Page 377 - Design and Operation of Heat Exchangers and their Networks

360 Design and operation of heat exchangers and their networks

N 0 M
X X
000
00
e 00 000 Δτ s e 0 00 Δτ s e 00
li θ i x , s ¼ 0
s
θ ðÞ g e lk θ sðÞ g e
l lk k li i
k¼1 i¼1
00
ð τ > 0; l ¼ 1, 2, …, N Þ (7.198)
The matrix expression of the equations earlier is presented as follows:
dΘ
e
¼ AΘ (7.199)
e
dx
0 0
0 00
e
e e
e e
Θ xðÞ ¼ G Θ + GΘ xðÞ (7.200)
00 000 0 00
00
e
e e
Θ ¼ G Θ + G Θ xðÞ (7.201)
e e
where A is an M M matrix whose elements are
2 3
!
6 7
M w M w
1 6 X U ik U jk X 7
a ij ¼ 6 δ ij sC i + U ik 7 ð i, j ¼ 1, 2, …, M Þ
_ 6 M 7
C i 4 k¼1 X k¼1 5
sC w,k + U mk
m¼1
(7.202)
0 00 000
and G, G , G , and G are the Laplace transforms of the matching matrices
e e e
e
0
00 Δτ li s
0
with time delay, whose elements are given by g ij e Δτ ij s , g ik e Δτ ik s , g li e 00 ,
000 Δτ lk s
and g lk e 000 , respectively.
The general solution of Eq. (7.199) can be obtained from Eq. (7.171) as
Rx
Θ xðÞ ¼ He D (7.203)
e
and the coefficient vector D in Eq. (7.203) is determined by the substitution
of Eq. (7.203) into the boundary condition (7.200):
0 0 00
0
V D ¼ G Θ + GV D (7.204)
e e
e
which yields the expression of D as
1 0 0
0
D ¼ V 2GV 00 G Θ (7.205)
e e
e
where

V xðÞ ¼ He Rx ¼ h ij e r j x (7.206)
M M
h 0 i
0
0
V ¼ V xðÞ ¼ h ij e r j x (7.207)
M M
h i
V ¼ V xðÞ ¼ h ij e r j x 00 (7.208)
00
00
M M