Summary of Heat Transfer 21 | Radiation Heat Transfer (II) | Mechanical Engineering | GATE Crash Course

Summary of "Heat Transfer 21 | Radiation Heat Transfer (II) | Mechanical Engineering | GATE Crash Course"

This video is a detailed lecture on Radiation Heat Transfer, focusing on fundamental laws, concepts, and practical applications relevant for mechanical engineering students preparing for GATE exams. The instructor revisits previous topics briefly and then dives deep into Planck’s Law, Stefan-Boltzmann Law, Wien’s Displacement Law, Kirchhoff’s Law, emissivity concepts, view factors (also called configuration or geometry factors), and radiation heat exchange between bodies.

Main Ideas, Concepts, and Lessons:

1. Review of Planck’s Law and Blackbody Radiation

Planck’s Law describes the monochromatic emissive power of a Blackbody as a function of wavelength and temperature.
Graphs of monochromatic emissive power vs. wavelength show peaks shifting with temperature.
As temperature increases, the peak wavelength (λ_max) shifts to shorter wavelengths.
λ_max corresponds to the wavelength at which monochromatic emissive power is maximum.

2. Wien’s Displacement Law

Derived by differentiating Planck’s Law with respect to wavelength and setting it to zero.
The product of peak wavelength and absolute temperature is constant:
- λ_max * T = 0.029 m·K (or 2900 μm·K)
This implies that as temperature increases, λ_max decreases (hyperbolic relationship).
Example: For a Blackbody at 400°C (~673 K), λ_max ≈ 4.31 μm.

3. Stefan-Boltzmann Law

Total emissive power of a Blackbody is proportional to the fourth power of its absolute temperature:
- E = σ T⁴
- Where σ (Stefan-Boltzmann constant) = 5.67 × 10⁻⁸ W/m²·K⁴
Units of emissive power: Watts per square meter (W/m²).
The law is derived by integrating Planck’s Law over all wavelengths.

4. Kirchhoff’s Law of Thermal Radiation

At thermal equilibrium, absorptivity (α) equals emissivity (ε) for a body:
- α = ε
A body in thermal equilibrium with its surroundings absorbs as much radiation as it emits, maintaining constant temperature.

5. Emissivity and Types of Bodies

Blackbody: Perfect absorber and emitter, emissivity ε = 1.
Grey body: Emissivity is constant but less than 1, independent of wavelength.
Real bodies: Emissivity varies with wavelength.
Monochromatic emissivity refers to emissivity at a single wavelength.

6. View Factor (Configuration or Geometry Factor)

Defines the fraction of radiation leaving one surface that strikes another.
Depends only on geometry, not on temperature or material properties.
Denoted as F_ij: fraction of energy leaving surface i that strikes surface j.
Important properties:
- Sum of view factors from a surface to all surfaces (including itself) equals 1.
- Reciprocity theorem: A_i F_ij = A_j F_ji, where A_i and A_j are surface areas.

7. Radiation Heat Exchange Between Surfaces

Consider two blackbodies at temperatures T₁ and T₂ exchanging radiation.
Radiation emitted from surface 1 striking surface 2:
- Q₁₂ = A₁ σ T₁⁴ F₁₂
Radiation emitted from surface 2 striking surface 1:
- Q₂₁ = A₂ σ T₂⁴ F₂₁
Net radiation exchange:
- Q_net = Q₁₂ - Q₂₁ = A₁ F₁₂ σ (T₁⁴ - T₂⁴)
For concentric cylinders example, the view factors and net exchange are calculated using geometry and temperatures.

Methodologies / Key Steps / Formulas:

Finding λ_max (Wien’s Law):
- Differentiate Planck’s Law w.r.t. λ, set derivative to zero.
- Solve for λ_max * T = constant (0.029 m·K).
Stefan-Boltzmann Law Derivation:
- Integrate Planck’s Law over all wavelengths.
- Result: E = σ T⁴.
Using Kirchhoff’s Law:
- At thermal equilibrium, absorptivity = emissivity.
- Body absorbs all radiation it emits, maintaining temperature.
Calculating View Factors:
- Define F_ij = fraction of radiation from surface i striking surface j.
- Use reciprocity theorem: A_i F_ij = A_j F_ji.
- For closed surfaces, sum of all F_ ...