Lesson 6.5: Radiation Heat Transfer (Stefan–Boltzmann Law, Shape Factor)

Radiation Heat Transfer is the transfer of energy through electromagnetic waves without the need for a medium. GATE and PSU exams often test laws, radiation exchange between surfaces, and view factors.

🔹 1. Introduction

• Definition: Heat transfer by electromagnetic waves due to temperature difference between surfaces

• Applications: Furnaces, solar collectors, radiative cooling of spacecraft, high-temperature industrial processes

• Key Concepts:

  1. Radiation does not require a medium

  2. Surface emissivity affects radiative heat transfer

  3. View factor (Shape factor) determines fraction of radiation exchanged

🔹 2. Stefan–Boltzmann Law

• Law Statement: Heat radiated per unit area of a blackbody is proportional to fourth power of absolute temperature

q=σT4q = \sigma T^4q=σT4

Where σ = 5.67 × 10⁻⁸ W/m²K⁴

• For real surfaces (emissivity ε):

q=ϵσT4q = \epsilon \sigma T^4q=ϵσT4

• Net Radiation Between Surface & Surroundings:

q=ϵσ(Ts4−T∞4)q = \epsilon \sigma (T_s^4 – T_\infty^4)q=ϵσ(Ts4​−T∞4​)

• Example: Black surface at 500 K in surroundings at 300 K, ε = 0.8 → q = ?

q=0.8⋅5.67×10−8⋅(5004−3004)≈3.68×104 W/m²q = 0.8 \cdot 5.67 \times 10^{-8} \cdot (500^4 – 300^4) ≈ 3.68 \times 10^4 \text{ W/m²}q=0.8⋅5.67×10−8⋅(5004−3004)≈3.68×104 W/m²

🔹 3. Shape Factor / View Factor

• Definition: Fraction of radiation leaving one surface that strikes another surface directly

q12=F12σ(T14−T24)q_{12} = F_{12} \sigma (T_1^4 – T_2^4)q12​=F12​σ(T14​−T24​)

Where F₁₂ = view factor from surface 1 to 2

• Properties:

  1. Reciprocity: A1F12=A2F21A_1 F_{12} = A_2 F_{21}A1​F12​=A2​F21​

  2. Summation: ∑Fij=1\sum F_{ij} = 1∑Fij​=1 for surface i

• Applications: Radiation exchange between parallel plates, concentric spheres, and enclosures

🔹 4. Combined Radiation & Convection

• When both modes are present:

qtotal=qconvection+qradiationq_\text{total} = q_\text{convection} + q_\text{radiation}qtotal​=qconvection​+qradiation​

• Example: Heated vertical plate losing heat to air by convection and radiation

🔹 5. Solved Examples (PYQ Style)

1. Radiation heat transfer from furnace wall to surroundings

2. Heat exchange between two concentric spheres

3. Calculate net radiation using view factors in enclosure problem

🔹 6. Practice Exercises

1. Compute radiative heat flux from black and gray surfaces

2. Determine view factor between rectangular and parallel plates

3. Calculate net radiation from a surface in an enclosure

4. Combine radiation and convection for total heat loss

5. Solve multi-surface radiation exchange problems

🔹 7. Summary

• Radiation: Energy transfer by electromagnetic waves, independent of medium

• Stefan–Boltzmann Law: q=ϵσT4q = \epsilon \sigma T^4q=ϵσT4

• View Factor (Shape Factor): Fraction of radiation exchanged between surfaces

• Applications: Furnaces, solar collectors, high-temperature industrial equipment