I n d . Eng. Chem. Res. 1990,29, 2252-2257
2252
Parametric Optimization of a Paper Machine Heat Recovery System Stefan G.Gustafsson and Goran Henriksson Valmet Paper Machinery Inc., Pansio, 20240 Abo, Finland
Henrik Sax6n* Heat Engineering Laboratory, Abo Akademi, Biskopsgatan 8, 20500 Abo, Finland
The design of heat recovery systems for paper machines is studied by means of a sequential modular flow-sheeting program. The program is used for economical optimization of the heat recovery system with respect to its design variables. In the optimization, the objective function, calculated by the flow-sheeting program, is mapped to an approximate model, which is repeatedly updated and optimized. The values of the design variables used in the mapping are provided by factorial experiments. Analyses have predicted that considerable steam savings can be obtained if the heat recovery systems that are designed by the use of current standard practice are replaced by optimized systems. The yearly venture profit of the system can usually be increased by about 5% by means of optimization.
Introduction A major part of the energy required by a paper machine is consumed in the dryer section, which means that a large potential of savings exists in this part of the process. The energy flows in the dryer section of a typical newsprint machine are illustrated in Figure 1: the energy enters the exhaust air, which can be utilized for heating the supply air for the dryer section, the process water, and the circulation water. The circulation water, in turn, may transfer its energy to the building ventilation air. If the heat recovery system is to be optimized, it is obvious that the system must be studied as an entirety because of the interplay between its units. A flow-sheeting program may provide such facilities. Several methods for process optimization using flow-sheeting programs as a basis have been proposed in the literature, and it is generally accepted that successive quadratic programming (SQP)is one of the most efficient methods for solving such problems (see, e.g., Berna et al., 1980). Generally, the flow sheet is not converged each time the objective function is calculated, but the torn variables are instead solved simultaneously as the optimization proceeds. Such a procedure can, at least theoretically, be implemented straightforwardly on an equation-oriented flow-sheeting package. However, considerable restructuring or reformulation of the simulation program may be needed in order to obtain reliable gradient information if the preprogrammed flow-sheeting program is based on the sequential modular formulation. Moreover, the requirements of storing and updating large matrices also cause problems. Process optimization by flow-sheet simulation is thus still a cumbersome task. Some investigators have therefore used an approach where simplified models for the different units (Jirapongphan et al., 1980) or sections of the plant (Parker and Hughes, 1981) are repeatedly created and optimized. This procedure generally results in considerable savings in the computational effort, since, e.g., thermochemical properties need not be evaluated as often, as is the case of optimization using the rigorous model. The present study rests on a small sequential modular flow-sheeting package for simulation of heat recovery systems for paper machines (Sundqvist, 1987). An engineering a p p r o a c h is made for “continuous” optimization of certain design variables in the process model. The procedure, which can be easily adapted to an existing sequential modular flow-sheeting program, makes use of an approximate model for the objective function, which is determined by the use of data from “designed” factorial experiments. The method differs from the approaches
made in the literature for optimization of heat recovery systems, which often focus on continuous optimization of design variables using equation-oriented flow-sheeting programs (Shah and Westerberg, 1980) or structural optimization (synthesis) using the pinch design method (Linnhoff and Hindmarsh, 1983; Cerda et al., 1983; Tjoe and Linnhoff, 1986). Two-phase synthesis and continuous optimization procedures, based on the solution of mixed integer nonlinear programming problems, have also been reported (see, e.g., Floudas et al., 1986; Kocis and Grossmann, 1988). However, even though some interesting recent papers (Ciric and Floudas, 1990; Colberg and Morari, 1990) have addressed a more general problem, the models of the process units have, as a rule, been quite simple.
Heat Recovery System The most important units of the paper machine heat recovery system are the heat exchangers, which, in general, are arranged in a towerlike configuration. A typical system is depicted in Figure 2. The conventional heat recovery (CHR) unit is an airto-air heat exchanger of channel design. This unit is primarily used for heat recovery from the hood exhaust air to the hood supply air but is also sometimes used to heat the building make-up air. The aqua heat recovery (AHR) unit is an air-to-water heat exchanger of plate design with combined counterflow and crossflow conditions. Heat in the AHR’s is transferred to process water or circulation water. The latter, in turn, transfers energy to the building make-up air. Air coils (AC) are water-to-air tube heat exchangers, where the heat-transfer surface is increased on the air side by fins. AC’s are mostly used to heat ventilation air with the circulation water from the AHR’s. Steam coils (SC), in turn, are used for final heating of the hood supply air, after its passage through the heat the recovery units. In the steam heat exchangers (SHE), temperatures of process or circulation water are finally raised on occasions where there is a shortage of recovered heat. Further components of the heat recovery system are s c r u b b e r s , m i x e r s , and s p l i t t e r s , but no design variables for these units have been optimized in this work. The mathematical model for the system is based on equations for mass and energy balances and for the heat-transfer phenomena formulated for the different process units (Forsbacka, 1981; Sundqvist, 1987). As moisture is present in the exhaust air, the heat-transfer expressions are rather complicated, requiring special modules in the simulation in order to describe both “dry” heat transfer and heat transfer with simultaneous mass
0888-5885f 90f 2629-2252$02.50f 0 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2253 30OC
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5-
O R Y I R SICTION V.07 L g l ~ O I s
SUPPLY AIR
32% L W
1
.IIV IO06 LY L I A K I G I AIR
,061 L W CONOiNSATf . W C
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Figure 1. Energy flows in the dryer section of a paper machine.
0
100
50
200
150
250
300
350 d
t WLL V w T U n o N UNITS
!a c*i
n
Figure 3. Annual duration curves for the outdoor air and water temperatures.
vestigation of the annual performance of a heat recovery system (i.e., simulation) requires a couple of CPU minutes on a DEC MicroVAX I1 minicomputer.
FgPPr lice0 SUPPLY
Figure 2. Typical configuration of the heat recovery system.
transfer (condensation of moisture). The output and the internal state of the modules are, in general, the unknowns; their values are obtained by simultaneously solving the balance and transport equations, hl(x,v)= 0. Here, the process variables (mass flow rates, temperatures, etc.) are included in x, while v holds the design variables (heattransfer areas, etc.). The layout of the recovery system must first be specified by entering a choice of units and a scheme for the information flow, which describes the physical flows. Second, the input flows, their states, and the design variables must be given. The order of calculation can now be determined automatically from the topology of the process. If recycles occur, the loops are torn, which gives equations, h2(x,v)= 0,for the unknown torn variables. Thus, the steady state of the process is given by
In the simulation of the heat recovery, the changes due to seasonal variation in the external conditions have been considered by the use of duration curves. Figure 3 depicts typical curves for the temperatures of the outside air, used for air conditioning of the plant, and for the freshwater. For instance, it can be seen that the temperature of the freshwater exceeds +10 "C during about 205 days (=365160 days) of the year. For variables showing seasonal trends, several points on the empirically acquired curves are used in the simulation instead of single mean values, and the results are then integrated over time. By this procedure, the model reflects the performance of the actual system better, but the computational cost greatly increases. The requirements of CPU time may be considerable if the performance of several configurations is studied. An in-
Background for the Optimization In order to verify the simulation program, some measurements were carried out on actual heat recovery systems of different paper mills. The results (Valmet Paper Machinery Inc., 1988) showed deviations of only a few percent between simulated and measured temperatures, moisture content, and recovered heat, which means that a meaningful optimization can be carried out by using the flowsheeting program. The heat recovery system reduces the steam costs of the mill mainly by replacing fresh steam by recovered heat. Naturally the investment and the utility costs must be considered in an economic evaluation of the system. By using annuities, an economic objective function to be minimized can be formulated as 4=
41 + 411 + 4111
i(i =
[
"(e
+
1)"e(e + 1)" s(s + 1)" 1)"- 1 P"B(S 1)"- 1 +
Tn
+ 1)"
+
+
]
(1)
where the three main terms express the investment (equipment) costs, the utility costs, and the value of the recovered heat. The first term includes the interest rate, i, for the investment, I. The latter two terms depend on T , the annual time of use of the system, while all three terms are affected by the project's lifetime in years, n.. The electrical effect, P, and the recovered heat flow, Q, are weighted by cost factors p e and p s , respectively. It should be pointed out that an increase in the steam consumption in the SC's and SHESwill lead to a corresponding decrease in the recovered heat flow. The annual increase in the prices of steam and electricity, s and e, are further parameters of the function. In the following, we will concentrate upon a continuous optimization of 4 with respect to the design variables, v, using a fixed structure of the heat recovery system. The problem optimally to connect and place the different units in the network, i.e., optimization of topology (see, e.g., Papoulias and Grossmann, 1983; Itoh et al., 1986), will not be considered. Neither will we deal with process synthesis. The reader is referred to HlaviEek (1978), Nishida et al. (1981), and Douglas (1988) for reviews of the work published in that area.
2254 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 Table I. Design Variables of the Heat Recovery System
variable 1 2 3
CHR gap between the plates on the exhaust air side gap between the plates on the supply air side total no. of plates
unit AHR water mass flow rate no. of cells type of cells
For design variables that only take on discrete values, a simplified treatment has been adopted. We will see that the values of these variables either can be treated as floating-point numbers or simply can be determined by shortcut procedures in the design. From the optimization point of view, the features of the different units of the heat recovery system are listed in Table I. The mass flow density in the CHR increases as the distance, d, between the plates decreases, yielding more recovered heat, but simultaneously the energy consumption of the fans increases. As the value of the third design variable, the number of plates, N , is comparatively big, it may be reasonable to treat it as a floating-point number in the optimization. In the AHR unit, an increased mass flow rate of water enhances the heat transfer a t the expense of greater pressure losses. The second item-the number of cells, M-is a variable that must be represented by an integer. Fortunately, experience has shown that the optimum solution practically always is obtained for a configuration where the mass flow density of air in the AHR is as close as possible to its upper limit, (nia/A)". This means that given the mass flow rate, ma, and the cross-sectional area of a cell, A,, the number of cells should be chosen as the smallest integer satisfying ma
M 1 ( 4 3
/ A ),*A
(2)
Finally, the type of cell can be selected on the basis of heuristics. All the properties listed for the AC assume only discrete values, but experience has shown that their values can be deduced from the operating conditions. A sparse distribution of the fins reduces fouling problems, while a dense arrangement yields a better heat recovery but greater pressure losses. As for the configuration, it may be noted that counterflow exchangers are efficient but may be unsuitable because of problems caused by the formation of ice in the wintertime. Therefore, different configurations are chosen depending on whether the heat recovery system is delivered to a paper machine in, e.g., northern or southern Europe. The dimensions of the SC's are fixed by the requirements that the temperature of the dryer air be raised from approximately 60 to 95 OC. Circulation and process water from the AHRs must be heated in the SHE during periods of production shutdown or when the temperatures of the fresh air and water are extremely low. The appropriate size of the SHE can be calculated from these conditions. Optimization Using Approximate Functions The most straightforward method for optimization of processes using sequential modular flow-sheeting programs is to let the simulation program provide values of the objective function to an external optimization routine, which manipulates the design variables. Considerable problems were, however, encountered when this approach was used in some early trials to optimize the heat recovery systems. Efforts to improve the poor gradient information indeed reduced the extent of some of the problems, but the computational effort still remained prohibitive.
AC ~
size and structure distribution (sparse/dense) co- or counterflow
SC size
SHE
~~
size
Nongradient search routines, like the polytope method (Nelder and Mead, 1965),could be used, but these methods are characterized by extremely slow convergence, and they are definitely not suited to solve problems with (design) constraints. The main disadvantages of "blackbox" optimization of parameters in flow-sheeting programs, which have been pointed out by several investigators in the literature, can thus be summarized: (1)The objective function values are corrupted by noise caused by iteratively determined solutions to the subproblems (modules or loops). This often leads to premature interruption of the search at a point far from the true optimum. (2) The procedure is computationally inefficient and time-consuming,as the steady state must be solved at each iteration. Various approximate methods for process optimization using flow-sheeting programs have therefore been proposed. Biegler and Hughes (1981) improved the approach made by Parker and Hughes (1981) by making use of a method for nonlinearly constrained optimization by Powell (1978) and were able drastically to reduce the CPU time. Berna et al. (1980) proposed a modification of the Powell method to make it possible to deal with large systems, simultaneously solving the optimization problem and the process flow sheet following an infeasible path. Hutchinson et al. (1986) applied Powell's method for optimization of processes by using the simultaneous modular flow-sheet package QUASILIN (Gorczynski et al., 1979). Similar subjects were treated by Biegler and Hughes (1982), Biegler and Cuthrell (1985), and Chan and Prince (1986). Chen and Stadtherr (1985) have developed a method for optimization of processes using simultaneous modular flowsheeting programs as a basis and pointed out the merits of this approach compared with applying the sequential modular formulation. In summary, a way of tackling process optimization by flow-sheet simulation is to solve min @(x,v) (3) V
subject to
h(x,v) = 0 g(x,v) 2 0 by some robust method for constrained nonlinear optimization, e.g., by SQP. These methods facilitate a simultaneous minimization of @ and a solution to the flowsheet equations and the (design constraint) inequalities. However, laborious reformulation may be required in order to obtain a robust method (e.g., reliable gradient information) especially if the flow-sheeting program at one's disposal is based on the sequential modular formulation (Kaijaluoto, 1984). A review of the use of flow-sheeting programs for process optimization, and the inherent problems, is given by Edgar and Himmelblau (1988). In the present work, a heuristic approach suited for straightforward application to process optimization by sequential modular flow-sheeting simulation is made. The method, which addresses the common problems listed in the beginning of this section and makes use of an approximate model for the objective function, is based on the following steps:
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2255 (1) Set the iteration index j = 0 and the convergence criteria parameters, f and tu. Assign initial guesses to the process variables .(IT, the design variables v(j),and the design constraints vminand vma. (2) Determine the “trust region”, Av(j),for the design variables. (3) Evaluate m values of the objective function, 4, using a set of design variables vectors (vk),k = 1,..., m,given by a factorial experiment. (4) Adapt a simplified model, @Yv,b),to the objective function of the rigorous model, 4, by estimating the former’s coefficients, b, from 1; = arg (min 11611~21 (4) where the m elements of the vector 6 are given by 8k = $(Xk,Vk) - @(Vk,b). (5) Optimize the approximate model by solving
I
3 = arg min @(v,b)
subject to
(6) Calculate 4(x,C). Check whether 116112 5 cQ and I13 tu. If they are, stop. If not, set v(j+l)= 3, j = j 1, and return to step 2. Naturally, the convergence of this algorithm cannot be guaranteed, partly because of the ad hoc method employed to adjust the extrapolation limits under step 2. At each iteration, the extrapolation interval is placed symmetrically with respect to the previous solution. Whenever the solution to step 5 stays well within the extrapolation limits, these are contracted. If, on the other hand, the solution lies on, or close to, the boundary of the feasible region, Av is not adjusted. It should be observed that the hard constraints of the design variables, vminand vma, must never be violated. Naturally, any constraints, g(x,v) 2 0, could be imposed instead of the box constraints used here. In step 3, a factorial experiment is “designed” in order to determine the coefficients of the approximate model, @, the structure of which has been heuristically chosen in the present work. By this measure, we obtain a simplified model that is expected to approximate the objective function of the rigorous model reasonably well in the region studied. Usually, four or five equally spaced factor levels for each design variable have been used in the program for designing the “experiments”, which provides a plan where the correlations between the vk’s in the set are satisfactorily low. This phase of the optimization requires m converged flow-sheet simulations, but the computational effort required is not very large, since m as a rule can be chosen as between 1.5 and 2.0 times the number of unknown parameters in the simplified model. A quasi-Newton method, using the DFP or the BFGS rank two updates of the inverse Hessian, has been applied to solving the problems under steps 4 and 5, while the constraints have been considered by the use of penalty functions. The fact that both the adaptation of the approximate model and the optimization of it with respect to the design variables can be carried out by the same routine makes the implementation of the method comparatively easy. It should be noted that it cannot be established whether the approximate model has converged to the true optimum of the rigorous model. However, as Biegler et al. (1985) have pointed out, the sufficient conditions for this would - v(j)l125
+
f
. . . . . . ..
g-+
Figure 4. Heat recovery system studied in the illustration example. The splitters and the mixers have been omitted from the figure for the sake of clarity.
be that the gradient of the simplified model should match the gradient of the rigorous model at all points, which can only hold if the two models are identical. Naturally, a further test of the convergence could be obtained by checking whether the estimates of the coefficients, b, in the approximate function have converged. This test has not been used in this work, but instead, the procedure has been started from several initial guesses in order to clarify whether the final results can be considered appropriate. In cases where optimization by the blackbox method was successful, we have found that the solutions of the two approaches agree well with each other. It has, however, been found that the new method requires only a fraction of the CPU time that was spent in the earlier blackbox optimizations. Another noteworthy fact is that the method is also much more robust, which means that a solution to the optimization problem is almost always found.
Illustration Example The configuration of a comparatively small heat recovery system for a modern newsprint machine is depicted in Figure 4. Supply air for the dryer section is first heated by heat from the exhaust air in a CHR unit (1). The temperature is further raised in the steam coil (2). Ventilation air for the machine room passes through air coils (6-a), where a circulating water-glycol mixture supplies the energy, which has been recovered in the AHR unit (3). If necessary, the temperature of the circulation water is further raised in a steam heat exchanger (4). In the objective function, eq 1,the annual time of use and the project lifetime were equated to r = 7860h/a and n = loa. The prices (expressed in Finnish Marks; FIM 4 = US. $1)of electricity and steam were set to equal p e = 170 FIM/MWh and p s = 90 FIM/MWh, respectively, with a 4 % yearly increase (s = e = 0.04) and an interest rate of 20% (i = 0.20). The approximate model of the objective function cocsists of three parts: (I) the investment costs, df’, which need not be approximated since they are fully known (Le., 4f =&); (11)the utility costs, &f; and (111)the value of the recovered heat, Process knowhow can be used when selecting expressions for the approximations. In this example, the following formula was used 4” = bj,o + bj,lw, + bj,zWE2 + bj,gWs + bj,4Ws2 + bj,5A,, + j = I1 or I11 (6) bj,sAtot2+ bj,7mw+ bj,sm,2 where w is the velocity of the air, which is a function of
2256 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 Table 11. Values of the Design Variables and Some Central Quantities for the Original and the Optimized Heat Recovery Svstems
CHR
AHR
design variables and results total no. of plates, N gap between plates on exhaust air side, d ~mm , gap between plates on supply air side, ds, mm mass flow density on exhaust air side, kg/(mz s) mass flow density on supply air side, kg/(m2 s) mass flow rate of water, ni,, kg/s no. of cells, M
energy flows recovered heat, Q,MWh/a electricity consumption, P, MWh/a steam consumption, MWh/a to the hood supply air to the AHR circulation water temp of the supply air after the CHR,"C objective function investment costs, $1, kFIM/a utility costs, $1~1, kFIM/a value of recovered heat, kFIM/a venture profit, I&[, kFIM/a
the mass flow rate, the gap between the plates, d, and the number of plates, N. Subscripts E and S respectively denote the exhaust and supply air sides, and A,, is the total heat-transfer surface in the CHR unit. The last two terms in eq 6 depend on the mass flow of water, mw,in the AHR unit. In this specific example, the equations have been simplified by setting bI16 = 0. The total number of parameters to be determined is thus 17, but as the contributions of &, and $Jmto the objective function have been discriminated, not more than about 15 (ma 15) converged flow-sheet calculations are required in the factorial experiment of step 3. In this context, it may be worth noting that there might be cases where the approximate functions chosen are simple enough to allow for an analytical solution of the problem in step 5. The initial values of the design variables, v(O), were chosen as those given by rules-of-thumb used in the design and correspond to the values used in operating systems. The order of magnitude of the extrapolation allowed ( A u J u J was 20-40%. The starting values of the design variables and those optimized are given in Table 11. The results were obtained after three passes through the steps presented above, requiring about 45 STE's (simulation time equivalents) of CPU time. The optimization results in a bigger heat-transfer surface of the CHR and smaller gaps between the plates. The increased mass flow density enhances the heat transfer, and about 2500 MWh/a of additional heat is recovered to the supply air. Consequently, the consumption of fresh stream in the SC decreases. However, higher pressure losses increase the consumption (+170 MWh/a) of electricity in the fans. The steam consumption in the SHE is not affected much, and the amount of heat recovered in the AHR thus remains practically constant. It may be noted that the program suggests that both the mass flow densities and the gap between the plates on the exhaust air side of the CHR should be at, or close to, their physical (hard) constraints imposed in the study. The venture profit of the heat recovery system is not very sensitive to the mass flow of water in the AHR unit, and this quantity only changes slightly from its starting value. The number of cells in the AHR was, in turn, determined to be 4 from eq 2 using ma = 54.5 kg/s and (ma/A)"AC = 15 (kg/s)/ cell. Figure 5 depicts how the utility costs, $JII, are affected by the gap between the plates, dE, on the exhaust air side of the CHR unit. The solid and dotted lines denote the results by the rigorous and the approximate models, re-
2:c
,
I
3300
kEIM/a
320c
I
c -
L
I
I
~
original 300 16.0 20.0 8.9 12.7 100 4
optimized 454 10.0 11.7 9.4 14.3 102 4
34 269 796 10412 10 254 158 60
36 728 968 7939 7765 174 68
530 170 3800 3100
650 200 4080 3230
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Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2257 The fact that the constraint is active at the solution is observed to affect the value of 6 only slightly. A consequence of this is that a system with even smaller investment costs, i.e., further reduced heat-transfer surface, can be chosen without significantly deteriorating the venture profit. In summary, there is a FIM 280000 (US. $70000) increase in the value of the recovered heat at the expense of a smaller increase in the energy consumption. Taking into account the investment costs and the interest rate, the yearly venture profit of the recovery system is increased by FIM 130000 (US.$32500). Closing Remarks This paper presents an engineering method for parametric optimization of design variables in sequential modular flow-sheeting programs and an application of it to design of heat recovery systems for paper machines. An approximate empirical model of the objective function is repeatedly adapted to the output of the flow sheet and optimized. A factorial experiment is used to assign values to the design variables, and the corresponding values of the objective function obtained by the rigorous model are mapped to the approximate model. By this approach, relatively few evaluations of the flow-sheeting program are required. The analysis carried out predicts that the consumption of fresh steam in the dryer section can be significantly reduced, which, for a specific illustration example, yielded an increase of 4.2% in the yearly venture profit of the heat recovery system. The results of the optimizations have already been considered in the practical design procedures. For instance, the (design) temperature of the supply air in the CHR has been raised because the results from the optimizations have indicated that this would be advantageous. Moreover, a merit of the approach with approximate functions is that the sensitivity of the venture profit with respect to the design variables at the optimum point can be easily analyzed on a pocket calculator once the optimization has been carried out. Acknowledgment We are grateful to Valmet Paper Machinery Inc. for giving us permission to publish this paper and also thank Dr. Jar1 Ahlbeck, Ab0 Akademi, who provided the program for designing factorial experiments. Literature Cited Berna, T. J.; Locke, M. H.; Westerberg, A. W. A New Approach to Optimization of Chemical Processes. AZChE J. 1980,26,37-43. Biegler, L. T.; Cuthrell, J. E. Improved Infeasible Path Optimization for Sequential Modular Simulators-11: The Optimization Algorithm. Comput. Chem. Eng. 1985,9,257-267. Biegler, L. T.; Hughes, R. R. Approximation Programming of Chemical Processes with Q/LAP. Chem. Eng. Prog. 1981, 77, 76-83. Biegler, L. T.; Hughes, R. R. Infeasible Path Optimization with Sequential Modular Simulators. AZChE J. 1982,28,994-1002. Biegler, L. T.; Grossmann, I. E.; Westerberg, A. W. A Note on Approximation Techniques used for Process Optimization. Comput. Chem. Eng. 1985,9, 201-206. Cerda, J.; Westerberg, A. W.; Mason, D.; Linnhoff, B. Minimum Utility Usage in Heat Exchanger Network Synthesis: A Trans-
portation Problem. Chem. Eng. Sci. 1983,38,373-387. Chan, W. K.; Prince, R. G. H. Application of the Chain Rule of Differentiation to Sequential Modular Flowsheet Optimization. Comput. Chem. Eng. 1986,10,223-240. Chen, H A . ; Stadtherr, M. A. A Simultaneous Modular Approach to Process Flowsheeting and Optimization. Part 111: Performance on Optimization Problems. AlChE J. 1985,31, 1868-1881. Ciric, A. R.; Floudas, C. A. A Mixed Integer Programming Model for Retrofitting Heat-Exchanger Networks. Ind. Eng. Chem. Res. 1990,29,239-251. Colberg, R. D.; Morari, M. Area and Capital Cost Targets for Heat Exchanger Network Synthesis with Constrained Matches and Unequal Heat Transfer Coefficients. Comput. Chem. Eng. 1990, 14,1-22. Douglas, J. M. Conceptual Design of Chemical Processes; McGrawHill Book Company: New York, 1988. Edgar, T. F.; Himmelblau, D. M. Optimization of Chemical Processes; McGraw-Hill Book Company: New York, 1988. Floudas, C. A,; Ciric, A. R.; Grossmann, I. E. Automatic Synthesis of Heat Exchanger Networks. AZChE J. 1986,32,276-290. Forsbacka, B. Optimization and Dimensioning of Heat Recovery Systems for Paper Machines. M.Sc. Thesis (in Swedish), Heat Eng Lab., Abo Akademi, Abo, Finland, 1981. Gorczynski, E. W.; Hutchinson, H. P.; Wajih, A. R. M. Development of a Modularly Organized Equation-oriented Process Simulator. Comput. Chem. Eng. 1979,3,353-356. HlavlEek, V. Synthesis in the Design of Chemical Processes. Comput. Chem. Eng. 1978,2,67-75. Hutchinson, H. P.; Jackson, D. J.; Morton, W. The Development of an Equation-oriented Flowsheet Simulation and Optimization Package-I & 11. Comput. Chem. Eng. 1986,10,19-47. Itoh, J.; Shiroko, K.; Umeda, T. Extensive Application of the T-Q Diagram to Heat Integrated System Synthesis. Comput. Chem. Eng. 1986,10,59-66. Jirapongphan, S.;Bmton, J. F.; Britt, H. I.; Evans, L. B. A Nonlinear Simultaneous Modular Algorithm for Process Flowsheet Optimization. 80th AIChE Annual Meeting, Chicago, IL, Nov 1980. Kaijaluoto, S. Process optimization by flowsheet simulation. Technical Research Centre of Finland, 1984,Publ. 20. Kocis, G. R.; Grossmann, I. E. Global Optimization of Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Problems in Process Synthesis. Ind. Eng. Chem. Res. 1988,27, 1407-1421. Linnhoff, B.; Hindmarsh, E. The Pinch Design Method for Heat Exchanger Networks. Chem. Eng. Sci. 1983,38,745-763. Nelder, J. A; Mead, R. A Simplex Method for Function Minimization. Comput. J. 1965,7, 308-313. Nishida, N.; Stephanopoulos, G.; Westerberg, A. W. A Review of Process Synthesis. AZChE J. 1981,27,321-351. Papoulias, S. A.; Grossmann, I. E. A Structural Optimization Approach in Process Synthesis-11: Heat Recovery Networks. Comput. Chem. Eng. 1983,7,707-721. Parker, A. L.; Hughes, R. R. Approximation Programming of Chemical Processes-1 & 2. Comput. Chem. Eng. 1981,5 , 123-141. Powell, M. J. D. A Fast Algorithm for Nonlinearly Constrained O p timisation Calculations. Lecture Notes in Mathematics; Springer-Verlag: Berlin, 1978; Vol. 630,pp 144-157. Shah, J. V.; Westerberg, A. W. EROS: A Program for Quick Evaluation of Energy Recovery Systems. Comput. Chem. Eng. 1980, 4,21-32. Sundqvist, H. Tailor-made Heat Recovery Systems. In Production and Utilization of Energy in the Pulp and Paper Industry; EUCEPA Symposium 87;Lisbon, 1987;Vol. 2,pp 7-20. Tjoe, T. N.; Linnhoff, B. Using Pinch Technology for Process Retrofit. Chem. Eng. 1986,28,47-60. Valmet Paper Machinery Inc. Optimization of Heat Recovery Systems of Paper Machines (in Finnish). Report for the Ministry of Trade and Industry, Abo, Finland, 1988.
Received for review January 3, 1990 Revised manuscript received June 11, 1990 Accepted June 29, 1990
