Page 257 - Design and Operation of Heat Exchangers and their Networks

246 Design and operation of heat exchangers and their networks

8
t 0 t 00 t 00 t 00
>
> HU,k lb,i HU,k i
> 0 00 00 00
< i, t t t t > 0
h HU,k lb,i HU,k i
Δt m,HU,ik ¼ ln t 0 t 00 = t 00 t 00
> HU,k lb,i HU,k i
>
>
: 0 00 00 00
1, t t t t 0
HU,k lb,i HU,k i
(6.45)

8
00
t t 00 t 00 t 0
> i
> CU,l ub,i CU,l
> 00 00 00 0
< i, t t t t > 0
h
i CU,l ub,i CU,l
00
00
00
Δt m,CU,il ¼ ln t t CU,l = t ub,i t 0 CU,l
i
>
>
>
1, t t t t 0
: 00 00 00 0
i CU,l ub,i CU,l
(6.46)
in which the indices “i” the ith process stream, “j” the jth process heat
exchanger, “k” the kth hot utility and, “l” the lth cold utility. φ is the penalty
factor against negative Q and negative Δt m . A large value of φ, for example,
3
10 , will yield a much larger heat transfer area.
The constraints applied to Eq. (6.38) consist of mass balance and energy
balance constraints for mixers, process heat exchangers, heaters, coolers and
the whole network, and the additional equality and inequality constraints
g(x) and h(x). To set up the relations between different constraint groups,
the necessary mapping tables are introduced for the formulation. For exam-
ple, the mapping table for the inlet of the N M mixers set the pointer of the
0
n th coming stream of the mth mixer to be the pointer of one of the five
sources (the ith stream at network entrance, hot stream out let of the jth heat
00
exchanger, cold stream out let of the jth heat exchanger, or the n th outlet of
00
the m th mixer). Similar mapping tables can be set for process heat
exchangers and network exits (excluding the utilities). These mapping tables
constitute a set of equality constraints. Other constraints are as follows:
(1) Mixers
We define a node as a mixer if N M,in process streams are mixed at the
node, or the stream leaving the node is splitted into N M,out substreams.
N M is the number of the mixers in the network.
Mass balance constraints
N 0 N 00
M,m
M,m
X _ 0 X _ 00
C 0 C 00 ¼ 0 m ¼ 1, 2, …N M Þ (6.47)
ð
M,mn M,mn
00
0
n ¼1 n ¼1
Energy balance constraints
N 0 N 0
M,m
M,m
X X
t 00 C _ 0 0 C _ 0 0t 0
M,mn 00 M,mn M,mn M,mn 0 (6.48)
n ¼1 n ¼1
0
0

00
¼ 0 m ¼ 1, 2, …, N M ; n ¼ 1, 2, …, N 00
M,m