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# Chapter 12 Radiation Heat Transfer Notes | EduRev

## : Chapter 12 Radiation Heat Transfer Notes | EduRev

``` Page 1

12-1
Chapter 12

View Factors

12-1C The view factor F
i j ?
represents the fraction of the radiation leaving surface i that strikes surface j
directly. The view factor from a surface to itself is non-zero for concave surfaces.

12-2C The pair of view factors F
i j ?
and F
j i ?
are related to each other by the reciprocity rule
A F A F
i ij j ji
? where A
i
is the area of the surface i and A
j
is the area of the surface j. Therefore,
A F A F F
A
A
F
1 12 2 21 12
2
1
21
? ? ? ? ?
12-3C The summation rule for an enclosure and is expressed as  F
i j
j
N
?
?
?
?
1
1
where N is the number of
surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all
surfaces of the enclosure, including to itself must be equal to unity.
The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum
of the view factors from surface i to the parts of surface j,  F F F
1 2 3 1 2 1 3 ? ? ?
? ?
( , )
.

12-4C The cross-string method is applicable to geometries which are very long in one direction relative to
the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as
F
i
i j ?
?
?
?
? ?
Crossed strings Uncrossed strings
string on surface 2

Page 2

12-1
Chapter 12

View Factors

12-1C The view factor F
i j ?
represents the fraction of the radiation leaving surface i that strikes surface j
directly. The view factor from a surface to itself is non-zero for concave surfaces.

12-2C The pair of view factors F
i j ?
and F
j i ?
are related to each other by the reciprocity rule
A F A F
i ij j ji
? where A
i
is the area of the surface i and A
j
is the area of the surface j. Therefore,
A F A F F
A
A
F
1 12 2 21 12
2
1
21
? ? ? ? ?
12-3C The summation rule for an enclosure and is expressed as  F
i j
j
N
?
?
?
?
1
1
where N is the number of
surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all
surfaces of the enclosure, including to itself must be equal to unity.
The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum
of the view factors from surface i to the parts of surface j,  F F F
1 2 3 1 2 1 3 ? ? ?
? ?
( , )
.

12-4C The cross-string method is applicable to geometries which are very long in one direction relative to
the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as
F
i
i j ?
?
?
?
? ?
Crossed strings Uncrossed strings
string on surface 2

12-2
12-5 An enclosure consisting of six surfaces is considered. The
number of view factors this geometry involves and the number of these
view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A seven surface enclosure (N=6) involves N
2 2
6 ? ? 36 view
factors and we need to determine 15
2
) 1 6 ( 6
2
) 1 (
?
?
?
? N N
view factors
directly. The remaining 36-15 = 21 of the view factors can be
determined by the application of the reciprocity and summation rules.

12-6 An enclosure consisting of five surfaces is considered. The
number of view factors this geometry involves and the number of
these view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A five surface enclosure (N=5) involves N
2 2
5 ? ? 25
view factors and we need to determine
N N ( ) (5 ) ?
?
?
?
1
2
5 1
2
10
view factors directly. The remaining 25-10 = 15 of the view factors
can be determined by the application of the reciprocity and
summation rules.

12-7 An enclosure consisting of twelve surfaces
is considered. The number of view factors this
geometry involves and the number of these view
factors that can be determined by the application
of the reciprocity and summation rules are to be
determined.
Analysis A twelve surface enclosure (N=12)
involves 144 ? ?
2 2
12 N view factors  and we
need to determine
N N ( ) ( ) ?
?
?
?
1
2
12 12 1
2
66
view factors directly. The remaining 144-66 = 78
of the view factors can be determined by the
application of the reciprocity and summation
rules.
2
1
4
5
3
6
5
4
3
2
1
2
1
3
9
11
12
10
4
5
8
6
7
Page 3

12-1
Chapter 12

View Factors

12-1C The view factor F
i j ?
represents the fraction of the radiation leaving surface i that strikes surface j
directly. The view factor from a surface to itself is non-zero for concave surfaces.

12-2C The pair of view factors F
i j ?
and F
j i ?
are related to each other by the reciprocity rule
A F A F
i ij j ji
? where A
i
is the area of the surface i and A
j
is the area of the surface j. Therefore,
A F A F F
A
A
F
1 12 2 21 12
2
1
21
? ? ? ? ?
12-3C The summation rule for an enclosure and is expressed as  F
i j
j
N
?
?
?
?
1
1
where N is the number of
surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all
surfaces of the enclosure, including to itself must be equal to unity.
The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum
of the view factors from surface i to the parts of surface j,  F F F
1 2 3 1 2 1 3 ? ? ?
? ?
( , )
.

12-4C The cross-string method is applicable to geometries which are very long in one direction relative to
the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as
F
i
i j ?
?
?
?
? ?
Crossed strings Uncrossed strings
string on surface 2

12-2
12-5 An enclosure consisting of six surfaces is considered. The
number of view factors this geometry involves and the number of these
view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A seven surface enclosure (N=6) involves N
2 2
6 ? ? 36 view
factors and we need to determine 15
2
) 1 6 ( 6
2
) 1 (
?
?
?
? N N
view factors
directly. The remaining 36-15 = 21 of the view factors can be
determined by the application of the reciprocity and summation rules.

12-6 An enclosure consisting of five surfaces is considered. The
number of view factors this geometry involves and the number of
these view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A five surface enclosure (N=5) involves N
2 2
5 ? ? 25
view factors and we need to determine
N N ( ) (5 ) ?
?
?
?
1
2
5 1
2
10
view factors directly. The remaining 25-10 = 15 of the view factors
can be determined by the application of the reciprocity and
summation rules.

12-7 An enclosure consisting of twelve surfaces
is considered. The number of view factors this
geometry involves and the number of these view
factors that can be determined by the application
of the reciprocity and summation rules are to be
determined.
Analysis A twelve surface enclosure (N=12)
involves 144 ? ?
2 2
12 N view factors  and we
need to determine
N N ( ) ( ) ?
?
?
?
1
2
12 12 1
2
66
view factors directly. The remaining 144-66 = 78
of the view factors can be determined by the
application of the reciprocity and summation
rules.
2
1
4
5
3
6
5
4
3
2
1
2
1
3
9
11
12
10
4
5
8
6
7
12-3
12-8 The view factors between the rectangular surfaces shown in the figure are to be determined.
Assumptions The surfaces are diffuse emitters and reflectors.
Analysis From Fig. 12-6,
24 . 0
5 . 0
2
1 1
5 . 0
2
1
31
3
?
?
?
?
?
?
?
?
? ?
? ?
F
W
L
W
L

and
29 . 0
1
2
2
5 . 0
2
1
) 2 1 ( 3
2 1
3
?
?
?
?
?
?
?
?
? ?
?
? ?
? ?
F
W
L L
W
L

We note that A 1 = A 3. Then the reciprocity and superposition rules gives
0.24 ? ? ? ? ? ?
31 13 31 3 13 1
A F F F A F
05 . 0 24 . 0 29 . 0
32 32 32 31 ) 2 1 ( 3
? ? ? ? ? ? ? ? ? ? ?
? ?
F F F F F
Finally, 0.05 ? ? ? ? ? ?
32 23 3 2
F F A A
W = 2 m
(2)
L2 = 1 m

L1 = 1 m
L3 = 1 m
A 3     (3)
A 2

A 1
(1)
Page 4

12-1
Chapter 12

View Factors

12-1C The view factor F
i j ?
represents the fraction of the radiation leaving surface i that strikes surface j
directly. The view factor from a surface to itself is non-zero for concave surfaces.

12-2C The pair of view factors F
i j ?
and F
j i ?
are related to each other by the reciprocity rule
A F A F
i ij j ji
? where A
i
is the area of the surface i and A
j
is the area of the surface j. Therefore,
A F A F F
A
A
F
1 12 2 21 12
2
1
21
? ? ? ? ?
12-3C The summation rule for an enclosure and is expressed as  F
i j
j
N
?
?
?
?
1
1
where N is the number of
surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all
surfaces of the enclosure, including to itself must be equal to unity.
The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum
of the view factors from surface i to the parts of surface j,  F F F
1 2 3 1 2 1 3 ? ? ?
? ?
( , )
.

12-4C The cross-string method is applicable to geometries which are very long in one direction relative to
the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as
F
i
i j ?
?
?
?
? ?
Crossed strings Uncrossed strings
string on surface 2

12-2
12-5 An enclosure consisting of six surfaces is considered. The
number of view factors this geometry involves and the number of these
view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A seven surface enclosure (N=6) involves N
2 2
6 ? ? 36 view
factors and we need to determine 15
2
) 1 6 ( 6
2
) 1 (
?
?
?
? N N
view factors
directly. The remaining 36-15 = 21 of the view factors can be
determined by the application of the reciprocity and summation rules.

12-6 An enclosure consisting of five surfaces is considered. The
number of view factors this geometry involves and the number of
these view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A five surface enclosure (N=5) involves N
2 2
5 ? ? 25
view factors and we need to determine
N N ( ) (5 ) ?
?
?
?
1
2
5 1
2
10
view factors directly. The remaining 25-10 = 15 of the view factors
can be determined by the application of the reciprocity and
summation rules.

12-7 An enclosure consisting of twelve surfaces
is considered. The number of view factors this
geometry involves and the number of these view
factors that can be determined by the application
of the reciprocity and summation rules are to be
determined.
Analysis A twelve surface enclosure (N=12)
involves 144 ? ?
2 2
12 N view factors  and we
need to determine
N N ( ) ( ) ?
?
?
?
1
2
12 12 1
2
66
view factors directly. The remaining 144-66 = 78
of the view factors can be determined by the
application of the reciprocity and summation
rules.
2
1
4
5
3
6
5
4
3
2
1
2
1
3
9
11
12
10
4
5
8
6
7
12-3
12-8 The view factors between the rectangular surfaces shown in the figure are to be determined.
Assumptions The surfaces are diffuse emitters and reflectors.
Analysis From Fig. 12-6,
24 . 0
5 . 0
2
1 1
5 . 0
2
1
31
3
?
?
?
?
?
?
?
?
? ?
? ?
F
W
L
W
L

and
29 . 0
1
2
2
5 . 0
2
1
) 2 1 ( 3
2 1
3
?
?
?
?
?
?
?
?
? ?
?
? ?
? ?
F
W
L L
W
L

We note that A 1 = A 3. Then the reciprocity and superposition rules gives
0.24 ? ? ? ? ? ?
31 13 31 3 13 1
A F F F A F
05 . 0 24 . 0 29 . 0
32 32 32 31 ) 2 1 ( 3
? ? ? ? ? ? ? ? ? ? ?
? ?
F F F F F
Finally, 0.05 ? ? ? ? ? ?
32 23 3 2
F F A A
W = 2 m
(2)
L2 = 1 m

L1 = 1 m
L3 = 1 m
A 3     (3)
A 2

A 1
(1)
12-4
12-9 A cylindrical enclosure is considered. The view factor from the side surface of this cylindrical
enclosure to its base surface is to be determined.
Assumptions The surfaces are diffuse emitters and reflectors.
Analysis We designate the surfaces as follows:
Base surface by (1),
top surface by (2), and
side surface by (3).
Then from Fig. 12-7 (or Table 12-1 for better accuracy)
38 . 0
1
1
21 12
2
2 2
1
1
1
? ?
?
?
?
?
?
?
?
? ?
? ?
F F
r
r
L
r
r
r
r
L

1 : rule summation
13 12 11
? ? ? F F F
62 . 0 1 38 . 0 0
13 13
? ? ? ? ? ? ? F F

? ?
0.31 ? ?
?
?
?
?
?
? ? ? ? ? ? ) 62 . 0 (
2
1
2 2
: rule y reciprocit
13
1 1
2
1
13
1
2
1
13
3
1
31 31 3 13 1
F
r r
r
F
L r
r
F
A
A
F F A F A

Discussion This problem can be solved more accurately by using the view factor relation from Table 12-1
to be
1
1
2
2 2
2
1
1 1
1
? ? ?
? ? ?
r
r
L
r
R
r
r
L
r
R

382 . 0
1
1
4 3 3 4
3
1
1 1
1
1
1
5 . 0
2
2
2
1
5 . 0
2
1
2 2
2
1
12
2
2
2
1
2
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
? ?
?
? ?
R
R
S S F
R
R
S

618 . 0 382 . 0 1 1
12 13
? ? ? ? ? F F
? ?
0.309 ? ?
?
?
?
?
?
? ? ? ? ? ? ) 618 . 0 (
2
1
2 2
: rule y reciprocit
13
1 1
2
1
13
1
2
1
13
3
1
31 31 3 13 1
F
r r
r
F
L r
r
F
A
A
F F A F A

(2)
(3)
(1)
L
D
Page 5

12-1
Chapter 12

View Factors

12-1C The view factor F
i j ?
represents the fraction of the radiation leaving surface i that strikes surface j
directly. The view factor from a surface to itself is non-zero for concave surfaces.

12-2C The pair of view factors F
i j ?
and F
j i ?
are related to each other by the reciprocity rule
A F A F
i ij j ji
? where A
i
is the area of the surface i and A
j
is the area of the surface j. Therefore,
A F A F F
A
A
F
1 12 2 21 12
2
1
21
? ? ? ? ?
12-3C The summation rule for an enclosure and is expressed as  F
i j
j
N
?
?
?
?
1
1
where N is the number of
surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all
surfaces of the enclosure, including to itself must be equal to unity.
The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum
of the view factors from surface i to the parts of surface j,  F F F
1 2 3 1 2 1 3 ? ? ?
? ?
( , )
.

12-4C The cross-string method is applicable to geometries which are very long in one direction relative to
the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as
F
i
i j ?
?
?
?
? ?
Crossed strings Uncrossed strings
string on surface 2

12-2
12-5 An enclosure consisting of six surfaces is considered. The
number of view factors this geometry involves and the number of these
view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A seven surface enclosure (N=6) involves N
2 2
6 ? ? 36 view
factors and we need to determine 15
2
) 1 6 ( 6
2
) 1 (
?
?
?
? N N
view factors
directly. The remaining 36-15 = 21 of the view factors can be
determined by the application of the reciprocity and summation rules.

12-6 An enclosure consisting of five surfaces is considered. The
number of view factors this geometry involves and the number of
these view factors that can be determined by the application of the
reciprocity and summation rules are to be determined.
Analysis A five surface enclosure (N=5) involves N
2 2
5 ? ? 25
view factors and we need to determine
N N ( ) (5 ) ?
?
?
?
1
2
5 1
2
10
view factors directly. The remaining 25-10 = 15 of the view factors
can be determined by the application of the reciprocity and
summation rules.

12-7 An enclosure consisting of twelve surfaces
is considered. The number of view factors this
geometry involves and the number of these view
factors that can be determined by the application
of the reciprocity and summation rules are to be
determined.
Analysis A twelve surface enclosure (N=12)
involves 144 ? ?
2 2
12 N view factors  and we
need to determine
N N ( ) ( ) ?
?
?
?
1
2
12 12 1
2
66
view factors directly. The remaining 144-66 = 78
of the view factors can be determined by the
application of the reciprocity and summation
rules.
2
1
4
5
3
6
5
4
3
2
1
2
1
3
9
11
12
10
4
5
8
6
7
12-3
12-8 The view factors between the rectangular surfaces shown in the figure are to be determined.
Assumptions The surfaces are diffuse emitters and reflectors.
Analysis From Fig. 12-6,
24 . 0
5 . 0
2
1 1
5 . 0
2
1
31
3
?
?
?
?
?
?
?
?
? ?
? ?
F
W
L
W
L

and
29 . 0
1
2
2
5 . 0
2
1
) 2 1 ( 3
2 1
3
?
?
?
?
?
?
?
?
? ?
?
? ?
? ?
F
W
L L
W
L

We note that A 1 = A 3. Then the reciprocity and superposition rules gives
0.24 ? ? ? ? ? ?
31 13 31 3 13 1
A F F F A F
05 . 0 24 . 0 29 . 0
32 32 32 31 ) 2 1 ( 3
? ? ? ? ? ? ? ? ? ? ?
? ?
F F F F F
Finally, 0.05 ? ? ? ? ? ?
32 23 3 2
F F A A
W = 2 m
(2)
L2 = 1 m

L1 = 1 m
L3 = 1 m
A 3     (3)
A 2

A 1
(1)
12-4
12-9 A cylindrical enclosure is considered. The view factor from the side surface of this cylindrical
enclosure to its base surface is to be determined.
Assumptions The surfaces are diffuse emitters and reflectors.
Analysis We designate the surfaces as follows:
Base surface by (1),
top surface by (2), and
side surface by (3).
Then from Fig. 12-7 (or Table 12-1 for better accuracy)
38 . 0
1
1
21 12
2
2 2
1
1
1
? ?
?
?
?
?
?
?
?
? ?
? ?
F F
r
r
L
r
r
r
r
L

1 : rule summation
13 12 11
? ? ? F F F
62 . 0 1 38 . 0 0
13 13
? ? ? ? ? ? ? F F

? ?
0.31 ? ?
?
?
?
?
?
? ? ? ? ? ? ) 62 . 0 (
2
1
2 2
: rule y reciprocit
13
1 1
2
1
13
1
2
1
13
3
1
31 31 3 13 1
F
r r
r
F
L r
r
F
A
A
F F A F A

Discussion This problem can be solved more accurately by using the view factor relation from Table 12-1
to be
1
1
2
2 2
2
1
1 1
1
? ? ?
? ? ?
r
r
L
r
R
r
r
L
r
R

382 . 0
1
1
4 3 3 4
3
1
1 1
1
1
1
5 . 0
2
2
2
1
5 . 0
2
1
2 2
2
1
12
2
2
2
1
2
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
? ?
?
? ?
R
R
S S F
R
R
S

618 . 0 382 . 0 1 1
12 13
? ? ? ? ? F F
? ?
0.309 ? ?
?
?
?
?
?
? ? ? ? ? ? ) 618 . 0 (
2
1
2 2
: rule y reciprocit
13
1 1
2
1
13
1
2
1
13
3
1
31 31 3 13 1
F
r r
r
F
L r
r
F
A
A
F F A F A

(2)
(3)
(1)
L
D
12-5
12-10 A semispherical furnace is considered. The view factor from the dome of this furnace to its flat base
is to be determined.
Assumptions The surfaces are diffuse emitters and reflectors.
Analysis We number the surfaces as follows:
(1):  circular base surface
(2):   dome surface
Surface (1) is flat, and thus  F
11
0 ? .
1 1   : rule S ummation
12 12 11
? ? ? ? F F F
0.5 ? ? ? ? ? ? ? ? ?
2
1
2
4
) 1 ( A  : rule y reciprocit
2
2
2
1
12
2
1
21 21 2 12 1
D
D
A
A
F
A
A
F F A F
?
?

12-11 Two view factors associated with three very long ducts with
different geometries are to be determined.
Assumptions 1 The surfaces are diffuse emitters and reflectors. 2 End
effects are neglected.
Analysis (a) Surface (1) is flat, and thus F
11
0 ? .
1 ? ? ? ?
12 12 11
1   : rule summation F F F
0.64 ? ?
?
?
?
?
?
?
? ? ? ? ? ?
? ?
2
) 1 (
2
A   : rule y reciprocit
12
2
1
21 21 2 12 1
s
D
Ds
F
A
A
F F A F
(b) Noting that surfaces 2 and 3 are symmetrical and thus
F F
12 13
? , the summation rule gives
0.5 ? ? ? ? ? ? ? ? ? ? ? ? ?
12 13 12 13 12 11
1 0 1 F F F F F F
Also by using the equation obtained in Example 12-4,
F
L L L
L
a b b
a
a
a
12
1 2 3
1
2 2 2
1
2
?
? ?
?
? ?
? ? ? 0.5
2b
a
? ?
?
?
?
?
?
? ? ? ? ? ?
2
1
A  : rule y reciprocit
12
2
1
21 21 2 12 1
b
a
F
A
A
F F A F
(c) Applying the crossed-string method gives
F F
L L L L
L
a b b
a
12 21
5 6 3 4
1
2 2
2
2 2
2
? ?
? ? ?
?
? ?
?
? ?
( ) ( )
a b b
a
2 2

(1)
(2)
D
(1)
(2)
D
(1)
(3)         (2)
a
L3 = b                                             L4 = b

L5                     L6
L 2 = a
L 1 = a
```
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