Page 392 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 375

The solutions of Eq. (7.247) have the same form as Eqs. (7.134), (7.137):
1
^
T xðÞ ¼ ^ V xðÞ V GV G T (7.249)
^ ^00
^0
^0 ^0

1
T ¼ G + G V V GV G T (7.250)
^0 ^0
^00 ^00 ^0
^ ^00
^ 000
^ 00
h i h i

^
^00
with V xðÞ ¼ h ij e , V ¼ h ij e ^ 0 , and V ¼ h ij e ^ 00 .
^0
^ ^r j x i
^ ^r j x i
^ ^r j x i
M M
M M M M
This equation provides the steady-state temperature distribution in a parallel
channel multistream plate-fin heat exchanger.
7.4.3 Linearized model for parallel channel multistream
plate-fin heat exchangers
^
^
By introducing the excess temperature vectors Θ ¼ T T, Θ p ¼ T p T p ,
^
and Θ f ¼ T f T f and by linearization, the products of the disturbances
and excess temperatures can be neglected for small disturbances around a
¯
new mean steady-state operation point denoted with “.” The governing
equation system (7.218)–(7.225) for section q (q¼1, 2, …, M p ; i¼1, 2,
…, n; j¼1, 2, …, m) can be linearized as follows:
0 1
1
ð
∂θ ij _ ∂θ ij U p, ij
C ij + C ij θ p, ij + θ p,i +1, j 2θ ij U f, ij @ θ f, ij dy θ ij A
∂τ ∂x 2
0
!
_
1 C ij
^
¼ U p, ij U p, ij ^ t p, ij + ^ t p,i +1, j 2^ t ij
2 ^ _
C ij
0 1
! 1
_ ð
C ij
+ U f, ij ^ U f, ij @ ^ t f, ij dy ^ t ij A (7.251)
^ _
C ij
0
∂θ p, ij 1 1
C p, ij ¼ U p, ij θ ij θ p, ij + U p,i 1, j θ i 1, j θ p, ij
∂τ 2 2

∂θ f, ij ∂θ f,i 1, j 1
^
+ K ij K i 1, j + U p, ij ^ U p, ij t ij ^ t p, ij
∂y ∂y 2
y¼0 y¼1
1
^
+ U p,i 1, j ^ U p,i 1, j t i 1, j ^ t p, ij (7.252)
2