Page 115 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 103

Example BA 5,5 2 2
B A
1 1
" #
N N
1 1 X 1 X
ð
ε 1 ¼ 1 α n F n +1 b A , a A Þ ¼ 1 β F n +1 a B , b B Þ (3.189)
ð
n
R 1 a A b B
n¼0 n¼0
where α n and β n are determined by
N
X
inv
A ð b A , b B , a A Þα n + B mn b B ð (3.190)
ðÞβ ¼ 1 m ¼ 0, 1, 2, …, NÞ
mn n
n¼0
N
X
inv
ðÞα n + A ð
ð
B mn a A a B , a A , b B Þβ ¼ 1 m ¼ 0, 1, 2, …, NÞ (3.191)
mn n
n¼0
ð m +1Þ! ð q q 0 m
inv 0 0
ð
A ð p, q, rÞ ¼ G n p 1 q =qÞ, rŠ dq
½
mn
q m +1 0 m!
" k m k ! ! #
m
m m +1ð Þ! X ð pÞ X m k r n + j +1 m
ð
¼ 1Þ m +1 F n + k +1 p, rð Þ
p k! j ð n + j +1Þ! k
k¼0 j¼0
(3.192)
m +1 x n
B mn xðÞ ¼ (3.193)
n + m +1 n!
Baclic et al. (1988) showed that the results with N¼1 are sufficiently
accurate for practical purposes in any combination of the NTU, R, and
ϕ values. N¼3 will yield ε values accurate to the sixth significant figure.
For the given data, R 1 ¼2, NTU 1,A ¼0.4, and NTU 1,B ¼0.6, we have
a A ¼0.4, b A ¼0.8, a B ¼0.6, and b B ¼1.2. We take N¼1 as an example and
inv
calculate A mn and B mn with Eqs. (3.192), (3.193), respectively, which yields

inv 0:24212 0:05505 10:6
ðÞ
A ð b A , b B , a A Þ ¼ , B mn b B ¼ ,
mn 0:26832 0:05641 10:8

1 0:2 inv 0:59887 0:41189
B mn a A ¼ , A ð a B , a A , b B Þ ¼ :
ðÞ
10:26667 mn 0:63079 0:43978

inv
ðÞ
A ð b A , b B , a A Þ B mn b B
mn
Let A ¼ inv ; its inverse matrix can be
ðÞ
B mn a A A ð a B , a A , b B Þ
mn
obtained as
2 3
0:42074 0:17161 4:12743 2:97951
6 0:12388 0:69757 14:8676 15:02479 7
1
A ¼ 6 7
4 4:06472 2:94894 0:18479 0:008104 5
4:94852 5:04292 0:10497 0:049999
Continued