Page 450 - Design and Operation of Heat Exchangers and their Networks
P. 450
Optimal control process of heat exchanger networks 433
+
s:t: 8u 2 R u u N δΔu u u N + δΔu 9ch d, c, uÞ ¼ 0, gd, c, uÞ 0
ð
½
ð
+
where Δu and Δu are the vectors of maximum possible deviations of the
uncontrolled operation variables in positive and negative directions, respec-
tively. Then, the feasibility can be stated as follows: A HEN is feasible, if the
flexibility index D 1. It indicates how large the deviation range of the
uncontrolled operation variables can be extended. This index might
be named as feasibility index rather than flexibility index because it does not
give any information about the variation of energy recovery (utility cost).
If the HEN is feasible, for a given u2R u , an optimal controlling
vector c2R c can be found such that the total utility cost reaches the
minimum:
n o
X X
ð
ð
min C d, c, uð Þ ¼ min a Q HU d, c, uÞ + b Q CU d, c, uÞ (9.5)
c2R c c2R c
s:t: hx, c, uÞ ¼ 0
ð
gx, c, uÞ 0
ð
If there is no solution of c2R c , which satisfies the constraints, the HEN
is not feasible at the point u, and we set C min (d,u)!∞.
A HEN is flexible if it is feasible, and for any possible disturbances in
u2R u , the running cost does not increase dramatically. To evaluate the
flexibility of a HEN quantitatively, Roetzel and Luo (2002) introduced a
flexibility factor defined as the ratio of the minimum total utility cost at
the nominal operation point C min (d, u N ) to the maximum value of the min-
imum total utility cost in the range of all possible values of the uncontrolled
operation variables:
C min d, u N Þ
ð
F dðÞ ¼ (9.6)
max C min d, uð Þ
u2R u
This factor can well describe the flexibility of the HEN: F¼1 means that
the disturbances have no influence on the utility cost; F¼0 indicates a non-
feasible HEN. If F falls below a given critical value F cr , then the exchanger is
considered to be nonflexible.
Example 9.1 Flexibility analysis of a heat exchanger network
This example is firstly given by Floudas and Grossmann (1987) and then
reanalyzed by Roetzel and Luo (2002). The problem data, network
construction, and heat transfer areas of the heat exchangers are given in
Table 9.1, Fig. 9.1,and Table 9.2, respectively. The hot utility cost (300°C)
