Area Target for Heat Exchanger Networks Using Linear Programming

A method is presented for calculating an area target for a heat exchanger network ... A heuristic rule has been developed on a number of temperature i...
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Ind. Eng. Chem. Res. 2003, 42, 1723-1730

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Area Target for Heat Exchanger Networks Using Linear Programming Jacek M. Jez3 owski,*,† Hiren K. Shethna, and Francisco J. L. Castillo AEA Technology Engineering Software (Hyprotech), Calgary, Alberta, Canada, and Department of Chemical Engineering and Process Control, Rzeszo´ w University of Technology, Al. Powstan´ co´ w Warszawy 6, 35-959 Rzeszo´ w, Poland

A method is presented for calculating an area target for a heat exchanger network (HEN). The approach is based on the solution of a linear programming problem modeled as an optimal transportation task. The transportation model uses temperature intervals and does not require knowledge of enthalpy intervals for the fixed heat recovery level. Restricted matches can be directly accounted for. A heuristic rule has been developed on a number of temperature intervals that keeps the number of variables and constraints in reasonable limits while ensuring accurate results. The targeting method can also be applied to HENs consisting of multipass (1-2) heat exchangers. The results from case studies have shown that solutions from the developed approach are very close to those calculated by complex nonlinear programming based methods. 1. Introduction Targeting is a valuable tool for assessing the economical benefits of heat integration. Moreover, the targeting stage forms an important step of sequential approaches for heat exchanger network (HEN) synthesis. In design mode, targets are applied to estimate an optimum value of the heat recovery level or an optimal heat recovery approach temperature (HRAT). This is commonly referred to as supertargeting.1,2 Also, an optimal heat load distribution (HLD) of matches can be calculated using targets.3 To calculate the total annual cost (TAC) of a HEN for a given HRAT value, one needs the total area target. In both insight- and optimization-based approaches to HEN design, calculation of the area target causes problems because it introduces nonlinearities. Simple technologies such as the Bath formula4,5 from pinch technology (PT) can yield inaccurate area targets and, as a consequence, an inaccurate initialization point for the structure development stage. More rigorous nonlinear programming (NLP) techniques require sophisticated solvers that do not ensure the global optimum. A short overview of available approaches for HENs area targeting is given in the following. We limit the presentation to approaches for the area target, i.e., a minimum total area that ensures a given level of heat recovery but does not account for number of matches or cost. Most likely, the most popular technique is the socalled Bath formula from PT. The crude form was first suggested by Townsend and Linnhoff,6 and the final method was presented by Ahmad et al.4,5 The method relies on vertical heat transfer in a composite curves plot with an imposed spaghetti structure. It is rigorous for strictly uniform heat-transfer coefficients (htc’s) of streams. Ahmad et al.5 claimed that it can be applied up to a threshold value of 10.0 of parameter R defined by

R ) hmax/hmin

(1)

* To whom correspondence should be addressed. E-mail: [email protected]. † Rzeszo´w University of Technology, Rzeszo´w, Poland.

where hmax/hmin denotes the maximum/minimum value of h in the problem. It is worth noting that the Bath formula is applicable to both 1-1 and 1-2 heat exchangers. One way to account for variable individual htc’s is to replace the global HRAT value applied in the Bath formula calculations by match-dependent temperature approaches (∆Tij). Rigorous formula (2) for ∆Ti contributions to approaches ∆Tij was developed by Nishimura,7 but it is limited to a single hot/cold stream and several cold/hot stream cases. Parameter κ in eq 2 is stream-

∆Ti ) κ/hiβ

(2)

dependent, and β equals 0.5, as was proven in ref 7. The diverse pinch concept of Rev and Fonyo8 extended its application for the general case of several hot to several cold streams. They calculated parameter κ by a simple iterative procedure, assuming the same heat recovery level as that for a fixed value of HRAT. A similar procedure was also used by Ahmad.4 Stream contributions ∆Ti were also applied by Briones and Kokossis9 to produce dispersed composites, which account for the influence of variable htc’s. Zhu et al.10 optimized parameters κ and β in eq 2 to reach more accurate area targets. Besides the Bath formula and its extensions, there exist several methods based on mathematical programming. Two NLP approaches were developed by Colberg and Morari11 and Yee et al.12 The rigorous model of Colberg and Morari11 combines both temperature (TIs) and enthalpy intervals. The spaghetti structure was imposed directly on matches, and a trans-shipment formulation was applied to model heat exchange. Yee et al.12 developed a stagewise superstructure that is a kind of the spaghetti structure based on TIs. The number of stages is determined by a simple heuristic rule though Daichendt and Grossmann13 found this to be too crude in some cases. Additionally, the assumption of isothermal mixing after each stage was imposed to ensure linear heat balances. Colberg and Morari11 as well as Yee et al.12 presented their models and examples of application for 1-1 heat exchangers only to avoid additional nonlinearities and

10.1021/ie020643i CCC: $25.00 © 2003 American Chemical Society Published on Web 03/18/2003

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numerical problems caused by the correction factor (Ft) in the design equation for the area of multipass heat exchangers. Besides NLP formulations, there are area targeting techniques applying linear programming (LP) models. They can be seen as hybrid approaches because they make use of PT concepts to simplify the optimization problem. Ahmad4 applied the transportation model to determine the area of matches in each enthalpy interval. Saboo et al.14 also applied a linear model solved sequentially by decreasing the size of the TIs until the area target converged. As noted in ref 11, this can increase the computation load up to an unmanageable size. Colberg and Morari11 suggested the possibility of application of the task assignment algorithm. They noted, however, that this requires a division of streams into equal-sized heat elements and can end in an overwhelming optimization task. The minimization of the HEN area is an important issue for sequential approaches for calculating optimal HLDs or for total cost targeting. It is, however, necessary to bear in mind that such approaches address the more general problem because in “typical” area targeting we do not account for the number of matches or total cost. Gundersen and co-workers3,15-17 presented in a series of papers mixed-integer LP optimization models for optimal HLD generation. Briones and Kokossis9 developed an area target model (ATM) to set the total cost targets for HEN synthesis. The model can be simplified to account for minimization of the area deviation, i.e., deviations from the vertical pattern. It accounts for 1-1 and 1-2 apparatuses. Despite the numerous approaches developed to date for the HEN area target, they do not adequately address industrial aspects of the problem. PT-based techniques such as the Bath formula are not rigorous in the case of variable htc’s of streams and fail for forbidden matches. NLP models are difficult to solve to the global optimum for large-scale HENs with a 1-2 apparatus. LP formulations may end up in overwhelming computations. The ATM model of Briones and Kokossis9 (in truncated form for area deviation minimization only) seems to be the best tool but requires complex data preparation. This work addresses area target calculation for HENs with true countercurrent (1-1) heat exchanger and multipass (1-2) apparatuses. A simple LP approach was developed. It accounts for restricted matches and can be extended for minimum TAC targeting by a simultaneous approach.18 The rest of the paper is organized as follows. Part 2 presents a formulation of the optimization model for the area target. In part 3, we describe the overall procedure and give an analysis of the approach. Several examples are presented in part 4 along with a comparison with the results from other approaches. Concluding remarks are given in the last part. 2. Linear Model for HEN Area Targeting Similarly to other works on the area target, we assume that heat loads of utilities are fixed and the number of matches minimization is not accounted for. To calculate the area target for a HEN, it is necessary to solve an optimization problem where the objective function is a sum of heat exchanger areas on matches

subject to heat balance constraints. To ensure linearity, one has to calculate log-mean temperature differences (LMTDs) and Ft’s in matches before optimization. Thus, the values of LMTDs and Ft’s of matches need to be given as data. To achieve possibly exact approximations of these parameters, Briones and Kokossis9 developed the procedure that requires enthalpy intervals, dispersed enthalpy curves calculated with the use of stream contributions ∆Ti, and a complex algorithm for final calculation of LMTDs. In our approach, we use TIs. They were applied in order to (i) account for restricted matches, (ii) further extend simultaneous total cost targeting,18 and (iii) simplify data preparation (enthalpy intervals are unnecessary). Additionally, we applied the transportation model formulation19 although it requires more variables than the commonly applied trans-shipment one.20 The reason is that the latter uses residual heat flows compounded from heats of process streams. The residuals cause serious problems with estimating temperature approaches and assigning htc’s to individual streams. Because in the transportation model matches between individual streams are used, there is a direct assignment of a htc to a stream. Also, the estimation of the temperature approaches for “minimatches” in the model is straightforward and can be performed even by hand calculations. As a consequence, data preparation is easy. The detailed procedure is given in part 3. It is worth noting that the transportation model was also chosen in a very recent work21 on HEN design. In the following the formulation of the optimization model for HEN area targeting is presented. Sets of streams are defined:

Hi {i/i ) 1, ..., NH ) hot streams, i.e., hot process and utility streams} Cj {j/j ) 1, ..., NC ) cold streams, i.e., cold process and utility streams} To include forbidden matches, we define set Fij of pairs (i, j) such that stream i ∈ Hi is not allowed to exchange heat with stream j ∈ Cj.

Fij {(i, j)/i ∈ Hi, j ∈ Cj and match between them is forbidden} Following a procedure given in section 3, streams have been divided into smaller elements for which we define the sets:

Him {i/i ∈ Hi and i is present in interval m} Cjn {j/j ∈ Cj and j is present in interval n} Let qim,jn denote the heat load in the match of hot stream i in interval m and cold stream j in interval n (see also Figure 1). Variables qim,jn for m, n ) 1, ..., M are defined only for m e n. Matches between a hot utility and a cold utility are excluded explicitly. The LMTDs in a match of load qim,jn are calculated from temperature differences calculated for the intervals m and n. Hence, all matches of i ∈ Him with j ∈ Cjn have identical temperature differences (at both sides) as

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Ft’s can be calculated by any standard method. Approximations of LMTD and Ft are rigorous if heat load qim,jn of the match is equal to the enthalpy change of i ∈ Him and j ∈ Cjn, i.e.

qim,jn ) ∆Him ) ∆Hjn; i ∈ Him; j ∈ Cjn; m, n ) 1, ..., M (6)

Figure 1. Illustration of the transportation model for HEN area targeting.

To reach such a condition, one has to create equal-sized heat elements, as suggested in ref 11. However, if TIs are sufficiently small, the approximation is good enough to calculate area targets (the procedure for creation of TIs is given in the following). The linear model P1 for area targets is as follows.

Area targeting model P1 M

min

M

∑ ∑ 1/(Ft × LMTD)m,n j∈C ∑ i∈H ∑ qim,jn/(hj + hi)

m)1n)1

jn

im

(7) M

s.t.

∑ ∑ qim,jn ) ∆Him;

i ∈ Hi; m ) 1, ..., M (8)

n)mj∈Cjn

M

∑ ∑ qim,jn ) ∆Hjn;

j ∈ Cjn; n ) m, ..., M

(9)

m)1i∈Hm

qim,jn ) 0; i ∈ Him; j ∈ Cjn; i, j ∈ Fi,j; m ) 1, ..., M; n ) m, ..., M (10)

Figure 2. Illustration for approximations of temperature approaches in area targeting model P1: (a) illustration for approximations of temperature approaches in the transportation model representation; (b) illustration for approximations of temperature approaches in the spaghetti structure representation.

defined by eq 3. Hence, for instance in Figure 2a match

{

hot ∆Tim,jn ≡ ∆Thot m,n ) Tm-1 - Tn-1 cold ∆Tim,jn ≡ ∆Tcold m,n ) Tm - Tn

}

i ∈ Him; j ∈ Cjn; m, n ) 1, ..., M (3) {i1m, jn} with load qi1m,jn and a match {i2m, jn} with load qi2m,jn have the same temperature approaches at both sides. This can be achieved in a totally splitted spaghetti structure illustrated by Figure 2b and also by Figure 4. As a consequence, the approximations of LMTD and Ft for matches are also calculated for intervals; thus

LMTDim,jn ≡ LMTDm,n; i ∈ Hi; j ∈ Cj; m, n ) 1, ..., M (4) Ftim,jn ≡ Ftm,n; i ∈ Hi; j ∈ Cjn; m, n ) 1, ..., M (5)

qim,jn g 0; i ∈ Him; j ∈ Cjn; i, j ∉ Fi,j; m ) 1, ..., M; n ) m, ..., M (11) Constraints (8) and (9) are heat balances. Equality (10) forces heat loads of forbidden matches to naught. At last, inequality constraint (11) ensures that heat loads have to take only positive values. 3. Algorithm of the HEN Area Targeting Method To solve model P1, one has to construct TIs and to calculate data for the model. Also, some postcalculations can be performed to increase an accuracy of targets. Hence, there are the following steps in the area targeting method: (A) creation of TIs; (B) solution of the model; (C) generation of matches structure from the results and calculation of its area. In the following we describe steps A and C because LP solvers are well-known. An analysis of the approach is also presented in this part. 3.1. Creating TIs. The size of a TI is a crucial point of model P1 for area targeting. Clearly, the smaller the size of the TI used, the closer the temperature approaches are calculated. This, however, leads to a very large number of variables. Notice that the transportation model is a polynomial time algorithm22 and, hence,

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Figure 3. Illustration of the spaghetti structure modifications: (a) illustration of the spaghetti structure for solutions from model P1; (b) standard presentation of parallel matches; (c) illustration of merged parallel matches in the modified structure.

Figure 4. Match types A and B.

the size of a problem influences largely the CPU time. We have developed a scheme of division of TIs that keeps their number within reasonable limits and ensures quite accurate area targets. The procedure is similar to that applied in PT. Preliminary division is performed based on inlet and outlet shifted temperatures of streams as described by Linnhoff and Flower.23 As for the choice of the approach temperature value required in constructing intervals, it is sufficient that a small number is used to meet thermodynamic feasibility constraints. Notice that this follows the concept of the double-temperature approach.3,9 Loads of utilities are calculated for HRAT, while the temperature approach applied in division for intervals is equivalent to the exchanger minimum approach temperature (EMAT). We then determine the mean size of the interval (dTmean) from eq 12, where dTmin is the smallest interval after preliminary division. Each interval of size higher

dTmean ) max[3 dTmin, 10]

(12)

than dTmean (e.g., dT) is divided into Nad intervals

Nad ) ceil[dT/dTmean]

(13)

where ceil[a] means a nearest integer number not less than a.

This scheme does not produce a large number of intervals and, thus, the number of variables in the area targeting model P1 is far less than that needed in the assignment task formulation.11 3.2. Structure of Matches. The approximation of temperature approaches in matches applied in model P1 imposed a superstructure that, in fact, is a totally splitted spaghetti structure within TIs similar to that from ref 11. Element i ∈ Him/j ∈ Cjn is splitted at as many branches and as many matches as it has in the solution of model P1; see the illustrative examples in Figures 2b and 3a. Such a structure can cause an overestimation of the area in some cases. As an example, let us consider a totally splitted structure from a solution of model P1 with two matches on adjacent intervals, with the same streams as shown in Figure 3a. This can be represented in a more familiar fashion shown in Figure 3b. However, a serial connection can also be built from the solution of Figure 3a, and matches can be merged, as shown in Figure 3c. The parallel connection of Figure 3b has an uneven distribution of driving forces, while merged matches from Figure 3c give a more even one in the entire network. Therefore, on the basis of the solution from model P1, a modified structure with merged matches such as that shown in Figure 3c can be generated in the last step of the area targeting method. Such a structure will be shown for example 5 in part 4. Our tests showed that usually the modified structure has an area up to 1015% less than the totally splitted spaghetti structure from the solution of model P1. The results of our method given in the following are for the modified structure. However, we consider this structure modification step as optional in the supertargeting procedure. To reduce calculations, one can omit it because the results from the solution of model P1 provide a correct profile of optimum area vs HRAT. 3.3. Analysis of the Method. To prove the validity of the suggested approach, we present here a qualitative analysis. It is clear that with infinitely small heat elements of equal enthalpy change model P1 will produce exact area targets because it becomes equivalent to the assignment task formulation.11 The questions are as follows: (1) Can model P1 produce good results with larger TIs of unequal enthalpy changes? (2) What should the sizes of the intervals be? To answer the first question, we show the two effects. To show the first effect, we make use of a case shown in Figure 2a as an illustrative example. The match with load qi1m,jn has an actual area higher than that calculated with the use of temperature differences from eq 3 because the actual LMTD is lower than that approximated by eq 3. However, the actual area of the second match with load qi2m,jn is lower than the area calculated for similar reasons. Hence, the sum of the actual areas should not differ largely from the sum of the calculated areas if the intervals do not differ largely. This is the first effect of canceling over- and underestimations. The second effect is of similar character. The division scheme into TIs is aimed more at equal-sized intervals in regards to temperature changes than to enthalpy changes. Balance equations of model P1 provide tight constraints such as for types of matches and force matches of types A and B (see Figure 4) at least at extreme hot- and cold-side TIs. Such matches give more

Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1727 Table 1. Data for Example 1 from Reference 11 (HRAT ) 20 K)

Table 4. Data for Example 3, Problem 4S1t in Reference 2

stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

H1 H2 C1 C2 steam cooling water

423 363 293 298 453 283

333 333 398 373 452 288

20 80 25 30

0.1 0.1 0.1 0.1 0.1 0.1

H1 H2 C3 C4 HU CU

448 398 293 313 453 288

318 338 428 385 452 298

10 15 20 15

0.2 2.0 0.2 2.0 4.0 2.0

Table 2. Data for Example 2 from Reference 12 (HRAT ) 20 °F) stream

TS (°F)

TT (°F)

CP (kBtu/h‚°F)

h (kBtu/h‚ft2‚°F)

H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 steam cooling water

320 480 440 520 390 140 240 100 180 200 456 100

200 280 150 300 150 320 431 430 350 400 456 180

16.67 20 28 23.8 33.6 14.45 11.53 16 32.76 26.35

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

Table 3. Comparison of Area Targets for the 1-1 Apparatus for Examples 1 and 2 our approach example 1 2

A1 (m2)

∆ (%)

solution from ref 11 A1 (m2)

∆ (%)

solution from ref 12 A1 (m2)

∆ (%)

Bath formula A1 (m2)

∆ (%)

2925.0 1.00 2896.0 0.0 2898.9 0.01 2896.0 0.0 2468.8 0.16 2490.0 1.02 2464.8 0.0

or less even distribution of driving forces that accounts for variable htc’s. Hence, they should be chosen by optimization. Furthermore, they ensure that our approximations are rigorous for hot- or cold-side temperature differences. Approximately evenly distributed sizes of TIs give, as a consequence, good approximations for LMTDs and Ft’s as well. Also, for different CP ratios of heat-exchanging stream matches of types A and B, our approximations under- and overestimate, respectively, the actual area. Because there is no incentive to prefer match A or B by optimization, these effects canceled each other. The two effects should result in good approximations of area targets, provided that TIs are sufficiently small. The answer to the second question requires an analytical expression, which we are not able to produce. We claim that our division scheme produces rather an upper limit for the number of intervals because the time of computation was of great care in our investigations. Taking into account the increasing power of the PC, a greater number can be used by decreasing the parameters in eq 12. 4. Examples 4.1. Problems with Uniform htc’s: Comparison with NLP11,12 and the Bath Formula.4,5 We start the presentation with examples featuring uniform htc’s. Notice that for such problems the Bath formula yields accurate results. Example 1 is taken from ref 11 (data in Table 1). Table 2 contains data for example 2 from Yee et al.12 The comparison of our area targets for 1-1 heat exchangers with those from refs 11 and 12 is given in Table 3. Relative errors were calculated by taking the solutions from ref 11 as accurate results. The results show perfect agreement of the Bath formula and the rigorous NLP method of the authors in ref 11. The target

Table 5. Comparison of Area Targets for Example 3 (1-1 Apparatus)

utility usage

Bath formulaa

diverse pinch methoda,8

NLP solution by the method in ref 11a

1580 1713.33 1774.44 1880 1980 2057.08 2232.78 2272.27 2330 2450 2737.5 2791.36 2811.67 2870.63 2874.44 2880 3280 3580 3780

484.53 (11.52) 442.76 (12.40) 428.69 (12.37) 408.64 (11.79) 393.26 (10.97) 383.21 (10.30) 364.69 (8.71) 361.20 (8.49) 356.49 (8.02) 347.96 (7.08) 332.92 (5.08) 330.77 (4.75) 330.00 (4.62) 327.90 (4.28) 327.78 (4.27) 327.59 (4.22) 316.82 (1.97) 313.56 (0.93) 313.43 (0.78)

482.34 (11.03) 440.08 (11.72) 426.44 (11.72) 408.04 (11.63) 393.71 (11.09) 384.92 (10.79) 368.229 (9.76) 365.09 (9.65) 360.84 (9.33) 353.14 (8.68) 340.11 (7.35) 339.05 (7.37) 336.07 (6.55) 330.31 (5.04) 329.93 (4.96) 329.40 (4.80) 312.78 (0.67) 311.05 (0.12) 313.90 (0.93)

434.42 393.90 381.48 365.51 354.39 347.41 335.46 332.94 330.03 324.94 316.82 315.77 315.41 314.45 314.34 314.31 310.68 310.68 311.00

a

our solution 432.43 (0.002) 394.06 (0.0406) 381.99 (0.134) 366.11 (0.164) 355.25 (0.243) 348.14 (0.210) 335.93 (0.140) 333.68 (0.222) 330.87 (0.254) 325.80 (0.264) 317.56 (0.233) 316.56 (0.250) 316.20 (0.250) 315.23 (0.248) 315.17 (0.264) 315.09 (0.248) 311.58 (0.289) 311.22 (0.174) 311.45 (0.145)

Values of area targets from Shenoy.2

area by the Yee et al.12 approach, which has some minor simplifications, differs by about 1%, and our results are of similar accuracy. 4.2. Problems with Varying htc’s: Comparison with NLP,11,12 the Bath Formula,4,5 and the Diverse Pinch Concept.8 Example 3 taken from Shenoy2 (data in Table 4) features high variations of htc’s, higher than the threshold value of 10.0 for parameter R. It is considered here because we can use the results from Shenoy2 for the Bath formula, the NLP approach from ref 11, and the diverse pinch method of the authors in ref 8 for varying values of heat recovery (i.e., various HRATs) and compare them with our results in Table 5. Area targets from our method are very close to those from the rigorous NLP method from ref 11. The relative error (numbers in brackets in Table 5) is less than 0.5% for all HRATs. The Bath formula and diverse pinch concept always overestimate the minimum targets. The deviations are higher for smaller HRATs (i.e., small values of utility loads) and decrease with an increase of the temperature approach. As a consequence, supertargeting with the Bath formula or diverse pinch concept can yield a bad value of the “optimal” HRAT for the design stage. As a result of “topology traps”,24 this will end in inferior HENs. Examples 4-6 are used to give a wider comparison with other approaches. The data are given in Tables 6-8, respectively. All of the examples feature variable htc’s, with the highest variations in examples 5 and 6. A comparison of the targets is presented in Table 9. Similar to previous cases, our method produces results very close to those of Colberg and Morari,11 even closer than the NLP method of the authors in ref 12. Example 5 is of special interest here because it was considered in detail also in refs 7 and 25 and is

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Figure 5. Modified structure of matches for the solution of example 5 (temperature is in kelvin). Table 6. Data for Example 4 from Reference 11 (HRAT ) 10 K, Qhu ) 620 kW, and Qcu ) 230 kW)

Table 10. Data for Example 7 from Reference 3 (HRAT ) 20 K)

stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

H1 H2 C1 C2 steam cooling water

395 405 293 353 520 278

343 288 493 383 519 288

4 6 5 10

2.0 0.2 2.0 0.2 2.0 2.0

H1 H2 H3 C1 C2 C3 steam cooling water

573 473 463 433 453 463 623 303

473 463 443 453 463 503 622 323

10 100 50 50 100 25 1000 50

0.1 1.0 1.0 0.1 1.0 1.0 4.0 2.0

Table 7. Data for Example 5 from Reference 11 (HRAT ) 10 K; There Is No Utility) stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

H1 H2 H3 H4 C1

443 416 438 448 273

293 393 408 423 434

0.5 2.0 0.5 1.0 1.0

2.0000 0.2857 0.0645 0.0408 2.0000

Table 8. Data for Example 6 from Reference 11 (HRAT ) 20 K) stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

H1 H2 H3 C1 C2 C3 C4 steam cooling water

626 620 528 497 389 326 313 650 293

586 519 353 613 576 386 566 649 308

9.802 2.931 6.161 7.179 0.641 7.627 1.690 -

1.25 0.05 3.20 0.65 0.25 0.33 3.20 3.50 3.50

Table 9. Comparison of Area Targets for the 1-1 Apparatus for Examples 4-6 our approach example 4 5 6

A1 (m2)

∆ (%)

solution from ref 11 A1 (m2)

∆ (%)

solution from ref 12 A1 (m2)

∆ (%)

Bath formula A1 (m2)

∆ (%)

260.6 0.69 258.8 0.0 263.6 1.85 295.7 14.2 30.20 1.21 29.84 0.0 31.3 4.89 47.71 59.88 174.7 0.63 173.6 0.0 179.9 3.63 227.0 30.7

advantageous for simple technologies (see large errors of the Bath formula). Nishimura7 reached the minimum area of 29.84 m2 with 20 minimatches, while the authors in ref 11 needed only 11 units to reach this area. The modified structure from our approach shown in Figure 5 requires 17 matches to reach an area of 30.20 m2. This shows that our modified spaghetti structure produces a reasonable number of matches. The influence of the match number on the area in the structure development stage can be additionally illustrated using the results from Zhu et al.25 They showed a solution with 9 units

and an area of 31 m2 and another with 6 units and an area of 36.58 m2. The next example from this group (no. 7; data in Table 10) presents a problem that was introduced in ref 3 to show the necessity of substantial pinch crossing to reach a design with the minimum TAC. The problem was also considered in refs 9 and 10. The authors in ref 9 proved that the use of the diverse pinch concept does not lead to a solution with a small area. Zhu et al.10 applied optimized parameters κ and β in eq 2 to design a network with a small area. Their cost optimum HEN features the total area of 494.12 m2. The area target from our method is 519.7 m2, i.e., about 5.2% larger than the minimum. The application of the Bath formula gives as much as 726.1 m2 of the total area, i.e., 47% above the optimum. It can be concluded that the suggested approach works well for both uniform and highly varying htc’s in the case of HENs with 1-1 heat exchangers. In the following, we will show that the quality of the results is also high for forbidden matches and 1-2 heat exchangers. 4.3. Problems with Forbidden Matches. Example 8 is considered here to show the effect of forbidden matches on area targets. We were able to find in the literature only one case of using NLP methods for forbidden matches. This was a problem from example 4 (see Table 6) with forbidden match H1-C2. Colberg and Morari11 reported an area target of 259.7 m2 for the 1-1 apparatus (see Table 11). The difference with the base case is only 0.9 m2. Our area targets for both cases are identical and such result was obtained by the use of the Bath formula. It is important to note that despite forbidden match H1-C2 the utility load does not change. We performed calculations for forbidden matches H2-C1 and H2-C2. In both cases, the utility load increases (Table 11). Area targets change in comparison with the base case (Table 11). It may be astonishing that the total area decreases with some

Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1729 Table 11. Comparison of Area Targets (A1 and A2) for Example 8 (Forbidden Matches) solution match Qhu/Qcu from ref 11 forbidden (kW) A1 base case H1-C2 H2-C1 H2-C2

620/230 620/230 840/450 662/272

258.8 259.7

our solution A1 A2 260.60 260.60 241.00 244.90

Bath formula A1 A2

349.4 349.4 322.4 326.6

295.70 295.78 147.68 240.59

295.7 385.9 180.7 385.9

Table 12. Data for Example 9 stream

TS (K)

TT (K)

CP (kW/K)

h (kW/m2‚K)

H1 H2 H3 C1 C2 steam cooling water

432 540 616 299 118 573 293

350 353 363 127 400 572 333

22.85 2.04 5.38 9.33 19.61 -

1.00 0.40 5.00 0.10 5.00 0.50 2.00

Table 13. Comparison of Area Targets for Example 9 Bath formula HRAT (K) 30 25 20 15 10 5

A1

(m2)

300 318.6 341.5 372.0 415.4 487.3

our approach

∆ (%)

A2

(m2)

∆ (%)

A1 (m2)

A2 (m2)

29.67 25.96 22.13 17.7 13.16 10.16

382.3 406.38 436.94 477.4 534.95 626.95

26.26 22.33 18.37 13.69 10.83 8.33

231.34 252.9 279.61 316.06 367.08 442.33

302.8 332.18 369.14 419.89 482.67 581.49

forbidden matches. However, it should be noticed that this is due to the increase of heat loads in utility matches with large driving forces and the decrease of loads on process stream matches with small driving forces. It can also be observed from the results in Table 11 that the Bath formula produces quite misleading targets if utility loads change as a result of restricted matches. Results from our approach agree well with the analysis of the forbidden match influence. 4.4. Area Target for the 1-2 Apparatus. The NLP approaches from refs 11 and 12 do not account for 1-2 heat exchangers. Other NLP optimization-based methods for HENs are aimed at designing total cost optimum HENs. A direct comparison of the TAC optimum area and the area target from our method is not justified. Hence, we were able to validate our results in an indirect manner. Example 9 is taken from refs 5, 10, and 26 (data in Table 12). The problem features the variation of htc’s larger than the threshold value of 10.0 of parameter R. A comparison of the area targets for 1-1 and 1-2 apparatuses for various HRATs calculated by the Bath formula and our method is presented in Table 13 (relative errors of the Bath formula were calculated by taking our results as accurate). Zhu et al.10 reported, for HRAT of 30 K and the 1-1 apparatus, an area target of 241 m2 calculated by the diverse pinch concept with optimized parameters κ and β in eq 2. After parametric optimization, they reached the final design of an area of 242 m2 while our result is 231.34 m2 (Table 13). Zhu et al.10 noted also that the minimum TAC lies at HRAT on the order of 38 K, while supertargeting with the Bath formula shows the optimal HRAT at 30 K. We have also found a similar effect for the 1-2 apparatus. Supertargeting with the Bath formula and our method for the area target gave an optimum HRAT in the region of 2527 K for the former technology and in the region of 3537 K for the latter. It is interesting to note that deviation of the Bath formula increases with an increase of HRAT. This is the opposite effect to that observed in example

3. The differences between the Bath formula and our approach for 1-1 and 1-2 apparatuses show a similar profile of changes with HRAT. However, they are smaller in the case of the 1-2 apparatus. A similar dependency was observed in other problems not presented here for brevity sake. We see the reason in the use of Shenoy’s method2 instead of the PT approach from refs 4 and 5 for calculating the number of shells in intervals in a version of software applied in solving the examples. The former, as was recently proved in ref 27, causes an overestimation of the total area by at least 10% in comparison with the PT procedure. 5. Conclusions A simple and robust method has been developed for calculating the minimum area targets of countercurrent and multishell HENs in grass-root design. It is based on solving the linear transportation model, for which the data preparation is very easy. The optimization model requires a moderate number of variables and can be solved in short CPU time for large-scale tasks. The approach allows accounting for constrained matches. The extensive computations showed that the solutions for problems with various htc’s differ by less than a few percent from the best results presented in the literature. Also, it gives a direct possibility of simultaneous optimization for both energy and area. Further extensions presented in ref 18 allow for total cost targeting. Acknowledgment The authors thank Hyprotech Ltd. for permission to use HX-Net software. Nomenclature A1 ) area target for HENs with 1-1 shell and tube heat exchangers A2 ) area target for HENs with multipass heat exchangers CP ) heat capacity flow rate Ft ) correction factor for a noncountercurrent flow pattern EMAT ) exchanger minimum approach temperature h, htc ) stream film heat-transfer coefficient HLD ) heat load distribution of matches in a HEN HRAT ) heat recovery approach temperature or global temperature minimum approach LMTD ) logarithmic-mean temperature difference m, n ) number of temperature intervals NC/NH ) total number of cold/hot streams, respectively Qcu/Qhu ) heat load of utility cu/hu TI ) temperature interval TS/TT ) stream inlet/outlet temperature, respectively R ) parameter defined by eq 1 to estimate applicability of the Bath formula ∆Thot,cold ) temperature approach at the hot, cold (respectively) sides of a match ∆ ) relative error ∆Ti ) stream i contribution to ∆Tij ∆Tij ) match-dependent temperature approach Superscripts max ) maximal value min ) minimum value Subscripts i/j ) stream i/j m/n ) temperature interval number

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Received for review August 20, 2002 Revised manuscript received December 10, 2002 Accepted January 20, 2003 IE020643I