Page 88 - Design and Operation of Heat Exchangers and their Networks

76 Design and operation of heat exchangers and their networks

The inlet and outlet coordinates are expressed as

0 1
0 00
z ¼ , z ¼ (3.29)
1 0
The eigenvalues of A are determined by

_
kA=C h r _
kA=C h ¼ 0

_
_ kA=C c r
kA=C c
which yields

_ _
r 1 ¼ 0,r 2 ¼ kA=C h kA=C c (3.30)
The eigenvector H is determined by
_
_
kA=C h h 11 + kA=C h h 21 ¼ 0
_
_
kA=C c h 11 + kA=C c h 21 ¼ 0

_
_
_
_
kA=C h + kA=C h kA=C c h 12 + kA=C h h 22 ¼ 0

_
_
_
_
kA=C c h 12 + kA=C c + kA=C h kA=C c h 22 ¼ 0
which yields

1 1
H ¼ _ _ (3.31)
1 C h =C c
The inlet matrix can be presented as
" r 1 z 0 r 2 z 0 #
h 11 e 1 h 12 e 1
0
V ¼ 0 0 (3.32)
h 21 e r 1 z 2 h 22 e r 2 z 2
Its inverse matrix is obtained as
" # 1
1 1
0 1
V ¼
_
_
ð
1 C h =C c e kA= _ C h kA= _ C cÞ
2 3
_ _ kA= _ C h kA= _ C cÞ
ð
C h =C c e 1
6 7

_
_
_
_
ð
ð
6 C h =C c e kA= _ C h kA= _ C cÞ 1 C h =C c e kA= _ C h kA= _ C cÞ 7
¼ 6 1 7
6 1 1 7
4 5

_ _ kA= _ C h kA= _ C cÞ _ _ kA= _ C h kA= _ C cÞ
ð
ð
C h =C c e 1 C h =C c e 1
(3.33)