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280
Analytical Heat Transfer
θ j
θ i
j
i
R
dG i
dA i
dA j
FIGURE 14.5
Elemental surface for radiation in an enclosure containing an isothermal gray gas.
gas emission = E bg /π = σT 4 /π, E bg the blackbody gas emission = σT 4 , dw j =
g
g
dA j cos θ j /R 2 , R the distance (beam length) between surface dA i and dA j , and
K the gas absorption coefficient.
Therefore,
cos θ i cos θ j
−KR
−KR
+ E bg (1 − e
)
dG i =
J j e
dA j dA i
πR 2
A j
1
G i =
dG i dA i
cos θ i cos θ j
−KR
J j e
−KR
+ E bg (1 − e
)
πR 2
dA j dA i
(14.12)
A i
1
= A i
A i A j
The distance (beam length) R varies over the surface. For convenience, we
define a mean surface (mean beam length) L ij , such that
−KL ij + E bg (1 − e
−KL ij )
1
cos θ i cos θ j
G i = J j e
A i
πR 2
dA j dA i
A i A j
= J j e
−KL ij + E bg (1 − e
−KL ij ) F ij
(14.13)
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� �
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280
Analytical Heat Transfer
θ j
θ i
j
i
R
dG i
dA i
dA j
FIGURE 14.5
Elemental surface for radiation in an enclosure containing an isothermal gray gas.
gas emission = E bg /π = σT 4 /π, E bg the blackbody gas emission = σT 4 , dw j =
g
g
dA j cos θ j /R 2 , R the distance (beam length) between surface dA i and dA j , and
K the gas absorption coefficient.
Therefore,
cos θ i cos θ j
−KR
−KR
+ E bg (1 − e
)
dG i =
J j e
dA j dA i
πR 2
A j
1
G i =
dG i dA i
cos θ i cos θ j
−KR
J j e
−KR
+ E bg (1 − e
)
πR 2
dA j dA i
(14.12)
A i
1
= A i
A i A j
The distance (beam length) R varies over the surface. For convenience, we
define a mean surface (mean beam length) L ij , such that
−KL ij + E bg (1 − e
−KL ij )
1
cos θ i cos θ j
G i = J j e
A i
πR 2
dA j dA i
A i A j
= J j e
−KL ij + E bg (1 − e
−KL ij ) F ij
(14.13)