Page 245 - Design and Operation of Heat Exchangers and their Networks

Optimal design of heat exchanger networks 235

in which
T
0
T ¼ t 0 t 0 ⋯ t 0
E E,1 E,2 E,N
T
0
0
¼ t 0 h,1 t 0 c,1 t 0 h,2 t 0 c,2 ⋯ t h,N E t 0 c,N E t M,1 t 0 M,2 ⋯ t 0 M,N M (6.14)
T
T ¼ t 00 t 00 ⋯ t 00
00
E E,1 E,2 E,N
T
¼ t 00 t 00 t 00 t 00 ⋯ t 00 t 00 t 00 t 00 ⋯ t 00 (6.15)
h,1 c,1 h,2 c,2 h,N E c,N E M,1 M,2 M,N M
2 3
V 1 0
⋱
6 7
6 7
6 7
V N E
V N N ¼ 6 7 (6.16)
1
6 7
6 7
⋱
4 5
0 1
To illustrate the interconnections among the heat exchangers, we use the
following four matching matrices (The first three have been introduced in
Section 3.6).
Interconnection matrix G: N N matrix whose elements g ij are defined as
the ratio of the thermal capacity rate flowing from channel j into channel
i to that flowing through channel i.
0
Entrance matching matrix G : N N matrix whose elements g ik are
0
0
defined as the ratio of the thermal capacity rate flowing from the entrance
k to channel i to that flowing through channel i.
00
00
Exit matching matrix G : N N matrix whose elements g li are defined as
00
the ratio of the thermal capacity rate flowing from channel i to the exit l
to that flowing out of exit l.
000
Bypass matrix G : N N matrix whose elements g 000 lk are defined as the
0
00
ratio of the thermal capacity rate flowing from entrance k to exit l to that
flowing out of exit l.
We can write the energy balances at the inlets of N channels and at the net-
00
work exits of N streams with these matrices as follows:
N 0 N
X X
t 0 ¼ g t + g ij t 00 ð i ¼ 1, 2, …, NÞ (6.17)
0 0
E,i ik k E, j
k¼1 j¼1
N 0 N
X X
00 000 0 00 00 00
t ¼ g t + g t ð l ¼ 1, 2, …, N Þ (6.18)
l lk k li E,i
k¼1 i¼1