16th European Symposiumon ComputerAided Process Engineering and 9th International Symposiumon Process SystemsEngineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by ElsevierB.V.
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A d i s c r e t e i n t e r a c t i v e g r a p h i c a l m e t h o d for h e a t e x c h a n g e r n e t w o r k synthesis E. S. Fraga ~, G. W. A. Rowe b aDepartment of Chemical Engineering, UCL (University College London), Torrington Place, London WC1E 7JE, UK bDivision of Applied Computing, University of Dundee, Dundee DD1 4HN, UK A discrete formulation for heat exchanger network synthesis (HENS) problems has been defined. The formulation provides the basis for a stochastic optimization procedure based on simulated annealing and also for visualisation and user interaction through a graphical interface. The discrete formulation is based on the concept of a quantum of heat flux used to discretize all the streams in the HENS problem. The design problem then consists of identifying matches of pairs of quanta. The simulated annealing procedure builds chains consisting of contiguous matches in the discrete search space. Results for a case study from the literature are presented highlighting the ease with which good solutions are obtained and the benefits from engineercomputer interaction. 1. I n t r o d u c t i o n Graphical representations for the heat exchanger network synthesis problem have a long history, including the use of cascade diagrams, enthalpytemperature diagrams and pinch point analysis [13]. The advantage of a graphical representation is the ease with which the engineer can gain insight into the key features of the design problem and the alternative solutions. However, these traditional graphical methods are less attractive when an engineer wishes to explore the tradeoffs inherent in HEN design problems between operating costs (utility consumption) and capital costs (the actual exchangers). The alternative is to use mixed integer nonlinear programming (MINLP) models to capture these tradeoffs. Unfortunately, the typical cost models used lead to nonconvex formulations and concomitant difficulties in finding globally optimal solutions. Previous work by the authors [4] demonstrated the potential of combining a graphical interface with stochastic optimization methods for the MINLP formulation. This paper presents a new combined approach which inherits all the positive properties of the previous approach but which provides the potential for solving more complex problems. The underlying model is based on the conversion of the MINLP to a fully discrete problem through the simultaneous discretization of heat flux and temperature. The discretization procedure defines a two dimensional grid as the space of possible solutions. This grid maps naturally to a representation suitable for use by a number of stochastic optimization procedures as well as to a graphical visualisation which caters for user in
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teraction. Potential and existing stream matches are indicated in the graphical interface and the user is able to explore alternative solutions by simple mouse clicks. The discrete formulation imposes no restrictions on the objective function used but does not attempt to find exact solutions. Instead, the tool is aimed at the early stages of design by encouraging exploration of alternatives, providing good starting points for a subsequent robust MINLP or NLP solution procedure. 2. T h e d i s c r e t e f o r m u l a t i o n Given a H E N S problem defined by a set of streams, we wish to create a discrete approximation to this problem. A stream, i, in the problem definition is specified by a heat duty, Qi, and a temperature range, [T/,in , T/,out] , assuming a constant rhCp heat flux. The conversion to a fully discrete formulation is based on the definition of a heat duty quantum, defined by a given value of AQ. Each stream, i, is converted into a set of quanta, Q{, each of which has the same heat duty, AQ, and the same temperature change: ATi = T/'°ut  T/'in
(1)
ni
where ni is the number of quanta for stream i defined by
(2) A stream defined by {Qi, [T/,in, T/,out]) is now replaced by a set of quanta, Qi j ._ Q i , 3  1 , . . . , n } where Q{ = {AQ, [TJI, TJ] } and Tj = T,i n + j A T . A solution of the discrete HENS problem now consists of a set of matches between quanta from complementary streams. Each match is a pairing between a quantum i from a hot Q'istream, h, and a quantum j from a cold stream, c, and is denoted a s M h'i  Qhi For a match , M ~,j, h'i the following constraints must be satisfied: Z~ :> Tcj1 and T~1 > T j
(3)
The assumption behind this constraint is that we are only considering countercurrent heat exchangers. Obviously, many individual pairings of hot and cold streams will not lead to good HEN designs and so the actual network designed will consist of sequences of pairings which match contiguous quanta from a given pair of hot and cold streams. For instance, if matches M c,j h'i and M c,j+l h'i1 both exist in a particular configuration, this pair of matches corresponds to a single exchange of 2 x AQ amount of heat with the hot stream temperature range [Tt~2, T~] and the cold stream temperature range [T~1, TJ+I]. The evaluation of a configuration, defined by a set of matches {rnk}, coalesces any valid sequence of matches into a single exchanger design. The optimization problem for the discrete formulation is to select a set of matches, {rnk}, a subset of all possible matches. This latter set is potentially large and depends on AQ.
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A Discrete Interactive Graphical M e t h o d f o r H e a t E x c h a n g e r N e t w o r k Synthesis
2.1. T h e g r a p h i c a l i n t e r f a c e An advantage of a discrete formulation can often be a straightforward mapping to computer screen coordinates for visualisation. For the discrete H E N S formulation, we have a set of potential matches defined by the sets of quanta from the hot and the cold streams. We can use a 2dimensional cartesian product between these two sets of quanta to define the space of potential matches and map this directly to a graphical representation. \
Cartesian grid
k___ Exchanger type Relative cost ~___.~ R~mo~n~m~,ch(lo 4)~E~Sss74.~7~.6s81S~//
Figure 1. Annotated illustration of the graphical interface for a 1 x 1 simple
HEN.
Figure 1 shows the example of this representation with the key elements of the graphical interface highlighted. These key elements include the main Cartesian grid with each hot quantum mapped to a row and the cold quanta mapped to columns. At the bottom of each column and to the right of each row are two visual cues about the type of exchanger (green indicating integrated, red indicating utility) and the relative cost of the individual exchanger. The Cartesian grid itself has four types of squares: a white square indicates an infeasible match between the two quanta involved, a red square indicates a match that is available for use, a green square indicates a match which is part of the current configuration and a grey square indicates a match which is feasible but not currently available due to one or the other half of the match being in use. The graphical interface provides for user interaction. The user can click on any square in the Cartesian grid to flip its status from available to "inuse" and vice versa. The interface updates immediately showing the effect of the change in configuration. 3. A s i m u l a t e d a n n e a l i n g p r o c e d u r e Although a graphical interface is available for investigating potential solutions, a solely interactive approach is not appropriate for most problems. Automation is and should be a key element in design, working in conjunction with the engineer whenever possible. The graphical representation, although very useful, is not enough and a computerbased optimization procedure is necessary. This section describes a simulated annealing procedure designed for the discrete formulation.
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The best configurations are those which consist of contiguous (in a diagonal sense in the visual representation, from top left to bottom right) matches of quanta. Therefore, we have developed a simulated annealing procedure based on identifying chains of matched quanta (links). A simulated annealing procedure is a descentbased optimiser which allows for uphill moves with a decreasing probability as the procedure progresses [5]. The procedure is based on the concept of neighbourhood to identify a new solution given an existing solution. The key to the development of a new simulated annealing procedure therefore is the definition of a neighbour. Given a solution, consisting of a set of chains, g = {c~, i = 1 , . . . , no}, neighbouring solutions come from the solutions generated by one of the following modifications to the current configuration: 1. Create a new chain with just one link (0.01). 2. Delete an existing chain, cj E g (0.01). 3. Delete a link from an existing chain (0.08). 4. Shift a chain either up, down, left or right in the Cartesian grid (0.60). 5. Extend an existing chain by one link at either end (0.30). The number at the end of each action description is the probability of this action being taken. If an action is not possible (e.g. it is impossible to shift any chains), then the next action in the list is attempted (in the order presented above). The probabilities given above are the defaults in the system but these can be easily modified in the problem deftnition. The system has been implemented using the Jacaranda system which is completely configurable using text based input files [6]. Likewise, all the standard parameters for a simulated annealing procedure (initial temperature, cooling factor, iterations, iterations at a given temperature level, etc.) can be specified by the user. The use of discretization means that the problem actually solved by the simulated annealing procedure is an approximation to the real problem. Therefore, the solutions obtained are also approximate. Furthermore, simulated annealing is a stochastic method and the solutions obtained differ from one attempt to the next. The solutions, however, have properties which should be shared by the real solutions. The solutions obtained are used to define a smaller nonlinear programme (NLP) which can subsequently be solved using any number of optimization procedures including direct search methods [7]. 4. R e s u l t s A number of case studies have been tackled using the hybrid method described above, consisting of the simulated annealing procedure followed by either user interaction or a subsequent N L P solution procedure for postprocessing. For this paper, we consider the case study presented by Morton [8] which consists of three hot streams and three cold streams, shown in Table 1. Please refer to [8] for full details. An initial investigation shows that one of the hot streams has significantly larger requirements than the other hot streams, and more importantly, has larger requirements
A Discrete Interactive Graphical Method for Heat Exchanger Network Synthesis
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Table 1 Heat exchanger network synthesis case study [8] Process Streams Stream H1 H2 H3 C1 C2 C3
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(°C) Q (kW) 40 60 50 180 210 160
6400 600 200 3100 3250 2250
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0.8 0.8 0.8 1.6 1.6 1.6
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(a) Screenshot (green exchanges replaced by (b) Reduced superstructure inferred from the simublack squares for printing) of a solution, lated annealing solution. Figure 2. Sample configuration for case study from [8] after solution by simulated annealing and the corresponding superstructure used as the basis for the subsequent mathematical programming step.
than any of the cold streams but covers appropriate temperature ranges for these cold streams. Therefore, we consider splitting the first hot stream in half and using this as the base HENS problem. Figure 2(a) shows a sample configuration achieved by one run of the simulated annealing procedure. The first two hot streams correspond to the two halves of the original first hot stream. The configuration shows integrated exchanges between the first cold stream and the third hot stream (cf. exchanger x13 in the superstructure shown in Figure 2(b)) and one half of the first hot stream (x11), from the second cold stream to the second hot stream (x22) and then to first hot stream before it splits (x21) and from the third cold stream to the other half of the first hot stream (x31). This configuration can then be used to define a reduced superstructure as the definition of an NLP formulation, as shown in Figure 2(b).
E.S. Fraga and G. W.A. Rowe
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The superstructure is solved using a hybrid direct search procedure combining the Hooke & Jeeves and Implicit Filtering methods with the implementations from Kelley [7]. The solution obtained has an objective function value of 1.66 × 106 with the following values of the decision variables: x l l = 2039, xl3 = 200, x21 = 2356, x22  391, x31 = 1846 and sill = 0.52 resulting in utility use of CUt = 159, CU2 = 209, CU3 = 0, HU1 = 861, IIlJ2 = 502 and HU3 = 404, all in kW. This compares favourably with the solutions obtained by Morton [8] not just in terms of the objective function (Morton obtains 1.6 × 106) but also in terms of complexity. Morton's solution requires 10 integrated exchangers and 4 splitters; ours has 5 exchangers. Note that the nonintegrated network has a cost of 6.6 × 106. With regards to computational efficiency, the simulated annealing step takes typically approximately 3 minutes (on a 3 GHz personal computer) including continual updating of the visualisation interface. The subsequent mathematical programming problem is easy to define and can be solved in a few seconds. These times are short enough to encourage experimentation and exploration of alternative superstructures and corresponding solutions for the purpose of gaining insights into the key aspects of the underlying problem. 5. C o n c l u s i o n s A novel formulation for heat exchanger network synthesis problems has been presented. The formulation is a discrete approximation to the original continuous problem definition and provides the basis for graphical interface for visualisation and user interaction. The discrete problem can be solved using a novel simulated annealing procedure based on the concept of chains of links where each link represents a match between a hot stream and a cold stream for a quantum amount of heat. The simulated annealing results provide the starting point for postprocessing which can include the user via the graphical interface and also other, possibly deterministic, optimization procedures. Discretization with visualisation provides an attractive combination for gaining insight into the key features of good solutions for a given H E N S problem. REFERENCES
1. 2. 3. 4. 5. 6.
7. 8.
Douglas, J. M. Conceptual Design of Chemical Processes. McGrawHill International Editions, (1988). Linnhoff, B. and Hindmarsh, E. Chem. Eng. Sci. 38(5), 745763 (1983). gravanja, Z. and Glavic, P. Computers chem. Engng 21(8), 833853 (1997). Fraga, E. S., Patel, R., and Rowe, G. W. A. Chem Eng Res Des 79(A7), 765776 (2001). van Laarhoven, P. J. M. and Aarto, E. H. L. Simulated annealing: Theory and applications. Kluwer Academic Publishers (Dordrecht), (1987). Fraga, E. S., Steffens, M. A., Bogle, I. D. L., and Hind, A. K. In Foundations of ComputerAided Process Design, Malone, M. F., Trainham, J. A., and Carnahan, B., editors, volume 96 of AIChE Symposium Series, 446449, (2000). Kelley, C. T. Iterative methods for optimization. SIAM, (1999). Morton, W. Proc. Inst. Mech. Eng. Part E 216(2), 89104 (2002).