Page 470 - Design and Operation of Heat Exchangers and their Networks

Optimal control process of heat exchanger networks 453

s:t: hx, c b , c e,ini , uÞ ¼ 0
ð
gx, c b , c e,ini , uÞ 0
ð
where c b,ini is the initial bypass fraction, c e,ini the initial valve opening for the
economic operation, c b the new bypass fraction, and w the weighting factor.
Then, the economic optimization described as follows is solved:

min C x, c b , c e , uð Þ ¼
c e 2R c e
h i
X X
min a Q HU x, c b , c e , uÞ + b Q CU x, c b , c e , uÞ (9.48)
ð
ð
c e 2R ce
s:t: zx, c b , c e , uÞ z tar ¼ 0
ð
hx, c b , c e , uÞ ¼ 0
ð
gx, c b , c e , uð Þ 0
In the one-step coordination control, the two controlling variables are
manipulated simultaneously by solving the dynamic optimization problem
with an optimization algorithm with pattern search (Hooke and Jeeves,
1961) combining a penalty function for the constraints. According to Sun
et al. (2018), we can express the optimization problem as

h
min C k x, c b , c e , uð Þ + wΔC k x, c b , c e , uÞ
ð
c b 2R c b ,c e 2R c e
X
+ φ h i x, c b , c e , u, τ k +1 Þj
j
ð
i
i
X i
+ φ min g j x, c b , c e , u, τ k +1 Þ,0 (9.49)
ð
j
j
where φ i and φ j are the penalty factors for different constraints and w is the
weighting factor. The utility costs C k and ΔC k are calculated with
Eqs. (9.50)–(9.53):
ð
1 τ k +1 X
h
C k x, c b , c e , uð Þ ¼ a Q HU x, c b , c e , u, τÞ
ð
τ k +1 τ k τ k
i
X
+ b Q CU x, c b , c e , u, τð Þ dτ (9.50)
ð
ð
ΔC k x, c b , c e , uð Þ ¼ C k,max x, c b , c e , uÞ C k,min x, uÞ (9.51)
h X
C k,max x, c b , c e , uð Þ ¼ max a Q HU x, c b , c e ,u, τð _ Þ
τ2 τ k, ∞Þ
½
i
X
+ b Q CU x, c b , c e ,u, τð _ Þ (9.52)