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Qwestrum Engineering360 · Mechanical Engineering · Heat & Mass Transfer

Thermal Radiation

A black body emits E = σT⁴ (σ = 5.67×10⁻⁸ W/m²K⁴); a grey body emits εσT⁴, and exchange between surfaces uses q = σA₁F₁₂(T₁⁴ − T₂⁴). View factors obey reciprocity A₁F₁₂ = A₂F₂₁, per RC Sachdeva.
Exam tip: never replace LMTD with an arithmetic mean of end ΔT unless ΔT_a ≈ ΔT_b; keep U and A in consistent SI units.

Key formulas & points

Skim these first — then read the full notes below.

  • Viewfactoralgebra:ΣF1j=1forenclosureView factor algebra: ΣF_{1j} = 1 for enclosure
  • Kirchhoff:ελ=αλatthermalequilibriumKirchhoff: \varepsilon_\lambda = \alpha_\lambda at thermal equilibrium
  • Radiation shields reduce heat transfer ∝ 1/(N+1)

Topic details

Introduction

Radiation is the mode that needs no medium and dominates at high temperature because of its T⁴ dependence. RC Sachdeva builds from the black-body Stefan-Boltzmann law to grey surfaces (emissivity ε) and then to enclosure exchange using view factors and the radiation network method.

Scope in B.Tech and GATE syllabus

View-factor algebra — the summation rule ΣF₁ⱼ = 1 and reciprocity A₁F₁₂ = A₂F₂₁ — lets students find unknown view factors in enclosures. The electrical-network analogy with surface resistances (1−ε)/(εA) and space resistances 1/(A₁F₁₂) parallels the conduction resistance idea.

Why this topic matters in practice

Radiation shields, which reduce exchange in proportion to 1/(N+1), and Kirchhoff's law (ε = α at equilibrium) are standard conceptual questions. The strong temperature sensitivity (fourth power) is the recurring theme.

Key relations & formulas

E=σT4E = \sigma T^{4}
(Stefan-Boltzmann, black body, σ = 5.67×10⁻⁸ W/m²K⁴)
q=εσA(T14−T24)q = \varepsilon\sigma A(T_{1}^{4} - T_{2}^{4})
(grey body to surroundings)
F12=F21F_{12} = F_{21}
(reciprocity, view factor)
q12=σA1F12(T14−T24)q_{12} = \sigma A_{1}F_{12}(T_{1}^{4} - T_{2}^{4})
(radiation between two surfaces)

Notation and sign conventions

Relation 1 —
E=σT4E = \sigma T^{4}
E=σT4E = \sigma T^{4}
(Stefan-Boltzmann, black body, σ = 5.67×10⁻⁸ W/m²K⁴)
Write this relation with symbols exactly as in Fundamentals of Engineering Heat & Mass Transfer — RC Sachdeva before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
q=εσAq = \varepsilon\sigma A
q=εσA(T14−T24)q = \varepsilon\sigma A(T_{1}^{4} - T_{2}^{4})
(grey body to surroundings)
Write this relation with symbols exactly as in Fundamentals of Engineering Heat & Mass Transfer — RC Sachdeva before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
F12=F21F_{12} = F_{21}
F12=F21F_{12} = F_{21}
(reciprocity, view factor)
Write this relation with symbols exactly as in Fundamentals of Engineering Heat & Mass Transfer — RC Sachdeva before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
q12=σA1F12q_{12} = \sigma A_{1}F_{12}
q12=σA1F12(T14−T24)q_{12} = \sigma A_{1}F_{12}(T_{1}^{4} - T_{2}^{4})
(radiation between two surfaces)
Write this relation with symbols exactly as in Fundamentals of Engineering Heat & Mass Transfer — RC Sachdeva before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

A black body is the ideal emitter/absorber; its total emissive power E_b = σT⁴ increases with the fourth power of absolute temperature. Real (grey) surfaces emit εσT⁴, with emissivity ε < 1.

Governing relations in practice

Net exchange between two surfaces depends on their temperatures, areas, emissivities, and geometric view factor F₁₂ (the fraction of radiation leaving 1 that reaches 2). For black surfaces q₁₂ = σA₁F₁₂(T₁⁴ − T₂⁴).

Design and analysis considerations

View factors satisfy reciprocity A₁F₁₂ = A₂F₂₁ and, in an enclosure, ΣF₁ⱼ = 1. These two rules solve most view-factor problems by algebra rather than integration.

Advanced theory and extensions

For grey surfaces the radiation-network method adds surface resistances (1−ε)/(εA) and space resistances 1/(A₁F₁₂), solved like an electrical circuit. A radiation shield inserts extra resistances, cutting heat flow to roughly 1/(N+1) of the unshielded value for N shields — the basis of multilayer insulation.

Assumptions and validity limits

State assumptions explicitly before using any relation for thermal radiation — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Heat Transfer viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Heat Transfer papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to thermal radiation.
4. Use equation 1:
E=σT4E = \sigma T^{4}
.
5. Use equation 2:
q=εσAq = \varepsilon\sigma A
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Thermal Radiation appears in heat exchangers, fins, and insulation. In Indian mechanical curricula this topic is tested because it connects theory to conduction, convection, and radiation.
GATE and semester exams often combine thermal radiation with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use thermal radiation?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using temperatures in Celsius rather than Kelvin in the T⁴ terms
• Forgetting emissivity for grey surfaces and treating them as black
• Violating the view-factor rules (reciprocity or the enclosure summation)
• Adding radiation and convection incorrectly instead of combining resistances/rates properly

Quick revision checklist

Before attempting thermal radiation problems, confirm you can:
1.
Viewfactoralgebra:ΣF1j=1forenclosureView factor algebra: ΣF_{1j} = 1 for enclosure

2.
Kirchhoff:ελ=αλatthermalequilibriumKirchhoff: \varepsilon_\lambda = \alpha_\lambda at thermal equilibrium

3. Radiation shields reduce heat transfer ∝ 1/(N+1)
Revise the solved examples in Fundamentals of Engineering Heat & Mass Transfer — RC Sachdeva and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Black-body emissive power

Problem

Find the emissive power of a black surface at T = 500 K. Take σ = 5.67×10⁻⁸ W/m²K⁴.

Solution

E = σT⁴ = 5.67e-8 × 500⁴ = 5.67e-8 × 6.25e10 = 3543.75 W/m².

Conceptual check — Thermal Radiation

Problem

In a Heat Transfer semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of thermal radiation." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Thermal Radiation, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    A black body emits E = σT⁴ (σ = 5.67×10⁻⁸ W/m²K⁴); a grey body emits εσT⁴, and exchange between surfaces uses q = σA₁F₁₂(T₁⁴ − T₂⁴). View factors obey reciprocity A₁F₁₂ = A₂F₂₁, per RC Sachdeva.
  2. 2
    State the relation E = σT⁴ and name each symbol.

    Model answer

    The governing relation is E=σT4E = \sigma T^{4}. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation q = εσA and name each symbol.

    Model answer

    The governing relation is q=εσAq = \varepsilon\sigma A. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation F₁₂ = F₂₁ and name each symbol.

    Model answer

    The governing relation is F12=F21F_{12} = F_{21}. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation q₁₂ = σA₁F₁₂ and name each symbol.

    Model answer

    The governing relation is q12=σA1F12q_{12} = \sigma A_{1}F_{12}. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: View factor algebra: ΣF₁ⱼ = 1 for enclosure

    Model answer

    Viewfactoralgebra:ΣF1j=1forenclosureView factor algebra: ΣF_{1j} = 1 for enclosure — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Kirchhoff: ε_λ = α_λ at thermal equilibrium

    Model answer

    Kirchhoff:ελ=αλatthermalequilibriumKirchhoff: \varepsilon_\lambda = \alpha_\lambda at thermal equilibrium — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Radiation shields reduce heat transfer ∝ 1/(N+1)

    Model answer

    Radiation shields reduce heat transfer ∝ 1/(N+1) — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Using temperatures in Celsius rather than Kelvin in the T⁴ terms?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Forgetting emissivity for grey surfaces and treating them as black?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Violating the view-factor rules (reciprocity or the enclosure summation)?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Adding radiation and convection incorrectly instead of combining resistances/rates properly?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Electrical network analogy for radiation between grey surfaces.
  • 2
    Avoid: Using temperatures in Celsius rather than Kelvin in the T⁴ terms
  • 3
    Avoid: Forgetting emissivity for grey surfaces and treating them as black
  • 4
    Avoid: Violating the view-factor rules (reciprocity or the enclosure summation)

📖 Standard books (India)

  • Fundamentals of Engineering Heat & Mass Transfer — RC Sachdeva

    Read: Syllabus unit

    Heat transfer and heat exchangers

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