Page 363 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 349

^
The elements of the coefficient matrix A are expressed as
0 1
B C
1 X B ^ U jn C
M w
a U in δ ij C ð i ¼ 1, 2, …, MÞ (7.127)
^ ij ¼
^ B
^ B M C
_
C i n¼1 @ X A
^ U mn
m¼1
where δ ij is the Kronecker symbol defined by

1, i ¼ j
δ ij ¼ (7.128)
0, i 6¼ j
The general solution of Eq. (7.126) is the same as Eq. (3.289):

^
^
^ Rx
T ¼ ^ He D ð 3:289Þ, (7.129)
r ^ i x
^
^ Rx
r
in which e ¼ diag e is a diagonal matrix, ^ i (i¼1, 2, …, M), and H are
^
^
the eigenvalues and eigenvectors of matrix A. The coefficient vector D is
determined by the boundary condition (7.121) or
0
^0 ^
T x ¼ G T + GT x (7.130)
^ ^ ^ 00
^ ^ 0
T T
x
00
0
x
where x ¼ ^x , ^x , …, ^x 0 and ^x ¼ ^ , ^ , …, ^ 00 are the initial
00
0
x
00
^ 0
1 2 M 1 2 M
dimensionless coordinate vectors of channel inlets and outlets at
T
0
0
^0
τ¼0, T ¼ ^ t , ^x , …, ^ t 0 is the supply stream temperature vector, and
1 2 N 0
T
00
T x ¼ ^ t 1 ^x , ^ t 2 ^x , …, ^ t M ^x 00 is the temperature vector at the
00
^ ^ 00
1 2 M
channel outlets.
Applying the general solution (3.289, 7.129) to Eq. (7.130), we obtain
1
^
^0 ^0
^ ^00
^0
D ¼ V GV G T (7.131)
^0
in which V and V are two M M matrices defined as
^00
hi h i
V ¼ ^v 0 ¼ h ij e ^ 0 (7.132)
^ ^r j x i
^0
ij
M M M M
hi h i
V ¼ ^v 00 ¼ h ij e ^ 00 (7.133)
^00
^ ^r j x i
ij
M M M M
and finally the initial steady-state temperature solution is presented as
1
^
^
^
^
^0
^ ^00
T xðÞ ¼ V xðÞD ¼ V xðÞ V GV G T (7.134)
^0 ^0