410
Ind. Eng. Chem. Process
Dittus, F. W., Boelter, L. M. K., Univ. Calif. Publ. Eng., 2, 443 (1930). Karamercan. 0. E., Gainer, J. L., Ind. Eng. Chem. Fundam., 18, 11 (1979). Keil, R. H., bird, M. H. I., Ind. Eng. Chem. ProcessDes. Dev., 10, 473(1971). Lemlich, R., Chem. Eng., 88, 171 (1961). Lemlich, R., Amour, J. C., Chem. Eng. Pmg. Symp. Ser., 81(57), 83 (1965). Lemlich, R., Hwu, C. K., AIChEJ.. 7, 102 (1961). Martinelli, R. C., Boelter, M. K., Weinberg, E. B., Yalsahi, S., Trans. ASME, 65, 789 (1943).
Des. Dev. 1980, 19, 410-420 Perry, R. H., Chilton, C. H., Ed., "Chemical Engineers' Handbook", 5th ed, pp 10-19, McGraw-Hill, New York, 1973. Sieder, E. N., Tate, G. E., Znd. Eng. Chem.. 28, 1429 (1936). West, F. E., Taylor, A. T., Chem. Eng. Prog., 48, 39 (1952).
Received f o r review August 3, 1978 Accepted April 1, 1980
Interactive Synthesis of Cascade Refrigeration Systems Wal Biu Cheng and Richard S. H. Mah" Department of Chemical Engineering, Northwestern University, Evanston, Illinois 6020 1
An interactive computational strategy has been developed for the evolutionary synthesis of minimal cost refrigeration cascades with multiple heat sources. Topological relationship as well as heuristics are used to evaluate candidate designs of increasing complexity. The factors considered by this strategy include choice of refrigerants and temperature approaches, use of intermediate temperatures and pressures, economizers and presaturators, different materials of construction, and all permissible intercycle and intracycle heat transfers. Modest but worthwhile design improvements were obtained on two previously published benchmark problems in relatively short computing times. Potential extension of this approach to mixed refrigerant cycles is briefly discussed.
Introduction In a previous paper (Cheng and Mah, 1978) we explored the potential of man-computer interaction as an effective approach to design synthesis. We enumerated a number of desirable characteristics of a potential application area and selected pipeline network synthesis as a candidate application for our investigation. As we pointed out in that earlier paper, the component behavior of a pipeline network is well characterized in terms of material conservation equations and pressure drop correlations, but the aggregate behavior of the network is complex enough that neither algorithmic nor heuristic methods prove adequate by themselves. In the present study we extend this investigation to a slightly more complex class of problems in which phase equilibria and thermodynamics as well as conservation relations play an important role in the system description. The area selected for study is refrigeration cascades. These systems are economically important in key processes such as natural gas liquefaction and ethylene manufacture, but they also occur as a component of numerous other chemical processes. Operating costs, and therefore energy efficiency, play a very significant role in these systems. Study of refrigeration systems is therefore of interest and importance not only in itself but also in the synthesis of more complex processes.
The Synthesis Problem Although there are many variations, the refrigeration systems used in the chemical industry are commonly based on the vapor recompression cycle, the simplest form of which is Figure la. Under normal circumstances, cooling water a t ambient temperature acts as the ultimate heat sink or cooling agent. For refrigeration temperature much below that of the cooling agent, a cascade of multiple cycles using different refrigerants as shown in Figure 17 is needed. In this paper we shall be primarily concerned with the synthesis of such cascades. Potential extension to the mixed refrigerant cycles (Figure 16a) which form a class of important and viable alternatives will be briefly discussed. 0196-4305/80/1119-0410$01.00/0
Several configurational modifications can be made to improve the efficiency of a single cycle or cascade system. Since refrigeration is provided mainly by the vaporization of the liquid in the evaporator, compression work is reduced if an intermediate level is added between two temperature levels with the vapor formed after the first pressure reduction separated from the liquid and fed directly into the high pressure compressor as shown in Figure 2a. The flash drum added for the phase separation is commonly known as an economizer. To standardize the terminology, a cycle with intermediate streams will be called a complex cycle. The constituent cycles, each containing only one compressor and one valve as shown in Figure la, will be referred to as simple cycles. The subsystem that contains the same refrigerant will be called a refrigerant cycle and it may be either a simple cycle or a complex cycle. Since compression work for a given pressure rise increases with the vapor temperature, changing a simple cycle into a complex cycle by adding an intermediate level may also allow the insertion of an intercooler, as shown in Figure 2a, to reduce the amount of superheat in the compressors for better performance. For lower temperature cycles where an intercooler is not feasible, an economizer may be transformed into a presaturator as shown in Figure 2c to reduce the superheat to zero. Presaturation, however, requires a higher refrigerant flow rate which may more than offset the improved performance of the compressor. Another factor which may nullify the advantage of adding intermediate levels is that compressor cost is a concave function of compressor power. Several small compressors may therefore cost more than a single large compressor even though the total power of the former is less than that of the latter. Another modification that may be considered to improve the efficiency of refrigeration systems is to transfer heat among system streams. There are two kinds of heat transfers: (1) intracycle heat transfer which involves streams in the same simple cycle and (2) intercycle heat transfer in which heat is transferred from one simple cycle to another. Figure 3 shows an example of an intercycle 0 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980 411 Condenser
Intercycle Hear Exchanger E
Evaporator
Prockss i o a d (0:
~/ l
/------. -
,
Stream PriortyValue
Figure 3. Intercycle heat transfer.
h, + d
\-
a
h
c
(4)
(3)
\
d
(2) ( I )
( C )
F i g u r e 1. Vapor recompression cycle: (a) flow diagram; (b) pressure-enthalpy diagram; (c) heat flow path.
(a’
(b,
Figure 4. Vapor-liquid heat transfer.
W
(c;
(d)
F i g u r e 2. Modifications of simple refrigeration cycles: (a) and (b) economizer and intercooler; (c) and (d) presaturator.
heat transfer from stream a to stream b by means of a heat exchanger as represented by the dashed line. The heat load of the intermediate cycle is reduced resulting in smaller compressors and less utilities required. A potentially beneficial intracyycle heat transfer is shown in Figure 4. With the operating enthalpy range expanded, less refrigerant will be required. However, as mentioned earlier, superheating increases the work required per unit flow. The aforementioned counteracting factors together with a large number of plausible heat transfers, the possible inclusion of economizlers and intercoolers, and the deter-
mination of process variables such as temperature approaches makes the search for an optimal design a formidable task. To summarize, refrigeration system design can be stated as follows. Given the heat loads, the original temperatures and target temperatures of a set of process streams as heat sources, and the specifications of compressors as well as a cooling agent, determine the configuration and parameters of a system that can achieve the required refrigeration at a minimal cost. For simplicity, we shall take the cost function to be the total annual cost of capital and utilities (Barnes and King, 1974). Other simplifying assumptions are: (1) the temperature spans of the heat sources are sufficiently narrow that no further load splitting is necessary, i.e., it is not worthwhile to introduce more temperature levels of cooling; (2) only countercurrent twostream single pass heat exchangers and multistage centrifugal compressors are considered; (3) heat capacities of the process streams and film heat transfer coefficients are assumed to be constant so that the driving force for heat transfer is given by the logarithm mean of the temperature differences at the ends of the heat exchanger; (4) valves are assumed to be isenthalpic; (5) compressors are assigned efficiencies based on isentropic compression; (6) minimum temperature approach for operational purposes is 2.8 K; (7) the costs of valves, flash drums, and connecting junctions are ignored as they are insignificant when compared to the costs of heat exchangers, compressors, and turbines; (8) individual equipment cost is given as an exponential expression, xSY, the parameters of which are given in Table Ib (Barnes and King, 1974); and (9) piping cost for the system is given as a linear function of the number of temperature levels and of the total equipment cost. The capital cost is therefore given as (Barnes and King, 1974) $COSt,,p = [0.95 + 0.15 ( N -
l)]Ef$i i
(1)
where N is the number of temperature levels, fi is the
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Table I. Equipment Data for Refrigeration System Design (a) Material Factor material of construction
re1 cost re1 cost lower ofcomofheat temp pressor exchangers limit, K
carbon steel killed carbon steel 3% nickel steel 9% nickel steel
1.00 1.29
1.00 1.43
24 5 225
1.43 1.65
2.12 3.00
170 115
H (0)
( b ) Equipment Cost equipment heat exchanger compressor steam turbine
S
area work work
units m2 kW kW
x
Y
600 2750 1050
0.6 0.5 0.6
P
( c ) Miscellaneous Data cooling water cost $16 /kW year compression energy cost $90/kW year cooling water temperature 300 K maximum cooling water temperature rise 5.6 K efficiency of compressor 0.8 efficiency of steam turbine 0.75 heat transfer coefficients, kW/mz K one phase 0.34 two phase 1.022
relative cost of material taken from Table Ia and Ci the individual equipment cost of equipment i. Other pertinent design data are given in Table IC. Among others, the design parameters to be determined include the types of refrigerants, the number of temperature levels, and the temperature approaches of the various heat exchangers. In this paper we will develop a systematic synthesis strategy which will include human interaction as well as heuristic and evolutionary elements. Dynamic Programming Method Of all the process synthesis problems, heat exchanger network synthesis has received the most attention (Hendry et al., 1973; King, 1976),probably because it is more limited in scope and better defined. All given hot streams are cooled and all cold streams are warmed either by transferring heat from one stream to another or by means of utilities. The only equipment involved is the heat exchanger and the only energy flow is the heat flow through the heat exchangers. On the other hand, the heat flow in a refrigeration system is from a low temperature to a high temperature, necessitating the use of compressors and valves. Although the subproblem of transferring heat among streams is similar to a heat exchanger network synthesis problem, the methods for solving the exchanger network problem are not applicable for solving the refrigeration system problem, as not all the streams in a refrigeration system must be involved, and if involved, it may not be necessary that as much heat as possible should be transferred as is assumed by all the currently available heat exchanger network synthesis methods. The refrigeration system synthesis problem has previously been tackled by Barnes and King (1974) using a procedure which combines dynamic programming with some heuristics. In their approach, the types of refrigerants, the number of temperature levels and temperature approaches are specified by the designer. The subproblem is then represented by the network as shown in Figure 5c which is derived from the vapor side of the pressure-enthalpy diagrams of the refrigerants as shown in Figure 5a and b. Each of the labeled nodes on the diagrams repre-
i H H
Ib)
!C)
Figure 5. Refrigeration system design using dynamic programming: (a) refrigerant 2; (b) refrigerant 1; (c) solution paths.
t
-
b
8
io
0 (2)
However, since ideal gas behavior is assumed in the above expression, candidates with only slightly positive transfer index should not be discarded. This kind of estimation can also be applied to the case of multiple cycles. If w’econsider the lower cycle, the units between streams b and h enclosed by dotted lines in Figure loa, as a single unit, and denote the difference in molar enthalpy of streams a and h by A, we have exactly the same configuration as shown in Figure 4. The only difference is that Th which correisponds to To in Figure 4 will now depend on the amount of heat transfer. With less flow in stream c, the temperature of stream h increases and thus makes the transfer index more negative. Since this change in T h is a complex function of many other variables, the effect of the heat transfer from stream a will not be estimated. Instead, the current value of Th is used and the heat transfer set up if the transfer index is found to be 180 g-mol/cal or less. Our experience shows that this works quite satisfactorily. Although the a-h heat transfer in Figure 10a does not change T b and Tg,a d--f heat transfer in the lower cycle does increase the X value of the upper cycle and change Tg and Th. Heat transfers in the lower cycles should therefore be tested first before those in the upper cycles are considered. With the user’s assessment of the topological relationship, this sequencing can be easily carried out. For intercycle matching, the a-g heat transfer in Figure 10b is usually not considered unless there is a water cooler in stream h, as shown to reduce the “priority value” of stream g to less than that of stream a. In such a case, a certain amount of heat load is diverted from compressor A to compressor B. Assuming ideal gas behavior, it can again be shown (Cheng, 1979) that the net change in compression work is
[Q&?A/(H~ - H2
+ Q ) l [ ( l k R ’ C ~ - 1) X (H3 - HZ)/1]- (H4 - HJ1 (3)
where H is the enthalpy, Q the amount of heat transferred, gA the refrigerant flow in compressor A, and k the compression ratio of compressor B. The intercycle matching is desirable if the above expression (diversion index) in cal/s is found to be negative.
where Tu,T,, and Ti are the temperatures of the upper level (valve inlet), lower level (valve output), and the new intermediate level, respectively, and B is an experimentally determined parameter with a recommended value of 0.5. Since the total cost of the refrigeration system is rather insensitive to this value when it is near the optimum, this formula with the recommended value is adopted in our program as a guide to the user who has the option to override it. As to the first question, we have developed two heuristic criteria (Cheng, 1979): (3a) If the averaged compression work required exceeds the initial compression work by a factor of 1.6, i.e.
(3b) if the vapor fraction of the outlet stream of the corresponding valve exceeds 0.5. A temperature level may be added if either of the above two criteria is met. An economizer between two adjacent valves (and compressors) is always beneficial. Its insertion is therefore assumed as part of the new design. Taken as they are, the two aforementioned criteria are hardly met by any cycle unless the comparison ratio is considerably higher than 12 which then exceeds the limit usually set by the manufacturers for mechanical reasons. There are, however, other factors involved which could lower the limits of these criteria and thus make intermediate levels more attractive: first, if intercooling by the cooling agent is feasible; secondly, if there is a decrease in compression it would mean a smaller heat load for all the upper (warmer) cycles and thus reduce their costs even though the cost of the cycle in question increases. The extent of the impact of these factors, however, cannot be conveniently established. Therefore, with the help of a computer to provide the work ratios and vapor fractions, human judgment is needed to make a better decision. One of the considerations that may affect the decision of the user is material factor. As seen in Table Ia, equipment cost rates are divided into several ranges based on the lowest temperature each type of material can be exposed to. A level may therefore not be added even though the above two criteria are met if the smaller heat load after adding a level brings one of the compressors in the upper cycles to a colder and thus more expensive range.
Heat Transfer, Presaturation, and Intercooling After taking step 1in which the refrigerants are selected, step 2 in which an initial design is set up, and step 3 in which the number of levels are fixed, we now proceed to step 4 of the design procedure and carry out the other configurational modifications. Streams are assigned “priority” labels and sorted and candidate heat transfers
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
302 8
7
@
k
J
382 8
Ethane I 85.0i b
___
Table IV. Process Stream Data process orig target heat example stream temp, K temp, K load, kW single 1 191 191 5000 load multiple 1 188 188 2000 load 2 207 207 2000 3 225 225 1000 4 238 238 4000 5 275 275 2500
3060r ;;
9:850
-----------7
272 0
247 2
G
I:;- 1 - -
-_-
247.2
250.0
2390 2162
I; I
q3730
I
Figure 12. Single load three-level design: cost = $1375638. (Barnes and King, 1974). Figure 11. Single load two-level design: (a) cost = $1 499297; (b) cost = $1 307 527; (c) cost = $1 277 612; (d) cost = $1 276 275.
identified as outlined earlier. Starting from the coldest cycle, each candidate heat transfer is assessed by a linearization method as previously described and, if found promising, is tried out by a detailed simulation. Unpromising candidates are usually bypassed except when a moderate increase in compression work could push some of the compressors to a higher and thus cheaper temperature range. If we discard a vapor-liquid heat transfer, we automatically transform any economizer downstream from the valve into a presaturator and try out the new design. Since presaturation increases the refrigerant flow and decreases the compressor superheat, it has exactly the opposite effect of a vapor-liquid heat transfer and could therefore reduce compression work if the vapor-liquid heat transfer could not. By the "priority" argument, an intercooler is always beneficial and is therefore added wherever the temperature approaches are feasible. Since the temperature of the cooling agent is relatively high, material factor is not a concern for this kind of modification. "Priority values", however, will be altered by intercooling and it is for this reason that intracycle heat transfers should be evaluated first before intercycle heat transfers are considered. Adjustment of Continuous Parameters After all the promising configurational modifications have been tested and the final system configuration decided upon, we can proceed to step 5 in which other continuous parameters are adjusted. One such kind of parameter is the heat duties of the selected heat transfers. Depending on their relative merits, the heat duties may be redistributed for further cost reduction. Another parameter which may be adjusted is the temperature approaches of the inter-refrigerant-cycle heat exchangers. The cost of the heat exchanger is compared with the cost
325 2
266 I
1 251 I 245 3 235 0
(b)
Figure 13. Single load three-leveldesign: (a) cost = $1 502040; (b) cost = $1 312 280.
of the compressor, and if found to be excessive, the temperature approach of the heat exchanger may be increased to reduce its cost a t the expense of the compressor cost. Still another parameter that may be adjusted is the temperature level, As already mentioned, the system cost is relatively insensitive to it. Other considerations such as the material factor may therefore be the only incentive for such adjustments. Sample Refrigeration System Designs 1. Single Load. With the design data shown in Table IV, the computer provided the system in Figure l l a as the
Ind. Eng. Chern. Process Des. Dev., Vol. 19, No. 3, 1980 417
ml
I
2837
2901
I
272 c
2350
m-!i a ‘F-i
2378
204.0
ETHANE
qF
2957
2901
2350
ac
2761
I 2172 Figure 15. Multiple load design: cost = $2 375 704 (Barnes and King, 1974).
(bl
Figure 14. Multiple load ‘designs: (a) cost = $2 323 019; (b) cost = $2 245 943.
initial design. The nunnbers beside each stream in Figures 11-15 are temperatures in K. Although ammonia and ethane were suggested by the program, propane and ethane were used instead to allow a comparison with the design reported by Barnes arid King (1974). Temperature approaches were assumed to be the operating limit of 2.8 K initially. The intermediate temperature of 243.3 K was determined by eq 4 wiith B = 0.5. T o carry out step 3 of the design procedure, the initial design was stimulated and the work requirements of the compressors checked. Since the work ratios of compressors A and B were 1.4 and 1,3,respectively, and since the vapor fractions in streams b and f were 0.35 and 0.37, respectively, no intermediate temperature level was needed. Step 4 was divided into three sub-steps for this problem: (a) intracycle heat transfer in the ethane cycle, (b) intracycle heat transfer in the propane cycle, and (c) intercycle heat transfer. For the first sub-step, the a-c heat transfer was the only candidate, The considerations pertaining to this candidate heat transfer were: (1)the transfer index was 648; (2) the compressor was in a very low temperature (high cost) range; and (3)temperature of stream d was high enough for a water cooler which would reduce the heat load for the propane cycle. Since compression work is much more sensitive to flow irate than to the amount of superheat, the cost of compressor B would be reduced at a faster rate than the cost of compressor A would increase. The last consideration therefore outweighed the first one. With the added benefit of pushing compressor A to a warmer range (the second consideration), the a-c heat transfer was deemed favorable and as much heat as possible was transferred, Le., until the limiting temperature approach was reached. A detailed simulation showed that the overall
cost decreased even though the cost of compressor A increased and thus confirmed the inferences made. Note that no specific guideline could have been laid down for the computer program to reconcile the conflicting heuristics. Human judgment was needed to make the decision. For the propane cycle, the only candidate proposed by the computer was the e-g heat transfer. Since the transfer index was -70 and since there was no upper cycle to be affected, we had no reason not to proceed with the design modification. A simulation again showed a decrease in cost. The intracycle transfers are shown in Figure l l b . The intercycle heat transfer from stream h to stream f was eliminated for topological reasons leaving the I-j heat transfer as the only candidate suggested by the computer. Although the diversion index had a positive value of 17, the closeness of the temperature of stream j to the next cheaper range made that candidate an attractive one. The heat transfer was therefore set up with just enough heat transferred to raise the inlet temperature of compressor A to 245 K-a partial transfer. The prediction that the increase in compression energy cost would be offset by the decrease in capital cost was confirmed for the simulated design as shown in Figure l l c . As there were no more candidates to be considered, the last step of the design process was carried out. Since the cost of exchanger C was 2 % of the cost of compressor B, and the costs of exchangers D and E were less than 1% of the cost of either compressor, the temperature approaches did not need to be increased. The only parameter to be considered was therefore the intermediate temperature level (streams f and g). Since lowering the intermediate temperature would reduce the heat flow of the profitable a-c heat transfer and increase the burden of the C-j heat transfer, such a move was deemed unproductive. To relieve the burden of the I-j heat transfer and in fact to eliminate the exchanger altogether, the intermediate temperature level was increased to 245 K resulting in the cheaper design in Figure l l d . A further increase to 247 K showed only a negligible 0.02% improvement. The last design was therefore taken as the final design.
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
The same problem was also solved by Barnes and King (1974) using the dynamic programming method. Figure 12 shows our closest duplication of their solution. Our design is therefore 7.2% cheaper ($1276275 vs. $1 375638). To demonstrate further the deficiency of the dynamic programming method, a three-level refrigeration system was also designed for closer comparison. The preliminary design was shown in Figure 13a. With the same argument as in the two-level design, the a-b heat transfer was set up even though the transfer index was 691. For the same reason and with a much lower transfer index of 92, the g-h heat transfer was also set up with a cooler added to stream i. The e-j heat transfer had a transfer index of 7 which, because of the approximate nature of the index, was too close to the negative range to be ignored. The heat transfer was therefore tried out and the design was found to be marginally better, being 0.04% cheaper. The intercycle heat transfer from stream g to stream b had a diversion index of 60. Notwithstanding the favorable effect it would have on the uppermost cycle, the index was too high for a complete transfer to be beneficial. A partial transfer was therefore made to increase the inlet temperature of the ethane compressor to only 245 K which was the lowest acceptable temperature range. By the argument of “priority”, the d-f heat transfer was always beneficial and was therefore incorporated into the final design as shown in Figure 13b. With a cost of $1312 280, this design is 4.6% cheaper than the one found by dynamic programming method. 2. Multiple Loads. With the design data as given in Table IV, the computer provided the system as shown in Figure 14a as the initial design. The compression ratios of the five compressors were so low that step 3 of the design procedure was actually bypassed. For intracycle heat transfer, the transfer indices of the d-e, c-g, and a-h heat transfers were 187, 274, and 203, respectively. Presaturation which has the opposite effect of intracycle heat transfer could therefore reduce the compression work. Because stream h was close to the lowest acceptable temperature of the material of construction, economizer F was in fact transformed into a presaturator. A surprising increase in capital cost resulted from the more expensive material needed for compressor C. With the designer directly involved in the design process, however, this kind of mistake could be easily detected and rectified. Learning from this, presaturating the inlet stream of compressor C was not tried out as it would bring the compressor to an even lower temperature range. Other candidates suggested by the computer, such as the f-b heat transfer, were similarly discarded. For the propane cycles, the feasibility of adding a cooler to stream C implied a favorable effect the j-k heat transfer would have on the upper cycle. With only a moderately high transfer index of 157 and the additional benefit of bringing compressor D into a cheaper range, a complete transfer was therefore set up. Finally, the transfer index of the i-m heat transfer was found to be only 27. It was therefore tried out and was shown by a simulation to be marginally beneficial, being 0.1 % cheaper. Compared with the solution given by Barnes and King as shown in Figure 15, our design again shows an improvement of 5.5% ($2 245 943 vs. $2 375 704) in spite of the fact that the design stipulations severely restricted the choice of process configurations. One should, however, sound a cautionary note. Since Barnes and King used their own version of Redlich-Kwong equation which may be different from the Soave-Redlich-Kwong equation (Reid et al., 1977; Soave, 1972) that we used, the temperature
of stream h could be below 245 K even without the two presaturators added. Presaturating stream g could therefore be a beneficial design. What we do doubt, however, is that the temperature could be below 225 K which would then justify the presaturation of stream h. As it is improbable that the two versions of RedlichKwong equation could differ by over 20 K, the difference in design could only be attributed to the different methods used.
Discussion Our work on INDER has again shown the effectiveness of interactive synthesis. Although a comparison with the previous dynamic programming method of refrigeration system design is difficult due to the lack of information on computational efficiency, the new method does appear to be faster, better, and more direct. As mentioned earlier, dynamic programming requires that the state of the candidate systems be known before the method can be applied. Since in design synthesis, the state of the system is determined by the outcome of the synthesis and not known a priori, a time-consuming iteration process must be employed. On the other hand, the new method does not assume the state of the candidate systems and no iteration is needed in this respect. The new method is faster also because of the efficiency of the evolutionary procedure. Alternative design configurations are largely eliminated by heuristics and human judgment. Fruitless design modifications are seldom encountered. As the optimal configuration is rather insensitive to the continuous design parameters, the iterations involved in step 5 of the design procedure are limited in scope and thus do not pose an excessive demand on computation time. The method is better because it does not make any assumption on the flow rates in the system and the temperature approaches of the heat exchangers and thus does not exclude as many promising designs. Indeed, the two sample problems showed that the optimal temperature approaches actually depended on the individual heat transfer and should not be generalized as was assumed by Barnes and King (1974). The method is also more direct because it deals directly with the refrigeration system and no translation is needed to get the design detail. In addition to the traditional heuristics of using the entire two-phase enthalpy range for refrigeration, several new heuristics have also been developed. Three heuristics were developed to select the refrigerants. Since a design using one set of refrigerants bears little relationship to the design using another set of refrigerants, the selection of refrigerants is too critical a step to be left entirely to heuristics. The selected refrigerants should therefore be regarded as a reminder of the types of refrigerants that should be used. Other design consideration must be carefully weighed by the user before the refrigerants are finally decided upon. Consider next the work ratio heuristics for adding intermediate temperature levels. Owing to the nature of the design problem, intermediate levels are seldom added unless the refrigeration temperature and cooling water temperature just happen to span the entire operating range of a refrigerant. Since most cases, especially the ones with multiple loads, are so specified that the compression ratios are usually not large enough for new levels to be beneficial, the threshold value of 1.6 has not been sufficiently tested. Human judgment is therefore needed to make the final decision. In general, the possibility of intercooling is a sufficient reason for testing the desirability of intermediate levels. The threshold value can therefore be somewhat lowered for warm refrigerant such as ammonia and butane
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
419
A
I
ibl
“ j
IC)
Figure 16. Metamorphosis of an MRC system. Table V. Core Reauirements of INDER subprogram functions main (input, initialization, etc. ) simulation and costing synthesis steps configuration change check and list labeled common total
core requirement, words 111468 161278 106548 64268 5528 44628 540338
by an extent the experience of the user may suggest. For intracycle and intercycle heat transfers, the heuristics of “priority value” as applied to a cascade system has been shown to be very successful as the number of candidates are significantly cut to a more manageable size. As to the accuracy of the transfer and diversion indices which are obtained by linearization, our experience shows that the estimated cost changes could be off by as much as 50% depending on the extent the heat transfer is set up. For indication purposes, however, this is an acceptable margin. With other topological and material factors to be involved in the decision, these indices have indeed served their purposes well. Performance statistics of program INDER are shown in Tables V, VI, and VII. Since the computation time spent on design simulation d.epends heavily on the complexity of the design, no generalized numbers can be given. Instead, the computing times €or the sample problems are listed in Tables VI and VII. It is notable that the computing time expended in interaction is generally quite small in comparison with the simulation times. A somewhat intangible but very important advantage of interactive synthesis is that being intimately involved in the design evolution, the designer gains a fuller understanding of the process. He completes the design with a feeling of confidence in the validity of that which he helped to create. Extensions to Mixed Refrigerant Cycles Cascade systems are riot the only solutions for the design of low-temperature refrigeration systems with multiple loads. Mixed-refrigerant cycles (MRC) such as the one depicted in Figure 16a are also feasible alternatives. In such a system, a mixture of refrigerants is successively
Table VI. Simulation Time Required by INDER on a CDC 6600 Computer system as depicted in Figure
simulation time, CP s
1l a llb llc lld 12 13a 13b 14a 14b 15
0.159 0.224 0.560 0.218 1.470 0.212 0.996 0.428 0.486 0.472
Table VII. Interaction Time Required by INDER on a CDC 6600 Computer type of interaction
time, CP s
design modification changing parameter stream labeling, sorting and matching
0.0 23 0.0 28 0.025-0.029
flashed at the same pressure and as the mixture becomes richer in colder refrigerants such as methane and leaner in warmer refrigerants such as propane, the temperature decreases providing the temperature range needed for the multiple loads. The economic advantage of MRC is that there are only two pressure levels and the refrigerants are mixed into one stream, and therefore only one compressor is needed. Unfortunately, the current technique for MRC design still relies heavily on case studies. One of the reasons for the lack of systematic synthesis approaches is that the individual cycles of the MRC system interact to a much greater extent than those of the cascade system. For example, if the temperature of stream a in Figure 16a is decreased, the flow rate of stream b will be increased and the flow rates of streams c and d will be decreased. By contrast a decrease in the temperature of stream a in Figure 17 has no effect on the flow rates in the two lower cycles. As shown in Figure 16a-c, with the corresponding heat transfers bearing the same labels hypothetical separation of cycles of an MRC system actually leads to a cascade system. Therefore, a plausible approach for synthesizing an MRC system is first to synthesize a cascade system for which synthesis techniques are available and then lump
420
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 420-426
*
a
Izr
TI
u
-
Figure 17. A cascade refrigeration system.
the cycles together to form an MRC system. There are, however, some significant differences between the two systems that hinder the applicability of this approach. First, because of the impurities in the coldest refrigerant at the coldest level after lumping the cycles, the temperature at that level will be higher than before lumping and thus cannot meet the coldest temperature requirement. Similarly, the temperature requirements of the colder levels will not be met while the temperature requirements of the warmer levels will be overly met. To avoid lowering the pressure and the possibility of going below the pressure limits of some of the refrigerants, a margin of say, 10 K below the temperature required by the heat sources must be provided for the colder levels in the cascade design. This provision, however, is contradictory to heuristics l a to ICoutlined earlier and thus invalidates the optimality of the cascade system. Another drawback is that in order to equalize the inlet temperatures of the three compressors in the cascade system so that they can be lumped into one, the amounts of superheat in the colder refrigerants are
excessively high and this again runs counter to the experience in optimal cascade system design. Despite these drawbacks the approach of lumping pure refrigerant cycles to form an MRC system does provide a feasible design for which further improvement can be made and is therefore worth exploring. Nomenclature E = parameter used in eq 4 C = individual equipment cost, $ C, = heat capacity, cal/g-mol K f = relative cost of material (carbon steel = 1.0) g = refrigerant flow, g-mol/s H = enthalpy, cal/g-mol k = compression ratio N = number of temperature levels P = pressure, N/m2 Q = amount of heat transferred, cal/g-mol R = gas constant, 1.987 cal/g-mol K S = cost parameter T = temperature, K x = coefficient of cost function y = exponent of cost function Greek Letters 17 =
compressor efficiency
X = operating enthalpy range, cal/g-mol
Literature Cited Barnes, F. J., King, C. J., Ind. Eng. Chem. Process Des. Dev., 13, 421-433 (1974). Cheng, W. E., Ph.D. Thesis, Northwestern University, Evanston, Ill., 1979. Cheng, W. B., Mah. R. S. H., Cornpot. Chem. Eng., 2, 133-142 (1978). Hendry, J. E., Rudd, D. F., Seader, J. D., AIChE J . , 18, 1-15 (1973). King, C. J., AIChE Monogr. Ser., 70(8), 3-31 (1976). Reid, R. C., Prausnitz, J. M., Sherwood, T. K., "The Properties of Gases and Liquids", 3rd ed, McGraw-Hill, New York, 1977. Soave, G., Chem. Eng. Sci., 27, 1191-1203 (1972).
Received for review August 23, 1979 Accepted March 31, 1980
The authors wish to acknowledge the support of this work in the form of a fellowship to W. B. Cheng from the AMOCO Foundation.
Nonisothermal Determination of the Intrinsic Kinetics of Oil Generation from Oil Shale S.-M. Shih and H. Y. Sohn' Departments of Metallurgy and Metallurgical Engineering and of Mining and Fuels Engineering, University of Utah, Salt Lake City, Utah 84 112
nonisothermal technique using various heating rates has been applied to the determination of the intrinsic kinetics of oil generation from oil shale. From an engineering standpoint the rate of oil generation c a n adequately be described by overall first-order kinetics with a constant activation energy of 199 kJ/mol. Various methods are applied to the determination of the kinetics parameters. The relative merits of these methods are discussed. The results are compared with data reported in the literature. The nonisothermal technique has the advantages of short experimental time and the elimination of difficulties due to the initial heat-up period accompanying the isothermal experiments. A
Introduction The kinetics of the decomposition of kerogen in oil shale, which is the precursor of oil, has been studied by a number of investigators (Hubbard and Robinson, 1950; Allred, 0196-4305/80/1119-0420$01.00/0
1966; Weitkamp and Gutberlet, 1970; Braun and Rothman, 1975; Johnson et al., 1975; Campbell et al., 1978). The complex nature of kerogen and its decomposition reaction has complicated the interpretation of data and led to
e 1980 American Chemical Society
