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MINLP retrofit of heat exchanger networks comprising different exchanger types
Computers and Chemical Engineering 28 (2004) 235 /251 www.elsevier.com/locate/compchemeng
MINLP retrofit of heat exchanger networks comprising different exchanger types Aleksander Sorsˇak, Zdravko Kravanja * Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova 17, PO Box 219, Maribor 2000, Slovenia
Abstract This paper describes a simultaneous MINLP optimization model for the retrofit of heat exchanger networks (HEN), comprising different exchanger types. This model is based on the stage-wise superstrucutre of HEN [Comput. Chem. Eng. 14 (1990) 1165], extended for retrofit and different exchanger types. Each potential match in the superstructure is now replaced by a match superstructure comprising: double pipe (DP), shell and tube (U tubes, ST), and plate and frame exchangers (PF). The operability constraints for the different exchanger types are modeled by the convex hull formulation of disjunctions, nonlinear terms are modeled only within the objective function and additional constraints are included to obtain feasible temperature distribution. Several examples are presented to illustrate the efficiency and advantages of the proposed multi-type model. # 2003 Elsevier Ltd. All rights reserved. Keywords: Heat exchanger types; Disjunctive modeling; MINLP; Retrofit of HEN
1. Introduction Like the earlier grass-root synthesis of heat exchanger networks (HEN), the retrofit of an existing HEN can be traditionally carried out using two main approaches: the pinch analysis approach based on thermodynamics and heuristics, and the mathematical programming approach, where HENs are presented as mathematical models. For the pinch analysis approach, Tjoe and Linhoff (1986), proposed a two step retrofit procedure and for the mathematical programming approach, Ciric and Floudas (1990), based on the transshipment model by Papoulias and Grossmann (1983a,b), developed an MINLP model for the retrofit of HEN allowing the reconstruction and relocation of existing HEs. In addition, based on the superstructure MINLP synthesis of HEN, Yee and Grossmann extended their synthesis model Yee and Grossmann (1990) for the retrofit of HEN Yee and Grossmann (1991). For other earlier important achievements occurred in the area of HEN retrofit see Jezowski (1994a,b). More recent research
* Corresponding author. Tel.: /386-2-229-4481; fax: /386-2-2527774. E-mail address: [email protected] (Z. Kravanja). 0098-1354/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00167-4
was directed towards more advanced methods, e.g. HEN retrofit considering pressure drops (Nie & Zhu, 1999), two-step approach by (Ma, Hui & Yee, 2000) using constant approach temperature in the first and MINLP to finalize design in the second step, combined mathematical programming and thermodynamic analysis approach by Zhu and Asante (1999) and a simultaneous process changes and HEN retrofit (Zhang & Zhu, 2000). However, the selection of different HE types (HET) has not yet been considered simultaneously during the retrofit of HEN, probably because the model size would be drastically increased. The objective of this work is to develop a tight model representation, comprising different exchanger types for the purposes of HEN retrofit, based on the synthesis model of HEN (Sorsˇak & Kravanja, 2001, 2002). The model is based on a stagewise superstructure (Yee & Grossmann, 1990) where each potential match between the hot and cold stream is replaced by a match superstructure (SM) comprising a double pipe HEs (DP), shell and tube HEs (ST), plate and frame HEs (PF) and a bypass (Sorsˇak & Kravanja, 2002). In order to be comprised of different HET and existing HEs, which may also be of different types and for the purposes of HEN retrofit, the HEN synthesis model (Sorsˇak & Kravanja, 2002) has been adapted:
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Nomenclature DlnTijk logarithmic mean temperature difference (K) DTijk temperature difference between the hot stream i and cold stream j within the stage k (K) DTmin minimum temperature difference (K) Aijkl heat transfer area (m2) Al heat transfer area of an existing heat exchanger in HEN (m2) BID bypass ID (/) crfijkl fixed relocation cost for existing exchangers in HEN ($/a) cfl Fixed annualized investment cost of exchangers ($/a) cnfl Fixed deinstallation cost of existing exchangers in HEN ($/a) CP set of cold streams (/) CU set of cold utilities (/) crvijkl variable relocation cost for existing exchangers ($/(m2a)) cvl variable annualized investment cost of exchangers ($/(m2a)) n cvl variable deinstallation cost of existing exchangers in HEN ($/(m2 a)) Ft correction factor for the temperature driving force (/) G heat capacity flowrate for process streams (W/K) Gvi heat capacity flowrate for hot utilities (W/K) Gvj heat capacity flowrate for cold utilities (W/K) HETl type of existing exchanger in HEN (/) HP set of hot streams (/) HU set of hot utilities (/) i hot stream index (/) j cold stream index (/) k stage index in the HEN superstructure ( /) kij overall heat transfer coefficient (W/(m2 K)) l heat exchanger type index ( /) Mnhe maximum number of new exchangers in HEN (/) Mrel maximum number of existing exchanger relocations in HEN ( /) POSijkl location of existing exchangers in HEN ( /) R temperature ratio (/) S thermal efficiency ( /) ST set of stages in HEN superstructure (/) T (i)A inlet temperature of hot stream for exchangers (K) ijkl T (i)Bijkl outlet temperature of hot stream for exchangers (K) Tik inlet temperature of hot stream for stage k (K) Ti (k1) outlet temperature of hot stream for stage k (K) TIN inlet temperatures of process streams (K) T (j)A outlet temperature of cold stream for exchangers (K) ijkl T (j)Bijkl inlet temperature of cold stream for exchangers (K) Tjk outlet temperature of cold stream for stage k (K) Tj (k1) inlet temperature of cold stream for stage k (K) TOUT outlet temperatures for process streams (K) wA penalty factor (/) yijkl binary variable of heat exchangers in HEN superstructure (/) ai connective heat transfer coefficient for process stream i . The match superstructure (SM) has been extended using existing HEs in the network, where each existing heat exchanger is described by its heat transfer area, HE type and position within the HEN. . External heaters and coolers, as used in the synthesis model (Sorsˇak & Kravanja, 2002), are omitted and replaced by two additional stages for the heat transfer between hot utility streams and cold process streams,
and for the heat transfer between cold utility streams and hot process streams. . The objective function comprises the operational costs of HEN, the investment costs for both, the existing HEs and the new HEs, reinstallation and relocation costs of existing HEs. Several examples are presented in this paper to illustrate the capabilities of the proposed model.
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237
Fig. 1. Four stage superstructure of HEN.
2. Retrofit model of HEN In order to determine the optimal retrofitted structure of the HEN, an extension of the simultaneous MINLP model for the HEN synthesis (Sorsˇak & Kravanja, 2002) is proposed. Several indexes have to be specified before presenting the model and superstructure
hot to the cold stream within the match superstructure (SM) comprising: double pipe HEs, ST HEs, plate and frame (PF) HEs, and bypasses. In the proposed exten-
. index i for hot streams given by the set HP; note that set of hot utilities (HU) is a subset of HP, . index j for cold streams given by the set CP; note that set of cold utilities (CU) is a subset of CP, . index k for the superstructure stages given by the set ST, . index l for the exchanger types, a bypass and existing exchangers given by the set HET. It should be noted that the proposed model can be used for the simultaneous MINLP synthesis of HEN when the specification of the existing heat exchanger is not presented. 2.1. Superstructure In the superstructure each cold stream (j) can be potentially matched with each hot stream (i) in several stages (k ). A vector of binary variables is assigned to the structural alternatives to determine the retrofitted HEN structures during the MINLP optimization. In the simultaneous MINLP synthesis model (Sorsˇak & Kravanja, 2002), the heat is potentially transferred from the
Fig. 2. Match superstructure.
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238
Table 1 Data of considered heat exchanger types Type
pmax (Mpa)
T (8C) range
A (m2) range
Cf ($/year)
Cv ($/(m2 year))
Double pipe ST PF
30.7 30.7 1.6
/100 /600 /200 /600 /25 /250
0.25 /200 10 /1000 1 /1200
1937 21 615 17 034
201 93 61
sion of the HEN superstructure (Fig. 1), the match superstructure (Fig. 2) is now updated with the existing HEs, each being described by:
XXX i
j
l
. heat transfer area (Al ), . type (HETl ), and . position within the network (POSijkl ).
× Max 0;
In order to establish the selection of different exchanger types for the heat transfer using utility streams, the external coolers and heaters, as presented in the original superstructure (Yee & Grossmann, 1990) are omitted, and replaced by two additional stages: a stage representing heat transfer between HU and cold process streams taking place within the first stage of the superstructure (k /1), and a stage representing the heat transfer between CU and the hot process streams taking place at the last stage of the HEN superstructure (k / K ). It should be noted that the introduction of the utility stages into the HEN superstructure enables the use of multiple utilities. The number of stages is determined by heuristics. If possible, K should take the number of the enthalpy intervals of the balanced composite curves. Generally, no improvements of HEN can be expected with the larger one than the number of the intervals. However, the size of the model drastically increases when the large number of stages is used and, very often, because of large K , the model cannot be solved to the optimal solution. In practical cases, the number of stages (K ) is generally smaller (halved or even smaller) than the number of enthalpy intervals. It also should be noted that different area, pressure and temperature ranges hold for different exchanger types (Hewitt, Shires & Bott, 1994). The basic recommendations are shown in Table 1. In addition, due to the leakage problem, using PF exchangers is not recommended when one of the streams in the network is toxic.
2.2. Objective function The objective function of the proposed model defines the annual cost of HEN and has been defined as: XXX XX [(cCU × Fijkl )j CU ]kK j i
j
l
X [(cHU × Fijkl )i HU ]k1 i l
i
k
X cfl × yijkl cvl
j
(1a)
XXX i
j
Fijkl kij × Ftijkl × Dln Tijkl
Al
(1b)
k
X (crfijkl × yijkl )crvijkl l
× Max 0; Al X
cnfl
× 1
kij × Ftijkl × Dln Tijkl (1c)
l fl:lBIDgPOSijkl 0
XXX i
l
Fijkl
j
k
yijkl
l fl;lBIDg
(1d)
i HP; j CP; k ST; l DHET where BID specifies a bypass index in the set of exchanger types and existing exchangers (DHET ). The objective function comprises the following terms: . cost of utility consumption (1a), . annualized investment costs for the HEs in the network (1b), . annualized costs for the relocation of existing HEs (1c), and . annualized costs for the deinstalation of those existing HEs (1d), which are not selected in the retrofitted structure of the adapted HEN. The fixed charge coefficients (cfl , cvl , crfijkl , crvijkl , cnfijkl , are specified by the linearization of Guthrie’s function (Guthrie, 1969). The cost of the PF exchanger has been specified as 70% (Walker, 1990) of the cost for the ST exchanger. In order to preserve the linearity of feasible space of the model, all nonlinear constraints (2) have been omitted by defining the corresponding charge coefficients (Sorsˇak & Kravanja, 2002). It should be noted that in the objective function (1b) only continuous area extensions to existing exchangers are allowed. Allowing discrete extensions would dramatically increase problem combinatorics and increase model complexity with many new discrete variables and constraints cnvijkl )
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239
Table 2 Ft correction factor coefficients i
ai
bi
0 1 2 3 4 5 6
15.210806 /1.603737 42.337642 / / / /
0.623393 /0.698824 /0.250187 0.434342 0.301423 /0.171389 /0.141410
Fig. 3. The Ft correction factor for the ST HEs with an even number of tube passes.
S
for discrete increments. Flijkl
Amin ijkl 5
kij × Ftijkl × Dln Tijkl 1 1 1 kij ai aj
5Amax ijkl
(2a) (2b)
In order to avoid the numerical difficulties of using logarithmic mean (LMTD ) at equal temperature differences, the temperature driving force has been specified by Chen’s approximation (Chen, 1987): (1=3) A B DTijkl DTij(k1)l A B Dln Tijkl DTijkl × DTij(k1)l × 2 (3) The approximation underestimates the logarithmic mean (LMTD ), which yields a ‘safe’ design of heat exchangers, because of the overestimation of the heat transfer area (A ) of the selected exchangers. This approximation can be directly used for the counter flow arrangement of streams in the exchangers. Since the ST exchanger type stream flow arrangement is a combination of the counter and co-current flow, the temperature driving force has to be corrected by the Ft correction factor (Fig. 3). The Ft has been defined by the Ft approximation (Sorsˇak & Kravanja, 1999), since the Underwood’s Ft equation (Hewitt et al., 1994) is an inconvenient form for optimization (Sorsˇak & Kravanja, 2002):
p × S Ft 1 2 × tan 6 X X 2 × ai × [lg(R)]i bj × [lg(R)]j 1
i0
B T(j)A ijkl T(j)ij(k1)l B T(i)A ijkl T(j)ij(k1)l
(6)
coefficients a and b are shown in Table 2. In (1b) and (1c) the smooth approximation Max(. . .) in the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Max(0; x)0:5× (xo)2 0:5 × x (7) has been applied, to account for the possible adaptation of the existing HEs in the network. 2.3. Feasible temperature distribution In order to avoid infeasible temperature distribution when the ST exchanger type is selected, additional constraints have been specified. Since the extended model comprises different exchanger types, the temperature distribution in HEN that holds for a pure counter flow exchanger may become infeasible if the ST exchanger is selected (Sorsˇak & Kravanja, 2002). This problem can be observed when we take a closer look at the temperature distribution of the counter flow and ST heat exchangers (Fig. 4). In counter flow heat exchangers the outlet temperature of the cold stream can be higher (Fig. 4a) because of the geometry of the transfer area. When ST exchangers are used, the flow arrangement combines the counter and co-current flows and the temperature distribution becomes infeasible when Tco /Tho (Fig. 4b). To overcome this problem, additional constraints have been specified for the ST exchanger type: (HETl /3):
j0
(4) where R defines the temperature ratio: R
B T(i)A ijkl T(i)ij(k1)l B T(j)A ijkl T(j)ij(k1)l
and S defines the thermal efficiency:
(5) Fig. 4. Temperature arrangement in (a) counter flow HE and (b) ST HE with U-tubes.
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Ti(k1) Tjk DT UP ×(1yijkl )HETl 3 ]DT min ;
(8)
i HP; j CP; k ST Furthermore, in order to obtain the Ft correction factor for the ST exchanger type, several constraints have been additionally specified: Tjk Tj(k1) DT UP ×(1yijkl )HETl 3 ] DT minFt ; i HP; j CP; k ST Tik Ti(k1) DT UP ×(1yijkl )HETl 3 ]DT minFt ; i HP; j CP; k ST Tik Tj(k1) DT UP ×(1yijkl )HETl 3 ] DT minFt ;
(9)
i HP; j CP; k ST
2.4. Energy balances of utility streams
stages. It should be noted that the energy balance constraints disallow heat transfer between HU and CU.
2.5. Disjunctive modeling of operational limitations for selected exchanger type and existing heat exchangers The selection of the exchanger types depends on the operating temperatures and pressures of the involved streams (Table 1). Since the pressures are not the optimization variables, they can be taken into account in a prescreening procedure before the optimization. However, all temperatures are modeled as optimization variables and affect the selection of exchanger types or existing exchangers during optimization. The temperature ranges have, therefore, been modeled by the convex-hull formulation of disjunctions. The following temperature constraints have been specified for hot streams: X Tik T(i)A i HP; j CP; k ST ijkl ; l HET
Since the inlet and outlet temperatures of the utility streams are constant, the heat capacity flowrates (Gi , Gj ) for the utilities have been modeled as variables (Gvi , Gvj ). Therefore, additional constraints have been specified:
Ti(k1)
Ti2 TiOUT ; i HU Tj(K1) TjOUT ; j CU XX (Fijk )CUTj 0 ; (TiIN TiOUT )×Gvi
T(i)Bij(k1)l 5Tlmax × yijkl ;
j
k
i HU XX (TjOUT TjIN )×Gvj (Fijk )HUTi 0 ; i
k
(10a) (10b)
X
T(i)Bij(k1)l ;
i HP; j CP; k ST
l HET max T(i)A × yijkl ; ijkl 5Tl
i HP; j CP; k ST; l HET
i HP; j CP; k ST; l HET (10c)
(10d)
j CU and TIN are inlet temperatures for hot and where TIN i j OUT CU, Ti and TOUT are outlet temperatures for hot and j CU and K defines the number of HEN superstructure stages. The outlet temperature of the first stage in the HEN superstructure for each hot utility (Ti 2) is equal to the outlet temperature of the utility (TOUT ). Therefore, i the HU were used in the first stage of HEN superstructure only. Furthermore, scalars for the inlet and OUT outlet temperatures of the HU (TIN ) were applied j , Ti directly into the energy balance equations (10c). Consequently, the energy balance constraints (10c) for the heat transfer between HU and cold process streams remain linear. The energy balance constraints for the heat transfer between CU and hot process streams (10b, d) remain linear in the same way. Thus, the implementation of the energy balance constraints (10) preserves the linearity of the feasible space of the HEN retrofit model. The use of CU is limited to the last stages of the superstructure of HEN. The first and the last stages of the HEN superstructure are, therefore, called the utility
min × yijkl ; T(i)A ijkl ]Tl
i HP; j CP; k ST; l HET
T(i)Bij(k1)l ]Tlmin × yijkl ;
(11)
i HP; j CP; k ST; l HET where Tik and Ti (k1) are the match temperatures at the B inlet and outlet of stage k , T (i )A ijkl and T (i)ij (k1)l are temperatures disaggregated over the set of different exchanger types, bypass and existing heat exchangers and y represents a vector of the binary variables. Similarly, we defined the temperature constraints for cold streams: X Tjk T(j)A i HP; j CP; k ST ijkl ; l HET
Tj(k1)
X
T(j)Bij(k1)l ;
i HP; j CP; k ST
l HET max T(j)A × yijkl ; ijkl 5Tl
i HP; j CP; k ST; l HET
T(j)Bij(k1)l 5Tlmax × yijkl ; i HP; j CP; k ST; l HET min × yijkl ; T(j)A ijkl ]Tl
i HP; j CP; k ST; l HET
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T(j)Bij(k1)l ] Tlmin × yijkl ;
(12)
i HP; j CP; k ST; l HET heat loads: X F(l)ijkl ; Fijk
2.7. Constraints to limit the number of new heat exchangers The maximum number of new heat exchangers in the HEN structure Mnhe were specified by: XXXX yijkl 5M nhe ; Öl fl:l BBIDg (18)
i HP; j CP; k ST
l HET
F(l)ijkl 5FUP × yijkl ;
241
i HP; j CP; k ST; l HET (13)
and temperature differences: X A DTijk DTijkl ; i HP; j CP; k ST
i
j
k
l
When the maximum number of new exchangers in the HEN superstructure is specified, the combinatorics is significantly reduced too, which additionally shortens the CPU times to obtain an optimal solution.
l HET
DTij(k1)
X
B DTij(k1)l ;
i HP; j CP; k ST
2.8. The single placement of existing heat exchangers
l HET A DTijkl 5 DT UP × yijkl ;
i HP; j CP; k ST; l HET
B DTij(k1)l 5 DT UP × yijkl ;
i HP; j CP; k ST; l HET
(14)
Additionally, the bypass constraints have been specified to allow for negative temperature difference when the bypass is selected (l /BID ): A ]DT LO DTijkBID
× yijk4 ;
The existing heat exchangers cannot be placed at more than one location within the HEN structure. However, since the existing exchangers are specified as additional exchanger types, constraints were specified to avoid multiple selection of the same existing exchanger in the structure of HEN: XXX yijkl 51; Öl:fl BIDg (19) i
j
k
i HP; j CP; k ST
B DTij(k1)BID ]DT LO × yijk4 ;
i HP; j CP; k ST (15)
LO
where DT is negative. Finally, only one heat exchanger or bypass in the match superstructure should be selected: X yijkl 1; i HP; j CP; k ST (16) l HET
In this way, match variables Tik , Tjk , Fijk , DTijk are distributed among different exchanger types and existing heat exchangers, which proves to be a very tight formulation of the HEN retrofit model. 2.6. Constraints to limit the number of relocations
2.9. Model resume Note that constraints described before are omitted in the Model resume. Only new equations and detailed are given: Objective function: Min annualized total retrofit costs (1). s.t. Heat balances for hot and cold streams: / process streams: (TiIN TiOUT )× Gi
X X
Fijk ;
j CP k ST
The maximum number of relocations Mrel for existing heat exchangers were formulated using the following constraints: XXXX (yijkl )POSijkl 0 5 M rel ; i j k l (17) Öl fl:l BIDg It should be noted that when the maximum number of relocations Mrel is small, the combinatorics is significantly reduced and the CPU time necessary to obtain the solution is drastically decreased. The maximum number of relocations (Mrel) should not exceed the number of existing HEs, because only existing HEs can be relocated.
i HP; HUTi 0 (TjOUT TjIN )× Gj
X X
Fijk ;
i HP k ST
j CP; CUTj 0 / utilities (10). Heat balances within the superstructure stage: / process streams: (Tik Ti(k1) )× Gi
X
Fijk ;
j CP
HUTi 0; i HP; k ST
(20)
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242 Table 3 First example data Hot streams
G (kW/K) a (kW/(m2K))
Tin (K)
Tout (K) p (Mpa) C ($/(kW a))
H1 H2 H3 (UT)
250 25 /
0.6 1.0 5.0
470 450 510
400 390 505
2.0 1.0 1.0
250.0
Cold streams C1 C2 C3 (UT) C4 (UT)
240 13 / /
0.7 1.0 1.0 1.0
330 410 300 330
390 500 320 350
1.0 1.0 1.0 1.0
21.0 14.0
Existing heat exchangers Match (i-j-k)
Type
A (m2)
SM index (l )
1-1-2 1-4-4 2-2-3 2-4-4 3-2-1 3-1-1
ST DP PF DP DP DP
621.5 267.0 163.8 11.5 29.9 56.7 should be omitted
5 6 7 8 9 10
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost for the retrofitted HEN: ($/a) Annual operating cost: ($/a)
X
(Tj(k1) Tjk )× Gj
1 265 750 118 800 0 1 384 550
DTij(k1) Ti(k1) Tj(k1) ;
Fijk ;
CUTj 0; j CP; k ST / utilities: (TiIN TiOUT )× Gvi
X
(25) Logical constraints: X Fijk FUP × yijkl 50;
(Fij1 )CUTj 0 ;
l HET l"4
j CP
HUTi 1; i HP (TjOUT TjIN )× Gvj
i HP; j CP; k ST
(21)
i HP
X
(26)
i HP; j CP; k ST (FijN )HUTi 0 ;
i HP
Feasible temperature distribution (8, 9). Constraints to limit the number of relocations (17). Constraints to limit the number of new heat exchangers (18). Selection of existing exchangers (19).
CUTj 1; j CP Inlet temperatures: TiIN Ti1 ;
i HP
TjIN Tj(NST 1) ;
j CP
(22)
A monotonic decrease of temperatures within the HEN: Tik ] Ti(k1) ;
i HP; k ST
Tjk ] Tj(k1) ;
j CP; k ST
TjOUT Tj1 ;
Temperature differences: DTijk Tik Tjk ;
i HP; j CP; k ST
Inlet and outlet temperatures of hot streams (11). Inlet and outlet temperatures of cold streams (12). Heat loads of exchangers (13). Temperature differences (14, 15).
Only one heat exchanger type or existing heat exchanger or bypass should be selected (16), where:
i HP
j CP
/ / / /
(23)
Outlet temperatures: TiOUT Ti(NST 1) ;
Disjunctive constraints of the operability limitations:
B A B Tik ; Tjk ; T(i)A ijkl ; T(i)ijkl ; T(j)ijkl ; T(j)ijkl ; Fijk ;
(24)
F(l)ijkl ; Gvi ; Gvj . . .] 0 DTijk ; DTjHU ; DTiCU ; . . .]DT LO HU yijkl ; yCU f0; 1gm i ; yj
(27)
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243
omitted from retrofitted solutions by fixing its binary variables to zero. Three different cases were examined: . a retrofit with a pre-estimated value of Ft , . a retrofit with Ft as an optimization variable, and . a retrofit where extensions of existing areas are prohibited.
Fig. 5. The HEN structure of first example before the retrofit.
3. Examples Three examples of increasing complexity are presented to illustrate the simultaneous MINLP retrofit of HEN. The first case study is a small numerical example while the second and the third examples are more complex case studies comprising streams from the process of HDA of toluene for which two initial nonintegrated networks are given such that they can mimic the scope of the proposed multi-type model for the retrofit of HEN. In addition a modified objective function of the HEN retrofit model is presented for fixed areas of existing exchangers when one would like to avoid the adaptation of exchangers with an additional heat transfer area (A ).
3.1. The first example The first example data are shown in Table 3. This example involves three hot streams, four cold streams with the four-stage superstructure of HEN. The HEN structure before the retrofit comprises six heat exchangers (Fig. 5) with an annual operating cost of 1 384 550 $/a. As an additional complication let us suppose that the last exchanger (l/10) due to some reason should be
3.1.1. Retrofit of the first example with a fixed value of Ft When the Ft correction factor for the logarithmic mean was estimated and fixed to 0.8 the annual HEN cost was reduced to 244 880 $/a. The main benefit comes from reduction of HU and CU for about 4.6 MW (Fig. 7). The optimal solution (Table 4, Fig. 6) comprises all existing heat exchangers but the last one, which should be omitted. One new DP heat exchanger (shadowed DP in Fig. 6) for the heat transfer between the hot stream H1 and cold stream C2 and one relocated ST heat exchanger (l/5) with additional 73 m2 of heat transfer area (Shadowed ST in Fig. 6) are added to the network. The model with 423 binary variables, 3042 continuous variables and 1950 equations was solved by MIPSYN (Sorsˇak & Kravanja, 2002) in the 130 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. The annual HEN cost is significantly reduced because of more efficient transfer of heat between the hot and the cold process streams. The temperature distribution within the proposed HEN structure remains feasible because the infeasible temperature distribution constraints are applied. 3.1.2. Retrofit of the first example with the variable value of Ft When the Ft correction factor was optimized, the annual HEN cost was additionally reduced, down to 235 000 $/a (Table 5). The topology of the solution, however, remains the same as in the case of the fixed value of Ft (Fig. 6). The additional reduction of the HEN cost was achieved because the Ft correction of the temperature driving force was determined during the
Table 4 Solution of the first example by the fixed value of the Ft Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-2-2-1 1-4-4-6 2-2-3-7 2-4-4-8 3-2-1-9
694.5 79.7 116.1 163.8 28.7 24.4
621.5 0.0 267.0 163.8 11.5 29.9
73.0 79.7 /150.9 0.0 17.2 /5.5
80,22 8.15 65.62 6.17 69.42 20.69
0,800 1.000 1.000 1.000 1.000 1.000
ST DP DP PF DP DP
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost for the retrofitted HEN: ($/a) Total annual cost: ($/a)
105 200 53 900 85 700 244 800
Relocated New HE
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tion could not be carried out in practice. Therefore, the Ft correction factor should be optimized simultaneously, when the HEN topology is determined.
3.1.3. Retrofit where the extensions of existing areas are prohibited In order to prohibit extension of the existing areas, the modified objective function was applied in the HEN retrofit model: XXX [(cCU × Fijkl )j CU ]kK j i
Fig. 6. Solution of the first example for the fixed and variable Ft correction.
j
l
XXX i
j
[(cHU × Fijkl )i HU ]k1 i
l
X X X X
optimization (Ft/0.88) which enabled an extension of the heat transfer area for the ST heat exchanger (l/5) by only 9 m2. It should be noted that the reduction of the annual HEN cost was obtained because the preestimated value of the Ft (0.80) was lower than the exact value of the Ft (0.88). On the other hand, if the Ft was overestimated (e.g. Ft /0.99) the resulted HEN cost would by underestimated and the corresponding solu-
i
j
k
cfl × yijkl (cvl wA l fl:lBIDg )×
l
Max 0;
Fijkl
Al
kij × Ftijkl × Dln Tijkl X X X X (crfijkl × yijkl )crvijkl × i
j
k
l
Table 5 Solution of the first example for the variable value of Ft Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-2-2-1 1-4-4-6 2-2-3-7 2-4-4-8 3-2-1-9
630.5 79.6 116.1 163.8 28.7 24.4
621.5 0.0 267.0 163.8 11.5 29.9
9.0 79.6 /150.9 0.0 17.2 /5.5
80.22 8.16 65.62 6.17 69.42 20.69
0.881 1.000 1.000 1.000 1.000 1.000
ST DP DP PF DP DP
Hot utility cost: ($/a) Cold utility cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
105 200 53 900 75 900 235 000
Relocated New HE
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245
Fijkl Max 0; kij × Ftijkl × Dln Tijkl Al l fl:lBIDgPOS
0
ijkl X XXX yijkl cnfl × 1
l
i
j
k
l fl;lBIDg
(28)
i HP; j CP; k ST; l DHET where wA denotes a high penalty for the adaptation of existing heat transfer areas. Therefore, rather than extending the existing very expensive areas, the existing exchangers are not extended. Note that the constraints remain untouched and that implementation of the penalty preserves the linearity of the feasible space, which improves the performance of the MINLP optimization. The example was solved by MIPSYN in the 270 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. Shadowed exchangers in the optimal solution (Fig. 8) correspond to one new, one relocated ST heat exchangers and one new double pipe exchanger. Note that the existing double pipe heat exchanger (l /8) is excluded from the optimal solution because of insufficient heat transfer area. The total annual HEN cost equals 265 400 $/a (Table 6). It should be noted that it is possible to establish selective prohibition for the extension of existing transfer areas. In this case the penalty factor wA is only specified for exchangers with fixed areas.
3.2. Second example The second example comprises five hot and six cold process streams (Table 7) with a six-stage superstructure of HEN. The stream data were taken from the HDA case study. The HEN structure contains 11 exchangers (one ST and ten DP exchangers) with annual operating costs of 614 610 $/a (Fig. 9). The use of PF exchangers is prohibited because some of the involved streams are toxic. Two different cases were carried out:
Fig. 8. Optimal solution of the first example with prohibited extension of the existing areas.
. only a double pipe heat exchanger type was assumed, and . different exchanger types were selected. 3.2.1. Retrofit of the second example by considering only double pipe heat exchanger type The first case with 2520 binary variables, 17 513 continuous variables and 8631 equations was solved by MIPSYN (Sorsˇak & Kravanja, 2002) in the 5300 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. Ten existing exchangers were required for the entire heat transfer, while one heater (l/15) was excluded from the optimal solution (Fig. 10). In addition, two existing exchangers were relocated (shadowed DP exchangers in Fig. 10). The total annual HEN cost was reduced to 285 330 $/a (Table 8). It should be noted that at match 1-1-3-5, which was originally ST exchanger, feasible temperature distribution was restored, so that the retrofit yielded feasible network although only DP exchangers were assumed. However, since the first match has ST exchanger, the cost has to be recalculated: the Ft correction factor is equal to 0.909 and the required heat transfer area equals to 437.3 m2 which is 29.5 m2 of heat transfer area with the annual adaptation costs of 2790 $/a more than
Table 6 Solution of the first example by the prohibition of extending existing areas Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-3 1-2-2-1 1-4-4-6 2-2-3-7 2-4-4-5 3-2-1-9
630.5 80.0 116.1 163.8 30.3 24.4
0.0 0.0 267.0 163.8 621.5 29.9
630.5 80.0 /150.9 0.0 /591.2 /5.5
80.21 8.14 65.62 6.17 69.42 20.69
0.881 1.000 1.000 1.000 0.950 1.000
ST DP ST PF DP DP
Hot utility cost: ($/a) Cold utility cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
105 200 53 900 106 300 265 400
New HE New HE Realocated
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Table 7 Second example data Hot streams
G (kW/K)
a (kW/(m2K))
Tin (K)
Tout (K)
p (Mpa)
C ($/(kW a))
H1 H2 H3 H4 H5 (UT)
49.27 27.54 1088.67 229.17 /
0.15 0.90 0.90 0.90 0.30
823.20 330.85 352.32 379.90 790.00
299.89 329.85 349.32 376.90 780.00
3.5 3.5 3.5 3.5 1.0
250.0
Cold streams C1 C2 C3 C4 C5 C6 (UT)
38.92 14.58 511.33 252.60 236.13 /
0.12 1.00 1.00 1.00 1.00 1.00
330.19 362.95 462.30 376.90 550.60 282.00
713.70 463.00 465.30 379.60 553.60 290.00
3.5 3.5 3.5 3.5 3.5 1.0
25.0
Existing heat exchangers Match (i-j-k)
Type
A (m2)
SM index (l )
1-1-3 1-1-5 1-2-5 1-3-2 1-4-4 1-5-4 1-6-6 2-6-6 3-6-6 4-6-6 5-1-1
ST DP DP DP DP DP DP DP DP DP DP
407.8 1949.4 81.4 34.2 17.4 47.1 914.2 1.3 106.4 15.7 171.4
5 6 7 8 9 10 11 12 13 14 15
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total operational cost: ($/a)
320 500 294 110 0 614 610
assumed DP exchanger. Therefore, the total annual costs slightly increases up to 288 120 $/a. 3.2.2. Retrofit of the second example by considering different exchanger types The topology of the HEN structure remains the same, when different exchanger types are taken into account (Fig. 10). The total annual HEN cost is now 288 120 $/a (Table 9). The heat transfer area of the existing ST exchanger has to be adapted with an additional 29.5 m2. It should be noted that the solution is the same as in the previous case, after the recalculation of the existing ST (l/5) exchanger. However, the heat transfer area in the first case is not large enough for the entire heat transfer. The example indicates how important it is to consider different exchanger types simultaneously during the optimization. 3.3. Third example The third example comprises the same process streams as the second example, but with slightly different HEN structure before the retrofit, containing 12 heat exchangers: two ST exchangers and ten double
Fig. 9. The HEN structure of second example before the retrofit.
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247
show that models based on single exchanger type cannot solve real HEN problems comprising different exchangers types correctly. Let us look the solution based on double pipe exchangers. The solution (Fig. 12) of the first case comprises 11 existing exchangers, two of them being relocated (l /13, l/16). The solution yields the total annual HEN cost of 380 790 $/a (Table 11). Note that resulted temperature distributions at the first and fourth (Fig. 12) exchangers, which are now assumed to be double pipe exchanger instead of existing ST exchangers (l/5, l/6), are feasible for double pipe but infeasible for ST exchangers. In the first exchanger (l/5) the outlet temperature of hot stream H1 is equal to 623.3 K while the outlet temperature of the cold stream C1 is equal to 713.7 K. The temperature distribution is infeasible for ST exchangers because the outlet temperature of the cold stream (C1) is higher than the outlet temperature of the hot stream (H1) and such temperature arrangement cannot take place within the ST exchangers with the U tubes. Similarly, in the second ST exchanger the outlet temperature of the hot stream is lower (515 K) than the outlet temperature of the cold stream (529.5 K). Consequently, the proposed solution is infeasible and cannot be carried out in practice. The example clearly indicates that models for double pipe exchangers may produce solutions that are infeasible for other types of exchangers.
Fig. 10. Solution of the second example with DP exchanger type.
pipe exchangers (Fig. 11) with a total operational cost of 647 060 $/a (Table 10). The retrofit of the six-stage HEN superstructure was carried out for two different cases: . only a double pipe heat exchanger type was assumed, and . different exchanger types were selected.
3.3.2. Retrofit of the third example by comprising different exchanger types When different exchanger types were considered during optimization, the whole resulting temperature distribution and all areas were feasible. The example was solved again by MIPSYN in the 11 000 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. The optimal solution contains ten existing
3.3.1. Retrofit of the third example by assuming the double pipe heat exchanger type The problem is solved first by assuming only double pipe exchangers although the original network comprises exchangers of different types. The objective is to Table 8 Solution of the second example by the use of DT exchanger type Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-1-5-6 1-2-5-7 1-3-4-8 1-4-5-9 1-5-4-10 1-6-6-11 2-6-6-12 3-6-6-13 4-6-6-14
397.9 1949.4 79.6 43.1 32.9 29.5 849.4 1.3 106.4 15.7
407.8 1949.4 81.4 34.2 17.4 47.1 914.2 1.3 106.4 15.7
/9.9 0.0 /1.8 8.9 15.5 17.6 /64.8 0.0 0.0 0.0
117.72 90.83 140.44 272.70 153.00 184.14 58.68 44.26 64.79 92.37
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DC DC DC DC DC DC DC DC DC DC
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
Virtual 0 262 060 23 270 285 330
Exact 0 262 060 26 060 288 120
Relocated Relocated
248
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Table 9 Solution of the second example by the use of different exchanger types Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-1-5-6 1-2-5-7 1-3-4-8 1-4-5-9 1-5-4-10 1-6-6-11 2-6-6-12 3-6-6-13 4-6-6-14
437.3 1949.4 79.6 43.1 32.9 29.5 849.4 1.3 106.4 15.7
407.8 1949.4 81.4 34.2 17.4 47.1 914.2 1.3 106.4 15.7
29.5 0.0 /1.8 8.9 15.5 /17.6 /64.8 0.0 0.0 0.0
117.72 90.83 140.44 272.70 153.00 184.14 58.68 44.26 64.79 92.37
0.909 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
CP DC DC DC DC DC DC DC DC DC
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
exchangers, two of them being relocated (l/8, l/9, shadowed DP exchangers in Fig. 13) and two new ST exchangers (shadowed ST exchangers in Fig. 13). The total annual HEN cost is now 430 020 $/a (Table 12). Note that the cost has increased by approximately 50 000 $/a, because of two additional ST heat exchangers for the heat transfer between the hot stream H1 and cold stream C1, necessary to avoid infeasible temperature distribution. This example shows the importance of comprising different exchanger types during the HEN optimization in order to avoid infeasible distribution of the temperatures within the proposed HEN structure.
Fig. 11. The HEN structure of the third example before the retrofit.
Relocated Relocated
0 262 060 26 060 288 120
4. Conclusions The use of the single-type DP model for the retrofit of HEN is perhaps more attractive than the proposed multitype model because it is simpler and computationally less exhaustive. However, as indicated by the examples, it may produce solutions that are not only suboptimal, but also infeasible when the existing HEN comprises different exchanger types. Thus, the examples clearly indicate the advantages of the proposed model. This model allows for a simultaneously heat-integrated retrofit of HEN, enables the selection of optimal and feasible heat exchanger types during the optimization,
Fig. 12. Solution of the third example with DP exchanger type.
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249
Table 10 Third example data Hot streams
G (kW/K)
a (kW/(m2K))
Tin (K)
Tout (K)
p (Mpa)
C ($/(kW a))
H1 H2 H3 H4 H5 (UT)
49.27 27.54 1088.67 229.17 /
0.15 0.90 0.90 0.90 0.30
823.20 330.85 352.32 379.90 790.00
299.89 329.85 349.32 376.90 780.00
3.5 3.5 3.5 3.5 3.5
250.0
Cold streams C1 C2 C3 C4 C5 C6 (PS)
38.92 14.58 511.33 252.60 236.13 /
0.12 1.00 1.00 1.00 1.00 1.00
330.19 362.95 462.30 376.90 550.60 282.00
713.70 463.00 465.30 379.60 553.60 290.00
3.5 3.5 3.5 3.5 3.5 3.5
25.0
Existing heat exchangers Match (i-j-k)
Type
A (m2)
SM index (l )
1-1-2 1-1-3 1-1-4 1-2-4 1-3-3 1-4-4 1-5-2 1-6-6 2-6-6 3-6-6 4-6-6 5-1-1
ST ST DP DP DP DP DP DP DP DP DP DP
774.8 736.8 53.9 156.0 103.6 47.1 28.6 919.8 1.3 106.4 15.7 184.2
5 6 7 8 9 10 11 12 13 14 15 16
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total operational cost: ($/a)
350 000 297 060 0 647 060
Table 11 Solution of the third example by the use of DP exchanger type Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-2-5 1-1-3-6 1-1-4-7 1-2-4-8 1-3-2-13 1-4-4-10 1-5-2-11 1-6-6-12 2-6-6-16 3-6-6-14 4-6-6-15
1013.9 736.9 254.3 186.3 46.6 56.5 34.6 849.4 1.3 106.4 15.7
774.8 736.9 53.9 156.1 1.3 47.1 28.6 919.8 184.3 106.4 15.7
239.1 0.0 200.3 30.3 45.3 9.4 6.0 /70.3 /182.9 0.0 0.0
106.0 117.6 117.0 60.0 252.2 89.1 156.1 58.7 44.3 64.8 92.4
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DP DP DP DP DP DP DP DP DP DP DP
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
0 262 060 117 730 380 790
Relocated
relocated
250
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Table 12 Solution of the third example by the use of different exchanger types Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-2-5 1-1-3-6 1-1-4-3 1-1-5-3 1-2-5-8 1-3-4-9 1-4-4-10 1-5-2-11 1-6-6-12 2-6-6-13 3-6-6-14 4-6-6-15
758.4 704.7 156.1 649.2 156.0 111.6 26.1 25.6 849.4 1.3 106.4 15.7
774.8 736.9 0.0 0.0 156.1 103.6 47.1 28.6 919.8 1.3 106.4 15.7
/16.5 /32.2 156.1 649.2 0.0 8.0 /21.0 /3.0 /70.3 0.0 0.0 0.0
114.7 133.8 128.8 106.2 71.7 105.3 192.6 212.5 58.7 44.3 64.8 92.4
0.799 0.836 0.952 0.816 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ST ST ST ST DP DP DP DP DP DP DP DP
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
Fig. 13. Solution of the third example with different exchanger types.
and yields retrofitted HEN with feasible temperature distributions.
References Chen, J. J. J. (1987). Letter to the editors: comments on improvement on a replacement for the logarithmic mean. Chemical Engineering Science 42 , 2488 /2489.
New HE New HE Relocated Relocated
0 262 060 167 960 430 020
Ciric, A. R., & Floudas, C. A. (1990). Comprehensive optimization model of the heat exchanger network retrofit problem. Heat Recovery Systems and CHP 10 , 407 /422. Guthrie, K. M. (1969). Capital cost estimating. Chemical Engineering 76 , 114. Hewitt, G. F., Shires, G. L., & Bott, T. R. (1994). Process heat transfer (pp. 155 /194). CRC Press. Jezowski, J. (1994a). Heat exchanger network grassroot and retrofit design. The review of the state-of-the art: part I. Heat exchanger network targeting and insight based methods of synthesis. Hungarian Journal of Industrial Chemistry 22 , 279 /294. Jezowski, J. (1994b). Heat exchanger network grassroot and retrofit design. The review of the state-of-the art: part II. Heat exchanger network synthesis by mathematical methods and approaches for retrofit design. Hungarian Journal of Industrial Chemistry 22 , 295 / 308. Ma, K.-L., Hui, C.-W., & Yee, T. F. (2000). Constant approach temperature model for HEN retrofit. Applied Thermal Engineering 20 , 1505 /1533. Nie, X. R., & Zhu, X. X. (1999). Heat exchanger network retrofit considering pressure drop and heat-transfer enhancement. American Institute of Chemical Engineering Journal 45 , 1239 /1254. Papoulias, S. A., & Grossmann, I. E. (1983a). A structural optimization approach in process synthesis */I. Utility systems. Computers and Chemical Engineering 7 , 695 /706. Papoulias, S. A., & Grossmann, I. E. (1983b). A structural optimization approach in process synthesis */II. Heat recover networks. Computers and Chemical Engineering 7 , 707 /721. Sorsˇak, A., & Kravanja, Z. (1999). Simultaneous MINLP synthesis of heat and power integrated heat exchanger network. Computers and Chemical Engineering 23 , 143 /147. Sorsˇak, A., & Kravanja, Z. (2001). Simultaneous MINLP synthesis of heat exchanger networks comprising different exchanger types. Computers and Chemical Engineering Supplement . In R. Gani & S. B. Jorgensen (Eds.), European Symposium on computer aided process engineering-11, Computer-aided chemical engineering, 9 (1st Ed., pp. 1095 /1100). Elsevier: Amsterdam, New York. Sorsˇak, A., & Kravanja, Z. (2002). Simultaneous MINLP synthesis of heat exchanger networks comprising different exchanger types. Computers and Chemical Engineering 26/4 /5 , 599 /615. Tjoe, T. N., & Linhoff, B. (1986). Using pinch technology for process retrofit. Chemical Engineering 28 , 47 /60.
A. Sorsˇak, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 235 /251 Walker, G. (1990). Industrial heat exchangers (pp. 93 /102). Hemisphere Publishing. Yee, T. F., & Grossmann, I. E. (1990). Optimization models for heat integration-II, heat exchanger network synthesis. Computers and Chemical Engineering 14 , 1165 /1184. Yee, T. F., & Grossmann, I. E. (1991). A screening and optimization approach for the optimal retrofit of heat exchanger networks. Industrial and Engineering Chemistry Research 30 , 146.
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Zhang, J., & Zhu, X. X. (2000). Simultaneous optimization approach for heat exchanger network retrofit with process changes. Industrial and Engineering Chemistry Research 39 , 4963 /4973. Zhu, X. X., & Asante, N. D. K. (1999). Diagnosis and optimization approach for heat exchanger network retrofit. American Institute of Chemical Engineering Journal 45 , 1488 / 1503.
MINLP retrofit of heat exchanger networks comprising different exchanger types Aleksander Sorsˇak, Zdravko Kravanja * Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova 17, PO Box 219, Maribor 2000, Slovenia
Abstract This paper describes a simultaneous MINLP optimization model for the retrofit of heat exchanger networks (HEN), comprising different exchanger types. This model is based on the stage-wise superstrucutre of HEN [Comput. Chem. Eng. 14 (1990) 1165], extended for retrofit and different exchanger types. Each potential match in the superstructure is now replaced by a match superstructure comprising: double pipe (DP), shell and tube (U tubes, ST), and plate and frame exchangers (PF). The operability constraints for the different exchanger types are modeled by the convex hull formulation of disjunctions, nonlinear terms are modeled only within the objective function and additional constraints are included to obtain feasible temperature distribution. Several examples are presented to illustrate the efficiency and advantages of the proposed multi-type model. # 2003 Elsevier Ltd. All rights reserved. Keywords: Heat exchanger types; Disjunctive modeling; MINLP; Retrofit of HEN
1. Introduction Like the earlier grass-root synthesis of heat exchanger networks (HEN), the retrofit of an existing HEN can be traditionally carried out using two main approaches: the pinch analysis approach based on thermodynamics and heuristics, and the mathematical programming approach, where HENs are presented as mathematical models. For the pinch analysis approach, Tjoe and Linhoff (1986), proposed a two step retrofit procedure and for the mathematical programming approach, Ciric and Floudas (1990), based on the transshipment model by Papoulias and Grossmann (1983a,b), developed an MINLP model for the retrofit of HEN allowing the reconstruction and relocation of existing HEs. In addition, based on the superstructure MINLP synthesis of HEN, Yee and Grossmann extended their synthesis model Yee and Grossmann (1990) for the retrofit of HEN Yee and Grossmann (1991). For other earlier important achievements occurred in the area of HEN retrofit see Jezowski (1994a,b). More recent research
* Corresponding author. Tel.: /386-2-229-4481; fax: /386-2-2527774. E-mail address: [email protected] (Z. Kravanja). 0098-1354/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00167-4
was directed towards more advanced methods, e.g. HEN retrofit considering pressure drops (Nie & Zhu, 1999), two-step approach by (Ma, Hui & Yee, 2000) using constant approach temperature in the first and MINLP to finalize design in the second step, combined mathematical programming and thermodynamic analysis approach by Zhu and Asante (1999) and a simultaneous process changes and HEN retrofit (Zhang & Zhu, 2000). However, the selection of different HE types (HET) has not yet been considered simultaneously during the retrofit of HEN, probably because the model size would be drastically increased. The objective of this work is to develop a tight model representation, comprising different exchanger types for the purposes of HEN retrofit, based on the synthesis model of HEN (Sorsˇak & Kravanja, 2001, 2002). The model is based on a stagewise superstructure (Yee & Grossmann, 1990) where each potential match between the hot and cold stream is replaced by a match superstructure (SM) comprising a double pipe HEs (DP), shell and tube HEs (ST), plate and frame HEs (PF) and a bypass (Sorsˇak & Kravanja, 2002). In order to be comprised of different HET and existing HEs, which may also be of different types and for the purposes of HEN retrofit, the HEN synthesis model (Sorsˇak & Kravanja, 2002) has been adapted:
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Nomenclature DlnTijk logarithmic mean temperature difference (K) DTijk temperature difference between the hot stream i and cold stream j within the stage k (K) DTmin minimum temperature difference (K) Aijkl heat transfer area (m2) Al heat transfer area of an existing heat exchanger in HEN (m2) BID bypass ID (/) crfijkl fixed relocation cost for existing exchangers in HEN ($/a) cfl Fixed annualized investment cost of exchangers ($/a) cnfl Fixed deinstallation cost of existing exchangers in HEN ($/a) CP set of cold streams (/) CU set of cold utilities (/) crvijkl variable relocation cost for existing exchangers ($/(m2a)) cvl variable annualized investment cost of exchangers ($/(m2a)) n cvl variable deinstallation cost of existing exchangers in HEN ($/(m2 a)) Ft correction factor for the temperature driving force (/) G heat capacity flowrate for process streams (W/K) Gvi heat capacity flowrate for hot utilities (W/K) Gvj heat capacity flowrate for cold utilities (W/K) HETl type of existing exchanger in HEN (/) HP set of hot streams (/) HU set of hot utilities (/) i hot stream index (/) j cold stream index (/) k stage index in the HEN superstructure ( /) kij overall heat transfer coefficient (W/(m2 K)) l heat exchanger type index ( /) Mnhe maximum number of new exchangers in HEN (/) Mrel maximum number of existing exchanger relocations in HEN ( /) POSijkl location of existing exchangers in HEN ( /) R temperature ratio (/) S thermal efficiency ( /) ST set of stages in HEN superstructure (/) T (i)A inlet temperature of hot stream for exchangers (K) ijkl T (i)Bijkl outlet temperature of hot stream for exchangers (K) Tik inlet temperature of hot stream for stage k (K) Ti (k1) outlet temperature of hot stream for stage k (K) TIN inlet temperatures of process streams (K) T (j)A outlet temperature of cold stream for exchangers (K) ijkl T (j)Bijkl inlet temperature of cold stream for exchangers (K) Tjk outlet temperature of cold stream for stage k (K) Tj (k1) inlet temperature of cold stream for stage k (K) TOUT outlet temperatures for process streams (K) wA penalty factor (/) yijkl binary variable of heat exchangers in HEN superstructure (/) ai connective heat transfer coefficient for process stream i . The match superstructure (SM) has been extended using existing HEs in the network, where each existing heat exchanger is described by its heat transfer area, HE type and position within the HEN. . External heaters and coolers, as used in the synthesis model (Sorsˇak & Kravanja, 2002), are omitted and replaced by two additional stages for the heat transfer between hot utility streams and cold process streams,
and for the heat transfer between cold utility streams and hot process streams. . The objective function comprises the operational costs of HEN, the investment costs for both, the existing HEs and the new HEs, reinstallation and relocation costs of existing HEs. Several examples are presented in this paper to illustrate the capabilities of the proposed model.
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237
Fig. 1. Four stage superstructure of HEN.
2. Retrofit model of HEN In order to determine the optimal retrofitted structure of the HEN, an extension of the simultaneous MINLP model for the HEN synthesis (Sorsˇak & Kravanja, 2002) is proposed. Several indexes have to be specified before presenting the model and superstructure
hot to the cold stream within the match superstructure (SM) comprising: double pipe HEs, ST HEs, plate and frame (PF) HEs, and bypasses. In the proposed exten-
. index i for hot streams given by the set HP; note that set of hot utilities (HU) is a subset of HP, . index j for cold streams given by the set CP; note that set of cold utilities (CU) is a subset of CP, . index k for the superstructure stages given by the set ST, . index l for the exchanger types, a bypass and existing exchangers given by the set HET. It should be noted that the proposed model can be used for the simultaneous MINLP synthesis of HEN when the specification of the existing heat exchanger is not presented. 2.1. Superstructure In the superstructure each cold stream (j) can be potentially matched with each hot stream (i) in several stages (k ). A vector of binary variables is assigned to the structural alternatives to determine the retrofitted HEN structures during the MINLP optimization. In the simultaneous MINLP synthesis model (Sorsˇak & Kravanja, 2002), the heat is potentially transferred from the
Fig. 2. Match superstructure.
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238
Table 1 Data of considered heat exchanger types Type
pmax (Mpa)
T (8C) range
A (m2) range
Cf ($/year)
Cv ($/(m2 year))
Double pipe ST PF
30.7 30.7 1.6
/100 /600 /200 /600 /25 /250
0.25 /200 10 /1000 1 /1200
1937 21 615 17 034
201 93 61
sion of the HEN superstructure (Fig. 1), the match superstructure (Fig. 2) is now updated with the existing HEs, each being described by:
XXX i
j
l
. heat transfer area (Al ), . type (HETl ), and . position within the network (POSijkl ).
× Max 0;
In order to establish the selection of different exchanger types for the heat transfer using utility streams, the external coolers and heaters, as presented in the original superstructure (Yee & Grossmann, 1990) are omitted, and replaced by two additional stages: a stage representing heat transfer between HU and cold process streams taking place within the first stage of the superstructure (k /1), and a stage representing the heat transfer between CU and the hot process streams taking place at the last stage of the HEN superstructure (k / K ). It should be noted that the introduction of the utility stages into the HEN superstructure enables the use of multiple utilities. The number of stages is determined by heuristics. If possible, K should take the number of the enthalpy intervals of the balanced composite curves. Generally, no improvements of HEN can be expected with the larger one than the number of the intervals. However, the size of the model drastically increases when the large number of stages is used and, very often, because of large K , the model cannot be solved to the optimal solution. In practical cases, the number of stages (K ) is generally smaller (halved or even smaller) than the number of enthalpy intervals. It also should be noted that different area, pressure and temperature ranges hold for different exchanger types (Hewitt, Shires & Bott, 1994). The basic recommendations are shown in Table 1. In addition, due to the leakage problem, using PF exchangers is not recommended when one of the streams in the network is toxic.
2.2. Objective function The objective function of the proposed model defines the annual cost of HEN and has been defined as: XXX XX [(cCU × Fijkl )j CU ]kK j i
j
l
X [(cHU × Fijkl )i HU ]k1 i l
i
k
X cfl × yijkl cvl
j
(1a)
XXX i
j
Fijkl kij × Ftijkl × Dln Tijkl
Al
(1b)
k
X (crfijkl × yijkl )crvijkl l
× Max 0; Al X
cnfl
× 1
kij × Ftijkl × Dln Tijkl (1c)
l fl:lBIDgPOSijkl 0
XXX i
l
Fijkl
j
k
yijkl
l fl;lBIDg
(1d)
i HP; j CP; k ST; l DHET where BID specifies a bypass index in the set of exchanger types and existing exchangers (DHET ). The objective function comprises the following terms: . cost of utility consumption (1a), . annualized investment costs for the HEs in the network (1b), . annualized costs for the relocation of existing HEs (1c), and . annualized costs for the deinstalation of those existing HEs (1d), which are not selected in the retrofitted structure of the adapted HEN. The fixed charge coefficients (cfl , cvl , crfijkl , crvijkl , cnfijkl , are specified by the linearization of Guthrie’s function (Guthrie, 1969). The cost of the PF exchanger has been specified as 70% (Walker, 1990) of the cost for the ST exchanger. In order to preserve the linearity of feasible space of the model, all nonlinear constraints (2) have been omitted by defining the corresponding charge coefficients (Sorsˇak & Kravanja, 2002). It should be noted that in the objective function (1b) only continuous area extensions to existing exchangers are allowed. Allowing discrete extensions would dramatically increase problem combinatorics and increase model complexity with many new discrete variables and constraints cnvijkl )
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239
Table 2 Ft correction factor coefficients i
ai
bi
0 1 2 3 4 5 6
15.210806 /1.603737 42.337642 / / / /
0.623393 /0.698824 /0.250187 0.434342 0.301423 /0.171389 /0.141410
Fig. 3. The Ft correction factor for the ST HEs with an even number of tube passes.
S
for discrete increments. Flijkl
Amin ijkl 5
kij × Ftijkl × Dln Tijkl 1 1 1 kij ai aj
5Amax ijkl
(2a) (2b)
In order to avoid the numerical difficulties of using logarithmic mean (LMTD ) at equal temperature differences, the temperature driving force has been specified by Chen’s approximation (Chen, 1987): (1=3) A B DTijkl DTij(k1)l A B Dln Tijkl DTijkl × DTij(k1)l × 2 (3) The approximation underestimates the logarithmic mean (LMTD ), which yields a ‘safe’ design of heat exchangers, because of the overestimation of the heat transfer area (A ) of the selected exchangers. This approximation can be directly used for the counter flow arrangement of streams in the exchangers. Since the ST exchanger type stream flow arrangement is a combination of the counter and co-current flow, the temperature driving force has to be corrected by the Ft correction factor (Fig. 3). The Ft has been defined by the Ft approximation (Sorsˇak & Kravanja, 1999), since the Underwood’s Ft equation (Hewitt et al., 1994) is an inconvenient form for optimization (Sorsˇak & Kravanja, 2002):
p × S Ft 1 2 × tan 6 X X 2 × ai × [lg(R)]i bj × [lg(R)]j 1
i0
B T(j)A ijkl T(j)ij(k1)l B T(i)A ijkl T(j)ij(k1)l
(6)
coefficients a and b are shown in Table 2. In (1b) and (1c) the smooth approximation Max(. . .) in the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Max(0; x)0:5× (xo)2 0:5 × x (7) has been applied, to account for the possible adaptation of the existing HEs in the network. 2.3. Feasible temperature distribution In order to avoid infeasible temperature distribution when the ST exchanger type is selected, additional constraints have been specified. Since the extended model comprises different exchanger types, the temperature distribution in HEN that holds for a pure counter flow exchanger may become infeasible if the ST exchanger is selected (Sorsˇak & Kravanja, 2002). This problem can be observed when we take a closer look at the temperature distribution of the counter flow and ST heat exchangers (Fig. 4). In counter flow heat exchangers the outlet temperature of the cold stream can be higher (Fig. 4a) because of the geometry of the transfer area. When ST exchangers are used, the flow arrangement combines the counter and co-current flows and the temperature distribution becomes infeasible when Tco /Tho (Fig. 4b). To overcome this problem, additional constraints have been specified for the ST exchanger type: (HETl /3):
j0
(4) where R defines the temperature ratio: R
B T(i)A ijkl T(i)ij(k1)l B T(j)A ijkl T(j)ij(k1)l
and S defines the thermal efficiency:
(5) Fig. 4. Temperature arrangement in (a) counter flow HE and (b) ST HE with U-tubes.
240
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Ti(k1) Tjk DT UP ×(1yijkl )HETl 3 ]DT min ;
(8)
i HP; j CP; k ST Furthermore, in order to obtain the Ft correction factor for the ST exchanger type, several constraints have been additionally specified: Tjk Tj(k1) DT UP ×(1yijkl )HETl 3 ] DT minFt ; i HP; j CP; k ST Tik Ti(k1) DT UP ×(1yijkl )HETl 3 ]DT minFt ; i HP; j CP; k ST Tik Tj(k1) DT UP ×(1yijkl )HETl 3 ] DT minFt ;
(9)
i HP; j CP; k ST
2.4. Energy balances of utility streams
stages. It should be noted that the energy balance constraints disallow heat transfer between HU and CU.
2.5. Disjunctive modeling of operational limitations for selected exchanger type and existing heat exchangers The selection of the exchanger types depends on the operating temperatures and pressures of the involved streams (Table 1). Since the pressures are not the optimization variables, they can be taken into account in a prescreening procedure before the optimization. However, all temperatures are modeled as optimization variables and affect the selection of exchanger types or existing exchangers during optimization. The temperature ranges have, therefore, been modeled by the convex-hull formulation of disjunctions. The following temperature constraints have been specified for hot streams: X Tik T(i)A i HP; j CP; k ST ijkl ; l HET
Since the inlet and outlet temperatures of the utility streams are constant, the heat capacity flowrates (Gi , Gj ) for the utilities have been modeled as variables (Gvi , Gvj ). Therefore, additional constraints have been specified:
Ti(k1)
Ti2 TiOUT ; i HU Tj(K1) TjOUT ; j CU XX (Fijk )CUTj 0 ; (TiIN TiOUT )×Gvi
T(i)Bij(k1)l 5Tlmax × yijkl ;
j
k
i HU XX (TjOUT TjIN )×Gvj (Fijk )HUTi 0 ; i
k
(10a) (10b)
X
T(i)Bij(k1)l ;
i HP; j CP; k ST
l HET max T(i)A × yijkl ; ijkl 5Tl
i HP; j CP; k ST; l HET
i HP; j CP; k ST; l HET (10c)
(10d)
j CU and TIN are inlet temperatures for hot and where TIN i j OUT CU, Ti and TOUT are outlet temperatures for hot and j CU and K defines the number of HEN superstructure stages. The outlet temperature of the first stage in the HEN superstructure for each hot utility (Ti 2) is equal to the outlet temperature of the utility (TOUT ). Therefore, i the HU were used in the first stage of HEN superstructure only. Furthermore, scalars for the inlet and OUT outlet temperatures of the HU (TIN ) were applied j , Ti directly into the energy balance equations (10c). Consequently, the energy balance constraints (10c) for the heat transfer between HU and cold process streams remain linear. The energy balance constraints for the heat transfer between CU and hot process streams (10b, d) remain linear in the same way. Thus, the implementation of the energy balance constraints (10) preserves the linearity of the feasible space of the HEN retrofit model. The use of CU is limited to the last stages of the superstructure of HEN. The first and the last stages of the HEN superstructure are, therefore, called the utility
min × yijkl ; T(i)A ijkl ]Tl
i HP; j CP; k ST; l HET
T(i)Bij(k1)l ]Tlmin × yijkl ;
(11)
i HP; j CP; k ST; l HET where Tik and Ti (k1) are the match temperatures at the B inlet and outlet of stage k , T (i )A ijkl and T (i)ij (k1)l are temperatures disaggregated over the set of different exchanger types, bypass and existing heat exchangers and y represents a vector of the binary variables. Similarly, we defined the temperature constraints for cold streams: X Tjk T(j)A i HP; j CP; k ST ijkl ; l HET
Tj(k1)
X
T(j)Bij(k1)l ;
i HP; j CP; k ST
l HET max T(j)A × yijkl ; ijkl 5Tl
i HP; j CP; k ST; l HET
T(j)Bij(k1)l 5Tlmax × yijkl ; i HP; j CP; k ST; l HET min × yijkl ; T(j)A ijkl ]Tl
i HP; j CP; k ST; l HET
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T(j)Bij(k1)l ] Tlmin × yijkl ;
(12)
i HP; j CP; k ST; l HET heat loads: X F(l)ijkl ; Fijk
2.7. Constraints to limit the number of new heat exchangers The maximum number of new heat exchangers in the HEN structure Mnhe were specified by: XXXX yijkl 5M nhe ; Öl fl:l BBIDg (18)
i HP; j CP; k ST
l HET
F(l)ijkl 5FUP × yijkl ;
241
i HP; j CP; k ST; l HET (13)
and temperature differences: X A DTijk DTijkl ; i HP; j CP; k ST
i
j
k
l
When the maximum number of new exchangers in the HEN superstructure is specified, the combinatorics is significantly reduced too, which additionally shortens the CPU times to obtain an optimal solution.
l HET
DTij(k1)
X
B DTij(k1)l ;
i HP; j CP; k ST
2.8. The single placement of existing heat exchangers
l HET A DTijkl 5 DT UP × yijkl ;
i HP; j CP; k ST; l HET
B DTij(k1)l 5 DT UP × yijkl ;
i HP; j CP; k ST; l HET
(14)
Additionally, the bypass constraints have been specified to allow for negative temperature difference when the bypass is selected (l /BID ): A ]DT LO DTijkBID
× yijk4 ;
The existing heat exchangers cannot be placed at more than one location within the HEN structure. However, since the existing exchangers are specified as additional exchanger types, constraints were specified to avoid multiple selection of the same existing exchanger in the structure of HEN: XXX yijkl 51; Öl:fl BIDg (19) i
j
k
i HP; j CP; k ST
B DTij(k1)BID ]DT LO × yijk4 ;
i HP; j CP; k ST (15)
LO
where DT is negative. Finally, only one heat exchanger or bypass in the match superstructure should be selected: X yijkl 1; i HP; j CP; k ST (16) l HET
In this way, match variables Tik , Tjk , Fijk , DTijk are distributed among different exchanger types and existing heat exchangers, which proves to be a very tight formulation of the HEN retrofit model. 2.6. Constraints to limit the number of relocations
2.9. Model resume Note that constraints described before are omitted in the Model resume. Only new equations and detailed are given: Objective function: Min annualized total retrofit costs (1). s.t. Heat balances for hot and cold streams: / process streams: (TiIN TiOUT )× Gi
X X
Fijk ;
j CP k ST
The maximum number of relocations Mrel for existing heat exchangers were formulated using the following constraints: XXXX (yijkl )POSijkl 0 5 M rel ; i j k l (17) Öl fl:l BIDg It should be noted that when the maximum number of relocations Mrel is small, the combinatorics is significantly reduced and the CPU time necessary to obtain the solution is drastically decreased. The maximum number of relocations (Mrel) should not exceed the number of existing HEs, because only existing HEs can be relocated.
i HP; HUTi 0 (TjOUT TjIN )× Gj
X X
Fijk ;
i HP k ST
j CP; CUTj 0 / utilities (10). Heat balances within the superstructure stage: / process streams: (Tik Ti(k1) )× Gi
X
Fijk ;
j CP
HUTi 0; i HP; k ST
(20)
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242 Table 3 First example data Hot streams
G (kW/K) a (kW/(m2K))
Tin (K)
Tout (K) p (Mpa) C ($/(kW a))
H1 H2 H3 (UT)
250 25 /
0.6 1.0 5.0
470 450 510
400 390 505
2.0 1.0 1.0
250.0
Cold streams C1 C2 C3 (UT) C4 (UT)
240 13 / /
0.7 1.0 1.0 1.0
330 410 300 330
390 500 320 350
1.0 1.0 1.0 1.0
21.0 14.0
Existing heat exchangers Match (i-j-k)
Type
A (m2)
SM index (l )
1-1-2 1-4-4 2-2-3 2-4-4 3-2-1 3-1-1
ST DP PF DP DP DP
621.5 267.0 163.8 11.5 29.9 56.7 should be omitted
5 6 7 8 9 10
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost for the retrofitted HEN: ($/a) Annual operating cost: ($/a)
X
(Tj(k1) Tjk )× Gj
1 265 750 118 800 0 1 384 550
DTij(k1) Ti(k1) Tj(k1) ;
Fijk ;
CUTj 0; j CP; k ST / utilities: (TiIN TiOUT )× Gvi
X
(25) Logical constraints: X Fijk FUP × yijkl 50;
(Fij1 )CUTj 0 ;
l HET l"4
j CP
HUTi 1; i HP (TjOUT TjIN )× Gvj
i HP; j CP; k ST
(21)
i HP
X
(26)
i HP; j CP; k ST (FijN )HUTi 0 ;
i HP
Feasible temperature distribution (8, 9). Constraints to limit the number of relocations (17). Constraints to limit the number of new heat exchangers (18). Selection of existing exchangers (19).
CUTj 1; j CP Inlet temperatures: TiIN Ti1 ;
i HP
TjIN Tj(NST 1) ;
j CP
(22)
A monotonic decrease of temperatures within the HEN: Tik ] Ti(k1) ;
i HP; k ST
Tjk ] Tj(k1) ;
j CP; k ST
TjOUT Tj1 ;
Temperature differences: DTijk Tik Tjk ;
i HP; j CP; k ST
Inlet and outlet temperatures of hot streams (11). Inlet and outlet temperatures of cold streams (12). Heat loads of exchangers (13). Temperature differences (14, 15).
Only one heat exchanger type or existing heat exchanger or bypass should be selected (16), where:
i HP
j CP
/ / / /
(23)
Outlet temperatures: TiOUT Ti(NST 1) ;
Disjunctive constraints of the operability limitations:
B A B Tik ; Tjk ; T(i)A ijkl ; T(i)ijkl ; T(j)ijkl ; T(j)ijkl ; Fijk ;
(24)
F(l)ijkl ; Gvi ; Gvj . . .] 0 DTijk ; DTjHU ; DTiCU ; . . .]DT LO HU yijkl ; yCU f0; 1gm i ; yj
(27)
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243
omitted from retrofitted solutions by fixing its binary variables to zero. Three different cases were examined: . a retrofit with a pre-estimated value of Ft , . a retrofit with Ft as an optimization variable, and . a retrofit where extensions of existing areas are prohibited.
Fig. 5. The HEN structure of first example before the retrofit.
3. Examples Three examples of increasing complexity are presented to illustrate the simultaneous MINLP retrofit of HEN. The first case study is a small numerical example while the second and the third examples are more complex case studies comprising streams from the process of HDA of toluene for which two initial nonintegrated networks are given such that they can mimic the scope of the proposed multi-type model for the retrofit of HEN. In addition a modified objective function of the HEN retrofit model is presented for fixed areas of existing exchangers when one would like to avoid the adaptation of exchangers with an additional heat transfer area (A ).
3.1. The first example The first example data are shown in Table 3. This example involves three hot streams, four cold streams with the four-stage superstructure of HEN. The HEN structure before the retrofit comprises six heat exchangers (Fig. 5) with an annual operating cost of 1 384 550 $/a. As an additional complication let us suppose that the last exchanger (l/10) due to some reason should be
3.1.1. Retrofit of the first example with a fixed value of Ft When the Ft correction factor for the logarithmic mean was estimated and fixed to 0.8 the annual HEN cost was reduced to 244 880 $/a. The main benefit comes from reduction of HU and CU for about 4.6 MW (Fig. 7). The optimal solution (Table 4, Fig. 6) comprises all existing heat exchangers but the last one, which should be omitted. One new DP heat exchanger (shadowed DP in Fig. 6) for the heat transfer between the hot stream H1 and cold stream C2 and one relocated ST heat exchanger (l/5) with additional 73 m2 of heat transfer area (Shadowed ST in Fig. 6) are added to the network. The model with 423 binary variables, 3042 continuous variables and 1950 equations was solved by MIPSYN (Sorsˇak & Kravanja, 2002) in the 130 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. The annual HEN cost is significantly reduced because of more efficient transfer of heat between the hot and the cold process streams. The temperature distribution within the proposed HEN structure remains feasible because the infeasible temperature distribution constraints are applied. 3.1.2. Retrofit of the first example with the variable value of Ft When the Ft correction factor was optimized, the annual HEN cost was additionally reduced, down to 235 000 $/a (Table 5). The topology of the solution, however, remains the same as in the case of the fixed value of Ft (Fig. 6). The additional reduction of the HEN cost was achieved because the Ft correction of the temperature driving force was determined during the
Table 4 Solution of the first example by the fixed value of the Ft Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-2-2-1 1-4-4-6 2-2-3-7 2-4-4-8 3-2-1-9
694.5 79.7 116.1 163.8 28.7 24.4
621.5 0.0 267.0 163.8 11.5 29.9
73.0 79.7 /150.9 0.0 17.2 /5.5
80,22 8.15 65.62 6.17 69.42 20.69
0,800 1.000 1.000 1.000 1.000 1.000
ST DP DP PF DP DP
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost for the retrofitted HEN: ($/a) Total annual cost: ($/a)
105 200 53 900 85 700 244 800
Relocated New HE
244
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tion could not be carried out in practice. Therefore, the Ft correction factor should be optimized simultaneously, when the HEN topology is determined.
3.1.3. Retrofit where the extensions of existing areas are prohibited In order to prohibit extension of the existing areas, the modified objective function was applied in the HEN retrofit model: XXX [(cCU × Fijkl )j CU ]kK j i
Fig. 6. Solution of the first example for the fixed and variable Ft correction.
j
l
XXX i
j
[(cHU × Fijkl )i HU ]k1 i
l
X X X X
optimization (Ft/0.88) which enabled an extension of the heat transfer area for the ST heat exchanger (l/5) by only 9 m2. It should be noted that the reduction of the annual HEN cost was obtained because the preestimated value of the Ft (0.80) was lower than the exact value of the Ft (0.88). On the other hand, if the Ft was overestimated (e.g. Ft /0.99) the resulted HEN cost would by underestimated and the corresponding solu-
i
j
k
cfl × yijkl (cvl wA l fl:lBIDg )×
l
Max 0;
Fijkl
Al
kij × Ftijkl × Dln Tijkl X X X X (crfijkl × yijkl )crvijkl × i
j
k
l
Table 5 Solution of the first example for the variable value of Ft Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-2-2-1 1-4-4-6 2-2-3-7 2-4-4-8 3-2-1-9
630.5 79.6 116.1 163.8 28.7 24.4
621.5 0.0 267.0 163.8 11.5 29.9
9.0 79.6 /150.9 0.0 17.2 /5.5
80.22 8.16 65.62 6.17 69.42 20.69
0.881 1.000 1.000 1.000 1.000 1.000
ST DP DP PF DP DP
Hot utility cost: ($/a) Cold utility cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
105 200 53 900 75 900 235 000
Relocated New HE
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245
Fijkl Max 0; kij × Ftijkl × Dln Tijkl Al l fl:lBIDgPOS
0
ijkl X XXX yijkl cnfl × 1
l
i
j
k
l fl;lBIDg
(28)
i HP; j CP; k ST; l DHET where wA denotes a high penalty for the adaptation of existing heat transfer areas. Therefore, rather than extending the existing very expensive areas, the existing exchangers are not extended. Note that the constraints remain untouched and that implementation of the penalty preserves the linearity of the feasible space, which improves the performance of the MINLP optimization. The example was solved by MIPSYN in the 270 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. Shadowed exchangers in the optimal solution (Fig. 8) correspond to one new, one relocated ST heat exchangers and one new double pipe exchanger. Note that the existing double pipe heat exchanger (l /8) is excluded from the optimal solution because of insufficient heat transfer area. The total annual HEN cost equals 265 400 $/a (Table 6). It should be noted that it is possible to establish selective prohibition for the extension of existing transfer areas. In this case the penalty factor wA is only specified for exchangers with fixed areas.
3.2. Second example The second example comprises five hot and six cold process streams (Table 7) with a six-stage superstructure of HEN. The stream data were taken from the HDA case study. The HEN structure contains 11 exchangers (one ST and ten DP exchangers) with annual operating costs of 614 610 $/a (Fig. 9). The use of PF exchangers is prohibited because some of the involved streams are toxic. Two different cases were carried out:
Fig. 8. Optimal solution of the first example with prohibited extension of the existing areas.
. only a double pipe heat exchanger type was assumed, and . different exchanger types were selected. 3.2.1. Retrofit of the second example by considering only double pipe heat exchanger type The first case with 2520 binary variables, 17 513 continuous variables and 8631 equations was solved by MIPSYN (Sorsˇak & Kravanja, 2002) in the 5300 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. Ten existing exchangers were required for the entire heat transfer, while one heater (l/15) was excluded from the optimal solution (Fig. 10). In addition, two existing exchangers were relocated (shadowed DP exchangers in Fig. 10). The total annual HEN cost was reduced to 285 330 $/a (Table 8). It should be noted that at match 1-1-3-5, which was originally ST exchanger, feasible temperature distribution was restored, so that the retrofit yielded feasible network although only DP exchangers were assumed. However, since the first match has ST exchanger, the cost has to be recalculated: the Ft correction factor is equal to 0.909 and the required heat transfer area equals to 437.3 m2 which is 29.5 m2 of heat transfer area with the annual adaptation costs of 2790 $/a more than
Table 6 Solution of the first example by the prohibition of extending existing areas Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-3 1-2-2-1 1-4-4-6 2-2-3-7 2-4-4-5 3-2-1-9
630.5 80.0 116.1 163.8 30.3 24.4
0.0 0.0 267.0 163.8 621.5 29.9
630.5 80.0 /150.9 0.0 /591.2 /5.5
80.21 8.14 65.62 6.17 69.42 20.69
0.881 1.000 1.000 1.000 0.950 1.000
ST DP ST PF DP DP
Hot utility cost: ($/a) Cold utility cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
105 200 53 900 106 300 265 400
New HE New HE Realocated
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Table 7 Second example data Hot streams
G (kW/K)
a (kW/(m2K))
Tin (K)
Tout (K)
p (Mpa)
C ($/(kW a))
H1 H2 H3 H4 H5 (UT)
49.27 27.54 1088.67 229.17 /
0.15 0.90 0.90 0.90 0.30
823.20 330.85 352.32 379.90 790.00
299.89 329.85 349.32 376.90 780.00
3.5 3.5 3.5 3.5 1.0
250.0
Cold streams C1 C2 C3 C4 C5 C6 (UT)
38.92 14.58 511.33 252.60 236.13 /
0.12 1.00 1.00 1.00 1.00 1.00
330.19 362.95 462.30 376.90 550.60 282.00
713.70 463.00 465.30 379.60 553.60 290.00
3.5 3.5 3.5 3.5 3.5 1.0
25.0
Existing heat exchangers Match (i-j-k)
Type
A (m2)
SM index (l )
1-1-3 1-1-5 1-2-5 1-3-2 1-4-4 1-5-4 1-6-6 2-6-6 3-6-6 4-6-6 5-1-1
ST DP DP DP DP DP DP DP DP DP DP
407.8 1949.4 81.4 34.2 17.4 47.1 914.2 1.3 106.4 15.7 171.4
5 6 7 8 9 10 11 12 13 14 15
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total operational cost: ($/a)
320 500 294 110 0 614 610
assumed DP exchanger. Therefore, the total annual costs slightly increases up to 288 120 $/a. 3.2.2. Retrofit of the second example by considering different exchanger types The topology of the HEN structure remains the same, when different exchanger types are taken into account (Fig. 10). The total annual HEN cost is now 288 120 $/a (Table 9). The heat transfer area of the existing ST exchanger has to be adapted with an additional 29.5 m2. It should be noted that the solution is the same as in the previous case, after the recalculation of the existing ST (l/5) exchanger. However, the heat transfer area in the first case is not large enough for the entire heat transfer. The example indicates how important it is to consider different exchanger types simultaneously during the optimization. 3.3. Third example The third example comprises the same process streams as the second example, but with slightly different HEN structure before the retrofit, containing 12 heat exchangers: two ST exchangers and ten double
Fig. 9. The HEN structure of second example before the retrofit.
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247
show that models based on single exchanger type cannot solve real HEN problems comprising different exchangers types correctly. Let us look the solution based on double pipe exchangers. The solution (Fig. 12) of the first case comprises 11 existing exchangers, two of them being relocated (l /13, l/16). The solution yields the total annual HEN cost of 380 790 $/a (Table 11). Note that resulted temperature distributions at the first and fourth (Fig. 12) exchangers, which are now assumed to be double pipe exchanger instead of existing ST exchangers (l/5, l/6), are feasible for double pipe but infeasible for ST exchangers. In the first exchanger (l/5) the outlet temperature of hot stream H1 is equal to 623.3 K while the outlet temperature of the cold stream C1 is equal to 713.7 K. The temperature distribution is infeasible for ST exchangers because the outlet temperature of the cold stream (C1) is higher than the outlet temperature of the hot stream (H1) and such temperature arrangement cannot take place within the ST exchangers with the U tubes. Similarly, in the second ST exchanger the outlet temperature of the hot stream is lower (515 K) than the outlet temperature of the cold stream (529.5 K). Consequently, the proposed solution is infeasible and cannot be carried out in practice. The example clearly indicates that models for double pipe exchangers may produce solutions that are infeasible for other types of exchangers.
Fig. 10. Solution of the second example with DP exchanger type.
pipe exchangers (Fig. 11) with a total operational cost of 647 060 $/a (Table 10). The retrofit of the six-stage HEN superstructure was carried out for two different cases: . only a double pipe heat exchanger type was assumed, and . different exchanger types were selected.
3.3.2. Retrofit of the third example by comprising different exchanger types When different exchanger types were considered during optimization, the whole resulting temperature distribution and all areas were feasible. The example was solved again by MIPSYN in the 11 000 s of CPU time on the computer system with 566 MHz CPU and 256 MB of RAM. The optimal solution contains ten existing
3.3.1. Retrofit of the third example by assuming the double pipe heat exchanger type The problem is solved first by assuming only double pipe exchangers although the original network comprises exchangers of different types. The objective is to Table 8 Solution of the second example by the use of DT exchanger type Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-1-5-6 1-2-5-7 1-3-4-8 1-4-5-9 1-5-4-10 1-6-6-11 2-6-6-12 3-6-6-13 4-6-6-14
397.9 1949.4 79.6 43.1 32.9 29.5 849.4 1.3 106.4 15.7
407.8 1949.4 81.4 34.2 17.4 47.1 914.2 1.3 106.4 15.7
/9.9 0.0 /1.8 8.9 15.5 17.6 /64.8 0.0 0.0 0.0
117.72 90.83 140.44 272.70 153.00 184.14 58.68 44.26 64.79 92.37
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DC DC DC DC DC DC DC DC DC DC
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
Virtual 0 262 060 23 270 285 330
Exact 0 262 060 26 060 288 120
Relocated Relocated
248
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Table 9 Solution of the second example by the use of different exchanger types Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-3-5 1-1-5-6 1-2-5-7 1-3-4-8 1-4-5-9 1-5-4-10 1-6-6-11 2-6-6-12 3-6-6-13 4-6-6-14
437.3 1949.4 79.6 43.1 32.9 29.5 849.4 1.3 106.4 15.7
407.8 1949.4 81.4 34.2 17.4 47.1 914.2 1.3 106.4 15.7
29.5 0.0 /1.8 8.9 15.5 /17.6 /64.8 0.0 0.0 0.0
117.72 90.83 140.44 272.70 153.00 184.14 58.68 44.26 64.79 92.37
0.909 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
CP DC DC DC DC DC DC DC DC DC
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
exchangers, two of them being relocated (l/8, l/9, shadowed DP exchangers in Fig. 13) and two new ST exchangers (shadowed ST exchangers in Fig. 13). The total annual HEN cost is now 430 020 $/a (Table 12). Note that the cost has increased by approximately 50 000 $/a, because of two additional ST heat exchangers for the heat transfer between the hot stream H1 and cold stream C1, necessary to avoid infeasible temperature distribution. This example shows the importance of comprising different exchanger types during the HEN optimization in order to avoid infeasible distribution of the temperatures within the proposed HEN structure.
Fig. 11. The HEN structure of the third example before the retrofit.
Relocated Relocated
0 262 060 26 060 288 120
4. Conclusions The use of the single-type DP model for the retrofit of HEN is perhaps more attractive than the proposed multitype model because it is simpler and computationally less exhaustive. However, as indicated by the examples, it may produce solutions that are not only suboptimal, but also infeasible when the existing HEN comprises different exchanger types. Thus, the examples clearly indicate the advantages of the proposed model. This model allows for a simultaneously heat-integrated retrofit of HEN, enables the selection of optimal and feasible heat exchanger types during the optimization,
Fig. 12. Solution of the third example with DP exchanger type.
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Table 10 Third example data Hot streams
G (kW/K)
a (kW/(m2K))
Tin (K)
Tout (K)
p (Mpa)
C ($/(kW a))
H1 H2 H3 H4 H5 (UT)
49.27 27.54 1088.67 229.17 /
0.15 0.90 0.90 0.90 0.30
823.20 330.85 352.32 379.90 790.00
299.89 329.85 349.32 376.90 780.00
3.5 3.5 3.5 3.5 3.5
250.0
Cold streams C1 C2 C3 C4 C5 C6 (PS)
38.92 14.58 511.33 252.60 236.13 /
0.12 1.00 1.00 1.00 1.00 1.00
330.19 362.95 462.30 376.90 550.60 282.00
713.70 463.00 465.30 379.60 553.60 290.00
3.5 3.5 3.5 3.5 3.5 3.5
25.0
Existing heat exchangers Match (i-j-k)
Type
A (m2)
SM index (l )
1-1-2 1-1-3 1-1-4 1-2-4 1-3-3 1-4-4 1-5-2 1-6-6 2-6-6 3-6-6 4-6-6 5-1-1
ST ST DP DP DP DP DP DP DP DP DP DP
774.8 736.8 53.9 156.0 103.6 47.1 28.6 919.8 1.3 106.4 15.7 184.2
5 6 7 8 9 10 11 12 13 14 15 16
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total operational cost: ($/a)
350 000 297 060 0 647 060
Table 11 Solution of the third example by the use of DP exchanger type Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-2-5 1-1-3-6 1-1-4-7 1-2-4-8 1-3-2-13 1-4-4-10 1-5-2-11 1-6-6-12 2-6-6-16 3-6-6-14 4-6-6-15
1013.9 736.9 254.3 186.3 46.6 56.5 34.6 849.4 1.3 106.4 15.7
774.8 736.9 53.9 156.1 1.3 47.1 28.6 919.8 184.3 106.4 15.7
239.1 0.0 200.3 30.3 45.3 9.4 6.0 /70.3 /182.9 0.0 0.0
106.0 117.6 117.0 60.0 252.2 89.1 156.1 58.7 44.3 64.8 92.4
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DP DP DP DP DP DP DP DP DP DP DP
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
0 262 060 117 730 380 790
Relocated
relocated
250
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Table 12 Solution of the third example by the use of different exchanger types Match (i-j-k-l)
A (m2)
Aob (m2)
Adod (m2)
DlnT (K)
Ft
Type
1-1-2-5 1-1-3-6 1-1-4-3 1-1-5-3 1-2-5-8 1-3-4-9 1-4-4-10 1-5-2-11 1-6-6-12 2-6-6-13 3-6-6-14 4-6-6-15
758.4 704.7 156.1 649.2 156.0 111.6 26.1 25.6 849.4 1.3 106.4 15.7
774.8 736.9 0.0 0.0 156.1 103.6 47.1 28.6 919.8 1.3 106.4 15.7
/16.5 /32.2 156.1 649.2 0.0 8.0 /21.0 /3.0 /70.3 0.0 0.0 0.0
114.7 133.8 128.8 106.2 71.7 105.3 192.6 212.5 58.7 44.3 64.8 92.4
0.799 0.836 0.952 0.816 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ST ST ST ST DP DP DP DP DP DP DP DP
Hot utility consumption cost: ($/a) Cold utility consumption cost: ($/a) Annualized investment cost of the retrofitted HEN: ($/a) Total annual cost: ($/a)
Fig. 13. Solution of the third example with different exchanger types.
and yields retrofitted HEN with feasible temperature distributions.
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0 262 060 167 960 430 020
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