- Email: [email protected]
RESHEX: an interactive software package for the synthesis and analysis of resilient heat-exchanger networks—II
Computers& Chem& Engmeermg,Vol IO,No 6, pp 591-599, 1986 Pnnted m Great Bntam
All nghts reserved
CopyrIght 0
0098-I354/86$3 00 + 0 00 Pergamon Journals Ltd
1986
RESHEX. AN INTERACTIVE SOFTWARE PACKAGE FOR THE SYNTHESIS AND ANALYSIS OF RESILIENT HEAT-EXCHANGER NETWORKS-II DISCUSSION NETWORK
OF AREA TARGETING AND SYNTHESIS ALGORITHMS
A K SAW? and M MORARI~ Department of Chemtcal Engmeenng, Umversrty of Wrsconsm, Madison, WI 53706, U S A R D COLBERG Chemrcal Engmeermg, 20641, Cahfomta Instttute of Technology, Pasadena, CA 91125, U S A (Recewed 5 December 1984, revrslon received 17 Aprrl 1986, recewed for pubbcatron 1 May 1986) Abstract-RESHEX 1s an mteracttve software package for syntheses and analysts of resilient heatexchanger networks (HENS) The algonthms used m RESHEX for area targetmg and for synthesis of the network structure are described Scope-Three Important steps m HEN design are the “targeting” of utthty and capital needs pnor to synthesis of the network, synthesis and opttmtzatton of the network and, finally, analysts of network performance under network structure changes and varymg operatmg condtttons (I e feastbthty and resrhence analyses) Penal-and-paper techniques have hmrted scope exact area targetmg becomes mfeastble when the film transfer coeffictents of the streams are not all equal, synthesis m the presence of constramts on sphts, and number of exchangers and matches, mvolves much tnal and error and expenence, and a ngorous performance analysts for a network wtth more than the mmrmum number of exchangers IS tmposstble for problems of practtcal size In thts paper effictent algonthms are presented for area targeting and for HEN syntheses m the presence of constramts on the number and locatton of stream splits and exchangers For utility targetmg, RESHEX uses the algonthms of Lmnhoff and Flower [l] and Cerda and Westerberg [2] For feastbthty and resthence testing, RESHEX uses the algonthms of Saboo and Moran [3] and Saboo et al [4-6] Conchts~ons and SigniScn~Thts paper descrtbes the algorrthms used m RESHEX [7] for area targeting and for HEN synthesis Among the techniques discussed are (1) a new LP formulatton for surface area targeting, and (11)an MILP formulatton for automattc synthesis of an HEN structure which allows the designer to influence the shape of the synthesized structure by constrammg the number and locatton of stream sphts and exchangers
2 AREA TARGETING
1 INTRODUCTION
RESHEX 1s an rnteracttve software package whtch has been developed at the Unrverstty of Wtsconstn for synthesis and analysts of resilient heat-exchanger networks (HENS) Part I [7, this issue, pp, 577-5891
of this set of papers outlined the features and functions of the program This paper describes the algorithms employed in RESHEX for area targetmg and for synthesis of the HEN structure These algonthms are either new or slgmficant modlficatlon/extenslons of available techniques and are presented here for the first time tcurrently managmg an energy consultancy busmess m India m collaboratron with ICI TENSA Technology Group $To whom all correspondence should be addressed, presently at California Instrtute of Technology, Pasadena, CA 91125, USA
Before an HEN 1s synthesized it 1s possible to compute a lower bound on the total exchanger area required for a problem tndependent of the network structure The area target can be calculated either for maximum energy recovery (MER) or for specified utility loads Under the assumption of ldentlcal overall heattransfer coefficients for all matches, graphlcal methods for esfabhshmg the total area target have been proposed by Raghavan [8] and Umeda et al [9] Townsend and Lmnhoff [lo] have extended these methods to approximately take mto account different film transfer coefficients The extended method 1s exact when all film transfer coefficients are equal When they are not, the area target can be overestimated Nevertheless, the method 1s simple and therefore it 1s implemented m RESHEX 591
A K
592
TABOO
In the case of slgmficantly different film transfer coefficients, the area target can be Improved by solvmg an LP transportation problem The temperature range IS dlvtded mto “temperature intervals (TI)” and the followmg LP 1s set up to mmlmlze the total heat-transfer area
,
0)
Energy balances on hot streams
ei
al 3 SYNTHESIS
CC
(14
qtk )I= QJ/,
kdk,
I
where aT = the area target, q,kJ, = the heat transferred
from hot stream 1 m TI k to cold stream J in TI 1, Q,k = the total heat duty for stream I m TI k, a constant which can be calculated a przon, h, = the film heat-transfer coefficient of stream I, the log mean temperature difference for A TL,,, an exchanger between stream I m TI k and stream J in TI 1, which IS also a constant since the TIs are determined before the LP 1s solved, S,k = the set of cold intervals 1 for which ATL,,k J/
=
J/
>
0
and & = the set of hot intervals A TLM,k J/
’
k for which
STRUCTURE
RESHEX uses the mixed-Integer linear program (MILP) transhlpment problem forrnulatlon by Papouhas and Grossmann [ll] for automatic synthesis of the HEN structure In its ongmal form the algorithm has some hmltatlons In this section modifications of the algonthm are proposed to mcorporate practical features, such as (1) allowmg the MER to be relaxed m order to reduce the number of exchangers, and (2) allowmg the user to influence the shape of the synthesized network by constrammg the number and location of exchangers and stream splits
(lb) Energy balances on cold streams
OF NETWORK
3 1 The Basic MILP Synthesis Algorithm The first step m the MILP formulation 1s to divide the temperature range mto k = 1, , K temperature intervals (TIs) defined by the supply and target temperatures and dlscontmultles (break points) m the heat capacity flowrates of each stream To motivate the MILP formulation, consider Fig 1 which shows a hot stream I and a cold stream J passing through a TI Tk 1s the orzgrnal temperature assigned to the boundary between TI k - 1 and TI k (e g the temperature assigned using the “problem table” algonthm [l] m the utility targeting section of RESHEX) Note that Tk IS a “shifted” temperature scale, m which mdlvldual AT, contrlbutlons have been subtracted for each hot stream and added for each cold stream The total heat duty of hot stream I m TI k (based on the temperatures otlgmally assigned to the TIs) 1s Q,k = w,(Tk - Tk+,), where W,k IS the heat capacity flowrate of hot stream I m TI k Likewise, the total heat duty of cold stream1 m TI k IS
0
The mam drawback of the above procedure 1s that the “proper” dlvlslon of the temperature range mto TIs has not yet been established Currently, the program starts with intervals based on the dlscontmultles m the composite heat duty curves The TIs are then made successively smaller and new LPs are solved until uT converges wlthm a specified tolerance or until the problem becomes prohlbltlvely large
Q/k= W,,(Tk-
where W, IS the heat capacity flowrate of cold stream in TI k Note that Qik and QJk can be calculated a prrorl, before any network synthesis The heat duty Qlk can be satisfied in two ways (1) by heat transfer q,Jkfrom hot stream I to cold stream J in TI k, and (2) by “cascading” a heat residual R, to the next TI, which
J
TI k
42
R,, k-l
Hot
stream
I
0 Q,k
R,k c
rIC 4 w a
5
Cold
stream
I
*
T,+1
?i Fig
I
Tk+,)r
Energy balance on TI k
Analysis of reslhent heat-exchanger networks-II
m essence allows hot stream I m TI k to transfer heat to cold streams which are m TIs colder than TI k Thus, by energy balance,
QA =
qvk +
(Rtk -
since residual R, k _ , may enter TI k In order to extend this idea to NH hot process streams and NC cold process streams, define the followmg addltlonal nomenclature #=
{2)1= 1, , NH} = set of hot process streams, , NC} = set of cold process w={(/IJ=L streams, s = hot utlhty stream (e g stream), w = cold utlhty stream (e g water), I? = total heating utlhty load allowed for the HEN (relaxed MER specified by the user or MER calculated by the energy targeting section of RESHEX), cf = total cooling utility load allowed for the HEN
and Rsk = residual TI k
cause of plant layout) The set of prohlblted matches 1s denoted by g=
R,k-,)
of hot utlhty stream s leaving
The following nomenclature IS defined to consider all possible stream matches (mcludmg heat transfer to a cold stream J m TI k from a hot stream I m any TI hotter than TI k)
593
streams z andj
{(z,J)~zE&',J&',
prohlblted
from matching)
Finally, to determine the existence of stream matches, integer variables Y, are defined as follows Yl,=
1, match exists between streams z and J -C 0, no match exists between streams z and
J
The y, satisfy
k=I
where U,, = arbitrary large number 2 mm
2 Q,, f
1 k-1
Q,k ,
it_,
1 e if there is a match between streams z and J in my TI k, then at least one q,,k IS nonzero, and y, must equal 1 m order to satisfy the mequahty Bmdarly, y,, and ys, can be defined to determine the existence of matches between hot process stream zand cold utlhty w, and cold process stream J and hot utility s, respectively Using this nomenclature, the followmg MILP can be formulated to mmlmlze the number of stream matches for any given utlhty consumption
tik = {I 1zE.S, hot process stream z present m TI k
TI k}, qllk = heat transferred q,,&=
from stream zc3EOk to
stream J &Sk, heat transferred from hot stream zeSk to cold utility w m TI k
where the ,8s are weights assigned to each match st Energy balances on hot streams RJ,--R,k-,+
1 9uk+9,wk=Q,k JE%
and
k=l,
ZEM,
qS,k= heat transferred from hot utlhty s to cold stream J in TI k
For slmphclty m deriving the constraints on number of exchangers and stream splits later m this paper, the synthesis MILP IS hmlted to one hot utlhty and one cold utlhty (the constraints and MILP are easily extended to multiple utlhtles) The hot utility enters the network m the hottest TI (Qs, = i?, Qsk= 0 for K) and can be cascaded (via residuals RJ k =2, to colders’TIs The cold utlhty occurs in the coldest TI
R,=R,,=O
,K
ZC.%?
(2b) Cw
Energy balances on hot utzhty Rsk-R,k-,+
1 ql/k=O
k=2,
,K
(24
JE@k
k=l,
R-6
(2e)
where 6 1s a tolerance allowing for roundoff error Energy balances on cold streams c qyk+%Jk=QJk IEJlli
J@%
k=lv
,K
(2f)
c 9rwK= CT ( 1s.V qlHk= 0
for k = 1,
,K-I
Vz >
Some streams may be prohlblted from matching (because of control or safety conslderatlons, or be-
Energy balances on cold utzlzty q,Hk=O
c-s<
zeYkr
k=l,
2 &k<<++6 rsxk
,K-1 k=K
(2g) (2h)
A K S~noo et al
594 Exzstence of heat -exchange matches
IEtik, VW Y,,
-
q,,,k >
0
despite the modlficatlons suggested here, there are still some problems with the procedure
JE?fk, k = 1, k
1 Exk,
=
, K
(21)
(21)
K
Prohlbzted matches Y,=
0
(21)
(l>J) Ep
Integer varrables y,=o,
1,
Y,, =o,
1,
y,=o,
1
Nonnegatlvlty constramts R&&O,
Rsk>O
q,,k 3
qrwk a
0,
leHk,
O, tEe%?k,
qs,k 3
,K
k=l,
(24
0
J&k,
k = 1,
,K
(20)
The objective function (2a) IS a weighted sum of streams which are matched If all j?, are equal, it IS equal to the number of pairs of streams which are matched Note that the independent variables over which the mmlmlzatlon IS performed are the heattransfer rates (q) m each stream match, the heat residuals (R) and the network structure as represented by the integer vanables (y) The weights & allow the designer to influence the synthesis For example, if the exchanger material cost for a particular match 1s very high or the streams are located far apart from each other, it may be desirable to give preference to other matches The designer can penalize this match by entering a large weight p, Constraints (21) can also be included to prohlblt certain matches altogether The two mam differences between the MILP formulation presented here and the one by Papouhas and Grossmann [l l] are (a) The problem 1s not divided into two parts, one below and one above the pinch This increases the size of the MILP but provides the flexlblhty crucial for solving practical problems It allows the designer to specify more than the minimum amount of heating and coohng With the MER requirement relaxed the synthesis algorithm can generate simpler networks, for example by removing network cycles (1e duplicate matches) which straddle the pinch [12] (b) A tolerance of + 6 IS allowed on the overall energy balances (2e) and (2h) It was found that unless the problem 1s perfectly scaled the MILP algonthm 1s generally unable to find a feasible solution with 6 = 0 The MILP solution identifies the pairs of streams which match m the network and the amounts of heat transferred between streams m each TI However,
(I) The MILP solution does not define the network structure automatically After the MILP 1s solved, further calculations are required to determine the streams which must be split and the location of these stream splits m the network Additional computations are also required to determine If more than one exchanger 1s needed between a pair of streams, i e if a cyclic structure is necessary (n) The MILP mmlmlzes the number of matches between streams This may or may not mmlmlze the number of heat exchangers, depending upon the method used to generate the network structure from the MILP solution Floudas et al [ 131use the MILP solution to generate a feasible network from a more general “superstructure” (A “superstructure” IS a network of heat exchangers with mathematical “plpmg” such that all senes and parallel combmatlons of heat exchangers can be represented, as well as sphttmg and bypassing of all streams ) In Floudas’ case, each stream match represents one and only one heat exchanger However, Floudas et al [ 131determine no mformatlon about the network structure until after the MILP IS completely solved We determine partial mformatlon about network structure whrle the MILP 1s being solved so that we can constrain both stream sphttmg and the number of heat exchangers on a particular stream (these constraints are described later m this paper) We use a method based upon the “shlftmg rules” of Lmnhoff and Flower [14] to generate a feasible network from the MILP solution In this method a stream match can represent one or more heat exchangers (m) The number of feasible solutions satisfying the constraints of the MILP (2a-o) 1s often very large The designer should be able to influence the network structure obtained by the MILP more directly Use of the weights & suggested for this purpose 1s not sufficient because even with extensive tnal and error it IS practically impossible to determine what the correct weights should be, for example, to mmlmlze the total cost of the network The computattons necessary to translate the MILP solution mto a network structure can be set up quite easily They also provide the insight needed to formulate the constraints to limit the number and location of exchangers and stream splits which can be added to the MILP (2a) and which largely ehmmate drawbacks (u) and (m) of the MILP algorithm These new constraints along with the weights /3, can then be used Judlclously by the designer to generate the most desirable network structure 3 2 Stream-spilt Constrarnts The temperatures Tk mltlally assigned to each TI boundary would be the actual temperatures of the
Analysis of reslhent heat-exchanger networks--II
595
TIk
Fig 2 One cold stream J exchanging heat with several hot streams m TI k
hot process streams if surplus heat m each TI was not cascaded but was lmmedlately sent to coolmg water The occurrence of nonzero residuals (&) corresponding to heat being cascaded alters the hot stream temperatures If the heat capacity flowrate of hot stream I IS assumed constant (W,, = W, for k = 1, K), then temperature T* of hot stream z at the mlet tb TI k (as indicated m Fig 1) 1s T,k= Tk +
T’1’ = T _ qllk Ik Ik w,’
T’“-T lk-
_!!& Ik Y’
p’,-T Jk-
lk
Tj;) = Tj;’
_!!& w’
_
(54
(ha)
J
t$ = J
5
(SC)
and
R, k-l
f$ (4) k-!, J A constraint to prevent sphttmg of cold stream/ m TI k IS important only when that stream exchanges heat with more than one hot stream Assummg that stream J 1s matched with the hot streams m order of increasing stream number, as shown m Fig 2, then the temperature at the cold end of each heat exchange match can be obtained by energy balance
(5b)
2
7
If W, ISnot constant, it 1s difficult to obtain a similar expression for TJ, before the network structure 1s synthesized The temperatures of cold process streams] m TI k depend upon the location of heaters on these streams If only one heating utility 1s used, then all utthty heat loads qsh on stream J m TIs colder than k can be shifted through TI k to the hottest TI (TI 1) Therefore the temperatures of the cold streams at the hot end of TI k are qk = Tk -
T\?= T+
qk
_
!$ _ !.$ J
J
(6’4 (64
The values of Rlkr qs&and qllk are generated by the MILP and all the temperatures can be computed using equations (3)-(6c) Cold stream J does not need to be spht m TI k if the closest approach temperature of each heatexchange match with stream J m TI k IS > AT,,,, 1 e if the approach temperature IS >AT,,, at the hot end of each match m TI k with W, 2 W,, T&- T$-‘)aO
for
{zIzEX~,
W, 2 W,}, Va)
and if the approach temperature 1s > AT, at the cold end of each match m TI k with W, < W,, T$)-
T$‘>O
for
(zIzEJEP~, W,< W,}
(7b)
Note that the r h s of these two mequahties 1s zero because AT,,, was accounted for m the defimtlon of the TIs The constraints to limit stream sphttmg on cold stream J in TI k are set up as follows New integer vanables nllk are defined to indicate whether con-
596
A K
SABOOetal
stramts (7a, b) are satisfied or not for the match between streams z and J m TI k T,,-T$-“+0n,,>O
for
{zIz~&
W,> W,} (84
and
where the streams are assumed to be matched m order of increasing stream number for both the hot and cold streams To hmlt stream z to N,k spht branches or less m TI k, and stream J to Nlk branches or less, the followmg two constraints are used NC 1
%k
6
&k
-
1
(124
J=I NH
where 0 1s a sufficiently large number Substltutmg for T,, T$’ and T$’ from equations (3)-(6c) transforms constraints (8a, b) mto
for
{~IzE&~,
W, 2 W,> (9a)
and
w, < w,} (9b)
{ZIZE&
for
Expressions (9a, b) assume that hot streams are matched with tge cold stream m an increasing order of stream indices, 1e stream 1 1s matched first and so on In actual pItactlce one would match m order of decreasing temperatures, which 1s done m RESHEX when the network 1s constructed from the MILP solution However, it 1s not possible to do that dunng synthesis because the temperatures of the streams are not known yet To prevent sphttmg of cold stream J m TI k, expreseons (9a, b) must be satisfied for every exchanger, 1 e n,,k must be zero for all hot streams z A bound on the number of branches m a spht can be imposed by Cnvk<~k-19
(10)
where N,k 1s the specified upper bound If sphttmg of stream J 1s prohibited m TI k, then N,k = 1 The stream-sphttmg constraints (9a, b) and (10) are easily extended to multiple hot and multiple cold streams
nvk
s
NJk
k-,+;‘i -,c,40
-
1
UW
Note that because an “ordered matching” of streams 1s assumed, the constraints introduced to prevent stream splits are onZy su$ictent but not necessary This means that mtroducmg the constraints will always prevent stream splits However, at times the split constraints might prevent the MILP from finding any feasible HEN, although the particular problem has a feasible solution without splits Thus the split constraints should be used cautiously and only after the MILP has generated a split solution m the absence of the constraints An alternate approach to hmlt stream sphttmg 1s to add a term of the form p,cnYk to the objective function This would prevent the occurrence of the MILP not finding a feasible HEN, but It might not give a network with the mmlmum number of matches nor might It prevent splits Furthermore, the weight j?. has to be selected by tnal and error The split constraints can even be formulated for problems with temperature-dependent heat capacltles Inequahtles (11 a, b) can be moddied to
;Ff
q3& +
Wien, k 0
(13a)
/ma. k =k
for the hot end of an exchanger, and
%Jk
J-1
R,
1 r=l
+
Fdn,Jk
>
0
(13b)
&,k
-I
,I
+
2
K
5
%Jk
+
Wtenvk
3
o
/k-k
R,k-,+;$
qr,k-
i
,‘-I
+
for
K W
K c Jk-k
%Jk
‘?q”k
, =I
+
((ir/)iiE%,
w!enjJk
2
jEwk,
o
w,< w,}
(lib)
for the cold end of an exchanger Extreme values W,msn and WJm,,are used because the actual values of W, and W, at any point m the network are not known before the MILP 1s solved Therefore mequahtles (13a, b) are only a sufficient condltlon to prevent stream splits Note that if wfm. 2 wJm..y then AT,,, constraint (13a) IS required only at the hot end of any exchanger matching streams z and J, If W,mx< W,,., then AT,,, constraint (13b) 1s required only at the cold end of any exchanger matching streams z and J, otherwise, AT, constraints (13a, b) are required on both ends of each exchanger
591
Analysis of reslllent heat-exchanger networks--II TI
k-l
TI k
R/,*-I
R
,,*-2
Hot streom
RLk
I
streams
Fig 3 Other matches with streams I or J occur between matches r/,_ , and uk If R, e _ , < qqk,then match ZJ~can nof be combined with match ~/k_, 3 3 Constramts on the Number of Exchangers Consider a match between streams I and J which IS repeated m TIs k - 1 and k-call these match ZJ~_ , and match ZJ~,respectively If no matches with either stream I or] occur between matches IJ~_ , and ZJ~(Fig 3). then the two matches can be combined to form one heat exchanger If any other matches with either stream I or J occur between matches ‘Jo_ , and IJ~,then It IS necessary to “shift” match ~/k “through” these other matches before match ZJ~can be combined with match ZJ~_, to form one exchanger Lmnhoff and Flower [14] provide rules when shlftmg 1s possible If It 1s not possible, then matches r/, _ , and ZJ~give nse to two exchangers which cannot be combined Along these ideas constraints can be set up to limit the number of matches which cannot be combined and thus restrlctmg the number of exchangers For slmphclty, combmatlon posslblhtles are checked only m adjacent TIs A match between streams z and J m TI k gives nse to a separate exchanger if the following logical expression is true A and [B or (C and D)],
(14)
where A is true if streams z and J are matched m TI k, B is true If streams I and J are not matched m TI k - 1, C 1s true if other matches with either stream I or J occur between matches ZJ~ _ , and ZJ~(Fig 3), D 1s true 1s residual R,k < qyk, the heat exchanged m match ZJ~
matches between ZJ~ _ , and IJ~ (condltlon C), which cannot be done because hot stream I has msufficlent residual at the mlet of TI k (condltlon D) Therefore a new exchanger 1s necessary (Fig 3) Integer variables are defined to indicate when condltlons A, B, C and D are satisfied (A) Integer variable yllk mdlcates If a match exists between streams z and J m TI k The constraint which defines yVk1s u,
Y,/k
-
q,,k
2
(15)
O,
where U, was defined earher as an upper bound on the amount of heat which can be transferred from Stream 2 t0 StKZUn J Whenever qrlk> 0, y,,k 1s equal to 1, otherwise yllk = 0 (B) Slmllar to the constraint correspondmg to condltlon A, mteger vanable y, k_, mdlcates If a match exists between streams z and J m TI k - 1 If such a match exists, then yYk_ , = 1, otherwise, YI/k-i=O
(C) Integer vanable plrk (defined by the following constraint) mdlcates when any other matches with either stream I or J occur between matches Ilk-1 and vk Fig 3) KP,,k-
2
Y,,
I”=,+1
k-
I -
‘i’ , =I
Yf,k
k-l
-
1-l
NC
-
1 f=j+l
Y,/
1
yy”k>o,
(16)
,=I
where K IS a sufficiently large positive number Whenever other matches occur between matches IJ~_ , and IJ~,P,,~= 1, otherwlse P,,~= 0 Note that If condltlon B 1s satisfied, then a new exchanger this constramt assumes that the streams are must be placed between streams I and J m TI k matched m order of mcreasmg stream numbers because no match ZJ_ , exists with which match IJ~ can be combined If both condltlons C and D are (D) Integer vanables sVkare mtroduced to indicate If residual R, k_, IS sufficient for match qk to be satisfied, match IJ~must be shlfted through the other
A K
598
SABOO et al
Table I SIX and CPU time reamred by the MlLP for the case studv m Part I 171 Stream-spht constramts for stream 7 m four
No addmonal constramts Number Number Number
a7 132 13
of rows of variables of Integer variables
Exchanger constramts
Both stream-spht and exchanger constramts
TIS
for stredm I
107 148 29
163 207 88
183 223 104 51 IO1
CPU requzremenr (s)
First Integer solution Fmal solution
4
9
34
5
13
49
shifted mto TI k - 1 (17)
Whenever the residual IS msuffiaent, wise sgk= 0 The followmg constraints expression (14)
sYk= 1, other-
are equivalent to logcal
+tl,-tl8=0, 4’~ G
Ke,
KU
-
UW U8b).
and ‘IS G
eukh
(184
where K 1s a sufficiently large number Integer vanable e,k equals 1 when logcal expression (14) is true (I e when match & cannot be combined with match I/k- I), e,,k= 0 when expression (14) is false The translation of logical expression (14) into constraints (18a-c) 1s explained m the Appendix An upper bound on the number of exchangers N, on a particular hot stream I can he imposed by (19) If there are no stream splits on stream I then the bound (19), together with constraints (15)-(18c), 1s suficlent to hmlt to N, the number of exchangers on stream z It 1s only sufficient because combmatlons among adJacent TIs alone are tested and because m a particular TI the matches are assumed to occur m a certain order This implies that constramts (15)-(19) can cause the MILP not to find a feasible HEN although a feasible solution with N, exchangers exists Just like the split constraints, the exchanger constraints should be used with caution and only after an unconstrained solution has been found to have an excessive number of exchangers on a particular set of streams Table 1 shows a comparison of the CPU requlrement for the MILP both with and wlthout the additional constraints for the seven-stream problem m the case study m Part I [7] The exchanger constraints increase the size of the problem considerably, m particular, the number of integer variables mcreases This IS reflected m the IO-fold increase in CPU time The stream-split constramts, however, do not increase the CPU reqmrement slgmficantly
An approximate method which tends to reduce the number of exchangers m the MILP solution while requiring much fewer variables IS to use only the constraints
(15) and (19) together
with
m selected TIs on specific stream pairs To reduce the CPU time requirements the MILP package LINDO [15] mcorporated m RESHEX was modified The solution of the MILP IS a two-step procedure (1) LP solution, (n) Branch-and-bound integer solution
technique
to determine
an
Most of the CPU time 1s consumed by Step (11) It was observed that the mltlal feasible integer solution 1s usually reached relatively quickly The search to improve it requires the bulk of the total CPU time, but often simply confirms the first integer solution to be the best This 1s verified by comparing the CPU requirements m Table 1 for the case study m Part I [7] Therefore, RESHEX displays the network generated from the first integer solution and then e;lves the user the option
to search for a better
solutions
Acknowledgement-Partial financial support from Department of Energy IS gratefully acknowledged
the
REFERENCES
B Lmnhoff and J R Flower, Synthesis of heat exchange networks I Systematic generatron of energy optimal networks AIChE JI 24, 663 (1978) J Cerda and A W Westerberg, Mmlmum utility usage m heat exchanger network synthesis-a transportation problem Chem Engng Scr 38, 373 (1983) A K Saboo and M Moran, Design of resilient processmg plants-IV Some new results on heat exchanger network synthesis Chem Engng Scl 39, 579 (1984) A K Saboo, M Moran and R D Colberg, Reslhence analysis of heat exchanger networks-I Temperaturedependent heat capacltles Comput them Engng (in press) A K Saboo, M Morarl and R D Colberg, Resilience analysis of heat-exchanger networks-II Stream sphts and flowrate varlatlons Comput them Engng (m press) A K Saboo, M Moran and D C Woodcock, Design of resihent processmg plants-VIII A Reslhence Index for heat exchanger networks Chem Engng SCI 40,1553 (1985) A K Saboo, M Morarl and R D Colberg, RESHEX an mteractlve software package for the synthesis and
Analysis of resilient heat-exchanger networks-II
8 9 10 11 12 13 14
15
analysis of reslhent heat-exchanger networks-I Program descrlptlon and apphcatlon Comput them Engng 10, 577 (1986) S Raghavan, Heat exchanger network synthesis A thermodynamic approach Ph D Thesis, Purdue Umv , West Lafayette, Ind (1977) T Umeda, T Harada and K Shlroko, A thermodynamic approach to the synthesis m chemical processes Comput them Engng 3, 373 (1979) D W Townsend and B LmnholT, Surface area targets for heat exchanger networks Paper presented at Inst them Engrs A Res Mtg, Bath, Avon (Apnl 1984) S A Papouhas and I E Grossmann, Structural opt)mlzatlon approach m process synthesis Part II heat recovery n&works Coiput che& Engng 7,707 (1983) B Lmnhoff and E Hmdmarsh. Pmch desuzn method for heat exchanger networks &em Engng &CI 38, 745 (1983) C A Floudas, A R Clrlc and I E Grossmann, Automatic synthesis of optimum heat-exchanger network configuratlons Chem Engng Commun (m press) B Lmnhoff and J R Flower, Synthesis of heat exchange networks II Evolutlonar; generation of networks with various crltena of optlmahty AIChE JI 24, 642 (1978) L Schrage, User’s Manual Lmear, Integer and Quadratrc Programmrng with LINDO Scientific Press, Palo Alto, Calrf (1984)
599
where condltlon B IS true If (1 - y,, Ir_, ) = 1 and condltlon E 1s true If m, = 1 (The “strange” Integer eqmvalent for condltlon B results from the fact that condltlon B 1s defined to be true when streams I andj are not matched m TI k - 1 ) An integer constramt correspondmg to expresslon (A 3) IS 05-(1-y,,-,)-m,+rl,--rl,=0,
(A 4)
where ‘I~Q Km, and q4
ml
0 0 I I
4
0 I 0 I
0 I I 1
ISthe same as the “truth table” for the loglcal OR expression (A 3) Next, the lolpcal expressions (A 1) and (A 3) can be combined to form F = B or (C and D) An integer constraint correspondmg
15--[2(1 -~,,-,)+@,,+s~)l+tls-16=0. APPENDIX
646)
where
This Appendix describes the translation of logical expressions mto Integer constraints, m particular, the translatlon of logcal expression (14) mto constraints (18ax) is explamed First, consider the logcal AND expression E=Cand
(A 5)
to expresslon (A 5) 1s
D,
(A 1)
where condltlon C IS true If P,,~= 1 and condltlon D IS true if s,,~= 1 (Conversely, condltlon C IS false If P,,~= 0 and condltlon D 1s false If sVk= 0 ) An integer constramt corresponding to expresslon (A 1) 1s I5-P,,-s,,+‘I,-%=O1
(A 2)
where
and t16d K(1 -m,) It IS easy to show that the “truth table” for Integer variable m, m constraint (A 6) 1s the same as the “truth table” for condltlon F m logical expresslon (A 5) In constraint (A 6), the quantity (1 -y,, k_ ,) IS multlphed by 2 so that It “balances” agamst the “double quantity” (p,,k+ @-Just hke the logical condltlon B m expressIon (A 5) must “balance” against the “double condltlon” (C and D) The number 1 5 m constramt (A 6) 1s chosen to make the “truth table” correct Finally, for logcal expression (14), G = A and [B or (C and D)],
and
the correspondmg lated and K is a sufficiently large number The followmg “truth table” can be tabulated for constramt (A 2) p*
$!i
ml
0 0 I I
0 I 0 I
0 0 0 I
integer constraint (18a) can be formu-
55-{4~,,+[2(1--~,-,)+~,,~+~~)1} +v7--q8=0,
(18a)
where and
This IS the same as the “truth table” for the logical AND expresslon (A 1) Now consider the logical OR expression F = B or E,
(14)
(A 3)
‘Is c K(1 -
e,,k)
Agam, It IS straightforward to show that the “truth table” for the integer variable e,, m constramt (18a) IS the same as the “truth table” for condltlon G m logcal expression (14)
All nghts reserved
CopyrIght 0
0098-I354/86$3 00 + 0 00 Pergamon Journals Ltd
1986
RESHEX. AN INTERACTIVE SOFTWARE PACKAGE FOR THE SYNTHESIS AND ANALYSIS OF RESILIENT HEAT-EXCHANGER NETWORKS-II DISCUSSION NETWORK
OF AREA TARGETING AND SYNTHESIS ALGORITHMS
A K SAW? and M MORARI~ Department of Chemtcal Engmeenng, Umversrty of Wrsconsm, Madison, WI 53706, U S A R D COLBERG Chemrcal Engmeermg, 20641, Cahfomta Instttute of Technology, Pasadena, CA 91125, U S A (Recewed 5 December 1984, revrslon received 17 Aprrl 1986, recewed for pubbcatron 1 May 1986) Abstract-RESHEX 1s an mteracttve software package for syntheses and analysts of resilient heatexchanger networks (HENS) The algonthms used m RESHEX for area targetmg and for synthesis of the network structure are described Scope-Three Important steps m HEN design are the “targeting” of utthty and capital needs pnor to synthesis of the network, synthesis and opttmtzatton of the network and, finally, analysts of network performance under network structure changes and varymg operatmg condtttons (I e feastbthty and resrhence analyses) Penal-and-paper techniques have hmrted scope exact area targetmg becomes mfeastble when the film transfer coeffictents of the streams are not all equal, synthesis m the presence of constramts on sphts, and number of exchangers and matches, mvolves much tnal and error and expenence, and a ngorous performance analysts for a network wtth more than the mmrmum number of exchangers IS tmposstble for problems of practtcal size In thts paper effictent algonthms are presented for area targeting and for HEN syntheses m the presence of constramts on the number and locatton of stream splits and exchangers For utility targetmg, RESHEX uses the algonthms of Lmnhoff and Flower [l] and Cerda and Westerberg [2] For feastbthty and resthence testing, RESHEX uses the algonthms of Saboo and Moran [3] and Saboo et al [4-6] Conchts~ons and SigniScn~Thts paper descrtbes the algorrthms used m RESHEX [7] for area targeting and for HEN synthesis Among the techniques discussed are (1) a new LP formulatton for surface area targeting, and (11)an MILP formulatton for automattc synthesis of an HEN structure which allows the designer to influence the shape of the synthesized structure by constrammg the number and locatton of stream sphts and exchangers
2 AREA TARGETING
1 INTRODUCTION
RESHEX 1s an rnteracttve software package whtch has been developed at the Unrverstty of Wtsconstn for synthesis and analysts of resilient heat-exchanger networks (HENS) Part I [7, this issue, pp, 577-5891
of this set of papers outlined the features and functions of the program This paper describes the algorithms employed in RESHEX for area targetmg and for synthesis of the HEN structure These algonthms are either new or slgmficant modlficatlon/extenslons of available techniques and are presented here for the first time tcurrently managmg an energy consultancy busmess m India m collaboratron with ICI TENSA Technology Group $To whom all correspondence should be addressed, presently at California Instrtute of Technology, Pasadena, CA 91125, USA
Before an HEN 1s synthesized it 1s possible to compute a lower bound on the total exchanger area required for a problem tndependent of the network structure The area target can be calculated either for maximum energy recovery (MER) or for specified utility loads Under the assumption of ldentlcal overall heattransfer coefficients for all matches, graphlcal methods for esfabhshmg the total area target have been proposed by Raghavan [8] and Umeda et al [9] Townsend and Lmnhoff [lo] have extended these methods to approximately take mto account different film transfer coefficients The extended method 1s exact when all film transfer coefficients are equal When they are not, the area target can be overestimated Nevertheless, the method 1s simple and therefore it 1s implemented m RESHEX 591
A K
592
TABOO
In the case of slgmficantly different film transfer coefficients, the area target can be Improved by solvmg an LP transportation problem The temperature range IS dlvtded mto “temperature intervals (TI)” and the followmg LP 1s set up to mmlmlze the total heat-transfer area
,
0)
Energy balances on hot streams
ei
al 3 SYNTHESIS
CC
(14
qtk )I= QJ/,
kdk,
I
where aT = the area target, q,kJ, = the heat transferred
from hot stream 1 m TI k to cold stream J in TI 1, Q,k = the total heat duty for stream I m TI k, a constant which can be calculated a przon, h, = the film heat-transfer coefficient of stream I, the log mean temperature difference for A TL,,, an exchanger between stream I m TI k and stream J in TI 1, which IS also a constant since the TIs are determined before the LP 1s solved, S,k = the set of cold intervals 1 for which ATL,,k J/
=
J/
>
0
and & = the set of hot intervals A TLM,k J/
’
k for which
STRUCTURE
RESHEX uses the mixed-Integer linear program (MILP) transhlpment problem forrnulatlon by Papouhas and Grossmann [ll] for automatic synthesis of the HEN structure In its ongmal form the algorithm has some hmltatlons In this section modifications of the algonthm are proposed to mcorporate practical features, such as (1) allowmg the MER to be relaxed m order to reduce the number of exchangers, and (2) allowmg the user to influence the shape of the synthesized network by constrammg the number and location of exchangers and stream splits
(lb) Energy balances on cold streams
OF NETWORK
3 1 The Basic MILP Synthesis Algorithm The first step m the MILP formulation 1s to divide the temperature range mto k = 1, , K temperature intervals (TIs) defined by the supply and target temperatures and dlscontmultles (break points) m the heat capacity flowrates of each stream To motivate the MILP formulation, consider Fig 1 which shows a hot stream I and a cold stream J passing through a TI Tk 1s the orzgrnal temperature assigned to the boundary between TI k - 1 and TI k (e g the temperature assigned using the “problem table” algonthm [l] m the utility targeting section of RESHEX) Note that Tk IS a “shifted” temperature scale, m which mdlvldual AT, contrlbutlons have been subtracted for each hot stream and added for each cold stream The total heat duty of hot stream I m TI k (based on the temperatures otlgmally assigned to the TIs) 1s Q,k = w,(Tk - Tk+,), where W,k IS the heat capacity flowrate of hot stream I m TI k Likewise, the total heat duty of cold stream1 m TI k IS
0
The mam drawback of the above procedure 1s that the “proper” dlvlslon of the temperature range mto TIs has not yet been established Currently, the program starts with intervals based on the dlscontmultles m the composite heat duty curves The TIs are then made successively smaller and new LPs are solved until uT converges wlthm a specified tolerance or until the problem becomes prohlbltlvely large
Q/k= W,,(Tk-
where W, IS the heat capacity flowrate of cold stream in TI k Note that Qik and QJk can be calculated a prrorl, before any network synthesis The heat duty Qlk can be satisfied in two ways (1) by heat transfer q,Jkfrom hot stream I to cold stream J in TI k, and (2) by “cascading” a heat residual R, to the next TI, which
J
TI k
42
R,, k-l
Hot
stream
I
0 Q,k
R,k c
rIC 4 w a
5
Cold
stream
I
*
T,+1
?i Fig
I
Tk+,)r
Energy balance on TI k
Analysis of reslhent heat-exchanger networks-II
m essence allows hot stream I m TI k to transfer heat to cold streams which are m TIs colder than TI k Thus, by energy balance,
QA =
qvk +
(Rtk -
since residual R, k _ , may enter TI k In order to extend this idea to NH hot process streams and NC cold process streams, define the followmg addltlonal nomenclature #=
{2)1= 1, , NH} = set of hot process streams, , NC} = set of cold process w={(/IJ=L streams, s = hot utlhty stream (e g stream), w = cold utlhty stream (e g water), I? = total heating utlhty load allowed for the HEN (relaxed MER specified by the user or MER calculated by the energy targeting section of RESHEX), cf = total cooling utility load allowed for the HEN
and Rsk = residual TI k
cause of plant layout) The set of prohlblted matches 1s denoted by g=
R,k-,)
of hot utlhty stream s leaving
The following nomenclature IS defined to consider all possible stream matches (mcludmg heat transfer to a cold stream J m TI k from a hot stream I m any TI hotter than TI k)
593
streams z andj
{(z,J)~zE&',J&',
prohlblted
from matching)
Finally, to determine the existence of stream matches, integer variables Y, are defined as follows Yl,=
1, match exists between streams z and J -C 0, no match exists between streams z and
J
The y, satisfy
k=I
where U,, = arbitrary large number 2 mm
2 Q,, f
1 k-1
Q,k ,
it_,
1 e if there is a match between streams z and J in my TI k, then at least one q,,k IS nonzero, and y, must equal 1 m order to satisfy the mequahty Bmdarly, y,, and ys, can be defined to determine the existence of matches between hot process stream zand cold utlhty w, and cold process stream J and hot utility s, respectively Using this nomenclature, the followmg MILP can be formulated to mmlmlze the number of stream matches for any given utlhty consumption
tik = {I 1zE.S, hot process stream z present m TI k
TI k}, qllk = heat transferred q,,&=
from stream zc3EOk to
stream J &Sk, heat transferred from hot stream zeSk to cold utility w m TI k
where the ,8s are weights assigned to each match st Energy balances on hot streams RJ,--R,k-,+
1 9uk+9,wk=Q,k JE%
and
k=l,
ZEM,
qS,k= heat transferred from hot utlhty s to cold stream J in TI k
For slmphclty m deriving the constraints on number of exchangers and stream splits later m this paper, the synthesis MILP IS hmlted to one hot utlhty and one cold utlhty (the constraints and MILP are easily extended to multiple utlhtles) The hot utility enters the network m the hottest TI (Qs, = i?, Qsk= 0 for K) and can be cascaded (via residuals RJ k =2, to colders’TIs The cold utlhty occurs in the coldest TI
R,=R,,=O
,K
ZC.%?
(2b) Cw
Energy balances on hot utzhty Rsk-R,k-,+
1 ql/k=O
k=2,
,K
(24
JE@k
k=l,
R-6
(2e)
where 6 1s a tolerance allowing for roundoff error Energy balances on cold streams c qyk+%Jk=QJk IEJlli
J@%
k=lv
,K
(2f)
c 9rwK= CT ( 1s.V qlHk= 0
for k = 1,
,K-I
Vz >
Some streams may be prohlblted from matching (because of control or safety conslderatlons, or be-
Energy balances on cold utzlzty q,Hk=O
c-s<
zeYkr
k=l,
2 &k<<++6 rsxk
,K-1 k=K
(2g) (2h)
A K S~noo et al
594 Exzstence of heat -exchange matches
IEtik, VW Y,,
-
q,,,k >
0
despite the modlficatlons suggested here, there are still some problems with the procedure
JE?fk, k = 1, k
1 Exk,
=
, K
(21)
(21)
K
Prohlbzted matches Y,=
0
(21)
(l>J) Ep
Integer varrables y,=o,
1,
Y,, =o,
1,
y,=o,
1
Nonnegatlvlty constramts R&&O,
Rsk>O
q,,k 3
qrwk a
0,
leHk,
O, tEe%?k,
qs,k 3
,K
k=l,
(24
0
J&k,
k = 1,
,K
(20)
The objective function (2a) IS a weighted sum of streams which are matched If all j?, are equal, it IS equal to the number of pairs of streams which are matched Note that the independent variables over which the mmlmlzatlon IS performed are the heattransfer rates (q) m each stream match, the heat residuals (R) and the network structure as represented by the integer vanables (y) The weights & allow the designer to influence the synthesis For example, if the exchanger material cost for a particular match 1s very high or the streams are located far apart from each other, it may be desirable to give preference to other matches The designer can penalize this match by entering a large weight p, Constraints (21) can also be included to prohlblt certain matches altogether The two mam differences between the MILP formulation presented here and the one by Papouhas and Grossmann [l l] are (a) The problem 1s not divided into two parts, one below and one above the pinch This increases the size of the MILP but provides the flexlblhty crucial for solving practical problems It allows the designer to specify more than the minimum amount of heating and coohng With the MER requirement relaxed the synthesis algorithm can generate simpler networks, for example by removing network cycles (1e duplicate matches) which straddle the pinch [12] (b) A tolerance of + 6 IS allowed on the overall energy balances (2e) and (2h) It was found that unless the problem 1s perfectly scaled the MILP algonthm 1s generally unable to find a feasible solution with 6 = 0 The MILP solution identifies the pairs of streams which match m the network and the amounts of heat transferred between streams m each TI However,
(I) The MILP solution does not define the network structure automatically After the MILP 1s solved, further calculations are required to determine the streams which must be split and the location of these stream splits m the network Additional computations are also required to determine If more than one exchanger 1s needed between a pair of streams, i e if a cyclic structure is necessary (n) The MILP mmlmlzes the number of matches between streams This may or may not mmlmlze the number of heat exchangers, depending upon the method used to generate the network structure from the MILP solution Floudas et al [ 131use the MILP solution to generate a feasible network from a more general “superstructure” (A “superstructure” IS a network of heat exchangers with mathematical “plpmg” such that all senes and parallel combmatlons of heat exchangers can be represented, as well as sphttmg and bypassing of all streams ) In Floudas’ case, each stream match represents one and only one heat exchanger However, Floudas et al [ 131determine no mformatlon about the network structure until after the MILP IS completely solved We determine partial mformatlon about network structure whrle the MILP 1s being solved so that we can constrain both stream sphttmg and the number of heat exchangers on a particular stream (these constraints are described later m this paper) We use a method based upon the “shlftmg rules” of Lmnhoff and Flower [14] to generate a feasible network from the MILP solution In this method a stream match can represent one or more heat exchangers (m) The number of feasible solutions satisfying the constraints of the MILP (2a-o) 1s often very large The designer should be able to influence the network structure obtained by the MILP more directly Use of the weights & suggested for this purpose 1s not sufficient because even with extensive tnal and error it IS practically impossible to determine what the correct weights should be, for example, to mmlmlze the total cost of the network The computattons necessary to translate the MILP solution mto a network structure can be set up quite easily They also provide the insight needed to formulate the constraints to limit the number and location of exchangers and stream splits which can be added to the MILP (2a) and which largely ehmmate drawbacks (u) and (m) of the MILP algorithm These new constraints along with the weights /3, can then be used Judlclously by the designer to generate the most desirable network structure 3 2 Stream-spilt Constrarnts The temperatures Tk mltlally assigned to each TI boundary would be the actual temperatures of the
Analysis of reslhent heat-exchanger networks--II
595
TIk
Fig 2 One cold stream J exchanging heat with several hot streams m TI k
hot process streams if surplus heat m each TI was not cascaded but was lmmedlately sent to coolmg water The occurrence of nonzero residuals (&) corresponding to heat being cascaded alters the hot stream temperatures If the heat capacity flowrate of hot stream I IS assumed constant (W,, = W, for k = 1, K), then temperature T* of hot stream z at the mlet tb TI k (as indicated m Fig 1) 1s T,k= Tk +
T’1’ = T _ qllk Ik Ik w,’
T’“-T lk-
_!!& Ik Y’
p’,-T Jk-
lk
Tj;) = Tj;’
_!!& w’
_
(54
(ha)
J
t$ = J
5
(SC)
and
R, k-l
f$ (4) k-!, J A constraint to prevent sphttmg of cold stream/ m TI k IS important only when that stream exchanges heat with more than one hot stream Assummg that stream J 1s matched with the hot streams m order of increasing stream number, as shown m Fig 2, then the temperature at the cold end of each heat exchange match can be obtained by energy balance
(5b)
2
7
If W, ISnot constant, it 1s difficult to obtain a similar expression for TJ, before the network structure 1s synthesized The temperatures of cold process streams] m TI k depend upon the location of heaters on these streams If only one heating utility 1s used, then all utthty heat loads qsh on stream J m TIs colder than k can be shifted through TI k to the hottest TI (TI 1) Therefore the temperatures of the cold streams at the hot end of TI k are qk = Tk -
T\?= T+
qk
_
!$ _ !.$ J
J
(6’4 (64
The values of Rlkr qs&and qllk are generated by the MILP and all the temperatures can be computed using equations (3)-(6c) Cold stream J does not need to be spht m TI k if the closest approach temperature of each heatexchange match with stream J m TI k IS > AT,,,, 1 e if the approach temperature IS >AT,,, at the hot end of each match m TI k with W, 2 W,, T&- T$-‘)aO
for
{zIzEX~,
W, 2 W,}, Va)
and if the approach temperature 1s > AT, at the cold end of each match m TI k with W, < W,, T$)-
T$‘>O
for
(zIzEJEP~, W,< W,}
(7b)
Note that the r h s of these two mequahties 1s zero because AT,,, was accounted for m the defimtlon of the TIs The constraints to limit stream sphttmg on cold stream J in TI k are set up as follows New integer vanables nllk are defined to indicate whether con-
596
A K
SABOOetal
stramts (7a, b) are satisfied or not for the match between streams z and J m TI k T,,-T$-“+0n,,>O
for
{zIz~&
W,> W,} (84
and
where the streams are assumed to be matched m order of increasing stream number for both the hot and cold streams To hmlt stream z to N,k spht branches or less m TI k, and stream J to Nlk branches or less, the followmg two constraints are used NC 1
%k
6
&k
-
1
(124
J=I NH
where 0 1s a sufficiently large number Substltutmg for T,, T$’ and T$’ from equations (3)-(6c) transforms constraints (8a, b) mto
for
{~IzE&~,
W, 2 W,> (9a)
and
w, < w,} (9b)
{ZIZE&
for
Expressions (9a, b) assume that hot streams are matched with tge cold stream m an increasing order of stream indices, 1e stream 1 1s matched first and so on In actual pItactlce one would match m order of decreasing temperatures, which 1s done m RESHEX when the network 1s constructed from the MILP solution However, it 1s not possible to do that dunng synthesis because the temperatures of the streams are not known yet To prevent sphttmg of cold stream J m TI k, expreseons (9a, b) must be satisfied for every exchanger, 1 e n,,k must be zero for all hot streams z A bound on the number of branches m a spht can be imposed by Cnvk<~k-19
(10)
where N,k 1s the specified upper bound If sphttmg of stream J 1s prohibited m TI k, then N,k = 1 The stream-sphttmg constraints (9a, b) and (10) are easily extended to multiple hot and multiple cold streams
nvk
s
NJk
k-,+;‘i -,c,40
-
1
UW
Note that because an “ordered matching” of streams 1s assumed, the constraints introduced to prevent stream splits are onZy su$ictent but not necessary This means that mtroducmg the constraints will always prevent stream splits However, at times the split constraints might prevent the MILP from finding any feasible HEN, although the particular problem has a feasible solution without splits Thus the split constraints should be used cautiously and only after the MILP has generated a split solution m the absence of the constraints An alternate approach to hmlt stream sphttmg 1s to add a term of the form p,cnYk to the objective function This would prevent the occurrence of the MILP not finding a feasible HEN, but It might not give a network with the mmlmum number of matches nor might It prevent splits Furthermore, the weight j?. has to be selected by tnal and error The split constraints can even be formulated for problems with temperature-dependent heat capacltles Inequahtles (11 a, b) can be moddied to
;Ff
q3& +
Wien, k 0
(13a)
/ma. k =k
for the hot end of an exchanger, and
%Jk
J-1
R,
1 r=l
+
Fdn,Jk
>
0
(13b)
&,k
-I
,I
+
2
K
5
%Jk
+
Wtenvk
3
o
/k-k
R,k-,+;$
qr,k-
i
,‘-I
+
for
K W
K c Jk-k
%Jk
‘?q”k
, =I
+
((ir/)iiE%,
w!enjJk
2
jEwk,
o
w,< w,}
(lib)
for the cold end of an exchanger Extreme values W,msn and WJm,,are used because the actual values of W, and W, at any point m the network are not known before the MILP 1s solved Therefore mequahtles (13a, b) are only a sufficient condltlon to prevent stream splits Note that if wfm. 2 wJm..y then AT,,, constraint (13a) IS required only at the hot end of any exchanger matching streams z and J, If W,mx< W,,., then AT,,, constraint (13b) 1s required only at the cold end of any exchanger matching streams z and J, otherwise, AT, constraints (13a, b) are required on both ends of each exchanger
591
Analysis of reslllent heat-exchanger networks--II TI
k-l
TI k
R/,*-I
R
,,*-2
Hot streom
RLk
I
streams
Fig 3 Other matches with streams I or J occur between matches r/,_ , and uk If R, e _ , < qqk,then match ZJ~can nof be combined with match ~/k_, 3 3 Constramts on the Number of Exchangers Consider a match between streams I and J which IS repeated m TIs k - 1 and k-call these match ZJ~_ , and match ZJ~,respectively If no matches with either stream I or] occur between matches IJ~_ , and ZJ~(Fig 3). then the two matches can be combined to form one heat exchanger If any other matches with either stream I or J occur between matches ‘Jo_ , and IJ~,then It IS necessary to “shift” match ~/k “through” these other matches before match ZJ~can be combined with match ZJ~_, to form one exchanger Lmnhoff and Flower [14] provide rules when shlftmg 1s possible If It 1s not possible, then matches r/, _ , and ZJ~give nse to two exchangers which cannot be combined Along these ideas constraints can be set up to limit the number of matches which cannot be combined and thus restrlctmg the number of exchangers For slmphclty, combmatlon posslblhtles are checked only m adjacent TIs A match between streams z and J m TI k gives nse to a separate exchanger if the following logical expression is true A and [B or (C and D)],
(14)
where A is true if streams z and J are matched m TI k, B is true If streams I and J are not matched m TI k - 1, C 1s true if other matches with either stream I or J occur between matches ZJ~ _ , and ZJ~(Fig 3), D 1s true 1s residual R,k < qyk, the heat exchanged m match ZJ~
matches between ZJ~ _ , and IJ~ (condltlon C), which cannot be done because hot stream I has msufficlent residual at the mlet of TI k (condltlon D) Therefore a new exchanger 1s necessary (Fig 3) Integer variables are defined to indicate when condltlons A, B, C and D are satisfied (A) Integer variable yllk mdlcates If a match exists between streams z and J m TI k The constraint which defines yVk1s u,
Y,/k
-
q,,k
2
(15)
O,
where U, was defined earher as an upper bound on the amount of heat which can be transferred from Stream 2 t0 StKZUn J Whenever qrlk> 0, y,,k 1s equal to 1, otherwise yllk = 0 (B) Slmllar to the constraint correspondmg to condltlon A, mteger vanable y, k_, mdlcates If a match exists between streams z and J m TI k - 1 If such a match exists, then yYk_ , = 1, otherwise, YI/k-i=O
(C) Integer vanable plrk (defined by the following constraint) mdlcates when any other matches with either stream I or J occur between matches Ilk-1 and vk Fig 3) KP,,k-
2
Y,,
I”=,+1
k-
I -
‘i’ , =I
Yf,k
k-l
-
1-l
NC
-
1 f=j+l
Y,/
1
yy”k>o,
(16)
,=I
where K IS a sufficiently large positive number Whenever other matches occur between matches IJ~_ , and IJ~,P,,~= 1, otherwlse P,,~= 0 Note that If condltlon B 1s satisfied, then a new exchanger this constramt assumes that the streams are must be placed between streams I and J m TI k matched m order of mcreasmg stream numbers because no match ZJ_ , exists with which match IJ~ can be combined If both condltlons C and D are (D) Integer vanables sVkare mtroduced to indicate If residual R, k_, IS sufficient for match qk to be satisfied, match IJ~must be shlfted through the other
A K
598
SABOO et al
Table I SIX and CPU time reamred by the MlLP for the case studv m Part I 171 Stream-spht constramts for stream 7 m four
No addmonal constramts Number Number Number
a7 132 13
of rows of variables of Integer variables
Exchanger constramts
Both stream-spht and exchanger constramts
TIS
for stredm I
107 148 29
163 207 88
183 223 104 51 IO1
CPU requzremenr (s)
First Integer solution Fmal solution
4
9
34
5
13
49
shifted mto TI k - 1 (17)
Whenever the residual IS msuffiaent, wise sgk= 0 The followmg constraints expression (14)
sYk= 1, other-
are equivalent to logcal
+tl,-tl8=0, 4’~ G
Ke,
KU
-
UW U8b).
and ‘IS G
eukh
(184
where K 1s a sufficiently large number Integer vanable e,k equals 1 when logcal expression (14) is true (I e when match & cannot be combined with match I/k- I), e,,k= 0 when expression (14) is false The translation of logical expression (14) into constraints (18a-c) 1s explained m the Appendix An upper bound on the number of exchangers N, on a particular hot stream I can he imposed by (19) If there are no stream splits on stream I then the bound (19), together with constraints (15)-(18c), 1s suficlent to hmlt to N, the number of exchangers on stream z It 1s only sufficient because combmatlons among adJacent TIs alone are tested and because m a particular TI the matches are assumed to occur m a certain order This implies that constramts (15)-(19) can cause the MILP not to find a feasible HEN although a feasible solution with N, exchangers exists Just like the split constraints, the exchanger constraints should be used with caution and only after an unconstrained solution has been found to have an excessive number of exchangers on a particular set of streams Table 1 shows a comparison of the CPU requlrement for the MILP both with and wlthout the additional constraints for the seven-stream problem m the case study m Part I [7] The exchanger constraints increase the size of the problem considerably, m particular, the number of integer variables mcreases This IS reflected m the IO-fold increase in CPU time The stream-split constramts, however, do not increase the CPU reqmrement slgmficantly
An approximate method which tends to reduce the number of exchangers m the MILP solution while requiring much fewer variables IS to use only the constraints
(15) and (19) together
with
m selected TIs on specific stream pairs To reduce the CPU time requirements the MILP package LINDO [15] mcorporated m RESHEX was modified The solution of the MILP IS a two-step procedure (1) LP solution, (n) Branch-and-bound integer solution
technique
to determine
an
Most of the CPU time 1s consumed by Step (11) It was observed that the mltlal feasible integer solution 1s usually reached relatively quickly The search to improve it requires the bulk of the total CPU time, but often simply confirms the first integer solution to be the best This 1s verified by comparing the CPU requirements m Table 1 for the case study m Part I [7] Therefore, RESHEX displays the network generated from the first integer solution and then e;lves the user the option
to search for a better
solutions
Acknowledgement-Partial financial support from Department of Energy IS gratefully acknowledged
the
REFERENCES
B Lmnhoff and J R Flower, Synthesis of heat exchange networks I Systematic generatron of energy optimal networks AIChE JI 24, 663 (1978) J Cerda and A W Westerberg, Mmlmum utility usage m heat exchanger network synthesis-a transportation problem Chem Engng Scr 38, 373 (1983) A K Saboo and M Moran, Design of resilient processmg plants-IV Some new results on heat exchanger network synthesis Chem Engng Scl 39, 579 (1984) A K Saboo, M Moran and R D Colberg, Reslhence analysis of heat exchanger networks-I Temperaturedependent heat capacltles Comput them Engng (in press) A K Saboo, M Morarl and R D Colberg, Resilience analysis of heat-exchanger networks-II Stream sphts and flowrate varlatlons Comput them Engng (m press) A K Saboo, M Moran and D C Woodcock, Design of resihent processmg plants-VIII A Reslhence Index for heat exchanger networks Chem Engng SCI 40,1553 (1985) A K Saboo, M Morarl and R D Colberg, RESHEX an mteractlve software package for the synthesis and
Analysis of resilient heat-exchanger networks-II
8 9 10 11 12 13 14
15
analysis of reslhent heat-exchanger networks-I Program descrlptlon and apphcatlon Comput them Engng 10, 577 (1986) S Raghavan, Heat exchanger network synthesis A thermodynamic approach Ph D Thesis, Purdue Umv , West Lafayette, Ind (1977) T Umeda, T Harada and K Shlroko, A thermodynamic approach to the synthesis m chemical processes Comput them Engng 3, 373 (1979) D W Townsend and B LmnholT, Surface area targets for heat exchanger networks Paper presented at Inst them Engrs A Res Mtg, Bath, Avon (Apnl 1984) S A Papouhas and I E Grossmann, Structural opt)mlzatlon approach m process synthesis Part II heat recovery n&works Coiput che& Engng 7,707 (1983) B Lmnhoff and E Hmdmarsh. Pmch desuzn method for heat exchanger networks &em Engng &CI 38, 745 (1983) C A Floudas, A R Clrlc and I E Grossmann, Automatic synthesis of optimum heat-exchanger network configuratlons Chem Engng Commun (m press) B Lmnhoff and J R Flower, Synthesis of heat exchange networks II Evolutlonar; generation of networks with various crltena of optlmahty AIChE JI 24, 642 (1978) L Schrage, User’s Manual Lmear, Integer and Quadratrc Programmrng with LINDO Scientific Press, Palo Alto, Calrf (1984)
599
where condltlon B IS true If (1 - y,, Ir_, ) = 1 and condltlon E 1s true If m, = 1 (The “strange” Integer eqmvalent for condltlon B results from the fact that condltlon B 1s defined to be true when streams I andj are not matched m TI k - 1 ) An integer constramt correspondmg to expresslon (A 3) IS 05-(1-y,,-,)-m,+rl,--rl,=0,
(A 4)
where ‘I~Q Km, and q4
ml
0 0 I I
4
0 I 0 I
0 I I 1
ISthe same as the “truth table” for the loglcal OR expression (A 3) Next, the lolpcal expressions (A 1) and (A 3) can be combined to form F = B or (C and D) An integer constraint correspondmg
15--[2(1 -~,,-,)+@,,+s~)l+tls-16=0. APPENDIX
646)
where
This Appendix describes the translation of logical expressions mto Integer constraints, m particular, the translatlon of logcal expression (14) mto constraints (18ax) is explamed First, consider the logcal AND expression E=Cand
(A 5)
to expresslon (A 5) 1s
D,
(A 1)
where condltlon C IS true If P,,~= 1 and condltlon D IS true if s,,~= 1 (Conversely, condltlon C IS false If P,,~= 0 and condltlon D 1s false If sVk= 0 ) An integer constramt corresponding to expresslon (A 1) 1s I5-P,,-s,,+‘I,-%=O1
(A 2)
where
and t16d K(1 -m,) It IS easy to show that the “truth table” for Integer variable m, m constraint (A 6) 1s the same as the “truth table” for condltlon F m logical expresslon (A 5) In constraint (A 6), the quantity (1 -y,, k_ ,) IS multlphed by 2 so that It “balances” agamst the “double quantity” (p,,k+ @-Just hke the logical condltlon B m expressIon (A 5) must “balance” against the “double condltlon” (C and D) The number 1 5 m constramt (A 6) 1s chosen to make the “truth table” correct Finally, for logcal expression (14), G = A and [B or (C and D)],
and
the correspondmg lated and K is a sufficiently large number The followmg “truth table” can be tabulated for constramt (A 2) p*
$!i
ml
0 0 I I
0 I 0 I
0 0 0 I
integer constraint (18a) can be formu-
55-{4~,,+[2(1--~,-,)+~,,~+~~)1} +v7--q8=0,
(18a)
where and
This IS the same as the “truth table” for the logical AND expresslon (A 1) Now consider the logical OR expression F = B or E,
(14)
(A 3)
‘Is c K(1 -
e,,k)
Agam, It IS straightforward to show that the “truth table” for the integer variable e,, m constramt (18a) IS the same as the “truth table” for condltlon G m logcal expression (14)