Page 46 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 33

peripheral wall temperature (H1) would be reasonable. Therefore, the
Nusselt number is evaluated with Eq. (2.31):

Nu H1 ¼ 8:235

4
2
1 2:0421γ +3:0853γ 2:4765γ +1:0578γ 0:1861γ 5
3
¼ 6:326
which yields the heat transfer coefficient as
Nu H1 λ 6:326 0:02688
2
α ¼ ¼ ¼ 27:69 W=m K
d h 0:00614
The fin efficiency can be determined by
ð
ð
m 1 sinh m 1 l f,1 Þcosh m 2 l f,2 Þ + m 2 cosh m 1 l f,1 Þsinh m 2 l f,2 Þ
ð
ð
η ¼
f
m 1 l f,1 + l f,2 Þ m 1 cosh m 1 l f,1 Þcosh m 2 l f,2 Þ + m 2 sinh m 1 l f,1 Þsinh m 2 l f,2 Þ½ ð ð ð ð Š
ð
(2.65)
in which “1” and “2” denote the first and second fin sections, respectively,
and the fin performance factor can be calculated with Eq. (2.59):
s ffiffiffiffiffiffiffiffiffiffiffiffi
α f P f
m ¼
λ f A c, f
where P f is the perimeter of the fin and A c,f is its cross-sectional area. In
this example,
r ffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2α 2 27:69
1
m 1 ¼ ¼ ¼ 9:814 m , l f,1 ¼ h fs ¼ 0:025 m
λ f δ f 230 0:0025
r ffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
α 27:69 1
m 2 ¼ ¼ ¼ 6:939 m , l f,2 ¼ s fs =2 ¼ 0:0035=2
λ f δ 230 0:0025
¼ 0:00175 m
Substituting these values into Eq. (2.65), we get the value of the fin
efficiency as
η ¼ 0:947
f
The overall fin efficiency can be determined with Eq. (2.50) as

A f 2h fs + s fs
η ¼ 1 1 ηð f Þ ¼ 1 1 η Þ
ð
0
f
A 2 h fs + s fs Þ
ð
2 0:025 + 0:0035
ð
¼ 1 1 0:947Þ ¼ 0:9503
ð
2 0:025 + 0:0035Þ
Now, we can express the local temperature difference as
Continued