Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol.17 n.47 São Paulo Dec. 2000
http://dx.doi.org/10.1590/S010466322000000400030
ENERGY AND CAPITAL TARGETS FOR CONSTRAINED HEAT EXCHANGER NETWORKS
L.C.Santos and R.J.Zemp*
Faculdade de Engenharia Química, DESQ, Unicamp,
C.P.6066, 13083970, Campinas  SP, Brazil
Fone: 0xx 19 788 3949 Fax: 0xx 19 289 4717
Email: zemp@desq.feq.unicamp.br
(Received:November 16, 1999 ; Accepted: April 6, 2000)
Abstract  A new procedure for estimating area and capital cost targets of constrained heat exchanger networks is presented. The method allows for match constrained networks and exchangers with more than one tube pass. The procedure is based on modelling the problem as a nonlinear formulation where the forbidden exchanger matches are included as constraints and the temperature difference correction due to multipass exchangers is included in the model. The difficulty of converging to a solution due to the additional nonlinear constraints imposed by the multipass exchangers required the use of a twolevel approach: at the inner level, the area targets for simple pass exchangers are obtained, and at the outer level the temperature difference required for multipass exchangers are computed and fed back to the inner level. The procedure is repeated until an appropriate tolerance between two iterations was achieved. A comparison between the estimated exchanger areas and costs estimated by the new procedure and the area and costs obtained from the final heat exchanger design shows a very good agreement.
Keywords: energy targets, cost targets, pinch analysis, heat exchanger networks.
INTRODUCTION
The targeting of utility and capital cost has been at the heart of the design of energy efficient chemical processes. The ability of estimating costs ahead of design allows the engineer to choose the most promising scenario based on desirable energy recovery and/or available investment money. Only after the most suitable process condition is decided for, the final design of the heat exchanger network is carried out. Pinch Technology is probably the best known set of tools available for doing industrial energy recovery systems design. However, most industrial processes present constraints that Pinch Technology cannot easily deal with, and alternative methods are now being searched for.
Irrespective of the methodology used to analyse and design the heat recovery system, the fundamental parameter that defines the amount of energy recovered is the minimum temperature approach in the system, DT_{min}. Large values of DT_{min }lead to small energy savings, as the possible energy exchange is limited due to the large temperature differences required for a match. On the other hand, a network with large values of DT_{min} will have small exchangers, reducing the capital cost. Overall, one can argue that there will be a range of optimal values of DT_{min} where the best tradeoff between savings and investment is achieved.
CONSTRAINTS IN HEAT RECOVERY TARGETS
The utility targeting for a given process using Pinch Technology is fundamentally based on the energy balance between hot and cold streams in each temperature interval of the process. Heat exchange between any pair of hot and cold streams is allowed. Surplus of heat in higher temperature intervals is then cascaded down to lower temperature intervals, while the deficit of energy in a given temperature interval is satisfied by using external hot utility. This procedure can be carried out both graphically, drawing the process composite curves (figure 1), or by using the problem table algorithm (as detailed by Smith, 1995).
However, in industrial problems it might not always be desirable to exchange heat between any given pair of hot and cold streams. The reasons could be the distance between the streams, or safety concerns. In this case, additional constraints have to be considered at each temperature interval when applying the problem table algorithm, turning this procedure into a very complex and tedious task (OYoung et al., 1988).
Papoulias and Grossmann (1983) proposed a procedure for utility targeting of match constrained heat exchanger networks by formulating the energy balances and constraints as an expanded transshipment model. In their model, surplus heat from a higher temperature interval (if available) flows into the lower interval, together with heat from the hot streams of the interval and hot utilities. On the cold side, heat flows out to the cold streams of this interval and the cold utilities. Any excess heat is transferred to the interval below (at a lower temperature). The obvious advantage is the possibility of including match constraints into the problem. However, much of the beauty of the graphical analysis presented by the composite curves is lost in this process.
CAPITAL COST TARGETING
Utility targets give a very good prediction of operating cost, but for a global picture, the cost of the heat exchanger network should also be predicted with a reasonable confidence. Townsend and Linnhoff (1984) proposed a procedure in which the composite curves (Figure 1) are divided into enthalpy intervals and the exchange area of interval computed by assuming a complex network with a match for each stream pair (spaghetti network). The total network area target is then given by the sum of the area over all intervals:
(1) 
Equation 1 presents a very simple and effective way to estimate network areas, and predicted area is rigorous for uniform heat transfer coefficients. However, for nonuniform heat transfer coefficients, the area predicted by equation 1 can be much higher than the actual one. This is due the imposed vertical match requirement of equation 1, whereas for nonuniform heat transfer coefficients, nonvertical matching (crisscrossing) can greatly reduce the required heat exchanger area (Figure 2).
Another disadvantage of equation 1 is the lack of possibility of considering forbidden exchanger matches.
The use of a mathematical approach for area targeting, in a similar way as proposed for utility targeting, can be used to include process constraints into the targeting procedures. The transshipment model proposed by Papoulias and Grossmann (1983) can be extended to target the heat exchanger area of the system (Colberg and Morari, 1990). The resulting model is nonlinear nature due to the equation required to compute the area. Their model can deal with unequal heat transfer coefficients as well as match constraints. In this approach, the utility targeting model is extended by dividing each temperature interval into enthalpy intervals, so that the area of each heat load (exchanger match) can be computed.
SHELL AND TUBE EXCHANGERS
Multipass shellandtube heat exchangers with even tube passes are probably the most common type of heat exchanger used in the chemical and process industries. Although requiring more area than the singlepass design, the advantages of higher tube side velocities, better cleaning and allowance for thermal expansion overcomes the additional area cost. The larger area of multipass shellandtube is due to the simultaneous countercurrent and concurrent flow. The design equation is given by:
(2) 
where F_{t}, for one shell pass and two or more even tube passes, is correlated to both the heat capacity flowrates ratio (R) and the thermal effectiveness P by:

(3) 
and

(4) 

(5) 
As the existence of multiple passes in the heat exchanger can lead to a undesired temperature cross, normally only heat exchanger designs that give a value of F_{t} above 0.75 – 0.8 are acceptable.
Multiple Shells
For cases where F_{t }< 0.8 (or 0.75) or if the area of a heat exchanger exceeds the maximum area allowed for a single shell, multiple shells in series need to be considered. A modification of the vertical area algorithm for multipass exchangers was proposed by Ahmad and Smith (1989), by including the F_{t} factor and number of shell in series into equation 1. This modified procedure produced a very good area estimate for multipass shell and tube heat exchangers. At this point, one is faced with a problem: Pinch Technology provides tools that can predict area of multipass heat exchanger networks, but without considering process restrictions, while the available LP/NLP models for constrained networks only consider singlepass heat exchangers.
A TARGETING PROCEDURE FOR CONSTRAINED MULTIPASS HEAT EXCHANGERS
The new targeting procedure that takes into account simultaneously match restrictions and multipass heat exchangers was implemented using a nonlinear programming (NLP) approach. The area targeting model proposed by Colberg and Morari (1990) was used as a departure point, together with the detailed description given by Shenoy (1995). This procedure models the network as a series of temperature and heat intervals, with heat exchanger matches placed between each pair of streams. The NLP then minimizes the total area of processprocess and processutility heat transfer, subject to:
(a) heat balance for each stream;
(b) heat balance for each match;
(c) feasible temperature difference for each match;
Since the temperature of the interval boundaries are not fixed, vertical area is not a constraint in this formulation, allowing for nonvertical matches if this leads to a lower heat transfer area.The procedure for singlepass heat exchanger area targeting was coded in GAMS (GAMS Corporation). Once tested for single pass heat exchangers, the changes for using multipass heat exchangers were implemented. The first step was to include multiple shell requirements into the NLP formulation, by computing the proper values of F_{t} and number of shells. However, the method of computing F_{t }based on equations 3 – 5 is not appropriate, as a search procedure has to used to identify the number of shells in series that leads to as acceptable value of F_{t} (Hewitt, 1994).
An Alternative Method for Multipass Heat Exchangers
The maximum temperature cross which can be tolerated is normally set by rules of thumb to be F_{t} ³ 0.8. The purpose of this limit is to avoid regions on the F_{t} plot where slopes are particularly steep, as any uncertainties or inaccuracies in design data also have a more significant effect in these regions. Consequently, for a good design, the regions of the F_{t} chart where slopes are steep should be avoided, irrespective of F_{t} ³ 0.8.
An alternative method to avoid steep slopes on the F_{t} chart is to accept values of P that are limited to some fraction of the maximum asymptotic value for P, say, P_{max}, which is given as F_{t} tends to ¥ :
(6) 
so that
0 < X_{p} <1 
(7) 
where X_{p} is a constant defined by the designer (Smith, 1995). A X_{p} of 0.9 leads to a limit of F_{t} = 0.8 for R = 1, and was used in this work.
So, if P for a given exchanger is smaller P_{lim}, the F_{t} value for this heat exchanger is computed simply using equation 4 or 5. However, if P is greater than P_{lim}, a series of multiple shells should be considered.
Ahmad et al. (1988) presented a procedure for computing the number of shells straight from the values of R, P and X_{p}, avoiding therefore the tedious task of trial and error. The number of shells for is so given by:
(8) 
where
(9) 
and for R = 1 by:
(10) 
Once computed the number of shells, the new F_{t} for these heat exchangers can be computed using equation 4 or 5, but using
(11) 
instead of P.
THE MODIFIED AREA TARGETING MODEL
All the additional equations required for computing the final temperature correction factor and the number of shells (equations 3 to 11) were added as constraints to the original NLP problem (based on the work of Colberg and Morari, 1990). Thus, it was expected that the new formulation would compute the proper area target for a multipass shell and tube heat exchanger network.
However, a major problem was posed by the fact that different equations are used for different values of R. The first step was to eliminate one equation from the model so that only one equation could be used for all values of R. An initial study showed that a very good agreement could be achieved by using the equations for with the approximation R = 1.001 for R = 1. Therefore, the set of equations for R = 1 could be eliminated from the NLP model, reducing the number of equations and avoiding increased complexity:
(12) 
The second step was to model equation 12 as a single arithmetic statement, because test clauses are not allowed for equations in the GAMS environment. Initially, it was proposed to overcome this limitation by using a dummy R variable, given by:
(13) 
This equation returns for values of , and R=1.001 for , with a small and negligible error for the computed value of F_{t}.
However, due to the highly nonlinear nature of the equations for estimating F_{t} and the number of shells, the model would not converge even for very simple cases, reaching unfeasible states during the solution procedure (in the new model, most restrictions are nonlinear, compared to the singlepass heat exchanger model, where only the objective function is nonlinear). Choosing carefully the initial values for all variables improved the solution convergence, but this would be unfeasible to do for larger problems. Also, shifting all the nonlinear constraints into the objective function showed to be impractical due to the large number of equations and their complexity.
THE TWOLEVEL APPROACH
Due to the difficulties of including the equations for the multipass heat exchangers into the model, a problem decomposition was introduced. The task of the decomposition was to divide the problem into smaller subproblems, solved each sequentially and repeat the whole procedure until convergence is achieved.
At the inner level, the area targets for singlepass heat exchangers are computed, using the original NLP formulation. If estimated values of F_{t }and number of shells are available, these are used. Otherwise, a value of 1 is assumed for both F_{t }and the number of shells of each match. The result is the minimum exchange area for the system. In addition, the values of the heat load and the temperatures of each heat exchanger match used to compute the area is also given.
At the outer level, the heat load and the temperatures of each match are used to compute the correction factor F_{t} and the number of shells for each match. These are the fed back to the inner level, and used for correction of the area and the cost function.
The procedure was repeated until an appropriate tolerance between two iterations was achieved, that is, the difference of area between two iterations was below a given limit. Although resulting in an iterative procedure, convergence is readily achieved for the default initial values (F_{t} and number of shells equal to 1 for each match).
RESULTS AND DISCUSSION: CASE STUDIES
The new procedure was applied to a series of test cases. Two examples will be shown in this paper: a 4stream and a 9stream problem. The 4stream problem will be used to illustrate the area targeting procedure and synthesis of (un)constrained multipass heat exchanger networks. The 9stream problem shows the benefits of using the NLP approach over the vertical area procedure for processes with nonuniform heat transfer coefficients.
Heat Exchanger Network with Uniform Heat Transfer Coefficients (Unconstrained and Constrained)
The stream data of the 4stream process is shown in table 1. The utilities used are also shown in this table. All the streams have the same heat transfer coefficient. A value of X_{p} = 0.9 was used to avoid infeasible multipass heat exchangers.
The system was analysed for a set of minimum approach temperatures: 10, 15, 20 and 25^{o}C. Table 2 shows the area targets for both singlepass and multipass heat exchangers, with no constraints imposed on the possible heat exchanger matches. As expected, the area for multipass heat exchangers is greater than the one for singlepass, due to the F_{t} correction factor. Also, the average F_{t} increases with the increase of DT_{min} (from 0.9199 for DT_{min} = 10^{ o}C to 0.9563 for DT_{min} = 25^{ o}C), due to larger temperature differences at the exchanger terminals for higher values of DT_{min}, and dominance of the countercurrent heat exchange over the concurrent heat exchange. A comparison between the area targets using the NLP method and the vertical area approach (equation 1) showed these to be very close, as expected.
The same 4stream problem was studied again, with the additional restriction of a forbidden heat transfer between hot stream 2 and cold stream 1. Table 3 shows the energy targets for the restricted system, obtained by using the LP utility targeting formulation bay Papoulias and Grossmann (1983).
Compared to the nonconstrained process, restriction on match H2C1 increased the hot utility requirement from 300 to 1400 kW (for a DT_{min} of 10^{o}C). The significant increase of energy requirement due to a simple process restriction shows the importance of using targeting procedures that can take account of process constraints. This is clearly an advantage of the mathematical approach over Pinch Analysis.
Next, area targets and shell targets for each DT_{min} were estimated using the proposed decomposition procedure. The results are also shown in table 3. As before, increasing the process DT_{min} decreases the network area. However, the estimated areas are smaller than ones predicted using the vertical area approach: for a DT_{min} of 10^{o}C, the area given by equation 1 is 1935 m^{2}, while the restricted process requires 1155 m^{2}. This is due to the significant contribution of large driving force utility  process heat exchangers for the restricted process. As a sideresult, a simple cost analysis could provide enough motivation for the elimination of the restriction if this would lead to an overall lower cost.
Heat exchanger networks using the targeted utility requirements were then designed so that the individual heat exchanger area could be computed. Special care has to be used in the design so that the match restriction between H2 and C1 was properly considered for. The networks for DT_{min} = 10^{o} and 25^{o}C are shown in figure 3. The topology is the same, showing that for the temperature range used, the problem shows no topology trap.
The heat exchanger areas were computed using equations 2 to 11. The detailed results of the heat exchanger design are shown in table 4 (for DT_{min} = 10^{o} and 25^{ o}C). In each case, one or more heat exchangers required a multiple shell configuration, justifying the inclusion of the multipass approach into the targeting procedure.
The final network heat exchanger areas are compared to the target areas in table 5, together with the number of shells. In addition, investment cost estimates using the targeted area and the final heat exchangers cost area also shown. For the cost calculations, the following cost equation was used.
(14) 
The estimated network areas using the new targeting procedure is very close to the final areas, with an average difference of around 3%. The same applies to the investment cost comparison, with an maximum difference of 2%. Considering the intrinsic uncertainties in cost estimate procedures, the results are very good. The number of heat exchanger shells is also predicted correctly.
For all cases, no convergence difficulties where identified. The execution time for each problem was of the order of seconds.
Heat Exchanger Network with NonUniform Heat Transfer Coefficients
A well studied process with nine streams (Ahmad and Linnhoff, 1989) was chosen to study the behaviour of the proposed targeting procedure for systems with nonuniform heat transfer coefficients. The stream and utility data are shown in table 6. Three sets of heat transfer coefficients were used in this work: a set with uniform coefficients (apart from the utilities), a set with nonuniform heat transfer coefficients as optimised by Polley and PanjehShahi (1991) based on pressure drop considerations, and a set of arbitrary heat transfer coefficients, with difference of one order of magnitude (streams containing vapor and twophase flows were set to 0.5 kW/K.m^{2}, while liquid streams were set to 5.0 kW/K.m^{2}). Multipass shell and tube heat exchangers were considered with a X_{p} = 0.9.
The results of the NLP area targeting procedure for a DT_{min} = 20^{o}C are shown in table 7. For the set with uniform heat transfer coefficients, the area target based on the vertical area procedure (equation 1) agrees well with the area estimated by the NLP formulation. For the set with nonuniform heat transfer coefficients, varying in the range from 0.738 to 1.1465 kW/K.m^{2}, the NLP targeting gave a smaller area target than the vertical procedure. This is explained by a slight trend of nonvertical alignment of heat exchangers due to the nonuniform heat transfer coefficients. For the last set of heat transfer coefficients, the vertical area procedure overestimates the heat exchanger network area by 928 m^{2}, compared to the NLP procedure. This result shows that for widely different heat transfer coefficients, the traditional vertical area target procedure can lead to an unfeasible economic heat recovery analysis.
CONCLUSIONS
The scope of this work was to improve the quality of heat exchanger network area targeting procedures for constrained network, by including the possibility of using multiple pass shell and tube heat exchangers.
Due to the additional nonlinear constraints imposed by multipass heat exchangers, the simple addition of the new equations to the model lead to problems with infeasible starting points. Carefully choosing the initial values, which lead to a solution, is impractical for all but small problems.
A new procedure using a twolevel approach has been implemented, where at the inner level the singlepass heat exchanger network area is estimated using the wellunderstood NLP method. At the outer level, information about heat exchanger temperatures and heat loads are used to compute the area correction (F_{t}) and the number of shells, which are then used to update the NLP area target procedure. The whole procedure is repeated until the convergence criterion is satisfied. Avoiding nonlinear constraints in the NLP formulation has shown to be the key issue in targeting area of heat exchanger networks, and this has been successfully implemented by the twolevel approach.
The results for a number of processes showed that a very close agreement of targets and final values for area and investments. The procedure is very quick, and can be repeated for a number values of minimum approach temperatures, therefore identifying the best tradeoff between operating and investment costs.
Overall, the proposed procedure can target area and investment costs of processes with the following constraints:
(1) nonuniform heat transfer coefficients
(2) forbidden heat exchanger matches
(3) single and multipass heat exchangers
The ability of considering the above restrictions simultaneously undoubtedly justifies the use of more complex models (LP and NLP).
Work is underway to modify the procedure to allow for pressuredrop restrictions for selected streams, thus extending the targeting procedure to pumping cost considerations. The twolevel approach is also being used to improve the automatic network synthesis procedure being used by our research group, so that the topology and the singleshell/multiple shell arrangement is generated simultaneously.
ACKNOWLEDGEMENT
The authors would like to acknowledge CAPES for the financial support.
NOMENCLATURE
T_{in}  inlet temperature 
T_{out }  outlet temperature 
Mcp  heat capacity flow rates 
h  film heat transfer coefficient 
Q_{hot }  minimum hot utility consumption 
Q_{cold}  minimum cold utility consumption 
T_{h,i}  inlet temperature for hot stream 
T_{h,o}  outlet temperature for hot stream 
T_{c,i}  inlet temperature for cold stream 
T_{c,o }outlet  temperature for cold stream 
R  ratio of heat capacity flow rates 
P  thermal effectiveness 
N_{shells }  number of shells for exchanger 
X_{p}  fraction of maximum P allowed by design 
P^{* }  modified thermal effectiveness P for multiple exchangers in series 
F_{t }  multipass temperature correction factor 
REFERENCES
Ahmad S., Linnhoff B. and Smith R., Design of Multipass Heat Exchangers: an Alternative Approach", Transactions of the ASME, Journal of heat transfer, vol 110, 2 (1988). [ Links ]
Ahmad S., Linnhoff B. and Smith R., Supertargeting: Different Process Structures for Different Economics, Journal of Energy Resources Technology, 111, pp 131136 (1889) [ Links ]
Ahmad S. and Smith R., Targets and design for minimum number of shells in heat exchanger networks, IChemE, ChERD, 67 (1989) [ Links ]
Colberg R. D. and Morari M., Area and capital cost targets for heat exchanger network synthesis with constrained matches and unequal heat transfer coefficients, Computers and Chemical Engineering, Vol 14, No 1, pp 122 (1990) [ Links ]
Hewitt G.F., Shires G.L. and Bott T.R., Process Heat Transfer, CRC (1994) [ Links ]
OYoung D.L., Jenkins D.M. and Linnhoff B., The constrained problem table for heat exchanger networks, IChemE Symposium Series, 109, pp 75116 (1988) [ Links ]
Papoulias S.A., and Grossmann I.E., A strutural optimization approach in process synthesisII, Computers and Chemical Engineering, Vol 7, No 6, pp 707721 (1983) [ Links ]
Polley G. T. and Panjeh Shahi M. H., Interfacing Heat Exchanger Network Synthesis and Detailed Heat Exchanger Design. Trans IcemE 69, Part A, 445457 (1991) [ Links ]
Shenoy U. V., Heat exchanger network synthesis  optimization process by energy and resource analysis, Gulf Publishing Company (1995) [ Links ]
Smith R., Chemical Process Design, McGrawHill (1995) [ Links ]
Townsend D.W. and Linnhoff B., Surface area targets for heat exchanger networks, IChemE 11^{th} Annual Res. Meeting, April, Bath  UK (1984) [ Links ]
*To whom correspondence should be addressed