Chapter 12 Radiation Heat Transfer Notes | EduRev

: Chapter 12 Radiation Heat Transfer Notes | EduRev

 Page 1


Chapter 12 Radiation Heat Transfer 
 12-1 
Chapter 12 
RADIATION HEAT TRANSFER 
 
 
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


Chapter 12 Radiation Heat Transfer 
 12-1 
Chapter 12 
RADIATION HEAT TRANSFER 
 
 
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
 
Chapter 12 Radiation Heat Transfer 
 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


Chapter 12 Radiation Heat Transfer 
 12-1 
Chapter 12 
RADIATION HEAT TRANSFER 
 
 
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
 
Chapter 12 Radiation Heat Transfer 
 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 
Chapter 12 Radiation Heat Transfer 
 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


Chapter 12 Radiation Heat Transfer 
 12-1 
Chapter 12 
RADIATION HEAT TRANSFER 
 
 
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
 
Chapter 12 Radiation Heat Transfer 
 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 
Chapter 12 Radiation Heat Transfer 
 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) 
Chapter 12 Radiation Heat Transfer 
 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


Chapter 12 Radiation Heat Transfer 
 12-1 
Chapter 12 
RADIATION HEAT TRANSFER 
 
 
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
 
Chapter 12 Radiation Heat Transfer 
 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 
Chapter 12 Radiation Heat Transfer 
 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) 
Chapter 12 Radiation Heat Transfer 
 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 
Chapter 12 Radiation Heat Transfer 
 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 
Read More
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!