Radiation Viewfactor

8/9/2019 Radiation Viewfactor

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MEL 725

Power-Plant Steam Generators (3-0-0)

Dr. Prabal Talukdar

Assistant Professor Department of Mechanical Engineering

IIT Delhi

Radiation Exchange between

Surfaces



View/Configuration Factor

• Also called shape factors.

• Radiation exchange between two or more

surfaces depends strongly on the surface

geometries and orientations, as well as ontheir radiative properties and temperature.

• To compute radiation exchange between

any two surfaces, we must first introducethe concept of a view factor.



Definition• The view factor Fij is defined as the fraction of the

radiation leaving a surface i that is intercepted by j.

dA jcosθ j



• Rate at which radiation leaves dAi and isintercepted by dA j may be expressed as

dqi→ j=Ii cosθi dAi dω j-i

i = intensity of radiation leaving surface idω j-i = soild angle subtended by dA j when viewed from dAi

= cosθ j dA j /R2

dqi→ j= Ii(cosθi cosθ j/R2 )dAidA j

• Assuming surface i emits and reflects diffusely

dqi→ j=Ji(cosθi cosθ j/πR2 )dAidA j

Derivation



• The total rate at which radiation leaves surface i

and is intercepted by j may then be obtained byintegrating over the two surfaces. That is

where it is assumed that the radiosity Ji is

uniform over the surface Ai.

Fij = radiation that leaves Ai and is intercepted by A j/total

radiation leaving Ai

= qi→ j / Ai

Ji

jiA A

2

ji

i ji dAdAR

coscos

Jq i j∫ ∫ π

θθ

=→

Derivation (cont’d)



Final Expression

ji

A A

2 ji

i

ij dAdAR

coscos

A1F

i j∫ ∫ π

θθ=

ji

A A

2 ji

j

ji dAdAR coscos

A1F

i j

∫ ∫ π θθ=

Similarly, the view factor F ji

defined as the fraction of the radiation

that leaves A j and intercepted by Ai can be expressed as

Assn: diffuse emitters and reflectors and have uniform radiosity



View Factor Relations• Reciprocity Relation:

AiFij = A jF ji

• Summation Rule:

• What is Fii ?• Value of Fii for a convex or flat surface?

Fii = 0

1F N

1 j

ij =∑=



View Factor Relations (cont’d)• Radiation exchange in an enclosure of N surfaces: N2

view factors required

• Summation rule can be applied to get N equations which

gives N view factors• Application of Reciprocity relation for N(N-1)/2 timesgives N(N-1)/2 view factors

• So we need essentially N2-N-N(N-1)/2

⎥

⎥⎥⎥

⎦

⎤

⎢

⎢⎢⎢

⎣

⎡

−−

−−−−

−−−−

NN2 N1 N

N22221

N11211

FFF

FFF

FFF



Example• For a 3 surface enclosure, we need

32-3-3(3-1)/2=3

F12 =F11=

F21=

F22= 1

2

10

A1/A2

1-A1/A2



View Factor Relation• Additive nature of view factor

∑

∑∑

∑

=

=

=

=

=

=

=

n

1k

k

n

1k

kik

)i( j

n

1k

kik )i( j j

n

1k

ik ) j(i

A

FA

F

FAFA

FF

i

j



Problem



Black Body Radiation Exchange

ijii ji F)JA(q =→

bj ji ji j

biiji ji

EFAq

EFAq

=

=

→

→

For a black surface Ji = Ebi

Similarly,

Net radiative exchange between the two surfaces:

)TT(FAEFAEFAq q q 4

j

4

iiji bj ji j biijii j jiij −σ=−=−= →→

Rate at which radiation leaves surface i and intercepted by j



Problem



Chart





• To use the last equation, surface radiosity Ji must known. To

determine this quantity, it is necessary to consider exchange

between surfaces of the enclosure

• Total rate at which radiation reaches surface i from all surfaces,

• or from Reciprocity relation,

• Cancelling the area Ai and substituting in

qi = Ai(Ji – Gi)


Surfaces• To use the last equation, surface radiosity Ji must known. To

determine this quantity, it is necessary to consider exchange

between surfaces of the enclosure

• Total rate at which radiation reaches surface i from all surfaces,

• or from Reciprocity relation,

∑=

= N

1 j

j j jiii JAFGA

∑=

= N

1 j

jiijii JAFGA

∑=

−= N

1 j

jijiii )JFJ(Aq

∑∑ == −=

N

1 j jiji

N

1 jijii )JFJF(Aq

Summation rule




Surfaces (cont’d)

∑∑==

=−= N

1 j

ij j

N

1 j

iijii q )JJ(FAq

iii

i bii

A/)1(

JEq

εε−

−=

∑= −−=εε− − N

1 j1

iji

ji

iii

i bi

)FA(JJ

A/)1(JE

Preferable

when temperature is known

Preferable when qi is known

∑= −−= N

1 j1

iji

jii

)FA(JJq



Set of Algebraic Equation

N N NNi Ni11 N

i NiNiii11i

1 N N1ii1111

CJaJaJa

CJaJaJa

CJaJaJa

=+−−−++−−−+

−−−−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−−−−

=+−−−++−−−+

−−−−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−−−−

=+−−−++−−−+

[ ][ ] [ ]CJA =In Matrix form,



Problem



Problem (cont’d)



Problem (cont’d)

222

22 b2

A/)1(

JEq

εε−

−=

∑=

−

−=

εε−− N

1 j1

iji

ji

iii

i bi

)FA(

JJ

A/)1(

JE

1232

321

212

12

222

22 b

)FA(JJ

)FA(JJ

A/)1(JE −− −+−=εε− −

For the hypothetical surface, J3 = Eb3

So, only unknowns are J2

and J1

24

22 b

2433 b3

m/W7348TE

m/W459TEJ

=σ=

=σ==F21= ?

F23 = ?



Problem (cont’d)121212 FAFA =

Y/L = 10/1 = 10

X/L = 1/1 = 1

F12 = 0.39

F21 = (A1/A2)F12

= (1x10/15) x 0.39

= 0.26

F13 = ?

F13 = 1-F12= 1 – 0.39

= 0.61

F31 = (A1/A3) x F13

=(10/(2(10.1)0.61

= 0.305F23 = ? F23 = 0.41



Problem (cont’d)7536J67.1J26.0 21 −=−

510002J39.0J10

)FA(

JJ

)FA(

JJ

A/)1(

JE

21

1131

31

1121

21

111

11 b

−=+−⇒

−+

−=

εε−−

−−

J2 = 12528 W/m2

kW7.77A/)1(

JEq 222

22 b2 −=

εε−−=



The Two-Surface Enclosure

22

2

12111

1

42

41

2112

A

1

FA

1

A

1

)TT(q q q

εε−

++ε

ε−−σ

=−==



Large (Infinite Parallel Plates)

A1,T1,ε1

A2,T2,ε2

q12 = ?

111

)TT(q

21

4

2

4

112

−ε

+ε

−σ=

Small convex object in a Large cavity q12

Radiation Viewfactor

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