Enclosure Radiation

Enclosure radiation analysis addresses heat exchange between multiple surfaces that form a closed or partially closed volume. This is fundamental to HVAC applications including building surfaces, duct interiors, radiant heating panels, and furnace chambers.

View Factors

View factors (also called shape factors or configuration factors) quantify the geometric relationship between surfaces exchanging radiation.

Definition

The view factor F₁₂ represents the fraction of radiation leaving surface 1 that is intercepted by surface 2:

F₁₂ = (1/A₁) ∫∫(cos θ₁ cos θ₂)/(πr²) dA₁ dA₂

Where:

• A₁ = area of surface 1 (ft² or m²)
• θ₁, θ₂ = angles between surface normals and line connecting differential areas
• r = distance between differential areas (ft or m)

Fundamental Properties

Reciprocity Relation:

A₁F₁₂ = A₂F₂₁

This allows calculation of one view factor from another when areas are known.

Summation Rule:

For an N-surface enclosure:

Σ(j=1 to N) Fᵢⱼ = 1

All radiation leaving surface i must reach some surface in the enclosure (including itself if concave).

Superposition:

View factors are additive. For composite surface (1+2) viewing surface 3:

A₁₊₂F₁₊₂,₃ = A₁F₁,₃ + A₂F₂,₃

View Factor Tables

Parallel Directly Opposed Rectangles:

Where X and Y are rectangle dimensions, D is separation distance.

Perpendicular Rectangles with Common Edge:

Two Infinite Parallel Plates:

F₁₂ = 1.0 (all radiation from one plate reaches the other)

Small Object in Large Enclosure:

F₁₂ ≈ 1.0 (nearly all radiation from small object reaches enclosure) F₂₁ ≈ A₁/A₂ (very small fraction of enclosure radiation reaches object)

Radiosity Method

Radiosity (J) represents the total radiant energy leaving a surface per unit area, including both emitted and reflected radiation.

Radiosity Definition

J = E + ρG = εEᵦ + ρG

Where:

• E = emitted radiation (Btu/hr-ft² or W/m²)
• G = irradiation (incident radiation)
• ρ = reflectivity
• ε = emissivity
• Eᵦ = blackbody emissive power = σT⁴

For gray, diffuse, opaque surfaces: ρ = 1 - ε

Therefore:

J = εσT⁴ + (1-ε)G

Surface Energy Balance

Net radiation leaving surface i:

qᵢ = Aᵢ(Jᵢ - Gᵢ)

The irradiation is the sum of radiosities from all surfaces:

Gᵢ = Σ(j=1 to N) FᵢⱼJⱼ

Combining:

qᵢ = Aᵢ[Jᵢ - Σ(j=1 to N) FᵢⱼJⱼ]

Radiation Network Method

The electrical network analogy simplifies enclosure radiation analysis by representing the problem as thermal resistances and potentials.

Surface Resistance

Resistance between blackbody emissive power and radiosity:

Rₛᵤᵣf,ᵢ = (1-εᵢ)/(εᵢAᵢ)

This represents the surface’s resistance to emitting radiation based on its emissivity.

Space Resistance

Resistance between surfaces due to geometry:

Rₛₚₐcₑ,ᵢⱼ = 1/(AᵢFᵢⱼ) = 1/(AⱼFⱼᵢ)

This is analogous to radiative exchange area.

Network Heat Transfer

Heat transfer from surface i to j:

qᵢⱼ = (Eᵦ,ᵢ - Eᵦ,ⱼ)/(Rₛᵤᵣf,ᵢ + Rₛₚₐcₑ,ᵢⱼ + Rₛᵤᵣf,ⱼ)

qᵢⱼ = (σT₁⁴ - σT₂⁴)/[(1-ε₁)/(ε₁A₁) + 1/(A₁F₁₂) + (1-ε₂)/(ε₂A₂)]

Gray Body Exchange Between Two Surfaces

For two gray, diffuse, opaque surfaces forming an enclosure:

q₁₂ = (σA₁(T₁⁴ - T₂⁴))/[1/ε₁ + (A₁/A₂)(1/ε₂ - 1) + (1-F₁₂)/F₁₂]

Special Cases

Infinite Parallel Plates (F₁₂ = 1, A₁ = A₂):

q₁₂/A = σ(T₁⁴ - T₂⁴)/(1/ε₁ + 1/ε₂ - 1)

Small Object in Large Enclosure (A₁ « A₂):

q₁ = ε₁A₁σ(T₁⁴ - T₂⁴)

The emissivity of the large enclosure becomes irrelevant.

Concentric Cylinders or Spheres:

q = (σA₁(T₁⁴ - T₂⁴))/[1/ε₁ + (r₁/r₂)(1/ε₂ - 1)]

Where r₁ and r₂ are inner and outer radii.

Three Surface Enclosures

For three-surface enclosures (common in HVAC: floor, ceiling, walls), the network method requires solving simultaneous equations.

Network Equations

For surface i at known temperature:

Eᵦ,ᵢ = Jᵢ + (1-εᵢ)/εᵢAᵢ · Σ(j=1 to 3) (Jᵢ - Jⱼ)/(1/AᵢFᵢⱼ)

For three surfaces:

• If all temperatures are known: solve for radiosities J₁, J₂, J₃
• If one heat flux is specified: solve for that temperature

Refractory Surfaces

A refractory (re-radiating) surface has zero net heat transfer (qᵢ = 0). It reaches an equilibrium temperature where absorbed radiation equals emitted radiation.

For refractory surface r:

Jᵣ = Σ(j=1 to N) FᵣⱼJⱼ

The refractory surface redistributes radiation but adds no thermal resistance.

Radiation Shields

Radiation shields are low-emissivity surfaces placed between hot and cold surfaces to reduce radiant heat transfer.

Single Shield Analysis

For infinite parallel plates with one shield:

q/A = σ(T₁⁴ - T₂⁴)/[(1/ε₁ + 1/ε₂ - 1) + (1/ε₃,ₐ + 1/ε₃,ᵦ - 1)]

Where ε₃,ₐ and ε₃,ᵦ are shield emissivities on sides facing surfaces 1 and 2.

For a shield with equal emissivities on both sides (ε₃,ₐ = ε₃,ᵦ = ε₃):

q/A = σ(T₁⁴ - T₂⁴)/[(1/ε₁ + 1/ε₂ - 1) + 2/ε₃ - 1]

Shield Effectiveness

Reduction factor with one shield:

q_with_shield/q_without_shield = (1/ε₁ + 1/ε₂ - 1)/[(1/ε₁ + 1/ε₂ - 1) + (2/ε₃ - 1)]

For low-emissivity shield (ε₃ = 0.05) between surfaces with ε₁ = ε₂ = 0.9:

Reduction factor = 0.026 (96% reduction in heat transfer)

Multiple Shields

With N shields between parallel plates:

q/A = σ(T₁⁴ - T₂⁴)/[(1/ε₁ + 1/ε₂ - 1) + N(2/ε_shield - 1)]

For identical shields with ε = 0.05:

Shield Temperature

The shield reaches an equilibrium temperature between the boundary surfaces. For single shield with equal emissivities:

T₃⁴ ≈ (T₁⁴ + T₂⁴)/2

More precisely, solve:

(T₁⁴ - T₃⁴)/(1/ε₁ + 1/ε₃ - 1) = (T₃⁴ - T₂⁴)/(1/ε₃ + 1/ε₂ - 1)

HVAC Enclosure Applications

Building Envelope Radiation

Room surfaces exchange radiation continuously. For a rectangular room:

• 6 surfaces (floor, ceiling, 4 walls)
• Surface temperatures vary with solar gains, heating/cooling
• Mean radiant temperature affects occupant comfort

Simplified calculation using area-weighted average:

MRT = Σ(AᵢTᵢ)/Σ(Aᵢ)

Radiant Panel Systems

Ceiling or floor radiant panels exchange with room surfaces:

q_panel = A_panel · ε_panel · σ · Σ Fₚ,ᵢ(T_panel⁴ - Tᵢ⁴)

Typical ceiling panel view factors to floor: 0.3-0.5 depending on ceiling height.

Duct Interior Radiation

In large rectangular or round ducts, radiation between interior surfaces affects heat transfer:

• Important at temperatures above 200°F (93°C)
• Combined convection-radiation heat transfer
• View factors approach 1.0 for long ducts

Furnace and Boiler Chambers

Combustion chambers involve high-temperature radiation:

• Flame radiation to furnace walls
• Multiple surface enclosures (burner, walls, tubes)
• Combination of gas and surface radiation
• Temperatures 1000-3000°F (540-1650°C)

Matrix Solution Methods

For N-surface enclosures, radiosity equations form a system of linear equations:

[A][J] = [B]

Where:

• [A] = coefficient matrix involving view factors
• [J] = radiosity vector
• [B] = vector of known quantities (blackbody emissive powers)

For surface i:

Jᵢ(1 - Σ ρᵢFᵢᵢ) - Σ(j≠i) ρᵢFᵢⱼJⱼ = εᵢσTᵢ⁴

This system is solved numerically for unknown radiosities, then heat fluxes are calculated:

qᵢ = εᵢAᵢ(σTᵢ⁴ - Jᵢ)/(1-εᵢ)

Design Considerations

Surface Selection:

• Low-emissivity surfaces (ε < 0.1) for radiation barriers
• High-emissivity surfaces (ε > 0.9) for radiant heat transfer
• Polished aluminum: ε ≈ 0.05
• Painted surfaces: ε ≈ 0.85-0.95

Geometric Configuration:

• Maximize view factors for desired heat transfer
• Minimize view factors where heat transfer is unwanted
• Use shields to redirect radiation paths

Temperature Levels:

• Radiation becomes dominant above 400°F (204°C)
• Fourth power temperature dependence makes high-temperature radiation critical
• Combined radiation-convection analysis required for moderate temperatures

Components

• Two Surface Enclosures
• Three Surface Enclosures
• N Surface Enclosures
• Radiation Network Method
• Matrix Solution Methods
• Electrical Analogy
• Gray Diffuse Surfaces
• Opaque Surfaces
• Refractory Surfaces
• Radiation Shields
• Multiple Shields
• Shield Optimization