A Forward Branching Scheme for the Synthesis of Energy Recovery

A Forward Branching Scheme for the Synthesis of Energy Recovery Systems Ram N. S. Rathore and Gary J. Powers* Department of Chemcal Engmeenng. Carnegfe-Mellon Unfversity, P/ttsburgh, Pennsylvan/a 752 73

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A forward branching scheme is developed for the synthesis of heat exchanger networks. Nearly optimal networks can be discovered in relatively few enumerations using this scheme. A larger number of enumerations, which is still much smaller than that required for exhaustive search, is necessary to establish optirnality. An upper bound on energy recovery is established and is used to measure the performance of each candidate network. An example is presented for a four-stream problem in which two streams are to be heated and two to be cooled.

One means for alleviating problems associated with fuel shortages is to conserve energy by carefully matching sources and sinks of energy within a process or between a number of processes. With this strategy, capital is expended to buy heat exchangers and control equipment so that fuel costs can be reduced. The control equipment is required to ensure that the more interconnected process which results from the energy recovery is controllable, safe, and easily started up and shut down. The problem is to synthesize the energy recovery network which maximizes the objective function for the system. A solution to the problem involves matching the hot and cold streams in the process in heat exchangers. The residual streams from a match can also be matched with other streams. Utilities such as steam or cooling water can be used within the network to change the temperature of residuals so that they meet the requirements a t their destination or to change the temperature of an inlet or intermediate stream so that an advantageous match can occur. Table I summarizes the previous work on this problem. Each solution method has been classified by the following. 1. Type of Matches Considered by the Method. Some systems consider only sensible heat change of fluids while others are more general and allow phase change. Some systems also consider the energy requirements associated with compression and heats of reaction. 2. Whether a Match with Utilities Is Restricted to the Last Match. A number of systems use this restriction and hence may exclude the optimal structure. 3. Whether Stream Splitting Is Allowed. Splitting streams can improve the performance of an energy recovery system. Most systems do not consider this aspect explicitly although the initial statement of the problem can be changed by manually splitting streams prior to energy recovery. 4. The Restrictions in the Heat Exchanger Model. In some systems the temperature of approach is fixed. In others the heat transfer coefficients are not a function of phase or flow rate or whether phase change is occurring. In addition, in some systems, the cost of the heat exchanger is assumed to vary linearly with area. A more accurate power law cost equation is used by others. 5 . The Type of Objective Function. Most systems consider only heat exchanger and utility costs. Most use the assumption that the rate of inflation matches the interest rate so that the sum of equipment cost (amortized over the equipment life) and utilities need not be corrected for the time value of money. Control costs and transportation costs (plumbing and pumping costs to move the material to the best exchanger locations) are not usually included. 6. The Search Method. Most of the work in this area has focused on developing efficient search techniques to overcome the combinatorial problem associated with ener-

gy recovery network synthesis. One heuristic technique has been developed. The key features of each of these systems are given in Table I. 7. Computer Program and Problem Size. Those methods for which programs have been written are indicated. The largest problem which has been solved by the method is also noted. The approach described in this paper is included in Table I for comparison. The major limitations on the size of the problem whicf! can be solved by these techniques are the storage space and the number of enumerations required to synthesize a feasible and near optimal energy recovery network. Much of the storage space and enumeration problem is due to the fact that the methods generate a large number of infeasible systems which are stored and then later detected during evaluation. This paper outlines a forward branching scheme which is designed to reduce the storage requirements and synthesize only feasible networks. The number of enumerations required using this approach can be drastically reduced by using an upper bound on the amount of energy integration which can occur in the system. This approach is motivated by the work of Ellwein (1974) and Arnold and Bellmore (1974). They present formal arguments to prove the efficiency of forward and backward branching schemes for large combinatorial optimization problems.

Problem Formulation Consider changing the temperature of m cold streams and n hot streams from given input to specified output temperatures. Heat exchangers and utilities can be purchased to solve the problem. The goal is to find the heat exchanger network which minimizes the total cost for the system. For this study, we have used

as the objective function. The variables f h and fc, are the heating and cooling costs per year and can be expressed as

where f e is the annual cost of heat exchange equipment and f u is the annual cost of utilities consumed. The variable f t is the annual cost of transporting a stream to the heat exchanger site. Stream Matching. The efficient synthesis of heat exchanger networks often involves: (a) the decomposition of the problem to the minimum set of matches which could occur in the final solution, (b) evaluation of each match to Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 2, 1975

175

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Ind. Eng. Chem., ProcessDes. Dev., Vol. 14, No. 2,1975


>t l - r

1i LL i

I

COLD STREAM

EXISTING TEMP (BEGINNING1

I

I

HOT STREAM B

I

I

DESIRED TEMP (END1

EXISTING TEMP (BEGINNING)

t

COOLING

DESIRED TEMP LEND1

AcoLD

WATER

TEMPERATURE

( 01

t

t

,,,A,

COOLi WG WATER

Figure 2. Heat exchanger networks for the matches shown in Figy.

U S U A L MATCH W I T H LARGER T E M P E R A T U R E D I F F E R E N C E S

F

TEMPERATURE

(b) STREAM


>

t 0

a a

"

-

c U

1 Y

MINIMUM APPROACH TEMPER A T U R E

RESIDUAL

3

0

L J

PREFERRED M A T C H W I T H S M A L L E R T E M P E R A T U R E DIFFERENCE

COLD STREAM

TEMPER AT U R E TEMPERATURE

( C )

(a)

Figure 1. Stream matches with one residual stream (Siirola, 1974).

HOT STREAM

t

" a U

determine its physical and economic feasibility, and (c) synthesis of a solution from the individual matches. I t is extremely important t o ensure that all possible matches are identified. Matches can occur in a number of ways as illustrated in Figure 1. Figure l a illustrates a simple two-stream problem in which a cold stream A may match with a hot stream B. Figures l b and IC illustrate two possible matches. In Figure lb, the cold stream is matched with the hottest portion of the hot stream and all of the energy required by the cold stream is transferred. This type of match produces a residual from the coldest portion of the original hot stream and all of the energy required by the cold stream is transferred. Following the original match, this remaining energy could be removed by matching the hot stream with cooling water or another cold stream. This match has the advantage of having a large temperature difference hence requiring a small, less expensive, heat exchanger. In Figure IC, the cold stream is matched with the coldest portion of the hot stream. The residual from this match is the hottest portion of the original hot stream. This match has a smaller temperature difference and requires a larger heat exchanger than the match illustrated in Figure l b . Figure 2 illustrates how these matches could be embodied in heat exchange equipment using cooling water as the utility. Figure 3 illustrates how a match can lead to two residual streams. In Figure 3a the hottest portion of the hot stream is matched with an intermediate point of the cold stream such that the minimum allowable temperature approach is just satisfied. This match results in the generation of two residuals from the cold stream. A similar match when the cold stream is small can lead to creation of two residuals from the hot stream. The second match of this type is shown in Figure 3b. In this case the match location is fixed by the limiting

a c

MINIMUM APPROACH TEMPERATURE

w 4 I

z

LL

COLD STREAM

stream and the minimum allowable temperature approach. In all of these matches it was assumed that the maximum amount of energy was transferred from the limiting stream. If this assumption is relaxed, it is possible to get matches with three or four residuals. A continuum of residuals results from this type of match. All previous workers have assumed maximum transfer to limit the size of the problem. Siirola (1974) has presented arguments supporting the maximum transfer assumption although no formal proof has been developed. Siirola was the first to point out matches of the types shown in Figures IC and 3b. Such matches can increase the level of energy recovery within 81 heat exchanger network by creating residuals of the hotter portion of the hot streams and the colder portion of the cold streams. In addition, these matches can lead to the use of less expensive utilities. All these points are illustrated in the following section.

Synthesis by Forward Branching The synthesis of energy recovery systems can be represented as a state-space search problem (Nilsson, 1971; Powers, 1973). The initial state is the hot and cold streams prior to any matches. The goal state is achieved Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 2 , 1975

177

Table 11. Description of 4SP1 and the Design Data


Steam pressure Cooling water temperature Maximum output temperature Minimum allowable approach temperatures: Heat exchanger Steam he at e r Water cooler Equipment downtime Over all heat transfer coefficients: Heat exchanger Steam heater Water cooler Heat exchanger cost in dollars Annual rate of return Cooling water cost Steam cost

962.5 psia (saturated) 100°F 180°F 20°F 25°F 20°F 380 hr/yr

Figure 4. First level decision nodes for problem 4SP1

150 Btu/hr ft2 OF 200 Btu/hr ft2 "F 150 Btu/hr ft2 "F 350 (area in ft2)Om6 0.1 5x $/lb 1x $/lb

Problem 4SP1 (all liquid streams) Steam A

B C D

Flow rate, Input Output lb/hr temp, "F temp, "F 20,643 27,778 23,060 25,000

140

320 240 480

320 200 500 280

Heat cap. 0.70 0.60 0.50 0.80

when all streams have been processed to their specified end conditions. Matches are operators which change one state into another. By repeatedly applying all match operations to the initial streams and all secondary (residual) streams, it is possible to generate all feasible energy recovery networks. Each sequence of matches which converts the initial state into the goal state constitutes a solution to the problem. The solutions can then be searched to determine the optimal solution. Forward branching is a depth-first tree search procedure for identifying feasible solutions. If the problem representation is carefully selected, it is possible to reduce duplication in the solution space. Ellwein (1974) provides a formal description of forward branching and offers proofs concerning the number of enumerations required to search the complete solution space. In the following section, we illustrate this method for a four stream problem which has been previously solved by Lee, et al. (1970), and Pho and Lapidus (1973). This is problem 4SP1 in Lee, et al. (1970). Table I1 describes the problem and the design data for heat exchangers. The seven possible first level decision nodes for the 4SP1 problem are shown in Figure 4. These nodes indicate that the streams denoted are used in energy matches. Within each decision node, it may be possible to match the streams in a number of different ways. The problem 4SP1 has one first level decision node which involves no energy matches (utilities are used eventually). Four decision nodes contain one primary energy match, and two decision nodes contain two primary matches. Each of these nodes can be further developed by considering all the types of matches possible for the streams involved. These matches can be evaluated by solving the relevant heat transfer, energy balance, and cost equations. The residual streams resulting from the matches are then considered as candidates for other matches. This matching and evaluation procedure is continued until all streams are processed to their desired end conditions (the goal 178

Ind. Eng. Chern., Process Des. Dev., Vol. 1 4 , No. 2 , 1975

state). Figure 5 illustrates the decision nodes when branching into the first level decision node for matches between (D and A) and (B and C). For the conditions given for streams B and C only one match is possible and the match results in two residual streams B (278.5, 200) and C (300, 500). The numbers following the stream name are the temperatures through which the residual must be heated or cooled. Hence matching B with C reduced the temperature of B from 320 to 2785°F and it still needs to be cooled to 200°F. The single match for B and C is due to the temperatures and flow heat capacities of the streams. (In addition, no stream splitting has been allowed and maximum allowable energy exchange is assumed. Without these constraints, a large number of residuals would result and the size of the problem would increase rapidly. This indicates that another problem representation is probably required if stream splitting and incomplete matches are allowed). Streams A and D can be matched in two ways: one match results in the creation of a residual D (349.9, 280) (cold portion of stream D), and the other results in creation of a residual D (480, 410.1). (Note that the residual D (480, 410.1) is the hot portion of stream D.) Hence these are three decision nodes a t the second level. (a) No matches between the residuals. (This gives rise to two heat exchanger networks which are not connected. The remaining energy loads are handled by utilities.) (b) Residuals C (300, 500) and D (349.9, 280) are matched. This match gives rise to the network shown in Figure 6c as no other matches can be made a t the third level. (c) Residuals C (300, 500) and D (480, 410.1) are matched. This results in the creation of the networks shown in Figures 6a and 6b. The annual costs are shown in Figure 6 and indicate the importance of the type of match shown in Figure 3c. By matching the colder portion of stream D with stream A, one obtains a hot residual from D which can heat a large segment of the residual stream C (300, 500). Hence more energy is recovered and a lower cost results. Figure 7 shows the six best networks and Figure 8 the three worst networks for the 4SPl problem. These were determined by developing all the first level nodes in Figure 4 as was done above. The cost of the worst structure is three times that of the optimal one. The total costs are dominated by the utility costs hence optimal systems will have a high level of energy recovery. All of the heat exchange networks in Figure 7 have the same high level of energy recovery. The small difference in their annual costs is due to the differences in the size of heat exchangers in the networks. There are over 20 networks for this problem that have annual costs within 10% ($13,573 to $14,900) of the optimal and achieve the same level of energy recovery. (There are over 200 networks that solve the problem.) The insensitivity of network cost to network structure for high energy recovery suggests an obvious search simplification. Compute an upper bound on energy recovery and synthesize a limited number of networks that approach within a given fraction of this bound. For example, it is often sufficient to generate 5 to 10 networks that come within 5% of the upper limit on energy recovery.


Figure 3. Stream matching schemes when primary matches (D,A) and (B.C) are allowed. Transportation costs are not included in the costs of matches

These networks can then be evaluated for control, start up, and shut down, and safety features and a satisfactory system selected. For a large industrial problem (containing over about 8 streams) developing any one of the decision nodes containing a primary match will probably lead to a number of networks which are close to the cost of the optimal one.

B

D

COSTIYR

Let Th be the temperature of the hottest stream of the m hot streams ( i e . , streams to be cooled). Heat can then be transferred to the cold streams up to the temperature (Th - A T n i ) where A T n i is the minimum approach temperature. All the heating requirements above this temperature must be met by utilities. Similarly all the cooling below ( T , A T n i ) , where T , is the coldest temperature of the n cold streams, must be met by utilities (cooling water, refrigerant, etc.). The remaining energy can be recovere d. Let Q,, be the energy content of all m hot streams above T , + AT,,,. Similarly, let Q, be the energy content of all n cold streams below TI, - A T n i . Then an upper bound on the energy that can be recovered Ql+is

:

$21469

(a)

An Upperbound on the Level of Energy Recovery B

D

?

+

The energy recovery computed by eq 4 can be used to measure the degree of optimality of candidate networks.

CCST'YR

= $43605

(C)

Comparison with Previous Work The networks shown in Figures 6a and 6b are less costly than those reported by Lee. e t al (1970). and Pho and Lapidus (11173). Their optimal structure is given in Figure 6c. They could not synthesize the networks in 6a and 6b because they did not consider matches of the type shown in Figure IC and Figure 3a. Siirola (1974) reported the same structures shown in Flgures 6a and 6b. The networks shown in Figures 7d and 7e have not been previously reported. The network in 'id uses cooling water to cool stream D a t two different temperature levels. Figure 7e illustrates a cyclic heat exchange network. The colder portion of stream A matches first with the colder portion of stream D. The intermediate portion of A matches

Figure 6. Three best structures synthesized by branching into the node containing t h e primary matches (B,C) and ( D , A ) .

with B and the remaining hot residual from A matches again with the hotter portion of stream D. These cyclic structures are a consequence of including matches of the type shown in Figure 3a in the matching operations. These types of structures can be significant when used in refrigeration systems.

Transportation Costs The total cost for the system should include the cost for transporting the materials to the heat-exchange site. For Ind. Eng. Chem., Process Des. Dev., Vol. 1 4 , No. 2 , 1975

179

D

B

320-F 1280'F COST/YR p0.F

$ 13645

3

(b) COST/YR =

$ 13573

(a) D

/200°F


/28O'F COSTIYR = $ I 3 6 9 5 28OOF

(C) COST/YR:

$13904

(d)

P

1480°F

279'F

353'F 231'F

i'oo""F COSTlYR = $14079

cw

$3*F 280'F

(f) COST/YR : $ I

4076

(e)

Figure 7. Six best heat exchange networks for 4SP1.

B

?

t COST/YR

:

280'F

$ 33176

(a)

c w 4 O ' F A-

320?*

140-

1200'F

3 50" F 240'F

280°F

/280'F COST/YR = $40236

(b)

COST/YR

:

$40605

(C)

Figure 8. Three worst heat exchange networks for 4SP1.

processing systems in which the energy recovery is within a single process the distances are not usually greater than 500 f t and the transportation costs (pumping, pipe, pipe 180

Ind. Eng. Chem., Process Des. Dev. Vol. 14, No. 2,1975

supports, etc.) are not usually significant when compared with the utility costs. However, for viscous or corrosive materials, the costs can be comparable to the heat-ex-


changer costs. For greater distances, the transportation costs for common fluids can also become significant. A general formulation of the heat exchanger network synthesis problem would introduce new variables for the coordinates (x, y , z ) for each heat exchange site. The source and destination of each stream undergoing heat exchange would also be included in the problem description. The problem is then to select both the structure of the heat exchanger network and the location of each exchanger in the network. The problem can be made discrete if the heat exchangers are constrained to be located a t the source or destination of one of the streams undergoing the match. With this scheme each energy match could occur a t one of four locations. This considerably expands the number of solutions possible. In addition, the residuals from each match now have a location associated with them. This location must be carried along in the analysis so that the next level of matching can be evaluated. The general trends expected when transportation costs are high is that the network will decompose into clusters. The local streams will be matched and the outlying streams will be serviced by utilities. Here we have assumed that the utilities are available everywhere in the space. If this is not the case, the utilities and their locations may be added to the stream list and the expanded problem solved in the same manner. Conclusions A tree hearch procedure is introduced for the solution of heat exchanger network synthesis problems. A general set of energy matches is defined so that all possible networks can be generated. Several previous investigators have not

included all possible matches and have overlooked less costly solutions. The expanded matches give rise to cyclic networks in which a stream can match twice with another stream. The cost of the networks is insensitive to changes in structure for systems that have high energy recovery levels. An upper bound on energy recovery is defined and used to limit the search for nearly optimal systems. Acknowledgments Part of this work was carried out a t Tufts University, Medford, Mass. Dr. J . J. Siirola of Tennessee Eastman Company made valuable suggestions regarding this work. Literature Cited Arnold, L. R . , Bellmore. M. Oper. Res., 22, 383 (1974). Ellwein. L. B., Oper. Res., 22, 144 (1974). Hwa, C. S.,AlChE Int. Chem. Eng. Sym., Ser.. No 4 (1965) Kesler. M. G., Parker. R . O., Chem. Eng Prog. Sym. S e i No. 92. 111 (1969) Kobayashi, S . , Umeda, T . , Ichikawa, A,, Chem. Eng. S o . , 26, 1367 (1971). Lee, K. F., Masso, A. H.,Rudd, D. F . , Ind. Eng. Chem., Fundam.. 9, 48 (1970). Masso, A. H . , Rudd. D. F.,AlChE J . , 15, 10 (1969) McGalliard, R. L., Westerberg. A. W., Chem. Eng. J . . 4, 127 (1972) Menzies, M. A., Johnson, A. I . , Can. J . Chern. Eng.. 50, 290 (1972) Nilsson, N . J.. "Problem Solving Methods in Artificial Intelligence." McGraw-Hill. New York, N.Y., 1971. Pho,T. K., Lapidus, L . , A I C h E J . , 19, 1182 (1973) Powers, G. J., "Non-Numerical Problem-Solving Methods in ComputerAided Design," in "Computer-Aided Design," p 327, Vlietstra and Wielinga, Ed.. North Holland, 1973. Siirola, J. J.. "Status of Heat Exchanger Network Synthesis," Paper No 42a, presented at 76th National AlChE Meeting, Tulsa. Okla., 1974.

Receiuedjor reciew July 31, 1974 Accepted December 16,1974

Reaction Variables in the Air Blowing of Asphalt Luke W . Corbett Exxon Research and Engineering Company, Linden, N e w Jersey 07036

In the manufacture of air-blown asphalts, the selection of flux source and the consistency of the flux are among the most important variables in determining the properties of the finished product. Reaction velocity is also dependent upon flux source as well as upon air rate and its dispersion. These variables, therefore, are significant when developing process designs or techniques of manufacture. The mechanism of air blowing is deduced by compositional analysis, in which naphthene-aromatics convert to polar-aromatics and they in turn to asphaltenes. Composition also shows that desirably higher penetrations result when the content of saturates plus unreacted naphthene-aromatics is relatively high, Differences due to flux source and in their blown products thus may be related to these compositional features.

Introduction The air blowing of asphalt, sometimes referred to as air conversion, provides products with properties that are unattainable by other means. These properties have made asphalt adaptable to roofing, waterproofing, adhesive, and sealing applications, as described by Abraham (1960) and Traxler (1961). This has resulted in the development of standard specifications for built-up roofing and waterproofing (ASTM, 1973), and specifications covering the use of blown asphalt in various industries (Krchma, 1965).

In order to meet these specifications, a manufacturer of blown asphalt initially concerns himself with the conversion qualities of the base stock used, hereafter termed flux. These qualities, in turn, are dependent to a large extent upon the crude source from which the flux is derived, as brought out by past investigations (Chelton, e t ai, 1959; Greenfeld, 1964; Hughes, 1962; Hoiberg, 1950; Thurston and Knowles, 1936). Other variables, such as flux consistency, blowing temperature, air rate, and catalysts if used, are also known to effect product characteristics as well as processing methods, but perhaps to a lesser Ind. Eng. Chem. Process Des. Dev., Vol. 1 4 , No. 2, 1975

181

A Forward Branching Scheme for the Synthesis of Energy Recovery

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