Page 457 - Design and Operation of Heat Exchangers and their Networks

440 Design and operation of heat exchangers and their networks

uncertainty by R u , and the set of the design variables by d. Let x represent
the state variables that describe the network operation; c represent control-
ling variables corresponding to the degrees of freedom at the design stage; z
represent the controlled variables, which is a subset of x; z ref is the set value
vector of z; and the deviation Δz¼z z ref . The synthesis and retrofit prob-
lems with structural controllability target were formulated by Papalexandri
and Pistikopoulos (1994a) as follows, respectively.
Synthesis problem of HEN:
(9.25)
min TAC synthesis
d,x,c
s:t: hd, x, c, uÞ ¼ 0
ð
gd, x, c, uÞ 0
ð
j f
8u 2 R u 9cj Δzj Δz max g
Retrofit problem of HEN:

min TAC retrofit (9.26)
Δd,x,c
s:t: hd, x, c, uÞ ¼ 0
ð
gd, x, c, uÞ 0
ð
8u 2 R u 9cj Δzj Δz max g
j f
where h(d, x, c, u) is the set of equality constraints and g(d, x, c, u) the set of
inequality constraints.
Halemane and Grossmann (1983) showed that the flexibility require-
ments in the previous problems are equivalent to the inequality constraint
that involves a max-min-max problem:

max min maxg j d, x, c, u 0 (9.27)
c
u2R u j2J g
s:t: hd, x, c, uÞ ¼ 0
ð
where J g is the index set for the inequalities g.
For the controlled variables z to be undisturbed from the disturbance
inputs u, it suffices

GR Mx, c, z, uފ AR Mx, c, z, uފ 1 (9.28)
½
½
ð
ð
where M(x, c, z, u) denotes the structural matrix of the network, GR is the
generic rank of M, and AR is the number of its active rows.