Page 460 - Design and Operation of Heat Exchangers and their Networks

Optimal control process of heat exchanger networks 443

Fig. 9.3 Time delay graph.

determined with the lump parameter model suggested by Mathisen and
Morari (1994). Then, the time delay matrix of the HEN and its control
structure can be expressed as a function of the network operating point.
To formulate the controllability requirements, Papalexandri and
Pistikopoulos (1994b) proposed two time delay–based criteria. The one is
for dynamic decoupling, which is expressed as

τ ij τ ij 1 m ij M 1 + α 1 0 8i, j, j 0 (9.31)
0
where τ ij is the response time of a controlled variable i to its pairing manip-
ulating variable j; τ ij is the response time of the controlled variable i to other
0
manipulating variables j (j 6¼j); m ij is the pairing variable that defines the
0
0
control of i through j; and α 1 is a positive variable, which shall be maximized
so that, for a control pair ij, the response time of the controlled variable i to
the manipulating input j is as small as possible compared with the response
0
time to any other manipulating input j . M 1 is a large positive number,
u
u
M 1 >α 1 where α 1 is a valid upper bound of α 1 , so that the criterion,
Eq. (9.31) becomes a redundant constraint when m ij ¼0.
The other criterion is the time effective control expressed by

τ ij τ ik 1 m ij M 2 + α 2 0 8i, j,k (9.32)
Differing from Eq. (9.31), here, k indicates a disturbance input, and the
index “2” refers to the parameters for the time effective control. This
criterion implies that the response time of the control variable i to the