Page 252 - Design and Operation of Heat Exchangers and their Networks

242 Design and operation of heat exchangers and their networks

n o
00
_
00
Q CU,i ¼ max C 00 t 00 t 00 00 ,0 (6.31)
00
i i ub,i
Q HU,i 00
A HU,i k ¼ (6.32)
00
F HU,i k k HU,i k max Δt m,HU,i k , Δt m,HU,i k =φg
f
00
00
00
00
Q CU,i 00
A CU,i l ¼ (6.33)
00
f
F CU,i l k CU,i l max Δt m,CU,i l , Δt m,CU,i l =φg
00
00
00
00

8
t 0 t 00 00 00
> 00 t t 00
> HU,k lb,i HU,k i
> 0 00 00 00
< h i, t HU,k t lb,i 00 t HU,k t 00 > 0
i
Δt m,HU,i k ln t 0 t 00 00 = t 00 t 00 00
00 ¼
> HU,k lb,i HU,k i
>
>
: 0 00 00 00
1, t t t t 00 0
HU,k lb,i 00 HU,k i
(6.34)

8
00
t 00 t 00 t 00 0
> 00 t
> i CU,l ub,i CU,l
> 00 00 00 0
< h i, t 00 t CU,l t ub,i 00 t CU,l > 0
i
Δt m,CU,i l ln t 00 t 00 = t 00 00 t 0
00
00 ¼
> i CU,l ub,i CU,l
>
>
: 00 00 00 0
1, t 00 t t 00 t 0
i CU,l ub,i CU,l
(6.35)
C E (A) and C U (Q) are investment cost function and utility cost function.
They are usually expressed as
C E AðÞ ¼ a + bA n (6.36)
C U QðÞ ¼ cQ (6.37)
where a, b, and c are cost constants. φ is the penalty factor against negative
3
Δt m . It is a large positive value, for example, 10 , yielding a much larger heat
transfer area.
Applying a constrained optimization algorithm to the sizing problem
(6.23) under the constraints (6.26) and (6.27), we can determine the optimal
heat transfer area of each exchanger in the network together with the opti-
mal thermal capacity rates of stream splits.
Example 6.2 Optimal sizing of the heat exchanger network.
Example 6.1 will be used here for the optimal sizing problem. We set the
heat transfer areas of the six process heat exchangers A E1 –A E6 and the
_
_
thermal capacity rates of cold stream C1 in exchangers C c,E2 –C c,E4 as
the variables to be optimized:
T
_
_
_
x ¼ A E1 A E2 A E3 A E4 A E5 A E6 C c,E2 C c,E3 C c,E4