5512
Ind. Eng. Chem. Res. 2008, 47, 5512–5528
Simultaneous Retrofit and Heat Integration of Chemical Processes Jose´ M. Ponce-Ortega,†,‡ Arturo Jime´nez-Gutie´rrez,† and Ignacio E. Grossmann§ Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya, Celaya, Guanajuato 38010, Me´xico; Facultad de Quı´mica, UniVersidad Michoacana de San Nicola´s de Hidalgo, Morelia, Michoacán 58060, Me´xico; and Chemical Engineering Department, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213
In this paper, a new formulation for the retrofit of heat exchanger networks considering process modifications is presented. The method accounts for the interactions between the process conditions and the heat integration options to provide an optimal structure for a redesigned heat exchanger network. The formulation is based on a superstructure that considers explicitly the plant layout and the piping arrangement, which yields a mixedinteger nonlinear programming model. The model presented here includes the treatment of isothermal process streams that exchange their latent heats, in addition to the streams commonly considered with sensible heat loads. The objective function consists of maximizing the total annual profit for the retrofit process, which includes the income from products sales and the expenses due to raw materials, capital cost for new units, utility costs, and the piping modification costs. The results for the cases of study show that significant improvement in the process profitability can be obtained with the simultaneous approach presented in this work for process retrofit with respect to the sole consideration of the heat exchanger networks. 1. Introduction Heat exchanger networks (HENs) have been widely applied in industrial projects over the past decades because they provide significant energy and economic savings. A good number of methodologies have been proposed for the HEN synthesis problem; for retrofit problems, however, the available methods are more limited, as has been noted in the review papers by Furman and Sahinidis,1 Jezowski,2,3 and Gundersen and Naess.4 The first works that considered the HEN retrofit problem were based on the pinch method.5,6 Within this approach, an energy recovery level is first obtained as part of a targeting procedure, and the basic rules of the pinch design method are then applied to review the existing network.7 Polley et al.8 extended the use of the pinch method to include pressure drop considerations for the retrofit of the HEN. Several mathematical programming methods have been proposed for the retrofit of HENs. Ciric and Floudas9 presented a mixed-integer nonlinear programming (MINLP) model, which was based on the transhipment model by Papoulias and Grossmann10 and an extension of the synthesis superstructure by Floudas et al.,11 to model the network configuration and provide the optimal structure of the revised network. Yee and Grossmann12 reported a general superstructure and formulated an MINLP model that represented explicitly existing and potential exchangers for the retrofit problem. Ma et al.13 proposed a two-step approach based on the superstructure by Yee and Grossmann.14 A methodology that combined the use of thermodynamics and mathematical programming techniques was proposed by Briones and Kokossis.15 Sorsak and Kravanja16 reported an MINLP model for the retrofit of HENs comprising different exchanger types. Bjork and Nordman17 proposed an MINLP model to solve large-scale HEN retrofit problems. Two methodologies for the retrofit of HENs considering the pressure drop effects were presented by Nie and Zhu18 and Lopes-Silva and Zemp.19 Athier et al.20 presented a hybrid method based on simulating annealing and non linear program†
Instituto Tecnolo´gico de Celaya. Universidad Michoacana de San Nicola´s de Hidalgo. § Carnegie Mellon University. ‡
ming techniques. Zhu and Asante21 proposed an approach that combined mathematical programming with a set of heuristic rules. Only a few works have attempted the simultaneous treatment of process modifications as part of HEN problems. Some ideas along these lines have been considered in the works by Duran and Grossmann,22 Lang et al.,23 and Grossmann et al.,24 in which the optimal flowsheet for the process is obtained by enforcing the minimum utility target. Zhang and Zhu25 addressed the problem of HEN retrofit considering process changes. However, these authors only considered the effects on utility consumption and did not take into account the capital cost associated with the retrofit process. Most of the methods described above for HEN retrofit assume that the process conditions (inlet and outlet temperatures, stream flow rates) are fixed, so that no interactions with process modifications are considered (see Figure 1). Allowing for potential adjustments in the operating conditions should provide the basis for more cost-effective heat integration. We should also note that none of the methodologies reported for the HEN retrofit has included the explicit treatment of isothermal streams, which are very common in the chemical industry, for instance, in the operation of distillation columns. In HEN synthesis problems, a fairly common approach to the treatment of isothermal streams consists in assuming a one degree change with a suitable adjustment of a pseudoheat capacity value to equal the enthalpy change. This approach, however, is prone to scaling problems during the problem numerical solution. Recent works have shown a more formal treatment of isothermal streams for HEN synthesis problems.26,27 In this paper, an MINLP model for the HEN retrofit that considers simultaneously the HEN structure and process modifications is presented. The proposed model considers the plant layout and modification costs through the superstructure by Yee and Grossmann12 in which operational and structural modifications of the process are added and considered simultaneously. In addition, the model considers explicitly the utility and the capital cost of the units and takes into account the isothermal process streams that may appear in the process using the approach reported by Ponce-Ortega et al.26
10.1021/ie071182+ CCC: $40.75 2008 American Chemical Society Published on Web 07/30/2008
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5513
Figure 1. Process interaction with the HEN.
Figure 2. Types of streams in the simultaneous HEN and process retrofit.
2. Outline For a proper retrofit analysis that incorporates the operating conditions of the process, it is necessary to consider the plant layout. Therefore, this work uses a two-dimensional superstructure for the HEN retrofit model in which the hot and cold process streams are identified in the retrofit process as part of the process conditions (see Figure 2). Figure 3 shows the superstructure for the HEN retrofit for two hot and two cold process streams based on the original formulation by Yee and Grossmann12 where one can see the complexity of the potential interconnections. Exchangers 1, 2, and 3 exist in the original network, whereas exchanger 4 is a new unit that may or may not exist in the optimal network configuration. The two-dimensional layout of the piping is explicitly taken into account in the superstructure, such that all possible interconnections between process streams and exchangers are considered. As a consequence, the model allows through repiping the reassignment of existing exchangers to different streams to maximize the use of existing area and exchangers. Also, additional area to existing exchangers is considered as a retrofit option. 3. Model Formulation The proposed mathematical formulation applies to the generalization of the superstructure of Figure 3 for arbitrary number of process streams, existing exchangers and potential new exchangers. Rules to establish the number of potential new exchangers are given in Yee and Grossmann.12 The following
sets are used for the model development. HPS1 contains the hot process streams that exchange sensible heat in the network, HPS2 contains the hot process isothermal streams (i.e., they exchange only latent heat and their temperatures remain constant), HPS contains all the hot process streams (HPS ) HPS1∪HPS2), HU corresponds to the hot utility streams and HS contains all the hot streams (HS ) HPS∪HU). HS1 is a set that contains the hot streams excluding isothermal streams (HS1 ) HPS1∪HU). Similarly, CPS1 is the set for the cold process streams that exchange sensible heat and CPS2 includes the cold process isothermal streams, whereas CPS, CU and CS represent the total cold process streams (CPS ) CPS1∪CPS2), the cold utility stream, and all the cold streams (CS ) CPS∪CU), respectively. CS1 contains the cold nonisothermal streams (CS1 ) CPS1∪CU). HCPS contains the hot and cold process streams (HCPS ) HPS∪CPS), and the set HCTS contains all hot and cold process and utility streams (HCTS ) HS∪CS). The set E represents all the exchangers in the superstructure, whereas the sets EE and NE represent the existing and the new exchangers, respectively. The detailed description of the symbols used in the model formulation is given in the Nomenclature section. For the sake of clarifying the presentation, we present first the 0-1 constraints for the topology selection followed by continuous variable equations and mixed integer constraints. Logical Assignment Constraints. For convenience, the following sets of binary variables are defined. wsk denotes that stream s is assigned to exchanger k. The variable yisk is used when the inlet of stream s is assigned to exchanger k, whereas yesk is used when stream s exits the HEN from exchanger k. xsk,l denotes the interconnection between exchangers k and l for stream s. The binary variables zhk,l and zck,l are used for hot or cold streams if piping segments between exchangers k and l exist, while zek is used for piping segments between exchanger k and the exit of the HEN. The variable Vk is used for new exchangers in the retrofitted HEN. Selection of Streams for Heat Exchanger Units. The assignment of a hot stream (eq 1) and a cold stream (eq 2) is done for each existing exchanger
∑ w )1
k ∈ EE
(1)
∑ w )1
k ∈ EE
(2)
k i
i∈HS
k j
j∈CS
For new exchangers, at most one hot and one cold stream can be assigned depending on whether or not the new exchanger gets selected for the retrofit network.
∑ w e1 k i
i∈HS
k ∈ NE
(3)
5514 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
∑ w e1
k ∈ NE
k j
yisk - wsk e 0 s ∈ HCTS
(4)
(8)
j∈CS
Every process stream must enter at least one heat exchanger of the superstructure. To model this situation the following equation is used
∑ yi g 1 k s
s ∈ HCPS
(5)
k∈E
For each heat exchanger no more than one hot or one cold inlet stream must be assigned,
∑ yi e 1 k s
k∈E
The previous equations also avoid mixing heat loads from different process streams. Logical Interconnection Constraints. Consistency constraints are necessary to allow a connection between two exchangers only if both exchangers service the same process streams. By definition, the variable xsk,l denotes that stream s is assigned to both exchangers k and l, and the variables zhk,l and zck,l denote the existence of piping sections.
(6)
If the interconnection variable xsk,l is equal to one, then exchangers k and l service that process stream s
(7)
xsk,l - wsk e 0 s ∈ HCPS, k ) 1, 2, ... , K - 1, and l ) k + 1, ... , K (9)
s∈HS
∑
yisk e 1
k∈E
s∈CS
Constraints (6) and (7) can be removed if mixing of different process streams is allowed. Fresh Streams Constraints. If an initial piping segment is selected for a given stream, then there is an exchanger that services that stream
xsk,l - wsl e 0 s ∈ HCPS, k ) 1, 2, ... , K - 1, and l ) k + 1, ... , K (10) When a piping segment between exchangers k and l exists, then one stream uses the piping. Therefore, for the hot side
Figure 3. Superstructure for the HEN retrofit for two hot and two cold process streams.
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5515
zhk,l -
∑
xk,l i e 0 k ) 1, 2, . . . , K - 1, and l ) k +
i∈HPS
1, . . . , K
(11)
Mass Balance for Initial Splitters. For each process stream (hot and cold) the following mass balance applies for the initial splitters in Figure 3
whereas for the cold side zck,l -
∑
xk,l j e0
FCpsIN )
k ) 1, 2, ... , K - 1, and l ) k + 1, ... , K
j∈CPS
(12) Similarly, for a piping section l and k (notice that for modeling purposed zl,k is different than zk,l, but they represent the same situation) zhl,k -
∑
xk,l i e 0 k ) 1, 2, ... , K - 1, and l ) k + 1, ... , K
i∈HPS
(13) zcl,k -
∑
∑ fCp
s ∈ HCPS
k s
(22)
k∈E
Mass Balance at Inlet Mixers for Each Exchanger. For convenience, two inlet mixers are modeled for each exchanger, one for the hot and one for the cold stream. At each inlet mixer, the mass flow that enters the exchanger is equal to the mass flow from inlet streams, plus the mass flow that comes from other exchangers, plus the mass flow of the utility. In this way, for the hot inlet mixers we have fCph,in k )
∑
fCpki +
i∈HPS
∑ fCp
h k l,k + FCpHUwHU
k∈E
l∈E l*k
xk,l j e 0 k ) 1, 2, ... , K - 1, and l ) k + 1, ... , K
(23)
j∈CPS
(14) Definition of an Exit Stream from a Heat Exchanger. To consider the outlet piping cost in the objective function, the binary variable yeik is activated if the piping segment for any hot stream at the exit of exchanger k exists and exchanger k services stream i. zehk + wki - yeki e 1 i ∈ HS, k ∈ EE
(15)
Similarly, for the cold streams, zeck + wkj - yekj e 1 j ∈ CS, k ∈ EE
(16)
These equations incorporate the cost of outlet piping segments according to the plant layout. Definition of New Heat Exchangers. If any pair of hot and cold streams are serviced by a new heat exchanger, then Vk ) 1, k ∈NE, and the following constraint must be satisfied
∑
i∈HS
wki +
∑
wkj - Vk e 1
k ∈ NE
j∈CS
∑
fCpkj +
j∈CPS
Vk + wkj - µkj e 1 j ∈ CPS2, k ∈ NE
(19)
Relocation of Heat Exchangers. Additional constraints can be written to account for possible relocations of heat exchangers within the plant. The assignment variable κl,k is used to denote the relocation of exchanger l with area EAMl to a new position k. Only one exchanger from a position l among K positions can be reassigned to a new position k K
k∈E
(20)
l)1
Similarly, each exchanger from original position l can be reassigned to only one new position k among K positions,
k)1
k∈E
l∈E l*k
(24) It is worth mentioning that FCpHU and FCpCU are constant (upper limits) and that the logical interconnection constraints prevent the mixing of different streams. Energy Balance at Inlet Mixers for Each Exchanger. To determine the inlet temperature to each exchanger, energy balances are needed for each inlet mixer. For the exchanger inlet mixer in the hot side h,in fCph,in k tk )
∑
∑ (fCp
(fCpki TIN i )+
h h,out )+ l,ktl
i∈HPS
l∈E l*k IN k FCpHUTHU wHU k ∈ E
c,in fCpc,in k tk )
∑
(25)
c c,out )+ l,ktl
l∈E l*k IN k FCpCUTCU wCU k ∈ E
(26)
Existing Piping Segment. The following inequality is required to define an existing piping segment fCpsk e FCpUPyisk k ∈ E, s ∈ HCPS
(27)
where FCpUP is an upper limit for the heat capacity flow rate. In this way, if fCpks is greater than zero, then the binary variable yisk must be 1 to satisfy eq 27. Mass Balance at Outlet Splitters for Each Exchanger. For any exchanger in the superstructure it is necessary to enforce the mass balance for the hot and cold outlet splitters. That is, the mass flow rate at the exchanger exit (which is equal to the inlet flow rate) is equal to the mass flow rate directed to other exchangers plus the mass flow rate discharged out of the HEN. For the hot side fCph,in k )
∑ fCp
h h k,l + fCpk,exit
l∈E
l∈E
∑ (fCp
(fCpkj TIN j )+
j∈CPS
K
k,l ) 1
c k l,k + FCpCUwCU
(18)
whereas for the isothermal cold process stream j, the binary variable µjk must be 1
∑κ
∑ fCp
and for the cold side
Vk + wki - µki e 1 i ∈ HPS2, k ∈ NE
k,l ) 1
fCpc,in k )
(17)
Definition of New Units for Isothermal Process Streams. If the new exchanger k processes an isothermal hot process stream i, then the binary variable µik must be equal to 1
∑κ
whereas for the cold inlet mixers,
l*k
(21) and for the cold side
k∈E
(28)
5516 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Chart 1
fCpc,in k )
∑ fCp
c c k,l + fCpk,exit
k∈E
(29)
l∈E l*k
Overall Heat Balance for Each Process Stream. To determine the target temperature for each process stream, an overall heat balance is required. The total heat exchanged by each stream is equal to the sum of the heat loads for the corresponding exchangers. Therefore, for the hot process streams that exchange sensible heats IN OUT FCpIN )) i (Ti - Ti
∑∑
Qi,j,k i ∈ HPS1
(30a)
k∈E j∈CS
whereas for the isothermal hot process streams Fλcond ) i
∑∑
Qi,j,k i ∈ HPS2
(30b)
k∈E j∈CS
where Fλicond is the condensation heat load for the hot stream i ∈HPS2. For the nonisothermal cold process streams OUT FCpIN - TIN j (Tj j ))
∑∑
Qi,j,k j ∈ CPS1
(31a)
k∈E i∈HS
and for isothermal cold process streams Fλevap ) j
∑∑Q
i,j,k
j ∈ CPS2
k∈E i∈HS
Figure 4. Original process for the motivational example.
(31b)
Here, Fλjevap is the evaporation heat load for the cold process stream j ∈CPS2. Equations for Heat Exchangers. The energy balance, design equation, and log-mean temperature difference for heat exchangers are applied as shown below. The heat balance equation around each heat exchanger (eqs 32 and 33) is used to determine the temperature change provided by the exchanger for nonisothermal process streams. One equation for each side of the exchanger is needed. The constraint is written in relaxed form as an inequality because the utility streams are not reassigned to a particular exchanger in the superstructure. As a result, the equality form of the heat balance equation would not be appropriate for utility streams with a prespecified temperature change. The relaxed form also follows from analyzing the Karush-Kuhn-Tucker conditions. The heat balance equation is only needed for the hot and cold streams that exchange sensible heats (i.e., streams of the sets HPS1 and CPS1). When streams with latent heat are involved (i.e., streams of the sets HPS2 and CPS2) eqs 34 and 35 apply, and they reflect that when exchanger k is used to process an isothermal stream the change of temperature must be zero. When utilities are needed, temperature changes are prespecified. Therefore, eqs 36 and 37 are used to determine the temperature differences when utilities are involved. Notice that if utilities are isothermal, then DT is set to zero.
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5517 Table 2. Piping Cost Data For Example 1
Table 1. Limits for the Operational Variables for Example 1 concept
lower value
upper value
FCp1IN (kW/°C) FCp3IN (kW/°C) T1IN (°C) T3IN (°C)
96 80 119.1 80
132 110 332.1 120
F CONP k,l
The heat exchanger design equation 38 is relaxed as an inequality to determine the additional area required in each match of the superstructure. In this way, when the existing area in a given match k (EAk) is not sufficient to meet the heat load, then the additional area required (AAk) is determined. The additional area is also limited by the objective function. If the existing area is equal to or greater than the required area, then additional area is not necessary. For the calculation of the log mean temperature difference (eq 39), Chen’s approximation is used to avoid singularities.28 Temperature difference constraints (eqs 40 and 41) are used to ensure thermodynamic feasibility between the hot and cold streams at the extremes of the heat exchangers. Notice that in the model formulation it is not necessary to fix the heat recovery level, and only the minimum temperature difference allowed in any match of the superstructure, ∆TMIN, is used. Temperature feasibility constraints (eqs 42 and 43) are needed to ensure the decrement of temperature for the hot process streams and the increment for the cold process streams. These equations are also used for a correct application of eqs 34 and 35 for isothermal streams. The set of equations for the above concepts is given below.
∑ ∑Q
h,in h,in h,out )e0 i,j,k - fCpk (tk - tk
k ∈ EE
(32)
i∈HS1 j∈CS
∑ ∑
c,out Qi,j,k - fCpc,in - tc,in k (tk k )e0
i∈HS j∈CS1
Figure 5. Superstructure for Example 1.
k ∈ EE
(33)
F CONPI s,k
F CONPE s,k
k,l
value ($/year)
s,k
value ($/year)
s,k
value ($/year)
1,2 1,3 2,3
1000 1200 1300
1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 4,1 4,2 4,3
500 0 500 0 250 650 0 450 720 350 0 640
1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 4,1 4,2 4,3
400 0 600 0 450 600 0 450 600 400 0 600
h,out e M1(1 - wki ) i ∈ HPS2, k ∈ EE th,in k - tk
(34)
k 1 tc,out - tc,in k k e M (1 - wj ) j ∈ CPS2, k ∈ EE
(35)
k h,out DTHUwHU - (th,in )e0 k - tk
k ∈ EE
(36)
k - (tc,out - tc,in DTCUwCU k k )e0
k ∈ EE
(37)
∑ ∑ (Q
(
i,j,k
i∈HS j∈CS
(
1 1 + hi hj
))
- (EAk + AAk)LMTDk e 0 k ∈ EE (38)
c,out h,out )(tk - tc,in LMTDk - (th,in k - tk k )
(
c,out h,out (th,in )(tk - tc,in k - tk k ) 2
))
1/3
e 0 k ∈ EE (39)
∆TMIN - (th,out - tc,in k k ) e 0 k ∈ EE
(40)
c,out ∆TMIN - (th,in ) e 0 k ∈ EE k - tk
(41)
th,out e th,in k ∈ EE k k
(42)
5518 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 6. Retrofitted HEN configuration for example 1. Table 3. Results Comparison for Example 1 concept
original process
HEN retrofit without process optimization
simultaneous HEN retrofit and process optimization
utility cost ($/year) capital cost ($/year) cost of row materials ($/year) sales of products ($/year) annual profit ($/year)
610,000 0 439,692,315 510,638,532 70,336,217
10,000 900 439,692,315 510,730,466 71,027,251
15,910 900 483,661,546 561,702,386 78,024,030
Figure 7. Original process for example 2. c,out tc,in k ∈ EE k e tk
(43)
For new exchanger units, the set of equations is applied only if the unit is selected for the retrofit network as part of the optimization process. Therefore, the disjunction shown in Chart 1 is written to account for new units in this chart, V and W are Boolean variables (that may be true or false), and the terms in
parentheses apply only if the Boolean variable is true. If we use the big-M formulation to model the disjunction, the following mixed-integer equations arise
∑ ∑Q
h,in h,in h,out ) e M3(1 - Vk) i,j,k - fCpk (tk - tk
k ∈ NE
i∈HS1 j∈CS
(44)
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5519 Table 5. Piping Cost Data for Example 2
Table 4. Specifications for Example 2 design basis
F CONP k,l
cost
product D feed purge gas generated steam working time payout factor utilities cooling water 320-290 K hot utility 690-690 K purchased electric power demineralized water fixed cost for new heat exchangers capital cost for new heat exchanger area (A in m2) compressor efficiency reactor conversion
$3.81/kmol $0.65/kmol $0.55/year $1.8537 × 10-5/kJ 8376 h/year 0.3/year $2.4642 × 10-6/kJ $5.5613 × 10-5/kJ $0.025/kW h $2.34 × 10-3/kmol $3000 $1650A γ ) 1.4, ηm ) 0.9, ηc ) 0.8 x ) 0.5 exp(-0.002T)(80/90) P[yAyB/(1 + yC + yD)]
Antoine Constants (P in mmHg) component A B C D
A
b
C
13.6333 14.3686 15.2243 18.5875
164.90 530.22 897.84 3626.55
3.19 -13.15 -7.16 -34.29
Film Heat Transfer Coefficients h (kW/(m2 K))
stream H1 H2 CU C1 C2 C3 CU
2.31 0.85 2.50 0.75 0.93 2.18 1.00 Constraints reactor
other
Toutlet g 320K e Tflash Tinlet e380K 0epurge e100% Toutlet e 690 K 450 K e Tinlet e670KproductD eproduct(0.96) 9atm epressure e29 atm 0 econversion e100%
∑ ∑Q
c,in c,out 4 k - tc,in i,j,k - fCpk (tk k ) e M (1 - V )
k ∈ NE
i∈HS j∈CS1
(45) h,out th,in e M1(1 - µki ) i ∈ HPS2, k ∈ NE k - tk
(46)
j ∈ CPS2, k ∈ NE
(47)
k 2 tc,out - tc,in k k e M (1 - µj )
k ∈ NE
(48)
k 6 k DTCUwCU - (tc,out - tc,in k k ) e M (1 - V ) k ∈ NE
(49)
k h,out - (th,in ) e M5(1 - Vk) DTHUwHU k - tk
∑ ∑ (Q
i,j,k
i∈HS j∈CS
(
1 1 + hi hj
))
- (EAk + AAk)LMTDk e
(
M7(1 - Vk) k ∈ NE (50)
c,out h,out )(tk - tc,in LMTDk - (th,in k - tk k )
(
c,out (th,in ) + (th,out - tc,in k - tk k k ) 2
))
F CONPI s,k
k,l
value ($)
s,k
value ($)
s,k
value ($)
1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,3 2,4 2,5 2,6 2,7 2,8 3,4 3,5 3,6 3,7 3,8 4,5 4,6 4,7 4,8 5,6 5,7 5,8 6,7 6,8 7,8
750 810 950 970 780 790 770 710 740 830 520 530 510 560 580 640 650 620 620 730 740 720 850 860 840 520 520 520
1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8
850 0 640 650 760 630 635 620 0 670 900 1200 1300 750 760 740 450 500 520 0 0 620 610 630 970 820 640 530 0 970 980 960 630 0 540 780 820 620 630 610 980 810 780 0 660 830 840 810 0 560 0 810 780 670 680 660
1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8
990 780 0 540 590 850 855 840 0 490 750 810 840 640 650 630 520 500 540 0 0 530 520 540 880 790 630 670 0 780 790 770 640 0 510 610 760 590 600 580 970 810 650 0 760 870 880 860 0 560 0 700 680 680 690 670
c,out ) e M10(1 - Vk) k ∈ NE ∆TMIN - (th,in k - tk k Qi,j,k e QUP i,j V
i ∈ HS, j ∈ CS, k ∈ NE
e M (1 - V ) k ∈ NE (51) 8
k
9 k ∆TMIN - (th,out - tc,in k k ) e M (1 - V ) k ∈ NE
(52)
(53) (54)
h,UP k V k ∈ NE fCph,in k e FCp
(55)
c,UP k fCpc,in V k ∈ NE k e FCp
(56)
h,in,UP k th,in V k ∈ NE k eT
(57)
th,out e Th,out,UPVk k
1/3
F CONPE s,k
k ∈ NE
(58)
c,in,UP k tc,in V k ∈ NE k eT
(59)
tc,out e Tc,out,UPVk k ∈ NE k
(60)
5520 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 8. Retrofit process for example 2.
AAk e AAUPVk k ∈ NE
(61)
LMTDk e LMTDUPVk k ∈ NE
(62)
where V and w are 0-1 binary variables used to model the Boolean variables V and W. Therefore, when a new heat exchanger is selected (Vk ) 1), eqs 44 to 53 are rigorously applied; otherwise, a big-M parameter is used to relax the constraints, and all the variables involved in the set of equations are set to zero by eqs 54 to 62. In eq 46 the binary variable µik is activated when a new exchanger k exists and the isothermal hot process stream i ∈HPS2 is present in the match. The binary variable µjk serves a similar function in eq 47 for the isothermal cold process streams. The relations used to activate the binary variables µik and µjk are given in eqs 18 and 19, respectively. Heat Loads for Heat Exchangers. To denote when a hot process streams is processed by a given exchanger k, the following inequality is used Qi,j,k - QUPwki e 0 i ∈ HS, k ∈ E
(63)
Similarly for the cold process streams Qi,j,k - QUPwkj e 0 j ∈ CS, k ∈ E
(64)
where QUP is an upper limit for the head load. Definition of Flows in Piping Segments. If there is any amount of flow of a hot stream between exchangers k and l, h then the piping segment zk,l must exist; otherwise, the piping segment does not exist fCphk,l - FCpUPzhk,l e 0 k, l ∈ E, k * l
(65)
fCphk,exit - FCpUPzehk e 0 k ∈ E
(67)
fCpck,exit - FCpUPzeck e 0 k ∈ E
(68)
Feasibility Constraint for Isothermal Streams. For isothermal process streams, an additional constraint is needed to ensure that the temperatures are the same in the interconnection between two exchangers that service the same isothermal stream. These additional constraints are needed because the heat balances for the heat exchanger (eqs 32 and 33) are relaxed for isothermal streams. Therefore, to avoid flows equal to zero when exchanger k services an isothermal stream, the following constraints must be applied for hot and cold isothermal process streams LO k i ∈ HPS2, k ∈ EE fCph,in k g FCp wi
(69)
LO k fCpc,in j ∈ CPS2, k ∈ EE k g FCp wj
(70)
where FCpLO is a lower limit for the heat capacity flow rate to avoid zero inlet mass flow to the heat exchanger. Existing Exchangers Moved to a Different Location. For the cases where exchangers can be physically moved to a different location, the existing area of exchanger l, EAMl, can be assigned to the position k by the following equation K
EAk )
∑ (EAM κ
l k,l)
k∈E
(71)
l)1
Similarly, for the piping segment for a cold stream fCpck,l - FCpUPzck,l e 0 k, l ∈ E, k * l
For any stream that leaves the HEN from a given exchanger k there is a piping segment. The binary variable zek is used in the following constraints for the hot and cold sides of the exchanger
(66)
It is worth mentioning that the cost associated to relocate a heat exchanger is usually higher than that for reconnecting new
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5521 Table 6. Results Comparison for Example 2 concept
original process
raw materials hot utility cold utility electricity demineralized water capital exchangers piping total costs
58,799,520.00 8,033,410.62 700,799.85 2,897,998.30 9,303,820.08 0 0 79,735,548.85
Costs ($/year) 58,799,520.00 233,851.98 453,231.67 2,897,998.30 9,303,820.08 76,083.69 1,905.00 71,766,410.72
58,799,520.00 0.00 756,332.01 2,278,500.00 3,165,612.90 7,399.73 849.00 65,008,213.64
135,942,969.55 11,138,588.39 9,240,876.40 156,322,434.34 76,586,885.49
Earnings ($/year) 135,942,969.55 11,138,588.39 9,240,876.40 156,322,434.34 84,556,023.62
158,950,000.00 4,482,500.00 3,144,196.39 166,576,696.39 101,568,482.75
product purge generated vapor total earnings annual profit
original process with HEN retrofit
streams to the heat exchanger. This situation is reflected in the objective function. Process Modeling Constraints. The types of process modifications depend upon each particular case. It is necessary to identify the types of process modifications allowed for each particular case. We identify two types of process modifications. The first type is associated with process conditions, for example conversion, pressure, and temperature of reactors, temperature and pressure of separation units, flow rates of purge, feed and product streams, and so forth. These variables are clearly restricted by feasibility conditions. A second type is related to structural modifications of the process, for example the addition or replacement of equipment. Process modifications may affect the temperatures and flow rates of the process streams and as a consequence the utility requirements.
Figure 9. Solution of example 2 for retrofit HEN without process optimization.
simultaneous retrofitted process
In this way, the constraints h and g represent material and energy balances, design specifications and structural relationships as follows h(x, z, y) ) 0
(72)
g(x, z, y) e 0
(73)
where x represents the continuous variables of the process that are involved in the HEN model (i.e., fCpsIN and TsIN for all the process streams), z corresponds to the continuous variables that affect the process but are not included in the HEN model (i.e., pressures and temperatures of equipments, equipment sizes, etc.), and y represents the binary variables for structural modifications in the flowsheet. Objective Function. The objective function maximizes the total annual profit of the process. The income in the objective function
5522 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 7. Stream Data for Example 3 stream
FCpsIN (kW/K)
TsIN (K)
TsOUT (K)
h (kW/(m2 K))
H1 H2 HU C1 C2 CU
30 15 20 40 -
443 423 450 293 353 293
333 303 450 408 413 313
1.6 1.6 4.8 1.6 1.6 1.6
Table 8. Piping Costs for Example 3 F CONPH k,l
F CONPC k,l
F CONPI s,k
k,l
value ($/year)
k,l
value ($/year)
1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6
0 4500 5700 3900 3800 3700 3900 3800 3950 0 3400 2800 3800 3400 3600
1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6
4100 4500 5700 0 3800 0 3900 3800 3950 3700 3400 2800 3800 3400 3600
Figure 10. Existing HEN for example 3.
depends on the sales of the products. The expenses depend on the raw materials, costs due to the modification in the process conditions, and the HEN retrofit annual cost. Additional expenses may be considered for the purchase of new process equipment. The HEN retrofit costs include the hot and cold utilities cost, the annualized capital costs and fixed charges for new heat exchangers, the fixed charge for the new piping segments (which depends on the distance between two exchangers k and l), the variable charge for new piping segments (which depends on the distance and the amount of flow), and finally the cost to relocate a heat exchanger from one position to another. max profit )
∑
COmsfs -
s∈produc
∑ ∑ ∑ CO
∑
s∈rowmat
∑ ∑ ∑ CO
HU(Qi,j,k) -
i∈HU j∈CPS k∈E
COAA
β k - CONE
k∈E
h F NPHk,lzk,l +
k
∑∑ k∈E l∈E
k∈E l∈E
V CONP (fh + fck,l) k,l k,l
F CONPE (yesk) s,k
∑ ∑ CO
s∈HCS k∈E
∑ (V ) - ∑ ∑ (CO
k∈NE
F zc ) CONPC k,l k,l
∑ ∑
CU(Qi,j,k) -
s,k
value ($/year)
s,k
value ($/year)
1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6
3200 3600 0 3800 3100 3400 0 3800 3800 4100 3400 3800 3700 3900 3200 3800 0 3400 3200 0 3500 3700 3900 4000 0 3100 3600 3800 3600 3900 3700 3400 2900 0 3700 2800
1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6
4100 3400 3100 0 3400 3100 3600 0 3200 2900 3900 3500 3900 3800 3400 3600 0 3400 3600 3400 0 3300 2900 2800 3500 3800 3300 3700 0 3100 3700 3300 2700 0 3200 3100
i∈HPS j∈CU k∈E
∑ AA
s∈HCS k∈E
COmsfs - r(x, z, y) -
F CONPE k,out
k F NPIs,k(yis ) -
∑ ∑C
s∈HCS k∈E
V NPEs,k -
∑ ∑ CO
s∈HCS k∈E
k V NPIs,k(fs ) -
∑ ∑ CO
MOV k,l (κk,l)
(74)
k∈E l∈E
where the term ∑s∈producCOmsfsrepresents the income for the products sale and ∑s∈rowmatCOmsfs corresponds to the expenses for the raw materials purchase. To represent the costs due to the process modifications the term r(x,z,y) is used. ∑i∈HU∑j∈CPS∑k∈ECOHU(Qi,j,k) and ∑i∈HPS∑j∈CU∑k∈ECOCU(Qi,j,k) correspond the hot and cold utility costs, respectively. The capital cost for additional area needed is given by COAA∑k∈EAAβk and the fixed cost for new heat exchanger units is CONE∑k∈NE(Vk). The fixed and variable costs for new piping
segments between exchangers k and l are represented by F h F c V h ∑k∈E∑l∈E(CONPH k,lzk,l + CONPCk,lzk,l) and ∑k∈E∑l∈ECONPk,l(fk,l + fck,l), respectively. ∑s∈HCS∑k∈ECOFNPEs,k (yeks ) and ∑s∈HCS∑k∈ECVNPEs,k represent the fixed and variable costs for the new piping segments for the streams that exit the HEN, respectively. To represent the fixed and variable costs for new piping segments for the streams at the inlet of the HEN the terms F k V k ∑s∈HCS∑k∈ECONPI s,k(yis ) and ∑s∈HCS∑k∈ECONPIs,k(fs ) are used. MOV Finally, the term ∑k∈E∑l∈ECOk,l (κk,l) is used to denote the cost to relocate a heat exchanger from one position to another. To determine the variable cost for the new piping segments from any exchanger at the exit of the network, the following disjunction is necessary,
[
Yesk s V V fk,outCONPEs,k - CNPE e0 s,k
][ ∨
¬Yesk V CNPEs,k ) 0
]
,
s ∈ HCS, k ∈ E
which can be formulated as a big-M constraint V V fsk,outCONPE - CNPE e M11(1 - yesk), s,k s,k
s ∈ HCS, k ∈ E (75)
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5523
Figure 11. Retrofitted HEN obtained for example 3. V CNPE e M11yesk, s,k
s ∈ HCS, k ∈ E
(76)
where M11 is an upper limit. Remarks 1. The MINLP model considers the plant layout and the piping structure explicitly. 2. The superstructure model allows complex piping configurations and does not require assumptions commonly used such as isothermal mixing, no bypass streams, and no splitting of streams. 3. The model formulation can consider the fixed and variable piping costs depending on the distance and the flow rate. In addition, the model formulation can consider the relocation of a given heat exchanger. 4. The model presented considers explicitly the interactions between the modifications in process conditions and the heat integration. 5. The model takes into account the existing units and the original conditions as well as the new units needed and the modifications of the operating conditions in the retrofit process. 6. The MINLP model includes the treatment of hot and cold process isothermal streams. 7. The MINLP model is nonconvex. Therefore, the optimal solution can only be guaranteed with global optimization methods.29 However, local methods like DICOPT30 can often find near optimal solutions that may be satisfactory for many practical applications. 4. Examples Example 1. This is a fairly simple problem that is used as a motivating example. Consider the process shown in Figure 4,
in which the feed must be heated until a given target temperature and then processed in a reactor with a final cooling process. The film heat transfer coefficients for all streams in this simplified example are assumed as 1 kW/(m2 °C). The heat capacity for components A and B are 4.17 kJ/(kg °C) and 5 kJ/(kg °C), respectively. The operation time of the plant is 8500 h/year. The hot and cold utilities cost are $110/kW-year and $10/kW-year, respectively. The prices for the components A and B (COmA and COmB) are $0.5987/kg and $0.6953/kg, respectively. The cost for new exchangers is given by $700A, where A is given in m2. The reaction is assumed to be exothermic with a heat of reaction equal to 818.94 kJ/kg. The reactor has a volume of 50 m3 and operates at a constant pressure of 10 kPa. The conversion of the reaction depends only on the inlet temperature to the reactor conversion ) 0.87 exp(-0.002Tinreac)
(77)
where Tinreac is in °C. The limits given in Table 1 are included to reflect the availability and demand for the materials, as well as limitations on the operation conditions. The superstructure shown in Figure 5 is used for the solution of this problem. It includes 3 heat exchangers, 2 of them are part of the original process and a new exchanger is considered as a potential unit for the optimal configuration. Table 2 shows the additional piping cost data for the retrofit problem. The additional process constraints for this example are the mass and energy balances in the reactor, which are related to the temperatures and flows of the process streams. Notice that for this simple problem the superstructure needed to account
5524 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
retrofit keeping the original process conditions unchanged, in which case the improvement is only 0.98%.
Table 9. Results Comparison for Example 3 concept 2
new area (m ) new units utility cost ($/year) capital exchangers cost ($/year) capital piping cost ($/year) total annual cost ($/year)
Yee and Grossmann12
Ma et al.13
this work
187.28 1 24667.00 56818.80 20600.00 102085.80
167.05 1 23300.00 51762.50 45900.00 120962.50
166.18 0 34277.79 41 544.00 23100 98921.79
for process modifications and energy integration gives rise to an MINLP problem that contains 145 constraints and 115 variables (with 29 binary variables). The solver DICOPT implemented in the General Algebraic Modeling Systems30 was used to solve the resulting MINLP model for this problem, requiring 0.53 s of CPU time. The retrofitted HEN is shown in Figure 6. The retrofitted configuration shows an improvement in the heat integration of the process without the need for additional area. Table 3 shows a comparison of the annual profit of the retrofitted process with respect to the original process. As can be seen, the simultaneous optimization and HEN retrofit yields a considerable improvement in the process profit (10.93% with respect to the original process). Table 3 also shows the results for the optimal HEN
Figure 12. Existing HEN for example 4.
Example 2. This example takes a retrofit problem from an extension of a process discussed by Duran and Grossmann22 (see Figure 7). The feed involves three chemical species A, B, and C, where C is an inert component. The feed mixture is treated with a two-stage compressor with intermediate cooling to raise its pressure, and then mixed with a recycle stream. The resulting stream is preheated with the reactor outlet stream and fed to the reactor where the components A and B react in an exothermic reaction to produce D. The effluent of the reactor is cooled and sent to a flash unit to recover the product D in the liquid stream. The product stream (liquid from the flash) is heated to deliver the product as saturated vapor. A fraction of the resulting vapor stream of the flash is purged to avoid the accumulation of inert C in the process. The purged stream is finally heated to a required temperature. The data for the process specifications are given in Table 4. The phase equilibrium in the flash is predicted with an ideal model, while isentropic compression corrected by efficiency factors is assumed for the compressors. For the process streams, heat capacities are assumed to be linear functions of composition.
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5525 Table 11. Piping Costs Information for Example 4
Table 10. Stream Data for Example 4 stream H1 H2 H3 HU C1 C2 C3 CU
TsIN (K)
TsOUT (K)
type
FCpsIN (kW/ K) or Fλs (kW)
480 420 390 627 330 391 349 303
320 420 390 627 490 391 349 315
HPS1 HPS2 HPS2 HU CPS1 CPS2 CPS2 CU
55 28090 11213 53 18320 12640 -
h (kW/ (m2 K)) 0.83 1.82 1.73 2.5 0.72 1.91 1.84 1.0
Table 5 shows the piping cost associated to the retrofit of the streams. A superstructure with five existing heat exchangers and three new units was used for this problem. The resulting MINLP model contained 1094 constraints with 852 variables, which included 296 binary variables. After solving this problem with GAMS/DICOPT software in 240.35 s of CPU time, we obtained the retrofit process shown in Figure 8. The new conditions in the retrofitted process are such that no hot utilities are required. In addition, new piping segments for the hot stream effluent to the reactor (H1 stream) are needed to obtain a better heat integration with the cold process streams C1, C3, and C2 in exchangers 5, 4, and 2, respectively, and a cold utility is used in exchanger 3 to provide the temperature needed at the inlet of the flash. None of the new exchangers formulated as part of the superstructure were selected. Only additional area in exchanger 1 of 14.95 m2 was needed. The total annual profit obtained from the simultaneous optimization of the process and the HEN retrofit is $101 568 482/year. Table 6 reports the results from the economics for the retrofitted process and for the original process. Notice how the retrofitted process provides a noticeable improvement in the annual profit. When the retrofit problem was formulated without process modifications, the solution shown in Figure 9 was obtained. A total annual profit of $84 556 338/year was obtained in this case. Notice in Figure 9 that if the simultaneous HEN retrofit-process conditions optimization is not conducted, a new heat exchanger and several piping modifications are needed to reduce the utility consumption. In addition, the simultaneous process and HEN retrofit optimization produces a better heat integration (there is in that case no need for hot utilities). The simultaneous retrofit optimization has a total profit 32.6% higher than the original process, which is significantly better that the improvement of 10.4% provided when only the HEN retrofit was considered. Example 3. This example was taken from Yee and Grossmann12 to show the application of the proposed methodology for HEN retrofit when the process conditions are fixed. The existing HEN is shown in Figure 10, with the stream data given in Table 7. The information for the costs for new piping segments is shown in Table 8. The hot and cold utility costs are $80/(kW-year) and $20/(kW-year), respectively. The fixed cost for new heat exchangers units is $10 000/year and the cost for additional heat transfer area is $250/(m2 year). The annual utility cost for the existing HEN is $158 000/year. To solve this problem, a superstructure with six heat exchangers was developed, five exchangers exist in the original HEN, and one new unit was considered. The model involved 578 constraints and 452 variables, with 180 binary variables and required 16.4 s of CPU time for its solution. Figure 11 shows the retrofitted network obtained using the proposed methodology. The few modifications needed for the existing process consist of piping modifications and additional area in exchangers 1, 3, and 4, which reduce the total annual cost to $98 921/year. Note that the new heat exchanger
F CONPH k,l
F CONPC k,l
k,l
value ($/year)
k,l
value ($/year)
1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,3 2,4 2,5 2,6 2,7 2,8 3,4 3,5 3,6 3,7 3,8 4,5 4,6 4,7 4,8 5,6 5,7 5,8 6,7 6,8 7,8
7400 8300 0 9200 3500 6700 6300 5200 5700 0 3950 5300 5400 4500 3900 5200 6900 5200 3600 5000 8700 4500 6500 7800 5600 4200 6200 5300
1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,3 2,4 2,5 2,6 2,7 2,8 3,4 3,5 3,6 3,7 3,8 4,5 4,6 4,7 4,8 5,6 5,7 5,8 6,7 6,8 7,8
7400 8300 4300 9200 0 6700 6300 5200 5700 3700 3950 5300 5400 4500 3900 5200 0 5200 3600 5000 8700 4500 6500 7800 5600 4200 6200 5300
F CONPI s,k
F CONPE s,k
s,k
value ($/year)
s,k
value ($/year)
1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 8,1 8,2 8,3 8,4 8,5 8,6 8,7 8,8
0 5700 5800 5800 6000 3400 4000 4200 3500 0 4700 5600 5700 3800 4300 4200 3700 3900 0 4200 5300 2400 4200 3300 3200 4300 3500 3700 3900 0 0 2900 0 3100 3050 3800 3600 3900 4500 3500 3700 0 2900 3800 3700 4200 4500 3900 4100 4050 0 3900 3800 4100 4300 3700 3500 3450 3400 0 0 4200 4300 5200
1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 8,1 8,2 8,3 8,4 8,5 8,6 8,7 8,8
4200 4100 3900 0 2800 5200 6000 5300 5200 5100 3800 2800 0 5000 5400 4800 5300 4800 0 3400 2900 5300 5500 4300 5600 5200 5100 4900 4800 0 0 3400 3700 4300 4500 5400 5400 0 2900 3600 4100 0 4400 4700 4800 3200 3100 3800 3900 4200 4100 5100 5300 2800 0 3100 4500 4400 4600 0 0 4300 4500 5500
considered in the superstructure was not needed. Table 9 shows a comparison of the results obtained in this work with the ones previously obtained by Yee and Grossmann12 and Ma et al.13 for the same stream data. The application of the proposed methodology yields a reduction of 37.4% in the total annual cost with respect to the original HEN, whereas the solutions of
5526 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 13. Retrofit HEN for example 4.
Yee and Grossmann12 and Ma et al.13 provide reductions by 35.4% and 23.4%, respectively. Notice that the piping modification costs turn out to be very significant for this problem because of the plant layout. It is important therefore to include such costs in model formulations for these types of problems. Example 4. This example considers the retrofit of a HEN with both isothermal and nonisothermal process streams. The process conditions are fixed in this case. The existing network shown in Figure 12 has a total annual cost of $529 800/year. Table 10 lists the stream data, and Table 11 gives the capital cost for new piping segments. The capital cost for new heat exchanger area is calculated by $380A/year (A in m2), and the hot and cold utility costs are $110/(kW year) and $20/(kW year), respectively. Notice that the process streams H1 and C1 exchange sensible heat, whereas H2, H3, C2 and C3 are isothermal process streams that exchange their latent heats. To solve this problem, a superstructure with seven existing heat exchangers and one potential new unit was formulated. The superstructure involves 1300 constraints and 872 variables, with 320 binary variables, and was solved with DICOPT in 104.2 s of CPU time. The retrofitted network obtained using the proposed MINLP model is shown in Figure 13. Minor piping modifications and additional exchanger areas are needed.
However, no new exchanger is required in the optimal solution provided by the model. The piping modifications require a capital cost of $30 700/year. The additional area is needed for units 4, 5, and 7 by 0.06, 13.88, and 73.716 m2, respectively, which requires an annualized capital cost of $33 310/year. The modifications in the retrofitted HEN reduce the utility consumption by 34.6%. This significant improvement was obtained because the formulation presented in this work includes a proper treatment for energy integration of both isothermal and nonisothermal types of process streams. Finally, the total annual cost of the retrofitted HEN is $410 530/year, which yields a reduction in the total annual cost of 22.5% with respect to the original HEN. Table 12 shows a summary of the costs for the original HEN and the retrofitted HEN. One may notice that the solution calls for what seems like an impractical low value of the additional area for exchanger 4. One could easily correct this situation by implementing a constraint to set this additional area to zero. Also, one could tailor the model implementation to some practical retrofit policy on this matter through the implementation of appropriate lower bounds for any additional area. It is worth mentioning that the utility consumption in the retrofitted HEN can be reduced even further if the heat load
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5527 CONPEs,k ) variable cost for piping segment for exchanger k to the exit of stream s depending upon the mass flow COMOV ) cost of moving one exchanger from position k to position k,l l CPS ) set containing the total cold process streams, CPS1∪CPS2 CPS1 ) set containing the cold process nonisothermal streams CPS2 ) set containing the cold process isothermal streams CS ) set containing the total cold streams, CPS∪CU CS1 ) set containing the cold process nonisothermal streams and the cold utilities, CPS1∪CU CU ) set containing the cold utilities DTCU ) prespecified temperature change for the cold utility DTHU ) prespecified temperature change for the hot utility E ) set containing any exchanger in the superstructure EE ) set containing the existing exchangers in the network EAk ) existing area of exchanger k EAMl ) existing area of exchanger l that can be relocated in the plant f ) mass flow rate FCpsIN ) total heat capacity flow rate for stream s FCpCU ) upper limit for the heat capacity flow rate for the cold utility FCpHU ) upper limit for the heat capacity flow rate for the hot utility fCpsk ) initial heat capacity flowrate of stream s to exchanger k fCpc,in ) inlet heat capacity flow rate for the cold side of exchanger k k c fCpl,k ) heat capacity flow rate for the cold side from exchanger l to exchanger k c fCpk,exit ) heat capacity flow rate for the cold side from exchanger k to the exit of the HEN fCpkh,in ) inlet heat capacity flow rate for the hot side to exchanger k h fCpl,k ) heat capacity flow rate for the hot side from exchanger l to exchanger k h fCpk,exit ) heat capacity flow rate for the hot side from exchanger k to the exit of the HEN g(x,z,y) ) set of inequality constraints for the process model h ) individual film heat transfer coefficient h(x,z,y) ) set of equality constraint for the process model HPS ) set containing the hot process streams, HPS1∪HPS2 HPS1 ) set containing the hot process nonisothermal streams HPS2 ) set containing the hot process isothermal streams HS ) set containing the total hot streams, HPS∪HU HS1 ) set containing the total nonisothermal streams plus utilities, HS1 ) HPS1∪HU HU ) set containing the hot utilities HCPS ) set containing the hot and cold process streams, HPS∪CPS HCT ) set containing the total streams, HS∪CS K ) number of exchangers in the superstructure LMTDk ) log-mean temperature difference for exchanger k M ) big-M parameter used for constraints NE ) set containing the new exchangers in the superstructure NP ) set containing the new piping segments NPE ) set to denote that stream s does not enter existing HEN at exchanger k Qi,j,k ) head load exchanged between streams i and j in exchanger k r(x,z,y) ) costs due to process modifications TsIN ) inlet temperature of stream s to HEN TsOUT ) outlet temperature of stream s from HEN tkc,in ) inlet temperature of cold stream to exchanger k tlc,out ) outlet temperature of cold stream from exchanger l tkh,in ) inlet temperature of hot stream to exchanger k V
Table 12. Results for Example 4 concept
original HEN
retrofitted HEN
new area (m2) new units utility cost ($/year) capital exchangers cost ($/year) capital piping cost ($/year) total annual cost ($/year)
0 0 529800 0 0 529800
87.66 0 346520 33310 30700 410530
exchanged between streams H1 and C1 in unit 1 is increased, but this utility reduction would require an additional area for exchanger 1 that would yield a higher total annual cost than the one reported here. It should be noted how the number of binary variables affected the computational effort for each problem. As the number of binary variables increases, the number of possible combinations grows exponentially, thus affecting the CPU time. 5. Conclusions This paper has presented an MINLP formulation for the retrofit of chemical processes considering simultaneously process modifications and heat integration. The model considers the plant layout and complex piping configurations. The superstructure used for the heat exchanger networks configuration is general and does not require imposing constraints such as no bypass or no splitting of streams. Also, the model formulation includes the treatment of isothermal process streams. For the economic assessment of alternatives, a simultaneous consideration is included for the capital cost of the new exchanger units, the additional area required, and the new piping segments, as well as the operating cost for hot and cold utilities. Fixed and variable piping costs can be considered in the model formulation, as well as the relocation of heat exchangers within the process. The examples presented here show that significant earnings can be obtained in the retrofit process when the process modifications and the heat integration retrofit are considered simultaneously, as opposed to the solution given by the consideration of heat integration restricted with unchanged process conditions. Acknowledgment This work was performed while A. Jime´nez was a FulbrightGarcı´a Robles Scholar at Carnegie Mellon University. Nomenclature AAk ) additional area required by exchanger k AEAk ) existing area assigned to existing exchanger k V CNPE s,k ) variable cost for new piping segment from exchanger k to the exit of stream s COms ) price of material s COAA ) unit cost for new area F CONPI s,k ) fixed cost for new initial piping segment of stream s to exchanger k V CONPI s,k ) variable cost for new initial piping segment of stream s to exchanger k CONE ) fixed cost for new heat exchangers F CONP k,l ) fixed cost for new piping segment between exchangers k and l COVNPk,l ) variable cost for new piping segment between exchangers k and l depending upon the mass flow F CONPE s,k ) fixed cost for piping segment for exchanger k to the exit of stream s
5528 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 tlh,out ) outlet temperature of hot stream from exchanger l Vk ) binary variable to denote the existence of a new heat exchanger unit in the retrofitted network wks ) binary variable to denote that stream s is assigned to exchanger k x ) vector of continuous variables in the HEN model xsk,l ) binary variable to denote that exchangers k and l service the same process stream s y ) vector of binary variables in the model for structural modifications in the flowsheet yisk ) binary variable to denote that inlet stream s is assigned to exchanger k yeks ) binary variable to denote the exit of stream s from exchanger k z ) vector of continuous variables of the process that are not included in the HEN model zk,l ) binary variable to denote the piping segment connecting exchangers k and l zek ) binary variable to denote the existence of piping segment from exchanger k to exit of HEN Greek Symbols Fλicond ) condensation heat load for stream i Fλjevap ) evaporation heat load for stream j ∆TMIN ) minimum temperature difference κk,l ) binary variable to assign the existing area of exchanger l to the location of exchanger k µsk ) binary variable to denote that isothermal stream s is assigned to exchanger k Indices i ) hot stream j ) cold stream k ) exchanger in the superstructure l ) exchanger in the superstructure Subscripts and Superscripts c ) cold side h ) hot side s ) any stream exit ) network exit point IN ) inlet to the HEN in ) inlet to an exchanger LO ) lower limit OUT ) outlet from the HEN out ) outlet from an exchanger UP ) upper limit
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ReceiVed for reView August 31, 2007 ReVised manuscript receiVed February 28, 2008 Accepted March 12, 2008 IE071182+
