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title: Concept of shape factor and radiation shield
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# Concept of shape factor and radiation shield
Unit 3, Lecture 26

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## Shape Factors and Radiation Shields

- Course: Heat and Mass Transfer (DI05019071)
- Unit 3: Radiation
- Lecture 26: Shape Factors, Radiation Shields, and Numerical Analysis

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Welcome. Today we complete our study of radiation by addressing geometry. How does the physical arrangement of surfaces affect radiation exchange? We will also look at how to block radiation and solve practical problems.
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## Lecture Agenda

- 1. Concept of Shape Factor (View Factor)
- 2. Shape Factor Algebra (Reciprocity & Summation)
- 3. Concept of Radiation Shields
- 4. Heat Transfer Reduction by Shields
- 5. Numerical Examples & Problem Solving

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We will start with Shape Factors to handle geometry, then introduce Radiation Shields, and finally put it all together in numerical examples.
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## What is a Shape Factor?

- Also known as View Factor or Configuration Factor (F_ij).
- Definition: The fraction of radiation leaving surface 'i' that directly strikes surface 'j'.
- Purely a geometric property.
- Depends on the sizes, shapes, orientation, and distance between the two surfaces.
- 0 ≤ F_ij ≤ 1

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If a surface emits 100 Watts, how much of that actually hits a second surface across the room? That percentage is the Shape Factor.
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## Self-Viewing Surfaces

- Can a surface radiate to itself? Yes, depending on its geometry.
- Flat or Convex surface: F_ii = 0 (cannot see itself).
- Concave surface: F_ii > 0 (can see itself).
- A portion of radiation leaving a concave cavity strikes another part of the same cavity.

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A flat wall cannot radiate to itself. But the inside of a bowl definitely radiates to other parts of the inside of the bowl.
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## Shape Factor Algebra: Reciprocity Theorem

- Relates the shape factor F_ij to F_ji.
- Equation: A_i * F_ij = A_j * F_ji
- Where A_i and A_j are the surface areas.
- Significance: If you know F_12, you can easily calculate F_21 without complex integration.

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This is a massive time saver. Finding a shape factor usually requires double area integration, but if you have one, reciprocity gives you the other for free.
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## Shape Factor Algebra: Enclosure Theorem

- Also known as the Summation Rule.
- For an enclosure of N surfaces, all radiation leaving one surface must strike some surface within the enclosure.
- Equation: Σ F_ij = 1 (summed from j=1 to N)
- Example for a 3-surface enclosure (leaving surface 1):
- F_11 + F_12 + F_13 = 1

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Conservation of energy dictates this. Everything leaving surface 1 has to go somewhere within a closed system. The fractions must add up to 100%.
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## Radiation Heat Exchange (Grey Surfaces)

- For two infinite parallel grey surfaces, the net heat transfer is:
- q_12 = (σ * (T_1^4 - T_2^4)) / [ (1/ε_1) + (1/ε_2) - 1 ]
- This formula uses thermal resistance concepts.
- - Surface resistances: (1-ε)/εA
- - Space resistance: 1/A*F_12

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To calculate actual heat transfer between real grey surfaces, we use an electrical circuit analogy with surface and space resistances.
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## Concept of Radiation Shields

- Radiation Shield: A highly reflective (low emissivity) material placed between two surfaces to reduce radiation heat transfer.
- They do not generate or remove heat; they add resistance to the radiation network.
- Commonly used in cryogenics, spacecraft (multi-layer insulation), and furnaces.

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If you want to stop conduction, use foam. If you want to stop radiation, use a highly reflective shield, like aluminum foil.
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## Effectiveness of a Radiation Shield

- Inserting 'n' shields with the same emissivity as the main plates reduces heat transfer by a factor of (n + 1).
- q_with_n_shields = q_without_shields / (n + 1)
- Example: One shield cuts heat transfer exactly in half (if ε_shield = ε_plates).
- To maximize effectiveness, shield emissivity should be as close to zero as possible.

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A single piece of foil can cut radiation heat transfer by 50% or more. This is why emergency space blankets work so well.
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## Numerical Example 1: Shape Factor

- Problem: A long concentric cylinder arrangement. Inner cylinder (1) radius r1, outer cylinder (2) radius r2.
- Find F_11, F_12, F_21, and F_22.
- Solution steps:
- 1. Inner cylinder is convex: F_11 = 0.
- 2. Summation rule for inner: F_11 + F_12 = 1 => F_12 = 1.
- 3. Reciprocity: A1*F12 = A2*F21 => F_21 = (A1/A2)*1 = r1/r2.
- 4. Summation for outer: F_21 + F_22 = 1 => F_22 = 1 - (r1/r2).

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Let's apply the algebra. See how we found all four shape factors without doing a single integral? Just geometry and rules.
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## Numerical Example 2: Heat Transfer

- Problem: Two large parallel plates at T1=800K (ε1=0.8) and T2=400K (ε2=0.6). Find heat flux (q/A).
- Formula: q/A = σ*(T1^4 - T2^4) / [(1/ε1) + (1/ε2) - 1]
- Calculation:
- σ*(800^4 - 400^4) = 5.67e-8 * (4096e8 - 256e8) = 21772.8 W/m²
- Denominator: (1/0.8) + (1/0.6) - 1 = 1.25 + 1.667 - 1 = 1.917
- Result: q/A = 21772.8 / 1.917 = 11,357 W/m².

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Here is a standard calculation for parallel plates. Plug in the temperatures and emissivities into the thermal resistance network formula.
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## Summary of Unit 3: Radiation

- Radiation requires no medium; it's governed by temperature and surface properties (α, ρ, τ, ε).
- Black bodies are ideal references; Grey bodies approximate real surfaces.
- Stefan-Boltzmann (E=σT⁴) is the core governing law.
- Shape factors (geometry) and Radiation shields (resistances) allow us to solve real-world system networks.

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This brings us to the end of Unit 3. You should now be able to calculate radiation heat transfer between various real-world geometric surfaces.
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## Looking Ahead

- Next Unit: Unit 4 - Heat Exchangers
- We will combine conduction, convection, and (rarely) radiation to analyze industrial equipment designed to transfer heat between two fluids.

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In Unit 4, we put all our knowledge together to design and analyze heat exchangers, the most common thermal equipment in industry.
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