Carnegie elln University Research Shwcase @ CU Department f Chemical Engineering Carnegie nstitute f Technlgy 98 inimum utility usage in heat exchanger netwrk synthesis : a transprtatin prblem Cerda Carnegie elln University Arthur W. Westerberg David asn Bd Linhff Fllw this and additinal wrks at: http://repsitry.cmu.edu/cheme This Technical Reprt is brught t yu fr free and pen access by the Carnegie nstitute f Technlgy at Research Shwcase @ CU. t has been accepted fr inclusin in Department f Chemical Engineering by an authrized administratr f Research Shwcase @ CU. Fr mre infrmatin, please cntact research-shwcase@andrew.cmu.edu.
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NU UTLTY USAGE N HEAT EXCHANGER NETWORK SYNTHESS - A TRANSPORTATON PROBLE by Cerda, Arthur W. Westerberg, David asn and Bd Linhf f DRC-6-25-8 September 98
NU UTLTY USAGE N HEAT EXCHANGER NETWORK SYNTHESS A TRANSPORTATON PROBLE by * Jaime Cerda Arthur W. Westerberg Carnegie-elln University Pittsburgh, PA 523 and David asn Bd Linnhff C Crprate Lab Runcrn, England This wrk spnsred in part by NSF Grant N. CPE-7889 * Current Address: NTEC Guemes 345 3 Santa Fe Argentina
ABSTRACT This paper frmulates the minimum utility calculatin fr a heat exchanger netwrk synthesis, prblem as a "transprtatin prblem" frm linear prgramming, thus allwing ne t develp an effective interactive cmputing aid fr this prblem. The apprach is t linearize cling/heating curves and partitin the prblem nly at ptential pinch pints. Thus frmulated bth thermdynamic and user impsed cnstraints are readily included, the latter permitting selected stream/stream matches t be disallwed in ttal r in part. By altering the frmulatin f the bjective functin, the paper als shws hw t slve a minimum utility cst prblem, where each utility is available at a single temperature level. A simple ne dimensinal search prcedure may be required t handle each utility which passes thrugh a temperature change when being used. Extending the partitining prcedure permits the frmulatin t accmmdate match dependent apprach temperatures, an extensin needed when indirect heat transfer thrugh a third fluid nly is allwed fr sme matches. < UNiVgRSlT PiJic&URGR PENNSYLVANA S
ntrductin Tw independently written manuscripts (Cerda and Westerberg (979) and asn and Linnhff (98)) were merged and significantly extended t prduce this paper. Bth had discvered the "transprtatin mdel" fr the minimum utility calculatin fr the heat exchanger netwrk synthesis prblem. n the last 3 years many papers have appeared which deal with the synthesis f cst effective heat exchanger netwrks t integrate chemical prcesses thermally. n the recent prcess synthesis review paper f Nishida et al (98) 2% f the 9 papers listed are n this tpic alne. As pinted ut in that and ther earlier papers, a mst significant cntributin f this entire wrk is the insight by Hhmann (97) and later by Linnhff and Flwer (978) which permits ne t establish the thermdynamic limit fr minimum required utilities t accmplish all the specified heating and cling fr such a prblem. This thermdynamic limit invlves lcating "pinch" pints within such netwrks where a minimum apprach temperature exists. This minimum utility limit is almst always attained by the better netwrk designs fund fr such prblems and thus is a very wrthwhile target. Unfrtunately, industry has typically implemented slutins using substantially mre than the minimum required utilities ften 3% r mre in excess (Linhff and Turner (98)). n this paper we shw hw t frmulate the minimum utility calculatin as a classical "transprtatin prblem" frm linear prgramming, a prblem fr which very efficient slutin algrithms exist. The apprach is t linearize heating and cling curves t any desired degree f accuracy. We will argue that nly crner pints and end pints can be ptential temperature "pinch pints. The temperatures f these pints
partitin the streams int substreams fr which ne can readily write the requisite thermdynamic cnstraints. Extending insights by Grimes (98) and Cerda (98), we shw that many. ften half r mre f the pints can be eliminated as pinch pint candidates, substantially reducing the size f the transprtatin prblem which must be slved. The designer frequently wishes t preclude matches being allwed between certain streams, and it wuld be useful fr him t discver if these cnstraints seriusly affect the minimum utility requirements fr a prcess. The. transprtatin prblem frmulatin readily accmmdates such cnstraints. The designer may have several utilities available at different temperature levels and csts. Simple adjustment f the csts used in the bjective functin and sme minr added partitining permit ne t find a slutin having a minimum ttal utility cst. We als shw that each utility which is nt available at a cnstant temperature level may require an added ne dimensinal search. Lastly we shw hw t generalize the temperature partitining task if ne wishes t assign a different minimum allwed apprach temperature t each stream/stream match. Limiting the transfer f heat between any tw streams t indirect transfer thrugh a third fluid requires this type f calculatin. The number f partitins can grw enrmusly. f the partitining is nt dne cmpletely, the calculatin will yield an upper bund (and prbably a gd ne) t the required minimum utilities. The paper gives an effective algrithm t find a first, and ften ptimal, slutin t the transprtatin prblem, ne which can be implemented by hand if desired. t als describes the classical transprtatin algrithm by Dantzig (963), principally t shw where in the slutin "tableau" ne discvers the thermdynamic pinch pint(s) fr all the prblems described abve.
The first tw authrs extend the use f transprtatin like mdels t aid in synthesizing minimum utility/minimum match netwrks in parts 2 and 3 f this paper. i Prblem Definitin We are given a set f ht and cld prcess streams amng which we wish t exchange heat t bring each frm its inlet t its target temperature. n general additinal heating and cling in the frm f utilities are needed t accmplish this task. Since the utilities used are cstly, we wish t calculate the least amunt needed which can then serve as a target t the design f a heat exchanger netwrk t accmplish ur task. We assume sufficient infrmatin is given fr each stream t allw us t calculate a heating r cling curve fr it as it passes thrugh the exchanger netwrk. We are given inlet and utlet temperatures; we must guess the likely pressure trajectry. Then we calculate enthalpy alng this trajectry, pltting T (rdinate) versus enthalpy flw (flw rate times specific enthalpy, abscissa). Als given fr the prblem is a minimum AT driving frce AT. t be allwed in any heat exchange. Example Prblem We shall illustrate the ideas thrughut this paper with the example fur stream prblem whse data are given in Table. Figure shws the cling and heating curves fr each f these streams.
Cld T interval Apparent S F C P Q - F C AT P Stream, c. Flw - 2-4 4-8 8-9 9-2 2-25.. 5. 2 phase 4. J regin.5 2. 2.2. 8.. 8 88 8 5 Ttal 398 Cld Stream, c. 4-8 8-225.3.5 3.9 4.5 56 22.5 358.5 Flw 3 Ht Stream, h. Flw - 3-2 2 + -2" 2"-4.6.6 * (phase change).2.2-6 - -72 232 Ht Stream, h 28-.8 3.2-576 Flw - 4 Table. Data fr 4 Stream Example Prblem. A T. is 2 fr the prblem.
35 T 3-25 2 5 C 2 2 4 6 8 H Cling Curves fr Streams h. and h and c fr Example Prblem. Figure and Heating Crves fr Streams c
Slutin Hhmann (97) presented a straightfrward methd t slve the minimum utility prblem. He develped tw curves ne the "super cling curve" frmed by merging the curves fr all the ht prcess streams and ne the "super heating curve" which merges the curves fr all the cld prcess streams. On a T versus enthalpy flw diagram, these curves can be mved arbitrarily t the right r left and thus placed s the super cling curve is belw the super heating curve. The cling curve is mved tward the heating curve until there is a minimum vertical distance ccurring between the curves which equals the minimum allwed AT driving frce the designer will permit in any heat exchanger. Figure 2 illustrates fr ur example prblem with AT. = 2. This pint f rainimin mum AT is termed a "pinch pint" fr the prblem. By cnstructin the curves are in exact heat balance where they are vertically abve and belw each ther. f these super streams existed and were placed in a cuntercurrent heat exchanger, the temperatures f each side wuld fllw the ppsing trajectries shwn. The pinch pint precludes further exchange. The heating f the cld streams yet t be dne, if any, represents the minimum ht utilities needed and the cling f the ht streams yet t be dne, minimum cld utilities. Bth are identified in Figure 2. Linnhff and Flwer (978) nte that n heat can pass acrss the pinch fr a minimum utility slutin. One can prve this bservatin easily by examining Figure 2. Suppse ne attempted t use heat frm the merged ht prcess stream abve the pinch t heat the merged cld stream belw the pinch. Such a mve wuld bring the merged cld, stream belw the pinch clser t the ht at the pinch, causing ne t have t mve the cld stream t the left t regain AT. in in as the driving frce at the pinch.
35-3- (3,88) 25 (25,924.5) NU COOLNG UTLTY =68 j(225,899.3) 2 5 (4,28) NU HEATNG UTLTY =6.5 Units (,) (,68) 2 <j 6 8 H Figure 2 erged Heating and Cling Super Curves fr Example Prblem.
8 By mving the streams in this manner relative t each ther, ne must be increasing the requirement fr utilities. We wish t autmate and generalize the Hhmann prcedure. Using their prblem table frmulatin, Linnhff and Flwer (978) shw hw t slve the minimum utility prblem if each stream is represented by segments f cnstant heat capacity* versus temperature. We take their ideas as ur starting pint, describing the task t be accmplished frm a smewhat different viewpint. This viewpint will give us significant prblem reductin insights. We t shall assume that the cling curve fr each stream can be apprximated by straight line segments. This assumptin is actually very realistic and can always be made in a safe manner by linearizing belw the curve fr ht streams and abve fr cld streams. Keeping the linearized curves at least AT. apart will guarantee" the actual streams are that min far apart. st streams, even thse underging phase change, require nly a few segments t apprximate their heating r cling curves reasnably. C/ineji Pint* and Pinch f the streams are all linearized as described, then the super curves f Hhmann are als built up f straight line segments as we see in Figure 2. Our gal will be t lcate the pinch pint fr any given prblem. Clearly we can state the fllwing: ) if it exists the pinch pint ccurs at a "crner" pint fr either f the tw merged super curves, 2) nt all crner pints can be pinch pints. Crner pints are where the super curves change slpe. Clearly nly a crner pint where ne curve appraches and then breaks away frm the ther curve can be a pinch pint candidate. We can write the fllwing relatinships t test a crner pint t see if it is a candidate pinch pint.
Cld Curve Crner Pint j Candidate nly if Ht Curve Crner Pint Candidate nly if Y (FC ) < Y (FC ) (2) La P i * * Pi where sets +., ~. are the cld streams cntributing t the merged heating curve just abve and belw crner pint j, respectively, and sets *. and ~. are similarly defined fr the merged ht cling curve at crner pint. The abve tests are generalizatins f an bservatin by Grimes (98), where he ntes that if all streams are represented as single straight lines, then nly stream inlet temperatures need be cnsidered t slve the minimum utility prblem. Fr this case crner pints alng a merged super curve will nly ccur where streams enter r leave the curve. Where a stream enters, the abve tests will keep that temperature as a candidate pinch pint; where it leaves, the pint will be rejected. Cerda (98) ntes that n temperature need be cnsidered if it is ut f range, i.e. if it is alng the merged stream and is mre than AT. mm abve r belw any f the temperatures spanned by the ther. We can use this test t reject crner pints as candidate pinch pints als. These tw rejectin tests will frequently eliminate abut half f the crner pints, which, as we shall see, will reduce ur prblem size t abut 25% f its apparent riginal size, a significant reductin.
T Dispsitin Ht 3 Reject. T ht. (alia Cerda) 28 Reject. T ht. 2 + 2" 4 3.8 4.4 3.2 " 4.4 3.2 Keep. Reject. Reject. Reject. Cld 4 8 9 2. 6. 4.5 2. 6. 4.5 2.5 Keep. Keep. Keep. Reject. 2 225 25 2.5 5.5. 5.5. Reject. Reject. Reject. Table 2. Crner Pints fr Super Curves in Figure 2 and their Dispsitin as Candidate Pinch Pints. Table 2 lists all crner pints fr ur example prblem and whether they need be accepted r can be rejected as candidate pinch pints. Nte nly ne ht and three cld crner pints ut f 3 ttal need be kept.
The prblem can nw be partitined at the candidate pinch pint temperatures. The ht candidate pints are first prjected nt the cld super stream and vice versa. As nted by Linnhff and Flwer (978), this prjectin is ffset by AT, thus the ht candidate pinch pints prject dwn AT. nt the cld stream and the cld prject up AT. nt the ht stream. Table 3 lists the ht stream and cld stream intervals created by this partitining. erval Ht Stream Cld Streams 2 3 4 j» 2 6 2 + t 2 t 6 t 2* t 4 8 t t 4 t 8 t Table 3. Temperature ntervals Created by Partitining at Candidate Finch Pints. AT. a 2. Temperatures nt underlined are caused by prjectin ram ther stream. Nte we prject the cld stream candidate pinch pint at nt the ht stream at 2, the 4 nt the ht at 6 and s frth. We nw shw that this partitining is dne as described t permit us t write thermdynamic cnstraints fr ur prblem. We nte that heat can be exchanged amng and within the intervals as fllws. ) Ht interval is abve (htter than) the cld interval - Heat can always be transferred frm a ht stream at a htter interval t a cld stream at a lwer ne. Fr example, heat in interval 4 fr a ht stream can always transfer t interval 3 r belw fr the cld stream.
2 2) Ht interval is belw (clder than) cld interval N heat can transfer frm the ht interval t the cld ne because the ht interval is everywhere t cld, except fr perhaps the httest pint which, after remval f an infinitesimal amunt f heat is mre than AT min clder than «v^*y temperature fr the cld interval. Fr example heat in ht interval 3 cannt transfer t cld interval 4. 3) Ht interval is the same as the cld interval Heat can always be transferred between the merged streams within the same interval t the extent it is available as needed, i.e. q in (heat available, heat needed) fr the interval with equality always pssible. slate the interval and mve the cld super stream t be belw the ht until it pinches. Frm the manner in which the intervals are defined, the ht end r the cld end f the interval must be pinched. At the pinch end, bth curves are vertically aligned i.e. bth start tgether at the pinch. ving away frm the pinch, the curves are in heat balance vertically and everywhere at least AT. apart. Thus ne can transfer heat until ne r the ther f the tw curves is satisfied. QED. TnxuiApnjtatLn 9n.bJLem F/unuUjutLri We can nw mdel the minimum utility calculatin as fllws. Let c. be cld stream i in interval k and h. a be ht stream j in interval i. Define a., as the heat needed by c.,, which can be readily calculated after partitining. Fr example the heat needed by cld stream c in interval 3 (4 t 8 ) is a 3 = 88 units (see Table ). Similarly define b as the heat available frm stream h. fl. Lfet q., be the heat
3 transferred frm h.. t c... The q, are t be calculated. Assume there are L intervals (equals 4 fr ur example prblem - see Table 3). Let there be C-l cld prcess streams and H-l ht prcess streams in ur prblem. Then the cld utility will be the C cld stream and the ht, the H ht stream. Assume the heat needed by the cld utility is at the lwest level in the prblem. Als assume it is in sufficient quantity t satisfy all the ht prcess stream cling needs, i.e. we require H-l L j- X-l Assume similarly that the ht utility is available at the highest level and is in sufficient quantity t satisfy by itself all the cld stream heating requirements. C-l L SL' X X'l i-l k-l \k Lastly assume the prblem is in heat balance verall. C-l L H-l L *~* + ) / *n. " * m + ) / b, (5) Cl ««ik HL fcj " JJ» i-l k-l j- X-l The abve simply say, chse bth a^. and b-.. t be large numbers. Then adjust them s the entire prblem is heat balanced. We can nw write ur transprtatin mdel fr the minimum utility prblem as fllws.
4 C L H L Subject t H L k-l t 2,--- f L (7) C L fr and where fr i and j are bth prcess streams and match is allwed, i.e. k ^ X # fr i and j are bth utility streams ( - C 9 J -H). nly i r nly j is a utility stream () therwise, where is a very large (think infinity) number. Equatin (7) says that the heat required by cld stream i in interval k must be satisfied by transferring heat frm smewhere amng the ht streams. Equatin (8) is a similar statement fr ht stream j in interval it must give lip its heat smewhere t ther streams. (9) says all heats transferred must be nnnegative, that is n heat can flw frm a cld stream t a ht ne. (6) is the bjective functin t be minimized, with cst cefficients defined by (). N cst is assciated with an allwed prcess stream - prcess stream match r frm the ht utility t
5 the cld utility (this latter match wuld never be implemented in a netwrk). Utility-prcess stream matches are given a nminal cst per unit f heat in the match s they will be used nly if the free matches d nt slve the prblem. Thermdynamical ly disallwed matches are given a near infinite cst t preclude their being part f any ptimal slutin. The abve is a classical transprtatin prblem fr which a very efficient slutin algrithm exists (see Dantzig (963) fr example). t is usually visualized by setting up a "tableau", as illustrated in Figure 3 fr ur example prblem. The clumns are fr the ht sub streams and the rws fr the cld substreams. Each entry is a "cell" which can cntain 3 numbers. The upper right is the cst cefficient, C,. The bttm number is the assigned <l-^ f r K, J fs tke match; the upper left we will discuss mmentarily. Fr each K, J* rw a,, ik is given t the far left and fr each clumn b j * t the very tp. We place the ht utility clumn (labeled H) t the far right and the cld utility clumn (labeled C) t the bttm. Cells have been marked "" if they are thermdynamically infeasible, i.e. if k > fr entry q..... K, jt Thz JnJutiaL SJjJutJjn The transprtatin prblem algrithm requires an initial feasible slutin. f we are careful, this initial slutin is frequently already ptimal. A rw and clumn rerdering algrithm has prved very effective t help get a gd initial slutin. Simply rerder all prcess stream rws such that the number f infeasible cells decreases frm tp t bttm and all prcess stream clumns such that they decrease frm right t left. Fr ties, place the higher temperature cells tward the tp and t the left. Figure 3 is rdered in that manner. f nly thermdynamic cnstraints are invlved, tie breaking is unnecessary.
a ik 23 22.5 c C 4 24 '3 56 23 8,5.5 V 8 4 inn - 6 h 4 6 h " h 256 h 24 7» 86 " ' 2 + 6 2-48 h 3 88 6 28 h 23 96 32 24 h 2 24 h 28 h 22 24 4 6/ \ h 2 64, H 6.5 9883.5 p ik -2-2 -2 - Pinch Ov 2 2 2 2 2 Pinch FGURE 3 Transprtatin Prblem Tableau fr Example Prblem. Tableau shws nitial feasible (and ptimal) slutin.
7 Once rerdered, we apply the fllwing slightly mdified "Nrthwest Algrithm t get ur initial feasible slutin.. Start in the upper left (nrthwest) crner. 2. Hve frm left t right in the uppermst rw t the first clumn having a cst less than, finding the cell crrespnding t rw c.,, clumn h. J * 3. Assign q ik.^ in(a ik> b ) t the cell. 4. Decrement bth a., and b by ' (?-i c -i«# 5. Crss ut the rw r clumn which has its heating r cling requirement a., r b reduced t zer. 6. Repeat frm step 2 until all rws and clumns are deleted. n Figure 3, we start with rw c., and clumn h.,. We assign q., = 6 = in(23o> 6) t the cell and crss ut clumn h ^. a., is nw equal t 7(= 23-6). Starting again at step 2, we identify rw c., again and clumn h~,. We assign 7 units t this cell, crss ut rw c., and reduce b~, t 86. The rest f the tableau is filled ut the same way. Nte rw 2 has t g all the way t the ht utility t cmplete its need fr heat. f nly thermdynamic cnstraints are invlved and if AT. is the same fr all matches, then ne can readily demnstrate the abve is repeating the same calculatins needed fr the prblem table f Linnhff and Flwer (978). Thus the initial slutin is always ptimal fr such a prblem. We can read ff the minimum utility requirements as 6.5 units f heating and 4 + 64 = 68 units f cling, which agrees with the Hhmann calculatin we did in Figure 2. The 9883.5 units f heating by the ht utility and assigned t the cld utility is a "dummy number and is ignred.
8 T lcate the pinch mst easily, we shuld first discuss hw t slve a transprtatin prblem, which we shall d mmentarily. We might nte the reductin f the prblem size resulting frm nly including the temperatures which are ptential pinch pints when partitining. The partitining f Linnhff* and Flwer (978) wuld have included every crner pint in the prblem, i.e. ht temperatures 3, 28, 2, 4 and and cld temperatures,, 4, 8, 9, 2, 225 and 25. The cmbined set f ht temperatures (after prjecting the cld nt the ht) gives the fllwing list:, 2, 4, 6, 2", 2*, 2, 22, 245, 27, 28 and 3. A crrespnding list 2 clder exists fr the cld streams. Fr ur example prblem we wuld create a tableau having 3 cld substreams plus the cld utility and 8 ht subst reams plus the ht utility t give a tableau with 4 x 9 ss 266 cells versus (see Figure 3) a tableau with 48 cells. Here the reduced prblem is nly 8% the size f the full ne. As we shall see a calculatin is needed fr every cell if we need t check fr ptimality s the reductin is real in terms f wrk required fr slving. Nn Thzxmdynamlc With a mathematical frmulatin fr the minimum utility prblem, we can add certain types f cnstraints trivially. One can readily. add cnstraints t preclude the exchange f heat between selected prcess streams, either in part r ttally. Fr example a match may be undesirable because the tw streams wuld be unsafe if mixed accidentally because f a leak in an exchanger. Other reasns fr rejecting a match are that the streams may be physically t far apart and bth vapr, thus requiring expensive piping t get them tgether, r the exchange may be a prblem fr cntrl r startup.
9 The engineer culd first slve the minimum utility prblem with nly thermdynamic cnstraints. He culd then selectively preclude matches r part matches and discver the impact, n the minimum utilities required. f the impact is t high, he can recnsider the validity f the cnstraint. T add user impsed cnstraints, we' repeat the same prcedure we used earlier. The difference is that we can nly merge ht r cld streams ver the temperature ranges where they are treated identically. Als the initializatin algrithm is n lnger guaranteed t yield an ptimal slutin. We illustrate these ideas by example. We shall slve ur example again but this time disallwing heat exchange between c. and tu abve the bubble pint (8 ) f c.. T be safe we disallw any exchange abve 75. We nw must treat c. and c~ differently (and thus unmerged) abve 75. The crner pints are fund fr c. and c~ merged up t 75 then fund individually fr c^ and c~ abve that pint. Als we must treat h. and h- differently here we culd limit this different treatment t abve 95 The resulting candidate pinch pints will be fund t be: cld, 4, 75 and 8 and ht 2 +, and 95. Prjecting the temperatures gives the final ht stream partitining temperatures f -, 2, 6, 95, 2 +, and Cld stream partitining temperatures are 2 clder. Figure 4 is the slutin tableau fr ur prblem, shwing the first feasible slutin fund by using the mdified Nrthwest Algrithm. Three cells are disallwed ver thse nt permitted because f thermdynamics, and they are marked with a "D" and given a cst f "". f this slutin is ptimal, and we shall see in a mment that it is, then minimum ht utilities are increased frm 6.5 t 7 (by 53.5 units). Cld utilities, by heat balance, must als increase by 53.5 units, which they d. Thus the restrictin causes a 37.6% increase in ttal utilities used. One can nw ask if it is wrth that increase.
a ik 23 22.5 9.5 77 36.5 8 5.5 Cld r C 5 c 25 4 24 c 7 3 c 23 2 C Y V" ^Ht it 8 75 4-2 -2-2 -2-2 -2-6 h 5 6 rr 2 256 h 25 D 22.5 *"" D 9.5 34 2 v 6 h 4 2 " 43 + 52 2 2 6 D 6 2 95 6 2 42 h 3 42 2 + 2 6 2 h 23 26.5 8 5.5 2 2 24 24 2 2 2 28 h 22 2 2 6 64 h 2 64 2 -OQ + 7( ) - - - -T - - H 983 p ik Pinrh -2-2 -2-2 -2-2 - FGURE 4 Transprtatin Prblem Tableau fr Example Prblem where N Heat Can Be Exchanged between c- and h- abve 75
2 We need t decide if the slutin is ptimal. T d s we give the steps fr slving a transprtatin prblem withut justificatin. The algrithm will be seen t be very simple, and we shall shw hw t find the pinch pints in the result. T slve a transprtatin prblem, given a first feasible slutin, prceed as fllws.. We must first establish fr each rw a "rw cst, P.j* and fr each clumn a "clumn cst", Y,.«We shw rw and clumn csts alng the right side and bttm f the tableau. Start with the tp rw and assign it a rw cst f zer. (We set y.- t zer.) 2. Fr any rw c, fr which a rw cst is already assigned, find an active cell (q. u.. > ) in that rw. Assign a clumn cst Y.. fr the clumn crrespnding t the active cell, such that (Set Y 5 t s + =.) 3. Repeat step 2 fr assigned clumns t set rw csts. 4. Repeat steps 2 and 3 as needed until all rw and clumn csts are set. (Set Y H t, set P C t -, set Y 23 t 2, etc.) Rw and clumn csts resulting using this algrithm are shwn in Figure 7. Cntinue as fllws. 5. Fr every cell (r at least every inactive cell) write - P ik int the upper left crner f the cell
22 6. f n cell exists where > C., exit. The current tableau is ptimal. Otherwise cntinue. Fr ur example, the tableau is fund t be ptimal. The steps needed if nt ptimal are as fllws. 7. Fr any cell with., > C, find a lp f active cells which this cell cmpletes by mving alternatively dwn rws and acrss clumns. Such a lp will exist. (Pretend cell (c 2,, h..) is a candidate cell. A lp wuld traverse the cells (clckwise) (c 2 4> h 5 )t (c 5, ^5 )> ( i5» H^» ^c» H^» ( > h 23^' 8. ark the first cell (i.e. cell (c 24, h 5 )) with a "+", the secnd cell with a "-", the third with a "+", alternating "+" with "-" arund the lp. Nte ne must have an even number f unique cells in such a lp s, when we reencunter the first cell, it will again be marked with a lf +". 9. Find q. f.. f minimum value assciated with a "- cell. Call it q.. ik,jt Tnin (Fr ur example q. = in(6, 983, 26.5, 43, 9.5) = 9.5.). Add q. t all "+" cells and subtract it frm all lf - tf cells. Ding this step assumes each rw and clumn remains in heat balance, that ur initially inactive cell is nw active and anther cell (the ne riginally set at q. ) i s nw inactive breaking the lp. We add 9.5 t all the "+" cells and subtract it frm all "-" cells. Cell (c^,, h«.) becmes inactive.
23 We wuld nw have a new and better slutin t ur prblem (if we had had t cntinue past step 7). Repeat frm step, establishing rw and clumn csts again, etc. J<ien*jJ!ying, the. Pinch Pint The rw and clumn csts identify the pinch pints fr ur prblem. f rw cst p., is different frm p.. fr stream i then the minimum ik i K+ utility prblem pinches at the temperature which partitins the prblem between cld intervals c. and c.. «Similarly we can spt the pinch pints by lking at the clumn csts, Y... Fr the prblem in Figure 4, the pinch pints are between c 5/ c 4 (i.e. at cld stream temperature 8 ) The change frm t 2 in., fr hh/h ic./h iy gives the same result a pinch at ht temperature 2 +. The prf fllws frm bserving as we did earlier that n heat crsses the pinch pint. All C.., are zer fr active matches amng prcess streams s where ne is zigzagging back and frth amng ht and cld substreams, the crrespnding p., and y becme the negative f ne anther and d nt change value. The pattern is brken at the pinch pint. One cannt carry the value f a rw r clumn cst directly acrss the pinch because n heat crsses the pinch. The rw and clumn csts n the ther side f the pinch pint must be generated by first passing thrugh the cell in the lwer right belnging t the interchange f heat between the ht and cld utilities. One then sets these rw and clumn csts by zigzagging back up t cells just belw the pinch. Passing thrugh this zer cst cell changes the p., and y by the sum f the csts assigned t the utility/prcess stream matches (here +=2). The rw and clumn csts have been develped in Figure 6 als; the pinch is between levels 4 and 3, crrespnding t a cld stream temperature f 8 and ht f 2*, the same as abve.
24 inimum Utility Cst Prblem Often several different ht and cld utilities will exist in a prblem. Fr example steam may be available at several different pressures and thus at several different cndensing temperatures. Aside frm cling water ne may als have brine r ne may prpse t "raise" steam with excess heat at prescribed pressures. We can deal directly with this prblem as a transprtatin prblem if all heating and cling can be treated as ccurring at pint temperature surces i.e. each perate at a single temperature. Cndensing steam is readily handled, therefre. Unfrtunately cling water is nt a pint surce in terms f temperature as it is heated when it passes thrugh the prcess. We shall first assume pint temperature surces fr all utilities and shw hw t set up a minimum utility cst prblem as a transprtatin prblem. We shall then discuss hw the prblem must be slved fr nnpint surces. Fr (temperature) "pint utility surces", add the temperatures fr the utilities t the candidate ht and cld pinch pints used t partitin the prblem. Change the csts C..., fr utility-prcess stream matches K,JX> t reflect the per unit cst f the utility invlved. When initializing using the Nrthwest Algrithm, always use the least expensive utility pssible when utilities are needed. The "left t right" search alng a rw and tp t bttm search alng a clumn will wrk if the least cst utilities are listed t the left r t the tp f the mre expensive nes. Otherwise, slve as befre. We nte that the actual C used fr utility csts need nly set ik f jx a rank rdering amng the ht utility stream csts r the cld utility stream csts. Assume utility streams cst us mney. Therefre, fr a minimum cst utility prblem, ne will never use mre than the minimum
25 ttal amunt f utilities fund in ur earlier frmulatin. The nly questin is hw t divide the utility heating and cling requirements amng the utilities available. Clearly we will use the least expensive ht utility until n mre ht utility is needed r until it can n lnger be used thermdynamically i.e. until it pinches with the cld prcess streams t which it is supplying heat. Being the least expensive is all we need t knw, nt its exact cst. The argument shuld nw be bvius. Thus we need nly assign relative csts t utilities, with these relative csts usually reflecting the temperature level. Htter ht utilities are generally mre expensive than clder nes; similarly, clder cld utilites are generally mre expensive than htter nes. The peculiar case f "raising 9 steam is handled by still assuming that the steam raising "utility csts mney but less than cling with cling water. f the cst is made less than zer (i.e. reflects making a prfit) the prblem slutin may n lnger invlve minimum ttal utility usage, and if it des nt, the slutin will in fact be unbunded. One will have unfrtunately set the csts s it is prfitable t turn a ht utility int a surce f heat t generate steam, an unlikely real wrld situatin r at least ne superfluus t the prblem at hand. Figure 5 shws the tableau fr ur example prblem if we have tw surces f heating ne at 25 and ne at 3 degrees. Only thermdynamic cnstraints are cnsidered. te, tw pinch pints exist, ne at (2O5 /85 ) and ne at (2OO + /8O ). Grimes (98) bserved that there must be ne pinch pint fr each utility past the first in a minimum utility cst prblem. Als nte that we use 63 units f the mre expensive utility, H-, and 53.5.f the less expensive clder utility, H. Csts assumed fr H and H~ were nly t rank rder them; i.e. H has a cst f and H. f 2.
a ik 8 8 5 22.5 56 8 5.5 C 5 C 25 C 4 C 24 C 3 C 23 C 2 C V ^5 8 4 S7 his 57 «25 c 2 + 6* 2 24 3 6 48 28 24 28 64 "25 23 7» h 4 3 h 24 6 l «h 3 l 88 l 6 l» ' "23." 96 32 h 2 l 24 h 22 l 24 4 h 2 N 64 5 H i 3 22.5 4883.5 5 H 2 2 2 63 2 2 2 2 2 5 p ik Pinch Pinch - -3-3 -3-2 3 3 3 3 3 2 2 Pinch Pinch FGURE 5 inimum Utility Cst Slutin Example. Hj^ B available at 25 and s less cstly than H,
27 A/n Pint Temp&i&tusie. CnsvOiairub* A utility stream which prvides its heat r cling in ttal r in part as sensible heat r is mu t i cmpnent and passes thrugh a phase change can significantly cmplicate the minimum utility cst slutin prcedure. Let us speak specifically abut cling water as ur example utility f this type. Nrmally nb uses cling water by heating it ttam sme available inlet temperature (say 37 C) t an allwable exit temperature (say 5 C). The prblem arises if cling can be dne at 37 C but 5 C is t ht. Then ne must use mre cling water until its exit temperature is lw enugh t d the cling needed. n the limit f a pint-temperature surce, ne wuld use an infinite amunt. f the cling water cst is prprtinal t the amunt used, then cst is affected by its exit temperature. Tw flws are significant fr such a* utility: ) the minimum flw which results if the entire temperature range (frm 37 C t 5 C) can be used and 2) the maximum, flw such that the cst per unit f cling makes it mre expensive than a clder utility, say brine. Fr such a utility, we can establish the flw per unit f heat as: F/Q «V J C p d9 T in and fr each we can plt cst versus T as shwn in Figure 6 where C^ is the cst per unit flw. f T fr cling water falls belw T, then ne shuld switch t brine as a clant.
Cst Unit f Heat CO ft ) i sr it (D 8, c 8 g
29 T slve a minimum utility cst prblem with a nn pint-temperature utility, first slve the minimum utility cst prblem as if its temperature everywhere were its inlet temperature i.e. treat as a pint-temperature surce utility. Use as its cst/unit f heat, the cst resulting frm allwing it t heat r cl thrugh its maximum temperature range - i.e. its least cst/unit f heat. Next set the flw t that at which it ceases t be less cstly than anther utility the flw crrespnding t exit temperature T f in Figure 6 fr cling water. Treat the utility as a required prcess stream with this flw, entering at its inlet and leaving at T f ; reslve the minimum utility prblem t see the impact when using such a prcess stream. f the use f ther utilities des nt increase, then this utility shuld be used as a heating r cling surce in a minimum utility cst slutin. f the usage increases fr the ther utilities, then it shuld be rejected as a utility; in ur example, brine shuld becme the cling utility instead. The reasn is bvius; its flw wuld have t increase beynd its maximum ecnmic flw t be part f a minimum utility usage slutin. t is thus t cstly per unit f heating r cling supplied. Repeat the abve fr every nn pint-temperature surce utility t select the active utilities. Then, ne at a time, we have t set their flwrates as fllws. The flws are bunded between F. (entire temperamm * ture range is used) and F, anther utility becmes less expensive. Figure 7 shws hw the minimum utility usage shuld change versus flwrate fr such a utility. Change the flw t its minimum, again treat as a required prcess stream and slve the minimum utility usage prblem. f the usage des nt increase, the minimum flw is the slutin. Otherwise we have t search fr the flw, F (see Figure 7). ncreasing the flw
3 t inimum Utility Usage Flw Figure 7 Effect f Varying Flwrate fr Bn Pint Temperature Utility n inimum Utility Usage.
3 will decrease ttal ther utility usage fr the prblem up t flw F ; it will then have n effect. We seek therefre the flw F as ur minimum utility cst slutin. The search shuld be dne at lw flws, e.g. at F. and F. + mm mxn AF. Assuming a linear behavir these tw slutins can be used t prject t F, ur next guess. The search can use a ne dimensinal secant methd tgether with an interval reducing methd; it will be rather quick. Frtunately each utility f this type can be dealt with separately, a significant prblem decmpsitin. atch Dependent AT. We nw cnsider the last tpic t be cvered in this paper: hw t slve the minimum utility usage r cst prblem when AT is nt the same fr every match allwed. We shall discver first why this prblem is an imprtant ne and then hw t slve it. Suppse we have tw streams we will nt allw in the same exchanger because a leak wuld lead t t dangerus a situatin r because the streams are bth vapr and far apart, leading t very cstly piping requirements. We may want t knw the impact f using a third fluid as illustrated in Figure 8 as a heating/cling lp between them n utility usage. We see that, if such a fluid culd be fund, it will exchange heat in tw exchangers, thus dubling the required AT. needed between ur tw riginal prcess streams. We culd thus mdel the minimum utility usage, where sme streams can nly exchange heat indirectly, by simply dubling the required AT. fr them. Nte there is a significant impact n exchanger area required ver a direct exchange at the larger AT, essentially increasing it by a factr f 4 since the driving frce is halved and tw exchangers are needed.
Redrculatlng ntermediate Stream Ht Stream Cld Stream t Figure 8 ndirect Transfer f Heat between a Ht and a Cld Stream.
33 T slve we shall discver we nly need t change the step where we prject ht stream candidate pinch pints nt cld streams and vice versa. The cnsequence is nt negligible as we will create an enrmus increase in the number f partitins fr ur prblem. T explain is best dne by example. Suppse we reslve ur prblem where c t and h~ were allwed t exchange heat nly belw 75. We hw state that they can indirectly exchange heat abve the cld stream temperature f 75. We shall mdel this pssibility by requiring a 4 minimum driving frce abve 75 fr c. between streams c. and iu. The candidate pinch pints fr the streams are almst the same as befre: c., 75, 8 ; c 2 4 ; i^ 2 + ; and in additin h 2 28 since h is nw less than 4 (= 2AT. ) htter n entry than c. is n L mm l exit (25 ) Figure 9 shws the required temperature prjectins fr this prbl- lf em. We break c«int c. and c. at 75 fr cnvenience. t is best t explain the prjectins ne at a time. We start with the inlet temperature fr c i at. Belw 75 fr c. the AT. between it and h is nly mm 2 2 s we prject the nt h 2 at 2. Next cnsider 4 n c^«this temperature prjects nt bth h and h- at 2 higher r at 6 The 6 n bth h. and h_ prject back nt c at 4. S much fr the easy nes. Nw cnsider 75 n c.. t prjects nt h at 95 and nt h- at 25 (i.e. 4 higher, nt 2 ). The 95 n h prjects nt c 2 at 75. The 25 n lu prjects back nt c 2 at 95 which prjects nt h at 25 which prjects nt c. at 95. Unfrtunately we are ff t the races nw because 95 n c. prjects nt h- at 235 which prjects nt c 2 at 25, back t \i at 235 and nt c at 25. The 25 n c cntinues: 255 n h-, 235 n c 2, 255 n h, and, panting, it stps
34 3- i 3 T ha "28 25-25 24 26 24 235 26 255 24 235 22 25 22 25 22 25 22 25 2-,2 95 2 95 2 95 2 95 X75 8 J.75 8 75 6 6 5-4 4 Ca +4 2 - C ci cv C 2 h, 2 2 2 h 2 2 4 2 rloo Figure 9 The Temperature Priecjfein Step fr Example Prblem Allwing' ndirect Heat Transfer between C. and C abve 75 ne.
35 since c~ has n prtin at 235 fr h t prject nt. The 28 inlet temperature fr h- prjects as fllws: 24 nt c., 26 nt h.. The + 2 temperature n h. prjects as: 8 n c. and c-, 8 n c. t 22 n h- t 2 n c^, 22 n h. t 2 n c-, etc. Figure shws the resulting intervals fr this prblem as well as an initial feasible slutin. The temperature levels are identified by their ranges rather than by a secnd subscript as labeling them by a secnd subscript is n lnger bviusly dne. Utility usage is back t the minimum fund fr.the uncnstrained prblem (Pigure 3) s this initial feasible slutin must als be ptimal. The use f indirect heat transfer has therefre returned ur utility requirements back t their riginal minimum value. The rw and clumn csts (p., and Y..) are als shwn s we can t K j* lcate the pinch pint fr this prblem. The p., change values when c. and c crss 8 and y. m when h. and h crss 2 ; thus this pint is the pinch pint fr the prblem. f ne chses t stp the prjecting f temperatures back and frth, say nly up t a single repeat reflectin n a stream, then, if ne is careful abut identifying infeasible cells in Figure as thse fr which at least a 2 driving frce is nt available, the slutin fund will be an upper bund n the minimum utility usage. This bunding fllws because mre partitining leads nly t mre chances fr heat exchange between streams. ff ft Discussin ^ - Three earlier wrks frmulated the heat exchanger netwrk synthesis prblem as a prblem invlving a linear prgramming mdel () These earlier frmulatins led t an "Assignment r "Set Cvering" prblem
* 4 36 c a" X O ' O O in C C i C l C i (OZl-OOl) 2^ i» - - C CN C (9^)^ -4 C C <>* C C *~ (56-9)^ * C t C C (S6l-9l) l C C (Z-S6l) Z «l (Z-S6t)4 H-4 9.5.5 t 4.5 C C ec (SZ-OOZ)^ (SZ-OOZ)4 *-4 h-4 (OZZ-SZ)^ fr-4 NO (zz-stz)^ (ssz-zz)^ mr C C UJ C s (S Z-ZZ)4 (OfrZ-SSZ)^ i i» H-4 (>Z-S Z) l *-* i i (SSZ-OtrZ)^ -H 46.5.5 Gwz-ssz)^ C (9Z-frZ)4-4 in s ( -9Z)4 22.5 in 22.5 T nterval 24-25 22-24 T 22-225 C u 25-22 OZZ-SZ C U 2-25 2-25 C U 2 in 2 vn C Ok r~i8-95! C 75-8 75-8 75 75 v 4-4- u - 4 C] C in C in C - ass s ^ m
37 rather than a "Transprtatin" prblem. The Assignment prblem is well knwn and als has a very efficient slutin algrithm available t slve it. The apprach was t partitin each stream int small equal prtins invlving "Q" units f heat each, rather like slicing a carrt int small equal sized bits. Cnstraints preclude matches nt pssible thermdynamically. The slutin has every ht bit f Q heat units matched t exactly ne cld bit f Q units fr anther stream. The ntin f a pinch pint was nt mentined in this apprach. Als the assignment prblems created are very large relative t thse created here, and it is unable t determine the precise minimum utility fr tw reasns: ) the inaccuracies caused by the "slicing and 2) the pinch pint will likely appear in the middle f a slice. Thus, while we can advcate slving mderately large prblems by hand, they cannt. The partitining generated here is caused by the crners in the cling curves admittedly sme are there due t apprximating the curves, but this partitining seems the mre natural ne. The handling f utilities which are nt available at a single fixed temperature fr the minimum cst prblem and the handling f match dependent AT. *s are new with this wrk.