- Email: [email protected]
Automated design method for heat exchanger network using block decomposition and heuristic rules
Computers chem. Engng Vol. 21, No. 10, pp. 1095-1104, 1997
Pergamon
Copyright © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0098-1354(96)00320-1 0098-1354/97 $17.00+0.00
Automated design method for heat exchanger network using block decomposition and heuristic rules Xin X. Zhu Department of Process Integration, University of Manchester Institute of Science and Technology, PO Box 88, Manchester M60 IQD, U.K. Abstract
The research in HEN synthesis has achieved significant progress over the last two decades, particularly through the discovery of the pinch concept and the research in targeting energy and capital. Normally, the HEN synthesis is carried out with two stages, targeting and design. In the past, many researchers mainly focused on developing manual design methods. Having realised the significance of saving engineering time and the possibility of using mathematical optimisation techniques to enhance the search for good designs, much research has been carried out to develop automated HEN design methods and significant progress has been achieved. However, there is still no design methods available which can fulfil the task of automated synthesis for practical applications. In the present paper, a simple method for automated synthesis of HEN is presented, which is based on the block concept proposed by Zhu et al. (1995a). The basic idea is to simplify a design problem by decomposing it into a number of blocks. In each block, piece-wise stream composites present similar profiles and hot streams are in energy balance with cold streams. After the block decomposition, design is carried out using area targeting principles and a number of newly developed heuristic rules. An MILP model and a simple MINLP model are used for selecting a best set of matches and determining the optimal split ratios, respectively. The initial designs produced by such a method can approach the energy and area targets closely coupling with a small number of units. Such good initial solutions are then provided for subsequent cost optimisation. The main advantage of this design strategy is that designs with good quality can be achieved efficiently and effectively. In addition, the block concept can be easily extended to handle problems with different film coefficients by combining the current method with the diverse pinch approach proposed by Rev and Fonyo (1991) [(Chem. Eng. Sci. 46, 1623-1634 (1991)] and thus the above strategy for HEN synthesis can be applied for such problems. © 1997 Elsevier Science Ltd
1. Introduction
Current versions of the Pinch Design Method (PDM) for HEN synthesis (Linnhoff and Ahmad, 1990) yield improved solutions by utilising the 'Driving Force Plot' (DFP) and 'Remaining Problem Analysis' (RPA). With these procedures, the designer provides significant input into the choice of matches and hence the solution process. However, the PDM is sequential in nature which can not consider match interaction and it only considers temperatures when selecting matches. As a consequence, it is not always clear which of a number of candidate matches should be chosen in terms of overall design, although some will be critical in the determination of the eventual cost of the completed network. The objective of this research is to develop a method which can solve problems associated with the current PDM and more importantly, to propose an algorithm for automated HEN synthesis. The most important part of these objectives is achieved by introducing a new concept, the block (Zhu et al., 1995a). The fundamental
idea of the block concept is to decompose a design problem into a number of blocks which feature significantly different characteristics for design requirement. This concept is derived from analysis of the Bath formula (Townsend and Linnhoff, 1984) which provides the physical insights: the minimum overall surface area can be achieved by transferring heat vertically based on the small enthalpy intervals and by following the composite profiles. A design following such an approach achieves the Bath area target but usually has a 'spaghetti structure'. However, if the condition with small enthalpy intervals can be relaxed to some extent so that a number of such intervals can be combined together to form a larger interval, namely the block. Then a design problem can be decomposed into a small number of blocks which can be used as the basis for design. It has been shown by Zhu et al. (1995a) that a design based on such a decomposition approach incurs a small penalty in area but gains a great benefit in network simplification. As a result, the Bath area target can be approached closely with a simple structure.
1095
1096
XIN X. ZHu
The block decomposition is determined based on the geometrical properties of the composite curves so that the small enthalpy intervals having similar profiles may be combined together to form a block. In each block, two straight lines, namely quasi-composite curves, can be used to approximate the actual composite curves, which themselves are piece-wise representation of a real problem (Fig. 1). The quasi-composite curves generated are used as the reference for match selection. After the block decomposition, the match and split choice are carried out based on the overall best fit to the quasicomposites while network simplification is also taken into account. When a problem has significantly different film coefficients, the diverse pinch approach (Rev and Fonyo, 1991) can be applied to exploit the difference in film coefficients by considering their individual ATcontributions. Thus the block decomposition can be applied to the modified composite curves and a design problem can be solved in the similar way as explained above.
2. The new method
2.1. The procedure
The new method can be explained by using a schematic diagram in Fig. 2. The procedure is similar to that of Zhu (1994). First, the optimal HRAT (ATtain) is
T
identified by using the Supertargeting approach (Linnhoff and Ahmad, 1986). Next, the corresponding composite curves are decomposed into a number of blocks and the true composite curves are approximated by the quasicomposite curves using the least square fit method. Then, the design is carried out one block by one and the match selection is determined by using the newly defined heuristic rules which are expressed in mathematical terms. Initial designs proposed will be provided for subsequent cost optimisation using mathematical programming techniques. 2.2. Heuristic rules for match selection
Following the philosophy of the PDM, the block method also considers how to follow the composite curves on the basis of overall best fit when determining matches. However, stepping further than the PDM, the heuristic proposed below considers the influence from both temperatures and heat loads. When a problem has substantially different film coefficients, the difference in film coefficients is exploited at the targeting stage by applying the diverse pinch approach by Rev and Fonyo (1991) and then the effects from both temperatures and heat loads are accounted at the design stage using following t~ equations. To calculate the fitness of a single match to the quasicomposites, the ~ function can be derived as
Blocks
1 /-!i
, i
,d
•
** I #
~
,o
~
~
i
ill
°,°o"
~
e o""
otlt
°
',
•
•
mlt
i I
0
~Q 0
Fig. 1. Quasi-composite curves and blocks.
Actual CompositeCurves "Quasi-Composites"
Automated design method for heat exchanger network
1097
Supertargeting
]optimal Composites{HBATo.,)I I BlockDecomposition I I Design One Block by One I I
Initial Design
I
Optimisation
Fig. 2. The design procedure.
8~ =
1
-
qij_2k Q~
1
1
cPi
cPj
approach. In detail, a set of matches within a block is determined simultaneously to minimise an objective function defined as
1
1
CP{Hot - qua~i)k
CP
minimise ~ 8~j
when
CP,n,,- qua~iIk =1=CP(col d_ quasi)k and
¢5ij-qijk CP(lt"t-q"~s°k l - CPI k-Qk CPi when
CP~,o,-quasi)k = CP~c,,ta-q.asi~k
( 1)
Where
CP~ = the heat capacity flowrate of hot stream i CPj = the heat capacity flowrate of cold streamj q0k = the heat load between hot stream i and cold stream j in kth block Qk = enthalpy change of kth block CP,uo,_q,,,~)k = the heat capacity flowrate of the hot quasi-composites in kth block CP~cold_quasi)k = the heat capacity flowrate of the cold quasi-composites in kth block.
2.3. Simultaneous match selection approach To take match interaction into account, the match choice by this method is done using a simultaneous
(2)
This minimisation problem can be formulated as a mixed integer linear program (MILP). The general problem statement can be written as following. For simplicity, first consider the case of hot and cold streams with an equal number (n) in a block. Let 8ij be defined as the fitness to the quasi-composites by a match between hot stream i and cold stream j in this block. Let binary variables x~j indicate the existence of a match between hot stream i and cold stream j, i.e. xzj = 1 declares the existence of a match and xlj = 0 non-existence of a match. Thus, the equation (2) can be converted into the following MILP problem as
minimise Lj= ~ I 8ij'xlj i=1
j=t
xij=l j = l ..... n x~j=l i=1 ..... n
x~j=0, 1 i,j= 1..... n
(3)
It should be noted that the match selection method by the MILP model (3) implies the tick-off rule (Linnhoff and Hindmarsh, 1983) when selecting a match. Thus, this method considers both minimisation of both overall
1098
XIN
surface area using the 8 function and the number of units. The solution from the MILP model (3) always gives the best set of matches rather than a single best match. If the total number of hot streams is not equal to that of cold streams, dummy elements with zero assignment are added to make a matrix square and then the MILP model (3) can be applied. It may be worth to consider energy import or export for a block aiming for network simplification. The detail discussions on above issues are given by Zhu (1994).
2.4. The strategy for stream splitting After a set of matches is selected by employing the MILP model (3), stream splitting may be required to handle a remaining problem. The method proposed for solving splitting problem is also based on the simultaneous approach and considers the best overall fit to the quasi-composites by splitting matches. From practical consideration, the split branches for a stream are restricted to two in this work. It has been proved (Zhu, 1994) that for a problem with two hot streams (i and k) to match one cold stream (j), the optimal split ratio can be determined as
CPi CPk CP) - CP~ CPj= CP) + CP~
(4)
The fitness of profiles of parallel splits to the quasicomposites can then be defined as
du
CPi
CP(not-quasi) I CP(coid_q.asi)
CP(H~t-q'asil I
I CPk do= CP]
(6)
CPicold_q,,,,i )
Note the equation (4) can only be used for determining the optimal split ratio for a problem with two streams against one stream of the other type. For a situation more complicated than this, we need to determine for all streams in a block that which stream is needed to be split with what ratio to match what two streams of the other type. Again, the objective is to achieve the overall best fit to the quasi-composites by splits. To give a clear illustration, consider a problem containing four hot streams with relatively small capacity flowrates and two cold streams with relatively large capacity flowrates (Fig. 3). For this problem, the optimal placement of matches and split ratios are not obvious. To achieve an optimal solution, we need to consider and compare as many possibilities as possible. First, consider all the possibilities for splitting cold stream 1 to match two of hot streams and calculate associated d values. For instance, consider hot streams 1 and 2 to match cold stream 1. Applying equation (4), the splits of cold stream 1 can be calculated as
CPnl CPm. CP~ = - 1+ CPm CPm CPcI
130
25 ~k3
=72.2 (kW/K)
-
1+ 25 20
2 1 CP ci = CPcl - CP cl = 57.8 (kW/K)
I
CP)
X. Z H u
(5)
I
CPm CPn, or " is 0.35. The ratio CP et
Thus, the match ratio ~ T
of the composite curves is calculated as 0.54. Thus the d
or
224
Block i
CP (kW/K)
140
E}
t,
25
~.
20
~,
45 55
•
Xi Ii
130
i
1i
14o I I
180
135
Fig. 3. Determining splits placement and split ratio using area targeting principle.
Automated design method for heat exchanger network value is 0.19 (9.54-0.35). If for a sub problem with only hot streams 1 and 2 against cold stream 1, this ratio is optimal. However, there are four hot streams in this case and we need to investigate all the possibilities to select the most suitable two hot streams to match with cold stream 1. After similar calculations for all possible splits for cold stream 1, it is found that cold stream 1 should be split with ratio of 46.4 and 83.6 to match hot streams 1
minimise ij,kd Y, dijk~xrk~
CPik dokt= CPjl
224
[]
E x,jkl= 1 ; jE xijk~=1 xi#t=0,1 i=l..n;
j=l..m;
K=I..N;
I=I..M;
(7)
Where CPi, and CPjl are defined as continuous variables and x~jk~defined as binary variables which are optimised in the optimisation process. Using the MINLP model for the above split problem by restricting split branches of a stream to two, the solution is that the cold stream 1 should be split to match hot streams 1 and 3, and that cold stream 2 is split to match hot streams 2 and 4. Both these parallel splits exactly equal to the ratio of the composites and mimic the quasi-composite curves perfectly (Fig. 4). The above strategy can be applied in a similar manner to solve splitting problems with any number of split branches allowed.
3. Examples
3.1. Example I To illustrate the principles and procedures of the proposed design method, a simple example is used. Consider the following four stream problem (Table 1). The stream data and cost data were introduced by Linnhoff and Ahmad (1990) and Ahmad et al. (1990), respectively.
140
Block i
:i
CP(Hot-q.asi) CP(cold-q,,a,,i )
CPik=ePi ; ~ CPfl=CPj
CPm CPH3 and 3, respectively. Thus the match ratio ~ or 2 CPc~ CP~j in such a splitting scheme is 0.54 which equals the ratio of the composites, Hence, the d value is 0.0. This d value is the minimum in all possible splits for cold stream 1, which indicates the cold stream 1 should be split to match against hot streams 1 and 3. If we consider cold stream 2, the similar procedure can be applied to determine which two hot streams should be matched against this cold stream with what split ratio. So far, cold streams 1 and 2 are considered independently. In other words, the match interaction between these two cold streams is not addressed. However, to determine an overall best fit to the composite curves when selecting matches involving stream splitting, a simultaneous approach should be considered. The general stream split problem can be stated as following. Assume there are n hot and m cold streams waiting for matches involving splitting. For generality, any number of splitting branches are allowed. Let d~#~indicate the fitness to the quasi-composites by a match between branch k of hot stream i and branch I of cold stream j in this block. Let binary variables xok~ indicate the existence of a match (ijkl), i.e. xi#t=l declares the existence of the match and xokt=0 nonexistence of the match. Thus, a simple MINLP model can be formulated to simultaneously determine both the set of matches and optimal split ratios:
1099
CP (kW/K) 25
[]
20
[]
II
45
i i
55
V
J
f i
/
I
2i i i I
13o
i
\
180 Fig. 4. Solution to the splitting problem shown in Fig. 3.
I
,
__/
, i i i
1~,5
14o
1100
XIN X . ZHU
Table 1. Steam and cost data for example 1 Heat Capacity flow~te kW/K 200 100 300 500
Steam I 2 3 4
Supply temperatu~ °C 150 170 50 80
Target temperature °C 50 40 120 110
(Zhu et al., 1995a), the number of blocks for this problem is determined as two (Fig. 5). Above the pinch, the quasi-composites are calculated as: Hot quasi-composites: T[°C] =90+0.0037AH [kW] Cold quasi-composites: T[°C]=80+0.00125AH [kW]
A H = H - 13,000; H [kW] el13,000 ... 33,000] Utility data Hot utility: Saturated steam: 180°C Cold utility: Supply temperature: 20°C; Target temperature: 40°C U = 100 W/m-'K for all streams
Cost data Cost of Hot utility: 110 $/kWyr; Cost of Cold Utility: 10 $/kW yr Installed heat exchanger cost: cost ($) =30,800+750 area 0.sl Plant lifetime =6 years; Rate of interest = 10% per annum First, use the supertargeting approach to find the optimal HRAT and then determine the block decomposition. The optimal HRAT is identified at 10 K and the corresponding composite curves are given in Fig. 5. According to the principles for block decomposition
(8)
where the enthalpy interval of this block is from 13,000 to 33,000 kW and H represents any enthalpy in this interval. The reciprocal of the slopes of equation (8) yields CPuot-quasi = 270 kW/°C and CPcot,t-quasi = 800 kW/ °C, respectively. Then calculate 6 values for all possible matches in this block and list results in a matrix. The results are summarised in the following matrix:
~,j
H,
H~
CI C2
0.74 0.27
0.02 0.31
The sum of 8 for the matches H1~22 and H2-C1 is 0.29 which is much less than 1.05, the sum of 8 for matches H1-C1 and H2~C2. Clearly, the matches H1C2 and H2-C 1 should be selected. For the block below the pinch, there are two hot streams (HI,H2) against one cold stream (C1). To fully utilise the available driving forces, it is necessary to split
170
T °C
,, I
170
150
130
110
90 I i
105
90 ,60
70
i i
50
80
•
"5O i i
30
I 0
I 5
I 10
I 15
Fig. 5. Composite curves----example I (HRAT=I0).
t 20
I 25
I 30
i 35
I 40
~ H - MW
Automated design method for heat exchanger network the cold stream to match with the two hot streams. The split ratio is determined using equation (4). The initial design is summarised in Fig. 6. The performance of details of the design is given in Table 2 to compare with the targets.
1101
K ATe= ~}
All stream temperatures are modified by the equation (9) (hot streams reduce their temperatures by minus the corresponding/iTs and vice versa cold streams increase their temperatures by plus the corresponding ATs. Then the composite curves are plotted based on these modified temperatures. By using the modified supertargeting approach (Rev and Fonyo, 1991; Zhu et al., 1995b) which incorporates the diverse pinch method, the optimal x and z are identified as 0.367 and 1.2, respectively for the problem of Table 3. The modified composite curves for the optimal K and z are given in Fig. 7. A two-block decomposition is then determined and the corresponding stream structure is shown in Fig. 8.
3.2. Example 2 To illustrate how the block method can handle problems with different film coefficients when combining the diverse pinch approach (Rev and Fonyo, 1991), use an example stream data by Rev and Fonyo who modified the original data of Ahmad et al. (1990) which is given in Table 3. In the diverse pinch approach, HRAT or AT,,i, is no longer used instead K and z are used as two optimisation variables (Rev and Fonyo, 1991; Zhu et al., 1995b). The individual/iT-contributions can be expressed as
CP (kW/K)
90
)
( )
IT] 15°
(9)
60
.-so
200
20O0
~] 170
C
120 ~
110 ~
100
107
~/~ 3OO0
104
so I-~
300
80 r ~
500
~0 12000
,80
O. = 7000 kW Area = 19690 m 2
Total cost = $1.60 x 106 yr "1
Fig. 6. Initial design----example 1.
Table 2. Designperformancefor example 1
Targets HRAT = 10K Initial Design Fig. 6
Hot utility
Area m "~ abovepinch
Area m2 belowpinch
Units
Total Cost M$/yr
7000
8852
10785
5
1.55
7000
8880
10810
8
1.60
Table 3. Streamand cost data for example2 Stream
Supplytemp. °C
Targettemp. °C
CP kW/K
h W/m2K
HI H2 H3 CI C2 Steam CW
159 267 343 26 I 18 300 20
77 80 90 127 265 300 60
2.285 0.204 0.538 0,933 1.961
100 40 500 10 500 50 200
Cost Data. Cost of hot utility: 110 $/kWyr;Cost of cold utility: 10 $/kWyr. Exchangercost modeh cost ($)=3,800+750 area 0.s3 Plant lifetime:6 years; Interestrate: 10% per annum.
1102
XIN X. ZHU
350 300 250 °~
200 150 100 50
i
0
I
I
I
I
t
I
r
1 O0
I
I
I
i
200
i
I
I
I
I
I
300
I
. . . .
400
I
. . . .
500
I
600
H (kW) Fig. 7. The modified Composite Curves for example 2 when K=0.367, z=l.2 (the utility consumption corresponds to HRAT = 36K). After the block decomposition for the modified composite curves, the methods for determining matches and splits discussed above can be applied and the initial design is produced given in Fig. 9 and the final design (Fig. 10) is achieved using the cost optimisation. The detail results for designs are listed in Table 4. As can be seen from the table, The initial design using the proposed method requires small overall area with a reasonably small number of units and the final design achieves the unit target with a 5.9% lower total cost than the cost target. It should be noted that the import/export rule is employed for the initial design (Fig. 9) and that is the
reason why the utility requirement in this design is relaxed in this design compared with the utility target.
4. Conclusions A simple but efficient algorithm for automated synthesis of HEN is proposed. The fundamental concept is the block which provides an efficient way to approach the area and energy targets simultaneously with a simple network. The significance of the concept is the simplification of a design problem which is achieved by decomposing it into a number of blocks with different
Block 2
Block 1 342.2
153.2
249.5
159
120.1
89.2 71.2 62.5
125.9 =
77 p,
,
i i b
'~
267
170.7
137.61
154
120.91
6o
= = i
90
= i i i i
127q .:102"9 6
:194.2
61 152.4
26 l~] i
118[~ z i i h
264.2 219.2 195.1
153.2
118.8 118.2
Fig. 8. Stream Structure for K=0.367 and z= 1.2 for example 2 (the utility consumption corresponds to HRAT=36K).
Automated design method for heat exchanger network
(,)
r~] 159
~-77
170.7
80
(,,
~154
I~1 ~
©
134.4
131.3
)
E~] 267
1103
©
120(
:- 90
16.1
(D
127q
19.7
/
~56.1
18.5
127.3 168.3
101.7
QH = 168.3 kW;
26
18.3 Area = 241 m2; Total cost = 49049 S/yr
Fig. 9. Initial design at K=0.367 and z= 1.2 for example 2.
134.5 @
r•159
131.3
(,,)
E~] 267
--77 80
121.6
©
90
17.0
116.5 (
127 q
%o 150.4 ( ) 38.2
265
169.2
26
/ 118 ~-~
119.1
QH --- 169.2 kW;
Network area = 242 mZ; Total cost = 46551 S/yr
Fig. 10. Optimized design at x=0.367 and z= 1,2 for example 2.
Table 4. Design results for example 2
Targets K=0.367, z= 1.2 Initial design (Fig. 9) Final design (Fig. 10)
Hot Utility kW
Area m 2
Units
Total Cost $/yr
161.2 168.3 169.2
278 241 242
6 8 6
49292 49049 46551
1104
XIN X. ZHu
characteristics of design requirement. Further more, the quasi-composite curves are used to replace the true composite curves to provide the reference for match selection in a design process. The qualitative analysis of driving forces is done at the stage of block decomposition while the quantity analysis of area requirement is performed by applying mathematical equations which embed the physical insights for achieving a minimum area design. More importantly, the effects from film coefficients, temperatures and heat loads are taken into accounted in this method in a systematic manner. By means of the t5 function for match selection and the d function for splits associated with the simultaneous approaches, the minimum area design can be achieved efficiently and effectively. In addition, the method is readily implemented as a software package for automated synthesis of HEN.
References
Ahmad, S., Linnhoff, B. and Smith, R. (1990) Cost optimum heat exchanger networks--2. Targets and design for detailed capital cost models. Comput. chem Engng. 14, 751-767. Linnhoff, B. and Hindmarsh, E. (1983) The pinch design
method for heat exchanger networks. Chem. Engng. Sci. 38, 745-763. Linhoff, B. and Ahmad, S. (1986) SUPERTARGETING, or the optimisation of heat exchanger networks prior to design. World Congress II, Chemical Engineering, Tokyo. Linnhoff, B. and Ahmad, S. (1990) Cost optimum exchanger networks--l. Minimum energy and capital using simple methods for capital cost. Comput. chem. Engng. 14, 729-750. Rev, E. and Fonyo, Z. (1991) Diverse pinch concept for heat exchange network synthesis: the case of different heat transfer conditions. Chem. Engng. Sci. 46, 1623-1634. Townsend D.W. and B. Linnhoff (1984) Surface area targets for heat exchanger networks. IChemE Annual Res. Mtg, Bath, U.K. Zhu Xin X (1994) Strategies for Optimisation in Heat Exchanger Network Design. PhD Thesis, The University of Adelaide, Australia. Zhu Xin X., B.K. O'Neill, J.R. Roach and R.M. Wood (1995a) A new method for heat exchanger network synthesis using area targeting procedures. Comput. chem. Engng. 19, 197-222. Zhu Xin X., B.K. O'Neill, J.R. Roach and R.M. Wood (1995b) Area-targeting methods for the direct synthesis of variable film coefficient heat exchanger networks. Comput. chem. Engng. 19, 223-239.
Pergamon
Copyright © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0098-1354(96)00320-1 0098-1354/97 $17.00+0.00
Automated design method for heat exchanger network using block decomposition and heuristic rules Xin X. Zhu Department of Process Integration, University of Manchester Institute of Science and Technology, PO Box 88, Manchester M60 IQD, U.K. Abstract
The research in HEN synthesis has achieved significant progress over the last two decades, particularly through the discovery of the pinch concept and the research in targeting energy and capital. Normally, the HEN synthesis is carried out with two stages, targeting and design. In the past, many researchers mainly focused on developing manual design methods. Having realised the significance of saving engineering time and the possibility of using mathematical optimisation techniques to enhance the search for good designs, much research has been carried out to develop automated HEN design methods and significant progress has been achieved. However, there is still no design methods available which can fulfil the task of automated synthesis for practical applications. In the present paper, a simple method for automated synthesis of HEN is presented, which is based on the block concept proposed by Zhu et al. (1995a). The basic idea is to simplify a design problem by decomposing it into a number of blocks. In each block, piece-wise stream composites present similar profiles and hot streams are in energy balance with cold streams. After the block decomposition, design is carried out using area targeting principles and a number of newly developed heuristic rules. An MILP model and a simple MINLP model are used for selecting a best set of matches and determining the optimal split ratios, respectively. The initial designs produced by such a method can approach the energy and area targets closely coupling with a small number of units. Such good initial solutions are then provided for subsequent cost optimisation. The main advantage of this design strategy is that designs with good quality can be achieved efficiently and effectively. In addition, the block concept can be easily extended to handle problems with different film coefficients by combining the current method with the diverse pinch approach proposed by Rev and Fonyo (1991) [(Chem. Eng. Sci. 46, 1623-1634 (1991)] and thus the above strategy for HEN synthesis can be applied for such problems. © 1997 Elsevier Science Ltd
1. Introduction
Current versions of the Pinch Design Method (PDM) for HEN synthesis (Linnhoff and Ahmad, 1990) yield improved solutions by utilising the 'Driving Force Plot' (DFP) and 'Remaining Problem Analysis' (RPA). With these procedures, the designer provides significant input into the choice of matches and hence the solution process. However, the PDM is sequential in nature which can not consider match interaction and it only considers temperatures when selecting matches. As a consequence, it is not always clear which of a number of candidate matches should be chosen in terms of overall design, although some will be critical in the determination of the eventual cost of the completed network. The objective of this research is to develop a method which can solve problems associated with the current PDM and more importantly, to propose an algorithm for automated HEN synthesis. The most important part of these objectives is achieved by introducing a new concept, the block (Zhu et al., 1995a). The fundamental
idea of the block concept is to decompose a design problem into a number of blocks which feature significantly different characteristics for design requirement. This concept is derived from analysis of the Bath formula (Townsend and Linnhoff, 1984) which provides the physical insights: the minimum overall surface area can be achieved by transferring heat vertically based on the small enthalpy intervals and by following the composite profiles. A design following such an approach achieves the Bath area target but usually has a 'spaghetti structure'. However, if the condition with small enthalpy intervals can be relaxed to some extent so that a number of such intervals can be combined together to form a larger interval, namely the block. Then a design problem can be decomposed into a small number of blocks which can be used as the basis for design. It has been shown by Zhu et al. (1995a) that a design based on such a decomposition approach incurs a small penalty in area but gains a great benefit in network simplification. As a result, the Bath area target can be approached closely with a simple structure.
1095
1096
XIN X. ZHu
The block decomposition is determined based on the geometrical properties of the composite curves so that the small enthalpy intervals having similar profiles may be combined together to form a block. In each block, two straight lines, namely quasi-composite curves, can be used to approximate the actual composite curves, which themselves are piece-wise representation of a real problem (Fig. 1). The quasi-composite curves generated are used as the reference for match selection. After the block decomposition, the match and split choice are carried out based on the overall best fit to the quasicomposites while network simplification is also taken into account. When a problem has significantly different film coefficients, the diverse pinch approach (Rev and Fonyo, 1991) can be applied to exploit the difference in film coefficients by considering their individual ATcontributions. Thus the block decomposition can be applied to the modified composite curves and a design problem can be solved in the similar way as explained above.
2. The new method
2.1. The procedure
The new method can be explained by using a schematic diagram in Fig. 2. The procedure is similar to that of Zhu (1994). First, the optimal HRAT (ATtain) is
T
identified by using the Supertargeting approach (Linnhoff and Ahmad, 1986). Next, the corresponding composite curves are decomposed into a number of blocks and the true composite curves are approximated by the quasicomposite curves using the least square fit method. Then, the design is carried out one block by one and the match selection is determined by using the newly defined heuristic rules which are expressed in mathematical terms. Initial designs proposed will be provided for subsequent cost optimisation using mathematical programming techniques. 2.2. Heuristic rules for match selection
Following the philosophy of the PDM, the block method also considers how to follow the composite curves on the basis of overall best fit when determining matches. However, stepping further than the PDM, the heuristic proposed below considers the influence from both temperatures and heat loads. When a problem has substantially different film coefficients, the difference in film coefficients is exploited at the targeting stage by applying the diverse pinch approach by Rev and Fonyo (1991) and then the effects from both temperatures and heat loads are accounted at the design stage using following t~ equations. To calculate the fitness of a single match to the quasicomposites, the ~ function can be derived as
Blocks
1 /-!i
, i
,d
•
** I #
~
,o
~
~
i
ill
°,°o"
~
e o""
otlt
°
',
•
•
mlt
i I
0
~Q 0
Fig. 1. Quasi-composite curves and blocks.
Actual CompositeCurves "Quasi-Composites"
Automated design method for heat exchanger network
1097
Supertargeting
]optimal Composites{HBATo.,)I I BlockDecomposition I I Design One Block by One I I
Initial Design
I
Optimisation
Fig. 2. The design procedure.
8~ =
1
-
qij_2k Q~
1
1
cPi
cPj
approach. In detail, a set of matches within a block is determined simultaneously to minimise an objective function defined as
1
1
CP{Hot - qua~i)k
CP
minimise ~ 8~j
when
CP,n,,- qua~iIk =1=CP(col d_ quasi)k and
¢5ij-qijk CP(lt"t-q"~s°k l - CPI k-Qk CPi when
CP~,o,-quasi)k = CP~c,,ta-q.asi~k
( 1)
Where
CP~ = the heat capacity flowrate of hot stream i CPj = the heat capacity flowrate of cold streamj q0k = the heat load between hot stream i and cold stream j in kth block Qk = enthalpy change of kth block CP,uo,_q,,,~)k = the heat capacity flowrate of the hot quasi-composites in kth block CP~cold_quasi)k = the heat capacity flowrate of the cold quasi-composites in kth block.
2.3. Simultaneous match selection approach To take match interaction into account, the match choice by this method is done using a simultaneous
(2)
This minimisation problem can be formulated as a mixed integer linear program (MILP). The general problem statement can be written as following. For simplicity, first consider the case of hot and cold streams with an equal number (n) in a block. Let 8ij be defined as the fitness to the quasi-composites by a match between hot stream i and cold stream j in this block. Let binary variables x~j indicate the existence of a match between hot stream i and cold stream j, i.e. xzj = 1 declares the existence of a match and xlj = 0 non-existence of a match. Thus, the equation (2) can be converted into the following MILP problem as
minimise Lj= ~ I 8ij'xlj i=1
j=t
xij=l j = l ..... n x~j=l i=1 ..... n
x~j=0, 1 i,j= 1..... n
(3)
It should be noted that the match selection method by the MILP model (3) implies the tick-off rule (Linnhoff and Hindmarsh, 1983) when selecting a match. Thus, this method considers both minimisation of both overall
1098
XIN
surface area using the 8 function and the number of units. The solution from the MILP model (3) always gives the best set of matches rather than a single best match. If the total number of hot streams is not equal to that of cold streams, dummy elements with zero assignment are added to make a matrix square and then the MILP model (3) can be applied. It may be worth to consider energy import or export for a block aiming for network simplification. The detail discussions on above issues are given by Zhu (1994).
2.4. The strategy for stream splitting After a set of matches is selected by employing the MILP model (3), stream splitting may be required to handle a remaining problem. The method proposed for solving splitting problem is also based on the simultaneous approach and considers the best overall fit to the quasi-composites by splitting matches. From practical consideration, the split branches for a stream are restricted to two in this work. It has been proved (Zhu, 1994) that for a problem with two hot streams (i and k) to match one cold stream (j), the optimal split ratio can be determined as
CPi CPk CP) - CP~ CPj= CP) + CP~
(4)
The fitness of profiles of parallel splits to the quasicomposites can then be defined as
du
CPi
CP(not-quasi) I CP(coid_q.asi)
CP(H~t-q'asil I
I CPk do= CP]
(6)
CPicold_q,,,,i )
Note the equation (4) can only be used for determining the optimal split ratio for a problem with two streams against one stream of the other type. For a situation more complicated than this, we need to determine for all streams in a block that which stream is needed to be split with what ratio to match what two streams of the other type. Again, the objective is to achieve the overall best fit to the quasi-composites by splits. To give a clear illustration, consider a problem containing four hot streams with relatively small capacity flowrates and two cold streams with relatively large capacity flowrates (Fig. 3). For this problem, the optimal placement of matches and split ratios are not obvious. To achieve an optimal solution, we need to consider and compare as many possibilities as possible. First, consider all the possibilities for splitting cold stream 1 to match two of hot streams and calculate associated d values. For instance, consider hot streams 1 and 2 to match cold stream 1. Applying equation (4), the splits of cold stream 1 can be calculated as
CPnl CPm. CP~ = - 1+ CPm CPm CPcI
130
25 ~k3
=72.2 (kW/K)
-
1+ 25 20
2 1 CP ci = CPcl - CP cl = 57.8 (kW/K)
I
CP)
X. Z H u
(5)
I
CPm CPn, or " is 0.35. The ratio CP et
Thus, the match ratio ~ T
of the composite curves is calculated as 0.54. Thus the d
or
224
Block i
CP (kW/K)
140
E}
t,
25
~.
20
~,
45 55
•
Xi Ii
130
i
1i
14o I I
180
135
Fig. 3. Determining splits placement and split ratio using area targeting principle.
Automated design method for heat exchanger network value is 0.19 (9.54-0.35). If for a sub problem with only hot streams 1 and 2 against cold stream 1, this ratio is optimal. However, there are four hot streams in this case and we need to investigate all the possibilities to select the most suitable two hot streams to match with cold stream 1. After similar calculations for all possible splits for cold stream 1, it is found that cold stream 1 should be split with ratio of 46.4 and 83.6 to match hot streams 1
minimise ij,kd Y, dijk~xrk~
CPik dokt= CPjl
224
[]
E x,jkl= 1 ; jE xijk~=1 xi#t=0,1 i=l..n;
j=l..m;
K=I..N;
I=I..M;
(7)
Where CPi, and CPjl are defined as continuous variables and x~jk~defined as binary variables which are optimised in the optimisation process. Using the MINLP model for the above split problem by restricting split branches of a stream to two, the solution is that the cold stream 1 should be split to match hot streams 1 and 3, and that cold stream 2 is split to match hot streams 2 and 4. Both these parallel splits exactly equal to the ratio of the composites and mimic the quasi-composite curves perfectly (Fig. 4). The above strategy can be applied in a similar manner to solve splitting problems with any number of split branches allowed.
3. Examples
3.1. Example I To illustrate the principles and procedures of the proposed design method, a simple example is used. Consider the following four stream problem (Table 1). The stream data and cost data were introduced by Linnhoff and Ahmad (1990) and Ahmad et al. (1990), respectively.
140
Block i
:i
CP(Hot-q.asi) CP(cold-q,,a,,i )
CPik=ePi ; ~ CPfl=CPj
CPm CPH3 and 3, respectively. Thus the match ratio ~ or 2 CPc~ CP~j in such a splitting scheme is 0.54 which equals the ratio of the composites, Hence, the d value is 0.0. This d value is the minimum in all possible splits for cold stream 1, which indicates the cold stream 1 should be split to match against hot streams 1 and 3. If we consider cold stream 2, the similar procedure can be applied to determine which two hot streams should be matched against this cold stream with what split ratio. So far, cold streams 1 and 2 are considered independently. In other words, the match interaction between these two cold streams is not addressed. However, to determine an overall best fit to the composite curves when selecting matches involving stream splitting, a simultaneous approach should be considered. The general stream split problem can be stated as following. Assume there are n hot and m cold streams waiting for matches involving splitting. For generality, any number of splitting branches are allowed. Let d~#~indicate the fitness to the quasi-composites by a match between branch k of hot stream i and branch I of cold stream j in this block. Let binary variables xok~ indicate the existence of a match (ijkl), i.e. xi#t=l declares the existence of the match and xokt=0 nonexistence of the match. Thus, a simple MINLP model can be formulated to simultaneously determine both the set of matches and optimal split ratios:
1099
CP (kW/K) 25
[]
20
[]
II
45
i i
55
V
J
f i
/
I
2i i i I
13o
i
\
180 Fig. 4. Solution to the splitting problem shown in Fig. 3.
I
,
__/
, i i i
1~,5
14o
1100
XIN X . ZHU
Table 1. Steam and cost data for example 1 Heat Capacity flow~te kW/K 200 100 300 500
Steam I 2 3 4
Supply temperatu~ °C 150 170 50 80
Target temperature °C 50 40 120 110
(Zhu et al., 1995a), the number of blocks for this problem is determined as two (Fig. 5). Above the pinch, the quasi-composites are calculated as: Hot quasi-composites: T[°C] =90+0.0037AH [kW] Cold quasi-composites: T[°C]=80+0.00125AH [kW]
A H = H - 13,000; H [kW] el13,000 ... 33,000] Utility data Hot utility: Saturated steam: 180°C Cold utility: Supply temperature: 20°C; Target temperature: 40°C U = 100 W/m-'K for all streams
Cost data Cost of Hot utility: 110 $/kWyr; Cost of Cold Utility: 10 $/kW yr Installed heat exchanger cost: cost ($) =30,800+750 area 0.sl Plant lifetime =6 years; Rate of interest = 10% per annum First, use the supertargeting approach to find the optimal HRAT and then determine the block decomposition. The optimal HRAT is identified at 10 K and the corresponding composite curves are given in Fig. 5. According to the principles for block decomposition
(8)
where the enthalpy interval of this block is from 13,000 to 33,000 kW and H represents any enthalpy in this interval. The reciprocal of the slopes of equation (8) yields CPuot-quasi = 270 kW/°C and CPcot,t-quasi = 800 kW/ °C, respectively. Then calculate 6 values for all possible matches in this block and list results in a matrix. The results are summarised in the following matrix:
~,j
H,
H~
CI C2
0.74 0.27
0.02 0.31
The sum of 8 for the matches H1~22 and H2-C1 is 0.29 which is much less than 1.05, the sum of 8 for matches H1-C1 and H2~C2. Clearly, the matches H1C2 and H2-C 1 should be selected. For the block below the pinch, there are two hot streams (HI,H2) against one cold stream (C1). To fully utilise the available driving forces, it is necessary to split
170
T °C
,, I
170
150
130
110
90 I i
105
90 ,60
70
i i
50
80
•
"5O i i
30
I 0
I 5
I 10
I 15
Fig. 5. Composite curves----example I (HRAT=I0).
t 20
I 25
I 30
i 35
I 40
~ H - MW
Automated design method for heat exchanger network the cold stream to match with the two hot streams. The split ratio is determined using equation (4). The initial design is summarised in Fig. 6. The performance of details of the design is given in Table 2 to compare with the targets.
1101
K ATe= ~}
All stream temperatures are modified by the equation (9) (hot streams reduce their temperatures by minus the corresponding/iTs and vice versa cold streams increase their temperatures by plus the corresponding ATs. Then the composite curves are plotted based on these modified temperatures. By using the modified supertargeting approach (Rev and Fonyo, 1991; Zhu et al., 1995b) which incorporates the diverse pinch method, the optimal x and z are identified as 0.367 and 1.2, respectively for the problem of Table 3. The modified composite curves for the optimal K and z are given in Fig. 7. A two-block decomposition is then determined and the corresponding stream structure is shown in Fig. 8.
3.2. Example 2 To illustrate how the block method can handle problems with different film coefficients when combining the diverse pinch approach (Rev and Fonyo, 1991), use an example stream data by Rev and Fonyo who modified the original data of Ahmad et al. (1990) which is given in Table 3. In the diverse pinch approach, HRAT or AT,,i, is no longer used instead K and z are used as two optimisation variables (Rev and Fonyo, 1991; Zhu et al., 1995b). The individual/iT-contributions can be expressed as
CP (kW/K)
90
)
( )
IT] 15°
(9)
60
.-so
200
20O0
~] 170
C
120 ~
110 ~
100
107
~/~ 3OO0
104
so I-~
300
80 r ~
500
~0 12000
,80
O. = 7000 kW Area = 19690 m 2
Total cost = $1.60 x 106 yr "1
Fig. 6. Initial design----example 1.
Table 2. Designperformancefor example 1
Targets HRAT = 10K Initial Design Fig. 6
Hot utility
Area m "~ abovepinch
Area m2 belowpinch
Units
Total Cost M$/yr
7000
8852
10785
5
1.55
7000
8880
10810
8
1.60
Table 3. Streamand cost data for example2 Stream
Supplytemp. °C
Targettemp. °C
CP kW/K
h W/m2K
HI H2 H3 CI C2 Steam CW
159 267 343 26 I 18 300 20
77 80 90 127 265 300 60
2.285 0.204 0.538 0,933 1.961
100 40 500 10 500 50 200
Cost Data. Cost of hot utility: 110 $/kWyr;Cost of cold utility: 10 $/kWyr. Exchangercost modeh cost ($)=3,800+750 area 0.s3 Plant lifetime:6 years; Interestrate: 10% per annum.
1102
XIN X. ZHU
350 300 250 °~
200 150 100 50
i
0
I
I
I
I
t
I
r
1 O0
I
I
I
i
200
i
I
I
I
I
I
300
I
. . . .
400
I
. . . .
500
I
600
H (kW) Fig. 7. The modified Composite Curves for example 2 when K=0.367, z=l.2 (the utility consumption corresponds to HRAT = 36K). After the block decomposition for the modified composite curves, the methods for determining matches and splits discussed above can be applied and the initial design is produced given in Fig. 9 and the final design (Fig. 10) is achieved using the cost optimisation. The detail results for designs are listed in Table 4. As can be seen from the table, The initial design using the proposed method requires small overall area with a reasonably small number of units and the final design achieves the unit target with a 5.9% lower total cost than the cost target. It should be noted that the import/export rule is employed for the initial design (Fig. 9) and that is the
reason why the utility requirement in this design is relaxed in this design compared with the utility target.
4. Conclusions A simple but efficient algorithm for automated synthesis of HEN is proposed. The fundamental concept is the block which provides an efficient way to approach the area and energy targets simultaneously with a simple network. The significance of the concept is the simplification of a design problem which is achieved by decomposing it into a number of blocks with different
Block 2
Block 1 342.2
153.2
249.5
159
120.1
89.2 71.2 62.5
125.9 =
77 p,
,
i i b
'~
267
170.7
137.61
154
120.91
6o
= = i
90
= i i i i
127q .:102"9 6
:194.2
61 152.4
26 l~] i
118[~ z i i h
264.2 219.2 195.1
153.2
118.8 118.2
Fig. 8. Stream Structure for K=0.367 and z= 1.2 for example 2 (the utility consumption corresponds to HRAT=36K).
Automated design method for heat exchanger network
(,)
r~] 159
~-77
170.7
80
(,,
~154
I~1 ~
©
134.4
131.3
)
E~] 267
1103
©
120(
:- 90
16.1
(D
127q
19.7
/
~56.1
18.5
127.3 168.3
101.7
QH = 168.3 kW;
26
18.3 Area = 241 m2; Total cost = 49049 S/yr
Fig. 9. Initial design at K=0.367 and z= 1.2 for example 2.
134.5 @
r•159
131.3
(,,)
E~] 267
--77 80
121.6
©
90
17.0
116.5 (
127 q
%o 150.4 ( ) 38.2
265
169.2
26
/ 118 ~-~
119.1
QH --- 169.2 kW;
Network area = 242 mZ; Total cost = 46551 S/yr
Fig. 10. Optimized design at x=0.367 and z= 1,2 for example 2.
Table 4. Design results for example 2
Targets K=0.367, z= 1.2 Initial design (Fig. 9) Final design (Fig. 10)
Hot Utility kW
Area m 2
Units
Total Cost $/yr
161.2 168.3 169.2
278 241 242
6 8 6
49292 49049 46551
1104
XIN X. ZHu
characteristics of design requirement. Further more, the quasi-composite curves are used to replace the true composite curves to provide the reference for match selection in a design process. The qualitative analysis of driving forces is done at the stage of block decomposition while the quantity analysis of area requirement is performed by applying mathematical equations which embed the physical insights for achieving a minimum area design. More importantly, the effects from film coefficients, temperatures and heat loads are taken into accounted in this method in a systematic manner. By means of the t5 function for match selection and the d function for splits associated with the simultaneous approaches, the minimum area design can be achieved efficiently and effectively. In addition, the method is readily implemented as a software package for automated synthesis of HEN.
References
Ahmad, S., Linnhoff, B. and Smith, R. (1990) Cost optimum heat exchanger networks--2. Targets and design for detailed capital cost models. Comput. chem Engng. 14, 751-767. Linnhoff, B. and Hindmarsh, E. (1983) The pinch design
method for heat exchanger networks. Chem. Engng. Sci. 38, 745-763. Linhoff, B. and Ahmad, S. (1986) SUPERTARGETING, or the optimisation of heat exchanger networks prior to design. World Congress II, Chemical Engineering, Tokyo. Linnhoff, B. and Ahmad, S. (1990) Cost optimum exchanger networks--l. Minimum energy and capital using simple methods for capital cost. Comput. chem. Engng. 14, 729-750. Rev, E. and Fonyo, Z. (1991) Diverse pinch concept for heat exchange network synthesis: the case of different heat transfer conditions. Chem. Engng. Sci. 46, 1623-1634. Townsend D.W. and B. Linnhoff (1984) Surface area targets for heat exchanger networks. IChemE Annual Res. Mtg, Bath, U.K. Zhu Xin X (1994) Strategies for Optimisation in Heat Exchanger Network Design. PhD Thesis, The University of Adelaide, Australia. Zhu Xin X., B.K. O'Neill, J.R. Roach and R.M. Wood (1995a) A new method for heat exchanger network synthesis using area targeting procedures. Comput. chem. Engng. 19, 197-222. Zhu Xin X., B.K. O'Neill, J.R. Roach and R.M. Wood (1995b) Area-targeting methods for the direct synthesis of variable film coefficient heat exchanger networks. Comput. chem. Engng. 19, 223-239.