Page 86 - Design and Operation of Heat Exchangers and their Networks

74 Design and operation of heat exchangers and their networks

0
where V is named as the inlet matrix:
" #
h 11 e r 1 z 0 1 h 12 e r 2 z 0 1
V 5 0 0 (3.17)
0
h 21 e r 1 z 2 h 22 e r 2 z 2
The coefficient matrix D can then be expressed with the inverse matrix
of the inlet matrix multiplied by the inlet temperature vector:

0 1 0
D ¼ V T (3.18)
The final solution can then be expressed as

Rz
T ¼ He V 0 1 T 0 (3.19)
Using the obtained eigenvalues and eigenvectors as well as the boundary
condition, we can explicit express the terms in Eq. (3.19) as
" #
r 1 z r 2 z kA= _ C h + kA= _ C cÞz
Rz h 11 e h 12 e 1 e ð
He ¼ r 1 z r 2 z ¼ kA= _ C h + kA= _ C cÞz (3.20)
_
_
h 21 e h 22 e 1 C h =C c e ð
" 0 0 # 1 " # 1
h 11 e r 1 z 1 h 12 e r 2 z 1 1 1
0 1
V ¼ 0 0 ¼ _ _
h 21 e r 1 z 2 h 22 e r 2 z 2 1 C h =C c
2 _ _ 3
C h =C c 1
6 _ _ _ _ 7
6 1+ C h =C c 1+ C h =C c 7
(3.21)
¼ 6 7
1 1
4 5

_ _ _ _
1+ C h =C c 1+ C h =C c
and finally obtain the temperature distributions:
_
_
ð
t h t 0 C h =C c + e kA= _ C h + kA= _ C cÞz
c
¼ (3.22)
0
t t 0 _ _
h c 1+ C h =C c
h kA= _ C h + kA= _ C cÞz i
_
_
ð
t c t 0 C h =C c 1 e
c
¼ (3.23)
t t 0 _ _
0
h c 1+ C h =C c
The outlet temperatures (z ¼ 1) are
_
_
ð
00
t t 0 C h =C c + e kA= _ C h + kA= _ C cÞ
h c
¼ (3.24)
0
t t 0 _ _
h c 1+ C h =C c