Page 351 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 337

The coefficient vector D in Eq. (7.75) is determined by the boundary con-
ditions, which yields

0 1 0 0 ^
D ¼ V Θ V D (7.81)
e
with
" # " 0 0 #
v 0 11 v 0 12 h 11 e r 1 x 1 h 12 e r 2 x 1
0
V ¼ ¼ (7.82)
v 0 v 0 h 21 e r 1 x 0 2 h 22 e r 2 x 0 2
21 22
" # " ^ 1 x 0 r ^ 2 x 0 #
r
v 0 v 0 h 11 e 1 h 12 e 1
0 11 12
V ¼ ¼ (7.83)
r
v 0 v 0 h 21 e ^ 1 x 0 2 h 22 e r ^ 2 x 0 2
21 22
Substituting Eq. (7.81) into Eq. (7.75), we obtain the Laplace transform
of the fluid temperature response to the disturbances in inlet fluid temper-
atures, thermal flow rates, and heat transfer parameters:
0 1
0
Rx
^
^
Rx
0
Θ ¼ He V 0 1 Θ + He ^ Rx He V 0 1 V V T 0 (7.84)
e
e
The real-time solution can be obtained with the FFT algorithm:
" #
M 1
e aτ n X 2iπnk=M 1
e
ð
ðÞ
f τ n ¼ Re fa + ikπ=τÞe faðÞ ð 2:178Þ, (7.85)
e
τ 2
k¼0
by which the temperature variation at all time points τ n ¼2nτ/M in the time
interval [0, τ] can be obtained simultaneously. The value of a in Eq. (2.178) is
11
taken as 4<aτ<5. M is an exponent of 2, usually M¼2 ¼2048.
7.1.3 Cell model
The so-called lumped-distributed parameter method combines the lumped
parameter characteristic and the distributed parameter characteristic to get
the dynamic behavior of two-stream heat exchanger. Actually, if the
exchanger is divided into many elements along its length, the lumped
parameter method can be applied to each element, and the total responses
of exchanger are accumulated numerically; thus, the whole exchanger will
also be characteristic of distributed parameter. Or the distributed parameter
method is first applied to whole exchanger, and then, the model is simplified
approximately into lumped parameter model. Li (1995), Du (1996), and Ye
(1998) changed the problem of describing dynamic behavior of multistream
heat exchanger into that of equivalent two-stream heat exchanger network.
With the help of transfer functions of two-stream heat exchanger, they set