SME
Heat Transfer · Problem type
View factors: reciprocity and the sum rule
Use this for “A 1 m² convex body sits inside a 4 m² enclosure, so F₁₂ = 1. Find F₂₁ and F₂₂.”
Diagram
Reciprocity:
F₂₁
=
A₁
F₁₂
A₂
=
1
×
1
4
=
0.25
Sum rule: F₂₂ = 1 − F₂₁ = 0.75; F₁₁ = 1 − F₁₂ = 0
A convex body can’t see itself (F₁₁ = 0): all it sends reaches the enclosure, which mostly sees itself.
This is an example. Type your own values to start a new problem.
A₁ (m²)
×
F₁₂
=
A₂ (m²)
×
F₂₁
F₁₁
+
F₁₂
=
1
F₂₁
+
F₂₂
=
1
Formulas
A₁
×
F₁₂
=
A₂
×
F₂₁
1 × 1 = 4 × 0.25
F₁₁
+
F₁₂
=
1
0 + 1 = 1
F₂₁
+
F₂₂
=
1
0.25 + 0.75 = 1
Assumptions
•
Two surfaces make a closed enclosure: what leaves one lands on itself or the other.
•
F₁₂ is the fraction of radiation leaving surface 1 that reaches surface 2.
•
A flat or convex surface cannot see itself (F₁₁ = 0); a concave one can.
Step-by-step
WE KNOW
F₁₂ = 1, A₁ = 1 m², A₂ = 4 m²
FIND
view factor 2 to 1 (F₂₁), view factor 1 to 1 (F₁₁), view factor 2 to 2 (F₂₂)
Step 1 · Find view factor 2 to 1 (F₂₁)
Use A₁ × F₁₂ = A₂ × F₂₁
Reciprocity:
the
exchange
each
way
is
the
same,
so
A₁
F₁₂
=
A₂
F₂₁
;
divide
by
A₂
.
F₂₁
=
A₁
×
F₁₂
A₂
F₂₁
=
1
×
1
4
F₂₁
=
0.25
Step 2 · Find view factor 1 to 1 (F₁₁)
Use F₁₁ + F₁₂ = 1
Everything leaving a surface lands somewhere in the enclosure: its view factors add to 1.
F₁₁
=
1
−
F₁₂
F₁₁
=
1
−
1
F₁₁
=
0
Step 3 · Find view factor 2 to 2 (F₂₂)
Use F₂₁ + F₂₂ = 1
Everything leaving a surface lands somewhere in the enclosure: its view factors add to 1.
F₂₂
=
1
−
F₂₁
F₂₂
=
1
−
0.25
F₂₂
=
0.75
Check
Put the numbers back in. Both sides should match.
1 × 1 = 4 × 0.25 ✓
0 + 1 = 1 ✓
0.25 + 0.75 = 1 ✓
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Radiation between two parallel gray plates
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The radiation heat transfer coefficient
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