m²)
• \(\rho\) = reflectivity
• \(G\) = irradiation, incident radiation per unit area (W/m²)

For gray, diffuse surfaces with ρ = 1 - ε: \[ J = \varepsilon \sigma T^4 + (1 - \varepsilon) G \]

Net Radiation Leaving a Surface

\[ q_i = A_i (J_i - G_i) \] Alternative form: \[ q_i = \frac{E_{bi} - J_i}{\frac{1-\varepsilon_i}{\varepsilon_i A_i}} \] Where:

• \(q_i\) = net radiation heat transfer from surface i (W)
• \(A_i\) = area of surface i (m²)
• \(E_{bi} = \sigma T_i^4\) = blackbody emissive power of surface i
• \(J_i\) = radiosity of surface i
• \(\varepsilon_i\) = emissivity of surface i

Note: The term \(\frac{1-\varepsilon_i}{\varepsilon_i A_i}\) is the surface resistance.

Radiation Exchange Between Surfaces

\[ q_{ij} = A_i F_{ij} (J_i - J_j) \] Where:

• \(q_{ij}\) = radiation heat transfer from surface i to surface j (W)
• \(F_{ij}\) = view factor from surface i to surface j

Alternative form using space resistance: \[ q_{ij} = \frac{J_i - J_j}{\frac{1}{A_i F_{ij}}} \] Note: The term \(\frac{1}{A_i F_{ij}}\) is the space resistance or geometric resistance.

Two-Surface Enclosure

\[ q_{12} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1-\varepsilon_1}{\varepsilon_1 A_1} + \frac{1}{A_1 F_{12}} + \frac{1-\varepsilon_2}{\varepsilon_2 A_2}} \] Where:

• \(q_{12}\) = net radiation from surface 1 to surface 2 (W)
• \(T_1, T_2\) = absolute temperatures (K)
• \(\varepsilon_1, \varepsilon_2\) = emissivities
• \(A_1, A_2\) = surface areas (m²)
• \(F_{12}\) = view factor from surface 1 to surface 2

Special cases: Large (infinite) parallel plates: \(A_1 = A_2 = A\), \(F_{12} = 1\) \[ q = \frac{A \sigma (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1} \] Small object in large enclosure: \(A_1 \ll A_2\), \(F_{12} = 1\) \[ q = \varepsilon_1 A_1 \sigma (T_1^4 - T_2^4) \]

Three-Surface Enclosure

For an enclosure with three gray, diffuse surfaces, the net heat transfer from each surface is: \[ q_1 = \frac{E_{b1} - J_1}{\frac{1-\varepsilon_1}{\varepsilon_1 A_1}} \] \[ q_2 = \frac{E_{b2} - J_2}{\frac{1-\varepsilon_2}{\varepsilon_2 A_2}} \] \[ q_3 = \frac{E_{b3} - J_3}{\frac{1-\varepsilon_3}{\varepsilon_3 A_3}} \] Conservation of energy for the enclosure: \[ q_1 + q_2 + q_3 = 0 \] Radiation exchange between surfaces: \[ q_{12} = \frac{J_1 - J_2}{\frac{1}{A_1 F_{12}}} \] \[ q_{13} = \frac{J_1 - J_3}{\frac{1}{A_1 F_{13}}} \] \[ q_{23} = \frac{J_2 - J_3}{\frac{1}{A_2 F_{23}}} \] For surface 1: \[ q_1 = q_{12} + q_{13} \] Solution method:

• Solve system of equations for radiosities \(J_1, J_2, J_3\)
• Calculate heat transfer rates using radiosity values
• Network analogy: surface resistances in series with space resistances

Reradiating (Adiabatic) Surface

For a surface that is perfectly insulated (adiabatic): \[ q_i = 0 \] \[ J_i = E_{bi} \]

• The reradiating surface absorbs and re-emits all incident radiation
• Its radiosity equals its blackbody emissive power
• Temperature adjusts to satisfy energy balance

Radiation Shields

Single Radiation Shield Between Parallel Plates

\[ q = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1 + \frac{1}{\varepsilon_{s1}} + \frac{1}{\varepsilon_{s2}} - 1} \] Where:

• \(q\) = heat transfer per unit area (W/m²)
• \(T_1, T_2\) = temperatures of the two plates (K)
• \(\varepsilon_1, \varepsilon_2\) = emissivities of the two plates
• \(\varepsilon_{s1}, \varepsilon_{s2}\) = emissivities of the two sides of the shield

If shield has identical emissivities on both sides (\(\varepsilon_{s1} = \varepsilon_{s2} = \varepsilon_s\)): \[ q = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1 + \frac{2}{\varepsilon_s} - 1} \] Reduction in heat transfer:

• With one shield, heat transfer is approximately halved compared to no shield (for similar emissivities)
• Multiple shields provide further reduction

Multiple Radiation Shields

For N shields between two parallel plates: \[ q = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1 + N\left(\frac{2}{\varepsilon_s} - 1\right)} \] Where:

• \(N\) = number of shields
• \(\varepsilon_s\) = emissivity of each shield (assuming all shields identical)

Radiation Combined with Convection

Total Heat Transfer

\[ q_{total} = q_{conv} + q_{rad} \] \[ q_{total} = hA(T_s - T_\infty) + \varepsilon A \sigma (T_s^4 - T_{surr}^4) \] Where:

• \(q_{total}\) = total heat transfer (W)
• \(q_{conv}\) = convection heat transfer (W)
• \(q_{rad}\) = radiation heat transfer (W)
• \(h\) = convection heat transfer coefficient (W/m²·K)
• \(A\) = surface area (m²)
• \(T_s\) = surface temperature (K)
• \(T_\infty\) = ambient fluid temperature (K)
• \(T_{surr}\) = temperature of surroundings for radiation (K)

Radiation Heat Transfer Coefficient

Linearized radiation heat transfer: \[ q_{rad} = h_r A (T_s - T_{surr}) \] Where the radiation heat transfer coefficient is: \[ h_r = \varepsilon \sigma (T_s + T_{surr})(T_s^2 + T_{surr}^2) \] Alternative form: \[ h_r = \varepsilon \sigma (T_s^2 + T_{surr}^2)(T_s + T_{surr}) \] Approximate form for small temperature differences: \[ h_r \approx 4 \varepsilon \sigma T_{avg}^3 \] Where:

• \(T_{avg} = \frac{T_s + T_{surr}}{2}\)

Combined Heat Transfer Coefficient

\[ q_{total} = (h + h_r) A (T_s - T_\infty) \] Where:

• \(h + h_r\) = combined (total) heat transfer coefficient

Assumption: \(T_{surr} \approx T_\infty\) for this simplification to be valid.

Gas Radiation

Gas Emissivity and Absorptivity

• Participating media: Gases like CO₂, H₂O, CO, SO₂, and hydrocarbons absorb and emit radiation
• Nonparticipating media: Monoatomic and most diatomic gases (O₂, N₂, H₂, air) are transparent to thermal radiation
• Gas radiation is a volumetric phenomenon, not a surface phenomenon
• Gas emissivity depends on temperature, partial pressure, and path length (beam length)

Gas Radiation Heat Transfer

Emission from a gas volume to a surface: \[ q = \varepsilon_g A \sigma T_g^4 \] Where:

• \(\varepsilon_g\) = gas emissivity (function of \(T_g\), \(p_a L\))
• \(A\) = surface area (m²)
• \(T_g\) = gas temperature (K)
• \(p_a\) = partial pressure of absorbing gas (atm)
• \(L\) = mean beam length (m)

Absorption by a gas from a surface: \[ q = \alpha_g A \sigma T_s^4 \] Where:

• \(\alpha_g\) = gas absorptivity (function of \(T_s\), \(p_a L\))
• \(T_s\) = surface temperature (K)

Net radiation exchange between gas and surface: \[ q = A \sigma (\varepsilon_g T_g^4 - \alpha_g T_s^4) \]

Mean Beam Length

• The mean beam length \(L\) is a characteristic dimension of the gas volume
• Accounts for the average path length of radiation through the gas

Common geometries:

• Sphere of diameter D: \(L = 0.65D\)
• Infinite cylinder of diameter D: \(L = 0.95D\)
• Semi-infinite cylinder of diameter D radiating to base: \(L = 0.65D\)
• Cube of side L radiating to a face: \(L = 0.66L\)
• Rectangular parallelepiped: Use charts or approximation formulas

General approximation: \[ L = 3.6 \frac{V}{A} \] Where:

• \(V\) = gas volume (m³)
• \(A\) = surface area enclosing the gas (m²)

Solar Radiation

Solar Constant

\[ G_{sc} = 1367 \text{ W/m}^2 \]

• Solar constant is the solar radiation incident on a surface normal to the sun's rays at the outer edge of Earth's atmosphere
• Average distance between Earth and Sun

Solar Radiation Components

Total solar radiation on a surface: \[ G_{total} = G_{beam} + G_{diffuse} + G_{reflected} \] Where:

• \(G_{beam}\) = direct (beam) radiation from the sun
• \(G_{diffuse}\) = diffuse radiation from the sky
• \(G_{reflected}\) = radiation reflected from surrounding surfaces

Solar Radiation on a Surface

\[ G = G_{beam} \cos\theta + G_{diffuse} \] Where:

• \(G\) = total solar irradiation on the surface (W/m²)
• \(\theta\) = angle of incidence (angle between sun's rays and normal to surface)

Solar Absorptivity and Emissivity

• Solar absorptivity \(\alpha_s\): fraction of incident solar radiation absorbed
• Infrared emissivity \(\varepsilon\): emissivity for thermal radiation at surface temperature
• For many surfaces: \(\alpha_s \neq \varepsilon\) (non-gray behavior)
• Selective surfaces: High \(\alpha_s\), low \(\varepsilon\) (e.g., solar collectors)

Heat Balance on a Solar-Irradiated Surface

\[ \alpha_s G A = \varepsilon A \sigma (T_s^4 - T_{surr}^4) + hA(T_s - T_\infty) \] Where:

• \(\alpha_s\) = solar absorptivity
• \(G\) = solar irradiation (W/m²)
• \(\varepsilon\) = infrared emissivity
• \(T_s\) = surface temperature (K)
• \(T_{surr}\) = surrounding temperature for radiation (K)
• \(h\) = convection heat transfer coefficient (W/m²·K)
• \(T_\infty\) = ambient air temperature (K)

Radiation Error in Temperature Measurement

Thermocouple Radiation Error

Energy balance on a thermocouple in a gas flow: \[ h(T_g - T_{tc}) = \varepsilon \sigma (T_{tc}^4 - T_{surr}^4) \] Where:

• \(T_g\) = true gas temperature (K)
• \(T_{tc}\) = measured thermocouple temperature (K)
• \(T_{surr}\) = temperature of surrounding surfaces (K)
• \(h\) = convection heat transfer coefficient between gas and thermocouple (W/m²·K)
• \(\varepsilon\) = emissivity of thermocouple

Radiation error: \[ T_g - T_{tc} = \frac{\varepsilon \sigma (T_{tc}^4 - T_{surr}^4)}{h} \] To minimize error:

• Increase gas velocity (increases \(h\))
• Use radiation shield around thermocouple
• Use low-emissivity thermocouple

Effective Sky Temperature

Sky Radiation Model

For surfaces exposed to clear sky: \[ q_{rad,sky} = \varepsilon A \sigma (T_s^4 - T_{sky}^4) \] Where:

• \(T_{sky}\) = effective sky temperature (K)
• Sky is modeled as a blackbody at temperature \(T_{sky}\)

Empirical correlation for clear sky: \[ T_{sky} \approx 0.0552 \, T_{air}^{1.5} \] Where:

• \(T_{air}\) = ambient air temperature (K)
• \(T_{sky}\) is typically 10-30 K below \(T_{air}\) for clear skies

Alternative approximation: \[ T_{sky} = T_{air} - 6 \text{ K (clear sky)} \]

Important Constants and Properties

Physical Constants

• Stefan-Boltzmann constant: \(\sigma = 5.670 \times 10^{-8}\) W/(m²·K⁴)
• Planck's constant: \(h = 6.626 \times 10^{-34}\) J·s
• Boltzmann constant: \(k = 1.381 \times 10^{-23}\) J/K
• Speed of light: \(c = 2.998 \times 10^8\) m/s
• Wien's displacement constant: \(C_3 = 2898\) μm·K
• First radiation constant: \(C_1 = 3.742 \times 10^8\) W·μm⁴/m²
• Second radiation constant: \(C_2 = 1.439 \times 10^4\) μm·K
• Solar constant: \(G_{sc} = 1367\) W/m²

Typical Emissivity Values

• Polished metals: 0.02-0.10
• Oxidized metals: 0.4-0.8
• Black paint: 0.95-0.98
• White paint: 0.85-0.95
• Concrete, brick: 0.85-0.95
• Wood: 0.80-0.90
• Glass: 0.90-0.95
• Water: 0.95-0.96
• Human skin: 0.95

Note: Actual values depend on surface condition, temperature, and wavelength. Consult tables for specific applications.