Extended Interval-Based Mixed Integer Nonlinear Programming

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Ind. Eng. Chem. Res. 2010, 49, 166–179

Extended Interval-Based Mixed Integer Nonlinear Programming Superstructure Synthesis of Heat and Mass Exchanger Networks Adeniyi J. Isafiade and Duncan M. Fraser* Chemical Engineering Department, UniVersity of Cape Town, 7700 South Africa

In this paper we present a method for extending the interval-based mixed integer nonlinear programming superstructure (IBMS) model of Isafiade and Fraser [Chem. Eng. Res. Des. 2008, 86, 245-257; 909-924] for heat and mass exchanger networks synthesis to handle split streams going through two or more exchangers in series. A major shortcoming of the IBMS generation approach is that only parallel heat/mass exchange can take place for split streams. Individual split stream branches cannot exchange heat/mass in more than one exchanger since split streams must be mixed immediately after exchanging heat/mass in order to define the next interval boundary temperature/composition. A major deficiency of this is the tendency for the search space to be reduced thereby limiting the number of possible network configurations which can be generated. A method for overcoming this shortcoming is presented in this paper. The approach involves splitting every stream into subsuperstructures just before it enters into the parent superstructure and subsequently mixing the split streams while exiting the subsuperstructures. The new method has been applied to literature example problems to demonstrate its benefits. 1. Introduction The synthesis of heat exchanger networks has been a vital tool for accomplishing energy reduction in chemical, petrochemical, and allied industries. The synthesis of mass exchanger networks has also been used to achieve pollution reduction in such plants. Techniques which have been developed for heat exchanger network synthesis (HENS) have also been adopted for mass exchanger network synthesis (MENS). These techniques involve conceptual and mathematical methods. Chief among the conceptual approach is pinch technology which was first applied to minimum utility targeting and design in HENS.1 Later, techniques were developed for targeting the corresponding heat exchange area and hence the capital cost with which to achieve the minimum utility targets.2 Also developed were methods through which the aforementioned targets can be met in design.1,3 The trend described above for the development of synthesis techniques for HENs has also been observed in MENs. The minimum mass separating agents (MSA) targets together with design methods with which to meet the MSA and minimum number of units targets were presentedby El-Halwagi and Manousiouthakis.4 Hallale and Fraser5,6 developed capital cost targeting tools for MENs. The authors also presented design methods for meeting the MSA and capital cost targets in design. The mathematical methods which have been applied to the synthesis of heat and mass exchange networks have been of two natures. The first method is an automated version of the pinch approach where linear programming (LP) and mixed integer linear programming (MILP) are used to determine the minimum energy/MSA targets and the corresponding minimum number of units.7-9 The second mathematical approach which is simultaneous in nature usually involves the use of nonlinear programming (NLP) or mixed integer nonlinear programming (MINLP) to optimize in one or two steps the competing costs. Examples of the simultaneous methods include the simplified stagewise superstructure (SWS) approach of Yee and Grossman10 for HENS, the flexible hyperstructure approach of * To whom correspondence should be addressed. E-mail: Duncan. [email protected]. Tel.: +27 21 650 2515. Fax: +27 21 650 5501.

Papalexandri and Pistikoupolos11 for both HENS and MENS, the IBMS method of Isafiade and Fraser12 for HENS, the mass exchange stagewise superstructure approach of Chen and Hung,13 the fairly linear mass exchange stagewise superstructure of Szitkai, et al.14 and the IBMS technique of Isafiade and Fraser15 for MENS. The HENS problem can be stated as follows:16,17 Given a number, NH, of hot process streams (to be cooled) and a number, NC, of cold process streams (to be heated), it is desired to synthesize a network of heat exchangers which can transfer heat from the hot streams to the cold streams while achieving a minimum total annual cost. Also given are the heat capacity flowrate (flowrate × specific heat), FCp, supply temperature, Ts, and target temperature Tt, of each process stream. Available for service are heating and cooling utilities whose costs, supply temperatures, and target temperatures are also given. The MENS problem can be stated as follows:16,18 Given a number NR of rich streams (sources) and a number NS of MSAs (lean streams), it is desired to synthesize a network of mass exchangers that can preferentially transfer certain species from the rich streams to the MSAs while achieving a minimum total annual cost. Given also are the flowrate of each rich stream, Gi, its supply (inlet) composition, yis, and its target (outlet) composition, yit, where i ) 1, 2, ..., NR. In addition, the supply and target compositions, xjs and xjt, are given for each MSA where j ) 1, 2, ..., NS. The mass transfer equilibrium relations are also given for each MSA. The flowrate of each MSA is unknown and is to be determined as part of the synthesis task. The candidate MSAs (lean streams) can be classified into NSP process MSAs and NSE external MSAs (where NSP + NSE ) NS). The process MSAs already exist on the plant site and can be used for the removal of the species at a low cost (often virtually free). The flowrate of each process MSA, Lj, that can be used for mass exchange is bounded by it availability in the plant and may not exceed a value of Ljc. On the other hand, the external MSAs can be purchased from the market and their flowrates are to be determined by economic considerations. This paper presents an improvement to the IBMS models for HENS and MENS developed by Isafiade and Fraser.12,15 The

10.1021/ie801823n CCC: $40.75  2010 American Chemical Society Published on Web 11/19/2009

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010

Figure 1. Structures excluded from the IBMS and SWS.17

new model incorporates structures where split streams can go through two or more exchangers in series prior to mixing. Interval-Based MINLP Superstructure. The interval based superstructure generation approach of Isafiade and Fraser12,15 involves defining the superstructure intervals by the supply and target temperatures/compositions of either the hot/rich or cold/ lean set of streams. The streams of the opposite kind are assumed to participate in all the intervals created in the superstructure subject to thermodynamic feasibility. Heat/mass can be exchanged by streams of opposite kinds, provided they exist in the same interval. Such heat/mass exchange can be accomplished through the splitting of the streams into equal numbers within each interval. The split streams recombine immediately at the cold/lean end of the interval for the hot/rich streams and at the hot/rich end of the interval for the cold/lean streams. The interval defining approach of the IBMS method is similar to the temperature/composition location defining method of the SWS for HENS and MENS by Yee and Grossmann10 and Szitkai et al.,14 respectively. The two superstructures are partitioned based on key variables such as temperature and composition. Based on these both, the IBMS and the SWS are fraught with similar shortcomings which, according to Floudas,17 are the exclusion of the following structures (shown in Figure 1): (i) structures like bypass streams which are only feasible with nonisothermal mixing and (ii) structures involving a split stream exchanging heat/mass in two or more exchangers in series. In the IBMS method, each temperature/composition interval boundary is defined by the supply and target temperatures/ compositions of either the hot/rich or cold/lean sets of streams. The intermediate temperatures/compositions of the opposite kinds of streams crossing these interval boundaries are treated as variables. In the SWS approach, only the first and last interval boundaries of the superstructure are fixed, they are defined by the supply and target temperatures/compositions of both the hot/ rich and cold/lean sets of streams. The number of superstructure intervals in this case is not problem dependent unlike in the IBMS. The number of intervals in the SWS can be selected as being equal to max{NH,NC} and max{NR,NL}for HENS and MENS respectively, where NH, NC, NR, and NL are the number of hot, cold, rich, and lean streams, respectively. The superstructure interval defining approach in the IBMS and SWS models is based on temperatures/composition which are key variables for determining optimum driving forces in heat/mass exchangers. Defining superstructures in this manner, according to Isafiade,19 is beneficial because setting initial points and bounds for the overall model is greatly simplified. This simplification is more pronounced in the IBMS model which defines each interval boundary with the stream supply and target parameters. A further benefit of the IBMS and SWS models is

167

that the nonlinear equations are restricted to the objective function thereby making for much shorter solution times. On the basis of the shortcomings discussed above for the IBMS model, an improvement to this model is presented in this paper for HENS and MENS. The new approach which is known as the extended interval-based MINLP superstructure (EIBMS) involves splitting each stream before it enters into the intervalbased superstructure and then mixing the split streams after exiting the superstructure. This approach will help to include structures like Figure 1a but not those in Figure 1b,c. The splitting approach requires that every stream is split into an equal number of substreams. Every substream is then allowed to go into subsuperstructures (which are determined by the number of substreams). In each subsuperstructure, each substream will exchange heat/mass just as it happens in the normal IBMS. Every substream has the same supply and target temperatures/ compositions which correspond to the supply and target temperatures/compositions of the parent superstructure. However the sub-stream flows (which are variables to be optimized) must add up to the flow of the parent stream. It is worth mentioning that each stream can be split into any number of substreams; however, the more the splits, the more difficult it would be to get near minimum solutions in shorter times. The multiperiod concept will be used to represent the participation of substreams in the subsuperstructures. In the multiperiod superstructure approach of Aaltola,20 Chen and Hung,21,22 Verheyen and Zhang,23 and Isafiade,19 the index p was used to represent each period of operation. In the EIBMS presented in this paper, the index p represents each substream and the subsuperstructure where it participates. It is worth mentioning that Szitkai et al.14 also used the multiperiod concept to synthesize MENs involving multiple components. 2. Extended Interval-Based MINLP Superstructure The construction of the EIBMS for HENS and MENS is similar to that of the multiperiod superstructure since the index p as mentioned earlier is used to represent the substreams in subsuperstructures. However for the EIBMS, the number of substreams for each parent stream is equal to the number of subsuperstructures. This is unlike the multiperiod superstructure where the participation of a stream in a subsuperstructure is dependent on the existence of such stream in the period concerned. The EIBMS, just like the IBMS can be generated on hot/rich or cold/lean basis for HENS and MENS, respectively. For a hot/rich-based EIBMS, the interval boundaries are generated for each subsuperstructure (p) using the supply and target temperature/compositions of the hot/rich set of streams. The cold/lean set of streams are assumed to participate in all of the intervals defined by the hot/rich streams for each subsuperstructure (p) subject to thermodynamic feasibility. This implies that each subsuperstructure will have the same number of temperature/ composition intervals and the substreams participating in the same set of intervals for the corresponding subsuperstructures. It should be known that the intermediate temperatures/ compositions of substreams in each subsuperstructure which are variables to be optimized are determined by the heat/ mass load exchanged by the substreams in such intervals. Further, the heat/mass loads to be exchanged by substreams in subsuperstructure intervals are different for the same pair of intervals. As an example, the heat/mass to be exchanged in interval 1 of subsuperstructures 1 and 2 will be different. For a cold/lean-based EIBMS, the reverse would be the case.

168


Figure 2. Hot-stream-based superstructure representing substreams p in subsuperstructure p.

It is worth mentioning that each subsuperstructure of the EIBMS still has the characteristics of the IBMS. This implies that substreams can still undergo interval stream split which is determined by the number of the substreams of the opposite kind of stream present in the interval concerned. The structure for a hot-stream-based superstructure which includes the index p (representing substreams and subsuperstructures) is shown in Figure 2 (the mass exchange analogue for a rich-stream-based superstructure is shown in Figure 3). Figure 4 illustrates the EIBMS with two subsuperstructures and hence two substreams for a hot-stream-based superstructure. It should be known that an analogous structure applies for the MEN EIBMS. Additional variables which include the flow and heat load of the substreams are shown in the model equations. The necessary indices, sets, parameters, and variables that are used to describe the EIBMS are given in the Appendix. 3. HENS EIBMS Model Equations Stream Overall Heat Balance Equations. Streams need to exchange heat with streams of the opposite kind for them to reach their target temperatures. Such heat exchange can occur in one or more temperature intervals. However each pair of substreams that will exchange heat can only do so in the subsuperstructures where the concerned substream exists. The sum of the heat exchanged over the intervals where heat transfer occurs is equated to the sum of the available heat for each stream

in all subsuperstructures. Equations 1-8 describe the stream and substream heat balances for hot and cold streams.

hqi )

(Tsi - Tti)Fi ) hqi

(1)

(Ttj - Tsj )Fj ) hqj

(2)

(Tsi - Tti)Fis,p ) aqis,p

(3)

(Ttj - Tsj )Fjs,p ) aqjs,p

(4)

∑ aq

is,p

i ∈ H is ∈ HS

(5)

j ∈ C js ∈ CS

(6)

p∈P

hqj )

∑ aq

js,p

p∈P

hqj )

∑ ∑q

is,js,m,p

is ∈ HS p ∈ P

(7)

js∈CS m∈M

aqjs,p )

∑ ∑q

is,js,m,p

js ∈ CS p ∈ P

(8)

is∈HS m∈M

Note that the stream overall flowrate, F, is modeled as a parameter for the process streams and as a variable for utilities. Equations 1 and 2 describe the overall heat load hq for every hot and cold stream. This heat load is split into the number of subsuperstructures, p, through the splitting of the stream flows, F. The heat load of each hot and cold substream, aq, is described by eqs 3 and 4. The sum of the substream heat load, aq, for each stream is equated to the overall heat load hq, this is shown


169

Figure 3. Rich-stream-based multiperiod interval superstructure.

in eqs 5 and 6 for the hot and cold streams. Equations 7 and 8 describe the sum of the heat exchanged for intervals in subsuperstructures where heat transfer takes place as being equal to the substream heat load aq for the stream concerned. Interval Heat Balance Equations. Heat balance needs to be carried out over each interval in every subsuperstructure so as to determine the interval boundary temperature, tis,m,p, necessary to calculate the exchanger driving forces. (tis,m,p - tis,m+1,p)Fis,p )

∑q

is,js,m,p

is ∈ HS m ∈ M p ∈ P

(9)

js∈CS

(tjs,m,p - tjs,m+1,p)Fjs,p )

∑q

is,js,m,p

js ∈ HS m ∈ M p ∈ P

(10)

is∈CS

Note that tis,m,p, tjs,m,p, Fis,p, Fjs,p, and qis,js,m,p are all variables to be optimized, and the value of the interval boundary temperature, tis,m,p, tjs,m,p, can be diffierent for substreams of the same parent stream even at the same temperature location in the overall superstructure. Superstructure Interval Boundaries (Temperature Locations). The temperature locations in the IBMS are defined by the supply and target temperatures of either the hot or cold set of streams. This implies that each temperature location is fixed. However the intermediate temperature of streams while crossing temperature locations which they do not define are treated as variables whose values determine the optimum driving forces and hence heat exchange area. This concept equally s and applies in the case of EIBMS where the parameters Ti,m t Ti,m are used to define the temperature locations in the

superstructure for a hot-stream-based superstructure, while the equivalent cold stream parameters are used for a cold-streambased superstructure. However the stream intermediate temperatures are variables (tis,m,p, tjs,m,p) which have index is and js representing the hot and cold substreams, respectively, in subsuperstructure p. Temperature Feasibility along the Superstructure. Along the superstructure, stream temperatures should decrease from left to right for both hot and cold substreams in all subsuperstructures. This is represented by the following equations: tis,m,p g tis,m+1,p

is ∈ HS m ∈ M p ∈ P

(11)

tjs,m,p g tjs,m+1,p

js ∈ CS m ∈ M p ∈ P

(12)

Logical Constraints. The existence of a match between substreams is and js in subsuperstructure p is modeled using binary variables yijm as done by Yee and Grossmann.10 The variables are used in logical constraints equations which will give a value of “1” to the heat load if the match exists in interval m and subsuperstructure p and a value of “0” if the match does not exist. An upper bound, ΩpH, is included in the logical constraint equation. The upper bound which can be set as the smaller of the substream heat load of each of the substreams participating in the match is used to set the maximum amount of heat that could be exchanged within a match. The logical constraint equation is qis,js,m,p - ΩpHYi,j,m e 0

(13)

Heat Exchange Area Calculation. The variables dtis,js,m,p which represents exchanger approach temperatures are also used

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Figure 4. Hot-stream-based extended superstructure showing subsuperstructures 1 and 2.

in a second set of logical constraint equations (14 and 15) to calculate the driving forces for the logarithmic mean temperature difference (LMTD) as done by Yee and Grossmann.10 yijm is equally used in the logical constraint equation. The binary variable variable will take on a value of “1” if the match is,js exists in interval m and subsuperstructure p in the optimal network so that the approach temperature is appropiately calculated. If the match is,js does not exist in the optimal network then the binary variable takes on a value of “0”. In this case the use of ΓpH in the logical constraint equation helps to inactivate the equation so that negative approach temperature will not be included for any match in the optimal network and hence prevent numerical errors. dtis,js,m,p e tis,m,p - tjs,m,p + ΓpH(1 - yi,jm) is ∈ HS js ∈ CS m ∈ M dtis,js,m+1,p e tis,m+1,p - tjs,m+1,p + ΓpH(1 - yi,jm) is ∈ HS js ∈ C m ∈ M

(14)

(15)

The value of Γp according to Shenoy24 can be set as the maximum of zero and each of the temperature differences between the hot and cold substreams in the matches concerned. To ensure that exchangers of infinite areas are not included in the solution network, an exchanger minimum approach temperature (EMAT), δH, is used as follows: dtis,js,m,p g δH

(16)

where δH is a small positive number. Objective Function. The objective function simultaneously minimizes the utility and capital costs where the capital cost

Table 1. Stream and Capital Cost Dataa for Example 124 stream

Ts (°C)

Tt (°C)

F (kW °C-1)

h (kW m-2 °C-1)

costs ($ kW-1 year-1)

H1 H2 C1 C2 HU1 CU1

175 125 20 40 180 15

45 65 155 112 179 25

10 40 20 15

0.2 0.2 0.2 0.2 0.2 0.2

120 10

a Capital cost ) 30 000 + 750[area (m2)]0.81 for al exchangers. Annualization factor ) 0.322.

includes the fixed exchanger costs and the area cost of each exchanger. Chen’s first approximation25 is used to calculate the LMTD as described by eq 17. LMTDis,js,m,p ) [(dtis,js,m,p)(dtis,js,m+1,p)(dtis,js,m,p + dtis,js,m+1,p)/2]1/3

(17)

The objective function is min

+ ∑ ∑ UHCq ∑ ∑ CUCq + ∑ ∑ ∑ CF ∑ ∑ ∑ AC [q /U (LMTD is,js,m,p

is∈HS m∈M

is,js,m,p

+

js∈CS m∈M

is,jsZi,j,m

is∈HS js∈CS m∈M is,js

is,js,m,p

is,js

is,js,m,p)]

AEis,js

(18)

is∈HS js∈CS m∈M

The first and second terms in eq 18 represent the total hot and total cold utilities annual operating costs, respectively. The third term represents the total fixed exchanger cost while the last term is the total area cost which involves the area index term.


171

Figure 5. EIBMS network for Example 1. Table 2. Comparison of Different Methods for Example 1

number of units hot utility (kW) cold utility (kW) TAC ($/year)

SWS

IBMS

EIBMS

6 333.88 253.88 235,400

6 332.04 252.04 237,800

6 338.901 258.903 235,841

Table 5. Stream and Capital Cost Data for Example 3 (Hallale, 1998)a rich stream

G (kg s-1)

Ys

Yt

R1 R2 R3 R4 R5

2 4 3.5 1.5 0.5

0.005 0.005 0.011 0.010 0.008

0.0010 0.0025 0.0025 0.0050 0.0025

Table 3. Stream and Capital Cost Data for Example 2a stream

Ts (K)

Ts (K)

F (kW K-1)

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

433 522 544 500 472 355 366 311 333 389 509 311

366 411 422 339 339 450 478 494 433 495 509 355

8.79 10.55 12.56 14.77 17.73 17.28 13.90 8.44 7.62 6.08

costs ($ kW-1 year-1)

37.67 18.12

Table 4. Comparison of TAC for Example 2

34

Linnhoff and Flower Papoulias and Grossmann8 Lewin et al.35 Lin and Miller36 Pariyani et al.37 Yerramsetty and Murty31 this work

Lc

Xs

Xt

m

b

cost ($ kg-1)

S1 S2 S3

1.8 1.0 ∞

0.0017 0.0025 0.0170

0.0071 0.0085 0.0170

1.2 1 0.5

0 0 0

0 0 0.001

a Kw ) 0.02 kg NH3/(s kg); annualization factor ) 0.225; annual operating time ) 8150 hours.

a U (overall heat transfer coefficent) for every match except those invoving steam ) 0.852 kW m-2 K-1. U for matches involving steam ) 1.136 kW m-2 K-1. Heat exchanger capital cost ) 145.63[area (m2)]0.6 $ year-1.

method

lean stream

stream splits

no of units

TAC ($/year)

0 0 0 3(2) 0 0

10

43,934 43,934 43,452 43,329 43,439 43,538 44,172

10 10 10 10 12

The equations defining the feasible space for the EIBMS are fairly linear except for the objective function equation, hence the overall model can be easily solved. Constraints for restricting matches can be easily included in the EIBMS model. Such restrictions can be in the form of preferred or forbidden matches. The restriction can be accomplished by fixing the binary

variables concerned or limiting the heat/mass to be exchanged by such streams. The solver DICOPT++, which uses CPLEX and CONOPT for solving MILP and NLP subproblems has been used to solve the EIBMS model equations. These solvers operate in the GAMS29 environment. An Intel Core 2 Duo Processor machine with a clock speed of 2.00GH was used to run the models. The EIBMS model is applied to two heat and three mass exchange network synthesis problem. Because of the parent stream splitting approach employed in the EIBMS, the model has the tendency to give solution networks having exchangers with negligible sizes. The installation costs for these infinitely small exchangers is still paid for in such cases. Hence a constraint is imposed on the model which excludes exchangers whose sizes are negligible. HENS Examples. The EIBMS for HENS is applied to two examples. The first example is a HENS problem which has been solved by Isafiade and Fraser12 using the IBMS model. The second example was adapted from Yerramsetty and Murty.31In the figures illustrating the EIBMS solution networks, the heat exchanged are boxed above the exchangers concerned while the intermediate temperatures are written in italics.

172


Figure 6. EIBMS initial subsuperstructure 1 network for Example 3.

Example 1. This is Example 1 of Isafiade and Fraser12 which was taken from Shenoy.24 Shenoy24 solved the problem using the SWS of Yee and Grossmann10 while Isafiade and Fraser12 applied the IBMS technique. The problem comprises two hot and two cold streams alongside one hot and one cold utility. The stream and capital cost data are shown in Table 1. Applying hot-stream-based EIBMS to this example gives a network with a TAC of 235,841 $/year with six units. Comparing the EIBMS solution with the IBMS solution presented by Isafiade and Fraser,12 it was found that the solution generation time of the EIBMS (1.96 s of CPU) was longer than that of IBMS (0.32 s). The initialization of the EIBMS model is more complex than that of the IBMS based on the fact that the EIBMS involves more variables which is dependent on the number of substreams. This implies that each substream that flows in the EIBMS has to be given initial points, lower and upper bounds. In terms of structure, the EIBMS structure (Figure 5) is different from the IBMS structure for this example (see Example 1 of Isafiade and Fraser12) in that it involves a 2-way split of H2 and C2 unlike the IBMS structure which involves a 2-way split of H2 and C1. This result is further compared with those of Shenoy24 and Isafiade and Fraser12 in Table 2. Example 2. This example was adapted from Yerramsetty and Murty.31 The problem has been solved by various authors such as Linnhoff and Flower,34 Papoulias and Grossmann,8

Lewin et al.,35 Lin and Miller,36 and Pariyani et al.37 The problem comprises five hot streams, five cold streams, and one hot utility and one cold utility. Table 3 presents the stream and capital cost data. A hot stream-based EIBMS with two subsuperstructures was used to solve this example problem. The EIBMS gives a TAC of 44,172 $/year with twelve units. The TACs of the differenet methods which has been applied to this problem are shown in Table 4. The TAC of the EIBMS is just about 1.9% higher than the lowest TAC in Table 4. 4. MENS EIBMS Model Equations The substream and subsuperstructure concepts described above for the EIBMS for HENS equally apply to MENS. The model equations are discussed next. Stream Overall Mass Balance Equations. Mass exchange can also occur in one or more composition intervals in subsuperstructures in order for streams to get to their target compositions. Summing the mass exchanged over every interval where mass transfer takes place is equated to the overall mass requirement of the stream concerned. The following equations illustrate stream and substream mass balances for rich and lean streams. (Yrs - Yrt )Gr ) omr

(19)


-

(Y*t l

omr )

Y*s l )Ll

) oml

(20)

(Yrs - Yrt )Grs,p ) amrs,p

(21)

*s (Y*t l - Yl )Lls,p ) amls,p

(22)

∑ am

rs,p

r ∈ R rs ∈ RS

(23)

∑ am

ls,p

l ∈ S ls ∈ SS

(24)

∑ ∑M

rs,ls,k,p

(25)

∑ ∑M

rs,ls,k,p

ls ∈ SS p ∈ P

rs,ls,k,p

rs ∈ RS k ∈ K p ∈ P

(27)

ls ∈ SS k ∈ K p ∈ P

(28)

(y*ls,k,p - y*ls,k+1,p)Lls,p )

rs ∈ RS p ∈ P

ls∈SS k∈K

amls,p )

∑M

ls∈SS

p∈P

amrs,p )

exchanged in subsuperstructure intervals for every rich and lean stream is shown in eqs 25 and 26 where it is equated to the substream mass load for the streams involved. Interval Mass Balance Equations. Interval mass balances are needed to calculate the interval boundary compositions for rich and lean streams. These boundary compositions are necessary to calculate the exchanger driving forces. (yrs,k,p - yrs,k+1,p)Grs,p )

p∈P

oml )

173

(26)

rs∈RS k∈K

The lean stream overall flowrate, L, is modeled as a variable for the process and external lean streams. Similar to the HENS EIBMS, eqs 19 and 20 describe the overall mass load, om, for each rich and lean stream. The flow rate of the rich and lean streams are split into the number of subsuperstructures, p. Equations 21 and 22 illustrate the mass load, am, of each rich and each lean substream. Equations 23 and 24 describe the summing of the substream mass loads for each stream to give the overall parent stream mass load, om. The sum of the mass

Figure 7. EIBMS initial subsuperstructure 2 network for Example 3.

∑M

rs,ls,k,p

rs∈RS

The values of yrs,k, p, y*ls,k, p can be different for substreams which belong to the same parent stream at the same composition location. Superstructure Interval Boundaries (Composition Locations). The supply and target compositions of either the rich or lean set of streams are used to define the EIBMS interval boundaries. The intermediate compositions which include the substream indices is and ls for rich and lean streams, respectively, and index p representing subsuperstructures are treated as variables to be optimized. Composition Feasibility along the Superstructure. The composition of the rich and lean substreams should decrease

174


Figure 8. EIBMS final structure for Example 3. Table 6. Comparison of the Composition Partitioned Superstructure Models FLM model

hybrid model

IBMS model

EIBMS model

number of units TAC ($/year) type and number of parallel splits

8 134,000 3-way split of S3

7 133,323 2-way split of R4 3-way split of S3

9 132,101 2-way split of R4, S3 (in two different intervals)

type and number of series splits

none

10 134,399 2-way splits of R2, R3, R4 and S2 3-way split of S3 none

none

1-way split of R3

along the superstructure from left to right. This is described by eqs 29 and 30. Yrs,k,p g Yrs,k+1,p y*ls,k,p g y*ls,k+1,p

rs ∈ RS k ∈ k p ∈ P ls ∈ SS k ∈ K p ∈ P

(29) (30)

Logical Constraints. The mass exchange analogue of the logical constraint equation is presented. The binary variable

Table 7. Stream Data for Example 4 rich stream

G (kg s-1)

Ys

Yt

R1 R2

2.00 1.00

0.05 0.03

0.010 0.006

lean stream

Lc (kg s-1)

Xs

Xt

m

b

cost ($ kg-1)

S1 S2 S3

5 3 ∞

0.005 0.010 0.0013

0.015 0.030 0.015

2.00 1.53 0.71

0.00 0.00 0.001

0 0 0.01


175

Table 8. Comparison of the Four Methods for Example 4 pinch technology

units splits TAC ($)

NLP superstructure

IBMS

supertarget

design at ∆ymin ) 0.001

option 1

option 2

option 1

option 2

EIBMS

333,312

7 S1 split 2-way 345,416

8 S1 split 2-way twice in series 332,000

7 S1 and S3 split 2-way 333,300

6 none 338,168

5 none 358,292

6 none 338,168

zr,l,k takes on a value of “1” if a match exists and “0” if otherwise. Ωph just like in HENS can be set as the smaller of the substream mass load of the two substreams participating in the match. Mrs,ls,k,p - ΩpMZr,l,k e 0

(31)

Exchanger Driving Forces Calculation. In the calculation of the exchanger driving forces, dyrs,ls,k,p, the binary variable, zr,l,k, is included in a second logical constraint equation where its value takes on a value of “1” if a match rs,ls exists in a subsuperstructure interval and “0” if the match does not exist. In this case, the presence of ΓpM helps to inactivate the equation so that negative approach compositions will not be included for any match in the optimal network. dyrs,ls,k e yrs,k - y*ls,k + ΓpM(1 zr,l,k) rs ∈ RS r ∈ R ls ∈ SS l ∈ S k ∈ K dyrs,ls,k g yrs,k - y*ls,k - ΓpM(1 zr,l,k) rs ∈ RS r ∈ R ls ∈ SS l ∈ S k ∈ K

LMCDrs,ls,k,p ) [(dyrs,ls,k,p)(dyrs,ls,k+1,p)(dyrs,ls,k,p + dyrs,ls,k+1,p)/2]1/3

(32)

(37)

For continuous contact columns, the exchanger mass capital cost calculation method by Hallale18 is used, while for stagewise columns, the per stage costing method of Papalexandri et al.,26 which employs the diameter of each column, is used. The new sizing formula of Fraser and Shenoy27 is used to size the stagewise exchangers. Equations 38 and 39 show the sizing equations. Continuous contact columns: min{

∑ (AC )(L ) + ∑ ∑ ∑ CB ls

rs,lszr,l,k

ls

ls

+

rs∈RS ls∈SS k∈K

∑ ∑ ∑ ACH

Drs.ls rs,ls[Mrs,ls,k,p /Kw{LMCDrs,ls,k}]

+ WT}


(38)

(33)

dyrs,ls,k+1 e yrs,k+1 - y*ls,k+1 + ΓpM(1 zr,l,k) rs ∈ RS r ∈ R ls ∈ SS l ∈ S k ∈ K

(34)

dyrs,ls,k+1 g yrs,k+1 - y*ls,k+1 - ΓpM(1 zr,l,k) rs ∈ RS r ∈ R ls ∈ SS l ∈ S k ∈ K

(35)

ΓpM can be set in the same way as described for HENS. An exchanger minimum approach composition (EMAC), δM, is also used to avoid the inclusion of exchangers of infinite height/stages in the optimal network. dyrs,ls,k,p g δM

(LMCD) which is included in the objective function for continuous contact columns.

where WT ) WF ∑rs∈RS ∑ls∈SS ∑k∈K(ewrs.ls,k + fwrs,ls,k). WF is a weighting coefficient. The first term in eq 38 represents the total annual operating cost. The second term is the total exchanger fixed cost. The third term in this equation describes the total exchanger annual capital cost per height together with the area cost index. Stagewise columns: min{

∑ (AC )(L ) + ∑ ∑ ∑ CB ls

ls∈SS

rs,lszr,l,k+

ls


ACTrs,ls

(36)

where δM is a small positive number. Objective Function. The MENS objective function just like that of HENS simultaneously minimizes the MSA and capital costs where the capital cost comprises the installation and mass exchanger costs. Chen’s first approximation25 is also used to calculate the logarithmic mean composition difference

Figure 9. NLP reducible superstructure solution of Comeaux33 for Example 4.

∑ ∑ ∑ [N

rs,ls]}

(39a)


Nrs,ls,kwhich is the number of stages is defined by Fraser and Shenoy27 as Nrs,ls.k )

(

∆yn + ∆y*n ∆yn1 + ∆yn2

)

1/n

where ∆yn ) rich stream concentration difference, ∆y*n ) lean stream equilibrium concentration difference, ∆y1n ) rich end of the exchanger driving force, ∆y2n ) lean end of the exchanger driving force, and n ) 1/3 by Underwood28 and 0.3275 by Chen.25 Similar to eq 38, the first term in eq 39a represents the total annual operating cost. The second term is the total exchanger fixed charge while the last term represents the total annual capital cost per stage of the exchangers. MENS Examples. Three MENS examples are presented in this paper. The first example (Example 3) is taken from Hallale,18 the second is adapted from Hallale and Fraser,5 while the third was taken from El-Halwagi.32 In the figures which describe the EIBMS solution networks for MENS, the mass exchanged are boxed above the exchangers concerned while the intermediate compositions are written in italics. Example 3. This MENS example was taken from Hallale.18 The problem has been solved by Szitkai, et al.14 where the

176


Figure 10. EIBMS final structure (subsuperstructure 2) for Example 4. Table 9. Stream and Capital Cost Data for Example 532 rich stream

G (kg s-1)

R1 R2

8.00 6.00

Ys

Yt

0.10 0.08

0.01 0.01

lean stream

Lc (kg s-1)

Xs

Xt

m

b

cost ($ kg-1)

S1 S2

10.00 ∞

0.01 0.00

0.02 0.11

2.00 0.02

0.00 0.00

0.00 0.08

authors used the fairly linear model (FLM) which is a mass exchange analogue of the SWS for HENS. Emhamed et al.30 and Isafiade and Fraser15 also solved this example using an hybrid version of the FLM and the IBMS models, respectively. The example comprises five gaseous streams from which ammonia needs to be removed. The available MSAs for the ammonia removal are water based, two (S1 and S2) of which are process MSAs while the third (S3) is an external MSA.The concentration ranges of the MSAs overlap, which means that the y-y* method of Hallale and Fraser5 needs to be used. The exchangers are continuous contact columns which are costed using the exchanger mass-based costing approach of Hallale.18 The stream and capital cost data are shown in Table 5. Applying a rich stream based EIBMS to this example and setting the index p equal to 2, gives a network with a TAC of 132,101 $/year. The network has nine units, eight of which exist in subsuperstructure p ) 2 while only one exists in superstructure p ) 1 (both subsuperstructures are shown in Figures 6 and 7). The only exchanger in subsuperstructure p ) 1 occurs as a result of R3 splitting into the two subsuperstructures with

Figure 11. EIBMS final structure for Example 5.

significant flows before exchanging mass. Substream 1 of R3 has a flow of 0.699 kg/s while substream 2 has a flow of 2.801 kg/s. The flowrates of the other set of rich process substreams in subsuperstructure 1 are insignificant. Figure 8 illustrates the combined superstructure where the exchangers with insignificant mass load have been excluded. The solution generation time (8.6 s) of the EIBMS for this example is also longer than that of the IBMS (0.73 s) presented by Isafiade and Fraser.15 The TAC of the previous techniques applied to this example are compared with the EIBMS approach in Table 6. The EIBMS model gives the lowest TAC although this cost difference is somewhat insignificant compared with the TACs of the other models. Example 4. This is Example 2 of Hallale and Fraser5 which was adapted from El-Halwagi.15 The problem also involves multiple MSAs whose concentration ranges overlap. Two phenol-rich aqueous streams (R1 and R2) are to be stripped of phenol. Available for this removal are two process MSAs and one external MSA. The MSAs are gas oil (S1), lube oil (S2), and light oil (S3) which is the external MSA. Hallale and Fraser5 assumed that the cost of the seive tray columns is $4552 per year per equilibrium stage, as presented by Papalexandri et al.26 The annual operating time is 8600 h. The stream data are presented in Table 7. This problem has also been solved by Comeaux33 and Isafiade and Fraser.15 Comeaux33 used NLP reducible superstructure while Isafiade and Fraser15 used the IBMS for MENS approach. Table 8 presents the solutions obtained by the different techniques which have been applied to this problem, together with the solution of the EIBMS. Hallale and Fraser5 only illustrated a design at a minimum composition difference in the rich phase (∆ymin) of 0.001. The TAC at this design is 345,416 $/year. The design involves seven units with S1 split two ways. The TAC after supertarget is 333,312 $/year. The NLP superstructure of Comeaux33 gave two options for solutions. The first option has a TAC of 332,000 $/year with eight units, with S1 split two ways twice, one after the other. The second option has a TAC of 333,300 $/year which involves seven units with S1 and S3 both split two ways. It should be pointed out that the present authors backcalculated the flow of S3 in the second solution of Comeaux,33 obatining 0.7153 kg/s and not 0.6528 kg/s as presented by Comeaux.33 The new S3 flow will give the annual operating cost (AOC) as 221,456 $/year and the TAC as 348,956 $/year. The structure is shown in Figure 9. The IBMS of Isafiade and Fraser15 also involves two solution options. The first solution has a TAC of 338,168 $/year. The network has six units with no stream splits. The second solution


has a TAC of 358,292 $/year with five units and no stream splits. The EIBMS which was generated on the lean basis gives the same TAC (338,168 $/year) as the first solution of the IBMS of Isafiade and Fraser.15 The optimum EIBMS structure also has six units with no stream splits. It should be known that the EIBMS solution for this example only involves subsuperstructure 2. Figure 10 presents the final structure for the EIBMS. The lean stream compositions are presented as the rich stream equivalent. Example 5. This example was taken from El-Halwagi.32 It is an oil refinery problem where two different streams (R1 and R2) of wastewater which are rich in phenolics are generated. Two processes are available to remove the phenol from the waste streams. The first involves the use of gas oil (S1; process MSA) in solvent extraction while the second process involves the use of activated carbon (S2; external MSA) for adsorption. It should be known that no capital cost information was given in El-Halwagi,32 in addition, the kinds of mass exchange process required by the streams were not stated. On the basis of the absence of capital cost data, the 4552 $ per year per equilibrium stage given by Papalexandri et al. (1994) is used in this paper. The operating hours were also assumed to be 8150 h per year. See Table 9 for the stream data. The superstructure comprises three composition locations. The solution gives a TAC of 2.22 × 107 $/year. The problem is operating cost dominant. Each of the mass exchangers requires only a single stage. The solution network for the problem is shown in Figure 11. It can be seen from this figure that R1 and S2 both split into superstructures 1 and 2 with S2 still splitting further in superstructure 1. 5. Conclusion This paper has presented an approach for improving the heat and mass exchange IBMS models of Isafiade and Fraser.12,15 Owing to the superstructure partitioning method employed in the IBMS, configurations involving split streams exchanging heat/mass in two or more exchangers cannot be included in the search for the optimum network. This shortcoming is overcome in the EIBMS model by splitting every stream before entering the superstructure. Each substream is made to go into a subsuperstructure after which it mixes with the other substreams at the exit point from the superstructure. Each subsuperstructure is modeled as an IBMS where the supply and targets temperatures/compositions of either the hot/rich or cold/lean set of streams are used to define the superstructure intervals. Although the TACs of the EIBMS compared with those of the IBMS are not significantly lower, the EIBMS offers the opportunity of generating network configurations which are different from those of the IBMS. A major shortcoming of the EIBMS is the increase in the number of variables and increased complexity in setting initial points and bounds for these variables. Acknowledgment This research was supported by the Claude Leon Foundation. Appendix Sets C ) cold process and utility streams H ) hot process and utility streams CS ) cold process and utility substreams HS ) hot process and utility substreams R ) rich process streams

177

S ) lean streams (process and external mass separating agents) RS ) rich process substreams SS ) lean substreams (process and external mass separating agents) K ) composition intervals in superstructure M ) temperature intervals in superstructure P ) subsuperstructures Indices i ) hot process or utility stream is ) hot process or utility substream j ) cold process or utility stream js ) cold process or utility substream r ) rich process stream rs ) rich process substream l ) lean stream (process or external mass separating agent) ls ) lean substream (process or external mass separating agent) m ) index for temperature interval location (m ) 1, ..., NOM and temperature location, 1, ..., NOM + 1) k ) index for composition interval location (k ) 1, ..., NOK and composition location, 1, ..., NOK + 1) p ) index for subsuperstructure (p ) 1, ..., NOS) Parameters NOM ) number of temperature intervals NOK ) number of composition intervals NOS ) number of substreams and subsuperstructures AC ) area cost coefficient for heat exchangers AE ) area cost index for heat exchangers CF ) fixed charge for heat exchangers CUC ) cost per unit of cold utility HUC ) cost per unit of hot utility ACls ) annual cost per unit of lean stream l ACTrs,ls ) column annual cost per stage involving rich substream rs and lean substream ls ACHrs,ls ) column annual cost per height involving rich substream rs and lean substream ls CBrs,ls ) column fixed charge involving rich substream rs and lean substream ls Drs,ls ) column area cost exponent involving rich substream rs and lean substream ls hc ) stream heat transfer coefficient m ) equilibrium constant for the transfer of component from rich stream r to lean stream l Kw ) lumped mass transfer coefficient b ) equilibrium line intercept Tsi,m ) supply temperature of hot stream i which starts from interval m t Ti,m ) target temperature of hot stream i which ends in interval m Tsj,m ) supply temperature of cold stream j which starts from interval m t Tj,m ) target temperature of cold stream j which ends in interval m ΩpH ) upper bound for heat exchanged in exchanger in subsuperstructure p ΩpM ) upper bound for mass exchanged in exchanger in subsuperstructure p ΓpH ) upper bound for driving force in heat exchangerin subsuperstructure p ΓpM ) upper bound for driving force in mass exchangerin subsuperstructure p Xsl,k ) supply composition of lean stream lwhich starts from interval k t Xl,k ) target composition of lean stream lwhich ends in interval k s Yr,k ) supply composition of rich stream r which starts in interval k t Yr,k ) target composition of rich stream r which ends in interval k

178


Y*s l,k ) equilibrium supply composition of lean stream l which starts in interval k *t Yl,k ) equilibrium target composition of lean stream l which ends in interval k εmin ) minimum composition difference Binary Variables yijm ) variable showing the existence of match i,j in interval m in the network zrlk ) variable showing the existence of match r,l in interval k in the network PositiVe Variables dtis,js,m,p ) heat exchanger driving force for match is,js in temperature interval m and subsuperstructure p Fi ) flow rate of hot stream i Fis,p ) flow rate of hot substream is in subsuperstructure p Fj ) flow rate of cold stream j Fjs,p ) flow rate of cold substream js in subsuperstructure p qis,js,m,p ) heat exchanged between hot substream is and cold substream js in temperature interval m and subsuperstructure p hqi ) overall heat load of hot stream i hqj ) overall heat load of cold stream j aqis,p ) heat load of hot substream is in subsuperstructure p aqjs,p ) heat load of cold substream js in subsuperstructure p tis,m,p ) temperature of hot substream is at hot end of interval m in subsuperstructure p tjs,m,p ) temperature of cold substream js at hot end of interval m in subsuperstructure p dyrs,ls,k,p ) mass exchanger driving force for match rs,ls in composition interval k and subsuperstructure p Gr ) rich stream flowrate Grs,p ) rich substream flowrate in subsuperstructure p Ll ) lean stream flowrate Lls,p ) lean substream flowrate in subsuperstructure p Mrs,ls,k,p ) mass exchanged between rich substream rs and lean substream ls in interval k and subsuperstructure p omr ) overall mass load of rich stream r oml ) overall mass load of lean stream l amrs,p ) mass load of rich substream rs in subsuperstructure p amls,p ) mass load of lean substream ls in subsuperstructure p Nrs,ls,k,p ) number of stages in staged column rs,ls,k in subsuperstructure p yrs,k,p ) rich substream rs composition at interval boundary k and subsuperstructure p xls,k,p ) lean substream ls composition at interval boundary k and subsuperstructure p y*ls,k,p ) lean substream ls equilibrium composition at interval boundary k and subsuperstructure p AbbreViations EIBMS ) ended interval based MINLP superstructure EMAC ) exchanger minimum approach composition EMAT ) exchanger minimum approach temperature FLM ) fairly linear model HENS ) heat exchanger network synthesis IBMS ) interval based MINLP superstructure LMCD ) logarithmic mean composition difference LMTD ) logarithmic mean temperature difference LP ) linear programming MENS ) mass exchanger network synthesis MILP ) mixed integer programming MINLP ) mixed integer nonlinear programming MSAmass separating agent

NLP ) nonlinear programming SWS ) stagewise superstructure

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ReceiVed for reView November 27, 2008 ReVised manuscript receiVed October 1, 2009 Accepted October 15, 2009 IE801823N

Extended Interval-Based Mixed Integer Nonlinear Programming

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