Page 85 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 73

0
z ¼ z : T ¼ T 0 (3.8)
in which

0 t 0
0 t h 0 h
z ¼ ,T ¼ ,T ¼ 0 (3.9)
0 t c t
c
and the coefficient matric
" #
_ _
kA=C h kA=C h
A ¼ (3.10)
_ _
kA=C c kA=C c
The general solution of Eq. (3.18) can be written as
Rz
T ¼ He D (3.11)
r 1 z
e 0
Rz
Rz
Here, we denote e as a diagonal matrix, e ¼ r 2 z , and r 1 and r 2
0 e
are the eigenvalues of the coefficient matrix A:

a 11 r a 12

j A rIj ¼ ¼ 0 (3.12)
a 22 r
a 21
Substituting Eq. (3.10) into Eq. (3.12) yields two eigenvalues:

_ _
r 1 ¼ 0, r 2 ¼ kA=C h + kA=C c (3.13)
The eigenvector H is determined by
ð A r i IÞ h i ¼ 0 (3.14)
where h i is the ith column of H. For the two eigenvalues given by Eq. (3.13),
Eq. (3.14) can be expressed as
_
_
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
Let h 11 ¼1, h 12 ¼1; it is easy to obtain

1 1
H ¼ _ _ (3.15)
1 C h =C c
The coefficient matrix D should be determined by the boundary condi-
tions. Substituting Eq. (3.11) into Eq. (3.8), we obtain
0
0
T ¼ V D (3.16)