Short Notes: View Factors | Short Notes for Mechanical Engineering PDF Download

 Page 1


View Factors 
Radiation heat transfer between surfaces depends on the orientation of the 
surfaces relative to each other as well as their radiation properties and 
temperatures. View factor (or shape factor) is a purely geometrical parameter that 
accounts for the effects of orientation on radiation between surfaces.
In view factor calculations, we assume uniform radiation in all directions 
throughout the surface, i.e., surfaces are isothermal and diffuse. Also, the medium 
between two surfaces does not absorb, emit, or scatter radiation.
• Fi_j or Fij = the fraction of the radiation leaving surface i that strikes surface j 
directly.
Note the following:
The view factor ranges between zero and one.
• F|j = 0 indicates that two surfaces do not see each other directly.
• F^ = 1 indicates that the surface j completely surrounds surface i.
• The radiation that strikes a surface does not need to be absorbed by that 
surface.
• F n is the fraction of radiation leaving surface i that strikes itself directly.
• F n = 0 for plane or convex surfaces, and Fii t 0 for concave surfaces.
Plane surface,
F ii = 0
Convex surface,
R i = 0
Concave surface,
F ii ^ ?
Page 2


View Factors 
Radiation heat transfer between surfaces depends on the orientation of the 
surfaces relative to each other as well as their radiation properties and 
temperatures. View factor (or shape factor) is a purely geometrical parameter that 
accounts for the effects of orientation on radiation between surfaces.
In view factor calculations, we assume uniform radiation in all directions 
throughout the surface, i.e., surfaces are isothermal and diffuse. Also, the medium 
between two surfaces does not absorb, emit, or scatter radiation.
• Fi_j or Fij = the fraction of the radiation leaving surface i that strikes surface j 
directly.
Note the following:
The view factor ranges between zero and one.
• F|j = 0 indicates that two surfaces do not see each other directly.
• F^ = 1 indicates that the surface j completely surrounds surface i.
• The radiation that strikes a surface does not need to be absorbed by that 
surface.
• F n is the fraction of radiation leaving surface i that strikes itself directly.
• F n = 0 for plane or convex surfaces, and Fii t 0 for concave surfaces.
Plane surface,
F ii = 0
Convex surface,
R i = 0
Concave surface,
F ii ^ ?
Fig. above shows View factor between surface and itself.
View Factor Relations
• Radiation analysis of an enclosure consisting of N surfaces requires the 
calculations of N2 view factors.
• Once a sufficient number of view factors are available, the rest of them can be 
found using the following relations for view factors.
The Reciprocity Rule
• The view factor Fjj is not equal to F y unless the areas of the two surfaces are 
equal. It can be shown that: Ai Fjj =Aj Fjj
The Summation Rule
• In radiation analysis, we usually form an enclosure. The conservation of 
energy principle requires that the entire radiation leaving any surface i of an 
enclosure be intercepted by the surfaces of the enclosure. Therefore,
• The summation rule can be applied to each surface of an enclosure by varying 
i from 1 to N (number of surfaces). Thus the summation rule gives N 
equations. Also reciprocity rule gives 0.5 N (N-1) additional equations. 
Therefore, the total number of view factors that need to be evaluated directly 
for an N-surface enclosure becomes
X
For eg.
The view factors F1 2 and F2i for the following geometries:
1. Sphere of diameter D inside a cubical box of length L = D
L = D
Solution:
• Sphere within a cube:
• By inspection, FI 2 = 1
• By reciprocity and summation:
F11+Fn = l^ F n = l - ^
o
2. End and side of a circular tube of equal length and diameter, L = D
Page 3


View Factors 
Radiation heat transfer between surfaces depends on the orientation of the 
surfaces relative to each other as well as their radiation properties and 
temperatures. View factor (or shape factor) is a purely geometrical parameter that 
accounts for the effects of orientation on radiation between surfaces.
In view factor calculations, we assume uniform radiation in all directions 
throughout the surface, i.e., surfaces are isothermal and diffuse. Also, the medium 
between two surfaces does not absorb, emit, or scatter radiation.
• Fi_j or Fij = the fraction of the radiation leaving surface i that strikes surface j 
directly.
Note the following:
The view factor ranges between zero and one.
• F|j = 0 indicates that two surfaces do not see each other directly.
• F^ = 1 indicates that the surface j completely surrounds surface i.
• The radiation that strikes a surface does not need to be absorbed by that 
surface.
• F n is the fraction of radiation leaving surface i that strikes itself directly.
• F n = 0 for plane or convex surfaces, and Fii t 0 for concave surfaces.
Plane surface,
F ii = 0
Convex surface,
R i = 0
Concave surface,
F ii ^ ?
Fig. above shows View factor between surface and itself.
View Factor Relations
• Radiation analysis of an enclosure consisting of N surfaces requires the 
calculations of N2 view factors.
• Once a sufficient number of view factors are available, the rest of them can be 
found using the following relations for view factors.
The Reciprocity Rule
• The view factor Fjj is not equal to F y unless the areas of the two surfaces are 
equal. It can be shown that: Ai Fjj =Aj Fjj
The Summation Rule
• In radiation analysis, we usually form an enclosure. The conservation of 
energy principle requires that the entire radiation leaving any surface i of an 
enclosure be intercepted by the surfaces of the enclosure. Therefore,
• The summation rule can be applied to each surface of an enclosure by varying 
i from 1 to N (number of surfaces). Thus the summation rule gives N 
equations. Also reciprocity rule gives 0.5 N (N-1) additional equations. 
Therefore, the total number of view factors that need to be evaluated directly 
for an N-surface enclosure becomes
X
For eg.
The view factors F1 2 and F2i for the following geometries:
1. Sphere of diameter D inside a cubical box of length L = D
L = D
Solution:
• Sphere within a cube:
• By inspection, FI 2 = 1
• By reciprocity and summation:
F11+Fn = l^ F n = l - ^
o
2. End and side of a circular tube of equal length and diameter, L = D
Aj
• Circular tube: with r2 / L = 0.5 and L / rl = 2, FI 3 » 0.17.
• From summation rule, F11 + FI 2 + FI 3 = 1 with F11 = 0, FI 2 = 1 - FI 3 = 0.83
• From reciprocity,
A. - nDL
x 0.S3 = 0.21
The Superposition Rule
• 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.
1 1 
Fig above shows The superposition rule for view factors. Fi _(2 i3 ) = Fi _ 2 + Fi _ 3 
The Symmetry Rule
• Two (or more) surfaces that possess symmetry about a third surface will have 
identical view factors from that surface.
For eg. the view factor from the base of a pyramid to each of its four sides. The 
base is a square and its side surfaces are isosceles triangles
From symmetry rule, we have: 
F1 2 = F -1 3 = Fm = Fig
Page 4


View Factors 
Radiation heat transfer between surfaces depends on the orientation of the 
surfaces relative to each other as well as their radiation properties and 
temperatures. View factor (or shape factor) is a purely geometrical parameter that 
accounts for the effects of orientation on radiation between surfaces.
In view factor calculations, we assume uniform radiation in all directions 
throughout the surface, i.e., surfaces are isothermal and diffuse. Also, the medium 
between two surfaces does not absorb, emit, or scatter radiation.
• Fi_j or Fij = the fraction of the radiation leaving surface i that strikes surface j 
directly.
Note the following:
The view factor ranges between zero and one.
• F|j = 0 indicates that two surfaces do not see each other directly.
• F^ = 1 indicates that the surface j completely surrounds surface i.
• The radiation that strikes a surface does not need to be absorbed by that 
surface.
• F n is the fraction of radiation leaving surface i that strikes itself directly.
• F n = 0 for plane or convex surfaces, and Fii t 0 for concave surfaces.
Plane surface,
F ii = 0
Convex surface,
R i = 0
Concave surface,
F ii ^ ?
Fig. above shows View factor between surface and itself.
View Factor Relations
• Radiation analysis of an enclosure consisting of N surfaces requires the 
calculations of N2 view factors.
• Once a sufficient number of view factors are available, the rest of them can be 
found using the following relations for view factors.
The Reciprocity Rule
• The view factor Fjj is not equal to F y unless the areas of the two surfaces are 
equal. It can be shown that: Ai Fjj =Aj Fjj
The Summation Rule
• In radiation analysis, we usually form an enclosure. The conservation of 
energy principle requires that the entire radiation leaving any surface i of an 
enclosure be intercepted by the surfaces of the enclosure. Therefore,
• The summation rule can be applied to each surface of an enclosure by varying 
i from 1 to N (number of surfaces). Thus the summation rule gives N 
equations. Also reciprocity rule gives 0.5 N (N-1) additional equations. 
Therefore, the total number of view factors that need to be evaluated directly 
for an N-surface enclosure becomes
X
For eg.
The view factors F1 2 and F2i for the following geometries:
1. Sphere of diameter D inside a cubical box of length L = D
L = D
Solution:
• Sphere within a cube:
• By inspection, FI 2 = 1
• By reciprocity and summation:
F11+Fn = l^ F n = l - ^
o
2. End and side of a circular tube of equal length and diameter, L = D
Aj
• Circular tube: with r2 / L = 0.5 and L / rl = 2, FI 3 » 0.17.
• From summation rule, F11 + FI 2 + FI 3 = 1 with F11 = 0, FI 2 = 1 - FI 3 = 0.83
• From reciprocity,
A. - nDL
x 0.S3 = 0.21
The Superposition Rule
• 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.
1 1 
Fig above shows The superposition rule for view factors. Fi _(2 i3 ) = Fi _ 2 + Fi _ 3 
The Symmetry Rule
• Two (or more) surfaces that possess symmetry about a third surface will have 
identical view factors from that surface.
For eg. the view factor from the base of a pyramid to each of its four sides. The 
base is a square and its side surfaces are isosceles triangles
From symmetry rule, we have: 
F1 2 = F -1 3 = Fm = Fig
• Also, the summation rule yields: F-n + F-|2 + F-|3 + Fi 4 + Fi 5 = 1
• Since, F-n = 0 (flat surface), we find
• F1 2 = F1 3 = F1 4 = F1 5 = 0.25
The Crossed-Strings Method
Geometries such as channels and ducts that are very long In one direction can be 
considered two-dimensional (since radiation through end surfaces can be 
neglected). The view factor between their surfaces can be determined by the 
crossstring method developed by H. C. Hottel, as follows:
^ ^ (crossed strings) - ^ (uncrossed strings)
I_ > J 2 x (string on surface i)
t,L;+L,)-(L;+L<)
Page 5


View Factors 
Radiation heat transfer between surfaces depends on the orientation of the 
surfaces relative to each other as well as their radiation properties and 
temperatures. View factor (or shape factor) is a purely geometrical parameter that 
accounts for the effects of orientation on radiation between surfaces.
In view factor calculations, we assume uniform radiation in all directions 
throughout the surface, i.e., surfaces are isothermal and diffuse. Also, the medium 
between two surfaces does not absorb, emit, or scatter radiation.
• Fi_j or Fij = the fraction of the radiation leaving surface i that strikes surface j 
directly.
Note the following:
The view factor ranges between zero and one.
• F|j = 0 indicates that two surfaces do not see each other directly.
• F^ = 1 indicates that the surface j completely surrounds surface i.
• The radiation that strikes a surface does not need to be absorbed by that 
surface.
• F n is the fraction of radiation leaving surface i that strikes itself directly.
• F n = 0 for plane or convex surfaces, and Fii t 0 for concave surfaces.
Plane surface,
F ii = 0
Convex surface,
R i = 0
Concave surface,
F ii ^ ?
Fig. above shows View factor between surface and itself.
View Factor Relations
• Radiation analysis of an enclosure consisting of N surfaces requires the 
calculations of N2 view factors.
• Once a sufficient number of view factors are available, the rest of them can be 
found using the following relations for view factors.
The Reciprocity Rule
• The view factor Fjj is not equal to F y unless the areas of the two surfaces are 
equal. It can be shown that: Ai Fjj =Aj Fjj
The Summation Rule
• In radiation analysis, we usually form an enclosure. The conservation of 
energy principle requires that the entire radiation leaving any surface i of an 
enclosure be intercepted by the surfaces of the enclosure. Therefore,
• The summation rule can be applied to each surface of an enclosure by varying 
i from 1 to N (number of surfaces). Thus the summation rule gives N 
equations. Also reciprocity rule gives 0.5 N (N-1) additional equations. 
Therefore, the total number of view factors that need to be evaluated directly 
for an N-surface enclosure becomes
X
For eg.
The view factors F1 2 and F2i for the following geometries:
1. Sphere of diameter D inside a cubical box of length L = D
L = D
Solution:
• Sphere within a cube:
• By inspection, FI 2 = 1
• By reciprocity and summation:
F11+Fn = l^ F n = l - ^
o
2. End and side of a circular tube of equal length and diameter, L = D
Aj
• Circular tube: with r2 / L = 0.5 and L / rl = 2, FI 3 » 0.17.
• From summation rule, F11 + FI 2 + FI 3 = 1 with F11 = 0, FI 2 = 1 - FI 3 = 0.83
• From reciprocity,
A. - nDL
x 0.S3 = 0.21
The Superposition Rule
• 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.
1 1 
Fig above shows The superposition rule for view factors. Fi _(2 i3 ) = Fi _ 2 + Fi _ 3 
The Symmetry Rule
• Two (or more) surfaces that possess symmetry about a third surface will have 
identical view factors from that surface.
For eg. the view factor from the base of a pyramid to each of its four sides. The 
base is a square and its side surfaces are isosceles triangles
From symmetry rule, we have: 
F1 2 = F -1 3 = Fm = Fig
• Also, the summation rule yields: F-n + F-|2 + F-|3 + Fi 4 + Fi 5 = 1
• Since, F-n = 0 (flat surface), we find
• F1 2 = F1 3 = F1 4 = F1 5 = 0.25
The Crossed-Strings Method
Geometries such as channels and ducts that are very long In one direction can be 
considered two-dimensional (since radiation through end surfaces can be 
neglected). The view factor between their surfaces can be determined by the 
crossstring method developed by H. C. Hottel, as follows:
^ ^ (crossed strings) - ^ (uncrossed strings)
I_ > J 2 x (string on surface i)
t,L;+L,)-(L;+L<)
Wien's Displacement Law
Wien's Displacement Law: Wien’s displacement law state that the product of Am a x 
and T is constant.
Am a x T = constant
Where, Am a x = Wavelength at which the maximum value of monochromatic 
emissive power occurs.