Page 360 - Design and Operation of Heat Exchangers and their Networks

346 Design and operation of heat exchangers and their networks

M
X
C w,k ∂t w,k U jk
¼ t j t w,k ð k ¼ 1, 2, …, M w Þ (7.109)
∂τ
L k L k
j¼1
with the initial conditions as
τ ¼ 0 : t i ¼ ^ t i xðÞ i ¼ 1, 2, …, Mð Þ, t w,k ¼^ t w,k xðÞ k ¼ 1, 2, …, M w Þ
ð
(7.110)
in which C i and C w,k are the thermal capacities of fluid i and wall k, respec-
_
tively; C i is the signed thermal flow rate of fluid i (positive for the flow in the
positive direction of the x-coordinate and negative for the counterflow); and
U ik is the heat transfer parameter between fluid i and wall k, U ik ¼(αA) ik .
To specify the boundary conditions and bypasses in a general form, a
general matrix method proposed by Strelow (2000) was developed by intro-
ducing four matching matrices defined as follows:
Interconnection matrix G
It is an M M matrix whose elements g ij are defined as the ratio of the
thermal flow rate flowing from channel j, to channel i, to that flowing
through channel i.
Entrance matching matrix G 00
0
It is an M N matrix whose elements g ik are defined as the ratio of
0
the thermal flow rate flowing from the entrance k, to channel i, to that
flowing through channel i.
Exit matching matrix G 00
00
00
It is an N M matrix whose elements g li are defined as the ratio of
the thermal flow rate flowing from channel i, to the exit l, to that flowing
out of exit l.
Bypass matrix G 000
0
It is an N N matrix whose elements g lk are defined as the ratio of
00
000
the thermal flow rate flowing from entrance k, to exit l, to that flowing
out of exit l.
The energy balance at the inlet of each channel yields a general form of the
boundary condition:
N 0 M
X X
0 g t τ Δτ 0 00
0 0
t i τ, x ¼ ik k ik + g ij t j τ Δτ ij , x j ð i ¼ 1, 2, …, MÞ (7.111)
i
k¼1 j¼1
00
0
where x i and x i are the coordinates of the inlet and outlet of channel i,
0
respectively; t k is the supply temperature of stream k;and Δτ ik and Δτ ij are
0
the time intervals for the streams flowing from entrance k to the inlet of