Page 469 - Design and Operation of Heat Exchangers and their Networks

452 Design and operation of heat exchangers and their networks

the HEN. The optimizer finds the optimal values of the bypass fraction, c opt ,
such that the target values of the controlled variables are satisfied. Assuming
that there is no mismatch between the static model of the optimizer and the
process (HEN), the remaining task, at this point, is to find how to implement
these optimal bypass fractions as a function of time. Since there is no offset
between the predictions of the algebraic and dynamic models at steady state,
the bypasses can be opened up to their optimal values either instantaneously
or as a function of time, for example, as a ramp starting from their nominal
values c N toward their optimal values c opt within the time interval Δτ,asis
represented by Eq. (9.44) for the ith bypass manipulation:
( τ

c N,i + c opt,i c N,i , τ < Δτ i
c i τðÞ ¼ Δτ i (9.44)
c opt,i , τ Δτ i
The choice of the time interval Δτ i depends on the dynamic character-
istics of the HEN and can be determined by experience, or by solving an
additional optimization problem:
ð ∞
min k zx, c τ=ΔτÞ, u, τŠ z tar kdτ (9.45)
ð
½
Δτ 0
s:t: hx, c, uÞ ¼ 0
ð
gx, c, uÞ 0
ð
For the optimal closed-loop control logic of HENs, there is a feedback of
the state variable x p pairing to the corresponding bypasses, x p x. The
function proposed for the implementation of the optimal controls is sug-
gested as

x p τðÞ x p,N
(9.46)
c τðÞ ¼ c N + c opt c N
x p,opt x p,N
Sun et al. (2018) proposed a methodology for two-stage coordination of
bypass control and economic optimization. Based on this methodology,
they developed a one-step coordination between the bypass control and
economic optimization. Two kinds of control manipulations are defined
in this method: bypass control c b and economic optimization control c e .
In the two-stage coordination control, firstly, the fraction of bypass is
adjusted quickly for meeting the control requirement:
" #
N c b
X
c b,ini, j c b, j
min + w zx, c b , c e,ini , uÞ z tar k (9.47)
ð
k
c b 2R c b c b,ini, j
j¼1