A 2 T 2 ε 2
A 2 T 2 ε 2
A R T R ε R
A 2 T 2 ε 2
A R
A R
A R
T R
T R
T R
q
1
2
1
q 2
ε R
ε R
ε R
A 1 T 1 ε 1
A 2 T 2 ε 2
A 1 T 1 ε 1
A 1 T 1 ε 1
A 1 T 1 ε 1
E bR
σT 1
4
J R E bR
=
J 2
J 1
1R
q
q 12
2
R
q
q R = 0
1– ε R
A R ε R
1
A 1 F 1R
1
A R F R2
σT
4
2
q 1 E b1
E b2
1– ε 1
A 1 ε 1
1
A 1 F 12
1– ε 2
A 2 ε 2
−q 2
∵ q R = 0, therefore, q 1 = −q 2
σT 1
4 − σT 4
q 1 = −q 2 =
2
(1 − ε 1 )/A 1 ε 1 + 1/(A 1 F 12 + 1/[(1/A 1 F 1R ) + (1/A 2 F 2R )])
+(1 − ε 2 )/A 2 ε 2
(13.11)
where T 1 , and T 2 are given
A R F R2 = A 2 F 2R
And surface emissivity, area, and view factors are also given or predetermined.
If q 1 = −q 2 is determined from above and if q R = 0, how to determine
reradiation surface temperature T R =?
From energy balance on the reradiation surface,
J 1 − J R
J R − J 2
q 1R =
= q R2 =
1/A 1 F 1R
1/A R F R2
and
E b1 − J 1
1 − ε 1
q 1 =
⇒ J 1 = σT 1
4
− q 1
(1 − ε 1 )/A 1 ε 1
A 1 ε 1
J 2 − E b2
1 − ε 2
q 2 =
⇒ J 2 = q 2
+ σT
4
(1 − ε 2 )/A 2 ε 2
A 2 ε 2
2
FIGURE 13.5
Electric furnaces with a reradiating surface.
262
Analytical Heat Transfer