Page 143 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 131

in which V is a (M+M M ) (M+M M ) matrix, whose nonzero elements are
r j x
v ij ¼ h ij e ð i ¼ 1, 2, …, M; j ¼ 1, 2, …, MÞ (3.316)
ð
v ii ¼ 1 i ¼ M +1, M +2, …, M + M M Þ (3.317)
The flow arrangement is defined by three matching matrices as follows:
Interchannel matching matrix G:Itis a (M+M M ) (M+M M ) 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.
Entrance matching matrix G :Itisa(M+M M ) N f matrix whose elements
0
0
g ik are defined as the ratio of the thermal capacity rate flowing from the
entrance of stream k to channel i to that flowing through channel i.
00 00
Exit matching matrix G :Itisan N f (M+M M ) matrix whose elements g ki
are defined as the ratio of the thermal capacity rate flowing from channel i to
the exit of stream k to that flowing out of the exit of stream k.
The energy balance at the entrances of M+M M channels yields
M + M
N f X M
X
_ 0 _ _ 00

C i tx ¼ C k i t in,k + C j i t i x ð i ¼ 1, 2, …, M + M M Þ
i i
k¼1 i¼1
(3.318)
_
Dividing Eq. (3.318) by C i , we obtain
M + M
N f X M
X

0 0 00
tx ¼ g t in,k + g ij t i x ð i ¼ 1, 2, …, M + M M Þ (3.319)
i ik i
k¼1 i¼1
or in the matrix form as
0 0 00
Θ x ðÞ ¼ G T in + GΘ xðÞ (3.320)
T
] is the inlet temperature vector
in which T in ¼[t in, 1 , t in, 2 , …, t in, N f
0 T
00 T
0
of the exchanger; x ¼[x 1 , x 2 , …, x M ] and x ¼[x 1 , x 2 , …, x M ]
0
0
00
00
00
are the coordinate vectors of the channel inlets and outlets, respectively;
Θ is the extended temperature vector;
T
0
0
0
Θ x ðÞ ¼ t 1 x , t 2 x , ⋯, t M x 0 (3.321)
1 2 M , t M +1 , t M +2 , ⋯, t M + M M
T
00
00
Θ x ðÞ ¼ t 1 x , t 2 x , ⋯, t M x 00 (3.322)
00
1 2 M , t M +1 , t M +2 , ⋯, t M + M M
0
t i (x i ) is the temperature of the fluid stream at the inlet of the ith channel
00
0
00
x i ; and t i (x i ) is that at the outlet of the ith channel x i , which can be expressed
from Eq. (3.312) as
M
X 0
0 r j x
i
t i x ¼ h ij e d j i ¼ 1, 2, …, MÞ (3.323)
ð
i
j¼1