Targeting Procedures for Energy Savings by Heat

8/12/2019 Targeting Procedures for Energy Savings by Heat

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Targeting Procedures for Energy Savings by HeaIntegration across PlantsHernanRodera andMiguel J . Bagajewicz

School of C hemical Engineering and Materials Science, U niversity of O klahoma, Norman, OK 73019

H eat integration across plants can be accomplished either directly using process streams or indi rectly using in termediate flui ds. By applying pinch analysis to a system of two plants, i t is fi rst shown that the heat transfer leading effecti ely to energy sa ings occurs at temperature le els between the pinch points of both plants. I n some cases,howe

er, heat transfer i n other r egions is required to attain maximum sa

in gs. A system -

atic procedure to identify energy-sa ing targets is discussed, as well as a strategy to determine the minimum number of intermediate-fluid circuits needed to achie e maxi -mum sa ings. A n M I L P problem i s proposed to determi ne the optimum location of the intermediate fluid circuits. The use of steam as intermediate fluid and extensions to a system of more than two plants are di scussed.

Introduction

Since the onset of heat integration as a tool for processsynthesis, energy-saving methods have been developed for thedesign of energy-efficient individual plants. Heat integration

across plants that is, involving streams from different plants.in a complex always has been considered impractical for var-

ious reasons. Among the a rguments used is the f act t hat plantsare physically apart from each other and, because of this sep-aration, pumping and piping costs are high. H owever, an evenmore powerful argument against integration is the fact thatdifferent plants have different startup and shutdown sched-ules: if integration is done between two plants and one of theplants is put o ut of service, the o ther plant ma y have to resortto an alternative heat-exchanger network to reach its targettemperatures. Plants may also operate at different produc-tion rates, departing from design conditions and needing ad-ditional exchangers to reach desired operating temperatures.All these discouraging aspects of the problem led practition-ers and researchers to leave opportunities for heat integra-tion between plants unexplored.

Notwithstanding the aforementioned objections, in severalpractical instances, these savings opportunities are actually

implemented either directly using process streams Siirola,.1998; Zecchini, 1997 , or indirectly t hrough the use of the

steam system in what has been called the steam belt Robert-

Correspondence concerning this article should be a ddressed to M. J . B agajewicz.

.son, 1998 . The first attempt to study the recovery of energythrough integration between processes wa s made by Morton

.and Linnhoff 1984 , who considered the overlap of grandcomposite curves to show the maximum possible heat recov-

.ery using steam. Later, Ahmad and Hui 1991 extended thisconcept to direct and indirect heat integration, also based onthe overlapping of grand composite curves. In addition, theyproposed a systematic approach to generate different heat-recovery schemes for interprocess integration.

The concept of total site was introduced by Dhole and .Linnhoff 1992 to describe a set of processes serviced by and

linked through a central utility system. U sing site-source andsite-sink profiles based on the combination of mod ified grand

.composite curves of the individual processes , they set targetsfor the generation and use of steam between processes. How-ever, the elimination of process-to-process heat-exchangezones, also ca lled pockets, from the grand composite curvesof the individual processes reduces in certain cases the op-portunities for energy recovery. This point is discussed in de-tail in the present article.

One of the questions in total-site integration is whether aprocess fluid should be used to perform the heat transfer or anintermediate fluid should be used. In addition, the questionof how to preserve energy efficiency when nonsimultaneousshutdowns take place needs to be addressed. The objectivethen is to have a dual design where both heat integration and

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independent operation are achievable. In this first approachto t he problem, energy ta rgets are established. The nonsimul-taneous operation of each of the plants is directly related tothe ability of the heat-exchanger network in both plants tooperate in either of the other modes. This multiobjective de-sign procedure will be presented in detail in a follow-up arti-cle.

In principle, direct transfer of heat from one plant to theother may involve many process streams, which results in

many heat exchangers. The incentives to use intermediatefluids are the following.

M ultiple pumps and compressors. Integration acrossplants may require the transferring of heat from a number ofstreams in one plant to a number of streams in the other.Thus, the cost of integration can be high because of the useof multiple pump and compressor units.

Pumpi ng and compression costs. A fluid with a largerheat capacity than the process streams will result in a smallerflow rate of liquids to pump among plants. When processstreams a re gases, then the installation of compressors tocover large distances can be more expensive than the equiva-lent pumping of an intermediate fluid.

Safety. Process streams being pumped large distances

may pose a hazard, should any spill occur.Control. Piping process streams long distances also in-

troduces long delays, which wo uld eventually make processcontrol more difficult. The use of intermediate fluids simpli-fies the problem.

There are, however, some disadvantages worth mention-ing.

The use of an intermediate fluid reduces the interval of .effective heat transfer that is, between pinches by a multi-

.ple of the minimum temperature difference T . There-minfore, compared with the direct integration case, smaller sav-ings can be obtained.

The number of heat exchangers involved in a setup thatuses intermediate fluids can also be higher than using directheat exchange. In this case, in the absence of other incen-tives, the trade-off is between the new number of heat ex-changers and the pumping costs.

In many cases, steam can be used as an intermediate fluid.This offers many possibilities, as the steam system is alreadyin place. Recent work regarding the use of the utility system

for the indirect integration of different processes Hui and.Ahmad, 1994 focuses on the generation a nd use of steam to

reduce utility costs. The present work introduces targets basedon fixed steam pressures that are usually available in theplants.

In this article, pinch analysis is used to establish targetmaximum energy savings for either direct or indirect integra-tion of the case of two plants. The article is organized asfollows: temperature intervals where heat transfer should takeplace are identified first, together with the identification ofwhich plant should be the source. An LP problem is set up todetermine these targets. The design of the intermediate fluidcircuits is considered next. The possibility of using a singleintermediate fluid circuit is evaluated. An MILP model is in-troduced to determine the location of the minimum numberof circuits needed to achieve the target savings. Extensions tothe problem of integration of a set of n plants are brieflydiscussed. To illustrate these concepts, examples using heat-

Figure 1. Possible location of combined plant pinchpoint.

integration problems from the literature are solved. In addi-tion, an example consisting of the integration between a crudeunit and an FCC plant is solved.

Maximum Transferable HeatConsider two plants and suppose that minimum utility tar-

gets are obtained independently for ea ch of the plants usingLP transportation or transshipment models Cerda et al.,

.1983; P apoulias and G rossmann, 1983 . When all t he streamsfrom both plants are included in the same set combined

.plant , the total minimum heating and cooling utility targetsare usually lower.

L ocation of the combined-plant pinch

Without loss of generality, assume that plant 2 has a pinchtemperature that is higher than the pinch temperature forplant 1. Therefore, the pinch of the combined plant can fa ll

.in any of these three regions: a above the pinch of plant 2; . .b between the pinches; or c below the pinch of plant 1 .Figure 1 .

Consider first the case in which all intervals above the pinchof plant 2 in each of the individual plants are sinks of heat. Ifthe corresponding intervals of both plants a re added, thecombined plant will also ha ve sinks in t hese intervals. There-fore, the system considered has a combined pinch located nohigher than the original pinch of plant 2. A similar analysiscan be made for the region below the pinch of plant 1. Whenthe preceding conditions are not met, the combined pinchcan be located in any place. The implications of this will beinvestigated further.

The first intuitive conclusion one can make is that heatshould be transferred at temperatures between the pinchpoints of the original plants. Indeed, this is the only regionwhere plant 2 is a heat source while plant 1 is a heat sink.This intuitive conclusion is in principle correct, but some-

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times heat transfer in the opposite direction from plant 1 to.plant 2 above or below both individual plant pinches is re-

quired to assist the realization of maximum savings.

Tr ansfer of heat outside the region between both pi nch points

Transferring heat either above the higher pinch or below the lower pinch does not decrease the utility usage. Only a n

equivalent amount of corresponding utility is shifted from oneplant to the other. Figure 2 illustrates the effect of an amountof heat Q transferred from plant 2 to plant 1 in the upperA

zone without loss of generality, intervals are lumped to allow .clarity of illustration . An increase of the heating utility in

the plant that releases the heat is followed by a reduction ofthe same utility in the plant that receives the heat. The sameeffect on the cooling needs is observed if a certain amount ofheat Q is transferred from plant 2 to plant 1 in the lowerB

.zone Figure 2 . In add ition, as the temperature level at whichthe heat transfer in the upper zone takes place is lowered, amaximum that can be transferred exists. For example, if thetransfer is made in the first interval, the amount that can betransferred is constrained by the o riginal utility usage o f plant

1, S I . If the heat is transferred in an interval below the firstminone, say interval i , the upper limit will be smaller, as some ofthe utility used by plant 1 is used to satisfy the heat demandof the first i y 1 intervals. Therefore, to compute this upperlimit one should subtract all the intervals that are heat sinks . Inegative values above the interval of transfer from S .minSimilar upper limits for the transfer of heat are found if thelower zone is considered.

In conclusion, no savings can be obtained by transferringheat in the regions above the higher pinch or below the lowerpinch. However, transfer from plant 1 to plant 2 in either oneof these regions is needed in some cases to facilitate thetransfer of heat in the region between both pinch tempera-tures. This is explored next.

Tr ansfer of heat between both pinch points

As illustrated in Figure 3, a certain a mount of heat Q isE transferred f rom plant 2 to plant 1 between pinch points. This

Figure 2. Effect of transferring heat outside the regionbetween both pinch temperatures.

Figure 3. Effect of transferring heat in the region be -tween both pinch temperatures.

transfer has the effect of reducing the heating utility in plant1 and cooling utility in plant 2. In addition, transferring heatfrom plant 2 to plant 1 has the effect of reducing the lowestlevel of the heating utility demand on plant 1, which is usu-ally the cheapest. Finally, it has no effect on the heating util-ity of plant 2 or the cooling utility of plant 1.

Assisted and unassisted heat t ransfer across plants

We now investigate the upper limits in the amount of heatthat plant 1 can accept and in the amount that plant 2 candeliver in the region between pinches. Consider the case inwhich there are only sink intervals above the pinch point ofplant 2 and only source intervals below the pinch point ofplant 1. The maximum heat that plant 1 can receive is theactual sum of the demands it has in the intervals betweenpinches. Similarly, the maximum amount that plant 2 can

transfer is the resulting available heat it has between pinches.Since any heat that is transferred to plant 1 at any tempera-ture interval can be cascaded down to lower temperatures,the real limitation on how much can be transferred is givenby the ability of plant 2 to fulfill the demand at each interval.Because all the intervals above the pinch point of plant 2 aresinks, the whole demand of the heat in plant 1 is only satis-fied by utility or by plant 2 from the intervals between pinches.Likewise, since all intervals below the pinch point of plant 1are sources of heat, plant 2 does not need to use heat fromthe intervals between pinches to satisfy any demand below the pinch point of plant 1. Therefore, the amount of heatthat can be transferred to plant 1 is not limited by such de-mand. This motivates the following definition:

Definition. U nassisted heat transfer across plants takesplace when only heat transfer between pinches is needed toachieve maximum savings.

A special case of unassisted heat transfer across plants iswhen plant 1 has only sink intervals above the pinch point ofplant 2 and only source intervals below the pinch point ofplant 1. U nassisted cases can also take place even thoughsome intervals in plant 1 are sources of heat above the pinchof plant 2 or some intervals in plant 2 are sinks of heat below the pinch of plant 1. I f a case is unassisted, the combined



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Table1. Example1

Temp.P lant 1 P lant 2 Combined P lantScale

I I I I I I I I I C P C P C P140 q S q S q S i i min i i min i i min120 y 12 y 12 30 y 19 y 19 20 y 31 y 31 38100 y 1 y 13 y 1 y 20 y 2 y 33

80 y 15 y 28 10 y 10 y 5 y 38I I I C PW W W min min min60 y 2 y 30 5 y 5 3 y 35

40 2 y 28 2 1 y 4 16 3 y 32 6Max. potential savings between Max. possible savingss 12

pinches s 12

pinch point of both plants lies in between pinches. Indeed,the addition of all intervals and the fact that in general thereis heat transfer across the location of the pinch of plant 2indicates that the pinch does not lie above this temperature.The same can be said for the region below the pinch point ofplant 1.

Assume now tha t some of the intervals in plant 1 above thelocation of plant 2 pinch are sources of heat. Furthermore,assume such heat sources are enough to produce a surplusthat in the absence of integration across plants is effectivelytransferred in plant 1 through the location of the pinch pointof plant 2. In o ther words, the surplus of heat above the pinchof plant 2 needs to be used to satisfy the heat demand ofplant 1 between pinches. In turn, this may limit the amountthat can be transferred from plant 2, and therefore limit themaximum savings that can be obtained. To prevent such limi-tation, one can transfer the surplus heat from plant 1 to plant2, reducing the heating utility of plant 2, and allowing maxi-mum heat transfer between pinches. The heat transfer out-side the region between pinches does not realize any savings,only shifts utility load from one plant to the other. In fact, ifthe surplus is larger than the heating utility of plant 2, theamount Q is limited by S II , and the surplus may becomeA minan effective limitation to realize all the potential for savings.

Figure 4. Cascadediagramsolutionfor Example1.

Similarly, if the heat demand of plant 2 in the correspondingintervals is not sufficiently large, the total surplus that can betransferred is limited. An exact symmetric case happens be-low the pinch of plant 1. Some of the surplus from plant 1below its pinch can eventually be used to satisfy this demand,thus freeing the heat from plant 2 to be completely availableto rea lize savings through transfer to plant 1 between pinches.

These two cases motivate the following definition.Definition. Assisted heat transfer across plants takes place

when heat transfer between pinches needs to be assisted byheat transfer outside this region to attain maximum savings.

The existence o f assisted cases has been overlooked b y Ah- .mad and Hui 1991 , who only showed that sometimes more

than one steam level is required for maximum indirect recov-ery between processes. However, they do not explore furtherthe significance of the a ssisted transfer in order to realize

.maximum savings. D hole and Linnhoff 1992 construct sitesource and site sink profiles based on the combination ofmodified grand composite curves of the individual processes.In these modified curves process-to-process hea t-integrationzones or pockets a re eliminated. Consequently, in the pres-ence of an assisted case, o pportunities for rea lizing ma ximumsavings are lost and only limited savings between pinches canbe pursued.

A model to predict the exact a mount of heat that needs tobe transferred in each region is presented later. First, someillustrative examples a re shown.

Example of unassisted case

Table 1 shows the interval balances, the heat cascade todetermine the utilities, and the actual value of these utilitiesfor each of the plants as well as for the combined plant. Sinkintervals are located above the pinch and source intervals arelocated below the pinch in either plant 1 or plant 2. There-fore, this is an unassisted case, a nd only transfer between

pinches is needed in order to obtain maximum savings. Thesesavings are obtained by subtracting from the sum of the indi-vidual utilities the combined utility. P inch locations are shownwith filled lines. As expected, the combined pinch is in be-tween the original plant pinches. Figure 4 shows the cascadediagram solution after the integration is conducted.

In order to compare the results obtained using the cascadediagram with methods that make use of grand composite

Figure 5. Countercurrent composite curve profiles forExample 1.



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Table2. Example2

Temp.P lant 1 P lant 2 Combined P lantScale

I I I I I I I I I C P C P C P140 q S q S q S i i min i i min i i min120 y 7 y 7 20 y 10 y 10 20 y 17 y 17 23100 y 5 y 2 y 10 y 20 y 5 y 22

80 y 15 y 17 14 y 6 y 1 y 23I I I C PW W W min min min60 y 3 y 20 10 4 7 y 16

40 3 y 17 3 5 9 29 8 y 18 15Max. potential savings between Max. possible savingss 17

pinches s 17

.curves, the approach of Ahmad and Hui 1991 is employed.Figure 5 shows the countercurrent profiles for the grandcomposite curves of the two plants. The grand compositecurve of plant 1 has been inverted to be able to establish themaximum amount of direct heat transfer. The extent of themaximum possible savings is rea ched whenever the profilescoincide in a point, as shown. Unassisted cases are thereforereadily tractable with the reported method.

Example of assisted cases

Table 2 presents the data corresponding to Example 2. Asource interval is located in the region above the pinch of

.plant 2 higher pinch in plant 1. This source interval pre-vents plant 1 from receiving all the potential heat available tobe transferred between pinches. However, a transfer of thenecessary amount from plant 1 to plant 2 above the higherpinch allows maximum potential savings to be realized. As-sisted cases below the two pinches are similar in nature andtherefore examples are omitted. The combined pinch lies be-tween pinches. This is a result of the fact that all the limita-tion for transfer between pinches can be completely re-moved. Figure 6 shows the cascade diagram solution aft er theintegration is conducted.



When a comparison with the method that uses grand com-posite curves is performed, the diagram of Figure 7a is ob-tained. The pocket present in the composite curve of plant 1has not been removed, since it makes possible the assistedtransfer to plant 2 and allows full transfer of heat betweenpinches. In the procedure introduced by Dhole and Linnhoff .1992 to indirectly integrate the tota l site through the utilitysystem, pockets a re eliminated prior to the construction of

the site-source and site-sink profiles. Therefore, whenever thepockets are eliminated, the possibility of realizing maximumsavings has been lost. This is illustrated in Figure 7b.

Table 3 presents the data for Example 3. A source intervalin plant 1 is found in the region above the pinch of plant 2.Thus, this is an assisted case. However, a limit imposed byplant 2 arises in the heat that plant 1 can transfer above thehigher pinch. Therefore, the limitation to obtain maximumpotential savings cannot be totally removed. Figure 8 shows

Table3. Example3

Temp.Plant 1 P lant 2 Combined P lant

Scale I I I I I I I I I C P C P C P140 q S q S q S i i min i i min i i mi n120 y 18 y 18 20 y 19 y 19 20 y 37 y 37 37100 5 y 13 y 1 y 20 4 y 33

80 y 5 y 18 10 y 10 5 y 28I I I C PW W W min min min60 y 2 y 20 5 y 5 3 y 2540 2 y 18 2 1 y 4 16 3 y 22 15

Max. potential savings between Max. possible savings s 3pinches s 7



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the cascade diagram solution after the integration is con-

ducted.Figure 9a shows the composite curves for this example. Inthis case, the presence of a pocket in plant 1 allows the par-tial removal of t he limitations in the transfer b etween pinches.The elimination of this pocket prevents the realization of themaximum possible savings as it is shown in Figure 9b.

Targeting Model for Heat IntegrationIn this section, a model that allows the automatic determi-

nation of unassisted and assisted cases is presented. Thismodel predicts the amount of heat that needs to be trans-ferred in ea ch interval t o achieve maximum savings. Applica-tion to either direct or indirect integration is possible. In

order to facilitate the computations, the temperature inter-vals are constructed using inlet and outlet temperatures of all

I II .streams from both plants that is, m s m s m .

M aximum energy sa ings Maximum savings that can be obtained by integration are

computed subtracting the combined plant minimum heatingutility f rom the summation of the individual plants minimumheating utilities. To obtain the amount of heat that has to betransferred in each interval, a model is constructed whereheat can be transferred independently within each interval .Figure 10 . A single direction of heat transfer is allowed:from plant 2 to plant 1 between pinches and from plant 1 toplant 2 outside this region.

The task is now to determine what amount is transferredat each interval to achieve maximum savings. To do that, an

I IILP model is proposed. Let and be the original mini-0 0mum heating utility of plant 1 and plant 2, respectively, whenno integration between plants is assumed. These values areS I and S II , the results obtained by solving the LP trans-min minportation or transshipment models for each of the plants sep-

I IIarately. In the same way, let an d be the original cool-m m I II . I IIing utilities W and W values . Also let and bemin min i i


the new heat transferred between intervals after integrationbetween plants is implemented. Finally, let q E be the heati transferred bet ween pinches in interval i from plant 2 to plant1, and q A and q B the heat transferred in the inverse direc-i i tion above the higher pinch and below the lower pinch, re-spectively.

The model tha t predicts the maximum possible energy sav-ings that effectively occur between pinches Q , and the even-

E tual minimum amount of heat Q and Q to be transferredA B

Figure 10. Splitting the heat transfer among intervals.



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in the regions outside the one between pinches is

Min I q II .0 m s.t.

I I s q Q y Q 0 0 A E II II s y Q 0 0 A

I s I q q I y q Ai i y 1 i i IIi s 1, . . . , p II I I I I A 5 s q q q q i i y 1 i i I s I q q I q q E i i y 1 i i I I Ii s p q 1 , . . . , p 1 . .II I I I I E 5 s q q y q i i y 1 i i I s I q q I y q B i i y 1 i i Ii s p q 1 , . . . , m .II I I I I B 5 s q q q q i i y 1 i i

I I s y Q m m B II II s q Q y Q m m B E

I I s 0, IIII s 0p p

I , II , q A , q E , q B G 0.i i i i i

In this formulation p I and p II are the respective pinchpoint levels, and it is assumed p I p II . The problem consid-ers the conditions of minimum utility usage for both plants asthe starting point. The objective function used needs someexplanation. Minimizing the utility needed in plant 1 servestwo purposes:

1. To reduce the utility in the amount transferred fromplant 2 between pinches.

2. To make sure that the amount of heat transferred fromplant 1 to plant 2 is strictly the minimum needed.

When a higher amount of heat than the minimum neededis transferred a bove pinches in the assisted ca se, the excessconsists of a simple shift of utility from plant 2 to plant 1.Such shifting requires equipment and therefore representsaddit ional investment without a benefit and should beavoided. The same result is obtained if one solves the prob-lem minimizing the amount of cooling utility of plant 2. Inthis case, the transfer to plant 1 is maximized, while thetransfer from plant 1 to plant 2 below both pinches is kept atits minimum necessary to assist in the savings. Finally, foreach unit of heat transferred between plants, both values arereduced by the same amount simultaneously. This implies thatindependent reductions of these utilities a re not possible.Hence, adding them to form the objective function of prob-lem 1 is possible. A simple balance around plant 1 proves

that the summation of the solutions heat transferred amountsE .q will represent the total possible amount of heat to bei

transferred between pinches Q .E Remark. The LP problem presented ha s degenerate solu-

tions. Indeed, to t ransfer heat surplus from an interval in plant2 to any interval in plant 1 between pinches, the heat can betransferred from plant 2 to plant 1 first and then transferreddown, or transferred down in plant 2 first, and then trans-

ferred to plant 1 at a lower interval. The same situations oc-cur in an inverse manner when the transfer takes place in anyof the regions outside the region between pinches. There-fore, many different paths are available. This degeneracy isactually a flexibility that can be exploited later when a designis attempted.

The results from the preceding models can now be usedas target values for models that will determine the heat-exchanger network needed to accomplish such savings. In

particular, the knowledge of what are the intervals at whichheat transfer from one plant to the other should take place .in addition to the d irection of such transfer is a useful inputfor these models. These models, which will be presented in afollow-up article will address the design of systems featuringthe minimum number of exchanger units to accomplish a d ual

.operation with and without integration .

Indirect Heat IntegrationThe focus is now on the case of indirect heat integration by

the use of intermediate fluid circuits. The design parametersfor these circuits, namely flow rate and inlet and outlet tem-peratures, have to be calculated.

Shift of scales

When an intermediate fluid is used, new streams appear ineach plant. C onsider the region between pinches first. In plant1, the intermediate fluid acts as a hot stream, whereas inplant 2, it acts as a cold stream. The temperature of the in-

.termediate fluid leaving plant 1 registered in its hot scaleshould be equal to the starting temperature of the same fluidin plant 2, requiring the coincidence between the respectivehot a nd cold scales. Thus, a shift consisting of moving the hot

.scale of plant 2 and with it, the cold scale too downwardT degrees in the region below its pinch is performed.min

Now consider the possibility of assisted cases. In the regionabove the higher pinch and below the lower pinch, the fluidcirculates in the inverse direction than between pinches.Therefore, a match between the cold scale of plant 1 and thehot scale of plant 2 is required in these two regions. To ac-

complish this, the hot scale of plant 2 and with it, the cold.scale is shifted upward T degrees in the zone above itsmin

pinch. Similarly, in the region below the lower pinch, a shift .of the hot scale of plant 1 and with it, the cold scale down-

ward is needed. However, the hot scale of plant 2 was al-ready shifted by T . Therefore, a shift of 2 T degreesmin min

downward of the hot scale of plant 1 and, with it, its cold.scale has to be performed. Finally, a s in the direct integra-

tion case, the temperature intervals are constructed using in-

let and outlet temperatures of all streams of both plants.As a result of these temperature shifts, smaller savings than

in the direct integration case may be achieved. If the use ofintermediate fluids is not mandatory due to safety or other

.considerations , then this reduction in savings potential mayor may not be compensated by the reduction in piping,pumping, a nd r or compression costs.

Summarizing, the scale shifts required are:1. A shift of both hot and cold scales downward by T min

degrees in plant 2 below its pinch.



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Figure 11. Shift of scales to allow the use of intermedi -ate fluids.

2. A shift of both hot and cold scales upward by T mindegrees in plant 2 above its pinch.

3. A shift of both hot and cold scales downward by 2 T mindegrees in plant 1 below its pinch.

These shifts create gaps in the scales, a s depicted in F igure11.

M aximum energy sa ings The maximum savings that can be obtained by integration

can be computed by subtracting the combined plant mini-mum heating utility using the shifted scales from the summa-tion o f t he individual plants minimum heating utilities. Next,to establish the regions in which heat tra nsfer should be ma deto accomplish the overall target, problem 1 is solved.

Remark. The solution to problem 1 can be implementedin practice. Indeed, a circuit can be established for each in-terval that has a nonzero heat transfer q E , q A , or q B . How-i i i ever, one is interested in performing the transfer with thesmallest amount of circuits possible. The issue is investigatedin the following sections.

Feasibility of a single-fluid circuit

Even though the target value for Q can always be trans-E ferred between pinches a nd the eventual target a mounts Q Aand Q can always be transferred outside this region, theB question is whether these transfers can be achieved with asingle circuit in each region. Figure 12a shows the unassistedcase with m intervals within the region between pinch tem-E peratures T II an d T I . The assisted cases are shown in Fig-p p ure 12b.

In the region between pinches, the heat released in eachinterval from plant 2 to the intermediate fluid does not have

to be the same as the heat released by the fluid to plant 1 inthe same interval. Likewise, in the region above the higherpinch and the region below the lower pinch, the heat re-leased from plant 1 in the same interval does not have to bethe same as the heat received by plant 2. A generalization forany of the regions comes next. The use of separate variablesfor the heat transferred to and from the intermediate fluidare defined as follows: q F C for the heat received by the fluidi in interval i , and q FH for the heat released by the fluid ini interval i . The total heat transferred Q is already given byF the targeting procedure, that is,

k F q m k F q m K K F C F H Q s q s q , 2 . F i i

F F i s k i s k

where k F is the first interval where the transfer betweenplants takes place and m represents the number of inter-K vals covered by a circuit in any of the regions.

Temperature constraints

Let us first explore the second law constraints regardingq F C an d q FH . A circuit covering m intervals betweeni i E pinches will receive heat from plant 2 and will deliver it toplant 1. In the assisted cases, a circuit covering either m orAm intervals will perform the inverse task, carrying heat fromB plant 1 to plant 2. Therefore, this can be generalized for a setof m intervals. The plant tha t is providing the heat is con-K sidered the heat-source plant, while the one receiving thatheat is considered the heat-sink plant. The following con-straints prevent the temperature of the fluid from going

Figure 12. Circuits of intermediate fluids in unassisted(and assisted cases gaps are omitted for

)simplicity .



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higher than the interval temperature T in the heat -sourcek y 1plant:

k F q m K F C F C F F F T y T G q k s k q m , . . . , k q 1 . . .k y 1 0 i K

i s k

3 .

F T F C F y T F C s Q . 4 . .k y 1 0 F

In t he heat -sink plant, similar constraints are introduced toprevent the temperature of the fluid going lower than theinterval temperature T :k

k F H F H F F F T y T G q k s k , . . . , k q m y 1 . .0 k i K

F i s k

5 .

F T F H y T F H F s Q . 6 . .0 k q m F K

In order to assure a closed circuit, it should be noticed that

the following has to be verified:

T F C s T F H F 7 .0 k q m K

T F H s T F C F . 8 .0 k y 1

Let us now examine what values the initial temperatures ofthe intermediate fluid can take. First, note that to guaranteefeasibility of heat transfer, T F C and T F H are equal to T 0 0 k y 1

and T , respectively, for some k that is, they are confined tok .be end-interval temperatures . Now consider the case where

the heat-source plant does not have any heat demand in the q F . FH F first set of k y k q 1 intervals, that is, q s 0, i s k ,i

. . . , k q . Then by increasing the flow rate of the intermediatefluid a nd without limitations in the transfer of heat from theheat-source plant, a new solution with an upper temperaturesmaller than the one considered initially is possible. If this isthe case, the heat-source plant will only be transferring heatto the intermediate fluid a t temperatures lower than T q . Tok find such a solution, heat can be cascaded down from the

q F .first k y k q 1 intervals in the hea t-source plant. The val-ues of q F C can be transformed to a new set q F C as follows:i i

q F C s 0 i s k F , . . . , k q 9 . i

k qF C FC F C q qq s q q q 10 . k q 1 k q 1 i

F i s k

q F C s q F C i s k q

q 2, . . . , k F q m , 11 . .i i K

where q F C is another degenerate solution. This solution al- i lows the circuit t o be established between the intervals k q q 1

F .and k q m . A similar argument can be made for the caseK where the last intervals in plant 2 do not transfer heat to theintermediate fluid.

Thus, one can assume without loss of generality that theinitial temperatures of the intermediate fluid in the heat-

source and sink plants are

T F C s T F H F s T F 12 .0 k q m k q m K K

T F H s T F C F s T F . 13 .0 k y 1 k y 1

With these equalities, the set of Eqs. 3 6 become

k F H F F

F F T y T G q k s k , . . . , k q m y 1 . .k y 1 k i K F i s k

14 .

k F q m K F C F

F F T y T G q k s k q 1, . . . , m . .k k q m i K K i s k

15 .

F T F y T F s Q 16 . .k y 1 k q m F K

Equations 14 16 constitute a feasibility test for a single

circuit transferring the hea t Q between plants. The flow rateF F can be calculated using Eq . 16. This value then can bereplaced in Eq s. 14 and 15. If any of these eq uations are notsatisfied, then a circuit between T F and T F cannotk y 1 k q m K transfer the maximum amount, but perhaps some smallervalue.

Candidate heat-tr ansfer sets

The flexibility at hand for defining the general variablesq FH and q F C is explored by a n adjusted heat-cascaded dia-i i

.gram Figure 13 . The target values Q , Q , and Q areE A B added and subtracted in the three defined zones of the cas-cade. This accounts for the supplies or demand each of the

(Figure 13. Adjusted heat-cascaded diagram gaps are)omitted for simplicity .



10/22

zones will experience when the respective single-circuit can-didates are considered. The values obtained for the differentintervals a re not realistic hea t-transfer amounts since someof them are negative, but rather a calculation aid. Moreover,because of these operations the adjusted heat-cascaded dia-gram of the heat-source plant may exhibit an induced pinchat each region where a circuit can be installed. In this in-stance, the transfer will only occur in the subzone delimitedby the real and induced pinches.

In the unassisted case, the solutions of Eq . 1 satisfy thefollowing relations:

k I I I E II s y Q q q q q G 0 .k p E i i

IIi s p q 1

k s p II q 1 , . . . , p I 17 . .k

I I I I E s q y q G 0 .k i i IIi s p q 1

k s p II q 1 , . . . , p I . 18 . .

The same set of equations can be written in terms of q E H i and q EC :i

k I I I E H II s y Q q q q q G 0 .k p E i i

IIi s p q 1

k s p II q 1 , . . . , p I 19 . .k

II I I E C s q y q G 0 .k i i IIi s p q 1

k s p II q 1 , . . . , p I . 20 . .

Similarly, for the assisted cases the relations for the upperand lower zones are:

k I I I A C s q Q y Q q q y q G 0 .k 0 A E i i

i s 1

k s 1 , . . . , p II 21 .k

II I I I I A H s y Q q q q q G 0 .k 0 A i i i s 1

k s 1 , . . . , p II 22 .

k I I B C s q y q G 0 .k i i

Ii s p q 1

k s p I q 1 , . . . , m 23 . .k

II I I I B H I s y Q q q q q G 0 .k p E i i IIi s p q 1

k s p I q 1 , . . . , m . 24 . .

Thus, any nonnegative set of generalized values q FH an di q F C that satisfies E qs. 19 20, 21 22, or 23 24, and also thei

.balance equation Eq . 2 is an acceptable candidate for a sin-gle circuit. If in addition, Eqs. 14 16 are satisfied, the candi-date set will be a feasible single-circuit solution in any of thezones. A few generalized candidate sets a re presented next.

One candidate set is given by the solution of problem 1,that is,

q FH s q F C s q F . 25 .i i i

Other sets can be found by making use of degeneracy. Inparticular, one can choose a set that prioritizes the heattransfer to the intermediate fluid over the heat transfer tothe interval below in the source plant. This solution is calledthe higher -circuit solution, because the circuit starts and endsat the higher possible intervals. A lower circuit solution ispresented later. The maximization of the heat delivered tothe intermediate fluid is the purpose of constructing a highercircuit solution.

Therefore, to establish the maximum a mount of heat thateach interval can provide to the intermediate fluid, the deficit

of heat in the intervals below it need to be taken into ac-count. The heat availability H at each interval is then de-k H H 4 H H 4fined as follows: Let s Min q , 0 a nd s Ma x q , 0k k k k

k s k F q 1, . . . , k F q m . In a ddition, let H F s 0, H F s q H F k k k k H H F q . Thus, the availability is given byk y 1 k

H F q H F q H F q H F 0k q m k q m k q m k q m K K K K H

F z s 26 .k q m K H H F F 0 q q G 0k q m k q m K K z H q H q H q H q z H q H 0k q 1 k k k k q 1 k H z sk H H H 0 q q z q G 0k k q 1 k

k s

k F

q1, . . . , k F

qm

y1. 27

.K

z H F q H F q H F q H F q z H F q H F 0k q 1 k k k k q 1 k H F z s 28 .k H H H F F F 0 q q z q G 0k k q 1 k

H s Ma x q H q z H q H , 0 k s k F , . . . , k F q m . 4k k k q 1 k K 29 .

In these equations, z H is an auxiliary variable that helpsk determine the amount of cumulative demand from the bot-

.tom at every interval. To illustrate this, consider first the H F situation depicted in Table 4 for which s 0, that is,k y 1

Table4. Determinationof HeatAvailabilityH H H H H H H H Interval q q q z q z i k k k k q 1 k k k

F k 12 0 12 8 0 8F k q 1 y 1 y 1 0 y 5 y 4 0F k q 2 y 1 y 1 0 y 4 y 3 0F k q 3 15 0 15 y 2 y 2 0F k q 4 y 15 y 15 0 y 32 y 17 0F k q 5 y 2 y 2 0 y 4 y 2 0F k q 6 20 0 20 20 0 20



11/22

either a higher circuit between pinches or a circuit above allpinches with an induced pinch.

U nder such conditions, all resulting heat flows cascadeddown are positive. On the source side, the higher-circuit solu-tion is given by

q F C F s Min H F , Q 30 . 4k k F k y 1

F C H F C F F

q s Min , Q y q

k s k q 1, . . . , k q m .k k F i K 5F i s k 31 .Equation 30 states that at the first interval, all the heat

available will be used provided it is lower than the overallmaximum. Eq uation 31 states that at every interval, the ma xi-mum that can be transferred is the surplus. In turn, for thesink plant, the higher solution is

q FH F s Q 32 .k F

q FH s 0 k s k F q 1, . . . , k F q m . 33 .k K

This means that if the sink plant can transfer all the maxi-mum possible heat Q in the first interval of transfer k F ,F then the higher-circuit solution will consist of this intervalonly.

At the other extreme, we have a solution that maximizesthe transfer of heat to the interval below in the heat-sourceplant, minimizing the transfer to the intermediate fluid. Thissolution is called the lower -circuit solution because it startsand ends at the lowest intervals possible. In such case, thesolutions for the sink and the source plants are somewhatrelated. Indeed, by transferring heat to lower intervals in thesource plant, one must make sure that the plant does notneed the heat at the same interval. The adjusted cascaded

heat values already account for this. Then, the lower-circuitsolution for the sink plant is given by

q FH F s Ma x y C F , 0 34 . 4k k k y 1

F H C F H F F q s Ma x y y q , 0 , k s k q 1, . . . , k q m .k k i K 5F i s k 35 .

Now let k q be the first interval with nonzero heat trans-ferred from the intermediate fluid to the sink plant, that is,

q F H F q . FH qk is such tha t q s 0, k s k , . . . , k y 1 ; q 0. Next,k k all the surplus heat in the source plant can be transferred

down until interval k q

is reached. From then on, only theminimum amount of heat should be transferred to the inter-mediate fluid. This minimum should be at least equal to q FH i to guarantee that temperature constraints have a chance ofbeing satisfied. Thus, the lower-circuit solution for the sourceplant is

q F C s 0 k s k F , . . . , k q

y 1 36 . .k

q F C s q FH k s k q , . . . , k F q m . 37 .k k K

By construction, no one-circuit solution can:1. Start at a lower interval;2. Transfer less heat fro m the intermediate fluid to the sink

plant at any interval defined by the lower solution.As a result of the calculation of the cumulative heat de-

mands, a limit for the starting point of the unique circuit isestablished. In view of the preceding, a second test for thefeasibility of a single circuit transferring the maximum sav-ings Q in any of the regions consists of constructing theF higher or lower solution and checking if the following equa-tions are satisfied:

k F T y T .k y 1 k F H Q G q F i

F F T y T . F k y 1 k q m K i s k k s k F , . . . , k F q m y 1 38 . .K

k F q m K F T y T .k k q m K E C Q G q F i F F T y T .k y 1 k q m i s k K

k s k F q m , . . . , k F q 1 . 39 . . .K

These equat ions have been o btained by substituting E q. 16 inEqs. 14 and 15.

It should be noted that the lower solution obtained by thepreceding procedure is not always feasible. Some other lowersolutions might exist, not necessarily covering the last inter-vals of the region between pinches, but covering a region thatends somewhere above it.

Example 4

In this example, Test Case 2 from Linnhoff and Hind- .marsh 1983 is plant 1 and problem 4sp1 is plant 2. The dat a

for the separate plants are shown in Table 5. Note that T minfor plant 1 is 20 C, while T for plant 2 is 10 C. Pinchmintemperatures and minimum utility consumption for each ofthe plants are shown in Table 6.

Di rect I ntegration Solution. Table 7 shows the results ofthe pinch analysis. The interval between pinches goes from

Table5. Datafor Example4

F T T Q s t . . . .Streams kWr C C C kW

Test Case 2 .H1 Hot 2.0 150 60 180.0 .C2 Cold 2.5 20 125 262.5 .H3 Hot 8.0 90 60 240.0 .C4 Cold 3.0 25 100 225.0

.S Stream 270 270 107.5 .CW Water 0.9 38 82 40.0T s 20 Cmin

Problem 4sp1 .C1 Cold 7.62 60 160 762 .H2 Hot 8.79 160 93 589 .C3 Cold 6.08 116 260 876 .H4 Hot 10.55 249 138 1171

.S Steam 270 270 128 .CW Water 5.68 38 82 250

T s 10 Cmin



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Table6. Individual Plant Pinch Analysis for Example4

Pinch Heat in g U t ilit y Cooli ng U t ilit y . . .P ro blem Temp. C kW kW

Test case 2 90 107.5 40.04sp1 249 128.0 250.0

90 C to 249 C. After considering all streams in a single set,the resulting combined pinch is located at 249 C upper limit

.of the interval between pinches . This is the consequence ofthe large availability of heat to transfer that plant 2 has in allthe intervals between pinches. This amount of heat is suffi-cient to supply the entire demand of plant 1. Therefore, themaximum possible heat savings for the direct integration arethe original minimum utility of plant 1, that is 107.5 kW. Thisis also the result obtained by solving problem 1.

In direct In tegration Solutions. After the hot temperaturescale in plant 2 is shifted down 10 C, the interval betweenpinches is from 239 C to 90 C. Table 8 shows the pinch anal-ysis for the indirect integration. The combined pinch then

.results at the upper bound 239 C . Fewer intervals than in

the case of direct integration are found, since some of theextreme temperatures now coincide due to the shift. The so-lution of problem 1 is Q s 107.5 kW, which is equal to theF maximum possible savings that can be obtained either withdirect or indirect integration. Therefore, in this case the shiftdoes not have any effect in reducing the amount that can betransferred between pinches. Table 8 shows that there is nodemand in the upper intervals of plant 1, and the shift doesnot appreciably decrease the large availability of heat in plant2 in the region between pinches.

An implementation of this indirect integration fo llows. Thetest of feasibility for a single circuit is applied first to a circuitcovering a ll the intervals between pinches. This solution isfeasible, and Table 9 shows the values obtained for the pa-

Table7. Pinch Analysisfor DirectIntegrationinExample4

Test Case 2 4sp1 Combined PlantI I I I I I I I I C C C P T q S q S q S i i min i i mi n i i min

. . . . . . . . . .C kW kW kW kW kW kW kW kW kW

270 0 0 107 .5 y 127.8 y 127.8 127 .8 y 127.8 y 127.8 127 .8249 0 0 353.1 225.3 353.1 225.3170 0 0 y 31.5 193.8 y 31.5 193.8160 0 0 56.4 250.2 56.4 250.2150 10.0 10.0 28.2 278.4 38.2 288.4

145 y 3.5 6.5 39.5 317.9 36.0 324.4138 y 6.0 0.5 y 58.9 259.0 y 64.9 259.5126 y 3.0 y 2.5 7.0 266.0 4.0 263.5120 y 94.5 y 97.0 31.6 297.6 y 62.9 200.6

93 y 10.5 y 107.5 y 22.9 274.7 y 33.4 167.290 90.0 y 17.5 y 152.4 122.3 y 62.4 104.8I I I C P W W W min min min70 45.0 27.5 0 122.3 45.0 149.8 . . .kW kW kW60 y 82.5 y 55.0 0 122.3 y 82.5 67.345 y 12.5 y 67.5 40 .0 0 122.3 250 .0 y 12.5 54.8 182 .5

Table8. Pinch Analysis for Indirect IntegrationinExample4

Test Case 2 4sp1 Combined PlantI I I I I I I I I C C C P T q S q S q S i i mi n i i min i i min

. . . . . . . . . .C kW kW kW kW kW kW kW kW kW

260 0 0 107 .5 y 127.8 y 127.8 127 .8 y 127.8 y 127.8 127 .8239 0 0 353.1 225.3 353.1 225.3160 0 0 y 31.5 193.8 y 31.5 193.8150 10.0 10.0 28.2 222.0 38.2 232.0145 y 8.5 1.5 95.8 317.8 87.3 319.3128 y 4.0 y 2.5 y 39.3 278.5 y 43.3 276.0120 y 14.0 y 16.5 y 19.6 258.9 y 33.6 242.4116 y 91.0 y 107.5 30.4 289.3 y 60.6 181.8

90 31.5 y 76.0 8.2 297.5 39.7 221.5I II C P W W W min min min83 103.5 27.5 y 175.3 122.2 y 71.8 149.7 . . .kW kW kW60 y 82.5 y 55.0 0 122.2 y 82.5 67.245 y 12.5 y 67.5 40 .0 0 122.2 250 .0 y 12.5 54.7 182 .5

Table9. Someof theIndirect SolutionstoExample4

No. of T T F up down . . .So lut io n I nterva ls C C kWr C

All intervals 7 239 90 0.721Lower circuit 5 150 90 1.792H igher circuit 2 239 150 1.208

rameters ending temperatures and rate heat-capacity prod-.uct . The higher-circuit solution is shown in Table 10. The

position of the resulting circuit is shown in Figure 14. Notethat Eqs. 38 and 39 are satisfied. Moreover, the intermediatesolutions between the circuit spanning all intervals and thehighest possible circuit a re feasible.

Finally, the lower-circuit solution is presented in Table 11,and its position is shown in Figure 15. The intermediate cir-cuits between the circuit spanning all intervals and the lowestpossible circuit a re proven feasible. Ot her solutions can befound each time that a certain amount of heat could be cas-caded and the one-circuit solutions that result are feasible.However, no solution will be able to start below the limitestablished by the lower solution. In this sense, the problemhas a large finite number of possible solutions that requirefurther analysis, taking into account the resulting heat-ex-changer network and the economic aspects.

Table10. Higher CircuitSolutiontoExample4Test Case 2 4sp1Q s 107.5 kWE

I E H II I I E C . . . . .Interval q kW q kW q kW kW q kWk k k k k IIp q 1s 2 0 107.5 353.1 321.6 107.5IIp q 2 s 3 0 0 y 31.5 0 0IIp q 3s 4 10 0 28.2 28.2 0IIp q 4s 5 y 8.5 0 95.8 36.9 0IIp q 5s 6 y 4.0 0 y 39.3 0 0IIp q 6s 7 y 14.0 0 y 19.6 0 0IIp q 7s 8 y 91.0 0 30.4 30.4 0



13/22

Figure 14. Higher single-circuit solution for Example 4.

Example 5 .In this case, an example taken from Trivedi 1988 is plant

.1 and example 1 from Ciric and Floudas 1991 is plant 2.The da ta for the separat e plants a re shown in Table 12. Pinchtemperatures and minimum utility consumption for each ofthe plants are shown in Table 13.

Direct Integration Solution. Table 14 shows the results ofthe pinch analysis. The interval between pinches goes from160 C to 200 C. After considering all streams in a single set,

the resulting combined pinch is located at 200 C upper limit.of the interval between pinches . The maximum possible sav-

ings for the direct integration are 104.4 kW. So lving problem1 gives the targeting values of the heat to be transferred in

.each of the zones. A minimum of 52.9 kW Q has to beAtransferred in the zone a bove both pinches in order to a ttain

the maximum possible savings.

In direct In tegration Solution s. In this case, the hot temper-ature scale in plant 2 below its pinch is shifted down 20 C,while above the pinch the same scale is shifted up 20 C. Agap of 40 C is then created in plant 2, and no integration ispossible in this zone. The interval between pinches is from

.160 C to 180 C hot scale of plant 1 . Table 15 shows thepinch analysis for the indirect integration. Again, the com-

.bined pinch is at the upper bound of this interval 180 C .

Table11. Lower Circuit SolutiontoExample4Test Case 2 4sp1Q s 107.5 kWE

I I E H II E C . . . . .Interval q kW kW q kW q kW q kWk k k k k IIp q 1s 2 0 0 0 353.1 0IIp q 2 s 3 0 0 0 y 31.5 0IIp q 3s 4 10 10 0 28.2 0IIp q 4 s 5 y 8.5 1.5 0 95.8 0IIp q 5s 6 y 4.0 y 2.5 2.5 y 39.3 2.5IIp q 6 s 7 y 14.0 y 16.5 14.0 y 19.6 14.0IIp q 7s 8 y 91.0 y 107.5 91.0 30.4 91.0

Figure 15. Lower single-circuit solution for Example 4.

The maximum possible savings from indirect integration, 65.2kW, are 38% lower than the savings from direct integration.Solving problem 1 gives the targeting values of the heat to be

.transferred in each of t he zones. A minimum of 13.7 kW Q Ais to be transferred in the zone a bove both pinches to attainthe maximum possible savings. This represents 26% of theamount to be transferred in the direct case.


F T T Q s t . . . .Streams kWr C C C kW

Tr i edi .H1 Hot 7.032 160 110 351.6 .H2 Hot 8.44 249 138 936.8 .H3 Hot 11.816 227 106 1,429.7 .H4 Hot 7.0 271 146 875.0 .C1 Cold 9.144 96 160 585.2 .C2 Cold 7.296 115 217 744.2 .C3 Cold 18 140 250 1,980.0

.S Steam 300 300 404.8 .CW Water 34.43 70 90 688.6

T s 20 CminCiric and Floudas

.H5 Hot 10 300 200 1,000 .H6 Hot 120 200 100 12,000 .C4 Cold 15 70 270 3,000 .C5 Cold 25 70 190 3,000 .C6 Cold 50 70 180 5,500

.S Steam 300 300 600

.CW Water 105 70 90 2,100T s 20 Cmin

Table13. Individual Plant Pinch Analysisfor Example5

Pinch Heating Ut i li ty Cooling Ut i li ty . . .P roblem Temp. C kW kW

Trivedi 160 404.8 688.6C iric and Floudas 200 600 2,100



14/22

Table14. Pinch Analysis for DirectIntegrationin Example5Trivedi C iric and Floudas Combined Plant

I I I I I I I I I C C C P T q S q S q S i i min i i min i i min . . . . . . . . . .C kW kW kW kW kW kW kW kW kW

300 0 0 404 .8 100.0 100.0 600 .0 100.0 100.0 900 .4290 0 0 y 95.0 5.0 y 95.0 5.0271 7.0 7 y 5.0 0.0 2.0 7.0270 y 231.0 y 224 y 105.0 y 105.0 y 336.0 y 329.0

249 y 3.07 y 254.7 y 60.0 y 165.0 y 90.7 y 419.7237 y 98.6 y 353.3 y 50.0 y 215.0 y 148.6 y 568.3227 33.3 y 320.0 y 85.0 y 300.0 y 51.7 y 620.0210 19.6 y 300.4 y 300.0 y 600.0 y 280.4 y 900.4200 39.2 y 261.2 600.0 0.0 639.2 y 261.2180 y 143.7 y 404.8 600.0 600.0 456.3 195.2160 249.9 y 155.0 420.0 1,020.0 669.9 865.0146 86.8 y 68.2 240.0 1,260.0 326.8 1,191.8138 7.2 y 61.0 90.0 1,350.0 97.2 1,289.0135 184.4 123.4 570.0 1,920.0 754.4 2,043.4116 113.1 236.5 180.0 2,100.0 293.1 2,336.5I II C P W W W min min min110 47.3 283.8 120.0 2,220.0 167.3 2,503.8 . . .kW kW kW106 0 283.8 180.0 2,400.0 180.0 2,683.8

100 0 283.8 688 .6 y 900.0 1,500.0 2,100 .0 y 900.0 1,783.8 2,684 .1

An implementation of this indirect integration follows.Table 16 shows the adjusted cascade heats. Only intervalsabove plant 1 pinch are shown since these intervals include

.the two zones of interest upper and between pinches . Sincean induced pinch appears in plant 1, a single interval is leftfor the transfer of the amount Q . In this case, a coincidenceAin the higher and lower circuits is found. A transfer of 13.7kW is required in the upper zone. Once this amount is trans-

ferred, a circuit is established in the interval b etween pinches.This circuit will transfer 65.3 kW. Then, the solution for theindirect integration in Example 5 is shown in Figure 16.

Use of Composite Cur es. For comparison, the methodthat uses countercurrent profiles for the grand compositecurves is also applied to the direct integration of the plants .Figure 17a . A pocket in plant 1 allows the transfer of therequired amount of heat above pinches that makes possible

Table15. Pinch Analysis for Indirect Integrationin Example5Trivedi C iric and Floudas Combined Plant

I I I I I I I I I C C C P T q S q S q S i i mi n i i min i i min . . . . . . . . . .C kW kW kW kW kW kW kW kW kW

300 0 0 404 .8 100.0 100.0 600 .0 100.0 100.0 939 .6310 0 0 y 195.0 y 95.0 y 195.0 y 95.0271 7.0 7 y 5.0 y 100.0 2.0 y 93.0270 y 231.0 y 224 y 105.0 y 205.0 y 336.0 y 429.0249 y 30.7 y 254.7 y 60.0 y 265.0 y 90.7 y 519.7237 y 69.0 y 323.7 y 35.0 y 300.0 y 104.0 y 623.7230 y 29.6 y 353.3 y 90.0 y 390.0 y 119.6 y 743.3227 13.7 y 339.6 y 210.0 y 600.0 y 196.3 y 939.6220 78.4 y 261.2 78.4 y 261.2GAP

180 y 143.7 y 404.8 600.0 0.0 456.3 y 404.8160 249.9 y 155.0 420.0 420.0 669.9 265.0146 86.8 y 68.2 240.0 660.0 326.8 591.8138 7.2 y 61.0 90.0 750.0 97.2 689.0135 184.4 123.4 570.0 1,320.0 754.4 1,443.4116 113.1 236.5 180.0 1,500.0 293.1 1,736.5I II C P W W W min min min110 47.3 283.8 120.0 1,620.0 167.3 1,903.8 . . .kW kW kW106 0 283.8 780.0 2,400.0 780.0 2,683.8

80 0 283.8 688 .6 y 900.0 1,500.0 2,100 .0 y 900.0 1,783 .8 2 ,723 .3



15/22

Table16. AdjustedCascadedHeatsfor Example5

Trivedi C iric and FloudasI I I I I I . . . .Interval q kW kW q kW kWk k k k

1 0 353.3 100.0 686.32 0 353.3 y 195.0 491.33 7.0 360.3 y 5.0 486.34 y 231.0 129.3 y 105.0 381.35 y 30.7 98.6 y 60.0 321.36 y 69.0 29.6 y 35.0 286.3

7 y 29.6 0 y 90.0 196.38 13.7 13.7 y 210.0 y 13.79 78.4 GAP

10 y 143.7 y 65.3 600.0 600.0

the realization of maximum savings. Figure 17b shows how the same method can be used for the indirect transfer of heatbetween the plants. The vertical line in the profile of plant 2represents the gap where no integration is possible. Still, anamount of heat can be transferred from plant 1 to plant 2inside the pocket to allow the realization of the maximumsavings for the indirect integration.

Example 6 This example consists o f a crude unit processing 150,000

.bblr d 24 MLr d and an FCC plant processing 40,000 bbl r d .64 MLr d . The crude unit is plant 1, while the FCC unit isplant 2. The data for the separate plants are shown in Table

.17. The T in this case is 5.6 C equivalent to 10 F f orminboth plants, and the downward shift of plant 2 during inter-mediate fluid integration is also 5.6 C. Pinch temperaturesand minimum utility consumption are shown in Table 18 andis the result of solving problem 1 for each of the plants.

Figure 16. Assisted-circuitsolutionfor Example5.


Di rect I ntegration Solution. Considerable energy recoveryis possible due to the large temperature difference between

.pinches 471.1 C to 143.5 C . The FCC unit needs a largeamount of cold utility below its pinch, due to the high tem-


F T T Q s t

. . . .Streams MWr C C C MWCrude Unit

.C1 Cold 0.6230 30.0 127.3 60.64 .C2 Cold 0.6945 127.3 239.3 77.78 .C3 Cold 0.7855 239.3 352.9 89.24 .H 1 Hot 0.0655 127.3 37.8 5.86 .H 2 Hot 0.3053 143.5 26.7 35.67 .H 3 Hot 0.1439 261.4 37.8 32.18 .H 4 Hot 0.0334 326.7 37.8 9.64 .H 5 Hot 0.3400 347.3 268.3 26.85 .H 6 Hot 0.2744 163.3 79.6 22.98 .H 7 Hot 0.1771 194.5 142.6 9.20 .H 8 Hot 0.2617 261.4 206.3 14.42 .H 9 Hot 0.1221 336.3 239.8 11.78

.F Fuel 124.2856 427.2 426.7 69.05 .CW Water 0.8968 15.6 26.7 9.96

T s

5.6 CminFCC Unit

.C4 Cold 0.0831 471.1 532.2 5.07 .H 10 Hot 0.0083 348.2 21.1 2.73 .H 11 Hot 0.0078 243.9 21.1 1.73 .H 12 Hot 0.0773 147.2 48.9 7.59 .H 13 Hot 0.0252 348.2 115.5 5.86 .H 14 Hot 0.0362 313.2 232.2 2.93 .H 15 Hot 0.1503 190.1 107.2 12.46

.F Fuel 9.1262 538.3 537.8 5.07 .CW Water 2.9990 15.6 26.7 33.32

T s 5.6 Cmin



16/22

Table18. Individual Plant Pinch Analysisfor Example6

Pinch Heat ing U t ilit y C o oling U t ilit y . . .Plant Temp. C MW MW

Crude unit 143.5 69.0 10.0FCC unit 471.1 5.1 33.3

peratures of the streams emanating from the reactor. On theother hand, the crude unit needs a large amount of hot utilityin order to heat up its streams during the fractionation proc-ess. Table 19 shows the results of the pinch analysis for thedirect integration. The intervals below the lower pinch havebeen merged since heat recovery is not performed here. Theresulting combined pinch is located at 163.3 C, and is theresult of a compensation for the heat in the first interval ofplant 1 by heat provided by plant 2. After this interval, theheat that plant 2 has available in the rest of the intervalsbetween pinches is not sufficient to supply the demand of thecorresponding intervals of plant 1. Therefore, the maximumpossible heat savings are 15.1 MW. Note that 13.8 MW aretransferred above the combined pinch, and 1.3 MW below the combined pinch.

In direct In tegration. Table 20 shows the results of the pinchanalysis for the indirect integration. After the hot tempera-ture scale in plant 2 is shifted down 18 C, the interval be-tween pinches is from 465.6 C to 143.5 C. The maximum pos-sible savings are now 13.9 MW. This a mount is 1.2 MWsmaller than the maximum possible heat found for the directintegration.

Table19. Pinch Analysis for DirectIntegrationinExample6

Crude U nit FC C U nit C ombined P lantI I I I I I I I I C C C P

T q S q S q S i i min i i min i i mi n . . . . . . . . . .C MW MW MW MW MW MW MW MW MW

532.2 0 0 69 .1 y 5.1 y 5.1 5.1 y 5.1 y 5.1 59 .0471.1 y 8.1 y 8.1 0.0 y 5.1 y 8.1 y 13.2348.2 y 0.7 y 8.8 0.0 y 5.0 y 0.7 y 13.9347.3 y 4.9 y 13.7 0.4 y 4.7 y 4.5 y 18.4336.3 y 3.1 y 16.8 0.3 y 4.4 y 2.8 y 21.2326.7 y 3.9 y 20.7 0.5 y 3.9 y 3.5 y 24.6313.2 y 13.0 y 33.7 3.1 y 0.8 y 9.9 y 34.5268.3 y 4.3 y 38.1 0.5 y 0.3 y 3.9 y 38.4261.4 y 3.7 y 41.8 1.2 0.9 y 2.6 y 40.9244.9 y 0.1 y 41.9 0.1 0.9 y 0.1 y 41.0243.9 y 0.5 y 42.5 0.3 1.2 y 0.2 y 41.2

239.8 y 1.9 y 44.4 0.6 1.8 y 1.3 y 42.6232.2 y 6.6 y 51.0 1.1 2.9 y 5.5 y 48.1206.3 y 6.1 y 57.1 0.5 3.4 y 5.6 y 53.8194.5 y 1.5 y 58.6 0.2 3.6 y 1.3 y 55.1190.1 y 9.1 y 67.8 5.1 8.7 y 4.0 y 59.0163.3 y 1.1 y 68.8 3.1 11.8 2.0 y 57.0I II C P W W W min min min147.2 y 0.2 y 69.1 1.0 12.8 0.7 y 56.3 . . .kW kW kW143.5 0.2 y 68.8 0.3 13.0 0.5 y 55.8

26.7 9.8 y 59.1 10 .0 15.2 28.2 33 .3 24.9 y 30.9 28 .2

Table20. Pinch Analysisfor IndirectIntegrationinExample6

C rude U nit FCC U nit C ombined PlantI I I I I I I I I C C C P T q S q S q S i i min i i min i i min

. . . . . . . . . .C MW MW MW MW MW MW MW MW MW

526.7 0 0 69 .1 y 5.1 y 5.1 5.1 y 5.1 y 5.1 60 .1465.6 y 8.8 y 8.8 0 y 5.1 y 8.8 y 13.9347.3 y 2.1 y 10.9 0 y 5.1 y 2.1 y 15.9

342.7 y 2.8 y 13.7 0.2 y 4.9 y 2.6 y 18.6336.3 y 3.1 y 16.8 0.3 y 4.5 y 2.8 y 21.3326.7 y 5.5 y 22.3 0.6 y 3.9 y 4.9 y 26.2307.7 y 11.4 y 33.7 2.7 y 1.2 y 8.7 y 34.9268.3 y 4.3 y 38.1 0.5 y 0.7 y 3.9 y 38.8261.4 y 3.7 y 41.8 1.2 0.5 y 2.6 y 41.3244.9 y 0.7 y 42.5 0.4 0.8 y 0.3 y 41.6239.8 y 0.4 y 42.8 0.1 0.9 y 0.3 y 41.9238.3 y 3.0 y 45.8 0.9 1.8 y 2.1 y 44.0226.7 y 5.2 y 51.0 0.8 2.7 y 4.4 y 48.3206.3 y 6.1 y 57.1 0.5 3.2 y 5.6 y 54.0194.5 y 3.4 y 60.5 0.4 3.6 y 3.0 y 56.9184.6 y 7.2 y 67.8 4.1 7.6 y 3.2 y 60.1I II C P W W W

min min min163.3 y 1.3 y 69.1 3.8 11.4 2.5 y 57.6 . . .kW kW kW143.5 0.2 y 68.8 0.2 11.6 0.4 y 57.2142.6 9.8 y 59.1 10 .0 16.6 28.2 33 .3 26.4 y 30.8 29 .3

Figure 18 shows the result of the higher-circuit solution.Without loss of generality, some of the intervals have beenlumped to clarify the illustration in the upper part of thezone between pinches and below the pinch of plant 1. This

Figure 18. Higher and lowercandidate single-circuit so -lutions for Example 6.



17/22

candidate solution covers all the intervals and in plant 2transfers all the heat in each interval except in the last one,where a lower amount of heat is transferred. In this last in-terval the value of the ma ximum possible heat Q is reached.E The lower-circuit solution is also shown in Figure 18. Onlyfour intervals are required to transfer the maximum possibleheat. In this case, the test with Eqs. 38 and 39 fails for bothcandidate solutions. Then, a single circuit will transfer asmaller amount of heat than the maximum Q . This is deter-E mined next.

MaximumAmountTransferred by a Single CircuitIn the case where a single circuit cannot realize the entire

target savings, one can still consider establishing a single cir-cuit and realize only a portion of these total savings. Con-sider the unassisted case first. The maximum amount of heat

transferred by a single circuit, for which its location starting.and ending intervals is known, can be obtained by solving

the following problem:

Min I q II .0 m

s.t.I I s y Q 0 0 E II II s 0 0

j s j q q j i s 1, . . . , k E y 1 ; j s I , I I .i i y 1 i

I s I q q I q q E H i i y 1 i i E E i s k , . . . , k q m .E I I I I I I E C 5 s q q y q i i y 1 i i j s j q q j i s k E q m q 1 , . . . , m ; .i i y 1 i E

j s I , I I

I I s m m II II s y Q m m E

I I s 0, IIII s 0 40 .p p k k

E H E E F T G q k s k , . . . , k q m y 1 . E i i E E E i s k i s k

k E q m k E q m E E E H F T s q E i i

E E i s k i s k

k E q m k E q m E E E C E E F T G q k s k q 1 , . . . , k q m . . E i i E

i s k i s k

k E q m k E q m E E E C F T s q E i i

E E i s k i s k

I , II , q E H , q E C G 0,i i i i

where T s T y T .i i y 1 i

The equalities that correspond to the heat balances in eachinterval have been split into two sets of equalities. The firstone considers only those intervals in which a ll the heat cas-cades down. The second set consists of the intervals in whichthe transfer is taking place. This problem is linear and offersno major d ifficulties.

For assisted cases, the set of equations for the balances ineach interval in problem 40 are replaced by the following new sets:

j s j q q j i s 1, . . . , k A y 1 ; j s I , I I .i i y 1 i

I s I q q I q q AC i i y 1 i i A Ai s k , . . . , k q m 41 . .AI I I I I I A H 5 s q q y q i i y 1 i i j s j q q j i s k A q m q 1 , . . . , k E y 1 ; . .i i y 1 i A

j s I , I I

j s j q q j i s k E q m q 1 , . . . , k B y 1 ; . .i i y 1 i E j s I , I I

I s I q q I q q BC i i y 1 i i B B i s k , . . . , k q m 42 . .B I I I I I I B H 5q s q q y q i i y 1 i i j s j q q j i s k B q m q 1 , . . . , m ; .i i y 1 i B

j s I , I I .

In addition to these equations, the following two new setsof constraints have to be added in order to account for thefeasibility of a single circuit in each of the aforementionedregions:

k k

AC A AF T G q k s k , . . . , k q m y 1 . A i i AA Ai s k i s k A Ak q m k q m A A

AC F T s q A i i A Ai s k i s k

A Ak q m k q m A AA H A AF T G q k s k q 1 , . . . , k q m . . A i i A

i s k i s k A Ak q m k q m A A

A H F T s q 43 . A i i A Ai s k i s k

k k B C B B F T G q k s k , . . . , k q m y 1 . B i i B

B B i s k i s k B B k q m k q m B B

Bc F T s q B i i B B i s k i s k

B B k q m k q m B B B H B B F T G q k s k q 1 , . . . , k q m . . B i i B

i s k i s k B B k q m k q m B B

BH F T s q . 44 . B i i B B i s k i s k



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Optimum Location of a Single CircuitProblem 40 is expressed for known fixed starting and end-

ing intervals. An MILP formulation to determine the opti-mum points of insertion of such a circuit in a ny of the regionsis presented next. The case of a circuit between pinches ispresented.

Consider the following general binary variables:

1 Interva l i q 1 is the starting interval .FH Y si 0 Otherwise1 Interva l i is the ending intervalF C Y si 0 Otherwise .

To guarantee that only an interval is a starting r ending one,the following inequalities are introduced:

p I y 1FH Y s 1 45 . i

IIi s p

p I

F C Y s 1. 46 . i IIi s p q 1

In addition, the following variables are defined:

1 Interva l i is in the circuitF Z si 0 Otherwise .These variables are related to Y F H an d Y F C by the follow-i i

ing eq ualities:

Z F II s Y F H II 47 .p q 1 p

Z F s Z F q Y F H y Y F C i s p II q 2 , . . . , p I . 48 . .i i y 1 i y 1 i y 1

These equalities are needed to restrict the values of theheat transferred to and from the intermediate fluid to zerofor intervals that are not in the circuit. For example, considera circuit starting in the first interval below the pinch temper-ature of plant 2. Then

Y F C II s 1 and Z F II s Y F H II s 1;p p q 1 p

otherwise, Y F H II s 0 and the first interval of transfer will bep located at a low level. That is, Z F II s 0. Lets now considerp q 1that the third interval is the starting one. Then

Y F C II s 1 a ndp q 2

Z F II s Z F II q Y F H II y Y F C II s 0q 1q 0s 1.p q 3 p q 2 p q 2 p q 2

In any case, Y F C II must be zero in the interval in which Y F H IIp p is one in order for the circuit to span at least an interval.Finally, consider that the circuit ends in the fifth interval.

Then

Y F C II s 1 andp q 5

Z F II s Z F II q Y F H II y Y F C II s 1q 0y 1s 0.p q 6 p q 5 p q 5 p q 5

Therefore, there will not be a transfer in the sixth interval.An optimization problem based on the preceding binary

variables to solve the unassisted case is then proposed.


I I s y Q 0 0 E II II s 0 0

j s j q q j i s 1, . . . , p II ; j s I , I Ii i y 1 i

I s I q q I q q E H i i y 1 i i II Ii s p q 1 , . . . , p .II I I I I E C 5 s q q y q i i y 1 i i j s j q q j i s p Iq 1 , . . . , m ; j s I , I I .i i y 1 i I I s m m II II s y Q m m E

I I s 0p

IIII s 0p k k

E E H II IF Z T G q k s p q 1 , . . . , p y 1 . . E i i i II I Ii s p q 1 i s p q 1I Ip p

E E H F Z T s q E i i i II I Ii s p q 1 i s p q 1I Ip p

E E C II IF Z T G q k s p q 2 , . . . , p . E i i i i s k i s k

I Ip p E E C F Z T s q E i i i

II I Ii s p q 1 i s p q 1 49 .

q E H y UZ E F 0 i s p IIq 1 , . . . , p I .i i q EC y UZ E F 0 i s p IIq 1 , . . . , p I .i i Z E II s Y E H IIp q 1 p

Z E s Z E q Y E H y Y E C i s p II q 2 , . . . , p I .i i y 1 i y 1 i y 1p I y 1

E H Y s 1 i IIi s p

p I

E C Y s 1 i IIi s p q 1

I , II , q E H , q E C , Z E G 0i i i i i E H E C 4Y , Y g 0 , 1 .i i



19/22

In these equations, U is an upper bound of the total heatthat can be transferred. This is a mixed-integer nonlinearproblem having a single nonlinearity that consists of theproduct of a continuous variable times a binary variable. Thefollowing constraints are introduced to eliminate this nonlin-earity:

E B y Z F 0i i E E B G 0

i B

sF Z

i i E E ~E F y B y 1y Z F 0 . .m 50 .i i F G 0 E E Z s 0, 1 . F y B G 0 .i i E Z s 0 , 1 , .i where is a sufficiently large number.

Assisted cases can also be solved by introducing similarconstraints in the appropriate temperature intervals.

( ) Example 6 continued

The linear formulation of problem 49 by introducing Eq. .50 is implemented in G AMS Bro oke et al., 1996 . The MIL P

model obtained is solved with the C PLE X solver. The twooptimum solutions found are shown in Table 21 and Figure19. Any of the circuits is capable of transferring 12.6 MW,which represents 91% of the total possible savings predictedby problem 1. Since a single circuit is not capable of transfer-ring the maximum possible heat, a new formulation is pre-sented that achieves this target with the minimum number ofcircuits.

OptimumLocation of Many CircuitsWhen a single circuit is not capable of realizing the maxi-

mum target savings, a step-by-step increase of the number ofcircuits seems a logical procedure to reach the minimum re-quired. Then at each step the optimal location of an increas-

ing number of circuits is to be found maximizing the overallheat transfer. The following modification of Eq. 49 is pro-posed in order to find the locaton of a number n of circuits:


I I l . q y Q 0 0 E II II s 0 0 j s j q q j i s 1, . . . , p II ; j s I , I Ii i y 1 i

n I I I E H s q q q q i i y 1 i i , l l s 1 II I

i

s p

q1 , . . . , p

.n II I I I I E C s q q y q i i y 1 i i , l l s 1Table21. SingleCircuit SolutionstoExample6

No. of T T F up down . . .S olut io n I nt erva ls C C MWr C

Optimum 1 4 226.7 163.3 0.199Optimum 2 3 206.3 163.3 0.293

j s j q q j i s p I q 1 , . . . , m ; j s I , I I .i i y 1 i I I s m m

II II l . s y Q m m E

I I s 0p

IIII s 0 51 .p

k k l . E E H F Z T G q E i , l i i , l II I Ii s p q 1 i s p q 1

II Ik s p q 1 , . . . , p y 1 . .I Ip p

l . E E H F Z T s q E i , l i i , l II I Ii s p q 1 i s p q 1

I Ip p l . E E C F Z T G q E i , l i i , l

i s k i s k II Ik s p q 2 , . . . , p .

I Ip p

l . E E C F Z T s q E i , l i i , l II I I l s 1, . . . , n i s p q 1 i s p q 1E H E I I Iq y UZ F 0 i s p q 1 , . . . , p .i , l i , l E C E II Iq y UZ F 0 i s p q 1 , . . . , p .i , l i , l E E H

II I IZ s Y p q 1, l p , l E E E H E C Z s Z q Y y Y i , l i y 1, l i y 1, l i y 1, l

I I Ii s p q 2 , . . . , p .Ip y 1

E H Y s 1 i , l IIi s p Ip

F C

Y s 1 i , l IIi s p q 1 I , II , q E H , q E C , Z E G 0i i i , l i , l i , l

E H E C 4Y , Y g 0 , 1 .i , l i , l

The strategy to find the optimum number of circuits con-sists of a trial procedure. At each step, the value obtained bysolving Eq. 51 is compared with the maximum heat possibleto be transferred. If the difference is not zero, then the valueof l is increased to approach the target. The minimum num-ber of circuits resulting from this procedure will be less than

or equal to the number of intervals between pinches.

( ) Example 6 continued

If the formulation just presented is applied to E xample 6, aminimum of two circuits is obtained. This set of two circuitswill transfer all the heat predicted by problem 1. The loca-

tion of both circuits for one of the possible solutions first.alternative to problem 51 is shown in Figure 20. As illus-

trated, this solution is the combination of one of the possible .solutions obtained f or a single circuit covering four intervals



20/22

Figure 19. Two alternative single-circuit solutions forExample 6.

and a circuit covering the last interval. Another possible solu- .tion second alternative is shown in Figure 21. In this alter-

native, the two circuits overlap. Due to degeneracy, there area large number of possible solutions. The two alternativesconsidered here were obtained using G AMS with CPL EX .

(Figure 20. Two-circuits solutions forExample6 first al -)ternative .

(Figure 21. Two-circuits solutions for Example 6 sec -)ond alternative .

Indirect Integration Using SteamThe use of steam for indirect integration imposes extra

restrictions on the maximum amount of heat that can betransferred. Consider that steam at fixed pressure levels is

generated by the plant having an excess of heat higher pinch.temperature . For simplicity, assume first that a single

steam-temperature level is specified between pinches, and .that is the only indirect fluid used Figure 22 . By computing

the cooling utility required for the source plant from its pinchdown to the level of steam generation, the heat load of thissteam can be established. This amount is the maximum heatthis steam will be able to transfer to the sink plant. Considernow the sink plant and the zone between pinches. This plantwill be able to use the steam coming from the source plant toreduce its heating utility demands only if this steam tempera-ture is above its pinch temperature. In addition, the maxi-mum load that the sink plant can accept is the deficit it pre-sents between the steam level and the pinch. Thus, the use oflatent heat of a single steam stream may reduce the opportu-nities of integration.

An alternative is the use of the utility system to balancethe steam supply and demand of source and sink plants, re-

.spectively Hui and Ahmad, 1994 . In any case, the difficul-ties arise when more than one steam level is considered. Hui

.and Ahmad 1994 consider the utility as a market, sellingand buying utilities at fixed prices from the processes. Treat-ing every single plant individually, they applied a procedure

.for multiple-utilities optimization Pa rker, 1989 .

Generalization for More than Two PlantsThe concepts explored up to this point can be extended to

the case in which a number n of plants is considered for



21/22

Figure 22. Indirect integration using a single-steamtemperature level.

integration. The increase in complexity is evident, since inprinciple, all possible combinations of two plants have to beevaluated. As a starting point, consider the example of inte-gration among three plants. First, the plants are ordered byincreasing pinch temperatures, and the highest and lowestpinches are identified. For the unassisted cases, the zone de-limited by these pinches is where all the integration can takeplace. The three possible ways of heat transfer are shown inFigure 23. In turn, assisted cases will require transfer heat inthe highest or lowest zones. To predict maximum possible

savings, problem 1 can be reformulated, accounting for eachone of the combinations of two plants and the respective di-rections in which the heat will be transferred. The results ofthis problem are useful targets for models that will determine

Figure 23. Integration amongthree plants.

the heat-exchanger network needed to accomplish the pre-dicted savings.

In the case of indirect integration, new shifts of scales arerequired. As a first-approach solution, three circuits can beestablished, accounting for the three possible combinationsof two plants in the unassisted cases. A downward shift isperformed in the scale of the second a nd third plants to fea-sible transfer heat from the first plant. In turn, a seconddownward shift is needed in the third plant in order to trans-

fer heat from the second plant. Other alternatives may con-sist of the use of a single circuit that splits in plant 2 and 3,picking up the required heat in each of these plants and per-forming a similar split when t he heat is to be released. There-fore, the concepts and tools developed for two plants are astepping stone for the generalized integration of a set of n plants, an issue that will be further investigated in follow-uparticles.

ConclusionThere is a large incentive to perform heat integration across

plants. Models that account for maximum energy savings bydirect and indirect heat integration, including in this last case

the location of the fluid circuits, were presented. Conse-quently, a strategy to capture these savings was developed.While all these studies determine the target savings, there isstill a need to determine a heat-exchanger network that canaccomplish minimum energy consumption while the plants areintegrated, as well as when they are functioning separately.This must take place at a minimum investment cost. Amongmany o ther o ptions, d ual-use heat-exchanger networks fea-turing the minimum number of units accomplish this goal.The design of such networks will be attempted in a follow-uparticle.

NotationF s product of heat capacity and flow ratek s auxiliary temperature intervalsk A s first transfer interval in the zone above both pinches

k B s first transfer interval in the zone below both pinchesk E s first transfer interval in the zone between pinchesp Is last interval above the pinch of plant 1

p II s last interval above the pinch of plant 2Q s total heat transferred in the generalized zoneF

q s heat surplus or heat demandq Is heat surplus or heat demand in plant 1

q II s heat surplus or heat demand in plant 2q C s heat demand in the heat-sink plant

q C P s heat surplus or heat demand in the combined plantq H s heat surplus in the heat source plant

S s minimum heating utilityminT s initial temperature of the intermediate fluid0T s supply temperatures T s target temperaturet

T s upper temperature of a fluid circuitupT s lower temperature of a fluid circuitdownY F H s binary variable starting interval of a fluid circuitY F C s binary variable starting interval of a fluid circuit

Z s binary variable denoting an interval that belongs to a fluidcircuit

s original minimum cascaded heat C s original minimum cascaded heat in the heat-sink plant H s original minimum cascaded heat in the heat-source plant

s variable used in the cascade of heat s cumulative heat demands

Is adjust cascaded heat in plant 1



22/22

II s adjust cascaded heat in plant 2H s heat demand in the heat source plant

Superscripts

AC s cold fluid stream in the zone above both pinchesAH s hot fluid stream in the zone above both pinchesBC s cold fluid stream in the zone below both pinchesBH s hot fluid stream in the zone below both pinchesEC s cold fluid stream in the zone of effective transfer of heatEH s hot fluid stream in the zone of effective transfer of heat

FC s cold fluid stream in the general caseFH s hot fluid stream in the general case j s chemical plant

Subscripts

j s chemical plantr s hot streams s cold stream

Literature CitedAhmad, S., and C. W. Hui, Heat Recovery Between Areas of In-

.tegrity, Comput . Chem . Eng ., 12 , 809 1991 .Brooke, A., D. Kendrick, and A. Meeraus, GAMS. A Users Guide ,

.G AMS D evelopment Corporation, Washington, D C 1996 .

Cerda, J., A. W. Westerberg, D. Mason, and B. Linnhoff, MinimumUtility Usage in Heat Exchanger Network Synthesis, Chem . Eng .

.Sci ., 38 , 373 1983 .Ciric, A. R., and C. A. Floudas, Heat Exchanger Network Synthesis

.Without D ecomposition, Comput . Chem . En g ., 15 , 385 1991 .D hole, V. R., a nd B . Linnhoff, Total Site Targets for Fuel, Co-G en-

eration, Emissions, and Cooling, Comput . Chem . En g ., 17 , S101 .1992 .

Hui, C. W., and S. Ahmad, Total Site Heat Integration Using the .Utility System, Comput . Chem . En g ., 18 , 729 1994 .

Linnhoff, B., and E. Hindmarsh, The Pinch Design Method for Heat

.Exchanger Networks, Chem . En g . Sci ., 38 , 745 1983 .Morton, R . J., and B . Linnhoff, Individual P rocess Improvements inthe Context of Site-wide Interactions, Inst . Chem . En g ., Annual

.Research Meeting, B ath, U K 1984 .Pa poulias, S., and I . E. G rossmann, A Structural O ptimization Ap-

proach in P rocess Synthesis II: Heat Recovery Networks, Com - .put . Chem . En g ., 7, 707 1983 .

Pa rker, S. J., Supertargeting for Multiple U tilities, PhD Thesis,Univ. of Manchester Institute of Science and Technology, Manch-

.ester, U K 1989 .Trivedi, K. K., The Pinch Design Method for the Synthesis of Heat

Exchanger Networks: The Constrained Case, AIChE Meeting, .Washington, DC 1988 .

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