Cheatsheet: Radiation

1. Fundamental Radiation Concepts

1.1 Basic Definitions

1.2 Key Relationships

• Energy balance: α + ρ + τ = 1
• For opaque surfaces: τ = 0, therefore α + ρ = 1
• Kirchhoff's Law: α = ε (at thermal equilibrium and same wavelength)
• Blackbody: ε = α = 1, ρ = τ = 0 (perfect emitter and absorber)
• Gray surface: ε and α are independent of wavelength
• Diffuse surface: radiation properties independent of direction

2. Blackbody Radiation Laws

2.1 Stefan-Boltzmann Law

2.2 Planck's Distribution Law

2.3 Wien's Displacement Law

2.4 Band Emission

• Fraction of blackbody radiation between 0 and λ: F_0-λ = f(λT)
• Fraction between λ₁ and λ₂: F_λ₁-λ₂ = F_0-λ₂ - F_0-λ₁
• Use radiation function tables with λT as parameter

3. Surface Radiation Properties

3.1 Real Surface Behavior

3.2 Typical Emissivity Values

4. View Factors (Configuration Factors)

4.1 Definition and Properties

4.2 View Factor Algebra

• Additive property: F_i(j+k) = F_ij + F_ik
• Subdivision: A_iF_i(j+k) = A_iF_ij + A_iF_ik
• For two-surface enclosure: F₁₂ = 1 - F₁₁ and F₂₁ = 1 - F₂₂

4.3 Common Configurations

5. Radiation Heat Exchange

5.1 Between Blackbodies

5.2 Between Gray, Diffuse Surfaces

5.2.1 Two-Surface Enclosure

5.2.2 Special Cases

5.3 Radiation Shields

• Thin, high-reflectivity surfaces placed between radiating surfaces to reduce heat transfer
• For N shields between parallel plates: Q_{with shields} = Q_{no shield}/(N+1)
• For identical shields (ε_s) between plates: Q/A = σ(T₁⁴ - T₂⁴)/[(1/ε₁ + 1/ε₂ - 1)(N+1) + 2N/ε_s]
• Low emissivity shields (polished aluminum, ε ≈ 0.05) most effective

5.4 Reradiating Surfaces

• Surfaces that are insulated or adiabatic (no net heat transfer)
• Surface temperature adjusts to satisfy Q_net = 0
• For three-surface enclosure with surface 3 reradiating: Q₁₂ = σ(T₁⁴ - T₂⁴)/R_total
• Network method used to determine equivalent resistance

6. Radiation Network Method

6.1 Network Elements

6.2 Solution Procedure

• Draw network with surface nodes (E_bi) and radiosity nodes (J_i)
• Connect surface node to radiosity node through surface resistance
• Connect radiosity nodes through space resistances
• For specified temperatures: E_bi known, solve for J_i and Q_i
• For specified heat fluxes: Q_i known, solve for J_i and T_i
• For reradiating surfaces: Q_i = 0, omit surface resistance
• Apply Kirchhoff's current law at radiosity nodes

7. Gas Radiation

7.1 Gas Properties

• Gases emit and absorb radiation in specific wavelength bands
• Non-participating gases (O₂, N₂, air): transparent to thermal radiation (ε ≈ 0)
• Participating gases (CO₂, H₂O, CO, SO₂, NH₃): emit and absorb radiation
• Gas emissivity depends on temperature, partial pressure, and path length

7.2 Mean Beam Length

7.3 Gas Emissivity and Absorptivity

• Use gas emissivity charts for CO₂ and H₂O with parameters p_gL_m and T_g
• p_g = partial pressure of gas (atm); L_m = mean beam length (m); T_g = gas temperature (K)
• For water vapor: multiply by correction factor C_w if p_w + p_a ≠ 1 atm
• Gas absorptivity: α_g = (T_s/T_g)^0.5 × ε_g(T_s, p_gL_mT_s/T_g)
• For mixture of CO₂ and H₂O: ε_g = ε_CO₂ + ε_H₂O - Δε (subtract overlap correction)

7.4 Radiation from Gas to Surface

8. Combined Heat Transfer Modes

8.1 Radiation with Convection

8.2 Linearized Radiation

• For small temperature differences: T_s⁴ - T_sur⁴ ≈ 4T_avg³(T_s - T_sur)
• Simplified radiation coefficient: h_rad ≈ 4εσT_avg³
• Valid when (T_s - T_sur)/T_avg <>

9. Solar Radiation

9.1 Solar Spectrum

• Solar constant: G_sc = 1353 W/m² (radiation intensity outside atmosphere)
• At Earth's surface: approximately 1000 W/m² on clear day at solar noon
• Peak wavelength: λ_max ≈ 0.48 μm (visible range)
• 98% of solar energy between 0.3-3.0 μm (short wavelengths)

9.2 Solar Angles

9.3 Solar Radiation on Surfaces

• Direct (beam) radiation: q_b = G_bcos(θ) where θ = angle of incidence
• Diffuse radiation: scattered by atmosphere, approximated as isotropic
• Total solar radiation: q_total = q_direct + q_diffuse + q_reflected
• Ground reflection: typically 20% for light surfaces, 10% for dark surfaces

9.4 Solar Absorptivity

• Solar absorptivity (α_s): fraction of incident solar radiation absorbed
• Thermal emissivity (ε): emission in infrared range (typical surface temperatures)
• For selective surfaces: α_s/ε ratio important (solar collectors want high ratio)
• Dark surfaces: α_s ≈ 0.9, ε ≈ 0.9
• White paint: α_s ≈ 0.2, ε ≈ 0.9
• Polished aluminum: α_s ≈ 0.1, ε ≈ 0.05

10. Practical Applications and Tips

10.1 Problem-Solving Strategy

• Identify surface temperatures or heat transfer rates (known vs. unknown)
• Determine surface properties (emissivities) and configuration
• Calculate view factors using geometry, reciprocity, and summation rules
• For two surfaces: use direct formula with appropriate simplification
• For multiple surfaces: use radiation network method
• Check if radiation dominates or combine with convection as needed

10.2 Common Assumptions

• Gray, diffuse surfaces (unless specified otherwise)
• Steady-state conditions
• Uniform surface temperatures and properties
• Opaque surfaces (τ = 0)
• Surfaces separated by non-participating medium (vacuum or transparent gas)

10.3 Unit Conversions

10.4 Key Insights

• Radiation heat transfer proportional to T⁴; very sensitive to temperature
• Radiation becomes dominant at high temperatures (> 500°C)
• Low emissivity surfaces (polished metals) effective for reducing radiation
• View factor determines geometric effectiveness of radiation exchange
• Radiation shields exponentially reduce heat transfer with number of shields
• For enclosed spaces, all surfaces exchange radiation with all others