B

HEAT TRANSFER: solutions to radiation tutorial problems (view factors)

(a) Add an imaginary third surface to complete the (isosceles) triangle and assume that no radiation escapes from the ends of the enclosure:

Rule of summation: F₁₁ + F₁₂ + F₁₃ = 1

Surface 1 is perfectly flat F₁₁ = 0
By symmetry F₁₂ = F₁₃
\ F₁₂ = 0.5

Rule of reciprocity: A₁F₁₂ = A₂F₂₁

Surfaces are of length "L" perpendicular to page:

(b) Assume that no radiation escapes from the ends of the enclosure. All radiation from 1 goes to 2, as 1 is flat and 2 is the only available destination for it. Surfaces are of length "L" perpendicular to page:

Rule of summation: F₁₁ + F₁₂ = 1

Surface 1 is perfectly flat F₁₁ = 0
\ F₁₂ = 1

Rule of reciprocity:

Let the disc be surface 1, the inside top and sides of the box be 2 and the remainder of the bottom inner surface 3.

Rule of summation: F₁₁ + F₁₂ + F₁₃ = 1
Bottom is perfectly flat F₁₁ = F₁₃ = 0
\ F₁₂ = 1.0

Rule of reciprocity: A₁F₁₂ = A₂F₂₁

Rule of summation: F₃₁ + F₃₂ + F₃₃ = 1
Bottom is perfectly flat F₃₁ = F₃₃ = 0
\ F₃₂ = 1.0

Rule of reciprocity: A₃F₃₂ = A₂F₂₃

Use the charts for view factors between two surfaces at right angles:

® from formula in module pack, F₁₂ = 0.132581156

Rule of reciprocity:

Here, we need to find F_1b2b
A₁F₁₂ = A₁F_12a + A₁F_12b

where, remembering that the graph can give figures for F₁₂, F_1a2, F_12a, F_1a2a
A₁ = A_1a + A_1b = 2A_1a
A₁F_12b = A_1aF_1a2b + A_1bF_1b2b
A_1aF_1a2 = A_1aF_1a2a + A_1aF_1a2b

Substituting back into first expression
2A_1aF₁₂ = 2A_1aF_12a + (A_1aF_1a2 - A_1aF_1a2a)+ A_1bF_1b2b
F_1b2b = 2F₁₂ - 2F_12a + F_1a2a - F_1a2
® from formula, F_12a = 0.101543970
® from formula, F_1a2 = 0.213712346
® from formula, F_1a2a = 0.179622642
F_1b2b = 2(0.132581156) - 2(0.101543970) + 0.179622642 - 0.213712346 = 0.027984668

By reciprocity, the other view factor can be found:

Use the formulae for view factors between two parallel discs, and treat cut-out areas as discs themselves:

Here, we need to find F_1b2b
A₁F₁₂ = A₁F_12a + A₁F_12b

where, remembering that the formulae can give figures for F₁₂, F_1a2, F_12a, F_1a2a
A₁ = A_1a + A_1b
A₁F_12b = A_1aF_1a2b + A_1bF_1b2b
A_1aF_1a2 = A_1aF_1a2a + A_1aF_1a2b

Substituting back into first expression
A₁F₁₂ = A₁F_12a + A_1a(F_1a2 - F_1a2a) + (A₁ - A_1a)F_1b2b
(A₁ - A_1a)F_1b2b = A₁F₁₂ - A₁F_12a - A_1a(F_1a2 - F_1a2a)

For large discs (1 and 2):

For small discs (1a and 2a):

® A_1a = p ´ 0.1² = 0.01p m²
® A₁ = p ´ 0.2² = 0.04p m²