Apply for surface 3:
where a 31 = −A 3 F 31 , a 32 = −A 3 F 32 , a 33 = (A 3 ε 3 /(1 − ε 3 ) + A 3 F 31 + A 3 F 32 ),
c 3 = (A 3 ε 3 /(1 − ε 3 ))σT 3
4 .
From the above three linear equations, the following matrix can be formed:
⎡
⎤
⎡ ⎤
⎡ ⎤
a 11 a 12 a 13
J 1
C 1
A = a 21 a 22 a 23
J = J 2
C = C 2
⎣
⎦
⎣ ⎦
⎣ ⎦
a 31 a 32 a 33
J 3
C 3
[A][J] = [C]
[ J] = [A]
−1
[C]
Alternatively, we can apply Equation 13.7 to each surface and get
J 1 = ε 1 E b1 + (1 − ε 1 )[F 11 J 1 + F 12 J 2 + F 13 J 3 ]
J 2 = ε 2 E b2 + (1 − ε 2 )[F 21 J 1 + F 22 J 2 + F 23 J 3 ]
J 3 = ε 3 E b3 + (1 − ε 3 )[F 31 J 1 + F 32 J 2 + F 33 J 3 ]
Similarly, the above three linear equations can be rearranged in order to
obtain the matrix relation as [A][J] = [C] and then solve for [J] = [A] −1 [C].
Once matrix [J] is determined, that is, J 1 , J 2 , and J 3 have been determined,
then use Equation 13.14 to find surface heat transfer rates as
σT 4
E b1 − J 1
1 − J 1
⇒ q 1 =
=
(1 − ε 1 )/(A 1 ε 1 )
(1 − ε 1 )/(A 1 ε 1 )
σT 4
E b2 − J 2
2 − J 2
⇒ q 2 =
=
(1 − ε 2 )/A 2 ε 2
(1 − ε 2 )/A 2 ε 2
σT 4
E b3 − J 3
3 − J 3
⇒ q 3 =
=
(1 − ε 3 )/A 3 ε 3
(1 − ε 3 )/A 3 ε 3
266
Analytical Heat Transfer